Algebra 1 Virginia SOL 2023 - Teacher Edition by Mathspace

Algebra 1 Teacher’s Edition

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Contents

Equations & Inequalities

1.01

Algebraic expressions (A.E0.1)

1.02

Properties of real numbers

1.03

Properties of equality (A.EI.1)

1.04

Multistep equations (A.EI.1)

1.05

Literal equations (A.EI.1)

1.06

2 3 4

Multistep inequalities (A.EI.1)

Topic 1 Assessment

114

Functions & Relations

120

2.01

Functions and relations (A.F.2)

124

2.02

Domain and range (A.F.1, A.F.2)

147

2.03

Evaluating functions (A.F.1, A.F.2)

164

2.04

Characteristics of functions (A.F.1, A.F.2)

180

Topic 2 Assessment

198

Linear Functions

204

3.01

Slope (A.F.1)

209

3.02

Transformations of linear functions (A.F.1)

230

3.03

Slope-intercept form (A.F.1)

254

3.04

Standard form (A.F.1)

287

3.05

Point-slope form (A.F.1)

319

3.06

Equations of parallel and perpendicular lines (A.F.1)

343

Topic 3 Assessment

363

Systems of Equations & Inequalities

370

4.01

Write and graph linear systems (A.EI.2)

374

4.02

Substitution method (A.EI.2)

401

4.03

Elimination method (A.EI.2)

425

4.04

Two variable linear inequalities (A.EI.2)

447

4.05

Systems of linear inequalities (A.EI.2)

475

Topic 4 Assessment

501

Contents mathspace.co

vii

5 6 7 8 viii

Exponents, Radicals, & Exponential Functions

508

5.01

Product rule (A.EO.3)

513

5.02

Power rule (A.EO.3)

525

5.03

Quotient rule (A.EO.3)

538

5.04

Zero and negative exponents (A.EO.3)

552

5.05

Rational exponents (A.EO.4)

569

5.06

Simplify radicals (A.EO.4)

589

5.07

Operations with numerical radicals (A.EO.4)

600

5.08

Characteristics of exponential functions (A.F.2)

614

5.09

Graphs of exponential functions (A.F.2)

630

Topic 5 Assessment

653

Polynomials & Factoring

658

6.01

Add and subtract polynomials (A.EO.2)

662

6.02

Multiply polynomials (A.EO.2)

681

6.03

Divide polynomials by a monomial (A.EO.2)

707

6.04

Factor GCF (A.EO.2)

718

6.05

Factor by grouping (A.EO.2)

734

6.06

Factor trinomials (A.EO.2)

748

6.07

Factor using appropriate methods (A.EO.2)

761

6.08

Divide polynomials (A.EO.2)

779

Topic 6 Assessment

790

Quadratic Functions

794

7.01

Characteristics of quadratic functions (A.F.2)

799

7.02

Quadratic functions in factored form (A.F.2)

827

7.03

Quadratic functions in vertex form (A.F.2)

854

7.04

Quadratic functions in standard form (A.F.2)

889

7.05

Compare linear, quadratic, and exponential functions (A.F.1, A.F.2)

916

Topic 7 Assessment

935

Quadratic Equations

942

8.01

Solve quadratics using graphs and tables (A.EI.3, A.F.2)

946

8.02

Solve quadratics by factoring (A.EI.3, A.F.2)

968

8.03

Solve quadratics using square roots (A.EI.3)

985

8.04

Solve quadratics using the quadratic formula (A.EI.3)

1008

8.05

Solve quadratics using appropriate methods (A.EI.3)

1034

Topic 8 Assessment

1047

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Data Analysis

1052

9.01

Data and sampling (A.ST.1)

1056

9.02

Scatterplots (A.ST.1)

1085

9.03

Linear regression (A.ST.1)

1114

9.04

Quadratic regression (A.ST.1)

1144

9.05

Analyze bivariate data (A.ST.1)

1168

Topic 9 Assessment

1193

Contents mathspace.co

1 Equations & Inequalities Big ideas • Expressions are the building blocks of algebra. They can be used to represent and interpret real-world situations. • The properties of real numbers can be applied to many types of expressions. • An equals sign indicates an equivalent relationship between two expressions. • A standard algorithm can be followed to solve a wide range of equations. This algorithm is reliable and useful in a variety of situations, but there is often a more efficient method that can be used based on the structure of the equation. • A solution set is the collection of all values that make an equation or inequality true.

Chapter outline 1.01 1.02 1.03 1.04 1.05 1.06

Algebraic expressions (A.E0.1) Properties of real numbers Properties of equality (A.EI.1) Multistep equations (A.EI.1) Literal equations (A.EI.1) Multistep inequalities (A.EI.1) Topic 1 Assessment

7 27 36 59 80 94 114

Equations are like a perfectly balanced seesaw - both sides must be equal.

1. Equations & Inequalities Topic overview Foundational knowledge Evaluating standards proficiency The skills book contains questions matched to individual standards. It can be used to measure proficiency for each. Students should be proficient in these standards. 7.PFA.2 — The student will simplify numerical expressions, simplify and generate equivalent algebraic expressions in one variable, and evaluate algebraic expressions for given replacement values of the variables. 7.PFA.4 — The student will write and solve one- and two-step linear inequalities in one variable, including problems in context, that require the solution of a one- and two-step linear inequality in one variable. 8.NS.2 — The student will investigate and describe the relationship between the subsets of the real number system.

8.PFA.1 — The student will represent, simplify, and generate equivalent algebraic expressions in one variable. 8.PFA.4 — The student will write and solve multistep linear equations in one variable, including problems in context that require the solution of a multistep linear equation in one variable. 8.PFA.5 — The student will write and solve multistep linear inequalities in one variable, including problems in context that require the solution of a multistep linear inequality in one variable.

Big ideas and essential understanding Expressions are the building blocks of algebra. They can be used to represent and interpret real-world situations. 1.01 — The structure of an expression can reveal important details about the situation it represents. The properties of real numbers can be applied to many types of expressions. 1.02 — The properties of real numbers can be applied to simplify and evaluate algebraic expressions more easily. An equals sign indicates an equivalent relationship between two expressions. 1.03 — The properties of equality allow an equation to be manipulated without changing the equivalence of the expressions on either side.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

A standard algorithm can be followed to solve a wide range of equations. This algorithm is reliable and useful in a variety of situations, but there is often a more efficient method that can be used based on the structure of the equation. 1.04 — A standard algorithm can be used to solve any linear equation accurately and efficiently. 1.05 — The same standard algorithms that are applied to equations with numerical coefficients can be applied to equations with coefficients represented by letters. A solution set is the collection of all values that make an equation or inequality true. 1.06 — Inequalities have an infinite number of solutions so their solution sets are often represented on a number line.

Standards A.EO.1 — The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables. A.EO.1a — Translate between verbal quantitative situations and algebraic expressions, including contextual situations. 1.01 Algebraic expressions A.EO.1b — Evaluate algebraic expressions which include absolute value, square roots, and cube roots for given replacement values to include rational numbers, without rationalizing the denominator. 1.01 Algebraic expressions A.EI.1 — The student will represent, solve, explain, and interpret the solution to multistep linear equations and inequalities in one variable and literal equations for a specified variable. A.EI.1a — Write a linear equation or inequality in one variable to represent a contextual situation. 1.03 Properties of equality 1.04 Multistep equations 1.06 Multistep inequalities A.El.1b — Solve multistep linear equations in one variable including those in contextual situations, by applying the properties of real numbers and/or properties of equality. 1.03 Properties of equality 1.04 Multistep equations

A.El.1c — Solve multistep linear inequalities in one variable algebraically and graph the solution set on a number line, including those in contextual situations, by applying the properties of real numbers and/or properties of inequality. 1.06 Multistep inequalities A.El.1d — Rearrange a formula or literal equation to solve for a specified variable by applying the properties of equality. 1.05 Literal equations A.El.1e — Determine if a linear equation in one variable has one solution, no solution, or an infinite number of solutions. 1.04 Multistep equations A.El.1f — Verify possible solution(s) to multistep linear equations and inequalities in one variable algebraically, graphically, and with technology to justify the reasonableness of the answer(s). Explain the solution method and interpret solutions for problems given in context. 1.03 Properties of equality 1.04 Multistep equations 1.06 Multistep inequalities

G.DF.1 — The student will create models and solve problems, including those in context, involving surface area and volume of rectangular and triangular prisms, cylinders, cones, pyramids, and spheres.

A.EI.3 — The student will represent, solve, and interpret the solution to a quadratic equation in one variable.

G.TR.2 — The student will, given information in the form of a figure or statement, prove and justify two triangles are congruent using direct and indirect proofs, and solve problems involving measured attributes of congruent triangles.

A.EO.2 — The student will perform operations on and factor polynomial expressions in one variable. A.EO.4 — The student will simplify and determine equivalent radical expressions involving square roots of whole numbers and cube roots of integers. A.F.1 — The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.

1. Equations & Inequalities mathspace.co

A2.EI.1 — The student will represent, solve, and interpret the solution to absolute value equations and inequalities in one variable.

A2.EI.4 — The student will represent, solve, and interpret the solution to an equation containing rational algebraic expressions.

A2.EI.2 — The student will represent, solve, and interpret the solution to quadratic equations in one variable over the set of complex numbers and solve quadratic inequalities in one variable.

A2.EO.3 — The student will perform operations on polynomial expressions in two or more variables and factor polynomial expressions in one and two variables.

Continuous Assessment Measure standards proficiency with check-ins Before starting a new topic, it’s a great time to go online and have students complete a Skills Check-in to measure their readiness for the topic.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

1.01 Algebraic expressions Subtopic overview Lesson narrative In this lesson, students will review important vocabulary terms related to algebraic expressions and use them to represent relationships and quantities that describe a real-world context. They will identify different parts of an expression, be able to evaluate the expressions at specific inputs, and explain what they mean in context. They will represent algebraic expressions pictorially using algebra tiles, including expressions with grouping and like terms. Then, they will use the structure of an expression to understand what it represents in order to create algebraic expressions to represent contextual problems. By the end of this lesson, students should be comfortable recognizing the different components that make up an algebraic expression and understand how to represent a real-world scenario as an algebraic expression.

Learning objectives Students: Page 4

Key vocabulary 

algebraic expression



base



coefficient

 constant term



exponent



expression



factor

 like terms



term



variable

Essential understanding The structure of an expression can reveal important details about the situation it represents.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving Teachers can incorporate this goal into their lesson by providing a variety of real-world situations related to algebraic expressions. Teachers can have students develop and solve problems that can be represented by algebraic expressions.

1.01 Algebraic expressions mathspace.co

MPG5 — Mathematical Representations Teachers can incorporate this goal into their lesson by having students represent algebraic expressions in multiple ways. For example, teachers can guide students to create both verbal and symbolic representations of the same situation. Teachers can also encourage students to visually represent the process of evaluating expressions with a variety of mathematical operations.

Content standards A.EO.1 — The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

A.EO.1b — Evaluate algebraic expressions which include absolute value, square roots, and cube roots for given replacement values to include rational numbers, without rationalizing the denominator.

A.EO.1a — Translate between verbal quantitative situations and algebraic expressions, including contextual situations.

Prior connections 7.PFA.2 — The student will simplify numerical expressions, simplify and generate equivalent algebraic expressions in one variable, and evaluate algebraic expressions for given replacement values of the variables.

8.PFA.1 — The student will represent, simplify, and generate equivalent algebraic expressions in one variable.

Future connections A.EO.2 — The student will perform operations on and factor polynomial expressions in one variable. A.EO.4 — The student will simplify and determine equivalent radical expressions involving square roots of whole numbers and cube roots of integers.

A.EI.1 — The student will represent, solve, explain, and interpret the solution to multistep linear equations and inequalities in one variable and literal equations for a specified variable.

Engage Activity That’s a number game

60 mins

Students will write mathematical expressions based on a number game where any input always produces the same answer.

Understanding and skills

Will use

Will develop

Writing an algebraic expression from a verbal description.

Representing a given scenario as an algebraic expression.

Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: Paper, pencil

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Support students with disabilities Support memory - solve multistep problems This task will require students to perform numerous calculations and then act on those results. To support them, provide resource sheets, templates, or organizers for recording information. Students may also break the problem into smaller chunks. Lastly, allow the use of calculators.

Support for English language learners Collect and display As pairs are working, listen for and collect vocabulary, phrases, and methods students use for writing an expression to represent the number game, along with creating and representing their own number game. Consider grouping language for each part of the process (observations about the given number game, forming an expression to represent the given number game, and creating and representing their own number games). Continue to update collected student language throughout the entire activity. Remind students to borrow language from the display as needed.

Classroom guide Hook Students choose one of the four numeric expressions with varying operations.

Which one doesn’t belong

•

5 mins

Which one doesn't belong? 5 + 18

5×4+3

28 − 5

14 + 11

Slide 1 from Student Engage Activity

Implementation details Encourage students to think about the number of terms, the value of the simpliﬁed expression, the operations, as well as the terms themselves.

Launch

5 mins

Provide students time to read the steps in the game individually before forming groups. Suggested grouping: Form groups of 3 or 4 and assign roles Think of a number. • Multiply it by 2. • Add 6. • Double it again. • Subtract 8. • Divide by 4. • Take away the number you first thought of. What number are you left with? Slide 2 from Student Engage Activity

1.01 Algebraic expressions mathspace.co

Continue when Students have read the Launch and understand the game.

Explore

Team roles

•

35 mins

Anticipated strategies Work backwards from the answer Students may choose an answer they wish to have as the result of their number game, and work backwards to choose operations which always result in that number.

Use inverse operations Students may recognize that by using some inverse operations or a combination of operations where the net result undoes all of them, will result in their expression always simplifying to the same number.

Create written description and corresponding expression Students will create a written description and a corresponding expression. They will show that their expression always produces the same result no matter which number is used as the input.

Misconceptions Incorrectly writing the order of operations in an expression Starting with your description, what order must the operations be completed in? Can we try your number game with an example? Do we get the value you anticipated would be the answer? Why or why not?

Purposeful questions Use the following questions to check for understanding and encourage critical thinking: • Which operations will you use? • How can you verify that your expression will always evaluate to the same number? How do you know? • If you perform an operation, say addition, how could you undo this operation in order to have no effect on your ﬁnal answer?

Continue when Students have created a written description and corresponding expression that always produces the same result no matter which number is used as the input.

Discuss

15 mins

Invite each groups’ Represent to share their number game and reasoning with the rest of the class. Consider starting with groups that wrote a written description and corresponding expression followed by other methods, such as working backwards from the answer or using inverse operations.

Discussion guide Have each group test their written description with the class and have the class play along by choosing their own starting number to plug in. Then, after the class veriﬁes that the description produces the correct output, ask the class to verify if their expression matches the description that was shared. After testing each group’s written description, call on several groups to share their strategies for guaranteeing that the output would always be the same number. If possible, have students that wrote written description and corresponding expression start, then have groups that worked backwards from the answer or use inverse operations share. 10

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 7 — 3.04 Evaluate algebraic expressions Grade 8 — 2.01 Represent algebraic expressions Grade 8 — 2.02 Simplify expressions and distributive property

Tools You may find this tool helpful: • Scientific calculator

Lesson supports The following supports may be useful for this lesson. More specific supports may appear throughout the lesson:

Connecting verbal and algebraic expressions Targeted instructional strategies In order to review key terms for algebraic expressions and practice evaluating expressions with replacement values, consider giving students the following set of steps and ask students to try it with a number of their choice: 1. Think of a number. 2. Double the number 3. Add 9 to the result 4. Subtract 5 from the result 5. Find half of the result. 6. Decrease the result by the original number. 7. You should get 2 every time. Show students the corresponding algebraic representations in each row of the table as they evaluate using their original number. Words Think of a number Double the number Add 9 to the result Subtract 5 from the result

Algebraic expression

Evaluate

x 2x 2x + 9 (2x + 9) − 5

Find half of the result Decrease the result by the original number Work through simplifying the last expression to show how the expression simplifies to 2.

1.01 Algebraic expressions mathspace.co

Student lesson & teacher guide Algebraic expressions Students are reminded of important vocabulary that arises when working with algebraic expressions before engaging in an exploration to uncover the meaning of the parts of an expression.

Students: Page 4

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Anatomy of an algebraic term Targeted instructional strategies Break down individual terms into their basic components so that students can understand them on a fundamental level. For any given term, show that it consists of: • The coeﬃcient • The variable(s) The coefficient has a sign and a magnitude or absolute value. Breaking down a term visually can assist with the explanation: Sign

Coefficient Variables

and can also be used to help explain less obvious cases like: Sign

Coefficient Variable

Collect and display English language learner support Create a chart with columns: “Context” and “Mathematical Representation.” Give students a story problem such as: A bucket fills with water at a rate of 100 mL per hour. The bucket initially has 20 mL in it. The mathematical phrases from the problem such as “rate of 100 mL per hour” and “initially has 20 mL” should go in the “Context” section of the chart, and the expression 100h + 20 should be written in the “Mathematical Representation” column. Write operations for key words such as “rate” meaning multiplication and “initially” meaning a constant value being added. Consider adding visual aids to show how terms relate to algebraic expression components. Work through additional story problems, adding additional key terms and their operations throughout the lesson.

Clarifications about algebraic terms Address student misconceptions Common misconceptions when first learning about algebraic expressions are: • Not including the negative sign as part of the coefficient • Equating the absence of a coefficient to mean that the coefficient is 0 (consider that x has a coefficient of 1, not 0) • Assuming that all symbols must be variables (consider that π is not a variable) Having students highlight and label key words that indicate operations and order in the problem will help ensure students are making sense of the problem as they work.

1.01 Algebraic expressions mathspace.co

Exploration Students: Page 5

Exploration In order to write an expression that can be used to model the total cost of buying new school supplies, Mr. Okware deﬁnes the following variables: Let x represent the cost of a folder, y represent the cost of a calculator, and z represent the cost of a pencil pack. 1.

What could the following expressions represent in this context? • x+y • 2x + 10z • x+y+z

• x + 3y + 4z

• 5y

• 4(4x + y + 2z)

In this context, what do the coeﬃcients describe?

What expressions could we write that wouldn’t make sense in this context?

Expressions and parts of expressions, like factors and coefficients, all have unique meanings in a given context. Viewing expressions in parts and as a whole while paying attention to the quantities represented by the variables can explain the relationships described by the expressions.

Suggested student grouping: In pairs can use algebra tiles to help us visualize algebraic expressions. The tile x represents an unknown number. StudentsWewill choose an algebraic expression that they think appropriately represents the presented scenario, The tile +1 represents adding one unit and −1 represents subtracting one unit. giving reasons for their choice and identifying why they think the other options are not suitable. Positive

Negative

Ideal student responses These ideal responses mayVariable differ from student responses can be −x Less −x or formal +x or +x responses. tiles other correct connected with the more precise mathematical language presented here. 1. What could the following expressions represent in this context? +1 −1 Unit tiles x+y cost of one folder and one calculator x + demonstrates y+z cost ofexpressions one folder, one calculator, This table how can be built using theand tiles:one pencil pack 5y 2x + 10z x + 3y + 4z 4 (4x + y + 2z)

cost of five calculators Algebraic Word expression

cost of two folders andexpression ten pencil packs

Representation with algebra tiles

+x four+1pencil +1 cost of one folder, three calculators, and packs

Five more than twice x

2x + 5

+1 +1 and +1 two pencil packs four times the cost of four folders, one+xcalculator,

2. In this context, what do the coeﬃcients describe? The sum of negative x, −x +item. 4 + 2x The coeﬃcients describe the quantity of each four, and double x

−x

3. What expressions could we write that wouldn’t make sense in this context? +x −1 −1because −1 −1 it is not possible to Expressions that don’t have integer coefficients would not make sense Three times the +x variables −1 −1 in−1a single −1 purchase only part of an item. Expressions the multiple term, for example 4xy, 3(x −include 4) difference of x and four would also not make sense because the product of two item +x costs −1does not represent anything meaningful. −1 −1 −1 Purposeful questions • What information is required to calculate the cost of some items? • What is being added together in these expressions? Possible misunderstandings • Students may misinterpret the question and think that the variables represent the quantities of each item. Point out to students that the coefficients for the variables are different across the different expressions, so the coefficient should correspond to something in the context that can change. In this case, it would be the 1.01 Algebraic expressions 5 quantity of items that needs to be purchased. mathspace.co

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Mr. Okware deﬁnes the following variables: Let x represent the cost of a folder, y represent the cost of a calculator, and z represent the cost of a pencil pack. 1.

What could the following expressions represent in this context?

• x + that y • 2xa+ meaning 10z Students are reminded each part of an algebraic expression has and can provide information about • x + y +Algebra z • x +to 3yrepresent + 4z the context it represents. tiles are used to show a visual way algebraic expressions, including 5y • 4(4x + y + 2z) distributing and like• terms. 2.

In this context, what do the coeﬃcients describe?

Students: Page 5 expressions could we write that wouldn’t make sense in this context? 3. What Expressions and parts of expressions, like factors and coefficients, all have unique meanings in a given context. Viewing expressions in parts and as a whole while paying attention to the quantities represented by the variables can explain the relationships described by the expressions. We can use algebra tiles to help us visualize algebraic expressions. The tile x represents an unknown number. The tile +1 represents adding one unit and −1 represents subtracting one unit. Positive +x

Variable tiles

Unit tiles

Negative −x

−x

−1

This table demonstrates how expressions can be built using the tiles: Algebraic expression

Word expression Five more than twice x

2x + 5

The sum of negative x, four, and double x

−x + 4 + 2x

Three times the difference of x and four

Representation with algebra tiles

3(x − 4)

−x

−1

Examples Students: Page 6 Example 1

1.01 Algebraic expressions mathspace.co

Create a model using algebra tiles for the following algebraic expression: −7 + 5(1 − 3x)

Create a strategy Algebra tiles use tiles with +1, −1, +x, and −x to represent the individual terms. The number outside of the parentheses for the distributive property represents how many groups of identical expressions will be made.

Apply the idea We will need the −x, +1, and −1 tiles to make our model. The 5 outside of the parentheses means we will need 5 groups of 1 − 3x. −1 −1 −1

−x

−1

−x

−1

−x

−1

−x

−1

−x

1.01 Algebraic expressions mathspace.co

−7 + 5(1 − 3x)

Example 1

Apply the idea

Create a model using algebra tiles for the following algebraic expression: We will need the −x, +1, and −1 tiles to make our model. The 5 outside of the parentheses means we will need 5 −7 + 5(1 − 3x) groups of 1 − 3x.

Create a strategy

−1

Algebra tiles use tiles with +1, −1, +x, and −x−1to represent the individual terms. The number outside of the parentheses for the distributive property represents how−1many groups of identical expressions will be made. +1 −x −x −x

Apply the idea

−1

−x

−x The 5 −x −xthe parentheses means we will need 5 We will need the −x, +1, and −1 tiles to make−1our+1model. outside of

groups of 1 − 3x.

−1

−x

−1 −1

−x

−1 −1

−1 +1 −x −x −x PurposeExample 2 −1 +1 −x −x −x Check that students can create a model using algebra tiles to represent an algebraic expression. This involves Vincenzo runs a removalist company that charges $37.50 per hour plus a one-off truck hire fee of $150.00. +1 to represent −x −x different −x understanding the distributive property and−1how terms using tiles.

Write an expression that models how much he charges for a job that lasts a hours. −1

−x

Students: Page 6

Create a strategy We need to look at the two values that affect the price of the job; the cost per hour of $37.50 and the truck hire fee of $150.00.

Example 2

For each hour worked, Vincenzo charges an additional $37.50. Let’s consider a few cases: Cost of working hour: $150company + $37.50that charges $37.50 per hour plus a one-off truck hire fee of $150.00. Vincenzo runs a1removalist Cost 2 hours: $150 + $37.50 + $37.50 Writeof anworking expression that models how much he charges for a job that lasts a hours. Cost of working 3 hours: $150 + $37.50 + $37.50 + $37.50

Createthat a strategy Notice for each additional hour worked, we add an additional $37.50. Multiplication is repeated addition, so we can multiply $37.50 by the of hours of adding repeatedly. We need to look at the twonumber values that affectinstead the price of the job; the cost per hour of $37.50 and the truck hire fee of $150.00.

Apply the idea Reflect check For each hour worked, Vincenzo charges an additional $37.50. Let’s and consider a few cases: Since the $150 is1 ahour: one-time this value will remain For this problem, we would replace a with the number of Cost of working $150 fee, + $37.50 constant. Next, we multiply $37.50 by the number of hours Vincenzo works on a particular job, and the result Cost of working 2 hours: $150 + $37.50 + $37.50 hours which is a. would be the amount of money he makes on that job. Cost of working 3 hours: $150 + $37.50 + $37.50 + $37.50 Cost: 37.5a + 150 Notice that for each additional hour worked, we add an additional $37.50. Multiplication is repeated addition, so we can multiply $37.50 by the number of hours instead of adding repeatedly.

Apply the idea

Reflect and check

Since the $150 is a one-time fee, this value will remain Mathspace Virginia SOL$37.50 Algebra by 1 the number of 6constant. Next, we multiply mathspace.co hours which is a.

For this problem, we would replace a with the number of hours Vincenzo works on a particular job, and the result would be the amount of money he makes on that job.

Cost: 37.5a + 150

Purpose Mathspaceto Virginia Algebra 1 6 students Challenge form SOL a linear algebraic expression representing a given scenario. mathspace.co

Expected mistakes Students may not recognize that the truck hire fee should only be included once, and incorrectly multiply by a to get the expression 187.50a. Point out that the cost per hour of $37.50 matches with the unit of hours for a, whereas the truck hire fee is a ﬂat cost with no reference to the number of hours.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Students: Page 7

Example 3 If the area of a square is given by the expression (2x − 1)2, explain what 2x − 1 represents in the context of the problem.

Create a strategy First, we need to remember the formula for the area of a square. We know that the area of a square is given by s2 where s represents the side length.

Apply the idea

Reflect and check Area = s

= (2x − 1)2

From here, we could easily use algebraic expressions to determine the perimeter or other useful measurements of the square.

Therefore, we have: s = 2x − 1 This shows that s which represents the side length is Example given by 2x 3 − 1. Thus, the side length of the square is 2x − 1. If the area of a square is given by the expression (2x − 1)2, explain what 2x − 1 represents in the context of the problem.

summary CreateIdea a strategy

PurposeFirst, weExpressions can be used to represent relationships. In an expression, sums often represent need to remember the formula for themathematical area of a square. totals, coefficients, and factors representof and exponents represent repeated Check students’ understanding of the concept area of a square and how to relate it withmultiplication. algebraic expressions. 2 multiplication, We know that the area of a square is given by s where s represents the side length. When interpreting an expression in context, we can use the units to help understand the meaning.

Expected mistakes Apply the idea Reflect and check Students may not recognize that2 2x − 1 represents the side length of the square, not the area.

From here, we could easily use algebraic expressions to determine the perimeter or other useful measurements of = (2x − 1) Reflecting with In life, the students order in which we do things is important. For example, we put on socks then shoes, rather than shoes and the square. then socks. It’s important to understand the connection between the algebraic expressions and geometric properties. Therefore, we have: Area = s Evaluate expressions 2

The order of operations the evaluate expressions with multiple operations, so that same property of Here, the expression 2x − 1 is a parttoof the formula for area, but also represents a the physical stells =not 2xus−just 1 steps numerical resultlength. is achieved. The order goes: the square - its side

ThisComplete shows that which represents the side symbols length is such as brackets […], parentheses (…), or absolute values ∣…∣. 1. all soperations within grouping given by 2x − 1. Thus, the side length of the square is symbols, do the innermost operation first. If there7are grouping symbols within other grouping Students:2xPage − 1. 2. Evaluate all exponents, such as squares and cubes. 3. Multiply and/or divide in order from left to right. 4. Add or subtract in order from left to right.

Idea summary

We often want to substitute values for the variables in an algebraic expression. That way we can evaluate the Expressions can be used to represent mathematical relationships. In an expression, sums often represent expression to yield a numerical result. totals, coefficients, and factors represent multiplication, and exponents represent repeated multiplication. As an example, a concession stand sells in bags of popcorn $2.50 eachto and hotunderstand dogs for $1.50 each. The total cost When interpreting an expression context, we canfor use the units help the meaning. of buying p bags of popcorn and h hot dogs can be represented by the expression 2.50p + 1.50h. Riley goes to the concession stand every Saturday and buys 2 bags of popcorn and 4 hot dogs for her friends, and wants to determine the total cost of her order 3 Saturdays in a row. Evaluate expressions In life, the order in which we do things is important. For example, we put on socks then shoes, rather than shoes and then socks.

Evaluate expressions The order of operations tells us the steps to evaluate expressions with multiple operations, so that the same

numerical result achieved. The order goes: Students review that theisorder of operations can be used to simplify expressions, and numbers can act as replacement inallalgebraic example introducing how(…), groups of operations 1. values Complete operationsexpressions. within groupingAn symbols such is as given brackets […], parentheses or absolute values ∣…∣. can be If therethe are distributive grouping symbols within other grouping symbols, do the innermost operation first. 1.01 Algebraic expressions 7 represented using property. 2. Evaluate all exponents, such as squares and cubes.

mathspace.co

3. Multiply and/or divide in order from left to right. 4. Add or subtract in order from left to right. We often want to substitute values for the variables in an algebraic expression. That way we can evaluate the expression to yield a numerical result. As an example, a concession stand sells bags of popcorn for $2.50 each and hot dogs for $1.50 each. The total cost of buying p bags of popcorn and h hot dogs can be represented by the expression 2.50p + 1.50h. 1.01 Algebraic expressions Riley goes to the concession stand every Saturday and buys 2 bags of popcorn and 4 hot dogs for her friends, and mathspace.co wants to determine the total cost of her order 3 Saturdays in a row.

Idea summary Expressions can be used to represent mathematical relationships. In an expression, sums often represent totals, coefficients, and factors represent multiplication, and exponents represent repeated multiplication. Students: Pages 7–8 When interpreting an expression in context, we can use the units to help understand the meaning.

Evaluate expressions In life, the order in which we do things is important. For example, we put on socks then shoes, rather than shoes and then socks. The order of operations tells us the steps to evaluate expressions with multiple operations, so that the same numerical result is achieved. The order goes: 1. Complete all operations within grouping symbols such as brackets […], parentheses (…), or absolute values ∣…∣. If there are grouping symbols within other grouping symbols, do the innermost operation first. 2. Evaluate all exponents, such as squares and cubes. 3. Multiply and/or divide in order from left to right. 4. Add or subtract in order from left to right. We often want to substitute values for the variables in an algebraic expression. That way we can evaluate the expression to yield a numerical result. As an example, a concession stand sells bags of popcorn for $2.50 each and hot dogs for $1.50 each. The total cost of buying p bags of popcorn and h hot dogs can be represented by the expression 2.50p + 1.50h. Riley goes to the concession stand every Saturday and buys 2 bags of popcorn and 4 hot dogs for her friends, and wants to determine the total cost of her order 3 Saturdays in a row. Since the same order will be repeated 3 times, the distributive property will be used to rewrite the expression. By substituting the values p = 2 and h = 4 into the new expression, we see that Total Spent = 3(2.50p + 1.50h)

Rewrite expression.

= 3(2.50 ⋅ 2 + 1.50 ⋅ 4)

Substitute values

= 3(5 + 6)

Evaluate the multiplication

= 3 ⋅ 11

Evaluate the addition

1.01 Algebraic expressions mathspace.co

The total cost of Riley’s purchase is $33.

Example 4 Evaluate: Advanced learners: Use coding for repeated substitution

Targeted instructional strategies

(u + v) (w − y)

when u = 5, v = 8, w = 2, and y = 10.

As an extension for advanced learners or for all students, show students how to use Google Colab or another computing resource to compile code for simple tasks like repeated evaluation using substitution. Create a strategy We will replace each variable in the expression with their given value. Then we will use the order of operations to For example, to evaluate 4x2 − 3x + c for a variety of values of c and x, a function could be defined, and then evaluate the numerical expression. different values could be put in the last line before running the program.

Apply the idea

def fun(c,x)

#Substitution

(u + v) (w − y) = (5 + 8) (2 − 10) = (13) (− 8)

−104*x+c ans=4*x**=2-3

print(ans)

Substitute u = 5, v = 8, w = 2, and y = 10 Evaluate the operations inside the parentheses Evaluate the multiplication

fun(5,8) Example 5

value of: how to use external resources to write code to increase independence or be shown in StudentsFind canthebe shown small group instruction for those developing coding skills. when x = −4 and y = 3.

Create a strategy We will replace each variable in the expression with their given value. Then we will use the order of operations to evaluate the numerical expression.

Apply the idea Substitute x = −4 and y = 3

Evaluate the exponents Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co Find a common denominator Evaluate the multiplication

Since the same order will be repeated 3 times, the distributive property will be used to rewrite the expression. By substituting the values p = 2 and h = 4 into the new expression, we see that Total Spent = 3(2.50p + 1.50h)

Examples Students: Page 8

Rewrite expression.

= 3(2.50 ⋅ 2 + 1.50 ⋅ 4)

Substitute values

= 3(5 + 6)

Evaluate the multiplication

= 3 ⋅ 11

Evaluate the addition

The total cost of Riley’s purchase is $33.

Example 4 Evaluate: (u + v) (w − y) when u = 5, v = 8, w = 2, and y = 10.

Create a strategy We will replace each variable in the expression with their given value. Then we will use the order of operations to evaluate the numerical expression.

Apply the idea (u + v) (w − y) = (5 + 8) (2 − 10)

Substitute u = 5, v = 8, w = 2, and y = 10

= (13) (− 8)

Evaluate the operations inside the parentheses

= −104

Evaluate the multiplication

PurposeExample 5 StudentsSince demonstrate thatwill they can substitute integers intoproperty a simple and evaluate. the same order be repeated 3 times, the distributive willexpression be used to rewrite the expression. Find the value of: By substituting the values p = 2 and h = 4 into the new expression, we see that

Expected mistakes Total Spent = 3(2.50p + 1.50h) Rewrite expression. Studentswhen mayx forget follow the order⋅ 2of+ operations and perform = 3(2.50 1.50 ⋅ 4) Substitute values multiplication before addition or subtraction. = −4 andtoy = 3. Emphasize the importance of the order in ensuring accurate results. = 3(5 + 6) of operations Evaluate the multiplication Create a strategy

= 3 ⋅ 11

Evaluate the addition

We replace variable in the expression with their given value. Then we will use the order of operations to Example 4 Step-by-step procedure use with Thewill total cost ofeach Riley’s purchase is $33. evaluate the numerical expression. Support students with disabilities

Example 4 beneﬁt from a step-by-step guide to solving problems like this one. Here’s a clear Some students may Apply the idea procedure they can follow: Evaluate:

Substitute x = −4 and y = 3 (u + v) (w − y) Evaluate the exponents

1. Identify the given values for each variable.

when u the = 5, given v = 8, wvalues = 2, andinto y = 10. 2. Substitute the expression.

Find a common 3. Perform the operations inside the parentheses ﬁrst.denominator Create a strategy

4. Finally, multiplication tothe get the result. We perform will replacethe each variable in the operation expression with their given value. Then we will use the order of operations to Evaluate multiplication evaluate the numerical expression.

Using this step-by-step guide, students can gradually understand the process of evaluating expressions with Evaluate the addition the idea multipleApply variables and operations. (u + v) (w − y) = (5 + 8) (2 − 10)

Students: Pages 8–9 8

Substitute u = 5, v = 8, w = 2, and y = 10

= (13) (− 8)

Evaluate the operations inside the parentheses

= −104

Evaluate the multiplication

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 5 Find the value of:

when x = −4 and y = 3.

Create a strategy We will replace each variable in the expression with their given value. Then we will use the order of operations to evaluate the numerical expression.

Apply the idea Substitute x = −4 and y = 3 Evaluate the exponents Find a common denominator

1.01 Algebraic expressions mathspace.co

when x = −4 and y = 3.

Create a strategy We will replace each variable in the expression with their given value. Then we will use the order of operations to evaluate the numerical expression.

Apply the idea Substitute x = −4 and y = 3 Evaluate the exponents Find a common denominator Evaluate the multiplication Evaluate the addition

Reflect and check We can apply the order of operations to problems with any type of real number such as integers, fractions, or decimals. 8

Mathspace Virginia SOL Algebra 1 mathspace.co

Example 6 Purpose x = 5 and y = 4, evaluate: StudentsFordemonstrate that they can evaluate expressions involving powers. Expected mistakes correct to two decimal places. Students may not include the parentheses around the −4 and as a result get −16 when they apply the exponent. Reflect aand check Create strategy

We can apply order of operations to problems with anygiven type value. of real Then number as integers, or decimals. Students:We Page 9 theeach will replace variable in the expression with their wesuch will use the orderfractions, of operations to evaluate the numerical expression.

Apply the idea Example 6 For x = 5 and y = 4, evaluate: correct to two decimal places.

Substitute x = 5 and y = 4 Evaluate the exponent Evaluate the multiplication Evaluate the addition

Create a strategy

Use a calculator to calculate the square root

We will replace each variable in the expression with their given value. Then we will use the order of operations to evaluate the numerical expression.

Idea summary

Apply the idea

Substitute the given value for each variable and then apply the order of operations: Substitute x = 5 and y = 4 1. Complete all operations within grouping symbols such as brackets […], parentheses (…), or absolute values Evaluate thesymbols, exponent ∣…∣. If there are grouping symbols within other grouping do the innermost operation first. 2. Evaluate all exponents such as squares and Evaluate cubes. the multiplication 3. Multiply and/or divide in order from left to right. Evaluate the addition 4. Add or subtract in order from left to right.

Use a calculator to calculate the square root

Practice

Idea summary

Purpose Substitute the given value for each variable and then apply the order of operations: do you remember? StudentsWhat demonstrate that they can evaluate an expression inside a square root. 1. Complete all operations within grouping symbols such as brackets […], parentheses (…), or absolute values ∣…∣. If there are grouping symbols within other grouping symbols, do the innermost operation first.

Match the following terms with their deﬁnitions: Expected1 mistakes 2. Constant Evaluate all exponents such as squares cubes. an unknown number i Symbol that and represents Students maya incorrectly order the steps of calculation, such as taking the square root before completing all 3. Multiply and/or divide in order from left to right. b Variable ii A purely numeric term in an algebraic expression multiplication and addition. Add or subtract in orderiiifromThe leftvalue to right. c4. Coeﬃcient that indicates how many of a variable in a term

d students Algebraic term iv Term including a variable Reﬂecting with After solving, teachers can facilitate a discussion around the importance of order of operations in mathematics. 2 Write the expression 9x in words. Practice Ask the students what would have happened if they had taken the square root before completing the other 3 State whether the following expressions are like terms: operations. andremember? 3p b 2p and 15p c 9p and 6q d 12 and 8 Whata do11pyou 4 3 4 4 3 k h

g 8h k and 9k h h 6h and 7k e 5h and 5hk f 4h and 3h Mathspace Virginia SOL Algebra 1 Teacher Edition 1 Match the following terms with their deﬁnitions: mathspace.co 1.01 Algebraic expressions a Constant i Symbol that represents an unknown number mathspace.co b Variable ii A purely numeric term in an algebraic expression c Coeﬃcient iii The value that indicates how many of a variable in a term

Substitute x = 5 and y = 4 Evaluate the exponent Evaluate the multiplication Evaluate the addition

Students: Page 9

Use a calculator to calculate the square root

Idea summary Substitute the given value for each variable and then apply the order of operations: 1. Complete all operations within grouping symbols such as brackets […], parentheses (…), or absolute values ∣…∣. If there are grouping symbols within other grouping symbols, do the innermost operation first. 2. Evaluate all exponents such as squares and cubes. 3. Multiply and/or divide in order from left to right. 4. Add or subtract in order from left to right.

Practice What do you remember?

Practice 1

Match the following terms with their deﬁnitions:

a Constant Students: Pages 9–12

Symbol that represents an unknown number

Variable

A purely numeric term in an algebraic expression

Coeﬃcient

iii

The value that indicates how many of a variable in a term

Term including a variable

What do youd remember? Algebraic term 1

the expression 9x in words. Match2theWrite following terms with their deﬁnitions:

3 State whether the following expressions arerepresents like terms: an unknown number Constant i Symbol that

a 11p and 3p Variable

b 2p and 15p and 6q d 12 and 8 A purely numeric term inc an9palgebraic expression 4 3 4 4 3 k

Coeﬃcient

iii

The value that indicates how many of a variable in a term

Algebraic term

Term including a variable

5h and 5hk

4h and 3h

Write the expression 9x in words.

State whether the following expressions are like terms:

8h k and 9k h

6h and 7kh

1.01 Algebraic expressions mathspace.co

11p and 3p

2p and 15p

9p and 6q

12 and 8

5h and 5hk

4h4 and 3h

8h3k4 and 9k4h3

6hk and 7kh

3x + 2y − 8x + 9

8p + 5

For the following algebraic expressions: i

State the number of terms.

Identify the numerical coeﬃcient of the ﬁrst term.

iii

Identify the constant term.

Determine if the expression contains like terms.

2x + 4

7y + 3 + 5x

Consider the expression x + 6. a

Find the value when x = 4.

Find the value when x = 9.

What happens to the value of x + 6 as we substitute in different values of x?

Evaluate the following expressions to one decimal place: a

14.5 − 4x when x = 4.2

18.6 − 3x when x = 4.1

m2 + 9n

If m = −3 and n = 4, evaluate the following expressions: a

mn − (m − n)

1.01 Algebraic expressions mathspace.co

Let’s practice 8

Write an algebraic expression for the following diagrams: a

−x −x +1

+x −1 −1 −1 −1

−x

−1

+x +x

Use the following algebra tiles to draw a diagram that represents each expression:

Variable tiles

Negative

x + 12 − 6x

−x

Unit tiles

−x

−1

−6 + 4(1 + 2x)

2(−x − 2) + 3x

7x + 8 + x − 11

Evaluate the following expressions: a

when m = 5

when x = 3

when a = 5 and b = 2

when a = 3 and

s∣−12 + t∣ when s = −3 and t = 10

when p = −7 and q = −8

g i k m

when a = 5 and b = 12 when a = −4 when r = 6 and s = 7 when x = 2 and y = 5 and b = 2

when

when y = −2 and

s 22

Positive

(a + b) (c − d) when a = 6, b = 9, c = 4 and d = 14

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

h j l n

when p = 6 and q = −2 when p = 2 and q = 4 when x = −2 and y = −2 when a = −9 and b = 10

5y − z2 + 9 when y = 7 and z = 4

y3 + 4yz − 2x2 when x = −4, y = 2 and z = 1

Dylan purchased 3 pens, 4 pencils and a single note pad. The total cost of all these items was 3x + 4y + 5 dollars. a

Interpret the variable y in context of the problem.

Interpret the variable x represent.

Interpret what the term 5 represents.

Explain how this expression could be revised to represent purchasing an unknown number of notepads.

Deborah earns $25 per hour of work and is paid double for every hour worked on the weekend. At the end of a week, Deborah calculates her pay for that week to be 25x + 50y dollars. a

Identify an expression that represents the number of hours Deborah worked during the weekdays.

Identify an expression that represents the amount of money Deborah earned from working on the weekend.

Mohamad and Valentina are throwing a party. Mohamad wants to reserve all street parking within a certain distance from their house. They are located 150 ft down a street. Write an expression for the endpoints of the parking area based on the distance on the road.

Brad and Patricia are making paper cranes to decorate their room. Brad can make m an hour while Patricia can make n an hour. Brad spends 6 hours making cranes while Patricia spends 5 hours over the weekend. Together they make a total of 6m + 5n paper cranes over the weekend. a

Interpret the meaning of the term 6m.

Interpret the meaning of the term 5n.

The side length of a square box is 3x + 1 yards. Write an expression for its area.

Roxy is 5 inches taller than Jane, while Katrina is 9 inches shorter than Jane. Write expressions that can be used to represent the height of each person.

Valerie is with a mobile phone provider that charges $0.26 per minute plus a connection fee of $0.45. Write an expression that models how much she will be charged for a call that lasts a minutes.

An expression for the surface area of cube shown in the image is 6s2. a

Interpret the meaning of the coeﬃcient.

Interpret the meaning of the s2.

Let’s extend our thinking 19

For each of the following, determine if the statement is true or false. Explain your reasoning. a

An expression must include a variable.

Any expression with more than one term can be simpliﬁed.

Tobias has two times as many books as Isabelle does, and Isabelle has four less than triple the number of books that William does. Explain a method that can be used to write an expression for the number of books that Tobias has.

The width of a rectangle is 19 cm less than double the length. a

Construct an expression that models the width of the rectangle.

Revise the expression in part (a) to create a model for the perimeter of the rectangle and a model for the area of the rectangle.

Determine whether there are any values of the length of the rectangle that are not viable. Justify your reasoning.

1.01 Algebraic expressions mathspace.co

Four friends share the cost of a pizza. The toppings for the pizza added $4.75 to its base price for plain cheese. a

Construct an expression that models the price for one friend’s share of the cost.

The following table has the prices of various pizza sizes: Small $13

Medium $15

Large $17

Determine which size pizza the friends could fairly split if friend 1 has $5, friend 2 has $5.50, friend 3 has $5, and friend 4 has $6.50. Justify your reasoning.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Answers

1.01 Algebraic expressions What do you remember? 1 a ii

b i

c iii

d iv

2 Example answer: 9 groups of x. 3 a Yes e No

b No

c No

d Yes

g Yes

h No

4 a i 2

ii 2

iii 4

iv No

b i 3

ii 7

iii 3

iv No

c i 4

ii 3

iii 9

iv Yes

d i 2

ii 8

iii 5

iv No

5 a 10 b 15 c T he value of the expression changes depending on the value of x.

−1

d 10

e –2

h –52

–12

k 7

o –30

p 28

q –2

r –16

s –150

10 a

11 a The price of a single pencil.

6 a −2.3

b 6.3

b The price of a single pen.

7 a −5

b 45

c The price of a single note pad.

Let’s practice 8 a −2x + 1

b 5x + 6 − 2x

c 6(x − 4)

d 7 + 3(1 + 2x)

9 a

−x

–8

d T he price of a single notepad is 5. Let n represent the number of notepads purchased, then the expression would be 3x + 4y + 5n. 12 a x

b 50y

13 |x – 150| 14 a T he number of paper cranes Brad made during the weekend. b T he number of paper cranes Patricia made during the weekend. 15 (3x + 1)2 or 9x2 + 6x + 1 16 Expressions will vary. For example: Let j represent Jane’s height in inches. Then ( j + 5) represents Roxy’s height and ( j – 9) represents Katrina’s height.

−1 −1 −1

−1

+x −x

−1

−x

−1

As an alternative, let r represent Roxy’s height. Then (r – 5) represents Jane’s height and (r – 5 – 9) or (r – 14) represents Katrina’s height. 17 (0.26a + 0.45) dollars 18 a Number of squares

b Area one square

Let’s extend our thinking 19 a False, for example 4 + 2 is an expression. b False, only like terms can be combined.

Answers mathspace.co

20 One possible solution: Let n represent the number of books William has. We know that Isabelle has four less than triple the number William has. So, if we were writing the number of book Isabelle has, the constant is -4 as she always has four less than triple William’s books. The coefficient will be 3 because Isabelle has tripled the number that WIlliam has. So, we can write the expression for Isabelle’s books in terms of William’s books: 3n – 4. Tobias has two times as many books that Isabelle has, so the coefficient is 2. Number of Tobias’s books is 2 times Isabelle’s books, which is 3n – 4, so the expression for the number of books Tobias has in terms of the number William has is 2(3n – 4) books. 21 a Let n be length of the rectangle. 2n – 19 cm b Let n be length of the rectangle. 2n – 19 + 2n – 19 + n + n cm or 6n – 38 cm Let n be length of the rectangle. n(2n – 19) cm or 2n2 – 19n cm

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

c A ny length that is 9.5 cm or less would not be possible because the width of the rectangle would then be 0 cm or a negative length, which does not make sense. 22 a Let p represent the cost of the pizza. One friend’s share would be

(p + 4.75) dollars.

b If each friend is paying

the total cost, a small pizza

with toppings would cost a total of $17.75, meaning each should contribute $4.44. Each friend can afford this. A medium pizza with toppings would cost a total of $19.75, meaning each should contribute $4.94. Each friend can afford this. A large pizza with toppings would cost a total of $21.75, meaning each should contribute $5.44. Since two of the friends only have $5, the friends couldn’t fairly order this pizza. If the friends were going to pay an equal amount, they should get the medium pizza. Instead, if each friend contributed all of their money they would have a total of $22 and could afford the large.

1.02 Properties of real numbers Subtopic overview Lesson narrative In this lesson, students will be reminded that real numbers include zero, the positive and negative whole numbers, the numbers that can be written as fractions, and every number in between. Students will learn about the real number properties of addition, subtraction, multiplication, and division for equations and inequalities. They will be provided with steps and solutions for equations and inequalities, and students will learn how to justify the work.

Learning objective

1.02 Properties of real numbers

Students: Page 13

After this lesson, you will be able to… • use the properties of real numbers to rewrite expressions or solve equations.

Review: Properties of real numbers Recall that the numbers that we use regularly for counting and measuring are called the real numbers. They include Key vocabulary zero, the positive and negative whole numbers, the numbers that can be written as fractions, and every number in 

between. additive identity



additive inverse



associative property of addition Natural numbers commutative property of addition



associative property of multiplication Rational numbers commutative property of multiplication



The counting numbers, starting from 1. distributive property



whole number Integers



The set property of numbers can be expressed in the identity of that addition (additive identity) form where a and b are integers and b ≠ 0.  identity property of multiplication (multiplicative identity)  integer Whole numbers  irrational number  natural number The counting numbers, starting from 0. Irrational numbers  rational number  real number The set of numbers that cannot be written in the where a and b are integers.

form

A set of numbers that include positive whole numbers (natural numbers), their negative counterparts, and zero.

Essential understanding

The properties of real numbers can be applied to simplify and evaluate algebraic expressions more easily. Real Number System Rational

Standards

Integers Whole

This subtopic addresses the following Virginia 2023 Mathematics Natural Standards of Learning standards.

Mathematical process goals MPG3 — Mathematical Reasoning

… −4 −3 −2 −1 0 1 2 3 4 …

−112 17

−3.12

−3 2

1 2

1.3

2.6

13 3

This goal can be integrated by asking students to justifyIrrational their use of specific real number properties when simplifying − 102 1+ 5 3 expressions. Teachers can ask students to explain why π 21are relevant for simplifying expressions. − 2certain properties 5

The real numbers are so familiar to us that we hardly notice that they have special properties that involve the addition and multiplication operations. For any real numbers a, b, and c, the following properties are always true. Commutative property Associative property

Addition a+b=b+a (a + b) + c = a + (b + c)

Multiplication a⋅b=b⋅a a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ c

1.02 Properties of real numbers mathspace.co

MPG5 — Mathematical Representations This goal can be integrated into the lesson by asking students to represent the properties of real numbers in different ways. For instance, teachers could have students visually represent the Commutative Property of Addition or Multiplication using number lines or arrays. In addition, students could be asked to create symbolic representations of real-world problems that require the use of real number properties.

Prior connections 8.NS.2 — The student will investigate and describe the relationship between the subsets of the real number system.

Future connections A.EO.2 — The student will perform operations on and factor polynomial expressions in one variable. A.EI.1 — The student will represent, solve, explain, and interpret the solution to multistep linear equations and inequalities in one variable and literal equations for a specified variable.

A.EI.2 — The student will represent, solve, explain, and interpret the solution to a system of two linear equations, a linear inequality in two variables, or a system of two linear inequalities in two variables.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 6.01 Properties of real numbers Grade 8 — 2.02 Simplify expressions and distributive property

Tools You may find these tools helpful: • Scientific calculator

• Index cards

Student lesson & teacher guide Review: Properties of real numbers Students review the classifications of the real number system and are reminded that numbers have special properties of how they can be combined and arranged in different ways. Students review the properties of real numbers for addition and multiplication. A table is provided to clearly lay out the properties using generalized expressions.

Students: Page 13

1.02 Properties of real numbers After this lesson, you will be able to… • use the properties of real numbers to rewrite expressions or solve equations.

Review: Properties of real numbers 28

Recall that the numbers that we use regularly for counting and measuring are called the real numbers. They include Mathspace Virginia SOL Algebra 1 Teacher Edition zero, the positive and negative whole numbers, the numbers that can be written as fractions, and every number in mathspace.co between.

1.02 Properties of real numbers After this lesson, you will be able to… • use the properties of real numbers to rewrite expressions or solve equations.

Review: Properties of real numbers Recall that the numbers that we use regularly for counting and measuring are called the real numbers. They include zero, the positive and negative whole numbers, the numbers that can be written as fractions, and every number in between. Natural numbers

Rational numbers

The counting numbers, starting from 1.

The set of numbers that can be expressed in the where a and b are integers and b ≠ 0.

form

Whole numbers The counting numbers, starting from 0.

Irrational numbers

Integers

The set of numbers that cannot be written in the

A set of numbers that include positive whole numbers (natural numbers), their negative counterparts, and zero.

form

where a and b are integers.

Real Number System Rational Integers Whole Natural … −4 −3 −2 −1 0 1 2 3 4 …

−112 17

−3 2

−3.12

1 2

2.6

13 3

1.3

Irrational − 102 5

−3 2

1+ 5 2

Addition a+b=b+a (a + b) + c = a + (b + c)

Inverse property

a + (−a) = 0

Identity property Distributive property

a+0=a

Multiplication a⋅b=b⋅a a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ c

a⋅1=a a (b + c) = a ⋅ b + a ⋅ c

Notice the distributive property only applies to distributing the operation of multiplication. If the positions of the addition and multiplication signs are swapped so that we have a + (b ⋅ c), a similar distributive rule is not true. This is one situation in which we must observe the correct order of operations when simplifying expressions. 1.02 Properties of real numbers mathspace.co

Create flashcards to help remember properties Targeted instructional strategies Encourage students to create flashcards with the name of the property on one side and the definition and an example on the other. Regularly reviewing these flashcards can help students remember these properties and recognize when to apply them in solving problems.

1.02 Properties of real numbers mathspace.co

Critique, correct, and clarify English language learner support Provide students with examples containing incorrect or incomplete explanations of each property. Students should first identify whether each example is incomplete or incorrect. Examples could include: • Commutative property: a + b = c + a + b • Associative property: a + (b + c) = a + b + c • Distributive property: a(b + c) = a ⋅ b + c • Identity property: a ⋅ 0 = 0 • Inverse property: a + (−a) = 1 Pair students and have them discuss and correct the errors. Encourage them to explain why the corrections are accurate. Have each pair share their corrected explanations with others. Discuss each correction, ensuring students understand why the original examples were incorrect or incomplete and how the corrected versions accurately represent the properties.

Use physical manipulatives to represent properties of real numbers Student with disabilities support Use manipulatives like colored blocks, counters, or algebra tiles to visually represent numbers and their properties. This hands-on approach helps students concretely understand abstract concepts. Ways to represent the properties with different physical manipulatives include: • Commutative property: To show 4 ⋅ 6 = 6 ⋅ 4, create 4 groups of 6 blocks and 6 groups of 4 blocks. Show both arrangements have 24 blocks. • Associative property: To show (2 + 3) + 4 = 2 + (3 + 4), group 2 red blocks and 3 blue blocks, then add 4 green blocks. Rearrange to show both groupings sum to 9. • Distributive property: To show 2(3 + 4), use 2 sets of blocks grouped into 3 and 4 blocks. Rearrange to show the sum of the groups is equal to the product of 2 times 7. • Identity property: To show 5 ⋅ 1 = 5, use 5 blocks in one group to show that multiplying by 1 keeps the number unchanged. • Inverse property: To show 6 + (−6) = 0, Pair 6 red blocks with 6 blue blocks to represent positive and negative values, showing they cancel out to zero.

The properties do not apply to subtraction and division Address student misconceptions Students might assume that since these properties are the properties of real numbers, that they apply to subtraction and division as well. However, we cannot generalize the rules for these operations. Show examples of when the properties do not apply to subtraction or division: • Commutative property: 5 − 3 ≠ 3 − 5 or 6 ÷ 3 ≠ 3 ÷ 6 • Associative property: (6 − 3) − 2 ≠ 6 − (3 − 2) or (8 ÷ 4) ÷ 2 ≠ 8 ÷ (4 ÷ 2)

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Examples Students: Page 14

Example 1 Verify the distributive property a(b + c) = a ⋅ b + a ⋅ c for the values a = 3, b = 13, and c = −1.

Create a strategy We need to substitute the values for a = 3, b = 13, and c = −1 into the distributive property equation a (b + c) = a ⋅ b + a ⋅ c. If both sides of the equation evaluate to the same number, then the property is true for those values.

Apply the idea Evaluate the expressions. a(b + c) = 3 (13 + (−1))

Example 1

Substitute a = 3, b = 13, and c = −1

= 3 (12)

Evaluate the subtraction inside the parentheses

= 36

Evaluate the multiplication

a ⋅ b + a ⋅ c = 3 ⋅ 13 + 3 ⋅ (−1) Substitute a = 3, b = 13, and c = −1 Verify the distributive property a(b + c) = a ⋅ b + a ⋅ c for the values a = 3, b = 13, and c = −1. = 39 − 3 Evaluate the multiplication

Create a strategy = 36

Evaluate the subtraction

So, expressions arethe thevalues same for for athese distributive propertyproperty is true inequation this case. We the need to substitute = 3, bnumbers = 13, andand c = the −1 into the distributive a (b + c) = a ⋅ b + a ⋅ c. If both sides of the equation evaluate to the same number, then the property is true for those Reﬂect and check values. How might you apply the same strategy to verify the other properties of real numbers?

Apply the idea Evaluate the expressions. a(b + c) = 3 (13 + (−1))

Substitute a = 3, b = 13, and c = −1

Example 2 = 3 (12) Evaluate the subtraction inside the parentheses Purpose = 36 Evaluate StudentsUsing demonstrate that they can apply the property the properties of real numbers, rewrite 3distributive (x + 4) the = 9 multiplication in another way.to speciﬁc values and understand the equality it represents. a ⋅ b + a ⋅ c = 3 ⋅ 13 + 3 ⋅ (−1) Substitute a = 3, b = 13, and c = −1 Create a strategy = 39 − 3

Evaluate the multiplication

Expected mistakes First identify the property the the addition and multiplication, the distributive property = 36we will be using. Given Evaluate subtraction Studentsa (bmight apply order + c) = not a ⋅ b correctly + a ⋅ c seems like athe good ﬁt. of operations, potentially calculating the expressions in the So, the expressions are the same for these numbers and the distributive property is true in this case. wrong order. Encourage students to check their work by both simplifying inside the parentheses first and by Apply idea distributing tothe see the two results are equivalent. Reﬂect and ifcheck

The distributive property tells us we can multiply each term in the parentheses individually by the 3 outside the How might you apply the same strategy to verify the other properties of real numbers? Students:parentheses Page 14as shown: 3 (x + 4) = 3 ⋅ x + 3 ⋅ 4 = 3x + 12

Therefore, the expression 3 (x + 4) is equivalent to 3x + 12 by the distributive property. So we may rewrite the equation Example 2 as 3x + 12 = 9. Using the properties of real numbers, rewrite 3 (x + 4) = 9 in another way.

Reﬂect and check

We will use the distributive property many times to expand and simplify expressions. Can you think of any ways this Create a strategy may be helpful for us? First identify the property we will be using. Given the addition and multiplication, the distributive property a (b + c) = a ⋅ b + a ⋅ c seems like a good ﬁt.

Apply the idea The distributive property tells us we can multiply each term in the parentheses individually by the 3 outside the parentheses as shown: 14

Mathspace Virginia SOL Algebra 1 mathspace.co

3 (x + 4) = 3 ⋅ x + 3 ⋅ 4 = 3x + 12

Therefore, the expression 3 (x + 4) is equivalent to 3x + 12 by the distributive property. So we may rewrite the equation as 3x + 12 = 9.

Reﬂect and check We will use the distributive property many times to expand and simplify expressions. Can you think of any ways this may be helpful for us? 1.02 Properties of real numbers

mathspace.co

The distributive property tells us we can multiply each term in the parentheses individually by the 3 outside the parentheses as shown: 3 (x + 4) = 3 ⋅ x + 3 ⋅ 4 = 3x + 12 Therefore, the expression 3 (x + 4) is equivalent to 3x + 12 by the distributive property. So we may rewrite the equation as 3x + 12 = 9.

Reﬂect and check We will use the distributive property many times to expand and simplify expressions. Can you think of any ways this may be helpful for us?

Purpose Students demonstrate how the distributive property can be used to simplify expressions. 14 mistakes Mathspace Virginia SOL Algebra 1 Expected mathspace.co Students might not distribute the 3 to both terms inside the parentheses, resulting in an incorrect expression like 3x + 4 on the left side of the equation. Encourage students to use a test value for x to see if the original expression is equivalent to the expression after distributing.

Reﬂecting with students Encourage students to provide their own examples where the distributive property can be applied and challenge them to rewrite these expressions.

Students: Page 15 Example 3 If 5 ⋅

= 1, use the properties of real numbers to solve for x.

Create a strategy First identify the property we will be using. Given the multiplication by an inverse, the inverse property of multiplication seems like a good ﬁt.

Apply the idea The inverse property of multiplication tells us that for every real number a, there exists a number b such that a ⋅ b = 1. The numbers a and b are called multiplicative inverses of one another. We usually write That is, we can rewrite the property as

instead of b.

This looks a lot like our problem. Let a = 5 for this property. Then, we have by the inverse property of multiplication.

. This tells us x = 5 in our example

Reﬂect and check See if you can challenge yourself to ﬁgure out what the value of x must be in this equation:

Idea summary Purpose The properties of real numbers can be used to simplify expressions and solve equations. Students demonstrate how to use the inverse property of multiplication to solve for a variable in an equation. Addition Multiplication Commutative property

a+b=b+a

a⋅b=b⋅a

Reﬂecting with students Associative property (a + b) + c = a + (b + c) a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ c Point out to students that recognizing the inverse property of multiplication can simplify the solution process in Inverse property a + (−a) = 0 this case. Identitytoproperty a+0 = a might apply this a ⋅ property 1=a Ask advanced learners consider how they in other, more complex, situations. Distributive property a (b c) = athe ⋅ b property +a⋅c An example problem that they may not be familiar with, but can+use to solve, is to identify what x must be in the equation ( y2 − 4) ⋅

= 1.

Practice 32

What do you remember?

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co 1 Determine whether each equation is true or false for every value of x. If the equation is true, state the property that proves it to be true. a x+1=1+x b x⋅3=3⋅x

Example 3

Misuse thetheinverse of real numbers If 5 ⋅ =of 1, use propertiesproperty of real numbers to solve for x.

use with Example 3

Address student misconceptions a strategy StudentsCreate may misuse the inverse property of multiplication and addition. For instance, they might incorrectly the property we will using. Given the multiplication an multiplicative inverse, the inverse property believe First that identify the additive inverse of be a number is always 1, or thatbythe inverse of of a number is always multiplication seems like a good ﬁt. 0. This is a misconception.

The additive Applyinverse the ideaof a number a is −a, which makes the sum of the number and its inverse equal to zero. property ofinverse multiplication us thatafor real number there exists a of number b such that a ⋅ its b = inverse 1. Similarly,The theinverse multiplicative of a tells number is every , which makesa,the product the number and The numbers a and b are called multiplicative inverses of one another. We usually write instead of b. equal to one. That is, we can rewrite the property as

To correct this misconception, provide clear deﬁnitions of the inverse properties of addition and multiplication, . This tells us inverse x = 5 in our This looks a lot examples like our problem. Let a = 5 for this property. Then,the we idea have that the and offer numerous and non-examples. Reinforce additive of example a number is by the inverse property of multiplication. what you add to the number to get zero, and the multiplicative inverse of a number is what you multiply by the number to get one. Reﬂect and check

See if you can challenge yourself to ﬁgure out what the value of x must be in this equation:

Students: Page 15

Idea summary The properties of real numbers can be used to simplify expressions and solve equations. Commutative property Associative property

Addition a+b=b+a (a + b) + c = a + (b + c)

Inverse property

a + (−a) = 0

Identity property Distributive property

a+0=a

Multiplication a⋅b=b⋅a a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ c

a⋅1=a a (b + c) = a ⋅ b + a ⋅ c

Practice What do you remember?

Practice 1

Determine whether each equation is true or false for every value of x. If the equation is true, state the property that proves it to be true.

Students: Pages a x 15–16 +1=1+x

x⋅3=3⋅x

1 ⋅ 8x = 18x

(x + y) + z = x + ( y + x)

(x ⋅ 7) ⋅ 3 = x ⋅ (7 ⋅ 3) What do youe remember? 2

What is the inverse of 3x?

Determine whether each equation is true or false for every value of x. If the equation is true, state the property 3 Use the distributive property to rewrite 5 (2x − y) in another way. that proves it to be true. a

x + 1 = 1 + x

x⋅3=3⋅x

1 ⋅ 8x = 18x

(x + y) + z = x1.02 + ( Properties y + x) of real numbers

(x ⋅ 7) ⋅ 3 = x ⋅ (7 ⋅ 3)

What is the inverse of 3x?

Use the distributive property to rewrite 5 (2x − y) in another way.

mathspace.co

1.02 Properties of real numbers mathspace.co

Let’s Practice 4

Is this number sentence true or false for every positive value of v? v÷3=3÷v

When asked to simplify the expression 7 + 6 (8x + 3), a student provides this work and gets the correct answer. 1

7 + 6 (8x + 3) = 7 + (48x + 18)

= 7 + (18 + 48x)

= (7 + 18) + 48x

= 25 + 48x

= 48x + 25

Identify the algebraic property used in: a 6

Step 1

Step 2

Step 3

Step 5

For each statement: i

Identify which property needs to be used to complete the statement.

Complete the statement by ﬁnding the missing value.

−7 + 8 = 8 + ⬚

(9 + 7) + 6 = 9 + (⬚ + 6)

b d

−6 ⋅ 5 = ⬚ ⋅ (−6)

6 ⋅ (9 ⋅ 7) = (6 ⋅ ⬚) ⋅ 7

Let’s extend our thinking 7

Is this equation true or false if a is not equal to b? Either explain why it is true using a property from the lesson or ﬁnd numbers for a and b that show it is false. a−b=b−a

Find the missing term that would make the number sentence true. a c e

9 10

0 + ⬚ = 9m

⬚ − 0 = 3d

d(4 + 3) = (⬚ + 5) ⋅ d

If 4( y + z) = 16, then z + y = ⬚.

b d f

4b ⋅ ⬚ = 1

a ⋅ 19 = 19 ⋅ ⬚

(4y2 − 3x) ⋅ ⬚ = −3x + 4y2

For this statement:

a + (−a) = 0 Explain why this is true for all values of a. 11

For this equation:

Use one of the properties in the lesson to ﬁnd the expression that y is equal to. 12

How could the distributive property help us ﬁgure out the solution to 396 ÷ 4 without using long division or a calculator?

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Answers 1.02 Properties of real numbers What do you remember? 1 a True by the commutative property of addition

Let’s extend our thinking 7 False. If a = 3 and b = 6, then 3 − 6 ≠ 6 − 3 because −3 ≠ 3 8 a 0 + 9m = 9m b c 3d − 0 = 3d

b True by the commutative property of multiplication

d a ⋅ 19 = 19 ⋅ a

c False

e d(4 + 3) = (2 + 5) ⋅ d

d True by the associative property of addition

e True by the associative property of multiplication f

True by the inverse property of multiplication

(4y2 − 3x) ⋅ 1 = 4y2 − 3x = −3x + 4y2

9 z+y=4 10 The sum of a real number and its additive inverse is equal to 0, the additive identity.

2 3 10x − 5y Let’s practice 4 False for all v ≠ 3 5 a Distributive property

11 Using the Inverse property of Multiplication, y = 3x2 − 2x + 1 12 We can rewrite 396 as 400 − 4. The distributive property tells us that 396 as 400 − 4 = 4(100 − 1). Therefore, 396 ÷ 4 = 4(100 − 4) ÷ 4 = (100 − 1) = 99

b Commutative property of addition c Associative property of addition d Commutative property of addition 6 a i Commutative property of addition ii −7 + 8 = 8 + (−7) b i Commutative property of multiplication ii −6 ⋅ 5 = 5 ⋅ (−6) c i Associative property of addition ii (9 + 7) + 6 = 9 + (7 + 6) d i Associative property of multiplication ii 6 ⋅ (9 ⋅ 7) = (6 ⋅ 9) ⋅ 7

Answers mathspace.co

1.03 Properties of equality Subtopic overview Lesson narrative In this lesson, students will apply their understanding of algebraic expressions and inverse operations to create and solve linear equations in both mathematical and real-world contexts. To ease students into this topic, the variables in this lesson are restricted to one side of the equation. Students will use the structure of equations to solve using inverse operations, make sense of quantities and their relationships in problem situations, and know and flexibly use different properties of operations, justify each step in the solution strategy, and check their answers for reasonableness and accuracy. By the end of this lesson, students should be able to create equations to solve problems, understand the use of inverse operations to solve equations, and select the appropriate properties of equality to justify solution steps.

Learning objective

1.03 Properties of equality

Students: Page 17

After this lesson, you will be able to… • justify the steps for solving an equation using the properties of equality, identity, and inverse operations.

Properties of equality An equation is a mathematical relation statement where two equivalent expressions and values are separated

Key by anvocabulary equal sign. The solutions to an equation are the values of the variable(s) that make the equation true.

Equivalent equations are equations that have the same solutions.  equivalent equations equation  inverse Equations, particularly in real-world contexts, are sometimes referred to as constraints as they describe restrictions or operations linear equation limitations given situation. One familiar type of equation is a linear equation.  solution (toof anthe equation) 

Linear equation An equation that contains a variable term with an exponent of 1, and no variable terms with exponents other than 1. Essential understanding

Example: 2x + 3 = 5 The properties of equality allow an equation to be manipulated without changing the equivalence of the expressions on either side. Equations are often used to solve mathematical and real-world problems. To solve equations, we use a variety of inverse operations to “undo” what was done to a variable. For instance, if a variable was multiplied by a number, we would use division to get the variable by itself.

Standards The distributive property is an important property that we use frequently to simplify and solve equations: This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

a (b + c) = ab + ac

Mathematical process goals

a, b, c

are real numbers

MPG2 — Mathematical Communication

Exploration

Teachers can foster mathematical communication by encouraging students to articulate their thought process and reasoning when applying the properties of equality and real numbers. This can be done through class discussions, Consider the undeniably true statement 5 = 5. Perform each of the following operations: written reflections, or short presentations. For example, teachers can ask students to explain why they chose a Add 3 to sidesto solve an equation. • Multiply −2 to both sides specific •property ofboth equality • Subtract 1 from both sides 36

• Divide both sides by 3

What do you notice about the equation that results from performing each operation?

Mathspace Virginia SOL Algebra 1 Teacher Edition 2. Suppose we started with the true equation x = 5. Would the equation still remain true after performing mathspace.co

each of the operations? Explain.

MPG3 — Mathematical Reasoning

MPG5 — Mathematical Representations

To integrate mathematical reasoning, teachers can provide activities that require students to apply and justify their use of the properties of real numbers and equality. For example, teachers can present equations where students need to justify why they would use the Division Property of Equality over the Multiplication Property of Equality.

This goal can be incorporated into the lesson by asking students to represent their understanding of the properties of equality and real numbers in different ways. For instance, teachers can ask students to visually represent the Symmetric Property of Equality using number lines or geometric shapes. Alternatively, students could use symbolic notation to express their understanding of these properties. Teachers could also incorporate technology such as graphing calculators or mathematical software to help students explore different representations of these properties.

A.EI.1f — Verify possible solution(s) to multistep linear equations and inequalities in one variable algebraically, graphically, and with technology to justify the reasonableness of the answer(s). Explain the solution method and interpret solutions for problems given in context.

A.EI.1b — Solve multistep linear equations in one variable including those in contextual situations, by applying the properties of real numbers and/or properties of equality.

Prior connections 8.NS.2 — The student will investigate and describe the relationship between the subsets of the real number system.

8.PFA.4 — The student will write and solve multistep linear equations in one variable, including problems in context that require the solution of a multistep linear equation in one variable.

Future connections A.EI.2 — The student will represent, solve, explain, and interpret the solution to a system of two linear equations, a linear inequality in two variables, or a system of two linear inequalities in two variables. A.EI.3 — The student will represent, solve, and interpret the solution to a quadratic equation in one variable. G.TR.2 — The student will, given information in the form of a figure or statement, prove and justify two triangles are congruent using direct and indirect proofs, and solve problems involving measured attributes of congruent triangles.

G.TR.3 — The student will, given information in the form of a figure or statement, prove and justify two triangles are similar using direct and indirect proofs, and solve problems, including those in context, involving measured attributes of similar triangles. A2.EI.1 — The student will represent, solve, and interpret the solution to absolute value equations and inequalities in one variable.

1.03 Properties of equality mathspace.co

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 8 — 2.06 Solve multistep equations Algebra 1 — 1.01 Algebra expressions Algebra 1 — 1.02 Properties of real numbers

Tools You may find this tool helpful: • Scientific calculator

Student lesson & teacher guide Properties of equality Students are reminded of the inverse operations and distributive property before engaging in an exploration to uncover the properties of equality.

Students: Page 17

1.03 Properties of equality After this lesson, you will be able to… • justify the steps for solving an equation using the properties of equality, identity, and inverse operations.

Properties of equality An equation is a mathematical relation statement where two equivalent expressions and values are separated by an equal sign. The solutions to an equation are the values of the variable(s) that make the equation true. Equivalent equations are equations that have the same solutions. Equations, particularly in real-world contexts, are sometimes referred to as constraints as they describe restrictions or limitations of the given situation. One familiar type of equation is a linear equation. Linear equation An equation that contains a variable term with an exponent of 1, and no variable terms with exponents other than 1. Example: 2x + 3 = 5 Equations are often used to solve mathematical and real-world problems. To solve equations, we use a variety of inverse operations to “undo” what was done to a variable. For instance, if a variable was multiplied by a number, we would use division to get the variable by itself. The distributive property is an important property that we use frequently to simplify and solve equations:

a (b + c) = ab + ac a, b, c

are real numbers

Exploration 38

Mathspace Virginia SOL Algebra 1 Teacher Edition Consider the undeniably true statement 5 = 5. Perform each of the following operations: mathspace.co • Add 3 to both sides • Multiply −2 to both sides • Subtract 1 from both sides

• Divide both sides by 3

Visualize inverse operations Targeted instructional strategies Ask students to consider how the expression including the variable was constructed, starting from just the variable. Once this is identiﬁed, students can solve the equation by applying the inverse operations in the reverse order. This can be demonstrated using a ﬂowchart, like so: +5

×2

2x + 5

2x −5

÷2

Critique, correct, and clarify English language learner support Break the students into pairs. Provide each pair with the incorrect examples of the properties of equality: • Symmetric property: If (a = b), then (b = b). • Transitive property: If (a = b) and (b = c), then (a = b). • Addition property: If (a = b), then (a + b = c + b). • Subtraction property: If (a = b), then (a − b = c − b). • Multiplication property: If (a = b), then (a ⋅ b = c ⋅ b). • Division property: If (a = b), then

• Substitution property: If (a = b) and (a = c), then (b = c). Instruct the students to identify the errors in each example and discuss with their partner why the given example is incorrect. Then, have them correct the example to reflect the accurate application of the property. After making corrections, each student should write a clarifying statement that accurately describes the property and its correct use. Facilitate a class discussion to review the corrections and ensure understanding. Encourage students to share their clarifying statements and provide feedback to one another. This process helps solidify their grasp of each property through collaboration and detailed explanation.

Provide a graphic organizer for examples and non-examples Student with disabilities support Provide a graphic organizer to help students categorize and generalize examples of each property. Create a table with rows for each property and columns for examples, non-examples, and generalizations. An example table is shown: Examples

Non-Examples

Generalization

Symmetric property Transitive property Addition property Subtraction property Multiplication property Division property Substitution property

1.03 Properties of equality mathspace.co

Address student misconceptions Linear equation

Students may apply an to onlyterm onewith side the equation. Explain students that any operation An equation thatoperation contains a variable an of exponent of 1, and no variableto terms with exponents other than 1. must Example: 2x 3 = 5sides of the equation. always be applied to +both Some students may ﬁnd it easier to think of it as applying an operation to the whole equation.

Equations are often used to solve mathematical and real-world problems. To solve equations, we use a variety of inverse operations to “undo” what was done to a variable. For instance, if a variable was multiplied by a number, we would use division to get the variable by itself.

Exploration The distributive property is an important property that we use frequently to simplify and solve equations: Students: Page 17

a (b + c) = ab + ac a, b, c

are real numbers

Exploration Consider the undeniably true statement 5 = 5. Perform each of the following operations: • Add 3 to both sides

• Multiply −2 to both sides

• Subtract 1 from both sides

• Divide both sides by 3

What do you notice about the equation that results from performing each operation?

Suppose we started with the true equation x = 5. Would the equation still remain true after performing each of the operations? Explain.

Suggested student grouping: Individual Symmetric property of equality If a = b, then b = a Students will perform operations on both sides of an equation with the purpose of noticing that a true statement Transitive property of equality If a = b and b = c, then a = c remains true Addition so longproperty as any of operations applied onea side equality If a = b,tothen + c = bof+ the c equation are also applied to the other. Subtraction property of equality Multiplication property of equality

If a = b, then a − c = b − c If a = b, then ac = bc

Substitution property of equality

If a = b, then b may be substituted for a in any expression

Ideal student responses

These ideal responses may differ from other correct student responses. Less formal responses can be If a = b and c ≠ 0, then Division property of equality connected with the more precise mathematical language presented here. 1. What do you notice about the equation that results from performing each operation? The equation remains true, since the values of both sides of the equation change equally because we are 1.03 Properties of equality 17 applying the same operations to both sides of the equation. mathspace.co 2. Suppose we started with the true equation x = 5 Would the equation still remain true after performing each of the operations? Explain. Yes, the equation remains true because, after substituting x = 5 into the new equations, the values on both sides are the same. Purposeful questions • After evaluating, are the values the same on both sides of the equation? • Since we were given x = 5 substitute x = 5 into each of the new equations. What do you notice? Possible misunderstandings • Students may not correctly apply the operations to both sides of the equation, resulting in an untrue statement. If this occurs, remind students that each operation needs to be applied to both sides of the equation, and to check if they have done this at each step. After conceptually uncovering the properties of equality, students are presented with their formal definitions. Students are also shown that properties of equality can also be represented by algebra tiles and balance scales, which shows a visual way of solving equations and keeping sides balanced.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

• Add 3 to both sides

• Multiply −2 to both sides

• Subtract 1 from both sides

• Divide both sides by 3

What do you notice about the equation that results from performing each operation?

Suppose we started with the true equation x = 5. Would the equation still remain true after performing

Students: Pageseach 17–18 of the operations? Explain.

Properties of equality are facts about equations. They describe different operations that can be performed on an equation that would maintain the truth of the equation statement. The following are the properties of equality and identity: If a = b, then b = a If a = b and b = c, then a = c If a = b, then a + c = b + c If a = b, then a − c = b − c If a = b, then ac = bc If a = b, then a + 0 = b and a = b + 0 If a a= = b, b and ·0,10=then If thenc a≠ + =b band andaa= =b b· 1+ 0

Symmetric property of equality Transitive property of equality Addition property of equality Subtraction property of equality Multiplication property of equality Additive identity Division property of equality Multiplicative identity Additive identity

Multiplicative identity If aproperties = b, then to a ·+show 10= =bbe aa= =b b· 1+of Additive identity band and 0 Substitution property ofrepresent equality these b may substituted forthe a in any expression Using algebra tiles, we can the balance two sides. The balance below represents −2 = 3x +we 2(−x + represent 1). Multiplicative identity If aproperties = b, then to a ·+show 10= =b band aa= =b b· 1+of Additive identity and 0 the two sides. The balance below Using algebra tiles, can these the balance Multiplicative identity If aproperties = b, then to a ·show 1 = b and a = b · 1 of the two sides. represents −2 = 3x +we 2(−x + represent 1). Using algebra tiles, can these the balance balance below 1.03 The Properties of equality +x

mathspace.co represents −2 = 3x +we 2(−x 1). Using algebra tiles, can+ represent these properties to show the balance of the two sides. The balance below +x +x −x +1 represents −2 = 3x + 2(−x + 1).

−1

+1 +1

−1

+x +x +x +x

−x −x

−1

−x

−1

+x +x

−x −x

+1 +1 +1 +1

−1

−x

Keeping the two sides balanced, we want to add or remove tiles to work towards a single x tile. We see from the balance that 2(−xsides + 1) represents + 2. to add or remove tiles to work towards a single x tile. We see from the Keeping the two balanced, −2x we want balance 2(−xsides + 1) represents + 2. to add or remove tiles to work towards a single x tile. We see from the Keeping that the two balanced, −2x we want By eliminating the zero pairs on the right side, we are left with −1 −1 +x +1 +1 balance that 2(−x + 1) represents −2x + 2. to add or −2 Keeping the two sides balanced, we want remove work towards a single x tile. Wewe see = x + tiles 2. tothe By eliminating zero pairs on the right side, arefrom left the with −1 −1 +x +1 +1 balance that 2(−x + 1) represents −2x + 2. −2 using = x + 2. eliminating the zero pairs on the right side, we are left with By the subtraction property of equality, we will make zero −1 −1 +x +1 +1 −2 = x + 2. pairs with the +2 on the right side the equation to isolate the eliminating the zero pairs on the right side, we are left with By using the subtraction property of equality, we will make zero −1 −1 +x +1 +1 variable. −2 = x + 2. pairs with the +2 on the right side of the equation to isolate the By using the subtraction property of equality, we will make zero variable. pairs withthe thesubtraction +2 on the right side of equality, the equation to isolate the By using property we will make zero variable. pairs with the +2 on the right side of the equation to isolate the A zero pair on the right side leaves us with x + 0 and using the additive identity we are left with just x. variable. A zero pair on the right side leaves us with x + 0 and using the additive identity we are left with just x. −1

−1

−1 −1 −1 −1

+1 +1 +1 +1

−1

A zero pair on the right side leaves us with x + 0 and using the additive identity we are left with just x. −1 −1 +1 −1 −1 +x −1 x + 0 and using the additive−1 −1identity −1 we are left with +x +1 A zero pair−1on −1 the right side leaves us with just x. +x

−2 − 2 = x + 0

−4 = x

−2 − 2 = x + 0

−4 = x

Solutions can be verified −2 = 3x + 2(−x + 1), we −2 − 2a=variety x + 0 of ways, including visually using algebra tiles. After −4 =solving x want to verify x−2 = −4. Solutions can that be verified −2 = 3x + 2(−x + 1), we − 2a=variety x + 0 of ways, including visually using algebra tiles. After −4 =solving x want to that xreplaced = −4. a variety Solutions can bebe verified of ways, including usingthe algebra tiles. Each +x verify tile will with four −1 tiles. A −x tilevisually will change four −1 tilesAfter to +1solving tiles. −2 = 3x + 2(−x + 1), we want to that xreplaced = −4. a variety Solutions can bebe verified of ways, including usingthe algebra tiles. Each +x verify tile will with four −1 tiles. A −x tilevisually will change four −1 tilesAfter to +1solving tiles. −2 = 3x + 2(−x + 1), we −1 −1 −1 −1 −1 −1 want to that = −4. with four −1 tiles. A −x tile will change the four Each +x verify tile will bexreplaced −1 tiles to +1 tiles. −1 −1 +1 +1

−1

−1 −1

−1 −1 +1 +1 −1 −1 +1 +1 +1 −1 +1 +1 +1 +1 −1 −1 −1 −1 +1 +1 +1 +1 −1 −1 −1

−1 −1

−1 −1 +1 +1 +1 +1 −1 −1

−1 −1

−1 −1 +1 +1

−1 −1

−1 −1 +1 +1

Each +x tile will be replaced with four −1 tiles. A −x tile will change the four −1 tiles to +1 tiles. −1 −1 −1 −1

18 18 18 18

Substituting x = −4 into −2 = 3x + 2(−x + 1)

Zero pairs keep the equation balanced at −2 = −2

Substituting x = −4 into −2 = 3x + 2(−x + 1)

Zero pairs keep the equation balanced at −2 = −2

Substituting x = −4 into −2 = 3x + 2(−x + 1) Mathspace Virginia SOL Algebra Substituting x = −4 into −2 = 3x1 + 2(−x + 1)

Zero pairs keep the equation balanced at −2 = −2 Zero pairs keep the equation balanced at −2 = −2

mathspace.co Mathspace Virginia SOL Algebra 1 mathspace.co Mathspace Virginia SOL Algebra 1 mathspace.co Mathspace Virginia SOL Algebra 1 mathspace.co

1.03 Properties of equality mathspace.co

Examples Students: Pages 19–20

Example 1 Viola and Akim were asked to solve the equation −10(3x + 7) + 4 = 54. Viola solved it like this: −10(3x + 7) + 4 = 54

Given equation

−30x − 70 + 4 = 54 −30x − 66 = 54 −30x − 66 + 66 = 54 + 66 −30x + 0 = 120

Evaluate the addition

−30x = 120 −30x ÷ (−30) = 120 ÷ (−30) 1 × x = −4

Evaluate the division

x = −4 Akim solved the equation like this: Given equation Evaluate the addition

Evaluate the division

Evaluate the addition

Evaluate the division

a Use properties of equality and identities to justify each missing step of their work.

Create a strategy From line to line, identify what changed and what operation was used, then ﬁnd the corresponding property or identity.

Apply the idea Viola’s work: −10(3x + 7) + 4 = 54

Given equation

−30x − 70 + 4 = 54

Distributive property

−30x − 66 = 54

Evaluate the addition

−30x − 66 + 66 = 54 + 66 −30x + 0 = 120 −30x = 120 −30x ÷ (−30) = 120 ÷ (−30)

Addition property of equality Evaluate the addition Additive identity Division property of equality

1 × x = −4

Evaluate the division

x = −4

Multiplicative identity

Akim’s work: Given equation Subtraction property of equality Evaluate the addition

Additive identity Mathspace Virginia SOL Algebra 1 Teacher Edition Division property of equality mathspace.co Evaluate the division Multiplicative identity

1.03 Properties of equality mathspace.co

−30x − 66 + 66 = 54 + 66 −30x + 0 = 120 −30x = 120 −30x ÷ (−30) = 120 ÷ (−30)

Addition property of equality Evaluate the addition Additive identity Division property of equality

1 × x = −4

Evaluate the division

x = −4

Multiplicative identity

Akim’s work: Given equation Subtraction property of equality Evaluate the addition Additive identity Division property of equality

Apply the idea

Evaluate the division

Viola’s work:

Multiplicative identity Given equation Subtraction property of equality

−10(3x + 7) + 4 = 54 −30x − 70 + 4 = 54 −30x − 66 = 54 −30x − 66 + 66 = 54 + 66 −30x + 0 = 120 −30x = 120 −30x ÷ (−30) = 120 ÷ (−30) 1 × x = −4

Distributive property Evaluate the addition Evaluate the addition Additive identity Addition property of equality Division of equality Evaluateproperty the addition Additive identity Evaluate the division Division property of equality Multiplicative identity Evaluate the division

x = −4 Multiplicative identity Akim’s work: Although Viola and Akim solved the problem in different ways, they both used inverse operations to arrive at the same answer. Given equation

Reﬂect and check

Subtraction property of equality b Compare their strategies.

Evaluate the addition Additive identity

Purpose Create a strategy Applyofthe idea property equality Provide an opportunity for advanced learners toDivision use the properties of equality justify each step of a solution We want to consider how their strategies are similar and Viola decided to expand and simplify the left side ﬁrst, method.how they are different. We can look at how manyEvaluate the division steps then use inverse operations and apply properties of were taken and which properties were used.

Multiplicative identityAkim solved the equation by directly using equality.

Expected mistakes Subtraction inverse propertyoperations of equalityfrom the start. They both arrived at the A common mistake is the misalignment of reasons and steps work,but usually withless thesteps. property of equality same of answer, Viola used Evaluate the addition preceding the step of work that uses it. The property of equality should be written next to the step of work in Additive identity Reflect and check which the property was applied. property of equality We can notice that Viola and Akim used differentDivision notation for simple operations. For example, Viola represented division the ÷ symbol, however, Akim used fractions. Also, Viola used explicit multiplication in their second last Reflecting withusing students Evaluate the division step of their working, whereas of Akim used implicit multiplication. both cases, the difference between notationHighlight has Ask students why both methods solving are valid, despite In using different properties of equality. for Multiplicative identity no impact on the properties or the results of the identities they use. students that since both methods properly applied operations to both sides of the equation, they could never result inReflect an untrue and equation. check 20 Mathspace Virginia Algebra Although Viola and AkimSOL solved the1 problem in different ways, they both used inverse operations to arrive at the

mathspace.co Students:same Page 20 answer.

b Compare their strategies.

Create a strategy

Apply the idea

We want to consider how their strategies are similar and how they are different. We can look at how many steps were taken and which properties were used.

Viola decided to expand and simplify the left side ﬁrst, then use inverse operations and apply properties of equality. Akim solved the equation by directly using inverse operations from the start. They both arrived at the same answer, but Viola used less steps.

Reﬂect and check We can notice that Viola and Akim used different notation for simple operations. For example, Viola represented division using the ÷ symbol, however, Akim used fractions. Also, Viola used explicit multiplication in their second last step of their working, whereas Akim used implicit multiplication. In both cases, the difference between notation has no impact on the properties or the results of the identities they use.

Mathspace Virginia SOL Algebra 1 mathspace.co

1.03 Properties of equality mathspace.co

Purpose Challenge students to consider the order in which we can use inverse operations to solve an equation involving multiple steps. Reﬂecting with students Ask students to discuss which strategy they preferred and why. Highlight the importance of keeping equations balanced and remind students that multiple methods can lead to the correct answer.

Students: Page 21 Example 2 Verify that the x = 3 is a solution to the equation 3 − 6x + 2x = −9.

Create a strategy A value is a solution if it can be substituted into the equation and make the equation true.

Apply the idea We will substitute x = 3 into the equation and see if both sides of the equation are the same after evaluating. 3 − 6 ⋅ (3) + 2 ⋅ (3) = −9

Substitute x = 3

3 − 18 + 6 = −9

Evaluate the multiplication

−15 + 6 = −9

Evaluate the subtraction

−9 = −9

Evaluate the addition

Since the two sides of the equation are equal, x = 3 is a solution.

Reﬂect and check Algebra tiles can also be used to verify that a value is a solution to an equation. The original equation 3 − 6x + 2x = −9 is shown. +1

−x

−1

Each +x tile will be replaced by three +1 tiles and −x tiles will change the +1 to −1. +1

−1

A solution will show the same value on both sides of the equals sign after zero pairs cancel out. +1

−1

Since the same number of −1 tiles are shown on each side, x = 3 is a solution to 3 − 6x + 2x = −9.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

1.03 Properties of equality mathspace.co

Purpose Check students can verify a solution to an equation by substituting the value into the equation and checking if both sides are equal.

Concrete-Representational-Abstract (CRA) Approach

use with Example 2

Targeted instructional strategies Concrete: Begin by engaging students with physical manipulatives to represent the equation 3 − 6x + 2x = −9. Provide sticky notes as placeholders for each term in the equation. Use different colored sticky notes to indicate positive and negative coefficients and another color for the constants. Students will need 6 “negative” sticky notes for the −6x term and 2 “positive” sticky notes for the 2x term, but only 1 sticky note for each constant term. Next, provide students with colored counters or integer chips and have them fill in the correct number of chips on each sticky note. For the sticky notes representing variables they should place 3 counters on each sticky note to represent the process of physically substituting the value 3 for each variable. Then, students will remove pairs of positive and negative chips, zero pairs, from the left side of the equation until no more pairs remain. They should see that there are now 9 negative chips on each side. Representational: Transition to drawing representations of the physical setup. Have students sketch the sticky notes and counters they used. Use symbols to represent the counters—circles with “+” signs for positive counters and circles with “−” signs for negative counters. Guide students to visually combine the counters in their drawings. They can pair one positive counter with one negative counter and cross them out. After simplifying, they should have nine negative counters remaining on both the left side and the right side. Abstract: Move on to try the problem algebraically. Have students write out the expression, replacing each variable with a set of parentheses. Then have students write the number 3 inside each set of parentheses. Finally evaluate the expression on the left side of the equation. Emphasize the importance of following the order of operations and performing each calculation carefully. Connecting the stages: Help students make connections between the concrete, representational, and abstract stages. Discuss how substituting counters for x on the sticky notes represents substituting algebraically into the equation. Highlight how pairing and removing counters in the physical and drawn representations correspond to adding and subtracting numbers when evaluating the expression. Ask guiding questions like: • “How did placing counters on the sticky notes help you understand substitution?” • “In what ways did your drawings reflect the physical manipulatives?” • “How does removing counters relate to combining like terms in the equation?”

Students: Page 22 Example 3 Solve the following equations and justify each step. a

Create a strategy Consider how the expression was constructed, starting from the variable. Once this is identiﬁed, we can solve the equation by applying the inverse operations in the reverse order.

Apply the idea Given equation Subtraction property of equality Evaluate the subtraction Additive identity Division property of equality

1.03 Properties of equality mathspace.co

Create a strategy Consider how the expression was constructed, starting from the variable. Once this is identiﬁed, we can solve the equation by applying the inverse operations in the reverse order.

Apply the idea Given equation Subtraction property of equality Evaluate the subtraction Additive identity Division property of equality Evaluate the division Multiplicative identity Multiplication property of equality Evaluate the multiplication Multiplicative identity

Reﬂect and check Sometimes, it is easier to simplify the problem before solving for a. In this case, we could have multiplied by 2 first, then solved the equation. Either way, we get the same answer. Given equation Evaluate the multiplication Subtraction property of equality Evaluate the subtraction Additive identity Division property of equality Evaluate the division Multiplicative identity

Purpose Mathspace Virginia SOL Algebra 1learners to demonstrate that they can apply inverse operations to solve Provide 22 an opportunity for advanced mathspace.co multistep equations and name the property of equality being used at each step of work. Expected mistakes Students might not distribute the 2 properly into

, resulting in an incorrect calculation. Remind them

that when a number is outside and next to a parenthesis, it should be multiplied to every term inside the parentheses. Reﬂecting with students Encourage students to check their solution by substituting the result back into the equation.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Students: Page 23 b

Create a strategy Since each term on the left hand side of the equation has the variable b in it, we can start by removing the fraction so we can collect like terms.

Apply the idea Given equation Multiplication property of equality Evaluate the multiplication Multiplicative identity Collect like terms Multiplication property of equality Evaluate the multiplication b Reflect and check You may find that the working could have been condensed to something like this:

Create a strategy

Given equation Since each term on the left hand side of the equation has the variable b in it, we can start by removing the fraction so Multiplication property of equality we can collect like terms. Collect like terms

Apply the idea

Multiplication property of equality

Often, steps such as evaluating sums or products, adding 0, or multiplying by 1, are combined into one step. Given equation Multiplication property of equality c

Evaluate the multiplication

Multiplicative identity Purpose Create a strategy Collect involving like terms fractions and variables in multiple terms using the Teach advanced learners how to solve equations It is easiest to work with whole numbers, so we can multiply everything by 3 to eliminate the fraction. properties of real numbers and properties ofMultiplication equality. property of equality Evaluate the multiplication Reflect and check

Apply the idea

Expected mistakes We can check our answer by substituting it back into the Given equation When multiplying 3, students may forget to distribute the multiplication across all terms on the left-hand side Reflect andby check original equation. Multiplication property of equality of the equation. Remind to apply the operation every term. You may find that thestudents working could have been condensed to to something like this: Combine like terms Given equation

Substitute Reﬂecting with students Subtraction property of equality Multiplication property of equality Ask students why we multiplied each side of the equation by 3 in the beginning. Discuss the idea of getting rid DivisionCollect property ofterms equality Evaluate the multiplication of fractions in equations to simplify thelike process of solving for the variable. Multiplication property of equality steps into one step. Students:Often, Page 23such as evaluating sums or products, adding 0, or multiplying by 1, are combined Evaluate the addition in the numerator c

Evaluate the division

Create a strategy

Evaluate the addition

It is easiest to work with whole numbers, so we can multiply everything by 3 to eliminate Evaluate the fraction. the division

Apply the idea

Reflect and check Given equation Multiplication property of equality Combine like terms Subtraction property of equality Division property of equality

1.03 Properties of equality

We can check our answer by substituting it back into the mathspace.co original equation. Substitute

Evaluate the multiplication

1.03 Evaluate theProperties addition in of theequality mathspace.co numerator Evaluate the division

Create a strategy It is easiest to work with whole numbers, so we can multiply everything by 3 to eliminate the fraction.

Apply the idea

Reflect and check Given equation Multiplication property of equality

We can check our answer by substituting it back into the original equation.

Combine like terms Subtraction property of equality Division property of equality

Substitute

Evaluate the multiplication Evaluate the addition in the numerator Evaluate the division Evaluate the addition Evaluate the division

1.03 Properties of equality mathspace.co

Purpose Advanced learners demonstrate that they can use the multiplication, subtraction, and division properties of equality to solve an equation involving fractions. Reﬂecting with students Ask students if using the multiplication property of equality is the only way to solve this equation. Point out that they could have separated the fraction into two terms. Ask students if they think this approach is better or worse. Why?

Students: Page 24 d 0.5x + 2(1.2x + 3) = 11.8

Create a strategy Before solving for x, we need to simplify the left side of the equation.

Apply the idea 0.5x + 2 (1.2x + 3) = 11.8

Given equation

0.5x + 2.4x + 6 = 11.8

Distributive property

2.9x + 6 = 11.8

Combine like terms

2.9x = 5.8 x=2

Subtraction property of equality Division property of equality

Reflect and check If you prefer working with whole numbers instead of decimals, you could have multiplied everything by 10 after the second step. 0.5x + 2(1.2x + 3) = 11.8

Given equation

0.5x + 2.4x + 6 = 11.8

Distributive property

5x + 24x + 60 = 118

Multiplication property of equality

29x + 60 = 118 29x = 58 x=2

Combine like terms Subtraction property of equality Division property of equality

Example 4 48

Yolanda works at a restaurant 5 nights a week and receives tips. On the first three nights, the total tips she received Mathspace SOL Algebra 1 Teacher was $32,Virginia $27, and $26. She earned twiceEdition as much in tips on the fourth night compared to the fifth night. The average mathspace.co amount of tips received per night for the week was $29. If the amount she received on the fifth night was $k, determine how much she received that night.

2.9x + 6 = 11.8 2.9x = 5.8 x=2

Combine like terms Subtraction property of equality Division property of equality

PurposeReflect and check If you prefer working with whole numbers instead decimals, you could have multiplied everything by 10 after the Advanced learners demonstrate that they can useofthe distributive property to solve an equation involving second step. parentheses. 0.5x + 2(1.2x + 3) = 11.8

Given equation

Expected mistakes 0.5x + 2.4x + 6 = 11.8 Distributive property Students may try to5xuse the division of equalityproperty by dividing the right-hand side of the equation by the + 24x + 60 = 118 property Multiplication of equality coeﬃcient in front of the Ask students whether 29xparentheses. + 60 = 118 Combine like terms they have applied the operation correctly to both sides of the equation. 29x = 58 Subtraction property of equality

Students: Pages 24–25

x=2

Division property of equality

Example 4 Yolanda works at a restaurant 5 nights a week and receives tips. On the first three nights, the total tips she received was $32, $27, and $26. She earned twice as much in tips on the fourth night compared to the fifth night. The average amount of tips received per night for the week was $29. If the amount she received on the fifth night was $k, determine how much she received that night.

Create a strategy The average is equal to the sum of values divided by the number of values. We can use this to build an equation to represent the constraint in terms of k about the average tips Yolanda received. Then, we can solve the equation for k.

Apply the idea We can represent her earnings on the fourth night as 2k, as it is twice her earnings on the ﬁfth night, k. Since $29 is equal to the average of her tips over ﬁve nights, we can write an equation representing the average of her tips in terms of k:

where 2k and k represent her earnings on the fourth and ﬁfth night, respectively. Now, we can solve the equation for k. Start with equation in terms of k Combine like terms in numerator Commutative property of addition 24

Mathspace Virginia SOL Algebra 1 mathspace.co

Multiplication property of equality Evaluate product on both sides of equation Multiplicative identity Subtraction property of equality Combine like terms on both sides of equation Additive identity Division property of equality Evaluate the division on both sides of the equation Multiplicative identity Symmetric property of equality

Yolanda received $20 on the ﬁfth night.

Reflect and check We can check that the solution, k = 20, is correct and makes sense in this context. Firstly, since Yolanda’s earnings from tips on the fifth night is equal to $20, her earnings on the fourth night is equal to $40. If we use these numbers to calculate the average over the five nights, we get: Calculate average over five nights Collect like terms on numerator Evaluate the division We can see that this gives us the same value for the average tips over the five nights as specified in the question. 1.03 Properties of equality Notice that both of the values for the tips earned on the fourth and fifth night are positive numbers, which makes mathspace.co sense within the context of the question. We may have questioned our solution if we returned a negative value for tips or the average.

Division property of equality Evaluate the division on both sides of the equation Multiplicative identity Symmetric property of equality Yolanda received $20 on the ﬁfth night.

Reflect and check We can check that the solution, k = 20, is correct and makes sense in this context. Firstly, since Yolanda’s earnings from tips on the fifth night is equal to $20, her earnings on the fourth night is equal to $40. If we use these numbers to calculate the average over the five nights, we get: Calculate average over five nights Collect like terms on numerator Evaluate the division We can see that this gives us the same value for the average tips over the five nights as specified in the question. Notice that both of the values for the tips earned on the fourth and fifth night are positive numbers, which makes sense within the context of the question. We may have questioned our solution if we returned a negative value for tips or the average.

Idea summary

Purpose To identify the properties of equalities needed to solve an expression, consider how an expression was Students understand how to use thetheconcept average in equation a real-world scenario to build and solve constructed, starting from variable. of Then, solve the by applying the inverse operations in thean equation. reverse order and matching the operation to the correct property.

Expected mistakes • Symmetric property of equality: if a = b, then b = a. Students may forget to multiply the fourth night’s tips by 2. Remind them that the problem states Yolanda earned • Transitive property of equality: if a = b and b = c, then a = c. twice as much on• the fourth night compared to the fifth. Addition property of equality: if a = b, then a + c = b + c. • Subtraction property of equality: if a = b, then a − c = b − c.

Reflecting with students • Multiplication property of equality: if a = b, then ac = bc. Ask students to consider how the problem would change if Yolanda worked a different number of nights, or if • Division property of equality: if a = b and c ≠ 0, then a ÷ c = b ÷ c. the ratio of her earnings between two nights was different. This can help them better understand the role of • Substitution property of equality: if a = b, then b may be substituted for a in any expression containing a. each part of the equation.

Check solutions and compare solving approaches Targeted instructional strategies

1.03 Properties of use equality 25 with Example 4 mathspace.co

Encourage students to check the clarity, reasonableness, and correctness of their own solution or of a worked example with steps missing or incorrect steps by: • Using substitution for an equation • Tracing through their procedure and checking the validity and accuracy at each step If students identify an error or lack of clarity, have students consider and answer questions like these to improve their solution: • How can you modify the solution to make it easier for others to follow? • What should this step be? How does that change the rest of the solution? • What misconception or misunderstanding do you think led to this error? Ask students with different approaches to compare solution steps side-by-side and discuss whether they believe they have the most eﬃcient solution method.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Collect like terms on numerator Evaluate the division We can see that this gives us the same value for the average tips over the five nights as specified in the question. Notice that both of the values for the tips earned on the fourth and fifth night are positive numbers, which makes sense within the context of the question. We may have questioned our solution if we returned a negative value for Students:tips Page or the25 average.

Idea summary To identify the properties of equalities needed to solve an expression, consider how an expression was constructed, starting from the variable. Then, solve the equation by applying the inverse operations in the reverse order and matching the operation to the correct property. • Symmetric property of equality: if a = b, then b = a. • Transitive property of equality: if a = b and b = c, then a = c. • Addition property of equality: if a = b, then a + c = b + c. • Subtraction property of equality: if a = b, then a − c = b − c. • Multiplication property of equality: if a = b, then ac = bc. • Division property of equality: if a = b and c ≠ 0, then a ÷ c = b ÷ c. • Substitution property of equality: if a = b, then b may be substituted for a in any expression containing a.

1.03 Properties of equality mathspace.co

Practice Students: Pages 26–29

What do you remember? 1

Solve the following equations. Justify each step using properties of equality. a

4x = 8

5m = −15

2.5k = −15

Determine the property that justiﬁes why the equations 9x = −27 and x = −3 are equivalent.

Solve the following equations. Justify each step using properties of equality. a

8m + 9 = 65

−10 + 3k = 5

7 − 8t = 15

3x − 4 = 2

8 = 2(1 − x)

Match the pictorial model with its equation a

−x +1 +1 +1 +1 +1 +1

−x

−x +1 +1 −x

−x

−1 −1

−1

1.03 Properties of equality mathspace.co

+x +x

−1

iii

−3x + 5x = −8

3(−x + 2) = −3

+x +x

−x

−1

−x

−1

Determine the property illustrated for each statement. a

If a = d, then a + 3 = d + 3.

Given that b = 16, then b − 5 = 11.

If f = g and g = h, then f = h.

Given

Subtracting 7 from both sides of x = 14

Dividing by 9 on both sides of 45x = −54

it is true that

Write the new equation produced for each scenario: a

Adding 1 to both sides of x = 9

Multiplying

to both sides of 5m = 3

Fill in the blank so that each resulting statement is true. a b

The Addition property of equality states that if a = b, then a + c = ⬚.

The multiplication property of equality states that if a = b and c ≠ 0, then ac = ⬚.

Let’s practice 8

Solve the following equations. Justify each step using properties of equality. a c

6x − 3 = 4x + 7

2.5x + 3x = 13.5 + x

Given the equation: x + 2(x + 3) = 18

Use algebra tiles to draw a model of the equation.

Solve the equation, using the model to justify steps.

A student incorrectly used the distributive property and wrote 7(4x + 3) = 28x + 3. Explain how to correct the error.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Consider the equation a

Dylan started to solve the equation as follows:

Determine the property that justiﬁes Dylan’s ﬁrst step of work. b

Dylan continued to solve the equation as shown: 8x + 48 = 144 8x = 96 Determine the property that justiﬁes this step of work.

Dylan finished solving the equation as follows: 8x = 96 x = 12 Determine the property that justifies Dylan’s final step of work.

For each statement: i

Write the statement as an equation in which x represents the number.

Solve the equation for x, justifying each step using properties of equality.

The sum of a number and 7 is 17.

Seven more than twice a number is 23.

Fifteen minus three-quarters of a number is 9.

he product of 5 and the sum of a number and T 7 equals 50.

The quotient of a number and −3 is −20.

Consider the given triangle which has a perimeter of 171 cm. Solve for the value of x.

8x cm

5x + 76 cm

6x + 19 cm

Consider the equation 21 = x + 13. a b

Solve for x. Kathleen started to solve the equation as follows: 21 = x + 13

Kathleen ﬁnished solving the equation as shown: 21 − 13 = x + 13 − 13 8=x+0

21 − 13 = x + 13 − 13

8=x

Determine the property that justiﬁes this step of work.

Determine the property that justiﬁes Kathleen’s ﬁnal step of work.

1.03 Properties of equality mathspace.co

Consider the equation a

Solve for x.

I rene started solving the equation as follows:

Determine the property that justiﬁes this step of work.

Determine the property that justiﬁes this step of work. iii

Irene continued solving the equation as follows:

iv Irene ﬁnished solving the equation as follows: 1x = 63 x = 63 Determine the property that justiﬁes this step of work

Determine the property that justiﬁes this step of work. 16

Ursula is solving for the missing leg length, x, of an isosceles right triangle with an area of 200 cm2. She solves the problem as follows:

Since 202 = 400, x = 20 cm Determine the property that justiﬁes Ursula’s ﬁnal step. 17

Consider the equation a b

Solve for x. Susana started solving the equation as follows:

Determine the property that justiﬁes this step of work.

Determine the property that justiﬁes Susana’s ﬁrst step of work. d

Susana continued to solve the equation as shown: 4(x + 6) = 4 x+6=1 Determine the property that justiﬁes this step of work.

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Susana continued to solve the equation as shown:

Susana finished solving the equation as follows: x+6=1 x = −5 Determine the property that justifies Susana’s final step of work.

Let’s extend our thinking 18

Athena and Emilio want to ride go-karts. It costs 50 cents per lap of the course. a

Write an equation to solve for the number of laps Athena and Emilio can afford if they have $12.

Solve for the number of laps they can afford. Justify each step using properties of equality.

Explain how that equation changes if they have to put a deposit of $1 on each cart used.

Give examples of the following properties using equations: a

Addition property of equality or Subtraction property of equality

Distributive property of equality

Transitive property of equality

Symmetric property of equality

Multiplication property of equality or Division property of equality

Use the Substitution property of equality to solve for x given y = x: a

14x − 2y = 36

−0.6x + 1.70y = 22

1.03 Properties of equality mathspace.co

Answers

5 a Addition property of equality b Subtraction property of equality

1.03 Properties of equality

c Transitive property of equality d Symmetric property of equality

What do you remember?

6 a x + 1 = 10

Given equation

1 a

Division property of equality

7 a b+c

Solution

8 a

Division property of equality

Division property of equality Solution

Multiplication property of equality Solution

Given equation Combine like terms

Given equation

Subtraction property of equality

Division property of equality

Solution

2 Multiplicative property of equality or Division property of equality

Solution

Given equation

3 a

Given equation

Subtraction property of equality

Given equation

b bc

Addition property of equality

Solution

d 5x = −6

Let’s practice

Given equation

b x–7=7

Given equation Subtraction property of equality

Subtraction property of equality Division property of equality

Multiplication property of equality

Solution

Given equation

Addition property of equality

Given equation

Division property of equality Multiplication property of equality

Solution Given equation

Distributive property

Subtraction property of equality Addition property of equality Division property of equality Addition property of equality

Solution

Given equation Division property of equality Subtraction property of equality Solution Division property of equality

Solution

4 a ii

b iv

c i

d iii

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9 a

+x +1

12 a i

Given equation Subtraction property of equality Solution

b i 2x + 7 = 23 ii

Given equation Subtraction property of equality

c i

Division property of equality Solution

Given equation Subtraction property of equality

Distributive property

+x +x +x

−1

Division property of equality Solution

d i 5(x + 7) = 50 ii

+x +x

Subtraction property of equality

Division property of equality

Given equation Division property of equality

Subtraction property of equality

Solution

e i ii

Given equation Multiplication property of equality Solution

13 x = 4 14 a x = 8

x = 4 10 The student should multiply the second part of the sum, 3, by the number outside the parentheses, 7 to get 21 as the second term on the right hand side. 11 a Multiplication property of equality b A ddition property of equality or Subtraction Property of equality c M ultiplication property of equality or Division property of equality

b A ddition property of equality or Subtraction property of equality c Addition identity 15 a x = 63 b i Division property of equality ii Addition property of equality iii Multiplicative property of equality iv Multiplicative identity 16 Substitution property of equality

Answers mathspace.co

19 a Example:

17 a x = 19 b Symmetric property of equality c M ultiplicative property of equality or Division property of equality d M ultiplicative property of equality or Division property of equality e A ddition property of equality or Subtraction property of equality

b Example:

c Example: x = y and y = z so x = z d Example:

Let’s extend our thinking 18 a 12 = 0.5 ⋅ L Where L is the number of laps. b

e Example:

Given equation Division property of equality Solution

c A ssuming both of them used their own cart, they would need to pay $2 as a deposit. So, the equation would instead be 12 = 0.5 ⋅ L + 2.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

20 a x = 3

b x = -8

c x = 20

1.04 Multistep equations Subtopic overview Lesson narrative In this lesson, students will analyze equations with a variable on both sides to determine the number of solutions and generalize when equations have one, none, or infinitely many solutions. Students will engage in an exploration where they investigate how both sides of an equation are equal using substitution. By the end of this lesson, students should have a solid understanding of how inverse operations work when solving equations, be comfortable translating mathematical and real-world problems into algebraic equations, and know how to check solutions for accuracy.

Learning objectives Students: Page 30

Key vocabulary 

equation



solution

Essential understanding A standard algorithm can be used to solve any linear equation accurately and efficiently.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving Teachers can incorporate this goal into their lesson by guiding students in the process of solving multistep linear equations, including providing real-world examples that require problem solving. For instance, when teaching the possible outcomes of solving linear equations, teachers can create complex problems that might end up with one solution, no solution, or an infinite number of solutions. Students can then apply problem-solving strategies to determine the outcome.

1.04 Multistep equations mathspace.co

MPG3 — Mathematical Reasoning

MPG5 — Mathematical Representations

Teachers can incorporate this goal into their lesson by emphasizing the importance of understanding and applying the properties of real numbers and equality. They can also ask students to justify their steps in solving multistep linear equations and to reason about the validity of their solutions. For example, when discussing the possible outcomes of solving linear equations, teachers can ask students to reason about why a given equation might have one solution, no solutions, or an infinite number of solutions.

Teachers can incorporate this goal into their lesson by utilizing different representations. For example, teachers could visually model the process of solving multistep linear equations using diagrams or graphical representations. They can also encourage students to use symbolic notation accurately when solving equations and verifying their solutions. When discussing the application of multistep linear equations in contextual situations, teachers can have students model these situations using equations, and then interpret the results in the context of the problem.

Content standards A.EI.1 — The student will represent, solve, explain, and interpret the solution to multistep linear equations and inequalities in one variable and literal equations for a specified variable. A.EI.1a — Write a linear equation or inequality in one variable to represent a contextual situation. A.EI.1b — Solve multistep linear equations in one variable including those in contextual situations, by applying the properties of real numbers and/or properties of equality.

A.EI.1e — Determine if a linear equation in one variable has one solution, no solution, or an infinite number of solutions. A.EI.1f — Verify possible solution(s) to multistep linear equations and inequalities in one variable algebraically, graphically, and with technology to justify the reasonableness of the answer(s). Explain the solution method and interpret solutions for problems given in context.

Prior connections 8.PFA.4 — The student will write and solve multistep linear equations in one variable, including problems in context that require the solution of a multistep linear equation in one variable.

A.EO.1 — The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

A.F.1 — The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships. A2.EI.1 — The student will represent, solve, and interpret the solution to absolute value equations and inequalities in one variable.

A2.EI.4 — The student will represent, solve, and interpret the solution to an equation containing rational algebraic expressions.

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Engage Activity Solar panels

60 mins

Students will determine how long various solar panel systems need to be used for their costs to be the same. Students will solve multistep linear equations with variables on both sides of the equation visually and algebraically.

Understanding and skills

Will use Solving multistep linear equations. Identifying the number of solutions to a linear equation.

Will develop Solving multistep linear equations visually with variables on both sides of the equation. Solving multistep linear equations algebraically with variables on both sides of the equation. Interpreting linear equations and their solutions in context.

Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: Calculator (optional) • Download and print copies of the student’s graphic organizer from the student Launch slide. • Access to the Google Project Sunroof savings estimator: https://www.google.com/get/sunroof (optional)

Support students with disabilities Support conceptual processing - apply concepts to new situations To help students apply balancing equations to the context of solar panels, use multiple representations of balancing equations. Concept maps can also be made, along with providing organizers for students to complete.

Support for English language learners Critique, correct, and clarify Before students share their responses, display the following incorrect statement: “I know that 2x + 4 = 5x − 5 can be rewritten as 2x = 5x + 9 because I have to move the numerals to one side of the equation to eventually solve for x. Invite students to identify the error, critique the reasoning, and write a correct explanation. Invite one or two students to share their critiques and corrected explanations with the class. Listen for and amplify the language students use to describe what should happen when +4 is moved to the right-hand side. Explain why the equation should be 2x = 5x − 9. This will help students understand how to solve for variables in an equation.

1.04 Multistep equations mathspace.co

Classroom guide Hook Students choose one of four equations that differ in structure, variable, number of steps to solve.

Which one doesn’t belong

•

5 mins

Which one doesn’t belong? Select one option. 50 x = 100

11 = 5 x − 4

3( x + 5) = −9

3 x − 1 = −2 x + 9

Slide 1 from Student Engage Activity

Implementation details The goal of the hook is to have students compare equations and justify why they think one equation is different. Students may choose an equation that doesn’t belong based on the solution, the number of steps required to solve, the structure of the equation, or which side of the equation the variable lies on. Ask students what they notice about each equation. Once students are able to identify a characteristic of the equation on their own, ask more about what they notice about the number of terms in the equation, the solution(s) to each equation, the coeﬃcients and constants in each equation, and so on.

Launch

5 mins

Ask students to share what they know about solar panels. Students may know about solar panels, if they Solar panels absorb and convert natural sunlight have some on their home or if they have studied their into a usable energy form. The cost of the solar properties in science class. Your school or nearby panels depends on the size of the system. buildings may have solar panels that you can point out if students don’t know much about them. Explain The size of solar systems are usually measured in that students will work in groups to investigate at least kilowatts (kW). A watt is the basic unit of power two different solar panel systems using the provided and a kilowatt is 1000 watts. solar panel info (students can also use Google Project Sunroof’s saving estimator, https://www.google.com/get/ sunroof) and solve for how many months the systems need to be used to cost the same. Note that Google Project Sunroof requires students to enter addresses (such as of a local building, school, restaurant, or store) and then review the available report. During the Launch, allow students to play around and investigate this resource for a few minutes if using or to review provided information in downloadable asset. If using the Google Project sunroof resource, the size of the solar panel system for the submitted address will be under “Your recommended solar installation size”. Students Slide 2 from Student Engage Activity can adjust the average monthly electric bill and this will change the size and cost. The cost of the solar panel system is under “Learn how to ﬁnance your solar panels”. Students should focus on the cost if they bought the solar panel systems. Important contextual information: Solar panels Suggested grouping: Form groups

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Continue when Students have read the Launch and understand the context of the problem.

Explore Anticipated strategies

Numbered heads together

System 1

Students may adjust the cost and size of the solar panel systems on the GeoGebra applet and record the possible months that the equation is balanced visually.

Creating a table

25 mins

Use the applet to balance the variables.

Balancing the equations visually

Encourage students to explain their reasoning algebraically as well as visually. Students may create a table of values to ﬁnd the exact month or realize that the visual scale in the GeoGebra applet is related to the two expressions provided. Students may set the expressions equal to each other and begin to solve for x. This is the ﬁrst time that students have solved equations with variables on both sides of equation, so let students explore. Make note of groups trying to or who have solved algebraically as well as visually for the class discussion.

•

7.5

12000

System 2 1.5

Size 1000 Cost Months = 1488

15.38x + 12 000

22.76x + 1000

Balanced Slide 5 from Student Engage Activity

Students may realize the GeoGebra applet provides a range of months that makes the equation “balanced” visually. To get an exact answer, students may substitute the value for x into the expressions on each side of the equation and solve. Students may repeat this process several times.

Solving equations algebraically Students may realize that they can set the two expressions equal to each other to solve for x. After solving for x, they can apply the properties of equality to variable terms to ﬁnd the desired month.

Misconceptions Determining the number of month without testing the value The app gives students a visual clue that the equation is close to being balanced, however a few months appear to be balanced when they are not. Encourage students to verify the answers in the applet are correct. Have you checked your solution in another way? How can you check if that month is correct? There are a range of solutions in this task based on the which solar panel systems each pair selects to investigate. Students will likely start by inputting their solar panel information into the GeoGebra app, then adjusting the month until the app appears to be balanced. Encourage students to check their answers, as the app appears to be balanced for various months and students will need to be verify the exact month using a more precise method of solving.

Performing an operation on a single side of the equation What happens when you perform [operation] to one side of an equation? Is the equation balanced? Why or why not?

Purposeful questions Use the following questions to check for understanding and encourage critical thinking: • What are the speciﬁcations for each of your chosen solar panel systems? What are their respective sizes? Costs? • What does x represent in this context? How do you know? • Is there another way to justify the number of months? Visually? Algebraically? Explain. 1.04 Multistep equations mathspace.co

Continue when Students have determined how many months the solar panel systems need to be used to cost the same and have justiﬁed their reasoning.

Discuss

25 mins

Have several pairs share their processes for solving the equations. Start with those that solved the problem visually before moving to those that solved the problem algebraically. Have the class compare and connect the processes students took. Consider making connections from the discussion to what solutions mean in the context of solar panels.

Discussion guide Have several pairs share their process for determining the month when the cost of the solar panel systems would be equal. Start with pairs that solved visually and then move to pairs that focused on using the expressions to solve. Build the discussion so that students can make a connection between the visual representation on the applet and any algebraic methods that were used. During the discussion it will be helpful to be able to display the applet so students can share their sizes and costs and process for solving for x. If no pairs chose to solve algebraically, ask students what they think the expressions represent and how they can create one equation using both expressions. Next, ask students what the unknown value is and ask students what strategies they could use to solve for x. Then ask a pair to share their equation and display it to the class. Work through how to solve for x, as a class and ask what the solution means in context of the solar panels. As an extension you may wish to give students the following prompt: Choose two new solar panel systems to compare and solve for how many months the solar panel systems need to be used to cost the same. Determine how many months a solar panel system would need to be used to save money compared to not having a solar panel system (set system size equal to 0 for one side). Discuss the impact of solar energy and the impact on climate change.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 8 — 2.04 Solve multistep equations Algebra 1 — 1.01 Algebraic expressions Algebra 1 — 1.02 Properties of real numbers Algebra 1 — 1.03 Properties of equality

Student lesson & teacher guide Multistep equations Students are reminded of the formal definition of a solution to an equation before engaging in an exploration to help them conceptually understand what a solution to an equation is. 64

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Students: Page 30

Generalize types of solutions using abstraction Targeted instructional strategies When solving multistep equations, it is important to recognize algebraically when an equation will have one solution, infinitely many solutions, or no solution. Use Abstraction to guide students to generalize multistep equations into simpler forms to identify the number of solutions. Each equation has the same expression on the left side of the equals sign, but different expressions on the right. Work through the steps to each of the following equations with students: 2(x − 5) = 4x + 10 2x − 10 = 4x + 10 2x − 10 = 4x + 10 −2x = 20 x = −10

2(x − 5) = 4x − 10 − 2x 2x − 10 = 4x − 10 − 2x 2x − 10 = 2x − 10 0x = 0 0=0

2(x − 5) = 4x − 2(x − 5) 2x − 10 = 4x − 2x + 10 2x − 10 = 2x + 10 0x = 20 0 = 20

If students work through the equations independently and notice equivalent coefficients on variables, encourage them to finish applying their inverse operations to isolate x. Tell students that equations can either have one solution, no solution, or infinitely many solutions, and ask students to predict which equation has which type of solution. Encourage students to then generalize to identify which type of solution exists for equations that simplify to: • ax + b = ax + b • ax + b = ax + c • ax + b = cx + d

1.04 Multistep equations mathspace.co

Information gap English language learner support Separate students into pairs and split the listed equations equally between the partners. Ask the partners to match the equations with their simplified forms and any solutions that make the equations true. • x=3 • 4x − 2 = 3x + 2 • 3(x + 2) − 4 = x + 2x + 2 • x=4 • 5x − 4 = 5x + 4 • 5(x − 2) + 6 = 6x + 4 − x • x=5 • 3x + 2 = 3x + 2 • 2(x − 1) + 2x = 3x + 2 The pairs must communicate effectively to match the equivalent equations and solutions. Students should notice that there are not an equal number of equations in each group. The pairs should describe their reasoning step-by-step, focusing on operations like distributive property, combining like terms, and determining which solutions match with their equations. Partners should ask clarifying questions to understand the process. They will then discuss and decide together which solutions and equations correspond.

Provide step-by-step checklist of finding and verifying solutions Support students with disabilities To support students with solving multistep equations, determining the number of solutions, and verifying solutions, provide them with a structured checklist. This checklist will help students organize their approach and ensure they follow each step systematically. Students should be reminded that not all steps in the checklist will be used in every problem. Consider providing the checklist in table form with additional details and workspace for students for students to work through problems, such as the one below: Simplify each side of the equation Move variables to one side Move constant terms to one side Simplify each side Isolate the variable Write the solution Verify the solution

Details Distribute and combine like terms Use addition or subtraction Use addition or subtraction Combine like terms Use division or multiplication Reread problem for formatting the solution Substitute solution for variable and evaluate

Workspace

Only looking at coefficients Address student misconceptions Students may sometimes determine the number of solutions by looking only at the coefficients of the variable. However, this can lead to mistakes when the coefficient is not obvious. For example, a student may: • Ignore the denominator’s effect on coefficients, as in

• Forget to distribute before comparing the coeﬃcients, as in 3x − 2 = 2(3x + 7) • Miss a difference in sign of the coeﬃcients, as in 4x + 3 = −4x + 3

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

While there are many ways to solve equations, encourage students who are having trouble comparing coefficients to compare once each side of the equation is in the following form: ⬚x + ⬚ or ⬚x − ⬚

Exploration Students: Page 30

Suggested student grouping: Small groups Students complete tables by evaluating algebraic expressions at different values for x to investigate the different types of solutions to an equation. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What are the differences between equations 1, 2, and 3? Equation 1 always has both sides of the equation equal. Equation 2 always has both sides of the equation not equal. Equation 3 has both sides of the equation equal only when x = 0. 2. How many solutions do you think there will be to each equation? Why? Equation 1 has infinitely many solutions. Equation 2 has no solutions. Equation 3 has a single solution. 3. How does the structure of an equation relate to the number of solutions it has? If both sides of the equation are identical, it will have infinitely many solutions. If the sides of the equations differ by a constant amount, it will have no solutions. Otherwise, a linear equation will have a single solution. 4. How many solutions do you think the equation 2(x − 8) = 3x + 4 will have? Why? The equation will have a single solution, because the sides of the equation have different coefficients for x after any distributing and simplifying. Purposeful questions • How is the number of solutions connected to the coefficients of the variables? • Will every equation have either a single, none, or infinitely many solutions?

1.04 Multistep equations mathspace.co

Possible misunderstandings • Students may not understand that using coeﬃcients to check the number of solutions should be done when both sides of the equation are fully simpliﬁed. Students are then shown a visual representation of determining the number of solutions to an equations using a balance scale and algebra tiles.

Students: Pages 30–31

If the equation simplifies to variables with the same coefficients and the same constants on each side of the equality, it will have infinitely many solutions. +x

+1 +1

+1 +1 +1

+1 +1

+1 +1 +1

The equation represented is 3(x + 2) = x + 2(x + 3), which has infinite solutions. When each side is simplified, the equation becomes 3x + 6 = 3x + 6, shown on the balance by three +x tiles and six +1 tiles. Regardless of what value of x is chosen for the +x tile, there are equal numbers of +x tiles on each side that will be added to the constants. The number of +1 tiles are also equal on each side. This creates a balanced scale regardless of the value chosen for x. If the equation simplifies to variables with the same coefficients but different constants, it will have no solution. The equation represented is 3(x + 2) = 3x − 2, which has no solution.

+1 +1

−1

+1 +1

−1

Similar to the previous example, regardless of the value of x chosen, the value represented by the +x tiles on each side of the equality will always be the same. However, the constants on either side of the equality are not the same. While the value of the +x tiles will always match, the constants keep the two sides from balancing.

If the equation simplifies to variables with different coefficients, there will be a unique solution regardless of constant values. +x

+1 +1

−1 −1 −1

+1 +1

The equation represented is 3(x + 2) = 4x − 3, which has a unique solution of x = 9.

+1 +1

When simplified, the equation is 3x + 6 = 4x − 3. In our other examples, attempting to add or remove tiles to isolate a variable would eliminate the variable entirely. Each side of this equation has a different number of +x tiles, so we can add and remove tiles equally to work towards isolating the variable to find the solution.

While there is more than one way to algebraically solve an equation, it is important to keep both sides of an equation equivalent by remembering that what you do to one side, you must do to the other.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Examples Students: Page 32 Example 1 Verify that the m = −4 is a solution to the equation 14 − 7m = −3(−2 + 3m).

Create a strategy Both sides of an equation should be equivalent after substituting a solution for the variable and evaluating.

Example 1 Apply the idea Substitute m =m −4 and usesolution the order of operations 14 to determine if both sides are equivalent. Verify Verify that that the the m = = −4 −4 is is a a solution to to the the equation equation 14 − − 7m 7m = = −3(−2 −3(−2 + + 3m). 3m). 14 − 7 ⋅ (−4) = −3(−2 + 3 ⋅ (−4)) Substitute m = −4 14 + 28 = −3(−2 − 12) Create a strategy

Evaluate the multiplication

14 equation + 28 = −3should ⋅ (−14) be Evaluate the subtraction Both substituting a Both sides sides of of an an equation should be equivalent equivalent after after substituting a solution solution for for the the variable variable and and evaluating. evaluating. 42 = 42 Evaluate the multiplication

Apply the sides idea are equivalent, m = −4 is a solution to 14 − 7m = −3(−2 + 3m). Since both Substitute Substitute m m= = −4 −4 and and use use the the order order of of operations operations to to determine determine if if both both sides sides are are equivalent. equivalent. 14 − 7 ⋅ (−4) = −3(−2 + 3 ⋅ (−4)) Substitute m = −4 14 − 7 ⋅ (−4) = −3(−2 + 3 ⋅ (−4)) Substitute m = −4 14 14 + + 28 28 = = −3(−2 −3(−2 − − 12) 12)

Evaluate Evaluate the the multiplication multiplication

PurposeExample 2 14 Evaluate 14 + + 28 28 = = −3 −3 ⋅⋅ (−14) (−14) Evaluate the the subtraction subtraction Check students can correctly substitute a valueEvaluate for a variable into an equation and verify the solution. 42 42 = = 42 42 Evaluate the the multiplication multiplication Use a balance scale and algebra tiles to solve 3(x − 4) = 2(−2x + 1). Since Since both both sides sides are are equivalent, equivalent, m m= = −4 −4 is is a a solution solution to to 14 14 − − 7m 7m = = −3(−2 −3(−2 + + 3m). 3m).

Students: Pages 32–34 Create a strategy

A balance scale represents the algebraic expressions on each side of an equation using +x, −x, +1, and −1 tiles. We work towards isolating a variable by adding and removing tiles on each side equally.

Example 2

Apply the idea

Use Use a a balance balance scale scale and and algebra algebra tiles tiles to to solve solve 3(x 3(x − − 4) 4) = = 2(−2x 2(−2x + + 1). 1). The original equation 3(x − 4) = 2(−2x + 1) is represented by the balance scale:

Create a strategy

−1 −1 −1 −1

A expressions on using +x −xof −x −1 −1 −1 −1 +1 A balance balance scale scale represents represents the the algebraic algebraic expressions on each each side side of an an equation equation using +x, +x, −x, −x, +1, +1, and and −1 −1 tiles. tiles. We and removing tiles side We work work towards towards isolating isolating a a variable variable by adding adding tiles−xon on each each −x side equally. equally. +xby −1 −1 removing −1 −1and +1

Apply the idea The The original original equation equation 3(x 3(x − − 4) 4) = = 2(−2x 2(−2x + + 1) 1) is is represented represented by by the the balance balance scale: scale: +x

−1 −1 −1 −1

+x +x +x +x +x +x

−1 −1 −1 −1 −1 −1 −1 −1

−x −x

+1 +1

−1 −1 −1 −1 −1 −1 −1 −1

−x −x

+1 +1

+x +x

+x Adding +4x to each side creates zero pairs−1for −1 the −1 −4x−1on the right side.

+x +x +x

−1 −1 −1 −1

−x

+x +x +x +x +x

−1 −1 −1 −1

−x

+x +x

−x −x

+1 +1

−x −x

+1 +1

Adding the −4x Adding +4x +4x to to each each side side creates creates zero zero pairs−1for for −1 the −1 −4x−1on on the the right right−xside. side. +x pairs

+x +x +x +x +x +x +x +x 32

+x Mathspace Virginia SOL Algebra 1+x

−1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1

mathspace.co

32 32

Mathspace Mathspace Virginia Virginia SOL SOL Algebra Algebra 11 mathspace.co mathspace.co

1.04 Multistep equations mathspace.co

Next, we remove the zero pairs on the right side, and the scale stays balanced. +x +x +x +x +x

−1 −1 −1 −1

Adding +12 to each side creates zero pairs for the −12 on the left side. +x +x

+1 +1 +1 +1

−1 −1 −1 −1

+1 +1 +1 +1

−1 −1 −1 −1

+1 +1 +1 +1

−1 −1 −1 −1

+1 +1 +1 +1

Remove the zero pairs from the left and the scale stays balanced. Since there are seven +x tiles on the left side, splitting the +1 tiles on the right into seven equal groups will tell us what each +x tile represents. +x +x

+1 +1 +1 +1 +1 +1 +1

Remove +x tiles from the left and two +1 tiles from the right at the same time to keep the scale balanced. Continue doing this until only one +x tile remains. +x

+1 +1

We can see that x = 2.

−1 −1

+1 +1

−1 −1

−1 −1 −1

−1 +1

+1 +1

−1 −1

−1 −1 −1

−1 +1

1.04 Multistep equations mathspace.co

After making zero pairs, a solution of an equation should show the same number of tiles left on each side of the balance.

−1 −1 −1

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Since there are six −1 tiles on each side of the balance, x = 2 is a solution to 3(x − 4) = 2(−2x + 1).

+1 +1

−1 −1

−1 −1 −1

−1 +1

+1 +1

−1 −1

−1 −1 −1

−1 +1

Reflect and check Balance scales can also be used to verify solutions to equations. For this example, to verify that x = 2 is the solution to 3(x − 4) = 2(−2x + 1), each +x will be replaced with two +1 tiles and each −x will be replaced with two −1 tiles. After making zero pairs, a solution of an equation should show the same number of tiles left on each side of the −1 −1 +1 +1 −1 −1 balance. +1 +1

−1 −1

−1 −1 −1

−1 +1

+1 +1

−1 −1 −1 −1 −1−1 −1

−1−1 −1−1 −1−1

−1 +1

−1 −1 −1

After making zero pairs, a solution of an equation should show the same number of tiles left on each side of the Since there are six −1 tiles on each side of the balance, x = 2 is a solution to 3(x − 4) = 2(−2x + 1). balance. −1 −1 −1

−1 −1 −1

PurposeExample 3 To ensure that students understand how to use a balance scale and algebra tiles to solve an equation. This skill many solutions the following equations have without solving. is crucialDetermine becausehow it helps students visualize the algebraic process and understand the principle of equality. a 4(x − 9) = x + 6 Since there are six −1 tiles on each side of the balance, x = 2 is a solution to 3(x − 4) = 2(−2x + 1).

Students: Page 34

Create a strategy Start by comparing both sides of the equation. We can see both sides are different and that there is an x on both sides. The coeﬃcient on the left side is 4, and the coeﬃcient on the right side is 1. Example 3

Apply the how ideamany solutions the following equations haveReﬂect check Determine withoutand solving. The equation a 4(x − 9) = xwill + 6have one unique solution.

We can verify our answer by solving the equation. 4(x − 9) = x + 6

Original equation

Create a strategy

4x − 36 = x + 6 Distributive property Start by comparing both sides of the equation. We can see both3x sides are different and that there is an x on both − 36 = 6 Subtraction property of equality sides. The coeﬃcient on the left side is 4, and the coeﬃcient on the right side is 1. 3x = 42 Addition property of equality

Apply the idea

x = 14 Reﬂect and check

The equation will have one unique solution.

There only one to solving this equation. We canisverify our solution answer by the equation.

Division property of equality

4(x − 9) = x + 6

Original equation

4x − 36 = x + 6

Distributive property

3x − 36 = 6

Subtraction property of equality

3x = 42

Addition property of equality

x = 14

Division property of equality

There is only one solution to this equation. 34

Mathspace Virginia SOL Algebra 1 mathspace.co

Purpose Students demonstrate that they can connect unequal coefficients to equations with a single solution. Expected mistakes Students may only consider the coefficients of x before expanding the expression and incorrectly determining that the equation has no solutions. 34

Mathspace Virginia SOL Algebra 1 mathspace.co

Students: Page 35

b 2x − 5 = 0.5(4x + 10)

Create a strategy The first thing we notice is that both sides seem to have different values. But if we distribute the coefficient on the right side, the first term would be 2x, and the second term would be 5. The equation now has the same variables with the same coefficients on both sides, but the constant values are different.

Apply the idea

Reﬂect and check

No solution

We can verify our answer by solving the equation. 2x − 5 = 0.5(4x + 10) 2x − 5 = 2x + 5 −5 = 5

1.04 Multistep equations Original equation mathspace.co Distributive property Subtraction property of equality

b 2x − 5 = 0.5(4x + 10)

Apply the idea

Reﬂect and check

No solution

We can verify our answer by solving the equation. 2x − 5 = 0.5(4x + 10)

Original equation

2x − 5 = 2x + 5

Distributive property

−5 = 5

Subtraction property of equality

The resulting statement is not true. This is how we know there is no solution to this equation. b 2x − 5 = 0.5(4x + 10)

Example 4 PurposeCreate a strategy first we notice is thatcan bothconnect sides seem to have different values.by But we distribute the coefficient thesolutions. Solve thething following equations: StudentsThe demonstrate that they expressions differing a ifcontext to equations withonno right side, the first term would be 2x, and the second term would be 5. The equation now has the same variables with

a mistakes 4(5x + 1) = −3(5x − 5) Expected the same coefficients on both sides, but the constant values are different. Students may distribute the multiplication correctly but not notice the difference in the signs of the constant Create a strategy Applyincorrectly the idea determining that the solution has Reflect and check values, thus infinitely many solutions. Looking at the equation, we see that the variables have different coefficients on both sides of the equation. The coefficient

No solution We can verify our answer by solving the equation. Reflecting with on the left students side is 4 ⋅ 5 = 20, and the coeﬃcient on the right side is −3 ⋅ 5 = −15. This means it will have a unique solution. 0.5(4x + to 10) make Original Ask students if they would expect a randomly chosen value2x for− a5 =variable theequation equation true. Could they Apply the idea 2xmany − 5 = 2x +5 Distributive property use this as a test for whether or not an equation has infinitely solutions? Is it mathematically rigorous to 4(5x + 1) = −3(5x − 5) Given equation −5 = 5 Subtraction property of equality do this? 20x + 4 = −15x + 15

Students: Page 35 35x + 4 = 15 35x = 11 x=

Example 4

Distributive property The resulting statement is not true. This is how we know Addition property equality there isofno solution to this equation. Subtraction property of equality Division property of equality

Reﬂect and check

Solve the following equations: We can verify the solution by substituting it back into the equation to see if it makes the equation true. a 4(5x + 1) = −3(5x − 5) Original equation

Create a strategy

Substitute x = Looking at the equation, we see that the variables have different coefficients on both sides of the equation. The coefficient on the left side is 4 ⋅ 5 = 20, and the coefficient on Multiply the right5side ⋅ is −3 ⋅ 5 = −15. This means it will have a unique solution.

Apply the idea 4(5x + 1) = −3(5x − 5) 20x + 4 = −15x + 15

Create common denominators Given equation

35x + 4 = 15

Distributive Evaluate theproperty addition Addition property of equality

35x = 11

Evaluate theproperty multiplication Subtraction of equality

x= What results is a true equation, so we know

Division property of equality is the correct solution.

Reﬂect and check We can verify the solution by substituting it back into the equation to see if it makes the equation true. Original equation

1.04 Multistep equations mathspace.co

Substitute x = Multiply 5 ⋅ Create common denominators Evaluate the addition Evaluate the multiplication What results is a true equation, so we know

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

is the correct solution.

Purpose Provide students with an example of solving an equation that has the variable on both sides. Reflecting with students Discuss with students the importance of correctly applying the distributive property and equality properties in order to maintain the balance of the equation. Additionally, review the steps to verify the solution by substituting it back into the original equation.

Students: Page 36 b

Create a strategy This equation has a mix of rational numbers, and the variables are in multiple terms. If we multiply both sides of the equation by the lowest common denominator, that will remove fractions and only leave us with decimals. Then, we can work on collecting all the variables.

Apply the idea The denominators are 2 and 4, so the lowest common denominator is 4. Original equation Multiplication property of equality Distributive property Distributive property Combine like terms Subtraction property of equality Division property of equality

Reﬂect and check There are many other ways we could have started solving this equation, but eliminating the fractions ﬁrst made it much easier to solve.

Example 5 Purpose Right now, Bianca’s father is 48 years older than Bianca. Challenge students to solve an algebraic equation involving the use of multiple properties of equality, as well as Two years ago, her father was 5 times as old as she was. applying an understanding of lowest common denominator to simplify work. Solve for y, Bianca’s current age.

Reflecting with students Createifathey strategy Ask students would still have reached the same solution if they had applied the multiplication property We want to write expressions representing Bianca’s age and her father’s age. Then we relate them with an equation twice (once for each denominator) rather than just once for the lowest common denominator. and solve for y.

Bianca’s father is currently y + 48 years old.

Advanced the old. properties toyjustify Two years ago,learners: Bianca was yUse − 2 years Her father was + 48 − 2 each = y + 46step years old.

use with Example 4

Targeted strategies Her father’sinstructional age was ﬁve times her age at this time, which produces our equation: y + 46 =of5(real y − 2)numbers and properties of equality, but not The standards require students to apply the properties necessarily to name the properties in each line of work. For advanced learners, having them justify each step of Apply the idea process by explicitly stating the properties Reﬂect and check their equation-solving of real numbers and properties of equality can Let’s check by referring back to the information given in y + their 46 = 5(understanding y − 2) Originalof equation help deepen why each step is valid. y + 46 = 5y – 10

Distributive property

the problem. Bianca’s father is 48 years older, so he is

Division property of equality

Yes, this is correct.

In addition, knowing the names of the properties and using toyears justify each line of work is good for 14 +them 48 = 62 old. Two years ago, he was 5 timespractice as y + 56 = 5y Addition property of equality proofs or for upper level mathematics courses that require deeper of reasoning or been justification for their old.a Two yearslevel ago, Bianca would have 12, and her 56 = 4y Subtraction property of equality father would have been 60. Is this 5 times Bianca’s age? work. 14 = y

Bianca is currently 14 years old.

Mathspace Virginia SOL Algebra 1 mathspace.co

1.04 Multistep equations mathspace.co

Subtraction property of equality Division property of equality

Reﬂect and check There are many other ways we could have started solving this equation, but eliminating the fractions ﬁrst made it easier Students:much Page 36to solve.

Example 5 Right now, Bianca’s father is 48 years older than Bianca. Two years ago, her father was 5 times as old as she was. Solve for y, Bianca’s current age.

Create a strategy We want to write expressions representing Bianca’s age and her father’s age. Then we relate them with an equation and solve for y. Bianca’s father is currently y + 48 years old. Two years ago, Bianca was y − 2 years old. Her father was y + 48 − 2 = y + 46 years old. Her father’s age was ﬁve times her age at this time, which produces our equation: y + 46 = 5( y − 2)

Apply the idea

Reﬂect and check

y + 46 = 5( y − 2)

Original equation

y + 46 = 5y – 10

Distributive property

y + 56 = 5y

Addition property of equality

56 = 4y

Subtraction property of equality

14 = y

Division property of equality

Let’s check by referring back to the information given in the problem. Bianca’s father is 48 years older, so he is 14 + 48 = 62 years old. Two years ago, he was 5 times as old. Two years ago, Bianca would have been 12, and her father would have been 60. Is this 5 times Bianca’s age? Yes, this is correct.

Bianca is currently 14 years old.

Mathspace Virginia SOL Algebra 1 mathspace.co

Purpose Show students how they can create an equation that models a real-world problem, and then solve it to find previously unknown contextual information. Expected mistakes Students may be able to construct two expressions, but not understand how to connect them into an equation. Ask students what each expression represents, and how they can adjust them so that they represent the same thing. Reflecting with students With any contextual problem, ask students what the mathematical solution represents in the context of the problem, and if they think that the solution is reasonable. This type of reflection will become increasingly important as students begin to consider valid and not valid solutions involving domains and ranges of variables.

Visualizing the problem

use with Example 5

Student with disabilities support For students who may struggle with abstract concepts, visualizing the problem can be very helpful. Here’s a strategy to do that: • Draw a timeline to represent time passing from two years ago to now. Mark Bianca’s age and her father’s age on the timeline. • Use different colors or symbols to represent the different pieces of information in the problem. For example, use one color to represent Bianca’s age and another to represent her father’s age. • Demonstrate how to translate the words in the problem into mathematical equations using the timeline. • Practice: Give students practice problems to solve using this visual strategy.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Students: Page 37

Idea summary A solution to an equation is any value that can replace the variable and make a true statement. A fully simplified equation in one variable will take one of the following three forms, corresponding to how many solutions the equation has: • x = a, where a is a number (a unique solution) • a = a, where a is a number (inﬁnitely many solutions) • a = b, where a and b are different numbers (no solutions) We can simplify an equation using the distributive property or the multiplication property of equality to eliminate fractions or decimals. After simplifying, we can continue using properties of equality to solve for the unknown.

Practice What do you remember?

Practice 1

Solve each of the equation: b

x+x+x+x+2+5=x+4+x+x

−k − k − k − 6 = −k − k − k − k + 6

7(7 + x) = 2x + 28 What do youa remember?

10(5 + x) = 10(x + 5)

−31 − 3(x − 9) = 2(x − 2) − 5x

m + m + 3 + 12 = 13 + m + m + m

Students: Pages 37–39 c p+p−3+5=p−p−p+1–5 2

Solve each of the equation: a c

Determine the number of solutions of each equation:

Consider the equation 9(n + 5) = 4n + 50 for n = 1.

m+m 3 + 12 13 +ofmthe+ left-hand m + m side of the equation when b nx= +1. x + x + x + 2 + 5 = x + 4 + x + x a + Find the=value

p + p b− 3Find + 5 the = pvalue − p of − pthe+ right-hand 1–5 d −k side of the equation when n =− 1. k − k − 6 = −k − k − k − k + 6 c

a c

7(7 + x) = 2x + 28 4

Find the value of the left-hand side of the equation when = 0.8. d n−31 − 3(x − 9) = 2(x − 2) − 5x Find the value of the right-hand side of the equation when n = 0.8.

your answers parts (b).1. Consider ctheConsider equation 9(n + 5) =from 4n + 50(a) forand n=

n = 0.8 a solution of the equation 0.28n − 0.4 = 2.3n − 3.03? Find the Is value of the left-hand side of the equation when n = 1.

5 the Solve each of of the Find value theequation: right-hand side of the equation when n = 1.

a 3f − 8 = f Consider your answers from parts (a) and (b). c

4w + 24 = w + 15

Is n = 1 a solution of 9(n + 5) = 4n + 50?

10r + 4 = 6r

−72 − 9p = −32 − p

10(5 + x) = 10(x + 5)

Consider the equation 0.28n − 0.4 = 2.3n − 3.03 for n = 0.8. b

Consider your answers from parts (a) and (b).

Determine the of solutions each equation: Is nnumber = 1 a solution of 9(n + 5)of = 4n + 50?

Consider the equation 0.28n − 0.4 = 2.3n − 3.03 for n = 0.8. g

3.2y + 17 = 52 − 1.8y

Find the value of the left-hand side of the equation when n = 0.8.

Find value the of the of the equation when n = 0.8. 6 the Determine valueright-hand that makes side each equation true.

a 6your − 7p =answers 41 −11 (a) = 2x − 5 (b). Consider from bparts and

35 = 5(n − 15)

f g − 3.03? Is n =e0.8 a solution of the equation 0.28n − 0.4 = 2.3n

−2(p − 8) = 16

1 = −( y + 10)

Solve each of the equation: a

3f − 8 = f

10r + 4 = 6r

4w + 24 = w + 15

equations −72 − 9p = −32 − 1.04 p Multistep mathspace.co

e g

3.2y + 17 = 52 − 1.8y

f h

1.04 Multistep equations mathspace.co

Determine the value that makes each equation true. a

6 − 7p = 41

−11 = 2x − 5

35 = 5(n − 15)

−2(p − 8) = 16

1 = −( y + 10)

Let’s practice 7

Solve each of the equation: a

4(x − 15) = x

5(x − 16) = x − 16

2(1.8w − 5) = 0.4w + 2

3(x − 3) = 2(x + 1)

3y = −13 − 4(2y + 5)

Determine whether the following equations are equivalent: a

5 − x − x = −x + 15

+ 5 = 3 − 2u and u + 5 = 6 − 4u

6y − 3( y + 4) = −12

5(x − 2) − 3x = 2(x + 5)

6 + 2x − 11 = −2x − 3

13 = 4m − (2m − 7)

Solve each of the equation: a

4y + 2 = y − 13 and 3y + 2 = −13

Solve each of the equation and justify each step using properties of equality: a

Determine the property that justiﬁes why the equations 2x + 4 = 3x − 6 and 4 = x − 6 are equivalent.

A student is in the process of solving an equation. The original equation and the ﬁrst step are as follows:

Determine the property used by the student for the ﬁrst step. 14

Consider the equation 3(x + 5) = 1.8x − 9. Peta solves the equation as follows:

3(x + 5) = 1.8x − 9

3x + 15 = 1.8x – 9

Distributive property

1.2x + 15 = −9

1.2x = −24

x = −20

⬚

Subtraction property of equality ⬚

Determine the property that justiﬁes Peta’s third step of work.

Determine the property that justiﬁes Peta’s last step of work.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

For each of the relation: i

Write an equation for the relation, using n to represent the unknown number.

Determine the number of solutions to the equation.

Three more than three times a number is equal to six more than four times the number.

Six more than two times a number is equal to ﬁve more than two times the number.

Two more than four times a number is equal to two added to quadruple the number.

A number is multiplied by 5 and then 2 is added. Then the result is multiplied by 6. This is equal to 10 times the number minus 8. a

Form an equation for this problem.

Solve the equation to ﬁnd the number.

Consider the given rectangle with a perimeter of 126 + 3y centimeters. 4y + 8

4y + 3

Write an equation and solve for the value of y. 18

A Payroll Oﬃcer has been told to distribute a bonus to the employees of a company worth 15% of the company’s net income. Since the bonus is an expense to the company, it must be subtracted from the income to determine the net income. If the company has an income of $120 000 before the bonus, then the Payroll Oﬃcer must solve the following for B: B = 0.15 (120 000 − B) Find the bonus, B.

A rectangle with a length of 2.5x + 4 cm and a width of x − 1 cm has the same perimeter as a square with side length

Write an equation and solve for the value of x.

Let’s extend our thinking 20

Three consecutive integers are such that the sum of the first and twice the second is 15 more than twice the third. a

Let x be the smallest of the integers. Solve for x.

State the three consecutive integers.

Are there any other sets of three consecutive integers that could fulfill the requirements? Explain how you know.

Determine whether the following statements are true or false. Explain your thinking. a

Any equation with variables on both sides must have multiple solutions.

Two equations will always have the same solution if one is a multiple of the other.

Explain the criteria for identifying the number of solutions that an equation has, and provide an example for each.

1.04 Multistep equations mathspace.co

Answers

Given

1.04 Multistep equations

Subtraction property of equality

What do you remember?

Multiplication property of equality

1 a m=2

b x = -3

c p = -2

d k = 12

Subtraction property of equality

2 a One solution

b Infinitely many solutions

Addition property of equality

c No solution

d Infinitely many solutions

Solution

3 a 54

b 54

c Yes

4 a -0.176

b -1.19

c No

5 a f=4

b r = -1

c w = -3

d p = −5

g y=7

Multiplication property of equality

b x = -3

c n = 22

d p=0

x = 60

g u=2

h y = −11

Addition property of equality

e y = -20 6 a p = -5 e y = 16

Given

Addition property of equality

Subtraction property of equality

Let’s practice 7 a x = 20 e y = -3

b x = 16 f

Division property of equality

d x = 11

x = 80

Solution

8 a Yes

b No

12 Addition property of equality or Subtraction property of equality

9 a x = −10

b y=0

13 Multiplication property of equality

c Infinitely many solutions

d No solution

14 a Subtraction property of equality

m=3

b Division property of equality

10 a y = 2 11 a

d No solution

15 a i 3n + 3 = 4n + 6

Given Addition property of equality Multiplication property of equality Addition property of equality Subtraction property of equality

ii One solution

b i 2n + 6 = 2n + 5

ii No solution

c i 4n + 2 = 4n + 2

ii Infinitely many solutions

16 a 6(5x + 2) = 10x − 8

b -1

17 2(4y + 8) + 2(4y + 3) = 126 + 3y, y = 8 18 B = $15 652.17 19

Division property of equality

Let’s extend our thinking

Solution

20 a x = 17 Given Create common denominators Combine like terms Multiplication property of equality Subtraction property of equality Addition property of equality Division property of equality Solution

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

b 17, 18, 19

c N o, because the equation used to solve for x in part (a) has only one unknown, x, to the power of one. Therefore, there is only one solution. 21 a F alse. If the variable terms do not add to zero, then the equation must mathematically have one solution. b T rue. For example, if a value of x satisfies 2x + 3 = 1, then it will also satisfy 4x + 6 = 2.

22 An equation that has no solutions will have the same x-term on both sides when simplified and different constant terms on both sides. This would lead to a false statement, meaning that no value of x will satisfy the equation. An example would be 4.5x − 7.8 = 4.5x + 9.3. An equation that has an infinite number solutions will have the same x-term on both sides when simplified and the same constant terms on both sides. This would lead to a true statement, meaning that any value of x will satisfy the equation. An example would be An equation that has exactly one solution will not have the same x-term on both sides when simplified. The constant term on both sides may be the same or different. This would lead to a equation where exactly one value of x will satisfy the equation. An example would be 2.7x + 8 = 3x + 8.

Answers mathspace.co

1.05 Literal equations Subtopic overview Lesson narrative In this lesson, students will use repeated reasoning of solving for specific values to analyze the structure of literal equations and solve for a particular variable. Students will engage in an exploration activity where they investigate solving for different variables of a literal equation. They will justify their answers using properties of equality and inverse operations. By the end of this lesson, students will be able to solve linear equations in one variable, including equations with coefficients represented by letters.

Learning objectives

1.05 Literal equations

Students: Page 40

After this lesson, you will be able to… • solve linear equations in one variable, including equations with coefficients represented by letters. • solve for a specific variable in a formula.

Literal equations

KeyLiteral vocabulary equation 

A formula or equation that consists primarily of formula variables.

Formula 

A typeequation of literal equation that describes a literal relationship between real-world quantities. Example: A = l ⋅ w

Essential understanding

where A is area, l is length, and w is width.

The same standard algorithms that are applied to equations with numerical coefficients can be applied to equations There are manyrepresented different formulas in science, mathematics, business, and other subjects that allow us to measure with coefficients by letters. quantities such as area, volume, speed, etc. We can use the properties of equality to isolate any variable in a literal equation or formula. The same variable might be used to represent different quantities across different formulas. For example, in the Standards

formula for the area of a rectangle, w is used to represent the width of the rectangle. However, in another context, This the following Virginia 2023 Mathematics Learning w subtopic might be addresses used to represent a weight or other value. To avoid anyStandards confusion,of formulas willstandards. always state what the variables represent.

Mathematical process goals

A variable can also act as a placeholder for other expressions when a formula applies in a variety of different situations. For example, the formula for the volume of both aMPG4 cone and a pyramid is Connections — Mathematical MPG2 — Mathematical Communication

Teachers can help students make connections between Teachers can encourage students to explain their their prior knowledge of solving linear equations thought process as they work through the problems. where B represents the area of the two-dimensional shape at the base and h is the vertical height from the base. and properties of equality and the new concept of They can have students share their solutions and To find the volume of each specific figure, we replace the B with the area formulas for the base shapes. rearranging literal equations. They can also show reasoning with the class, using the language of students how these mathematical concepts are related mathematics to express their ideas. Teachers can also and how they can be applied in different contexts, such use mathematical discussions to clarify and deepen height (h) as calculating the area of a rectangle or solving for speed students’ understanding of the concepts being studied. in the distance formula. h r

A = s2 80

Mathspace Virginia SOL Algebra 1 Teacher Edition s mathspace.co

A = π r2

A.EI.1d — Rearrange a formula or literal equation to solve for a specified variable by applying the properties of equality.

A.EO.1 — The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

Future connections A.F.1 — The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.

A2.EO.3 — The student will perform operations on polynomial expressions in two or more variables and factor polynomial expressions in one and two variables.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Algebra 1 — 1.02 Properties of real numbers Algebra 1 — 1.03 Properties of equality Algebra 1 — 1.04 Multistep equations

Tools You may find this tool helpful: • Scientific calculator

Student lesson & teacher guide Literal equations Students are introduced to the concepts of literal equations and formulas and why they are important. They are also shown that variables can be used as placeholders in formulas with multiple possible values, such as B in the volume formula for a cone or pyramid. Then they engage in an exploration to uncover the benefit of solving an equation for a specific variable.

1.05 Literal equations mathspace.co

Students: Page 40

1.05 Literal equations After this lesson, you will be able to… • solve linear equations in one variable, including equations with coefficients represented by letters. • solve for a specific variable in a formula.

Literal equations Literal equation

Formula

A formula or equation that consists primarily of variables.

A type of literal equation that describes a relationship between real-world quantities. Example: A = l ⋅ w where A is area, l is length, and w is width.

There are many different formulas in science, mathematics, business, and other subjects that allow us to measure quantities such as area, volume, speed, etc. We can use the properties of equality to isolate any variable in a literal equation or formula. The same variable might be used to represent different quantities across different formulas. For example, in the formula for the area of a rectangle, w is used to represent the width of the rectangle. However, in another context, w might be used to represent a weight or other value. To avoid any confusion, formulas will always state what the variables represent. A variable can also act as a placeholder for other expressions when a formula applies in a variety of different situations. For example, the formula for the volume of both a cone and a pyramid is

where B represents the area of the two-dimensional shape at the base and h is the vertical height from the base. To find the volume of each specific figure, we replace the B with the area formulas for the base shapes.

height (h) h r

A = s2 s

A = π r2

Moving terms with the correct operations Address student misconceptions When working with equations with multiple variables, students may get confused with moving multiple variables at the same time with the correct operation. For example, when solving for y in the equation xz + wy = v, students may assume that the x and z must be 40 Mathspace Virginia SOL Algebra 1 moved separately rather than being thought of as the single term xz. mathspace.co Advise students to highlight or box terms and label them with their operation to help them identify the inverse operations needed to isolate the desired variable.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Connect literal equations to single-variable equations Targeted instructional strategies Point out to students that solving literal equations is equivalent to solving an equation in one variable. This realisation can be prompted by first having students work through different equations and identify the inverse operations needed to solve for x: Equation x + 5 = −4 2x − 3 = 9

Inverse Operation(s)

−3(1 + 4x) = 33 Once students have identified the steps and inverse operations needed to solve each equation, add a third column to the table and ask students to apply the same inverse operations to solve for x: Equation x + 5 = −4 2x − 3 = 9

Inverse Operation(s)

−3(1 + 4x) = 33

Literal Equation x + y = −4 3x − 8 = w

2(5y + 3x) = −42

Students can then be given other literal equations to solve without being given the similar multistep equation first.

Information gap English language learner support Divide students into pairs and give them an equation such as: P = 2l + 2w Their goal is to answer the following prompt: How can we solve for a variable other than P? Each student will receive a different variable to isolate. • Solve for l. P = 2l + 2w • Solve for w. P = 2l + 2w Each student must communicate their piece of information without showing their paper. They need to discuss and piece together the information to write a set of steps that apply the non-isolated variables. Students should first clarify what each variable means. Each should solve their equations for their given variable. Each pair presents their solution and explains the steps taken to reach it. They should discuss any discrepancies or different methods used and write any common steps followed. They should then rewrite their list of steps to describe how to solve for any variable other than the isolated variables.

1.05 Literal equations mathspace.co

Algorithmic thinking - provide a step-by-step checklist for solving Student with disabilities support Provide students with a detailed checklist with space next to each description for students to work out their problems. This approach can help to encourage and develop computational thinking as they see that a complex problem can be decomposed into smaller steps. If possible, place the checklist in a plastic sleeve for students to use dry erase marker. A checklist could include what is shown below: 1. Identify the variable to isolate: Circle or highlight the target variable in the equation. 2. Simplify both sides of the equation if needed: Combine like terms, distribute any factors. 3. Move terms with the target variable to one side: Use addition or subtraction to shift terms. 4. Move all other terms to the opposite side: Use addition or subtraction to isolate the variable term. 5. Isolate the variable by dividing or multiplying: Divide or multiply to get the variable alone.

Exploration Students: Page 41

Exploration Consider the formula for distance:

d = rt d

distance

rate

time

Use the distance formula to solve each of the following: • The speed at which a person travels if they drive 100 mi in 75 min • The distance traveled when walking 5 km in 30 min • The distance traveled by a car driving 65 mph for 5 hr • The time it will take to travel 1000 km if you walk 80 m/min • How fast a plane is traveling if it can ﬂy 2789 mi in 6 hr • How long it will take to walk to the store 2 mi away if you walk 176 ft/min 1.

Which ones took the most effort to solve?

How could we reduce the effort when making repeated calculations for the same variable?

Using the division property of equality, we can rearrange the equation relating distance, rate, and time to be or

Suggested student grouping: In pairs By rearranging a formula for a variable of interest, we can reduce the number of repeated calculations needed, Studentsdepending will use on thewhich distance formula to solve a collection of problems, with some requiring students to variable is unknown. rearrange the formula to isolate different variables in order to reduce the diﬃculty of solving.

Example 1 Ideal student responses These ideal Ohm’sresponses law states: may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. V = IR 1. Which ones theI most effort where V istook voltage, is current, and to R issolve? resistance. It is expected that students will ﬁnd it easiest to solve for distances, and harder to solve for rates and times. Write the formula for current. The problems requiring unit conversions are expected to take the most effort. strategy Apply the idea 2. HowCreate could awe reduce the effort when making repeated calculations for the same variable? The formula for current is Ohm’s law with I isolated. We can rearrange the distance formula to isolate the variable weGiven wantequation to solve for. We use inverse operations and properties of equality to get the solution.

Division property of equality Symmetric property of equality

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Example 2 Solve for x in the following equation:

Exploration

Purposeful questions

Consider the formula distance: • How can we express thefor rate in terms of distance and time? • How can we express the time in terms of distancedand rate? = rt • When can we perform algebraic operations with variables different units? d distance r ratethat have t time Use the distance formula to solve each of the following: Possible misunderstandings

Exploration •may The not speed at which‘rate’ a person travels if theyTo drive 100 mi in up 75 min • Students connect with ‘speed’. help clear this misunderstanding, ask students for an distanceand traveled when 5 km in 30 mina rate of distance per time. example •ofThe a speed, point outwalking that the units are Consider the formula for distance: • The distance traveled by a car driving 65 mph for 5 hr

• The time it will take to travel 1000 km if you walk d 80 = m/min rt

Students should confident in identifying whatmivariable shouldt betime isolated in order to solve a given problem, • aim How to fastbe a plane is traveling ifdit can ﬂy 2789 in 6 distance r hrrate • How long applying it will take to walk to operations the store 2 mito away if youthat walkvariable 176 ft/min on one side of the equation. and then be comfortable inverse isolate

Use the distance formula to solve each of the following: 1. Which ones took the most effort to solve? • The Students: Page 41speed at which a person travels if they drive 100 mi in 75 min 2. How could we reduce the effort when making repeated calculations for the same variable? • The distance traveled when walking 5 km in 30 min

• The distance traveled by a car driving 65 mph for 5 hr Using• the division we km canifrearrange them/min equation relating distance, rate, and time to be The time itproperty will take of to equality, travel 1000 you walk 80 •orHow fast a plane is traveling if it can ﬂy 2789 mi in 6 hr • How long it will take to walk to the store 2 mi away if you walk 176 ft/min By rearranging a formula for a variable of interest, we can reduce the number of repeated calculations needed, depending on which unknown. 1. Which onesvariable took theismost effort to solve? 2.

How could we reduce the effort when making repeated calculations for the same variable?

Example 1 Using the division property of equality, we can rearrange the equation relating distance, rate, and time to be

Ohm’s law states: Examples or

V = IR

ByPage rearranging a formula for a variable of interest, we can reduce the number of repeated calculations needed, Students:where V is41 voltage, I is current, and R is resistance. depending on which variable is unknown. Write the formula for current.

Example 1 Create a strategy

Apply the idea

The formula for current is Ohm’s law with I isolated. Ohm’s law states: We use inverse operations and properties of equality to V = IR get the solution. where V is voltage, I is current, and R is resistance. Write the formula for current.

Create a strategy Example 2

The formula for current is Ohm’s law with I isolated. We use and properties of equality to Solve forinverse x in theoperations following equation: get the solution.

Given equation Division property of equality Symmetric property of equality

Apply the idea Given equation Division property of equality Symmetric property of equality

Create a strategy We need to rearrange the equation to isolate x. We can use the properties of equality and inverse operations to solve Example 2 for a variable, just as we would for linear equations. literal equations

Purpose for x in thecan following equation: Checks Solve that students identify which variable they need to isolate, and then isolate it using inverse operations. 1.05 Literal equations mathspace.co

1.05 Literal equations mathspace.co

Reﬂecting with students Create a strategy Ask students if they think it is necessary to rearrange the formula, given that it is already quite simple. We need to rearrange the equation to isolate x. We can use the properties of equality and inverse operations to solve Ask students to consider they would rearranging literal equations for a whether variable, just as we wouldbother for linear equations. the equation if they only had to calculate the current for one electrical circuit. What if they had to do this for hundreds of circuits?

1.05 Literal equations mathspace.co

or By rearranging a formula for a variable of interest, we can reduce the number of repeated calculations needed, depending on which variable is unknown.

Example 1

Working with contextual formulas

use with Example 1

Ohm’s law states:

Student with disabilities support

V = IR

When presented with a contextual formula, students may get overwhelmed by information and not know where where V is voltage, I is current, and R is resistance. to start. Write Advise students to start with just the formula, then steadily connect the variables to the context, one at the formula for current. a time. Create strategy It can also helpato highlight common visual and languageApply cues the likeidea “where V is voltage...” following a formula, The formula for current is Ohm’s law with I isolated. which usually explains what each variable in the formula represents.Given equation We use inverse operations and properties of equality to get the solution.

Division property of equality Symmetric property of equality

Students: Pages 41–42 Example 2 Solve for x in the following equation:

Create a strategy We need to rearrange the equation to isolate x. We can use the properties of equality and inverse operations to solve literal equations for a variable, just as we would for linear equations.

Apply the idea

Reﬂect and check Given equation Division property of equality Subtraction property of equality

Remember, when rearranging an equation, we reverse 1.05 Literal equations 41 the operations acting on the variable wemathspace.co want to isolate, in the reverse order of operations. Whatever is done to one side of the equation, must be done to the other to keep the equation balanced.

Multiplication property of equality Symmetric property of equality

Example 3 Purpose find the sum, S, of the interior angles of any polygon with n sides, the following equation can be used: StudentsTodemonstrate that they can apply their equation solving skills to isolate a named variable. S = 180(n − 2)

a mistakes A polygon’s angles sum to 900°. Use the sum of interior angles formula to determine its number of sides. Expected

Students may believe that they cannot apply inverse operations to variables like x. Let students know that Create a strategy they can apply inverse operations to anything, as long as the operation is applied properly to both sides of Since S is the sum of the angles, replace the S in S = 180(n − 2) with 900 and solve. the equation. Apply the idea

Reﬂect and check

Targeted instructional repleaced with 900 and westrategies will solve for n.

replaced with 900 is to distribute first then solve.

Simple questions with 2 Since the numbers angles sum toand 900°,purposeful the S in the formula will be An alternative but equivalent way to solve foruse n once S Example is Replace S the withlesson’s 900 Replace S with 900 Select worked examples that address learning target with relatively simple numbers. As a class Distribute Divide both sides by 180 consider questions like: Add 360 to both side • What is this question asking us to do? • What strategies can we use to 2complete this problem? Add to each side • What are the relationships Simplify between the quantities in this equation? How will the placement Divide by 180of the quantities and the operations impact what I do ﬁrst? A polygon whose angles sum to 900° has 7 sides • What have we seen before that might help us with this problem? b Write an equation to solve for n using the properties of equality.

Create a strategy 86

We need to rearrange the equation to isolate n. We can use the properties of equality and inverse operations to get the solution. Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Apply the idea

Reﬂect and check Given equation

Equations can be represented multiple ways, and the

Have students write a set of steps for the example which can be used for other questions with more involved numbers. In particular, students can use a Step-by-step graphic organizer like this one: Procedure Steps Details Step 1: Step 2: Step 3: Step 4: Step 5:Apply the idea

Reﬂect and check

Once students have writtenGiven their equation steps, have them consider questions Remember, whenlike: rearranging an equation, we reverse the operations acting on the variable we want to isolate, • How can we check that our answer is correct? in the reverse order of operations. Whatever is done to • Could these steps be done in aproperty different order? Division of equality one side of the equation, must be done to the other to • Can we think of a different set of steps that could be used? keep the equation balanced. Subtraction property of equality • Do we need any additional steps for questions that have different values? • How can we make our mathematical language more precise? Multiplication property of equality

Students: Page 42

Symmetric property of equality

Example 3 To find the sum, S, of the interior angles of any polygon with n sides, the following equation can be used: S = 180(n − 2) a A polygon’s angles sum to 900°. Use the sum of interior angles formula to determine its number of sides.

Create a strategy Since S is the sum of the angles, replace the S in S = 180(n − 2) with 900 and solve.

Apply the idea

Reﬂect and check

Since the angles sum to 900°, the S in the formula will be An alternative but equivalent way to solve for n once S is repleaced with 900 and we will solve for n. replaced with 900 is to distribute first then solve. Replace S with 900

Replace S with 900

Divide both sides by 180

Distribute Add 360 to both side

Add 2 to each side Divide by 180

Simplify A polygon whose angles sum to 900° has 7 sides b Write an equation to solve for n using the properties of equality.

PurposeCreate a strategy Challenge to identify which variable needs touse bethe solved for inoforder solve theoperations problem,toand We students need to rearrange the equation to isolate n. We can properties equalitytoand inverse get then solve forthe it solution. after subsituting in the given value. Reflecting with Apply thestudents idea Reﬂect and check Students can divide the total Given number of degrees in the shape by the of sides for aways, regular polygon to Equations can number be represented multiple and the equation determine how many degreesDistributive each angle has. Have students try equation drawing regular polygons with the desired is also equivalent to the simplified property number of sides and measuring the angles reinforce the formula. Addition propertyto of help equality equation found in the second to last step, Division property of equality Reflexive property

After applying the reflexive property, represents an equivalent and correct way to solve for n.

The equation S = (n − 2) solved for n is

Mathspace Virginia SOL Algebra 1 mathspace.co

1.05 Literal equations mathspace.co

Distribute

Divide both sides by 180

Add 360 to both side

Add 2 to each side Divide by 180

Simplify

Students:A Page 42 polygon whose angles sum to 900° has 7 sides b Write an equation to solve for n using the properties of equality.

Create a strategy We need to rearrange the equation to isolate n. We can use the properties of equality and inverse operations to get the solution.

Apply the idea

Reﬂect and check Given equation

Equations can be represented multiple ways, and the

Distributive property

simplified equation

Addition property of equality

equation found in the second to last step,

Division property of equality Reflexive property

is also equivalent to the

After applying the reflexive property, represents an equivalent and correct way to solve for n.

The equation S = (n − 2) solved for n is

Mathspace Virginia SOL Algebra 1 mathspace.co

Purpose Show students how they can isolate a variable in a given equation to determine other values in that equation. Expected mistakes Students may see that they can divide 360 by 180 and forget that the S also gets divided by 180. Students should write out each term using division before fully simplifying.

Students: Page 43 c Use the equation from part (b) to find the number of sides of a polygon whose angles sum to 1440°.

Create a strategy Replace S with 1440 and evaluate.

Apply the idea

Reﬂect and check Original equation

Using the original equation equation S = 180(n − 2) can also be used to find the number of sides, and will give the

Substitute S = 1440

same value as

Divide

Original equation

Simplify

Replace S with 1440

A polygon whose angles sum to 1440° has 10 sides.

Distribute Add 360 to both sides Divide both sides by 180 Simplify Using the original equation helps confirm the correct steps were taken to isolate the needed variable.

Idea summary

Purpose In the same way we solve one-variable equations, we can use inverse operations and the properties of Show studentsequality that isolating variable, substituting and solving for values, can be a more efficient to isolate aa variable in abefore literal equation. method.

Practice What do you remember? 88

Mathspace Virginia SOL Algebra 1 Teacher Edition 1 What beneﬁts come from rewriting literal equations? mathspace.co 2 Which of the following are literal equations? A

3x + 6y = 18

A = 1 5π

Create a strategy Replace S with 1440 and evaluate.

Apply the idea

Reﬂect and check

Reflecting with students Using the original equation equation S = 180(n − 2) can Original equation be usedforms to find the number of sides,might and will the useful. Ask advanced learners to describe situations in which thealso different of the equation begive most same value as given the number of sides of the polygon, For example, they should Substitute say that S 180(n − 2) is most useful when S ==1440 while

is most useful polygon. Divide when given the sum of the interior angles of theOriginal equation

Simplify S with 1440each of the For all students, summarize this example by asking which equation they would Replace use when given A polygon whose angles sum to 1440° has 10 sides. Distribute following pieces of information: Add 360 to both sides • The polygon has 5 sides • The sum of the interior angles of the polygon is 540° Divide both sides by 180 • There are 6 interior angles of the polygon, each with a measure of 120° Simplify Using the original equation helps confirm the correct steps were taken to isolate the needed variable.

Students: Page 43

Idea summary In the same way we solve one-variable equations, we can use inverse operations and the properties of equality to isolate a variable in a literal equation.

Practice What do you remember?

Practice 1

What beneﬁts come from rewriting literal equations?

Which43–45 of the following are literal equations? Students: 2Pages A

3x + 6y = 18

12 = 4x − (−6 + 2x) What do youD remember?

C = 2π r

List the variables in each equation: 2

a P come = 4s from rewriting b literal V = π requations? h What beneﬁts

The formula for the of aequations? triangle is Which4 of the following arearea literal

A D 3

, where b is the base length of the triangle, and h is the height.

Find the area of the triangle with a base length of 6 cm and a height of 14 cm.

Find the height of a triangle with an area of 45 cm2 and a base length of 15 cm.

180(n − 2) = s

3x + 6y = 18

A = 1 5π

c −Find base length ofEa triangle 12 = 4x (−6 the + 2x) C = 2with π r a height of 12.4 cm and an area of 62 cm2.

x−7=y

P = 4sa

b = −2x V π r2=h4m

cc 8y = x + 2z

180(n − 2) = s

, where b is the base length of the triangle, and h is the height.

Find the area of the triangle with a base length of 6 cm and a height of 14 cm.

1.05 Literal equations Find the height of a triangle with an area of 45 cm2 and a base length of 15 cm.

Find the base length of a triangle with a height of 12.4 cm and an area of 62 cm .

mathspace.co

Solve for x in each of the equation: a

The formula for the area of a triangle is

c 5

5 variables Solve for xinineach each of the equation: List the equation:

a 4

A = 1 5π

x−7=y

−2x = 4m

8y = x + 2z

Solve for the mentioned variables in each of the equation: a

E = mc2, solve for m

A = P (1 + r)t, solve for P

P = I2R, solve for R

V = π r2h, solve for h

, solve for z

g = hx2k, solve for h

1.05 Literal equations mathspace.co

Let’s practice 7

Solve for x in each of the equation: a

3y = −8(7 + 5x)

b f

c 14y = 9xz − 5

Solve for k in

Solve for R in 16V = 20IR − 4E

Solve for the mentioned variables in each of the equation:

10y = −2k(5n + 3x)

, solve for h

D = b2 − 4ac, solve for c

, solve for c

d = uv4 − t, solve for u

12kx = 3m + 6nx

7m = 3k(4n + 8x)

Solve for y in the following equations, and justify each step using properties of equality: b

9x2 + 3ky = 12k

4x − 9 = 2y + my

Determine the property that justiﬁes why the equations 5(kx + 4) = 10x − 6 and kx + 4 = 2x − 1.2 are equivalent.

Consider the formula Nadeem rearranges the formula to isolate r as follows: 1 2

Multiplication property of equality

⬚

4 5

Subtraction property of equality ⬚

Determine the property that justiﬁes Nadeem’s third step of work.

Determine the property that justiﬁes Nadeem’s last step of work.

Harrison also rearranged the equation, but obtained the formula method may have differed.

Explain how Harrison’s solution

The formula for the force of an object is F = ma, where F is the force, m is the mass of the object, and a is the object’s acceleration. Write an equation that could be used to solve for the mass of the object.

The formula for the perimeter of a rectangle is P = 2l + 2w, where P is the perimeter, l is the length, and w is the width. Write an equation that could be used to ﬁnd the length of the rectangle.

Betsy knows the formula to convert Celsius into Fahrenheit is

She wants to convert a temperature

from Fahrenheit to Celsius. Find an equation that she could use to do this conversion. Write an equation that could be used to ﬁnd v0.

The displacement of an object is given by

Write an equation that could be used to solve for r2 for each of the following formulas:

The area of a circle is given by the formula A = π r2.

The volume of a cone is given by the formula

The area of a sector of a circle is given by the formula

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

When the heater in a house is on a setting of s, the temperature, T, of the house within the ﬁrst 30 minutes can be estimated by using the formula: Where a is the initial temperature and t is the number of minutes since turning the heater on.

Write an equation that could be used to solve for s.

Find the temperature setting required to reach a room temperature of 70 °F after 25 minutes when the initial temperature is 50 °F.

The surface area of a rectangular prism is given by formula S = 2(lw + wh + lh), where l, w and h are the dimensions of the prism. a

Write an equation that could be used to solve for l.

Find the length of a rectangular prism with a width of 4 cm, a height of 3.5 cm and a surface area of 73 cm2.

Let’s extend our thinking 21

The resistance of a parallel circuit is given by the formula a

Write an equation that could be used to solve for b.

Find the value of b, correct to two decimal places, if a = 8 and c = 23.

Solve for k in each of the equation: a

Determine the similarities and differences of solving 3x + y = z and 3x + 8 = 2. Explain your thinking.

Ruﬁno has submitted the following work to write the equation for the mass of an object, m, given the kinetic energy, KE, and velocity, v:

Determine whether Rufino is correct. Explain each correct step of his work, otherwise fix his error.

1.05 Literal equations mathspace.co

Answers

Given

1.05 Literal equations

Division property of equality

What do you remember? Subtraction property of equality

1 We can have multiple equations solving for various unknowns. Once these equations have been rewritten, we do not have to do this manipulation again.

Division property of equality

2 A, B, E

Solution

3 a P, s

b V, r, h

c F, C

4 a 42 cm2

b 6 cm

c 10 cm

Factor

b x = −2m

Division property of equality

5 a x=y+7 c 8y − 2z =x

d n, s

Given

d x = 15c − 35

Solution

6 a

12 Division property of equality or multiplication property of equality

e z = x2y2

13 a Distributive property b Division property of equality c H arrison may have used the division property of equality to divide by L at step 3, or individually divided the terms on the right side of the equation in step 5.

Let’s practice 7 a

14 15 16 17 8 18 a

10 a

b 2

c c = sn + 6

19 a

b s=8

20 a

b l = 3 cm

d Let’s extend our thinking

11 a

Given

21 a

b b = 12.27

22 a

Multiplication property of equality Division property of equality Solution

Given Subtraction property of equality Division property of equality Solution

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

23 Both equations can be rearranged for x. When 3x + 8 = 2 is rearranged for x the answer is numerical. Whereas when 3x + y = z is rearranged for x the answer has unknowns, y and z, in it. However, if you substitute y = 8 and z = 2, you get the same numerical answer as solving 3x + 8 = 2.

24 Rufino has an error in his work. The equation for KE as given Multiplying both sides by 2

Now he should have divided by v2

Answers mathspace.co

1.06 Multistep inequalities Subtopic overview Lesson narrative In this lesson, students will solve linear inequalities in one-variable. They will begin by recognizing that we solve these in the same way, by using the same properties, as equations. They will use problem-solving skills to understand inequality changes (flips) when multiplying or dividing by a negative number. Then, they will learn how to solve the inequalities algebraically and graphically. By the end of the lesson, students will make sense of the relationship between quantities in order to create inequalities for real-world and mathematical problems. They will also make a plan for solving and be able to explain if a solution is viable or nonviable.

Learning objectives Students: Page 46

Key vocabulary 

inequality



interval notation



linear inequality



set notation



solution set



viable solution

 non-viable solution

Essential understanding Inequalities have an infinite number of solutions so their solution sets are often represented on a number line.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving Teachers can incorporate this goal into their instruction by presenting students with various real-world problems that require multistep linear inequalities. Students can be guided to apply the mathematical concepts and skills they’ve learned to solve these problems. For example, teachers could present a problem scenario related to economics or science where students have to create and solve a multistep linear inequality.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

MPG2 — Mathematical Communication

MPG5 — Mathematical Representations

This goal can be incorporated by encouraging students to explain the process of solving multistep linear inequalities. This could involve verbal explanations, writing down their process, or even presenting to the class. Teachers could also have students discuss and justify their methods and solutions, fostering a classroom environment of mathematical discourse.

This goal can be incorporated by having students represent their solutions to multistep linear inequalities in different ways, such as algebraically and graphically on a number line. Teachers can also engage students in activities that require them to interpret and make connections between different representations. For example, students could be asked to solve a multistep inequality and then represent their solution graphically, discussing how the graph reflects their algebraic solution.

MPG4 — Mathematical Connections To incorporate this goal, teachers could relate the concepts and procedures used in solving multistep linear inequalities to solving multistep linear equations. This could be done by providing students with opportunities to compare and contrast the two.

A.EI.1c — Solve multistep linear inequalities in one variable algebraically and graph the solution set on a number line, including those in contextual situations, by applying the properties of real numbers and/or properties of inequality.

Prior connections 7.PFA.4 — The student will write and solve one- and two-step linear inequalities in one variable, including problems in context, that require the solution of a one- and two-step linear inequality in one variable.

8.PFA.5 — The student will write and solve multistep linear inequalities in one variable, including problems in context that require the solution of a multistep linear inequality in one variable.

A2.EI.1 — The student will represent, solve, and interpret the solution to absolute value equations and inequalities in one variable.

A2.EI.2 — The student will represent, solve, and interpret the solution to quadratic equations in one variable over the set of complex numbers and solve quadratic inequalities in one variable.

1.06 Multistep inequalities mathspace.co

Engage Activity Multistep inequalities

60 mins

Students will play a game in pairs and use multistep inequalities to determine winning numbers.

Understanding and skills

Will use Solving one- and two-step inequalities.

Will develop Solving multistep linear inequalities in one variable. Writing inequalities from a mathematical context.

Could extend Solving three-step inequalities.

Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: Paper, pencil, calculator

Support students with disabilities Support perseverance - play math games that involve winning and losing Minimize the competitiveness of the game by repositioning pairs as a team investigating different winning strategies. Ask these pairs to write a runthrough guide for students who are asked to play the game in other classes.

Support for English language learners Compare and connect Ask students to prepare a visual representation of the numbers that allow the “Adder” to win in one move and the numbers that allow the “Doubler” to win in one move. Once students have a visual representation, invite them to consider the following prompts: • Where is there overlap between these visual representations? What does that mean for the game? • Which position has more starting numbers to win in one move from? How do you know? • Based on your visual display which role would you prefer to play? Why?

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Classroom guide Hook

Which one doesn’t belong

•

5 mins

Students choose one of four inequalities. Which one doesn't belong? x + 5 > 10

3x − 6 ≤ 9

2 x + 10 > 30

40 < 2(2 x + 1) + 18 D

Slide 1 from Student Engage Activity

Implementation details

Highlight student responses about the inequality symbol used, the relative position of numbers and variables, the numbers and operations involved. Encourage students to discuss the solution to each inequality as well as the number of steps required to solve each inequality.

Launch

5 mins

Provide students time to read the instructions of the game before forming pairs. Suggested grouping: Form pairs

You and your partner are going to play a game. One of you will be the “Adder” (adds 50 to the number) The other will be the “Doubler” (doubles the number)

ADDER

+50 When I get a number I add 50 to it

DOUBLER

x2 When I get a number I double it

Slide 2 from Student Engage Activity

Explore

Think-pair-share

•

35 mins

Students will be playing a game where they take turns manipulating values. The ﬁrst person to reach or exceed 1000 wins. These are the rules: • One person starts the game by choosing a number between 0 and 999 and giving it to their partner. • The partner will perform their operation, • If you are the Adder, add 50 to it. • If you are the Doubler, double it. If the number is now 1000 or more, you win. • Otherwise, continue the game by passing it back to your partner for them to double or add.

ADDER

DOUBLER

+50

Slide 5 from Student Engage Activity 1.06 Multistep inequalities mathspace.co

Anticipated strategies Help students discuss the fairness of the game based on the starting roles. The starting player can always win if they pick the right starting number, though students who initially choose numbers at random may perceive the Doubler as being “better” than the Adder. The next element of this activity is ﬁguring out which numbers are best for a quick win for both roles. Walk around the room and encourage a sequence of events to be listed out for each role. If students are stuck, have them verbalize the game while they play an example round. Encourage students to think about the edge case examples (like 499 for the Adder) if they are confused by the question. The ﬁnal element of this activity is students describing their explanations and thinking with mathematical inequalities. Students will describe the results of different choices and how it relates to their roles. Make sure they address if they start or their partner starts. Since each student will be describing their own role, encourage discussion amongst the pairs so that they can see the similarities and differences between each role. If students are stuck, have them look to the sequence of events once again.

Testing multiple values Students play the game many times, try to test as many numbers as possible, and record who wins.

Making a table Students may start to record ranges of numbers in table form, choosing values between the ranges for the next test.

Modeling with inequalities Students generalize a gameplay strategy by building inequalities. Let the number that the starting player chooses be x. If the adder starts the game, they will pass it to the doubler, and the sequence will proceed as follows: x → 2x → 2x + 50 → 2(2x + 50) → … If the adder wants to win straight away, they want to choose a number where 2x < 1000, but 2x + 50 ≥ 1000. If the doubler starts the game, they will pass it to the adder, and the sequence will proceed as follows: x → x + 50 → 2(x + 50) → 2(x + 50) + 50 → … If the doubler wants to win straight away, they want to choose a number where x + 50 < 1000, but 2(x + 50) ≥ 1000. To describe the desirable choices for a player who is starting with inequalities, the above inequalities do just that. To describe the undesirable number choices when a player is starting would be the opposite of the above inequalities. For example, an undesirable number choice when the adder starts would be 2x ≥ 1000 or 2x + 50 < 1000.

Misconceptions Confusion about the game rules Could you explain the rules to me in your own words?

Not recognizing the need for a variable How can we represent the initial number to be choosen so that we can use inequalities to represent our choices?

Poor interpretation of overlapping inequalities Does every solution to this inequality represent a good choice?

Purposeful questions Use the following questions to check for understanding and encourage critical thinking: • Who is winning? • Do you want to swap roles? Why or why not? Explain your reasoning. • How could we represent “the number you pick to start the game” mathematically?

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Continue when Students have come up with a description of the desirable and undesirable starting values for their role. Pairs have shared their ideas with one another so that the numbers for each role, the adder and doubler, are addressed. Students have also discussed the fairness of the game by addressing the advantages and disadvantages of the roles.

Discuss

15 mins

Select a few pairs to share the strategies they used in the task. Pick pairs that have used varying strategies, and if no pairs produce an inequality, encourage the class to generalize their observations. Consider making connections from the discussion to representing real-life situations with inequalities.

Discussion guide Invite students to share their strategies. Some groups may have only written a strategy based on their observations, while others may have recorded their work in a table. If any groups were able to generalize and produce inequalities to represent their strategy, have these groups share last and ask students to make a connection between the different representations. If no groups produce an inequality, encourage students to generalize their observations. In other words, if they notice boundaries on the desirable and undesirable starting values for a particular player in the game, ask if they think this can be represented with an equation or an inequality. If they know it can be represented with an inequality, allow students several minutes to work on writing one that represents the same information they came up with during the task. Have students test their inequality on several starting values to verify that it works as they intend. As an extension you may wish to give students the following prompt: Ask students to think about the good and bad numbers after a certain amount of rounds (for example, 3 times of passing a number back and forth).

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 8 — 2.06 Solve multistep inequalities Algebra 1 — 1.01 Algebraic expressions Algebra 1 — 1.02 Properties of real numbers

Tools You may find these tools helpful: • Scientific calculator • Number line

1.06 Multistep inequalities mathspace.co

Student lesson & teacher guide Multistep inequalities Students are reminded of the properties that can be applied when solving inequalities as well as what the solution set of an inequality looks like algebraically and graphically. This includes a connection to set builder notation and interval notation. Examples are shown with the different notations and solutions on a number line. Students then engage in an exploration to discover how the properties of equality can be extended to inequalities.

Students: Pages 46–47

100

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Writing solutions of inequalities in different forms Targeted instructional strategies Inequalities can be solved and represented in many ways, and a useful exercise is to have students finding the solution of a given inequality, then connect different representations to this solution. Work through the following problem with students or have students solve independently: Distribute

Combine like terms

Move variables and constants

Divide by −6

Reverse the inequality sign

Discuss with students whether 0 is included as a possible solution. Graph the solution x < 0 on a number line. −5

−4

−3

−2

−1

Highlight that since this arrow continues infinitely in the negative direction, we can use −∞ to represent the values past the shown number line. In addition, we will need a way to show that while the solutions do not extend past 0, it is not included as a solution. Interval notation uses brackets and parentheses to show whether values are included or not included. Our solution x < 0 would be written as (−∞, 0). A useful graphic to connect representations of solutions of inequalities on a number line to interval notation is shown. Not included

Included

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Compare and connect English language learner support Students will focus on comparing and connecting different methods of solution and representations of multistep inequalities. Break students up into small groups, and give students different multistep inequalities and representations that all have the same solution, such as inequalities that solve to x ≤ 2. This could look like: • Solve 7x − 2 ≤ 4(x + 1) with steps shown, and write your solution as an inequality. • Solve 4x + 4 ≥ 3x + 2(2x − 1) and graph your solution on a number line. • Solve 5x + 2(x + 3) ≥ 10x and write your solution in interval notation. Divide students into groups, assigning each group a different method to solve the inequality. Once the groups have completed their solutions, facilitate group presentations where students explain their methods and representations, highlighting similarities and differences in approaches. Next, guide the students to notice that the solutions in all inequalities were equivalent. Encourage a discussion on how these notations represent the same solution differently. To wrap up, conduct a class reflection on the advantages and challenges of each method and representation. Encourage students to articulate why one might choose a particular method over another in different contexts.

Representing inequalities with notation and number lines handout Student with disabilities support A handout with visual examples of the different representations of the solutions of inequalities including number lines, inequalities, and interval notation can effectively support students’ visual-spatial processing by illustrating how inequalities are represented visually. The handout should display different representations of solutions to inequalities, including: • Inequality • Set builder notation • Interval notation • Number line An example is shown:

102

Inequality

Set builder notation

Interval notation

Number line

x>3

{x|x > 3}

(3, ∞)

−5 −4 −3 −2 −1 0 1 2 3 4 5

x>1

{x|x < 1}

(-∞, 1)

−5 −4 −3 −2 −1 0 1 2 3 4 5

x ≥ -1

{x|x ≥ -1}

[-1, ∞)

−5 −4 −3 −2 −1 0 1 2 3 4 5

x ≤ -2

{x|x ≤ -2}

(-∞, -2)

−5 −4 −3 −2 −1 0 1 2 3 4 5

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Don’t forget to reverse signs and fill in points Address student misconceptions Two common errors that students make are: • Forgetting to reverse the inequality sign when multiplying or dividing both sides of the inequality by a negative value, and • Forgetting to ﬁll or leave unﬁlled the point on the number line to match the inequality symbol. Both can be addressed by reminding the student of these steps as they work through questions.

Exploration Students: Page 47

Suggested student grouping: In pairs Students will complete the table by applying operations to both sides of a given inequality and assessing whether or not the inequality remained true. Through this, students should aim to connect the properties of equality to the properties of inequality, and notice any special cases where the properties of inequality differ. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. The student is asked to complete the following ﬁgures: The table of operations in inequalities. Consider the inequality 1<4 6 > −2 3 < 10 1 > −7 4>2 −8 < 12

Perform the operation on the inequality Add 2 to both sides Subtract 2 from both sides Multiply by 2 on both sides Multiply by − 2 on both sides Divide by 2 on both sides Divide by − 2 on both sides

Write the new inequality True or false? 3<6 True 4 > −4 True 6 < 20 True −2 > 14 False 2>1 True 4 < −6 False

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2. Did any operations cause the given inequality to become a false inequality? When either multiplying or dividing by −2, the new inequality was false. 3. Can you think of something to change about a false inequality without changing the operation performed? When multiplying or dividing both sides of an inequality by a negative value, we need to also change the direction of the inequality symbol so that the inequality remains true. Purposeful questions • What is common between the operations that keep the inequality true? • What is common between the operations that make the inequality false? • Would these rules be the same or different if the inequality symbols were inclusive (≤ and ≥)? Possible misunderstandings • Students may already be aware that they are supposed to change the direction of the inequality symbol when multiplying or dividing by a negative value, and thus make all the new inequalities true (bypassing the purpose of the exploration). These students will already have the desired understanding, so there is nothing to be corrected, but these students may need to be informed that they are ahead and will need to wait for the class to catch up. After exploring the various properties of inequality, students are presented with their precise deﬁnitions.

Students: Page 47

Examples Students: Page 48

Example 1 Consider the inequality

a Solve the inequality.

Create a strategy We want to isolate x on one side of the inequality and a number on the other.

Apply the idea Original inequality Multiplication property of inequality Addition property of inequality Division property of inequality The solution can be written as x ≥ −6, {x | x ≥ −6}, or [−6, ∞).

Reﬂect and check

104

Mathspace Virginia SOL Algebra 1 Teacher Edition Solving an inequality is similar to solving an equation. However, we need to reverse the direction of the inequality mathspace.co when multiplying or dividing by a negative number. b Plot the inequality on a number line.

Consider the inequality

a Solve the inequality.

Create a strategy We want to isolate x on one side of the inequality and a number on the other.

Reﬂect and check Solving an inequality is similar to solving an equation. However, we need to reverse the direction of the inequality when multiplying or dividing by a negative number.

Example 1 b Plot the inequality on a number line. Consider the inequality .

the idea PurposeApply a Solve the inequality. Plot the solution of thean inequality x ≥ −6. Note the that since we include −6 the point should be ﬁlled. Show students how tosetsolve inequality using properties of inequality. Create a strategy

Expected Wemistakes want to isolate −10 x on−9 one−8side and−1 a number −7 of−6the−5inequality −4 −3 −2 0 1 on 2 the 3 other. 4 5 6 7 8 9 10 Students may forget to change the direction of the inequality symbol when applying the division property with Apply the idea a negative value. will result in the incorrect solution of x ≤ −6. Ask students to choose a value that satisfies Reﬂect andThis check x ≤ −6 and it into original inequality to see if it is a solution. If it does not satisfy Original inequality Whatsubstitute if the solution was the x > −6? they need to go back and check their Endpoints included in the solution aresteps. ﬁlled points.

, then

Multiplication property of inequality Endpoints not included in the solution are unﬁlled points. Addition property of inequality

Reflecting with students Division property inequality Ask advanced learners to justify each step of their of equation-solving process by explicitly stating the properties cTheIssolution x = 3 a viable or nonviable solution to the inequality? can be written as x ≥ −6, {x | x ≥ −6}, or [−6, ∞). of inequality. Although it is not required by the standards, it can help all students deepen their understanding of why each step is valid. Create a strategy Reﬂect and check

We can determine if x = 3 is viable or non-viable by using the number line or algebraically substituting the solution

inequality is similar to solving an equation. However, we need to reverse the direction of the inequality Students:Solving Pagean48 into the inequality. when multiplying or dividing by a negative number. b Plot the inequality on a number line.

Apply the idea Plot the solution set of the inequality x ≥ −6. Note that since we include −6 the point should be ﬁlled. −10 −9 −8 −7 −6 −5 −4 −3 −2 −1

Reﬂect and check Virginia SOL Algebra 1 What ifMathspace the solution was x > −6? 48

mathspace.co

Endpoints included in the solution are ﬁlled points. Endpoints not included in the solution are unﬁlled points. c Is x = 3 a viable or nonviable solution to the inequality?

PurposeCreate a strategy We can determine if x = 3 is viable or non-viableon byausing the number Show students how to represent an inequality number line. line or algebraically substituting the solution into the inequality.

Expected mistakes Students may forget to ﬁll in the point at x = −6 to match the inequality symbol, which is inclusive.

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105

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1

Reﬂect and check What if the solution was x > −6?

Students:Endpoints Pagesincluded 48–49in the solution are ﬁlled points.

Endpoints not included in the solution are unﬁlled points. c Is x = 3 a viable or nonviable solution to the inequality?

Create a strategy We can determine if x = 3 is viable or non-viable by using the number line or algebraically substituting the solution into the inequality.

Apply the idea Original inequality Substitute x = 3 Evaluate the multiplication Evaluate the subtraction Evaluate the division

Mathspace

Virginia SOL Algebra 1

Since mathspace.co x = 3 leads to a true statement, we can conﬁrm that x = 3 is a viable solution to the inequality.

Reﬂect and check By using the number line, we can see that the point x = 3 is in the solution set of the inequality, meaning it is a viable solution to the inequality. Any points that are not in the solution set are considered non-viable and will lead to a false statement when substituted into the inequality algebraically.

Example 2 Purpose Calandra how charges to style hair, asawell as an additional foil. Paulineby would like the total costthe for her Show students to $37.72 evaluate whether solution is viable$6orper nonviable subtituting it into original styling to be no more than $95.86. inequality. a Write an inequality that represents the number of foils Pauline could get.

Reﬂecting with students Createwhether a strategy Ask students or not x = 3 is within the solution they plotted on the number line in part (b). Point out to has no moreinthan to spend. “No more than” equal to.” constraints to consider. studentsPauline that any value this$95.86 solution will be viable, sincemeans there“less arethan no or contextual Apply the idea Simplify the number line

use with Example 1

We can write an inequality in words that represents the cost to style Pauline’s hair:

Student with disabilities support

cost of styling + cost per foil ⋅ number of foils ≤ total Pauline can spend

For students who have diﬃculty drawing, allow them to simplify their number lines to a single point on the line Translating that into an algebraic expression we get: 37.72 + 6N ≤ 95.86 where N represents the number of foils. with an indicated direction: b How many foils could Pauline get and still afford the styling?

−5

Createwho a strategy For students have diﬃculty differentiating between the number line and the line indicating the solution set, Solve the inequality then write solution set.the number line (and adapt questions to have a similar format): allow them to draw their and solution set the lines above Apply the idea 37.72 + 6N ≤ 95.86 6N ≤ 58.14 N ≤ 9.69

Original inequality −5 Subtraction property of inequality Division property of inequality

According to the solution, Pauline could get 9.69 foils or fewer. However, since she can’t get a partial foil, a more realistic solution is that she can get 9 foils or fewer.

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Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Reﬂect and check By using theidea number line, we can see that the point x = 3 is in the solution set of the inequality, meaning it is a viable Apply the solution to the inequality.

Original inequality Any points that are not in the solution set are considered non-viable and will lead to a false statement when into the inequality algebraically. Students:substituted Page 49 Substitute x = 3 Evaluate the multiplication

Example 2

Evaluate the subtraction

Calandra charges $37.72 to style hair, as wellthe asdivision an additional $6 per foil. Pauline would like the total cost for her Evaluate styling to be no more than $95.86. Since x = 3 leads to a true statement, we can conﬁrm that x = 3 is a viable solution to the inequality. a Write an inequality that represents the number of foils Pauline could get.

Reﬂect and check Create strategy By usingathe number line, we can see that the point x = 3 is in the solution set of the inequality, meaning it is a viable Pauline has no inequality. more than $95.86 to spend. “No more than” means “less than or equal to.” solution to the Any points that are not in the solution set are considered non-viable and will lead to a false statement when Apply the idea substituted into the inequality algebraically. We can write an inequality in words that represents the cost to style Pauline’s hair: cost of styling + cost per foil ⋅ number of foils ≤ total Pauline can spend

Example 2

Translating that into an algebraic expression we get: 37.72 + 6N ≤ 95.86 where N represents the number of foils. Calandra charges $37.72 to style hair, as well as an additional $6 per foil. Pauline would like the total cost for her styling to be no more than $95.86. b How many foils could Pauline get and still afford the styling? a Write an inequality that represents the number of foils Pauline could get.

Purpose Create a strategy Show students to write an inequality that represents a real-world problem. Create ahow strategy

Solve the inequality and then write the solution set. Pauline has no more than $95.86 to spend. “No more than” means “less than or equal to.”

Expected mistakes Apply the idea StudentsApply might forget the idea to include the initial cost of hairstyling in their inequality, focusing only on the cost per foil. 37.72 + 6N ≤ 95.86 Original inequality StudentsWecan or labelintheir to help all needed canhighlight write an inequality wordsnumbers that represents the them cost toinclude style Pauline’s hair: values as they work through the 6N ≤ 58.14 Subtraction property of inequality problem. cost of styling + cost per foil ⋅ number of foils ≤ total Pauline can spend N ≤ 9.69

Division property of inequality

that an algebraic we get: + 6NHowever, ≤ 95.86 where N represents number foils. According to theinto solution, Paulineexpression could get 9.69 foils37.72 or fewer. since she can’t get athe partial foil, aofmore Students:Translating Page 49 realistic solution is that she can get 9 foils or fewer. b How many foils could Pauline get and still afford the styling?

Create a strategy Solve the inequality and then write the solution set.

1.06 Multistep inequalities mathspace.co

Apply the idea 37.72 + 6N ≤ 95.86

Original inequality

6N ≤ 58.14

Subtraction property of inequality

N ≤ 9.69

Division property of inequality

According to the solution, Pauline could get 9.69 foils or fewer. However, since she can’t get a partial foil, a more realistic solution is that she can get 9 foils or fewer.

1.06 Multistep inequalities 49 Purpose mathspace.co Students demonstrate that they can solve an inequality and interpret the solution to solve the real-world problem.

Expected mistakes Students may round up, or not round at all, which is incorrect if we consider the context of the problem. Remind students that they should always interpret the number in the context of the problem to check whether or not they need to round their answer.

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107

Students: Page 50 c Determine whether N = −2 is a viable solution to the inequality in the context of the question.

Create a strategy Keep in mind it is not realistic to get part of a foil or a negative number of foils.

Apply the idea Pauline can get a maximum of 9 foils and a minimum of 0 foils, so while −2 is mathematically part of the solution set for the inequality N ≤ 9.69 it is not a viable solution in this context.

Reﬂect and check Unlike a value that is not in a solution set of an inequality, this is an example of a solution that was mathematically valid and part of the original solution set but when considering the context we have found that it is non-viable. c Determine whether N = −2 is a viable solution to the inequality in the context of the question.

Create a strategy in mind Example 3it is not realistic to get part of a foil or a negative number of foils. PurposeKeep Check that students can identify the contextual constraints of a real-world problem and apply them to determine Apply theinequality idea 4(x + 5) < 3(2 − x). Solve the whetherPauline a solution is viable or nonviable. can get a maximum of 9 foils and a minimum of 0 foils, so while −2 is mathematically part of the solution set for the inequality N ≤ 9.69 it is not a viable solution in this context. Create a strategy

Expected mistakes To solve this inequality, we need to simplify each side, isolate x on one side, and then use the inequality to find the StudentsReﬂect may assume that since N = −2 makes the inequality true, it will be a viable solution. Ask students if check solution. and We can start by distributing the 4 and 3 on each side of the inequality. Pauline Unlike can have −2 foils. Is this a viable in the this context of the problem? a value that is not in a solution setsolution of an inequality, is an example of a solution that was mathematically valid and part of the original solution set but when considering the context we have found that it is non-viable. Apply the idea

Students: Page 50 4(x + 5) < 3(2 − x) 4x + 20 < 6 − 3x

Original inequality Distributive property

Example4x3+ 3x < 6 − 20

Add 3x to both sides and subtract 20 from both sides

Solve the inequality 4(x + 5) < 3(2 − x). x < −2

Divide both sides by 7

7x < −14

Simplify both sides

The solution to the inequality can be written as x < −2, {x | x < −2}, or (−∞, −2).

Create a strategy To solve and this inequality, Reflect check we need to simplify each side, isolate x on one side, and then use the inequality to find the solution. start by the a 4 value and 3 less on each of the To check We ourcan solution, wedistributing can substitute thanside −2 into theinequality. original inequality and see if it holds true. Let’s take x = −3 as an example.

Apply the4idea (− 3 + 5) < 3(2 − (−3)) 4(x + 5) 4 ⋅<23(2 < 3−⋅ x) 5

4x + 20 <86<−153x

Substitute x = −3 into the original inequality Original inequality Simplify the expressions

Distributive property the expressions Continue simplifying 4x + 3x < 6 − 20 Add 3x to both sides and subtract 20 from both sides The inequality 8 < 15 is true, so our solution x < −2 is correct. 7x < −14 Simplify both sides x < −2

Divide both sides by 7

The solution to the inequality can be written as x < −2, {x | x < −2}, or (−∞, −2).

Reflect and check To check our solution, we can substitute a value less than −2 into the original inequality and see if it holds true. Let’s take x = −3 as an example. 4 (− 3 + 5) < 3(2 − (−3)) 4⋅2<3⋅5 8 < 15

Substitute x = −3 into the original inequality Simplify the expressions Continue simplifying the expressions

The inequality 8 < 15 is true, so our solution x < −2 is correct. 50

Mathspace Virginia SOL Algebra 1 mathspace.co

Purpose Show students how to solve linear inequalities that involve the distributive property and variables on both sides of the inequality. 108

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

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Virginia SOL Algebra 1

Reﬂecting with students Encourage students to always check their solution by substituting a value from the solution set back into the original inequality. This will verify whether the solution is correct or not.

Students: Page 51

Idea summary Just like the properties of equality, the properties of inequality can justify how we solve inequalities. The multiplication and division properties of inequality change the meaning of an inequality when multiplying or dividing by a negative number, meaning we have to reverse the inequality symbol when applying the property: • If a > b and c < 0, then a ⋅ c < b ⋅ c • If a > b and c < 0, then The solution set of an inequality is the set of values that makes the inequality true. Because inequalities can have infinitely many solutions, inequalities used to represent real-world situations often include solutions that are unreasonable in context and therefore non-viable.

Practice What do you remember?

Practice 1

Plot these inequalities on a number line: a

x > −9

Students: Pages 51–53 2

x ≤ 15

x < −29

Describe the range of values that satisfy each inequality. a

x ≥ 29

What do you remember? 3 Which of these inequalities represents the solution for x in 10 ≤ 6 − 4x? A

a 2

x ≤ −4

x≥4

C x ≥ 29 1

x ≤ 15 −5 −4 −3 −2 −1 0

2 3 4 5

x < −29 −5 −4 −3 −2 −1 0

Which of these inequalities represents the solution for x in 10 ≤ 6 − 4x? A

Write each of the relation as an inequality using mathematical symbols:

x ≤ −4

x ≤ −1

The sum of 3 groups of p, and 9, is less than 24.

x≤1

x≥4

b Thenumber sum of 5 lines times x, and 3 is at the leastsolution 23. Which of these represent for 4x − 7 < 5?

Six more than the value of x is at least seven.

Half of x is no more than ﬁve.

−5 −4 −1 0 of 1negative 2 3 four 4 and 5 x is at most three. e −3 The−2 product

6 For each of the inequality: i

Solve for x.

3x − 7 < 8

−5 −4 −3 −2 −1 0

x≤1

Which of these number lines represent the solution for 4x − 7 < 5?

x > −9 A

−5 −4 −3 −2 −1 0

−5 −4 −3 −2 −1 0 1 2 3 4 5 Describe the range of values that satisfy each inequality.

a 3

x ≤ −1

Plot these inequalities on a number line:

−5 −4 −3 −2 −1 0

2 3 4 5 b

4 < 6x − 2

Plot the solutions on a number line.

−6x − 7 ≤ 5

−5 −4 −3 −2 −1 0

Write each of the relation as an inequality using mathematical symbols: e 1.5x + 8 > 12.5 f g 2 − 3.6x < 20.9

The sum of 3 groups of p, and 9, is less than 24.

7 sum Which the23. solution for r in “5 more than 2r is less than 39”? The ofof5these timesinequality x, and 3represents is at least

Six more than the value of x is at least seven.

Half of x is no more than ﬁve.

The product of negative four and x is at most three.

r > 17

r < 17

r > 22

2 3 4 5

2(x − 3) < −16

2 3 4 5

r < 22

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109

For each of the inequality: i

Solve for x.

Plot the solutions on a number line.

3x − 7 < 8

−6x − 7 ≤ 5

1.5x + 8 > 12.5

2 − 3.6x < 20.9

4 < 6x − 2

2(x − 3) < −16

Which of these inequality represents the solution for r in “5 more than 2r is less than 39”? A

r > 17

r < 17

r > 22

r < 22

x=4

x=0

x=2

Let’s practice 8

Consider the inequality: 5(x + 3) ≤ 35. a

Solve for x.

State whether these solutions are viable or nonviable: i

x = −4

iii

Consider the inequality: 4 − 2x < 3x − 2. a

Solve for x.

State whether these solutions are viable or nonviable: ii

i 10

x=8

x=1

iii

Which of the following are values in the solution set of the inequality shown? Select all that apply. −3.5(2 − 6x) < −27 + x

−1

5x − 40 ≥ 50

SOL

−8 − m > 3

−8(x + 8) ≥ −40

d h

Find the solution to each inequality, writing each answer in set notation. a

Solve each of the inequality and justify each step using properties of inequality: a

−m + 7(−3m + 4) ≥ 4 + 2m

−1 + 4b ≤ 2b − 18 + b

Solve each inequality, writing each answer in interval notation. a

42 − 3y ≥ 74 − y

9x − (12 − x) ≥ − 6x

27 + 3a < −5 + 2(7a − 6)

Consider the situation: “3 less than 3 groups of p is no more than 24”. a

Write the relation as an inequality.

Solve the inequality.

Find the largest value p can take.

Graph the solution to the inequality on the number line provided. 3 − (x + 11) ≥ −5x −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

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2 3 4 5 6 7 8 9 10

−2

Skye has a budget for school stationary of $43, but has already spent $21.14 on books and folders. Let p represent the amount that Skye can spend on other stationery. Write the budget constraint as an inequality and solve for p.

James is saving up to buy a laptop that is selling for $550. He has $410 in his bank account and expects a nice sum of money for his birthday next month.

Write the inequality that models the situation in which James can afford the laptop. Let x represents the amount he is to receive for his birthday.

Plot the solution to the inequality on a number line.

To get a grade of C, Uther must obtain an average score of at least 75 over his four exams. So far, he has taken the ﬁrst three exams and achieved scores of 68, 60, and 86. a

Write the inequality that models the situation for the score, x, Uther must obtain on his last exam to get a C or better.

Solve for x.

Describe the solution regarding Uther’s score.

Let’s extend our thinking 19

Ryan wants to save up enough money so that he can buy a new sports equipment set, which costs $40.00. Ryan has $22.10 that he saved from his birthday. In order to make more money, he plans to wash neighbors’ windows for $2 per window. a

Let w be the number of windows that Ryan washes. Write an inequality to represent the situation.

Solve for w, correct to two decimal places.

State whether each statement is correct. Explain your thinking. i

Ryan must wash more than 9 windows to be able to afford the equipment.

Ryan must wash at least 8 windows to be able to afford the equipment.

iii If Ryan washed 8 windows, and 95% of another window, he could afford the equipment. iv The number of windows Ryan must wash to be able to afford the equipment must be greater than or equal to 9. 20

Rochelle tried to solve the following inequality but made a mistake in her work: Step 0: −4 − 2x > 10 Step 1:

−2x > 14

Step 2:

x > −7

Determine which step is incorrect and explain the error. 21

Percy tried to plot the solution to the inequality 4x + 28 ≥ −8 on a number line, however, his answer is incorrect.

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1

Identify the errors and explain how to rectify them.

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Answers

8 a x≤4 b i Viable

1.06 Multistep inequalities

iii Viable

What do you remember?

9 a

−20 −18 −16 −14 −12 −10 −8 −6 −4 −2 0 2 4 6 8 10 12 14 16 18 20

1 a

−20 −18 −16 −14 −12 −10 −8 −6 −4 −2 0 2 4 6 8 10 12 14 16 18 20

b 2 a x can be equal to 29 or any number greater than 29.

ii Nonviable iv Viable

b i Viable

ii Nonviable

iii Nonviable

iv Viable

10 D, F 11 a

Given Addition property of inequality

b x can be equal to any number less than -29, not including 29.

Division property of inequality

3 B

Solution

4 D

5 a 3p + 9 < 24 c x+6≥7

b 5x + 3 ≥ 23

Addition property of inequality

Division property of inequality

e −4x ≤ 3

Solution

Let’s practice

Given Distributive property

6 a i x<5

Addition property of inequality

ii −1

Division property of inequality Solution

b i x>1 ii

d −4 −3 −2

−1

Given

Multiplication property of inequality

c i x ≥ -2

Subtraction property of inequality

ii −5 −4 −3 −2 −1

Solution

d i x < -5 ii

Given

e i x>3

Division property of inequality

ii −1

Addition property of inequality

Multiplication property of inequality

ii −8 −7 −6 −5 −4 −3 −2 −1

g i x > −5.25 ii −7 −6 −5 −4 −3 −2 −1

Addition property of inequality 1

Division property of inequality Solution

7 B

112

Given Multiplication property of inequality

h i x ≥ 7.5 ii

Solution

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Given

16 a p + 21.14 ≤ 43, p ≤ $21.86 17 a x + 410 ≥ 550

Addition property of inequality

b 0

100

200

300

400

500

Distributive property

18 a Subtraction property of inequality

Division property of inequality

b x ≥ 86 c T o get an overall grade of C Uther must score at least 86 on the last exam. Let’s extend our thinking

Solution

Given Subtraction property of inequality Multiplication property of inequality Subtraction property of inequality Division property of inequality Solution

12 a {x∣x > 2}

b {b∣b ≤ −17}

c {m∣m < 1}

d {n∣n > −8}

13 a (−∞, −16]

b (4, ∞)

c [ ,∞)

d (−∞, 11)

14 a 3p − 3 ≤ 24

b p≤9

c p=9 15

19 a 2w + 22.10 ≥ 40

b w ≥ 8.95

c i N o, washing exactly 9 windows would generate $18 which, with the $22.10 he already has, is enough money to afford the equipment. ii No, washing 8 windows would only generate $16 which would give him a total of $38.10, which is not enough money to afford the equipment. iii No, he gets paid for washing the whole window, not parts. iv Yes, if Ryan washed 9 windows, he would earn enough to buy the equipment. If he washed more than 9 windows, he would have more than enough money. 20 Step 2 is incorrect. At this step she divided both sides by a negative number, and so she should have reversed the inequality symbol. 21 The minimum should be at -9 instead of -8 and the dot at that point should be filled as -9 is included in the solution.

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

Answers mathspace.co

113

Topic 1 Assessment: Equations & Inequalities 1

Maria works 6 hours per week at a part-time job, 8 hours per week dog walking, and she gets a weekly allowance. Her weekly earnings can be represented by the expression 6a + 8b + 15. a

What does each variable represent?

What does the constant term represent?

What does the term 6a represent?

What does 6a + 8b represent?

The Hawks scored 30 points less than the Dolphins in their most recent game. Let d be the number of points scored by the Dolphins. Write an algebraic expression to represent how many points the Hawks scored.

SOL

, y = −7, and z = 64? Show your work.

What is the value of the expression

Evaluate the expression

Determine the property that justiﬁes why the equations 9x = −72 and

A rectangular pool is enclosed by a fence that uses 500 feet of fencing. The length of the enclosed region is 50 feet longer than the width. Write an equation that can be used to determine the length of the enclosed region.

Stickers cost $1.25 each. You buy 3 more stickers than your friend buys. You and your friend spend a total of $11.25 on stickers.

when

if a = 13 and b = −5. Show your work. are equivalent.

Write an equation that can be used to determine the number of stickers your friend buys.

Find the number of stickers you and your friend buy.

What value of y makes this equation true? 3y − 16 = −5y a

y=2

Solve the equation

Solve the equations:

11 − 4x = 39

y = −2

c 11

y = −8

= 6 and justify each step.

SOL

2b + 7 (b − 4) = 3b − 10

Solve for x: 7x − 7 − 9x > 3x − 17

SOL

The formula for the volume, V, of a cone is shown:

where r is the radius and h is the height. Which equation can be used to find h? A SOL

3π r2 = h

Solve for m.

114

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

3V − π r2 = h

Zaya is buying a computer for school. She must pay $100 up front and can pay the rest off in 12 equal payments. If the total cost of the computer is $1600, ﬁnd the amount of a single monthly payment.

Liam correctly solved an equation using the steps shown: 5(y + 2) = 5y + 7 5y + 10 = 5y + 7 5y + 10 − 5y = 5y + 7 − 5y 10 = 7 10 − 7 = 7 − 7 3 = 0 Which of these statements is true about the solution of the equation Liam solved? A

The solutions are 3 and 0

The solution is only 3

The solution is only 0

The equation has infinitely many solutions

The equation has no solution

Determine the number of solutions to the equation: 2 (m − 2) + 2m = 4 (m − 1)

The velocity, v, that an object r units distant from the center of the Earth must have in order to escape the Earth’s gravity is given by

where G is gravitational constant and M is the mass of an object. Solve for M. 18

Solve the inequalities and justify each step: a

9x − 27 ≤ 36

Solve for x. Write your answer in both set notation and interval notation.

Plot the solutions on a number line. b

Solve for x.

Determine whether the solutions are viable or nonviable: x=1

x=2

iii

Which expression represents three more than twice a number, x? A

SOL

4 (4x − 5) ≥ 28 + 4x

Consider the inequality: 5 − 3x < 2x − 4.

i SOL

For the inequalities:

a 20

3 + 2x

2x + 3

2(3 + x)

2(x + 3)

Blaise plans to attend the spring carnival at his school. • The cost per game is $0.50. • The cost of entry to the carnival is $5.50. • He can spend no more than $13 to stay within budget. What is the maximum number of games that Blaise can play? A

Topic 1 Assessment: Equations & Inequalities mathspace.co

115

SOL

Based on the transitive property, complete this statement.

If 8 − 5x < −x + 6, and −x + 6 < 2 + 7x, then 8 − 5x < ⬚

Alex tried to solve the following inequality but made a mistake in his work: Step 0: Step 1: Step 2: Step 3: Step 4:

State between which two steps did Alex made a mistake and describe the error.

Performance task 25

Ernestina is participating in a fun run to raise money that will support the fostering and adoption of pets at the local animal shelter. She is asking her family and friends to sponsor her by pledging to donate a certain amount of money for every mile she runs. Some people are donating this way and others have decided to instead just give a one time donation no matter how many miles she runs. After the fun run, Ernestina reports back to her donors that she ran 8 miles. The shelter is offering small tokens of thanks based on how much money each person raises. Amount raised $25 − $49 $50 − $99 $100 − $199 $200 − $299 $300+

116

Token of thanks I love animals magnet Picture of a fostered pet Puppy or kitten stuffed animal Access to live video stream of fostered puppies Meet and greet with pets at the shelter

Ernestina’s friend Noah donated a ﬁxed amount of $20 and her cousin Urbana donated a certain amount per mile. If together the two donated $38, how much did Urbana pledge to donate for each mile run? Explain.

Some of the teachers at Ernestina’s school have come together to support her fundraising efforts. They decide that they will each donate $15 and a group of students from their classes offers to donate $0.10 for each mile Ernestina runs. The number of teachers donating is two less than half the number of students. Write and solve an inequality to ﬁnd the possible number of students who could donate in order for Ernestina to get the picture of the fostered pet. Explain.

Ernestina wishes she would have run a little farther so that she could have gotten access to the live video stream of the puppies. Write and solve a compound inequality to ﬁnd how far she would have needed to run for Noah and Urbana’s donations to get her access to the video feed, but not the meet and greet. Do you think this would be a realistic goal? Explain.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Answers

Given equation Multiplication property of equality

Topic 1 Assessment: Equations & Inequalities

Subtraction property of equality

1 a The variable a represents the hourly rate for the part-time job and b represents the hourly rate for the dog walking job. b T he constant term 15 represents Maria’s weekly allowance. c T he term 6a represents the money Maria earns from her part-time job. d T he expression 6a + 8b represents the total money Maria earns from both her part-time and dog walking jobs.

Division property of equality

A.EI.1b, A.EI.1f 10 a x = −7

b y = 13

c a = −4

d b=3

A.EI.1b 11 x < 2 A.EI.1c 12 C A.EI.1d

A.EO.1a

13 m = −6

2 d − 30

A.EO.1b

A.EO.1a

14 $125

3 Substitute

A.EI.1a, A.EI.1b

, y = −7, z = 64

15 E Evaluate grouping symbols

A.EI.1e

Evaluate the exponents

16 Inﬁnitely many solutions Evaluate the cube root Evaluate the multiplication Evaluate the subtraction

A.EI.1e 17 A.EI.1d

A.EO.1b Given

18 a Substitute a = 13 and b = −5

Addition property of inequality

Evaluate the exponents

Division property of inequality

Evaluate the subtraction

Solution

Take the square root

A.EO.1b

Given

5 Division property of equality

Multiplication property of inequality

A.EI.1b 6 Let the width of the rectangular region be w feet. Then the length of the rectangular region is w + 50w + 50 feet. The perimeter of a rectangle is given by 2 ⋅ (length + width). Therefore, the equation that represents this situation is 2 ⋅ (w + w + 50) = 500. Simplifying this equation gives 4w + 100 = 500.

Subtraction property of inequality Solution Given

Subtraction property of inequality Multiplication property of inequality

A.EI.1a 7 a Let x be the number of stickers your friend buys. Then, the equation is 1.25x + 1.25(x + 3) = 11.25.

Subtraction property of inequality Division property of inequality

b You buy 6 stickers and your friend buys 3 stickers.

Solution

A.EI.1a, A.EI.1b A.EI.1c, A.EI.1f 8 A A.EO.1b

Topic 1 Assessment: Equations & Inequalities mathspace.co

117

19 a i {x∣x > 2}, (2, ∞)

Performance task

25 a Let x represent the amount of money Urbana pledged to donate for each mile run. The expression 20 + 8x represents the $20 donated by Noah combined with the amount Urbana donated for the 8 miles Ernestina ran. This amount needs to equal the 38 they donated in total giving the equation 20 + 8x = 38. Solving this equation gives x = $2.25 per mile.

−2 −1 0 1 2 3 4 5 6 7 8

b i {x∣x ≥ 4}, [4, ∞) ii −2 −1 0 1 2 3 4 5 6 7 8

c i {x∣x ≤ −6}, (−∞, −6] ii

b If x represents the number of students who donate, the −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

20 a b i Nonviable

ii Viable

iii Nonviable

iv Viable

A.EI,1c, A.EI.1f 21 A, B A.EO.1a 22 B A.EI.1a, A.EI.1c, A.EI.1f 23 2 + 7x A.EI.1c 24 Steps 0 and 1. Alex did not reverse the inequality symbol when multiplying by a negative number. A.EI.1c, A.EI.1f

118

number of teachers is

The amount of money donated by teachers is

A.EI.1c

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

and the amount donated by students for the

8 miles is 0.10 (8) x. Since Ernestina needs to raise at least $50 to get the picture the inequality is 15

+ 0.10 (8) x ≥ 50. Solving gives x ≥ 9.64.

But since 9.64 is not a valid number of students, we must round up to the nearest whole student. So at least 10 students must donate for Ernestina to get the picture. c T he expression 20 + 2.25m represents the amount of money Urbana and Noah would donate if Ernestina ran m miles. Since the video feed requires donations totaling between $200 and $299 the compound inequality is 200 ≤ 20 + 2.25m ≤ 299. Solving gives 80 ≤ m ≤ 124. This is not a realistic goal because that would require Ernestina to run at least 80 miles which is approximately 3 marathons. A.EI.1a, A.EI.1b, A.EI.1c, MP1, MP2, MP3, MP4

2 Functions & Relations Big ideas • There are many ways to represent a function (equation, table, graph, written description, etc.). The way a function is represented can affect what conclusions can be made. • Functions provide a representation for how related quantites vary. This makes functions a good way to represent many real world situations.

Chapter outline 2.01 2.02 2.03 2.04

Functions and relations (A.F.2) Domain and range (A.F.1, A.F.2) Evaluating functions (A.F.1, A.F.2) Characteristics of functions (A.F.1, A.F.2) Topic 2 Assessment

124 147 164 180 198

Functions help predict real-life events like tracking a car’s rental cost!

2. Functions & Relations Topic overview Foundational knowledge Evaluating standards proficiency The skills book contains questions matched to individual standards. It can be used to measure proficiency for each. Students should be proficient in these standards. 8.PFA.2 — The student will determine whether a given relation is a function and determine the domain and range of a function.

Big ideas and essential understanding There are many ways to represent a function (equation, table, graph, written description, etc.). The way a function is represented can affect what conclusions can be made. 2.01 — A relation is a function when each input only has one output. This can be determined algebraically or graphically.

Functions provide a representation for how related quantites vary. This makes functions a good way to represent many real world situations. 2.02 — The domain and range of a function can provide insight into the context it models but a real-world context can also be a limiting factor on the domain and range of a function.

2.04 — The characteristics of a function provide information about the real-world situation it represents; making it easier to understand, interpret, and analyze.

2.03 — For a function that represents a real-world situation, analyzing the output for a given input can provide valuable information for understanding the situation.

Standards A.F.1g — For any value, x, in the domain of f, determine f (x), and determine x given any value f (x) in the range of f, given an algebraic or graphical representation of a linear function. A.F.1a — Determine and identify the domain, range, zeros, 2.03 Evaluating functions slope, and intercepts of a linear function, presented algebraically or graphically, including the interpretation of these characteristics in contextual situations. 2.02 Domain and range 2.04 Characteristics of functions A.F.1 — The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.

122

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

A.F.2 — The student will investigate, analyze, and compare characteristics of functions, including quadratic and exponential functions, and model quadratic and exponential relationships. A.F.2a — Determine whether a relation, represented by a set of ordered pairs, a table, a mapping, or a graph is a function; for relations that are functions, determine the domain and range. 2.01 Functions and relations 2.02 Domain and range 2.04 Characteristics of functions

A.F.2b — Given an equation or graph, determine key characteristics of a quadratic function including x-intercepts (zeros), y-intercept, vertex (maximum or minimum), and domain and range (including when restricted by context); interpret key characteristics as related to contextual situations, where applicable. 2.02 Domain and range 2.04 Characteristics of functions A.F.2g — For any value, x, in the domain of f, determine f (x) of a quadratic or exponential function. Determine x given any value f (x) in the range of f of a quadratic function. Explain the meaning of x and f (x) in context. 2.03 Evaluating functions

Future connections A2.F.2 — The student will investigate and analyze characteristics of square root, cube root, rational, polynomial, exponential, logarithmic, and piecewise-defined functions algebraically and graphically.

2. Functions & Relations mathspace.co

123

2.01 Functions and relations Subtopic overview Lesson narrative In this lesson, students will learn about the difference between a relation and a function. They will investigate and analyze relations and functions in graphs, lists of ordered pairs, tables, and mapping diagrams. Additionally, students will learn about the vertical line test and how to apply it. By the end of the lesson, students will be able to determine if a relation is a function.

Learning objective

2.01 Functions and relations

Students: Page 56

After this lesson, you will be able to… • determine whether a relation is a function.

Relations A relation is a set of ordered pairs which represent a relationship. Key vocabulary For example, we can think of the names of people in a math class and their ages as ordered pairs, like (Bob, 13).  function  input-output table  mapping diagram coordinate plane These pairs of information represent a relation.  ordered pair  relation  vertical line test If we chose a speciﬁc age (like 13), we could list all the names of the people who are this age. It could be one person, Bob, or it could be multiple. If a teacher wanted to look for the person who was 13 years old, that description might ﬁt four people which means there’s not one clear answer. 

Essential understanding We can express the same relation in several different ways: as a mapping, a set of ordered pairs, an input-output

table, a is graph in the coordinate as an in terms x and/or y that describes a graph.or graphically. A relation a function when eachplane, input or only hasequation one output. Thisofcan be determined algebraically A mapping diagram shows how the input values are assigned one or more output values. Consider the mapping below:

Standards −1

A mapping of a relation 0

This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards. 0 2

1 Mathematical process goals 4

MPG3 — Mathematical Reasoning MPG1 — Mathematical Problem Solving We cancan write an input-output from thethe mapping, making sure that each pair is mathematical represented. Remember Teachers can promote reasoningthat through Teachers integrate problemtable solving into the ﬁrst value of a relation is an input value and the second value is the output value. The input is the value of x that activities that require students to use logical thinking. lesson by creating activities where students apply is applied to the relation. The output is the y, or the answer that is received as a result of putting x into the relation. For example, teachers can ask students to justify why their understanding of relations and functions to solve A table can be laid For out horizontally (like thecan oneask shown below) or vertically. a relation is or is not a function by examining different real-world problems. instance, teachers students determine a2 real-world scenario can representations such as graphs, tables, or mappings. x to −1 0 whether 1 be represented such y 2 as 0a function, 2 4 as the cost of items in a store or temperature changes throughout a day. This also corresponds to the set of ordered pairs {(−1, 2), (0, 0), (1, 2), (2, 4)}, which can be graphed in the coordinate plane, as shown below. 4

A graph of the relation represents the (x, y) pairs in the coordinate plane.

3 2 124

Mathspace Virginia 1SOL Algebra 1 Teacher Edition x mathspace.co −4 −3 −2 −1 1 2 3 4 −1

−2

MPG5 — Mathematical Representations Teachers can integrate representations by asking students to represent relations and functions in a variety of forms, including graphs, tables, or ordered pairs. Teachers can also encourage students to connect the relations to realworld contexts. For instance, they can ask students to create a graph or a table that represents a relation from a given scenario, such as the relationship between the distance traveled by a delivery service and the delivery fee.

Content standards A.F.2 — The student will investigate, analyze, and compare characteristics of functions, including quadratic and exponential functions, and model quadratic and exponential relationships.

A.F.2a — Determine whether a relation, represented by a set of ordered pairs, a table, a mapping, or a graph is a function; for relations that are functions, determine the domain and range.

Prior connections 8.PFA.2 — The student will determine whether a given relation is a function and determine the domain and range of a function.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 8 — 3.01 Review: Plot points and represent relations Grade 8 — 3.02 Identify functions

Student lesson & teacher guide Relations Students will review the concept of relations and the different ways that relations can be represented from 8th grade. Previously, the represented relations as a set of ordered pairs, an input-output table, a graph and an equation. In Algbera 1, they will also represent relations with mapping diagrams.

Students: Page 56

2.01 Functions and relations After this lesson, you will be able to… • determine whether a relation is a function.

Relations

2.01 Functions and relations mathspace.co For example, we can think of the names of people in a math class and their ages as ordered pairs, like (Bob, 13). These pairs of information represent a relation. A relation is a set of ordered pairs which represent a relationship.

125

After this lesson, you will be able to… • determine whether a relation is a function.

Relations A relation is a set of ordered pairs which represent a relationship. For example, we can think of the names of people in a math class and their ages as ordered pairs, like (Bob, 13). These pairs of information represent a relation. If we chose a speciﬁc age (like 13), we could list all the names of the people who are this age. It could be one person, Bob, or it could be multiple. If a teacher wanted to look for the person who was 13 years old, that description might ﬁt four people which means there’s not one clear answer. We can express the same relation in several different ways: as a mapping, a set of ordered pairs, an input-output table, a graph in the coordinate plane, or as an equation in terms of x and/or y that describes a graph. A mapping diagram shows how the input values are assigned one or more output values. Consider the mapping below: A mapping of a relation

−1

We can write an input-output table from the mapping, making sure that each pair is represented. Remember that the ﬁrst value of a relation is an input value and the second value is the output value. The input is the value of x that is applied to the relation. The output is the y, or the answer that is received as a result of putting x into the relation. A table can be laid out horizontally (like the one shown below) or vertically. −1 2

x y

0 0

1 2

2 4

This also corresponds to the set of ordered pairs {(−1, 2), (0, 0), (1, 2), (2, 4)}, which can be graphed in the coordinate plane, as shown below. A graph of the relation represents the (x, y) pairs in the coordinate plane.

4 3 2 1

−4 −3 −2 −1 −1

−2 −3 −4

Examples Mathspace Students:56Page 57 Virginia SOL Algebra 1 mathspace.co

Example 1 Write the relation {(2, 2), (4, 4), (6, 3), (7, 5)} in the table below. x y

Create a strategy Write the second coordinate of each ordered pair in the y row, below the x-value it corresponds to.

Apply the idea x y

2 2

4 4

6 3

7 5

PurposeExample 2 Check that students can express a relation given as a set of ordered pairs in a table of values. 126

Consider the relation: {(−9, −5), (−5, −10), (−5, −4), (−3, 7), (−2, −4), (−1, 1)}. Represent the relation on the coordinate plane. Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Create a strategy The ﬁrst value of each ordered pair tells us how to move along the x-axis, while the second value tells us how to

Apply the idea 8

Highlight inputs and outputs in each representation

use with Example 1

Student with disabilities support Students might struggle to identify x and y-values in representations of relations, making it difficult for them to convert between the representations. With a set and a table of values specifically, use color coding to Example 1 distinguish between x and y-values. Write the relation 2), (4,x-values 4), (6, 3), (7, in the table below. in yellow, as shown. For example, you might{(2, write in5)} blue and y-values

{(2, 2), (4, 4), (6, 3), (7, 5)}

x 4 6 7 Additionally, you2 can provide extra practice with matching x and y-values using y

more tables and ordered pairs. This can help build confidence and reinforce the concept. Create a strategy

Students: Page 57

x y

2 2

4 4

Apply the idea

Write the second coordinate of each ordered pair in the y row, below the x-value it corresponds to.

6 3

7 5

Example 2 Consider the relation: {(−9, −5), (−5, −10), (−5, −4), (−3, 7), (−2, −4), (−1, 1)}. Represent the relation on the coordinate plane.

Create a strategy

Apply the idea

The ﬁrst value of each ordered pair tells us how to move along the x-axis, while the second value tells us how to move along the y-axis.

6 4 2 −8

−6

−4

−2

x 2

−2 −4 −6 −8

−10

Example 3 Purpose A relation is deﬁned follows:ay relation = −4 if x isgiven positive y = 4ofif ordered x is 0 or negative. Check that students can as express asand a set pairs on the coordinate plane. a Complete the table. −4 −3 −2 −1 0 the 1 representations 2 3 4 Makex connections between all

use with Example 2

y instructional strategies Targeted

In Algebra 1, students are expected to interpret relations represented as a set of ordered pairs, a table, a Create a strategy Apply the idea mapping, or a graph. Use this example to help them make connections between all four representations. For each positive x-value: y = −4, otherwise y = 4. When x = 1, 2, 3, 4: y = −4. • A set of ordered pairs When x = 0, −1, −2, −3, −4: y = 4. {(−9, −5), (−5, −10), (−5, −4), (−3, 7), (−2, −4), (−1, 1)} x y

• A table x y

−9 −5

−5 −10

−5 −4

−3 7

−2 −4

−4 4

−3 4

−2 4

−1 4

0 4

1 −4

2 −4

3 −4

4 −4

−1 1

2.01 Functions and relations mathspace.co

127

Example 1 Write the relation {(2, 2), (4, 4), (6, 3), (7, 5)} in the table below. x 2 • A mapping

−9 Write the second coordinate of each ordered pair−5 in the y row, below the x-value it corresponds to. −3 −2 −1

Create a strategy

Apply−10 the idea −5 x 2 4 y −4 2 4 1 7

6 3

7 5

Example 2 • A graph Consider the relation: {(−9, −5), (−5, −10), (−5, −4), (−3, 7), (−2, −4), (−1, 1)}. Represent the relation on the y coordinate plane. 8

Create a strategy The ﬁrst value of each ordered pair tells us how to move along the x-axis, while the second value tells us how to move along the y-axis. −8 −6 −4

6 Apply 4 the idea 2 −2

8 6

−2

4 2

−4 −6 −8

−8

−6

−4

−2

−10

x 2

−2 −4 −6 −8

−10

Students: Page 57 Example 3 A relation is deﬁned as follows: y = −4 if x is positive and y = 4 if x is 0 or negative. a Complete the table. x y

−4

−3

−2

−1

Create a strategy

Apply the idea

For each positive x-value: y = −4, otherwise y = 4.

When x = 1, 2, 3, 4: y = −4. When x = 0, −1, −2, −3, −4: y = 4. x y

−4 4

−3 4

−2 4

−1 4

0 4

1 −4

2 −4

3 −4

4 −4

Purpose Check that students can express a relation given using specific criteria as a table of values. 2.01 Functions and relations

mathspace.co

Expected mistakes Students might get confused by a relation represented as a description or may misread the criteria. It may be helpful for them to label the table with the criteria before adding the values. For example, drawing a line starting at the positive values in the table (starting at 1) could help students visually see where to start writing −4. x

−4

−3

−2

−1

y x is 0 or negative 128

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

x is positive

Students: Page 58

Purpose Check that students can graph ordered pairs using a given table of values. Reﬂecting with students Students may connect the points to create two horizontal lines. 4

3 2 1 −4 −3 −2 −1 −1 −2

x 1

Ask them to consider what the lines represent. In this case, the line at y = 4 represents all x-values that are less than or equal to zero. Since all of those values would be negative, the y-values would have an output of y = 4, connecting these points with a line is a valid representation of the relation. Highlight the importance of having an unfilled point at x = 0 for the line at y = −4.

−3 −4

Students: Page 58

2.01 Functions and relations mathspace.co

129

Functions Students will review the criteria that makes a relation a function. A relation is a function if and only if each element in the domain is paired with a unique element of the range.

Students: Pages 58–60

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Graphs can also show one input (x-value) paired with one output (y-value ). By graphing the previous set of ordered pairs, we see the graph of a function and the graph not representing a function. 4

−4 −3 −2 −1 −1

−2

−3

−4

Function

Not a function

1 −4 −3 −2 −1 −1

−4 −3 −2 −1 −1

−2

x 1

−4 −3 −2 −1 −1

−2

−3

−4

Function

x 1

Not a function

Example 4

Concrete-Representational-Abstract (CRA) Approach This mapping shows the relation F.

Targeted instructional strategies

1 −1 2 0 1 3 Use a Concrete-Representational-Abstract (CRA) approach to help students explore the concept of mappings 2 4

and functions.

a Find the output when x = 1.

Concrete: Use index cards or sticky notes to represent inputs and outputs. Write numbers (such as 1, 2, 3, 4) on cardsCreate for inputs and other numbers or letters for outputs. Have students work in pairs to create relations a strategy Apply the idea by connecting cards output yarn or string. they can place a string from input We start input from the inputto oval labeledcards x and using follow the When xFor = 1, yexample, = −1. line(s) from 1 to the output value(s). 1 to output A, input 2 to output B, and so on. Encourage them to make different connections, including cases where an input connects to multiple outputs. Do a gallery walk as a class and discuss the different connections b Determine if F is a function. emphasizing how some have inputs connected to multiple outputs. Create a strategy For a function, each x-value maps to a unique y-value.

2.01 Functions and relations mathspace.co

131

Representational: Guide students to draw mapping diagrams that represent the relations they created with the manipulatives. Have them draw two ovals on paper, one labeled “Domain (Inputs)” and the other “Range (Outputs).” Inside the ovals, they can write the numbers or letters from their cards. Students should draw arrows from each input to its corresponding output, mirroring the strings they used earlier. Discuss with them how to identify if a relation is a function by looking at the mapping diagram—specifically, if any input has more than one arrow pointing to different outputs. Abstract: Transition students to working with abstract symbols and notation. Show them how to represent relations as sets of ordered pairs, such as (1, A), (2, B), (3, C). Introduce the concept of a function formally, explaining that in a function, each input has exactly one output. Connecting the stages: Refer back to the physical manipulatives when discussing the mapping diagrams, highlighting how the strings correspond to the arrows. When working with ordered pairs and graphs, point out how multiple strings or arrows going from an input to multiple outputs relates in multiple ordered pairs with the same y-value graph with ordered located the same x position. Graphsor canaalso show onemultiple input (x-value) pairedpairs with one outputat(y-value ). By graphing the previous set of ordered pairs, we see the graph of a function and the graph not representing a function. 4

Misunderstanding functions with multiple inputs sharing the same output 3

Address student misconceptions 2

1 1 Students may think that a relation is not a function if multiple inputs are associated with a single output. x ξ They might mistakenly believe to be a function, each output to only one −4 −3 −2 −1must1 correspond 2 3 4 −4 −3 −2 −1 that1 for 2 a3relation 4 −1 −1 input, misapplying the concept that functions have unique outputs for each input. −2

−2

−3 clarify that the definition of a function requires −3 To address this misconception, each input to have exactly one −4 such as the function f (x) = x2, −4 output, but allows for multiple inputs to share the same output. Provide examples where both x = 2 and x = −2 produce the same output f (x) = 4.

Function

Not a function

Use mapping diagrams to visuallyy illustrate this concept by showing multiple arrows y from different inputs 4 4 pointing to the same output. Encourage students to create their own examples of functions where different 3 discuss why these still satisfy the definition of3 a function. This approach helps inputs yield the same output, and 2 2 students focus on the uniqueness of outputs for each input, reinforcing the correct understanding of what 1 1 defines a function. x x −4 −3 −2 −1 −1

−4 −3 −2 −1 −1

−2

Examples Students: Page 60

−2

−3

−4

Function

Not a function

Example 4 x

This mapping shows the relation F.

1 2 3 4

a Find the output when x = 1.

Create a strategy

Apply the idea

We start from the input oval labeled x and follow the line(s) from 1 to the output value(s).

When x = 1, y = −1.

b Determine if F is a function.

PurposeCreate a strategy For astudents function, each maps to a unique Check that can x-value use a mapping to findy-value. the output of a relation given the input.

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y −1 0 1 2

3 4

a Find the output when x = 1.

Create a strategy

Apply the idea When x = 1, y = −1.

start from the input oval labeled x and follow the Students:WePages 60–61 line(s) from 1 to the output value(s). b Determine if F is a function.

Create a strategy For a function, each x-value maps to a unique y-value.

Apply the idea

Reflect and check

If we input x = 2 into the mapping, we get both y = 0 and y = 1. This means that F is not a function.

To be a function, each input value should only map to one output value. For example, G would be a function: x 1 2 3 4

Mathspace Virginia SOL Algebra 1 mathspace.co

y −1 0 2

Example 5 Purpose The that pairs ifofan values in the table a relation between x and Recognize input maps torepresent two or more outputs, then it y. is not function. Ensure students can identify x −8 −7 from −6a mapping −3 2 7 9 9 10 functions and relations y

−18

−16

−15

−2

−4

−9

Expected mistakes theythe represent function? Apply idea Reflect StudentsDo may think thatathis is not a function because both x = 3and andcheck x = 4 map to y = 2. x we input x = 2definition into the mapping, we get both 0 andthem To be a function, inputfor value should only map to one Remind Ifthem of the of a function, andy =make aware that each it is fine 1 Create a means strategy Apply the idea = 1. Thisto F is not a output. function. output value. For example, G would be a function: differenty inputs havethat the same The relation is a function if for every x-value, there is The x-value of 9 yields the y-values of −4 and 11.2So, the x G y To checkexactly for understanding, ask students which single arrow removed to make one y-value. pointscould do notbe represent a function. 3 this a function. This will highlight that the issue is that x = 2 is mapped to two1 different−1 2 0 4 y-values and not that x = 3 and x = 4 both map to y = 2. 3 4

Example 6

Students: Page 61

y −1 0 1 2

Oprah makes scarves to sell at the market. It costs her $2 to produce each one, and she sells them for $5. a Complete the graph of the points representing the relation between Example 5 of scarves she manages to sell and her total proﬁt for when 1, 2, 3, the number 4 and 5 scarves are sold. The ﬁrst point has been plotted for you. The pairs of values in the table represent a relation between x and y. x y

−8 8

−7 13

−6 −18

−3 −16

2 −15

7 −2

9 −4

9 11

10 −9

Profit

16 14 12 10 8

Do they represent a function?

6 4

Create a strategy

Apply the idea

The relation is a function if for every x-value, there is exactly one y-value.

The x-value of 9 yields the y-values of −4 and 11. So, the −8 −6 −4 −2 2 4 6 8 10 points do not represent a function.

Quantity

Create a strategy

Example 6 can be found using the formula: The total profit Purpose Total profit = Total revenue − Total cost Oprah makes scarves to sell atifthe market. It shown costs herin$2 produce each one, and she sells them for $5. Check that students can identify a relation a to table of values represents a function. a Complete the graph of the points representing the relation between Profit Expected mistakes 18 the number of scarves she manages to sell and her total profit for when 1, 2, 3, For this input (x = 9), Students might for repeating 16 4 and look 5 scarves are sold. They-values first point has been plotted for you. rather than repeating x-values with 147 x −8 −7 −6 −3 −2 9 9 10 12 different outputs. Remind students that a y 8 13 −18 −16 −15 −2 −4 11 −9 10 function has one output for every input, 8there are two different meaning that every x-value should map to 6 and relations 2.01 Functions outputs (y = −4 61 and y = 11) only one y-value. It it fine for different 4 mathspace.co inputs to map to the same output. 2 Quantity

−8 −6 −4 −2

2 4 6 8 10

2.01 Functions and relations mathspace.co

Create a strategy The total proﬁt can be found using the formula:

133

y −1 0

1 2 3 4

Use computational thinking

use with Example 5

Targeted strategies Exampleinstructional 5 As an extension have students explain how they could determine if a very large, unordered table of values The pairs of values in the table represent a relation between x and y. represents a function or relation. A large set could be created using the =RANDBETWEEN(low, high) function in x −8 −7 −6 2 7 digitally 9 10 Google Sheets and shared with −3 the students or9printed. y

−18

−16

−15

−2

−4

−9

Encourage the use of technology including code, sorting tools, graphing technology, or duplicate checking Do they represent a function? functions on spreadsheets. Create a strategy

Apply the idea The x-value of 9 yields the y-values of −4 and 11. So, the points do not represent a function.

The relation is a function if for every x-value, there is Students:exactly Pages one61–62 y-value.

Example 6 Oprah makes scarves to sell at the market. It costs her $2 to produce each one, and she sells them for $5. a Complete the graph of the points representing the relation between the number of scarves she manages to sell and her total proﬁt for when 1, 2, 3, 4 and 5 scarves are sold. The ﬁrst point has been plotted for you.

Profit

16 14 12 10 8 6 4 2 −8 −6 −4 −2

Quantity 2 4 6 8 10

Create a strategy The total proﬁt can be found using the formula: Total profit = Total revenue − Total cost

Apply the idea Total revenue = Number of scarves sold ⋅ $5 Total cost = Number of scarves sold ⋅ $2 Profit for 1 scarf = 1 ⋅ 5 − 1 ⋅ 2

Substitute the number of scarves

= $3

Evaluate

Profit for 2 scarves = 2 ⋅ 5 − 2 ⋅ 2

Substitute the number of scarves2.01 Functions and relations

= $6

mathspace.co

Evaluate

Profit for 3 scarves = 3 ⋅ 5 − 3 ⋅ 2

Substitute the number of scarves

= $9

Evaluate

Profit for 4 scarves = 4 ⋅ 5 − 4 ⋅ 2

Substitute the number of scarves

= $12

Evaluate

Profit for 5 scarves = 5 ⋅ 5 − 5 ⋅ 2

Substitute the number of scarves

= $15

Evaluate

Plot the pairs of values found. 18

Profit

(5, 15)

14 12

(4, 12)

(3, 9)

8 6 4 2

134

(2, 6) (1, 3) Quantity

Mathspace Virginia SOL Algebra 1 Teacher Edition1 2 3 4 5 6 7 8 9 mathspace.co b Is this relation a function?

= $9

Evaluate

Profit for 4 scarves = 4 ⋅ 5 − 4 ⋅ 2

Substitute the number of scarves

= $12

Evaluate

Profit for 5 scarves = 5 ⋅ 5 − 5 ⋅ 2

Substitute the number of scarves

= $15

Evaluate

Plot the pairs of values found.

Apply the idea

Profit

Total revenue = Number of scarves sold ⋅ 16 $5

(5, 15)

Total cost = Number of scarves sold ⋅ $2 14 Profit for 1 scarf = 1 ⋅ 5 − 1 ⋅ 2 = $3 Profit for 2 scarves = 2 ⋅ 5 − 2 ⋅ 2 = $6

Apply the idea

(4, 12) Substitute the number of scarves

Evaluate (3, 9)

8 6

(2, 6) Substitute the number of scarves

(1, 3)

Evaluate

Profit for 3 scarves = 3 ⋅ 5 − 3 ⋅ 2 Substitute theQuantity number of scarves Total revenue = Number of scarves sold ⋅ $5 1 2 3 4 5 6 7 8 9 = $9 Evaluate Total cost = Number of scarves sold ⋅ $2 Profit for 4 scarves = 4 ⋅ 5 − 4 ⋅ 2 Substitute the number of scarves Profit for 1 scarf = 1 ⋅ 5 − 1 ⋅ 2 Substitute the number of scarves b Is this relation a function? = $12 Evaluate = $3 Evaluate for 5 scarves = 5 ⋅ 5 − 5 ⋅ 2 Substitute the number of scarves PurposeCreateProfit a strategy Profit for 2 scarves = 2 ⋅ 5 − 2 ⋅ 2 Substitute the number of scarves = $15a relation on a graph Evaluate Check that express the context of a real-world problem. This students relation is acan function if =for isusing exactly one total profit. $6every quantity sold, there Evaluate Plot the pairs of values found. Profit for 3 scarves = 3 ⋅ 5 − 3 ⋅ 2 Substitute the number of scarves Expected mistakes Profit

Apply the idea

= $9 StudentsChecking may use thepair cost of making each16each quantity ofEvaluate scarves the y-values, with rather the profit. each of values in the graph, quantity of scarvesas sold is associated onlythan one total profit, soMake (5, 15)the number studentsthis aware that profit =(y-values) the difference between the cost of making the scarves and what she Profit forthe 4represent scarves ⋅ 5 − 4 ⋅ 2is 14 Substitute of scarves relation does a4function. sells them for. = $12 Evaluate 12 (4, 12) Profit for 5 scarves = 5 ⋅ 5 − 5 ⋅ 2

Substitute the number of scarves

(3, 9) Reflecting with students 8 Idea summary= $15 Evaluate 6 (2, 6)by looking at the ordered pairs. Some students may recall Ask students what type of pattern the relation follows relation is a function if and only if4each element in the domain is paired with a unique element of the range. Plotwith theApairs of values found.from their work linear functions 8th grade. (1, 3)

Students: Page 62

2 Profit Quantity 18 1 2 3 4 5 6 7 8 9 16 (5, 15) 14 12

b Is this relation a function?

(4, 12)

(3, 9)

Create a strategy

(2, 6)

4 sold, there is exactly one total profit. Mathspace Virginia SOL Algebra 62 This relation is a function if for every1 quantity (1, 3) mathspace.co 2 Quantity

Apply the idea

2 3 4 5 6

7 8 9

Checking each pair of values in the graph, each quantity of scarves sold is associated with only one total profit, so this relation does represent a function. b Is this relation a function?

Create a strategy

Idea summary

PurposeThis relation is a function if for every quantity sold, there is exactly one total profit. A relation is a function if and only if each element in the domain is paired with a unique element of the range. Check that students can identify if a relation shown on a graph represents a function. Apply the idea

each pair of values in the graph, each quantity of scarves sold is associated with only one total profit, so Students:Checking Page 62 this relation does represent a function.

Idea summary 62

A relation Virginia is a function if and1 only if each element in the domain is paired with a unique element of the range. Mathspace SOL Algebra mathspace.co

Mathspace Virginia SOL Algebra 1 mathspace.co

2.01 Functions and relations mathspace.co

135

The vertical line test Exploration Students: Page 63

The vertical line test Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 2.01 to answer these questions. 1.

What similarities did you notice in the graphs that were labeled as functions?

How did the vertical line help you determine which graphs represented functions?

Would a horizontal line be useful in determining if a relation is a function?

Sometimes it is easier to investigate the graph of a relation to determine whether or not it is a function. When looking at a graph, if you can draw a vertical line anywhere so that it crosses the graph of the relation in more than one place, then student it is not a function. Suggested grouping: In pairs

Students will use a GeoGebra applet to explore whether various relations are functions. The aim of the line test to discover that a relation is a function if the vertical line passes through the function explorationVertical is for students The graph at only one point. of a relation is a function if a vertical line intersects the graph of a relation at exactly one point across the entire graph.

Ideal student responses Here responses are two examples relations being checked withstudent the vertical line test. A Less function is saidresponses to “pass the can vertical These ideal mayofdiffer from other correct responses. formal be line test” while a relation that is not a function “fails the vertical line test.” connected with the more precise mathematical language presented here. y

7 1. What similarities did you notice in the graphs that were labeled as functions? 4 6 In the graphs that were labeled as functions, the vertical line crossed the graph at only one point regardless 5 3 of where it was positioned. 43 2

2 1 2. How did the vertical line help you determine which graphs represented functions? 1 x x The vertical line helped to determine the graphs that represented functions by highlighting where there was −4 −3 −2 −1 1 2 3 4 −7−6−5−4−3−2 −1 −1 1 2 3 4 5 6 7 −1 only one output for every input. If a vertical line crossed the graph at more than one point, then there was −2 −3 −2 more than one output for an input. −4 −3

−5 3. Would a horizontal line be−6 useful in determining if a relation is a function? −4 A horizontal line would not−7be useful because a function can have the same output (y-value) for different inputs (x-values). Fails the vertical line test (is not a function) Passes the vertical line test (is a function)

Purposeful One questions pair of points is enough to decide that a relation is not a function, but it is not enough to decide that a relation is function. We must keep points on the graph until it either fails the test or we have checked for all x-values. • Whata is the definition of a checking function? When classifying, remember that every function is a relation, but not every relation is a function. • How does this vertical line relate to the definition of a function? • What do you notice about the relations that are not functions? How does the vertical line help show that it is not a function?

Possible misunderstandings • For graphs with discrete domains, students might not see the points turn orange when the line intersects them. This could cause them to misunderstand the vertical line test. Have them drag the slider very slowly and stop when the line intersects the points to show them how the color of the points change. Students will learn how to use the vertical line test to determine if the graph of a relation is a function or not. 2.01 Functions and relations mathspace.co

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Explore online to answer the questions

mathspace.co Use the interactive exploration in 2.01 to answer these questions. 1. What Students: Page 63 similarities did you notice in the graphs that were labeled as functions? 2.

How did the vertical line help you determine which graphs represented functions?

Would a horizontal line be useful in determining if a relation is a function?

The vertical line test Sometimes it is easier to investigate the graph of a relation to determine whether or not it is a function. When looking at a graph,Interactive if you can draw exploration a vertical line anywhere so that it crosses the graph of the relation in more than one place, Explore online to answer the questions then it is not a function.

mathspace.co

Vertical line test

The graph of a relation is a function if a vertical line intersects the graph of a relation at exactly one point across Use the interactive exploration in 2.01 to answer these questions. the entire graph. 1. What similarities did you notice in the graphs that were labeled as functions? Howexamples did the vertical line help you determine graphsline represented functions? Here 2. are two of relations being checked withwhich the vertical test. A function is said to “pass the vertical line test” while a relation that is not a function “fails the vertical line test.” 3. Would a horizontal line be useful in determining if a relation is a function? y y 7 4 6 Sometimes it is easier to investigate the graph of a relation to determine whether or not it is a function. When looking 5 3 at a graph, if you can draw a4 vertical line anywhere so that it crosses the graph of the relation in more than one place, 2 3 then it is not a function. 2 1 1 x x Vertical line test −4 −3 −2 −1 1 2 3 4 −7−6−5−4−3−2 −1 −1 1 2 3 4 5 6 7 −1 at exactly one point across The graph of a relation −2 is a function if a vertical line intersects the graph of a relation −3 −2 the entire graph. −4 −3 −5 Here are two examples of −6 relations being checked with the vertical line test. A function −4is said to “pass the vertical −7

line test” while a relation that is not a function “fails the vertical line test.” Fails the vertical line test y (is not a function)

Passes the vertical line y test (is a function) 7 4 6 One pair of points is enough to decide that a relation is not a function, but it is not enough to decide that a relation is 5 a function. We must keep checking points on the graph until it either fails the test or we3 have checked for all x-values. 4 2 3 When classifying, remember that every function is a relation, but not every relation is a function. 2 1 1 x x −4 −3 −2 −1 1 2 3 4 −7−6−5−4−3−2 −1 −1 1 2 3 4 5 6 7 −1 −2 −3 −4 −5 −6 −7

Examples Students: Page 64

−2 −3 −4

Fails the vertical line test (is not a function)

Passes the vertical line test (is a function)

One pair of points is enough to decide that a relation is not a function, but it is not enough to decide that a relation is Example 7

a function. We must keep checking points on the graph until it either fails the test or we have checked for all x-values. 2.01 Functions and relations mathspace.co

2.01 Functions and relations mathspace.co

Determine whether the following graphsfunction show functions. When classifying, remember that every is a relation, but not every relation is a function. a

6 5 4 3 2 1 −6 −5 −4 −3−2 −1 −1

x 1 2 3 4 5 6

−2 −3 −4 −5 −6

Create a strategy Draw vertical lines throughout the graph and check whether each line crosses at only one point on the graph at a time.

Apply the idea 6 5 4 3 2 1 −6 −5 −4 −3−2 −1 −1

2.01 Functions and relations mathspace.co x 1 2 3 4 5 6

137

−5 −6

Create a strategy Draw vertical lines throughout the graph and check whether each line crosses at only one point on the graph at a time.

Apply the idea 6 5 Example 7 4 3 Determine whether the following graphs show functions. 2 1 a y −6 −5 −3−2 −4 −1 6 −1 5 −2 4 −3 3 −4 2 −5 1 −6 x −6 −5 −4 −3−2 −1 −1

x 1 2 3 4 5 6

1 2 3 4 5 6

Each vertical line passes through the graph only once, so the graph is a function. −2

−3 −4 −5 −6

y 4

Purpose 3 Check that students can use2 the vertical line test to determine if a graph can be described as a function or not. Create a strategy

x Draw vertical lines throughout the graph and check whether each line crosses at only one point on the graph at a time. Expected mistakes −8 −6 −4 −2 2 4 −1 the concept of the vertical line test. They may need help connecting that the Students may misunderstand Apply the idea vertical line test checks for−2x-values that have more than one output. If the vertical line passes through multiple y −3 6 points, it would show an x-value that has more than one output. 5 −4

3 Reflecting with students 2 Challenge advanced learners to explain their answer without using the vertical line test as their justification. 1 x Create a strategy Refer them back to the definition of a function if they need help started. −6 −5 −4 −3−2 −1 5 6 1 2 3 4getting −1 Draw vertical lines throughout the graph and check whether each line crosses at only one point on the graph time. An example explanation could be, “This function is defined by four discrete points. Each point hasataa distinct −2 −3 x-value, which means that there is only one output for each of the four distinct inputs. Therefore, this graph −4 represents a function.” −5 −6

Mathspace Virginia SOL Algebra 1 mathspace.co Each vertical line passes through the graph only once, so the graph is a function. 64

Students: Pages 64–65 b

y 4 3 2 1 −8 −6 −4 −2

−1

x 2

−2 −3 −4

Create a strategy Draw vertical lines throughout the graph and check whether each line crosses at only one point on the graph at a time.

138

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Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Apply the idea 4

3 2 1 −8 −6 −4 −2

x 2

−1

−2 −3 −4

The vertical lines shown cross the graph at multiple points, so the graph is not a function.

Apply the idea c 4

Purpose 3 2 Check that students can2 use the vertical line test to determine if a graph can be described as a function or not. 1 x

x −8 −6 −4 −2 2 4 Reflecting with students −1 −4 −3 −2 −1 1 2 3 4 Discuss with students whether they need to draw multiple−2vertical lines, as shown in this solution. According −1 to the definition of a function, finding only one vertical line−3that touches multiple points is enough to show this −2 −3 relation is not a function. −4 −4

vertical lines shown cross the graph at multiple points, so the graph is not a function. Students:The Page 65

Create a strategy

c y Draw vertical lines throughout the graph and check whether each line crosses at only one point on the graph at a time. 4 3

Apply the idea

2 1

−4 −3 −2 −1 −1 −2 −3 −4

x 1

3 2 1

−4 −3 −2 −1 −1

−2 −3

Create a strategy

−4

Draw vertical lines throughout the graph and check whether each line crosses at only one point on the graph at a time. Each vertical line passes through the graph only once, so the graph is a function.

Apply the idea 4

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3

2.01 Functions and relations mathspace.co

−4

Each vertical line passes through the graph only once, so the graph is a function.

2.01 Functions and relations mathspace.co 2.01 Functions and relations mathspace.co

139

Purpose Check that students can use the vertical line test to determine if a graph can be described as a function or not.

Stronger and clearer each time

use with Example 7

English language learner support Use this routine to help students clarify their reasoning as to why a relation is or is not a function. Give students a few minutes to individually justify their answers to each part in Example 3. Challenge students to write explanations that do not use the vertical line test as their only justification. Next, provide time for students to meet in pairs to discuss their responses. Each pair will have 2–3 minutes to critique their reasoning, and students should meet with 2–3 different partners. In these pairs, students should ask their partner clarifying questions such as: • Does it pass or fail the vertical line test? How do you know? • Why can we use the vertical line test to say this is a function? • How does the vertical line test verify that this is not a function? After meeting with a few different partners, give students a few minutes to adjust and refine their initial justifications. Then, invite students to share their responses with the class. Highlight responses that use the definition of a function as justification.

Students: Page 66

Idea summary While all functions are relations, not all relations are functions. The vertical line test for functions: 5 4 3 2 1

When looking at a graph, if you can draw a vertical line anywhere so that it crosses the graph of the relation in more than one place, then it is not a function.

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

Practice What do you remember?

Practice

−3

Consider the following mapping:

Complete the following table:

Students: Pages 66–70 x −3

−2

−1

If the input is −1, what is the corresponding output?

Complete the table: −3

−2

State the inputs. 3

140

State the outputs.

2 Represent Consider the following mapping: of the inputs and outputs as a set of ordered pairs. each relationship

x y

What do youb remember? State the inputs. a

−1

−2

−1

Output

−2

3−1

State the outputs.

−3

Input

If the input is −1, what is the corresponding output?

Determine whether the following statements are true or false:

When SOL working with a1 Teacher function, Edition substituting a certain value of x into the formula gives only one value of y for Mathspace a Virginia Algebra mathspace.co that value of x. b All relations are functions. c Some relations are functions.

Represent each relationship of the inputs and outputs as a set of ordered pairs.

Input

Output

Determine whether the statements are true or false: a

When working with a function, substituting a certain value of x into the formula gives only one value of y for that value of x.

All relations are functions.

Some relations are functions.

All functions are relations.

No functions are relations.

No relations are functions.

A relation always passes the vertical line test.

The graph of every straight line is a function.

The graph of every non-vertical straight line is a function.

There is no straight line graph that is a function.

The graph of every straight line is a relation.

The graph of every non-horizontal straight line is a function.

Identify the name used to describe a graph where for some value of x, there exists two or more different values of y.

Express the following relations in a table of values: a

For each of the relation represented by a table, represent the relation on a coordinate plane: a

{(2, 2), (4, 4), (6, 3), (7, 5)}

x 5 10 15 20 y 15 30 45 60

25 75

x y

−4 −4

−3 3

−2 −2

−1 1

0 0

1 1

2 −2

3 3

4 −4

Express each of the relation as a series of points on the coordinate plane: a

{(2, 5), (2, 7), (−3, −4), (−9, 13)}

{(1, 5), (7, −2), (−5, −10), (13, −13)}

Let’s practice 8

Determine whether or not each set of points represents a function: a

{(2, 5), (7, −3), (5, 2), (−4, −9)}

{(2, 5), (2, 7), (−3, −4), (−9, 13)}

{(1, 5), (1, 1), (7, −2), (−5, −10)}

{(1, 5), (7, −2), (−5, −10), (13, −13)}

{(1, 5), (1, 7), (−2, −5), (−5, −10)}

{(−1, −9), (0, 0), (1, 9), (2, 18)}

{(−2, 4), (−1, 1), (2, 4), (6, 36)}

The pairs of values in the table represent a relation between x and y. Determine whether they represent a function: a

x −4 −3 −2 −1 0 y −4 −3 −2 −1 0

1 −1

x −9 −7 −6 −5 −3 −2 y 10 10 10 10 10 10

2 3 4 −2 −3 −4

3 10

5 10

10 10

x y

−8 −7 −6 −3 2 7 9 8 13 −18 −16 −15 −2 −4

9 3

9 11

10 −9

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141

Consider the points in the table: −4 4

x y a 11

−3 3

−1 1

0 2

0 0

1 1

2 2

3 3

Plot the points on a coordinate plane.

4 4 Do they represent a function?

A relation is deﬁned as follows: y = 1 if x is positive and y = −1 if x is 0 or negative. a

Complete the table for this relation: x y

−4

−3

−2

−1

Plot the points on a coordinate plane.

Do these values represent a function?

Determine whether each of the graph shows a function: a

4 2

x −6

−4

−2

−6

−4

−2

−4

−6

2 x

−6

−4

−2

−4

−2

−4

−6

2 −8 −6 −4 −2 −2

142

x −6

x 2

2 −8 −6 −4 −2 −2

−4

−6

−8

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x 2

h 8

x −6

−4

−2

−8

x 1

x −6

−4

−2

−4

−3

−6

−4

−6

−1

−4

−4 −3 −2 −1

−8 −6 −4 −2 −2

−2

Determine whether each mapping is a function or not. a

5 9

−5 1

10 −4

c 8

4 5 6 7 8

−3 −2 4 5 6

−2 −1 3 8

Let’s extend our thinking 14

A particular store is offering one free t-shirt for every two t-shirts purchased. The relationship between the total cost and the number of t-shirts is shown in the table. a

Is the relationship between the total cost and the number of t-shirts a function?

Based on the table, how much will you pay for one t-shirt?

Based on the table, how much will you pay for 3 t-shirts?

Using this relation, how much will you pay for 6 t-shirts?

Number of t-shirts 1 2 3 4 5

Total cost (dollars) 19 38 38 57 76

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143

The following shows the relationship of the the total distance traveled (in kilometers), by Valentina in t hours.

Distance (km) 575

Express the relation as a set of ordered pairs.

Is this relation a function?

460

If Valentina’s car is traveling at a constant speed what might be the total distance traveled after 6 minutes?

345 230 115 t 1

The following sets of points represent a relation between x and y. Find a value of k so that the relation does not represent a function: a

{(8, 6), (5, k), (4, 9), (2, 1)}

{(k, 3), (9, 8), (1, 4), (5, 7)}

{(9, k), (6, 2), (5, 3), (1, 4)}

{(2, 9), (7, 6), (k, 4), (3, 1)}

{(k, 5), (8, 9), (1, 4), (3, 2)}

{(6, 9), (k, 4), (2, 8), (7, 3)}

{(k, 6), (2, 8), (7, 4), (5, 9)}

{(5, k), (1, 7), (2, 9), (3, 8)}

Consider the ordered pairs: {(−9, −5), (−5, −10), (−5, −4), (−3, 7), (−2, −4), (−1, 1)}

Plot the ordered pairs on a coordinate plane.

Which ordered pair would need to be removed from the set so that the remaining ordered pairs represent a function?

Crazy Mobile charges $2.80 a minute plus a connection fee of 90c for an international call. a

Complete the table.

Is this relation a function?

International call cost (dollars)

Tracy makes scarfs to sell at the market. It costs her $3 to produce each one, and she sells them for $6. a

Consider when 1, 2, 3, 4 and 5 scarfs are sold. Plot the points representing the relation between the number of scarfs she manages to sell and her total proﬁt.

State whether the relation is a function. Explain your reasoning.

A vending machine in a college cafeteria offers different types of coffee ﬂavors like espresso, cappuccinos, mochas, and lattes. The table shows the selected ﬂavor and the vended coffee. Selected Vended

144

Call length (minutes) 1 2 3 4 5

espresso espresso

cappuccino mocha

Is this machine operating correctly?

Is this relation a function? Explain your reasoning.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

latte latte

espresso espresso

cappuccino cappuccino

latte mocha

Answers

7 a

14 12 10 8 6 4 2

2.01 Functions and relations What do you remember? 1 a

−1

b −1, 0, 1, 2

c 0, 2, 4

−9−8−7−6−5−4−3−2−1 −2 −4

d 2

2 {(1, 7), (3, 5), (6, 2), (9, 9)} 3 a True

b False

c True

d True

e False

False

g False

h False

True

False

k True

y 3

−6 −3 −3

False

12 15

−9

−12 −15

3 5 Let’s practice 6 8

8 a Yes e No

c No

d Yes

Yes

g Yes

h Function

b Yes

c Yes

d No

9 a Yes

b No

10 a y

b No

−4 −3 −2 −1

−1

x 1

−2

x 5

−6

6 a

x 1 2 3

4 Relation 5 a

15 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

−3

−4

11 a

−4

−3

−2

−1

b y 4

1 2 3 4 5

c Yes

3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

Answers mathspace.co

145

12 a Function

b Not a function

c Function

d Not a function

e Function

g Not a function i

Function Not a function

j b No

c No

d Yes

b $19

c $38

d $76

15 a {(1, 115), (2, 230), (3, 345), (4, 460) (5, 575)} b yes

c 690 km

9.30

12.10

14.90

19 a

Profit

16 14 12

6 4

c No such value of k exists.

d 3 or 2 or 7

e 8 or 1 or 3 6 or 2 or 7

No. of scarfs 4 5

b Y es. There is exactly one input related to one output based on the table of values.

g 2 or 7 or 5 h No such value of k exists. 6

20 a No y

4 2 −9 −8 −7 −6 −5 −4 −3 −2 −1 −2

x 1

−4 −6 −8 −10

b (−5, −10) or (−5, −4)

146

3.70 6.50

b 5 or 9 or 1

17 a

1 2

16 a No such value of k exists.

International call cost (dollars)

b Yes

Let’s extend our thinking 14 a Yes

Call length (minutes)

h Function

Function

13 a Yes

18 a

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

b N o. A function is a rule that assigns to each input exactly one output. Based on the table, when cappuccino and latte are selected, two different ﬂavors were vended, showing a one to two relation which is not a function.

2.02 Domain and range Subtopic overview Lesson narrative In this lesson, students will learn how to identify the domain and range of discrete, continuous, and step functions. They have previously learned about inequality, so in this lesson they will be introduced to set-builder notation. The lesson will also discuss different factors to consider when determining an appropriate domain and range for a given context. By the end of this lesson, students will be able to identify and describe the domain and range, including any algebraic or contextual constraints, for mathematical and real-world problems.

Learning objectives Students: Page 71

Key vocabulary 

continuous



dependent variable



discrete



domain constraint



independent variable



range

 domain

Essential understanding The domain and range of a function can provide insight into the context it models but a real-world context can also be a limiting factor on the domain and range of a function.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG2 — Mathematical Communication

MPG5 — Mathematical Representations

Teachers can integrate this goal into their instruction by having students communicate their thinking and reasoning process when determining the domain and range of functions. They can encourage students to use mathematical vocabulary and notation to express their ideas precisely. Teachers can ask students to explain the differences between continuous and discrete domains and ranges to their peers.

Teachers can incorporate this goal by having students represent the domain and range of functions using a variety of methods such as graphs, set notation and inequalities. They can also have students make connections between these different representations. For instance, teachers can demonstrate how a continuous graph represents a continuous domain and range, and how set notation can be used to express these domains and ranges. 2.02 Domain and range mathspace.co

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Content standards A.F.1 — The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships. A.F.2 — The student will investigate, analyze, and compare characteristics of functions, including quadratic and exponential functions, and model quadratic and exponential relationships. A.F.1a — Determine and identify the domain, range, zeros, slope, and intercepts of a linear function, presented algebraically or graphically, including the interpretation of these characteristics in contextual situations.

A.F.2a — Determine whether a relation, represented by a set of ordered pairs, a table, a mapping, or a graph is a function; for relations that are functions, determine the domain and range. A.F.2b — Given an equation or graph, determine key characteristics of a quadratic function including x-intercepts (zeros), y-intercept, vertex (maximum or minimum), and domain and range (including when restricted by context); interpret key characteristics as related to contextual situations, where applicable.

Prior connections 8.PFA.2 — The student will determine whether a given relation is a function and determine the domain and range of a function.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 8 — 3.03 Domain and range Grade 8 — 3.04 Independent and dependent variables Algebra 1 — 1.06 Multistep inequalities

Tools You may ﬁnd these tools helpful: • Ruler • Highlighter

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Student lesson & teacher guide Domain and range Students are reminded of the concepts of domain and range and how to identify them from a graph. They are also shown how to represent domain and range using set and interval notation.

Students: Pages 71–72

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A domain made up of a single connected interval of values is said to be a continuous domain. The function shown has a continuous domain. It is deﬁned for every x-value in an interval. 4

The domain and range of this function can be written in interval or set-builder notation. Shown below is the set-builder notation:

Domain: {x−∞ < x < ∞}

Range: {y−3 ≤ y < ∞} A domain made1 up of a single connected interval of values is said to be a continuous domain. The function shown x notation is similar to set builder notation but we only has a continuous domain. It is deﬁned for every Inequality x-value in an interval. −4 −3 −2 −1 1 2 3 4 include the inequalities. −1 The domain and range of this function can be written in interval or y −2 Domain: −∞ < x < ∞ 4 set-builder notation. Shown below is the set-builder notation: −3 Range: −3 ≤ y < ∞ 3 Domain: {x−∞ < x < ∞} −4 2 1 −4 −3 −2 −1 −1

Example 1

x 1

Range: {y−3 ≤ y < ∞} Inequality notation is similar to set builder notation but we only include the inequalities. Domain: −∞ < x < ∞

−2

Examples −3 Consider the function shown in the graph. Students: Page 72

Range: −3 ≤ y < ∞

−4

y 8 6 4 2

Example 1

−8 −6 −4 −2 2 −2 y 8 −4

Consider the function shown in the graph.

6 −6 4 −8 2

a State whether the function has a discrete or continuous domain.

−8 −6 −4 −2 −2

Apply the idea

−4

The function is deﬁned at every value of x across an interval, so it has a continuous domain.

−6

x 2

−8

b Determine the domain of the function using set-builder notation. a State whether the function has a discrete or continuous domain.

Apply the idea Apply the idea

Reﬂect and check

We can see that the function is deﬁned for every x-value The domain of the function written in inequality notation is The function is deﬁned at every value of x across an interval, so it has a continuous domain. between −6 and 8, including −6 but not including 8. Domain: −6 ≤ x < 8 So the domain of the function can be written as b Determine theDomain: domain {x of the −6 function ≤ x < 8} using set-builder notation.

Purpose Apply the idea Reﬂect and check To assess students’ understanding of the difference between discrete and continuous domains, and how to We can see that the function is deﬁned for every x-value The domain of the function written in inequality notation is identify between them from a graph. −6 and 8, including −6 but not including 8. Domain: −6 ≤ x < 8

So the domain of the function can be written as Expected mistakes Domain: {x−6 the ≤ x <function 8} Students may mistakenly consider as having a discrete domain because of the endpoints marked on the graph. It’s important to stress that the line segment represents all points between the marked points, including decimal values, thereby making the domain continuous. Mathspace Virginia SOL Algebra 1 72 In contrast, amathspace.co discrete domain would only include specific, separate points.

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a State whether the function has a discrete or continuous domain.

Apply the idea function Students:The Page 72is deﬁned at every value of x across an interval, so it has a continuous domain. b Determine the domain of the function using set-builder notation.

Apply the idea

Reﬂect and check

We can see that the function is deﬁned for every x-value between −6 and 8, including −6 but not including 8.

The domain of the function written in inequality notation is Domain: −6 ≤ x < 8

So the domain of the function can be written as Domain: {x−6 ≤ x < 8}

Purpose Check students can determine the domain of a function from a graph and express it using set notation.

Students: Page 73 72

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Virginia SOL Algebra 1

c Determine the range of the function using set-builder notation. mathspace.co

Apply the idea

Reﬂect and check

We can see that the function reaches every y-value between −6 and 4, including 4 but not including −6.

The range of the function written in inequality notation is Range: −6 < y ≤ 4

So the range of the function can be written as Range: {y−6 < y ≤ 4}

Example 2 Purpose To checkConsider students’ understanding of set notation and their ability to determine the range of a given function. the function shown in the graph. y 5

a Determine the domain of the function using interval notation.

Expected mistakes 4 Students may include the lower bound in the range. It is important to remind them that3 it is not included in this case. 2 1

Draw a box around the function Student with disabilities support

−1

use with xExample 1 1

To help students to visualize the domain and range, use a highlighter with a straight edge to draw a box −2 containing the function. If no side can be drawn to contain it, then that bound will be −3 ∞ or −∞. If it goes through an unfilled point, use a dashed line. y 8 Reﬂect and check

Apply the idea

We can see that the function is defined for every x-value 6 The domain could have been written in inequality between negative infinity and positive infinity. notation as: So the domain of the function can be written as Domain: (−∞, ∞)

Domain: −∞ < x < ∞

This domain is also referred to as the set of all real

−8 −6 −4 −2 numbers. 2 4 6 −2

−4

b Determine the range of the function using interval notation.

−6

Apply the idea We can see that the function is defined for every y-value between, and including, −2, to positive infinity.

−8 Reﬂect and check The range could have also been written using inequality notation:

In this case we can see that the left side of the box is at x = −6 and the right side is at x = 8 (and not included), y ≥ −2and the top of the box is so the domain is {x ∣ −6 ≤ xRange: < 8}.[−2, The∞)bottom of the box is at y = −6 (and notRange: included) at y = 4, so the range is {y ∣ −6 < y ≤ 4}.

Idea summary Different notations help us represent discrete and continuous functions: Set notation (discrete): {1, 2, 3, 4, 5} Set notation (continous): {x− 4 ≤ x < 10} Inequality notation: −4 ≤ x < 10

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Steps for identifying domain and range in set notation

use with Example 1

Targeted instructional strategies Offer students the following steps to use when identifying the domain and range in set notation. Domain: 1. Check the left side of the graph to see if the graph has an endpoint or continues infinitely. • If it has an endpoint, identify the x-coordinate and whether it is a filled or unfilled point. This will determine the lower bound value and the inequality symbol. • If it continues infinitely, the lower bound will be −∞ and the inequality symbol is < 2. Check the right side of the graph to see if the graph has an endpoint or continues infinitely. • If it has an endpoint, identify the x-coordinate and whether it is a filled or unfilled point. This will determine the upper bound value and the inequality symbol. • If it continues infinitely, the upper bound will be ∞ and the inequality symbol is < 3. Put together the set notation Range: Same process, but check the bottom and top of the graph. Recall that an unfilled endpoint or infinity will use <, while a filled endpoint will use ≤. 8

−8 −6 −4 −2 −2

x 2

−8 −6 −4 −2 −2

−4

−6

−8

c Determine the range of the function using set-builder notation.

Apply the idea

Reﬂect and check

We can seeDomain: that the function {x∣−6 ≤reaches x < 8}every y-value between −6 and 4, including 4 but not including −6.

The range of the function written in inequality notation is Domain: {x∣−6 < x < ∞} Range: −6 < y ≤ 4

So the range of the function can be written as

Students: Page 73

Range: {y−6 < y ≤ 4}

Example 2 Consider the function shown in the graph.

y 5

a Determine the domain of the function using interval notation.

4 3 2 1 −1

−1

x 1

−2 −3

Apply the idea

Reﬂect and check

We can see that the function is defined for every x-value between negative infinity and positive infinity.

The domain could have been written in inequality notation as:

So the domain of the function can be written as Domain: (−∞, ∞)

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Domain: −∞ < x < ∞ This domain is also referred to as the set of all real numbers.

b Determine the range of the function using interval notation.

between −6 and 4, including 4 but not including −6.

Range:−1−6 < y ≤ 4 1

So the range of the function can be written as

−1

Range: {y−6 < y ≤ 4}

x 2

−2 −3

Example 2 Apply the idea

Reﬂect and check

We can see the function for every x-value The domain could have been written Consider thethat function shown is in defined the graph. y in inequality between negative infinity and positive infinity. notation as: 5 a Determine the domain of the function using interval notation. So the domain of the function can be written as Domain: −∞ 4< x < ∞ Domain: (−∞, ∞)

This domain is also referred to as the set of all real 2 numbers. c Determine the range of the function using set-builder notation. 1 b Determine the range of the function using interval notation.

Apply the idea

Reﬂect and check

−1

x 3

can see that the function reaches every y-value The range of the function written −2 in inequality notation is PurposeWe Apply the idea Reﬂect and check between −6 and 4, including 4 but not including −6. Range: −6 < ywritten ≤ interval 4 usingnotation. −3 StudentsWedemonstrate that they iscan writeforthe domain of aThe function fromhave a graph using can see that the function defined every range could also been inequality So the range of the function can be written as y-value between, and including, −2, to positive infinity. notation: Range: {y−6 < y ≤ 4} Reflecting with students Range: [−2, ∞) Range: y ≥ −2

Apply the idealearners to the set notation for real numbers, Reﬂect and Introduce advanced .check Explain that the set of integers and the We can see that the function is defined for every x-value The domain couldtohave written in inequality set of rational numbers also have their own symbols, and it is common seebeen these symbols in college math between negative infinity and positive infinity. notation as: courses.Example WhenIdea they reach university, they will most likely describe this domain using set notation: {x∣x∈} 2 So the domainsummary of the function can be written as Domain: −∞ < x < ∞ where the symbol ∈ can be read as “belongs to”.

Different notations help us represent discrete and continuous functions: Consider the functionDomain: shown in(−∞, the∞) graph. y set of all real This domain is also referred to as the Set notation (discrete): {1, 2, 3, 4, 5} 5 Determine Students:a Page 73 the domain of the function using interval notation. numbers. Set notation (continous): {x− 4 ≤ x < 10} 4 Inequality notation: −4 ≤ x < 10 b Determine the range of the function using interval notation. Interval notation: (−5, 6]

3 2 1

Apply the idea

Reﬂect and check

We can see that the function is defined for every y-value between, and including, −2, to positive infinity.

The range could have also−1been written1 using2 inequality 3 −12.02 Domain and range 73 notation: −2

Range: [−2, ∞)

Range: y ≥ −2

mathspace.co

−3

Idea summary

Reﬂect and check PurposeApply the idea Different notations help us represent discrete and continuous We can see that the function is defined for every x-value domain could have written in inequality Check students’ understanding of how to determine and The write thefunctions: range of abeen function from a graph using between and positive infinity. notation as: Setnegative notationinfinity (discrete): {1, 2, 3, 4, 5} interval notation. So the domain of the(continous): function can as Set notation {xbe − 4written ≤ x < 10} Inequality notation: −4 ≤ x <∞) 10 Domain: (−∞,

Mixing up domain and range Interval notation: (−5, 6] Address student misconceptions

Domain: −∞ < x < ∞

This domain is also referred to as the set of alluse realwith Example 2 numbers.

Students may mix up the domain and range. Proper notation and detail can help with this. For example, just b Determine the range of the function using interval notation. writing [−2, ∞) can lead to confusion as to whether it is x or y is being talked about. Instead, we can write 2.02 Domain and range 73 Range: [−2, ∞) to help remind students that we are looking at which y-values are included.mathspace.co Apply the idea

Reﬂect and check

We can see that the function is defined for every y-value between, and including, −2, to positive infinity.

The range could have also been written using inequality notation:

Students: Page 73

Range: [−2, ∞)

Range: y ≥ −2

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Domain and range in context Students are introduced to the concepts of constraints and how to identify domain and range from a context.

Students: Page 74

Domain and range in context Understanding the limitations on the domain and range of a function in context are important for interpreting situations. Depending on the context, a discrete function may be appropriate for a situation or a continuous function could be better suited to the scenario. The choice of whether rational numbers or speciﬁcally integers or whole numbers should alsorange be considered when given a real-world situation for interpretation. Domain and in context Understanding the limitations on the domain and range of a function in context are important for interpreting variable DomainDepending constrainton the context, a discrete function may Dependent situations. be appropriate for a situation or a continuous function could be betteror suited to theof scenario. The choice of whether The rational numbers or speciﬁcally integers or whole output of a function whose value depends on A limitation restriction the possible x-values, numbers be equation, considered when given for interpretation. the independent variable usuallyshould writtenalso as an inequality, or ina real-world situation set-builder notation Domain constraint Independent variable A limitation or restriction of the possible x-values, The input of a function whose value determines usually written as an equation, inequality, or in the value of other variables set-builder notation

Dependent variable The output of a function whose value depends on the independent variable

Independent variable The input3of a function whose value determines the Example value of other variables Students: Page 74

Consider the relationship between the cost of a hotel stay and the length of the stay. Suppose the hotel charges $75 per night and the stay last 7 nights. a State the3independent and dependent variables. Example Consider the relationship between the cost of a hotel stay and the length of the stay. Suppose the hotel charges $75 Apply the idea per night stay last hotel 7 nights. Since the and totalthe cost of the room depends on the number of nights at the hotel, the number of nights is the a State the independent andcost dependent variables. variable. independent variable and the is the dependent

Apply the idea

b Determine an appropriate domain and range and explain your reasoning. Since the total cost of the hotel room depends on the number of nights at the hotel, the number of nights is the independent variable and the cost is the dependent variable.

Create a strategy

The domain and range may be discrete or continuous, and the types of real numbers in the domain and range also b Determine an appropriate domain and range and explain your reasoning. need to be considered.

Purpose Apply idea Createthe a strategy To help students identify and understand the distinction between independent and dependent variables in a An appropriate for the be basedand on the types number nights a person to stay the hotel, The domain anddomain range may be function discrete would or continuous, of of real numbers in theplans domain andat range also practicaland context. makes sense for the domain to be discrete whole numbers in set notation because payment is counted in full needit to be considered. days.

Expected mistakes Apply the idea Domain: {1, 2, 3, 4, 5, 6, 7} Students may incorrectly assign the cost as the independent variable and the number of nights as the An appropriate domain for the function would be based on the number of nights a person plans to stay at the hotel, Based on the choice for the domain, the range will that also have cost discrete whole number dependent Ensure students onbecause thevalues. number ofisnights, the other and itvariable. makes sense for the domain tounderstand be discrete whole the numbers independs set notation payment countednot in full Range: {75, 150, 225, 300, 375, 450, 525} way around. days. Domain: {1, 2, 3, 4, 5, 6, 7}

Reﬂect and check

Based on the choice for the domain, the range will also have discrete whole number values. It doesn’t make sense to determine the cost of staying at the hotel for 1.5 nights for instance, because the hotel would Range: {75, 150, than 225, a300, 375, 450, 525} need to be booked for 2 nights in order to stay longer night.

Reﬂect and check It doesn’t make sense to determine the cost of staying at the hotel for 1.5 nights for instance, because the hotel would need to be booked for 2 nights in order to stay longer than a night.

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a State the independent and dependent variables.

Apply the idea Since the total cost of the hotel room depends on the number of nights at the hotel, the number of nights is the

Students:independent Page 74variable and the cost is the dependent variable. b Determine an appropriate domain and range and explain your reasoning.

Create a strategy The domain and range may be discrete or continuous, and the types of real numbers in the domain and range also need to be considered.

Apply the idea An appropriate domain for the function would be based on the number of nights a person plans to stay at the hotel, and it makes sense for the domain to be discrete whole numbers in set notation because payment is counted in full days. Domain: {1, 2, 3, 4, 5, 6, 7} Based on the choice for the domain, the range will also have discrete whole number values. Range: {75, 150, 225, 300, 375, 450, 525}

Purpose Check students can identify appropriate domains and ranges for a given situation, and explain their reasoning. 74

Mathspace Virginia SOL Algebra 1 mathspace.co

Reflecting with students Ask students to explain their reasoning for choosing a discrete domain and range. Discuss the importance of understanding the context when determining the domain and range of a function.

Make connections with other representations

use with Example 3

Student with disabilities support For students who benefit from visual aids, drawing a graph or creating a table of values can help understand the problem better. In this case, a graph showing the number of nights on the x-axis (independent variable) and the total cost on the y-axis (dependent variable) can provide a visual representation of the problem. 900

Total cost

800 700 600

The points on the graph represent the total cost for each possible night. For example, the point (1, 75) represents one night costing $75, while the point (7, 525) represents staying 7 nights costing $525.

500 400 300 200 100 1

Number of nights 2 3 4 5 6 7 8 9

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Students: Page 75 Example 4 Dog Weight

Consider the graph of the function of a dog’s weight over time:

Weight (pounds) 80 60 40

Example 4 20

Consider the graph of the function of a dog’s weight over time:

Dog Weight Weight (pounds)

a Determine if this graph represents a function.

Time (months) 10

Create a strategy The graph of a relation represents a function if it passes the vertical line test.

Apply the idea

40 20

While the graph appears almost vertical close to the y-axis, it still passes the vertical line test. Dog Weight

Weight (pounds) a Determine if this graph represents a function.

Create a strategy

Time (months) 10

The graph of a relation represents a function 60 if it passes the vertical line test.

Apply the idea

While the graph appears almost vertical close to the y-axis, it still passes the vertical line test. 20

Dog Weight Weight (pounds)

Time (months) 10

Since it passes the vertical line test, the graph is a function. 60

40 b Explain why the domain of this function is continuous.

Purpose 20 Create a strategy To assess students’ ability to recognize whether a given graph represents a function by applying the vertical line (months) The domain of a function is either continuous or discrete, and theTime context of the problem gives us information test. about which type of domain makes the most sense.5 10 15

Students:Since Page 75 itthe passes Apply ideathe vertical line test, the graph is a function. Since a dog’s weight can be measured at any time, such as 4.5 months as opposed to only at each month mark based on the graph, the function is continuous. b Explain why the domain of this function is continuous.

Create a strategy The domain of a function is either continuous or discrete, and the context of the problem gives us information about which type of domain makes the most sense.

2.02 Domain and range mathspace.co

Apply the idea Since a dog’s weight can be measured at any time, such as 4.5 months as opposed to only at each month mark based on the graph, the function is continuous.

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Purpose To check students’ understanding of continuous and discrete domains and their ability to apply these concepts to real-world scenarios.

Students: Page 76 c State an appropriate domain and range of the function based on the information in the graph. Justify your solution.

Create a strategy Since the graph is continuous, the domain and range may be given in interval or set-builder notation.

Apply the idea An appropriate domain for the relationship would be from 0 to around the age when the dog stops growing because the graph will eventually ﬂatten. Domain: {x  0 ≤ x ≤ 24} An appropriate range for the function would also begin at 0 up to the maximum weight when it stops growing. It doesn’t make sense for a puppy to be born weighing 0 pounds exactly so we will exclude 0 from our interval, however, a puppy could weigh between 0 and 1 pound (think of puppies whose weight is measured in ounces). Based on the graph, the dog may grow up to 80 pounds. Range: {y  0 < y ≤ 80}

Idea summary

Purpose Discrete domains apply to problems where the independent variable only includes certain values in an interval, Check students’ understanding of the concept of domain and and their ability to interpret from a graph. whereas continuous domains apply to problems where therange independent variable includes all valuesthese in an interval. Expected mistakes StudentsPractice might erroneously include zero in the range, not considering that it is practically impossible for a puppy to weigh exactly zero. Ask students why zero is excluded from the range and how practical considerations can affect What to dodiscuss you remember? mathematical representations. 1

State the deﬁnition for the:

a Domain Stronger and clearer each time

Range

use with Example 4

English language 2 State when thelearner domain ofsupport a function is: a Continuous Discrete Use this routine to help students clarify their reasoning as tob why this function has a continuous domain. Give students aan few minutes to individually their each part in Example 4. andyour c 3 State appropriate and range of the function based the information in the graph. Justify solution. Determine whetherdomain each relation is a justify function. If theanswers relationonisto a function, determine its domain range in set notation.

Next, provide time for students to meet in pairs to discuss their responses. Each pair will have 2–3 minutes to a a strategy b {(−2, 3), (4, 3), (6, 0), (0, 6)} Create x 2 3 2 should 3 2 critique their reasoning, and students meet with 2–3 different partners. In these pairs, students should Since the graph range may be given in interval or set-builder notation. y is continuous, −1 −2 the−3domain −4 and −5 ask their partner clarifying questions such as: cthe idea d specific y • Can Apply we measure weight x times? x a dog’s 5 7 1at any 0 time4or only at very • DoesAna appropriate dog liveyforever? How will this impact the domain that we choose? 2 8 2 −4 1 domain for the relationship would be from 0 to around the0 age when the 5dog stops growing because • Is it possible for eventually a puppy ﬂatten. to weigh nothing (0 pounds and 0 ounces)? the graph will Domain: {x 0 ≤ x ≤ 24}

 9 After meeting with a few different partners, give students a few minutes to adjust−8and refine their initial An appropriate range for the function would also begin at 0 up to the maximum weight when it stops growing. justifications. Then, invite students to share their responses with the class. 2 0 from our interval, It doesn’t make sense for a puppy to be born weighing 0 pounds exactly 7so we will exclude however, a puppy could weigh between 0 and 1 pound (think of puppies whose weight is measured in ounces). Based on the graph, the dog may grow up to 80 pounds.

Students: Page 76

Range: {y  0 < y ≤ 80}

Idea summary 76

Discrete domains apply to problems where the independent variable only includes certain values in an interval, whereas continuous domains apply to problems where the independent variable includes all values in an interval.

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Practice What do you remember? 1

State the deﬁnition for the:

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157

Practice Students: Pages 76–81

What do you remember? 1

State the deﬁnition for the: a

Range

Discrete

State when the domain of a function is: a

Domain

Continuous

Determine whether each relation is a function. If the relation is a function, determine its domain and range in set notation. a

x 2 3 2 3 2 y −1 −2 −3 −4 −5 x 5 7 1 0 y 2 8 2 −4

4 1

{(−2, 3), (4, 3), (6, 0), (0, 6)}

5 10

−8

For each of the graph: i

State the domain of the relation using interval notation.

State the range of the relation using interval notation.

2 −8 −6 −4 −2 −2

−4

−4 −6

−8

4 x 2

−8 −6 −4 −2 −2

−6

−8 −6 −4 −2 −2

158

2 −8 −6 −4 −2 −2

−4

−6

−8

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For each of the graph: i

State the domain of the relation using set notation.

State the range of the relation using set notation.

2 −8 −6 −4 −2 −2

−4

−6

−8

x 2

−8 −6 −4 −2 −2

−4

−8 −6 −4 −2 −2

x 2

−4

−6

−8

−8 −6 −4 −2 −2

For each of the graph, identify whether the domain of the function is discrete or continuous: a

1 −4 −3 −2 −1

−1

x 1

−4 −3 −2 −1

−1

−2

−3

−4

x 1

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159

The graph represents the amount of gas in a fuel tank over time during a roadtrip: a

State the domain and range of the relation using set notation.

Interpret the meaning of the domain and range within the context of the problem.

13 12 11 10 9 8 7 6 5 4 3 2 1

y (gallons)

x (hrs) 7

Let’s practice 8

State the domain and range of the collection of points shown on the graph.

6 4 2

−8 −6 −4 −2 −2

−4 −6 −8

Determine whether each relation is a function. If the relation is a function, determine its domain and range in set notation. a

b 4

1 −4 −3 −2 −1

x 1

−1

−4 −3 −2 −1

−2

−3

−4

d 4

−2

−4 −3 −2 −1

−1

−2

−3

−4

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x 1 2 3 4 5

−6 −5 −4 −3 −2 −1 −1

x 1

−1

−2

160

x 1

Determine for each graph whether or not it has a domain consisting of all the real numbers. If it does not, state the domain. a

2 −8 −6 −4 −2 −2

x 2

−4

−6

Determine for each graph whether or not it has a range consisting of all the real numbers. If it does not, state the range. a

2 −8 −6 −4 −2 −2

−8 −6 −4 −2 −2

−8

x 2

−8 −6 −4 −2 −2

x 2

−4

−6

−8

For each of the following: i

Identify an independent variable and a dependent variable.

State the domain and range for the scenario.

Uma earns between $200 and $450 selling between 35 and 50 burritos, inclusive.

Jermaine starts reading his book from page 112. Over the course of 17 days, he reads up to and including page 237.

Brigid Kosgei completes a 26.2 mile marathon in 134 minutes.

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161

Let’s extend our thinking 13

State the domain and range for each of the graph: a

−8 −6 −4 −2 −2

8 10

2 −4 −2

−4

−2

x 2

−4

−6 −8

−6

−10

−8

y 8 6 4 2 x −8 −6 −4 −2

−2 −4

The graph of a function is shown. a

State the range of the function.

A ball is thrown from an apartment window in a high-rise building. The height of the ball above ground over time can be modelled by the function shown in the graph, where the ball is thrown at x = 0. State the range of the function for this context.

y 125 100 75 50 25 x −6 −4 −2

−25

162

A comet is travelling through the solar system. It passes by the Earth before curving around the Sun and heading back out into deep space. The distance of the comet from Earth, y, is tracked by satellite telescopes and expressed as a function of time, x, since the comet was ﬁrst spotted. a

State the domain of this function.

The satellites detect that, at its closest point, the comet was a distance of 0.14 AU (astronomical units) from Earth. State the range of the function.

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Answers

8 Domain: x∈{−2, −1, 0, 1, 2}, Range: y∈{−4, −1, 2, 5, 8}

2.02 Domain and range

9 a Not a function.

What do you remember? 1 a T he domain of a function is the complete set of possible values of the input of the function. b T he range of a function is the complete set of possible values of the output of the function. 2 a T he domain of a function is continuous if the possible inputs of the function include all the values within an interval. b T he domain of a function is discrete if the possible inputs of the function are not all connected. 3 a Not a function. b F unction. The domain is {−2, 0, 4, 6}. The range is {0, 3, 6}. c F unction. The domain is {0, 1, 4, 5, 7}. The range is {−4, 1, 2, 8}. d Not a function. 4 a i Domain: [−4, 7]

ii Range: [−5, 7]

b i Domain: (−2, 4]

ii Range: [−6, 5)

c i Domain: (−4, 6)

ii Range: (−5, 3)

d i Domain: [−1, 3]

ii Range: [−1, 6]

5 a i Domain: {x∣−2 ≤ x ≤ 3}

b F unction. The domain is {x∣−∞ ≤ x ≤ ∞}. The range is {y∣y = −2}. c F unction. The domain is {x∣−6 ≤ x ≤ 5}. The range is {y∣−4 ≤ y ≤ 2}. d Not a function. 10 a The domain is (−∞, 8] b The domain consists of all the real numbers. 11 a The range consists of all the real numbers. b The range is [2, 2] 12 a i T he number of burritos Uma sells is an independent variable. The amount of money Uma earns is a dependent variable. ii Domain: 35 ≤ x ≤ 50, Range: 200 ≤ y ≤ 450 b i T he number of days Jermaine has been reading for is an independent variable. The page number Jermaine is currently on is a dependent variable. ii Domain: 0 ≤ x ≤ 17, Range: 112 ≤ y ≤237 c i T he number of minutes that Brigid Kosgei has been running the marathon for is an independent variable. The number of miles Brigid Kosgei has run in the marathon is a dependent variable. ii Domain: 0 ≤ x ≤ 134, Range: 0 ≤ y ≤ 26.2

ii Range: {y∣−6 ≤ y ≤ 4} b i Domain: {x∣−2 < x ≤6}

Let’s extend our thinking

ii Range: {y∣−5 < y ≤ 7}

13 a Domain: {x∣x ≥ −5} Range: All real numbers

c i Domain: {x∣−3 < x < 3} ii Range: {y∣−2 < y < 6}

b Domain: x = 4 Range: {y∣−6 < y < 6}

d i Domain: {x∣−6 ≤ x ≤4}

c Domain: {x∣−5 < x < −1 and 0 < x < 9} Range: {y∣y = 5, 3, 8}

ii Range: 6 a Continuous

Let’s practice

b Discrete

7 a Domain: {x∣0 ≤ x ≤ 6} and Range: {y∣0 ≤ y ≤ 12}

14 a y ≤ 125

b 0 ≤ y ≤ 125

15 a x ≥ 0

b y ≥ 0.14

b T he domain of x goes from 0 to 6, meaning that the hours spent driving on the roadtrip is 6 altogether, as well as the amount of time it takes to run out of gas. The range of y goes from 0 to 12, meaning that the amount of gas in used during the roadtrip is 12 gallons. We also know that the gas runs out since it equals 0 at the end of the time spent driving.

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2.03 Evaluating functions Subtopic overview Lesson narrative In this lesson on evaluating functions, students will learn how to calculate the output of a function given specific input values. They will explore function notation and understand how it represents the relationship between inputs and outputs. The lesson includes examples of evaluating linear functions at various points, constructing tables of values, and interpreting graphs to find function values. By the end of the lesson, students should be able to evaluate functions given an input value and determine input values given an output, using both algebraic and graphical methods.

Learning objectives

2.03 Evaluating functions

Students: Page 82

After this lesson, you will be able to… • evaluate functions for valid inputs. • interpret statements that use function notation in a real-world context.

Evaluating functions

Key vocabulary Recall that a function maps each input of a relation to exactly one output. Functions are typically represented in  

function notation, so the relationship between inputs and outputs are clear.  function  function notation evaluate

output Input



relation

Function notation

The independent variable of a function; usually the x-value

Essential understanding

 input

A notation that describes a function. For a function f when x is the input, the symbol f (x) denotes the corresponding output.

Output For a function that represents a real-world situation, analyzing the output for a given input can provide valuable The dependent variable of a function; usually the information for understanding the situation. y-value We have seen the equation for a linear function y = mx + b. By naming a linear function f, you can also write the Standards

function using function notation: f (x) = mx + b. This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards. This is a way of saying that mx + b is a function of x. This is useful because it quickly tells us that we are working with a function whereprocess y can represent Mathematical goalsa relation that is not a function. The notation f (x) is another name for y. If f is a function, and x is in its domain, then f (x) represents the output of f MPG4 — Mathematical Connections corresponding to the input x. You can use letters other than f to name a function, such as g or h. Teachers can show students how the process of evaluating functions is connected to students’ prior knowledge of evaluating algebraic expressions. Teachers can alsof create connections by linking the concept of (x) = mathematical y functions to other real-world topics and situations. examples, f isReal-world the name of function such as the cost of renting a car based on the number of days, can be used to illustrate the practical application of functions, thus establishing a connection x is the input between mathematics and the real world. y is the output To evaluate a function at a point is to calculate the output value at a particular input value: If f (x) = −7x + 9, then determine the value of f (1). This is the same as stating to evaluate the function y = −7x + 9 when x = 1. 164

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Therefore, f (1) = 2 for the function f (x) = −7x + 9.

f (1) = −7(1) + 9 f (1) = −7 + 9 = 2

MPG5 — Mathematical Representations Teachers can reach this goal by teaching students how to represent functions in various forms such as algebraic expressions, tables, and graphs. Students can be guided to the understanding that different representations can convey the same mathematical idea. Furthermore, teachers can show how to interpret these representations in realworld contexts, thus enabling students to see representation as both a process and a product.

A.F.1g — For any value, x, in the domain of f, determine f (x), and determine x given any value f (x) in the range of f, given an algebraic or graphical representation of a linear function. A.F.2g — For any value, x, in the domain of f, determine f (x) of a quadratic or exponential function. Determine x given any value f (x) in the range of f of a quadratic function. Explain the meaning of x and f (x) in context.

Prior connections 8.PFA.3 — The student will represent and solve problems, including those in context, by using linear functions and analyzing their key characteristics (the value of the y-intercept (b) and the coordinates of the ordered pairs in graphs will be limited to integers).

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Algebra 1 — 1.01 Algebraic expressions Algebra 1 — 2.01 Functions and relations Algebra 1 — 2.02 Domain and range

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Student lesson & teacher guide Evaluating functions Students are introduced to the concept of function notation and shown how the different parts relate to inputs and outputs, as well as how the notation connects to graphs. Students are shown how to evaluate function notation.

Students: Page 82

2.03 Evaluating functions After this lesson, you will be able to… • evaluate functions for valid inputs. • interpret statements that use function notation in a real-world context.

Evaluating functions Recall that a function maps each input of a relation to exactly one output. Functions are typically represented in function notation, so the relationship between inputs and outputs are clear. Input

Function notation

The independent variable of a function; usually the x-value

A notation that describes a function. For a function f when x is the input, the symbol f (x) denotes the corresponding output.

Output The dependent variable of a function; usually the y-value

We have seen the equation for a linear function y = mx + b. By naming a linear function f, you can also write the function using function notation: f (x) = mx + b. This is a way of saying that mx + b is a function of x. This is useful because it quickly tells us that we are working with a function where y can represent a relation that is not a function. The notation f (x) is another name for y. If f is a function, and x is in its domain, then f (x) represents the output of f corresponding to the input x. You can use letters other than f to name a function, such as g or h.

f (x) = y f x y

is the name of function is the input is the output

To evaluate a function at a point is to calculate the output value at a particular input value: If f (x) = −7x + 9, then determine the value of f (1). This is the same as stating to evaluate the function y = −7x + 9 when x = 1. f (1) = −7(1) + 9 f (1) = −7 + 9 = 2 Therefore, f (1) = 2 for the function f (x) = −7x + 9.

166

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Think of functions in terms of inputs and outputs Targeted instructional strategies Emphasize to students that f (x) is just new notation and does not change what the function does. When a function is written in the form y = …, we can replace y with f (x) as they both represent the output or dependent variable. A linear function still has a y-intercept (initial value) and a rate of change (slope), but we are just writing this in a different way. Input: x

Input: x

Function: f

Output: y = f (x)

General visual of functions

Function: f Multiply input by 2 and add 5

Output: f (x) = 2x + 5

Visual of f (x) = 2x + 5

This new notation is helpful because it allows us to write just f (2) instead of “substitute 2 in for x and evaluate” and allows us to name different graphs on the same coordinate plane more easily.

Collect and display English language learner support As students are working, note how students describe the concepts of “input value”, “output value”, and “function notation”. Collect the different ways that students find to understand these concepts and display them in a common place for the students to access. If students do not come up with alternative ways to word these concepts and are confused by them, suggest some of your own. For example: • Output value • Input value • Often y or f (x) • Often x • Dependent variable • Independent variable • What we end up with • What we start with • What we get after we simplify/evaluate • What we substitute in • Value of the range • Value of the domain • Function notation • y and f (x) are swapped • A way to name the function • Makes it easier to substitute • Usually f, but can be other letters Take care to address any rewordings that contradict or are too similar to other concepts that the student will learn in the future, such as ensuring students know we can only “swap” y and f (x) when the relation is a function.

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Breaking down function notation Student with disabilities support Break down the different parts of function notation to help students understand how to read it and relate it equations written in terms of y. Consider using different colors for the different parts, like the example shown. Name of function ( f of x)

f (x) = 2x −5 Input (domain)

Output (range)

Make students aware that f (x) = 2x − 5 is the same as writing y =2x − 5, but function notation simply highlights the input value. Then, provide an example that shows them how the notation highlights the input values.

f (x) = 2x −5 Evaluate for f (2)

Function notation is not multiplication Address student misconceptions Student may confuse function notation with multiplication when written with parentheses. For example, if f (x) = x, they may think f (2) = 2x. This may be further compounded by the wording of “f of x” as students may recall that “of” often signified multiplication. Emphasize that going from f (x) to f (2) we have replaced the x with 2, so we are substituting in x = 2.

Examples Students: Page 83 Example 1 Consider the function

where x is the independent variable. a Construct a table of values for the function at x = −3, 0, 9, 12, 27.

Create a strategy In order to construct a table of values, we will need to evaluate the function at the given values of x.

Apply the idea Substituting x = −3, we have:

Substituting x = 0, we have:

Substituting x = 9, we have:

Substituting x = 12, we have:

Substituting x = 27, we have:

168

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co A completed table for the function at the given values of x is:

Consider the function Substituting x = 9, we have:

Substituting x = 12, we have:

where x is the independent variable. a Construct a table of values for the function at x = −3, 0, 9, 12, 27.

Create a strategy

Substituting x = 27, we have: In order to construct a table of values, we will need to evaluate the function at the given values of x.

Apply the idea A completedxtable function at the given values of x is: Substituting = −3,for wethe have: Substituting x = 0, we have: x f (x)

−3 −6

0 −5

9 −2

12 −1

27 4

Substituting x = 9, we have:

Substituting x = 12, we have:

b Evaluate the function for f (2).

Purpose the idea that they can complete a table by evaluating a function at different input values. StudentsApply demonstrate Substituting x = 2, we have: Substituting x = 27, we have: Reﬂecting with students Point out to students that patterns might exist within the function. Encourage students to use Computational Thinking with Pattern Analysis to see if they can predict the next few values based on the patterns they observe. They may need to build out tableatusing consistent For example, as each input value increases by 3, A completed table for thethe function the given values of values. x is: we see the output values increasing by 1. x

f (x)

−3 −6

Students: Page 83

0 −5

9 −2

12 −1

27 4

b Evaluate the function for f (2).

Apply the idea Substituting x = 2, we have:

2.03 Evaluating functions mathspace.co

Purpose Students demonstrate that they can evaluate a function expression for a given function. Reflecting with students Challenge advanced learners to convert the values they found in parts (a) and (b) to coordinate pairs, and plot the points on a coordinate plane. Then, have them connect the points to graph the function. Additionally, ask the students to explain why this is a function to connect to the previous lessons. 2.03 Evaluating functions mathspace.co

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Students: Page 84 Example 2 Consider the graph:

5 4 3 2 1

a Evaluate the function for f (−1).

Example 2

a Evaluate the function for f (−1).

We can find the value of the function on the graph when x = −1 by moving down the y-axis until we intersect the curve.

Create a strategy We can find the value of the function on the graph when x = −1 by moving down the y-axis until we intersect the curve.

−3 −2 −1 1 −1 −2 f (x) −3 5 −4 4 −5 3 2 1

Consider the graph:

Create a strategy

f (x)

−3 −2 −1 1 2 3 4 5 −1 From x = −1, move down until we intersect the curve. −2 −3 f (x) −4 5 −5 4 3 2 1 Apply the idea x

Apply the idea

From x = −1, move −3 −2down −1 until1 we2intersect 3 4 the 5 curve. −1 −2 f (x) 5 −3 4 −4 3 −5 2 1

When x = −1, −3 we −2 can −1 see on the graph 1 2 3 that 4 f5(x) = −2. −1

b Determine the value of x when f (x) = 4.

−2 −3

−4 Purpose −5 Create a strategy Apply the idea Check students’ ability to evaluate a function from a given x-value using a graph.

To find the value of x on the graph where f (x) = 4, move horizontally to the left from f (x) = 4 until you intersect the Students:curve, Page 84 then move vertically downward until you reach the x-axis. b Determine the value of x when f (x) = 4.

Create a strategy To find the value of x on the graph where f (x) = 4, move horizontally to the left from f (x) = 4 until you intersect the curve, then move vertically downward until you reach the x-axis.

f (x)

When x = −1, we can see on the graph that f (x) = −2. 4 3 2 1

Apply the idea −3 −2 −1

x 1

−1 −2 f (x) 5 −3 4 −4 3 −5 2 1

When f (x) = 4,−3we−2can−1see on1 the2 graph 3 4that5 x = −3. −1

Mathspace Virginia SOL Algebra 1 mathspace.co

−2 −3 −4 −5

When f (x) = 4, we can see on the graph that x = −3.

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Purpose Check that students’ can find an input from a graph when given an output. Expected mistakes Students may forget that f (x) = y, and therefore not realize that f (x) = 4 means y = 4. Point out to students that the question asks us to find the x-value, meaning the given value must be the y-value.

Students: Page 85 Example 3 Consider the function f (x) = 3x − 5 to answer the following: a Find the range when the domain is {−3, 0, 11}.

Create a strategy We need to find the range (the values of f (x)) when the domain (the x-values) are {−3, 0, 11}. Substitute x-values and solve for f (x).

Apply the idea Substitute x = −3 for x in the function. f (x) = 3x − 5

Original function

f (−3) = 3(−3) − 5

Substitute x = −3

= −9 − 5

Evaluate the multiplication

= −14

Evaluate the subtraction

Substitute x = 0 for x in the function. f (x) = 3x − 5

Original function

f (0) = 3(0) − 5

Substitute x = 0

=0−5

Evaluate the multiplication

= −5

Evaluate the subtraction

Substitute x = 11 for x in the function. f (x) = 3x − 5

Original function

f (11) = 3(11) − 5

Substitute x = 11

= 33 − 5

Evaluate the multiplication

= 28

Evaluate the subtraction

When the domain is {−3, 0, 11}, this function has a range of {−14, −5, 28},

Reflect and check This could be written in function notation as {f (−3), f (0), f (11)} = {−14, −5, 28}. b Find the domain when the range is {−2, 4, 7}.

Purpose a strategy StudentsCreate demonstrate that they can find the range given domain values.

We need to find the domain (the x-values) when the range (the values of f (x) is {−2, 4, 7}. Substitute the f (x) values

andmistakes solve for x. Expected Students may be confused by the set notation and not realize that the set is a list of x-values. Ask students Apply the idea which variable denotes the domain. Remind them that the curly brackets represent a set of values, and this set Replace f (x) with −2 in the equation and solve for x. includes domain values (or x-values). f (x) = 3x − 5

Original function

−2 = 3x − 5

Substitute f (x) = −2

3 = 3x

Add 5 to both sides

1=x

Divide both sides by 3

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171

= 33 − 5

Evaluate the multiplication

= 28

Evaluate the subtraction

When the domain is {−3, 0, 11}, this function has a range of {−14, −5, 28},

Reflect and check

Students:This Pages 85–86 could be written in function notation as {f (−3), f (0), f (11)} = {−14, −5, 28}. b Find the domain when the range is {−2, 4, 7}.

Create a strategy We need to find the domain (the x-values) when the range (the values of f (x) is {−2, 4, 7}. Substitute the f (x) values and solve for x.

Apply the idea Replace f (x) with −2 in the equation and solve for x. f (x) = 3x − 5

Original function

−2 = 3x − 5

Substitute f (x) = −2

3 = 3x

Add 5 to both sides

1=x

Divide both sides by 3

Replace f (x) with 4 in the equation and solve for x. f (x) = 3x − 5

Original function

4 = 3x − 5

Substitute f (x) = 4

9 = 3x

Add 5 to both sides

3=x

Divide both sides by 3

Replace f (x) with 7 in the equation and solve for x. f (x) = 3x − 5 Original function Replace f (x) with 4 in the equation and solve for x. 7 = 3x − 5 Substitute f (x) = 7 f (x) = 3x − 5 Original function 12 = 3x Add 5 to both sides 4 = 3x − 5 Substitute f (x) = 4 4=x Divide both sides by 3 9 = 3x Add 5 to both sides When this function has a range of {−2, 4, 7}, the domain is {1, 3, 4}. 3=x Divide both sides by 3 Replace f (x) with 7 in the equation and solve for x. Reflect and check

f (x) = 3x − 5 function notation Original We can write this using asfunction {f (−2), f (4), f (7)} = {1, 3, 4}. 7 = 3x − 5 Substitute f (x) = 7 12 = 3x c Evaluate f (7) − f (2) 4=x

Add 5 to both sides Divide both sides by 3

PurposeWhen function has a range of {−2, 4, 7}, the domain is {1, 3, 4}. Createthis a strategy StudentsFind demonstrate they can find the domain range the values of that f (7) and f (2), then subtract the value given of f (2) from thevalues. value of f (7). Reflect and check

canthe write this using function notation as {f (−2), f (4), f (7)} = {1, 3, 4}. Students:We Page 86 Apply idea First, let’s evaluate f (7). c Evaluate f (7) − f−(2)5 f (x) = 3x

Original function

f (7) = 3(7) − 5

Substitute x = 7

Create a strategy = 21 − 5

Evaluate the multiplication Find the values= of subtractthe thesubtraction value of f (2) from the value of f (7). 16f (7) and f (2), then Evaluate Now evaluate f (2). Apply the idea f (x) = 3x f−(7). 5 First, let’s evaluate ff (2) = 3(2) − (x) = 3x − 55 = 3(7) 6 − 5− 5 f (7) = = 1 −5 = 21

Original function Substitute x=2 Original function Evaluate the Substitute x =multiplication 7

Evaluate the the multiplication subtraction Evaluate Now we can find f (7) − f (2) by substituting their values. = 16 Evaluate the subtraction f (7) − f (2)

Now evaluate f (2). f (x) = 3x − 5

Original function

f (2) = 3(2) − 5

Substitute x = 2

= 16 − 1 = 15

=6−5

Evaluate the multiplication

Evaluate the subtraction

Now we can find f (7) − f (2) by substituting their values. f (7) − f (2) = 16 − 1

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= 15

2.03 Evaluating functions mathspace.co

Now evaluate f (2). f (x) = 3x − 5

Original function

f (2) = 3(2) − 5

Substitute x = 2

=6−5

Evaluate the multiplication

Evaluate the subtraction

Now we can find f (7) − f (2) by substituting their values. f (7) − f (2) = 16 − 1 = 15

Purpose Students demonstrate that they can evaluate and subtract functions. Expected mistakes Students might think they can subtract the terms immediately because they look alike, thus getting the expression f (5). Ask them to evaluate f (5) so they can see that f (7) − f (2) ≠ f (5).

Students: Page 87 86

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Example 4 Let f (x) represent the height of a growing plant, f, in inches, where x represents the time since it was planted in days.

Plant Growth 20 18 16 14 12 10 8 6 4 2

a Interpret the meaning of f (10) = 8.

Height (inches)

f (x)

Time (days) 2 4 6 8 10 12 14 16 18 20

Create a strategy We can use the units of the given information and the graph to help with the interpretation.

Apply the idea We’re given that f (x) represents the height of a growing plant in inches, so to interpret f (10), we need to determine what an input of x = 10 means. We know that x represents the time in days since the plant was planted. So this means that 10 days have passed since the plant was planted. We also know that all of this is equal to 8. This is the output, or what our function f (x) is equal to. Since our function represents the height of a growing plant in inches, this means that our plant is 8 inches tall. Based on the graph, when x = 10, y = 8 so f (10) = 8 is represented by the ordered pair (10, 8) on the graph. The plant has a height of 8 inches 10 days after being planted. b Interpret the meaning of f (6).

Reflect and check PurposeApply the idea We knowhow that xto represents in daysvalue since in thethe context Using the we can find the actual height of the plant Show students interpretthea time function of graph, a real-world scenario. plant was planted and x = 6. So this means that 6 days

after 6 days.

have passed since the plant was planted. Expected mistakes Plant Growth represents the height a growing plantinterpreting in StudentsSince mayf (x) mix up the input andofoutput when the function notation. inches, f (6) represents the height of the plant 6 days after being planted.

20 18 16 14 12 10 8 6 4 2

Height (inches)

f (x)

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Time (days) 2 4 6 8 10 12 14 16 18 20

173

what an input of x = 10 means. We know that x represents the time in days since the plant was planted. So this means that 10 days have passed since the plant was planted. We also know that all of this is equal to 8. This is the output, or what our function f (x) is equal to. Since our function represents the height of a growing plant in inches, this means that our plant is 8 inches tall. Based on the graph, when x = 10, y = 8 so f (10) = 8 is represented by the ordered pair (10, 8) on the graph.

Students: Page 87

The plant has a height of 8 inches 10 days after being planted. b Interpret the meaning of f (6).

Apply the idea

Reflect and check

We know that x represents the time in days since the plant was planted and x = 6. So this means that 6 days have passed since the plant was planted.

Using the graph, we can find the actual height of the plant after 6 days. Plant Growth

Since f (x) represents the height of a growing plant in inches, f (6) represents the height of the plant 6 days after being planted.

20 18 16 14 12 10 8 6 4 2

Height (inches)

f (x)

Time (days) 2 4 6 8 10 12 14 16 18 20

Purpose 2.03 Evaluating functions Students demonstrate that they can interpret the meaning of a function expression.

mathspace.co

Reflecting with students Point out to students that f (x) is equivalent to y on the graph. So f (6) can be interpreted as the value on the y-axis when x = 6.

Students: Page 88 c Interpret the meaning of f (x) = 12.

Apply the idea We know that f (x) represents the height of a growing plant in inches, so if f (x) = 12, then the height of the plant is 12 inches x days after being planted.

Reflect and check By using the graph, we can find the number of days when the height of the plant is 12 inches. Plant Growth 20 18 16 14 12 10 8 6 4 2

Height (inches)

f (x)

Time (days) 2 4 6 8 10 12 14 16 18 20

Idea summary

Purpose An equation where the output variable is isolated like y = mx + b can be written as a function in the form, Students demonstrate that they can interpret the meaning of a function expression and corresponding value. f (x) = mx + b. We evaluate a function, written in function notation as f (c), by replacing all values of x with c and evaluating the expression.

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f (x) = y f is the name of function x is the input y is the output

8 6 4 2

Time (days) 2 4 6 8 10 12 14 16 18 20

Students: Page 88

Idea summary An equation where the output variable is isolated like y = mx + b can be written as a function in the form, f (x) = mx + b. We evaluate a function, written in function notation as f (c), by replacing all values of x with c and evaluating the expression.

f (x) = y f is the name of function x is the input y is the output

Practice What do you remember?

Practice 1

Evaluate each expression for the following values of x:

i x=2 Students: Pages 88–92 a

x = −3

4x + 1

6x − 2

x − 3x − 2 What do youe remember? 2

SOL

(x + 1) (x + 2)

For the function notation y = f (x), identify:

Evaluate each expression for the following values of x: b Dependent variable a Independent variable d Output i x = 2 c Input ii x = −3 a

4x + 1

x2 − 3x − 2

6x − 2

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(x + 1) (x + 2)

For the function notation y = f (x), identify: a

Independent variable

Dependent variable

Input

Output

Suppose that (7, −6) is an ordered pair that satisﬁes the function g. Write this situation using function notation.

For the function

, where x is the independent variable:

Construct a table of values for the function at x = −6, −3, 0, 18, 24, 72.

Evaluate the function for x = −27.

What is the value of this expression when a = 27, b = 25, and c = −7?

A SOL

−12

154

−55

−125

What is the value of this expression when z = −18? −5z + 7 A

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175

Let’s practice 7

For each function, find the value of f (x). a

If f (x) = −6x + 4, find: i

f (0)

f (2)

f (−5)

g(−4)

f (−5)

, find: f (4)

f (−4)

Evaluate the functions for each given value. a

If f (x) = 2x2 − 2x + 5, evaluate

If p(x) = x2 + 8, evaluate p(20).

If f (x) = x2 + 8x, evaluate f (−11).

y − 4x2 = 5 + x

Consider each of the equation: i

Rewrite the equation using function notation f (x).

Find the value of f (3).

x + 3y = 6

−6x + 5y = 7

y + 6x2 = 3 − x

For each graphed function, find the value of f (x) for each value of x. a

f (1)

f (3)

7 6 5 4 3 2 1

5 4 3 2 1 −2 −1

x 1

−1

f (−1)

y 4 3 2 1 −3 −2

−1

x 1

−3 −2 −1 −1 −2 −3

−2

−2 −3 −4

176

g(5)

If i

f (3)

, find:

If i

If f (x) = 3x − 1, find: i

If f (x) = 4x + 4, find: i

f (4)

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

f (0) 5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

For each function, find the value of x for each value of g(x). a

g(x) = 3x − 8, g(x) = 7

For each graphed function, find the value of x for each value of f (x). a

f (x) = −5

y 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

f (x) = 6

1 2 3 4 5

f (x) = −4

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

−5 −4 −3 −2 −1 −1 −2 −3

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

9 8 7 6 5 4 3 2 1

5 4 3 2 1

1 2 3 4 5

Given the graph of p(x), find each input of the corresponding outputs. 100

p(x)

80 60 40 20 −20

x 1

p(x) = 110

−40 −60 −80 −100

p(x) = −100

p(x) = −60

For the function f (x) = 5x2 − 2x + 6, find the range given the domain is {−5, 0, 4}.

For the function f (x) = −4x − 4, find the domain when the range is {−8, 16, 24}.

p(x) = 50

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Sharon works at a pizza shop and makes $15 per hour. Her wages can be represented by the function W(t) = 15t, where t represents the number of hours that she works. a b

Explain what the expression W(19) represents. Find her wages after working 26 hours.

Mindy can model her tomato plant’s height in inches over time using the function h(t) = 2t, where t is the number of weeks after Mindy planted it. a

Explain what the equation h(4) = 16 represents.

Find the height of Mindy’s tomato plant after 6 weeks.

Let’s extend our thinking 18

You are designing a square garden with a quadratic function representing its area, A(x) = x2, where x is the length of one side. a

What does A(x) = 36 mean?

If the area of the garden is 36 square meters, find the length of each side.

What is the significance of x in A(x) = 36?

Consider the function f (x) = x2 − 17x + 4. a

Form an expression for f (b).

Simplify the expression for f (2).

If f (2) = 15, find the value of k.

Consider the functions f (x) = 7x + 2 and g(x) = 15 − 2x. Find the value of the following: a

Consider the function f (x) = x2 + 2x + k. a

Form an expression for f (a).

f (1) + g(2)

f (−2) − g(7)

Han is painting a mural which requires

f (0) ⋅ g(0)

gallons of paint per square foot. Han also needs 3 additional gallons

of paint to go over the outline of the mural once they are ﬁnished.

178

Express the relationship of the scenario using function notation, letting x represent the square footage of the mural.

Find the number of gallons of paint that Han will need if the mural has a size of 24 square feet.

A function describing the strictly increasing relation between temperature and a person’s average resting heart rate has a domain of 50 ≤ T ≤ 105 (Fahrenheit) and a range of 40 ≤ f (T ) ≤ 90 (beats per minute), based on experimental results. a

Determine the average resting heart rate at a temperature of 50° F.

State the temperature that is expected if a person has an average resting heart rate of 90 beats per minute.

Explain whether or not the function output f (10) be a reliable estimate for the corresponding real life scenario.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Answers 2.03 Evaluating functions What do you remember? 1 a i f (2) = 9

ii f (−3) = −11

b i f (2) = 10

ii f (−3) = −20

c i f (2) =

ii f (−3) = 0

d i g (2) = 12

ii g (−3) = 2

e i h(2) = −4

ii h(−3) = 16

i k(2) = 13

ii k(−3) = −12

2 a x

b y

–6

f (x)

–4

–3

b 7

c 2

d −4

11 a x = 5

b x = −64

d x = −12

12 a x = −2

b x=5

c x = −4, 4

d x = −3

13 a x = 18.5

b 10 ≤ x ≤ 12, x = 17.75, 19

c x = 14

d x = 2, 6, 7.5, 13.5, 15.5

14 Range: {141, 6, 78} 15 Domain: {1, −5, −7}

c x

16 a W(19) represents the amount of money Sharon makes if she works 19 hours. d y

3 g(7) = −6 4 a

10 a 2

b W(26) = $390 17 a The equation h(4) = 16 represents that the tomato plant will be 16 inches tall after 4 weeks. b 64 inches Let’s extend our thinking 18 a A (x) = 36 represents the area at side length of x meters.

b x = 6 meters

5 B

c In A(36), x represents the side length if the area is 36 square meters.

6 C Let’s practice

19 a a2 − 17a + 4

b b2 − 17b + 4 b k=7

7 a i −20

ii 4

20 a f (2) = 8 + k

b i 12

ii −16

21 a 20

c i 8

ii −13

d i

ii −7

e i 1

8 a

b 33

c 408 ii 1

9 a i

ii 5

b i 2

c i f (x) = 4x + x +5

ii 44

d i f (x) = 3 − x − 6x2

ii −54

b −13

c 30

d 10

22 a b f (24) = 7, so Han will need 7 gallons of paint. 23 a 40 beats per minute

b 105 °F

c T he function output f (10) is the average resting heart rate of a person at 10 °F according to the model. However, since T = 10 falls well outside of the domain of the function, we do not have enough information to consider this a reliable estimate.

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2.04 Characteristics of functions Subtopic overview Lesson narrative In this lesson, students will learn the definitions of a variety of key characteristics and features of functions. Students will make sense of problems and analyze context to interpret key features, and reason abstractly and quantitatively to connect key features to the quantities they represent in order to create a graph to model a contextual situation. This lesson extends from the content in prior lessons where students were introduced to domain, range and average rate of change. By the end of the lesson, students should be confident in identifying, interpreting, and describing key features of functions represented in graphs and tables, as well as creating graphs to represent contextual situations.

Learning objective Students: Page 93

Key vocabulary 

characteristic (of a function)



maximum



minimum



range



x-intercept



y-intercept



zero (of a function)

Essential understanding The characteristics of a function provide information about the real-world situation it represents; making it easier to understand, interpret, and analyze.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving Teachers can integrate this goal by providing students with problems that require them to apply the concepts of x-intercepts, y-intercepts, maximum or minimum points, and domain and range. For example, a problem could involve a graph of a function, and students could be asked to identify these key characteristics. Alternatively, students could be given a real-world problem, such as a business scenario, and asked to identify these characteristics in the context of the problem.

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MPG3 — Mathematical Reasoning

MPG5 — Mathematical Representations

Teachers can integrate this goal by asking students to justify their answers and reasoning. For example, if a student identifies a point as a maximum or minimum point of a function, the teacher could ask the student to explain why they believe this is the case. This could involve discussing the shape of the graph, the values of the function at different points, and so on.

Teachers can integrate this goal by showing students how to represent functions and their key characteristics in multiple ways. For instance, teachers can demonstrate how to find the x intercepts, y intercepts, maximum or minimum points, and domain and range of functions from graphs, sets of ordered pairs, and tables. In addition, teachers can encourage students to use different representations and understand the connections among them. For example, teachers can ask students to represent a function using a table and then translate that into a graph.

Engage Activity Identifying characteristics of functions from a graph

60 mins

Students will choose a graph and a context from a set of choices. They will then write a ‘story’ which connects the key features of the graph to their chosen context. Then, students will share their completed ‘story graphs’ with the class and try to determine what the story was that other groups were trying to tell.

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Understanding and skills

Will use

Will develop

Identifying increasing, decreasing, and constant intervals from a graph or written description of a function.

Interpreting key features of a function in a given context. Sketching a graph from speciﬁc key features.

Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: None. • Download and print copies of the students graphic organizer from the student Launch slide.

Support students with disabilities Support visual-spatial - create and interpret visual representations Provide handouts of the graphs so that students can add labels directly on the sketch.

Support for English language learners Collect and display As pairs are working, listen for and collect vocabulary, phrases, and methods students use for describing the key features of their graph. Consider grouping language for both the context speciﬁc features as well as key graph features. Continue to update collected student language throughout the entire activity. Remind students to borrow language from the display as needed. Some terms and phrases may include: origin, x-intercept, y-intercept, minimum, maximum, rate of change, increasing/decreasing, positive, negative, and point of inﬂection.

Classroom guide Hook Students write observations about a graph depicting the height of an amusement park ride over time.

Notice and wonder

•

5 mins

What do you notice? What do you wonder?

Speed/Velocity (km/h)

Space Mountain Ride Profile 80 70 60 50 40 30 20 10 10

30 40 50 60 Ride Time (seconds)

Slide 1 from Student Engage Activity

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Implementation details The graph is designed in such a way that the context is labeled. This encourages students to make a connection between the context provided and the key features of the graph. For example, students might notice that the interval between t = 10 and t = 20 is a gap in the function, and wonder what is happening to the ride during that time. Students might also notice the maximum height of the rollercoaster is about 70 feet high, and the steepest intervals are from t = 20 to t = 30 and t = 60 to t = 70. Here are some questions to ask during the Launch: • What is the maximum of the graph? • How does this connect to the rollercoaster scenario? • What other features of the graph do you notice? • What are the intercepts of the function? • Are there any parts of the graph you have questions about?

Launch

5 mins

You and your partner will be using one of the provided graphs to tell a story. However, when shared with the class, they will only be able to use your story to sketch the graph. Read the information carefully. Your teacher will tell you when to move on

Time Time

Time

Slide 2 from Student Engage Activity

Ask your students, “What if the graph did not have a label? Would this change anything you noticed or wondered in the hook?” Explain that today’s goal is about telling a story about a graph through identifying and interpreting key features in a context. When telling your group’s story about the graph, it is important to be speciﬁc so classmates can eventually try to replicate the graph from the story. Important mathematical concepts: Origin, x-intercept, y-intercept, minimum, maximum, rate of change, increasing/ decreasing, positive, negative, point of inﬂection Suggested grouping: Form pairs

Continue when Students have read the Launch slide and understand the context of the problem.

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Explore

Think-pair-share

•

25 mins

Anticipated strategies Determining units and deciding on reasonable scaling and values Students will need to reason about what kind of units to use (i.e for time: seconds, minutes, hours, days) and how to appropriately scale each axis. The values that students choose should be reasonable to the context. For example, if students select ‘karaoke’ they may choose the x-axis to be time in minutes or seconds, since these are reasonable units to measure the length of a song. In this instance, a student might have the maximum value on the x-axis be 8 minutes, since most songs reasonably fall within that length. For a sky diving scenario, it would be reasonable for the x-axis to be time in seconds and the y-axis to be height off the ground in feet or meters, with the maximum value on the axis being in the thousands of feet and scaling appropriately.

Connecting key features with context Students will look for ways to connect the features of their graph to the context which they’ve selected. This could mean beginning with a speciﬁc feature (i.e intercept) and determining what must be true in the situation for this feature to hold a certain value. Students should have descriptions of at least three key features; such as domain, range, intercepts, and rate of change from a graph, and how the key features relate to their context. The story should have enough information for other groups to sketch a graph from the description.

Misconceptions Labels and axes of graph that do not align with story What units are used in the story so far? What are the units and labels on the axes of the graph? How are these connected?

Describing key feature incorrectly in context What does a positive or negative output mean in your context? What does the average rate of change mean in your context?

Purposeful questions Use the following questions to check for understanding and encourage critical thinking: • What are the key features you are including in your story? • Why did you choose those values? • Would someone be able to re-create this graph from your story? Why or why not? • How did you interpret these key features when trying to write your story?

Continue when Students have a story and a graph. Each group’s graph should have the following: a title, appropriate scaling and labels on each axis, and descriptions of at least three key features of the graph and how they relate to the context.

Discuss

25 mins

Have a group discussion for students to share there stories. Consider making connections from the discussion to graphs in general being representative of any type of story.

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Discussion guide Ask groups to read their story for the class. For each group, the class should work together to sketch a graph that matches the story. Encourage student discussion here. The goal is to create a sketch as close as possible to the group’s graph (including scaled axes, labels, and title). The group that is displaying or reading their story should not correct their classmates and should reveal their graphs after the class agrees on their guess for a sketch of the graph. This can turn into a fun competition and can motivate students to make their stories as descriptive as possible for key features of their graphs. Alternatively, groups could swap stories and try to recreate another group’s story with their partner. This could be completed as a class or in partners.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 8 — 3.05 Characteristics of linear functions Algebra 1 — 2.02 Domain and range Algebra 1 — 2.03 Evaluating functions

Student lesson & teacher guide Characteristics of functions Students learn about the various key features and characteristics of a function and how to express them.

Students: Pages 93–94

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9 8 7 6 5 4 3 2 1 −9−8−7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7 −8 −9

1 2 3 4 5 6 7 8 9

• Domain: {x∣ − 3 ≤ x ≤ 4} • Range: {y∣ − 6 ≤ y ≤ 8} • x-intercept: (1, 0) • Zeros: (1, 0) • y-intercept: (0, 2) • Maximum: (−3, 8) • Minimum: (4, −6)

Example 1

Using ‘Capturing Quantities’ to Interpret Key Features of Functions Consider the function shown in the graph: Targeted instructional strategies

14 12

To support students in connecting key features of functions to the quantities they 10 represent, use the ‘Capturing 8 Quantities’ strategy. 6

Begin by posing a real-world problem, such as tracking the height of a balloon as4 it is released into the air over 2 time. Encourage students to identify and list all relevant quantities like time and height, and discuss how these x quantities relate to each other. −2 −1 1 2 3 4 5 6 7 8 −2

Guide students to create diagrams or graphs representing the situation, focusing −4 on key features like domain, range, intercepts, and intervals where the function is increasing or decreasing. −6 Facilitate a class discussion where students share their diagrams and explain how each key feature a Identify whether the function has a maximum or minimum value and state this value. corresponds to aspects of the real-world scenario. Create a strategy 186

We need to find the lowest point on the graph and use the y-value to indicate how low it is. Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co Apply the idea 14

Collect and Display: Vocabulary exercise English language learner support As you discuss concepts like domain, range, intercepts, and increasing or decreasing functions, listen for the language students use to describe these ideas. For example, a student might say, “The graph starts at some point and keeps going forever,” which relates to the concept of domain. Another student might mention, “The highest point the graph reaches...” connecting to the maximum value of the function. Capture their words, phrases, and any associated symbols or diagrams on a visible chart or whiteboard. This evolving display serves as a collective reference that students can continually refer to, helping them connect new vocabulary to their existing understanding. • Domain:Encourage {x∣ − 3 ≤ x ≤ 4}students to use this shared language 9 during discussions and when working on problems. For example, them by saying, “When you mention 8 • Range: {y∣ − 6 ≤ y ≤prompt 8} 7 • x-intercept: (1, 0) ‘where the graph crosses the x-axis,’ remember we call those the x-intercepts or zeros.” 6 5

• Zeros: (1, 0)

As the lesson progresses, 4update the display with new insights • y-intercept: (0, 2)and refined definitions to deepen their 3 2 • Maximum: (−3, 8) students’ mathematical vocabulary but also comprehension and language skills. This strategy not only builds 1 • Minimum: (4, −6) validates their contributions, making them feel heard and involved in the learning process. −9−8−7−6−5−4−3−2−1 1 2 3 4 5 6 7 8 9

Examples Students: Page 94

−1 −2 −3 −4 −5 −6 −7 −8 −9

Example 1 Consider the function shown in the graph:

14 12

10 8 6 4 2 −2 −1 −2

x 1

2 3 4 5 6 7 8

−4 −6

a Identify whether the function has a maximum or minimum value and state this value.

Create a strategy We need to find the lowest point on the graph and use the y-value to indicate how low it is.

Apply the idea 14 12

10 8 6 4 2 −2 −1 −2 −4 −6

x 1

2 3 4 5 6 7 8

This function has a minimum value of −4.

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187

Purpose Show students how to determine whether a function has a minimum or maximum value, and then find that value. Expected mistakes Students may argue that the function also has a maximum value of ∞. Explain to students that ∞ is not a fixed value, so it cannot be a maximum value. Using the infinity symbol when describing domain and range is a simple way for us to explain that the parabola increases infinitely (rather than “to infinity”).

Students: Page 95 b State the range of the function.

Create a strategy

Apply the idea

In part (a), we identiﬁed that the function has a minimum value of −4, so we know that the function can’t take values smaller than −4.

Looking at the function, we can see that it stretches up towards inﬁnity on both sides of the minimum point, so the function can take any value greater than or equal to −4. That is, the range of the function is:

b State the range of the function.

Range: {y  y ≥ −4}

Create a strategy

Apply the idea

cIn part State ofthe thefunction function.has a minimum (a),the wex-intercept(s) identiﬁed that Looking at the function, we can see that it stretches up value of −4, so we know that the function can’t take towards inﬁnity on both sides of the minimum point, so PurposeCreate a strategy values smaller than −4. the function can take any value greater than or equal to Show students how to determine and express the range −4. of aThat function. The x-intercept(s) of a function are the points where the function crosses x-axis. In this case, by is, thethe range of the function is: looking at the graph, we can see that there are two x-intercepts. Range: {y y ≥ −4}



Students: Page 95

Apply the idea c State the x-intercept(s) of the function.

Create a strategy

14 12

10 The x-intercept(s) of a function are the points where the function crosses the x-axis. In this case, by looking at the 8 graph, we can see that there are two x-intercepts. 6 4

Apply the idea

2 −2 −114 −2 12 −4 10 −6 8 6

x 1

2 3 4 5 6 7 8

The x-intercepts of this function are the points (1,4 0) and (5, 0). 2

−2 −1 −2 −4 −6

Example 2

x 1

2 3 4 5 6 7 8

A penguin is tagged with a tracker to record its height above sea level when hunting. The height of the penguin is graphed against of time. The x-intercepts this function are the points (1, 0) and (5, 0). h(ft) 50 40 30 Example 2 20 Purpose 10 Show students how to identify the x-intercepts of a function from a graph.

t(mins) A penguin is tagged with a tracker to record its height above sea level when hunting. The height of the penguin is 5 10 15 20 25 graphed against time. −10 −20 h(ft) −30 50 −40 40 −50 30 20 10

188

t(mins)

Use the key features of the graph to describe the penguin’s time spent hunting. Be as detailed as possible. 5 10 15 20 25 Mathspace Virginia SOL −10 Algebra 1 Teacher Edition mathspace.co −20 2.04 Characteristics of functions −30 −40 −50

mathspace.co

b State the range of the function.

Create a strategy

Apply the idea

In part (a), we identiﬁed that the function has a minimum value of −4, so we know that the function can’t take values smaller than −4.

Looking at the function, we can see that it stretches up towards inﬁnity on both sides of the minimum point, so the function can take any value greater than or equal to Include scaffolding for identifying key points use with Example 1 −4. That is, the range of the function is:

Student with disabilities support

Range: {y  y ≥ −4}

For students who face challenges identifying key points, scaffold by labeling key points with letters and asking questions like, “What key feature is represented by the points A and C ?” or “Is point B included or excluded c State the x-intercept(s) of the function. from the range?” Create a strategy

The x-intercept(s) of a function are the points 14 where the function crosses the x-axis. In this case, by looking at the 12 graph, we can see that there are two x-intercepts.

10 8 6 y 414 212

Apply the idea

x A C 10 −2 −1 8 1 2 3 4 5 6 7 8 −2 6 −4 4 B −6 2

x −2 −1 1 2 3 4 5 6 7 8 −2 Once students are comfortable with this, remove the scaffolding and ask for general key features like “identify −4 the x-intercepts” or “write the range in interval −6 notation.”

The x-intercepts of this function are the points (1, 0) and (5, 0).

Students: Pages 95–96

Example 2 A penguin is tagged with a tracker to record its height above sea level when hunting. The height of the penguin is graphed against time. 50 40 30 20 10 −10 −20 −30 −40 −50

h(ft)

t(mins) 5

Use the key features of the graph to describe the penguin’s time spent hunting. Be as detailed as possible.

Create a strategy

2.04 Characteristics of functions mathspace.co

We can see that the graph has key features like intercepts, a minimum point, a domain, and a range.

To interpret the graph in context, we can use the axes of the graph to match key features to their real-world meaning.

Apply the idea The y-intercept is (0, 30), meaning that the penguin is 30 ft above sea level at 0 minutes into its hunting time. The x-intercepts are (5, 0) and (20, 0), meaning that the penguin is exactly at sea level at 5 and 20 minutes into its hunting time. The minimum of the graph is approximately (12.5, −41), so the penguin’s lowest point is about 41 ft below sea level at about 12.5 minutes into its hunting time. If we combine this information, we can make a description of the penguin’s hunting time. For example: When the penguin needs to hunt, it leaves its nest, which is 30 ft above sea level. The penguin makes its way down to the water and dives into the water 5 minutes after leaving the nest. The penguin swims down to a depth of around 41 ft below sea level, reaching its deepest point around 12.5 minutes into its hunting time before returning to the water’s surface at 20 minutes. The penguin spends 15 minutes underwater in total. The penguin then spends the last 5 minutes of its hunting time climbing back up to its nest, ﬁnishing a bit higher than where2.04 it started. Characteristics of functions mathspace.co

Reﬂect and check We only need to make sure that the description matches the key features of the graph, so there are many possible

189

Create a strategy We can see that the graph has key features like intercepts, a minimum point, a domain, and a range. To interpret the graph in context, we can use the axes of the graph to match key features to their real-world meaning.

Reﬂect and check We only need to make sure that the description matches the key features of the graph, so there are many possible examples. For example, it is completely valid to say that the penguin dives into the water using a submarine as long as the deepest point is still 41 ft below sea level.

PurposeExample 3 Show students how to identify key features on a graph and relate them to a real-world context. A hiker’s elevation over a given period of time is graphed:

Elevation

a What the zeros of the function represent in this situation? Reflecting withcould students Remind students that we only need to make sure that the description matches the key features of the graph, so there are many possible examples. For example, it is completely valid to say that the penguin dives into the water using a submarine as long as the deepest point is still 41 ft below sea level.

Misinterpreting Increasing and Decreasing Intervals

Time

use with Example 2

Address student misconceptions Create a strategy increasing and decreasing intervals on a function’s graph by assuming that any Students may misinterpret The zeros of aon function represent where f (x) 0. upward movement the graph indicates an=increasing interval, without considering the progression along the x-axis. Apply the idea

To address this misconception, remind students that wetime. read graphs from left to right, just like reading text. The y-axis represents elevation and the x-axis represents Emphasize that on an function increasing interval, as the x-values increase, the y-values The zeros of this represent the times where the hiker reached ground level. also increase; on a decreasing interval, as the x-values increase, the y-values decrease. Use visual aids by providing graphs with clear examples of increasing and decreasing intervals. For instance, show a graph of a function with both rising and falling sections, and work with students to identify and label these intervals. Encourage students to trace the graph with a finger or pointer from left to right, noting how the y-values change in relation to the x-values. 96

190

Mathspace Virginia SOL Algebra 1 mathspace.co

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Students: Page 96 Example 3

A hiker’s elevation over a given period of time is graphed:

Elevation

a What could the zeros of the function represent in this situation?

Time

Create a strategy The zeros of a function represent where f (x) = 0.

Apply the idea The y-axis represents elevation and the x-axis represents time. The zeros of this function represent the times where the hiker reached ground level.

Purpose Check if students understand the zeros of a function in a real-world context. Mathspace Virginia SOL Algebra 1 Students:96Page 97 mathspace.co

b Write a description of the meaning of the maximum of the function in relation to the hiker’s elevation over time.

Create a strategy

Apply the idea

The maximum is the highest point of the function.

The function relates the elevation of the hiker over a period of time. The maximum represents the time when the hiker reached their highest elevation on their journey.

c Would the domain or range tell us how long the hiker traveled?

Purpose a strategy Apply the idea Check ifbCreate students understand the maximum a function a real-world context. Write a description of the meaning of theof maximum of theinfunction in relation to the hiker’s elevation over time. The domain represents the inputs, or x-values of a

The x-axis is labeled as time.

function, range represents the y-values. The domain represents how long the hiker traveled. Reflecting withawhile students Create strategy Apply the idea Have students thinkisabout the minimum this graph. Ask students what the negativeofy-axis would The maximum the highest point of theof function. The function relates the elevation the hiker over arepresent in context if the graph were extended. period of time. The maximum represents the time when the hiker Idea summary Students: Page 97 reached their highest elevation on their journey. The key features of a function and how to describe them are as follows: • We can write intercepts, zeros, minimums, and maximums as values or ordered pairs. • Domain and range: write in inequality, set notation, and interval notation. c Would the domain or range tell us how long the hiker traveled?

Create a strategy

Apply the idea

The domain represents the inputs, or x-values of a function, while range represents the y-values.

The x-axis is labeled as time.

Practice

The domain represents how long the hiker traveled.

What do you remember? 1

State deﬁnition for the following function characteristics: Ideathesummary

aTheMaximum b and Minimum key features of a function how to describe themc arex-intercept as follows: 2 3

y-intercept

Describe thewrite connection between and x-intercepts. • We can intercepts, zeros,zeros minimums, and maximums as values or ordered pairs. 2.04 Characteristics of functions • Domain and range: write in inequality, set notation, and interval notation. For each graph of the function y = f (x): mathspace.co i Identify whether the function has a maximum or minimum and state its value. ii

State the range of the function.

iii

State the domain of the function

191

b Write a description of the meaning of the maximum of the function in relation to the hiker’s elevation over time.

Create a strategy

Apply the idea

The maximum is the highest point of the function.

The function relates the elevation of the hiker over a

period of time. Purpose The maximum represents the time when the hiker Check if students understand domain and range in a real-world context. reached their highest elevation on their journey.

Reflecting with students Challenge advanced learners to think of an appropriate unit for the x-axis. Since it represents time, what could c Would the domain or range tell us how long the hiker traveled? be an appropriate unit for hiking time? Once they decide on a unit, encourage them to label the x-axis using a reasonable scale for the units and context. Create a strategy Apply the idea The domain represents the inputs, or x-values of a

Students:function, Pagewhile 97 range represents the y-values.

The x-axis is labeled as time. The domain represents how long the hiker traveled.

Idea summary The key features of a function and how to describe them are as follows: • •

We can write intercepts, zeros, minimums, and maximums as values or ordered pairs. Domain and range: write in inequality, set notation, and interval notation.

Practice What do you remember?

Practice 1

State the deﬁnition for the following function characteristics:

a Maximum Students: Pages 97–101 2

Minimum

x-intercept

y-intercept

Describe the connection between zeros and x-intercepts.

3 For each graph of the function y = f (x): What do you remember?

Identify whether the function has a maximum or minimum and state its value.

State the range of the function.

iii

Maximum

Minimum

Describe the connection8 between zeros and x-intercepts. 6

y-intercept

x 2

10 12

−4 Identify whether the2function has a maximum or minimum and state its value. x

State the range −6 −4 of −2the function. 2 4 6 −2

y x-intercept 4

−4 −2 −2

For each graph of the function y = f (x): 4 i

y −4 10

−6

State −6 the domain of the function −8

−10 −12

−4 −2 −2

4 2 −6 −4 −2 −2

iii b

192

State the domain of the function

State the deﬁnition for the following function characteristics: a

x 2

10 12

−4 2.04 Characteristics of functions −6 −8

−4

−10

−6

−12

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

x 2

mathspace.co

d 8

4 −10 −8 −6 −4 −2

x 2

−8 −6 −4 −2

−4

−2

−8

−4

−12

−6

For each function: i

State the coordinates of the x-intercept.

−5 −4 −3 −2 −1

State the coordinates of the y-intercept. 10

3 2

6 4

−1

x 1

−5 −4 −3 −2 −1 −2

2 3 4 5

−2

−4

−3 −4

−6 −8

−5

−10

x 1

2 3 4 5

For each table: i a

State the coordinates of the x-intercept(s).

x −2 −1 0 1 2 3 y 8 6 4 2 0 2 x 0 1 2 3 y 5 0 −3 −4

4 −3

5 0

State the coordinates of the y-intercept(s).

x y

−3 −4

−2 −1

−1 0

0 2

1 5

2 8

x y

−2 10

−1 5

0 0

1 5

2 10

3 15

2.04 Characteristics of functions mathspace.co

193

Let’s practice 6

For each graph, identify the number of: x-intercepts

Zeros

iii

y-intercepts

8 6

3 2

−5 −4 −3 −2 −1 −1

x 1

−8 −6 −4 −2

2 3 4 5

−4

−3

−6

−4

−8

d 3

3 2 1 −1

2 3 4 5

−3

−4

For each graph, describe the location of any maximum and minimum points. a

4 −6 −4 −2 −4

x 2

−8

194

−2

x 3

−5 −4 −3 −2 −1 −1

x 2

−2

−4 −3 −2 −1

6 4 2

−12

−4 −3 −2 −1 −2

−16

−4

−20

−6

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

x 1

For each graph, state the following: i

Domain

iii

x- and y-intercepts

Range

1 −2 −1

x 1

−1

−2

−3

−4

−5

−1

−6

−1

x 1

1 −4 −3 −2 −1

−3 −2 −1

x 1

−1

−1 −2

−2

−3

−4

Grover counts the number of people in his class, x, and the number of empty seats, y. He puts his results into a table: x y

22 0

20 2

17 5

13 9

11 11

8 14

3 19

0 22

Identify and describe the following features of the table and explain what each means in this context: a

x-intercept

y-intercept

Domain

Range

Let’s extend our thinking 10

The graph shows the height (in meters) of a soccer ball against the time (in seconds) that has passed after it has been kicked.

Height 12

Find the coordinates of the y-intercept.

Interpret the y-value of the y-intercept in this context.

Find the coordinates of the x-intercept.

Interpret the x-value of x-intercept in this context.

6 4 2 Time 1

2.04 Characteristics of functions mathspace.co

195

Two construction workers are competing to see who can lay the most bricks in one hour. The graph shows the number of bricks laid and the time, in minutes.

320

Bricks laid

280

Determine who can lay more bricks in 60 minutes.

240

Explain why the y-intercept of both lines is 0.

200

Charlie

160

Neville

120 80 40

Minutes 10 20 30 40 50 60 70 80

Immanuel records his speed as he runs around the neighborhood and graphs the results as speed, y, against time, x. 9

y (mph)

8 7 6 5 4 3 2 1

x (mins) 2

196

Identify the maximum point and describe what it means in the context of Immanuel’s run.

What do the domain and range tell us about Immanuel’s run?

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Answers

8 a i Domain: ii Range:

2.04 Characteristics of functions

iii x-intercept is 3 y-intercept is −3

What do you remember?

b i Domain: all real values

1 a The largest or highest y-value of a function.

ii Range: all real values

b The smallest or lowest y-value of a function.

iii x-intercept is 3

c T he value of x at the point where a line or graph intersects the x-axis. The value of y is 0 at this point.

y-intercept is 6

d T he value of y at the point where a line or graph intersects the y-axis. The value of x is 0 at this point.

ii Range: y = 2

2 Zeros and x-intercepts both represent the points of a function where f (x) = 0. 3 a i Minimum: 0

d i Domain: −2 ≤ x ≤ 4

ii Range: y ≤ 0

y-intercept is -1

ii Range: y ≤ 16

9 a The x-intercept is (22, 0). It tells us that when there are no empty seats the number of people in the class is 22.

ii Range: y ≥ −4

b The y-intercept is (0, 22). It tells us that when there are no people in the class the number of empty seats is 22.

iii Domain: x ∈  d i Minimum: −4

y-intercept is 2

ii Range: y ≥ 0

iii Domain: x ∈  c i Maximum: 16

iii There is no x-intercept

ii Range: −2 ≤ y ≤ 1

iii Domain: x ∈  b i Maximum: 0

c i Domain: all real values

c T he domain is from x = 0 to x = 22. It tells us that there are between 0 and 22 people in the class.

iii Domain: x ∈  4 a i (3, 0)

ii (0, −3)

b i (−1, 0)

ii (0, −2)

iii x-intercept is 2

d T he range is from y = 0 to y = 22. It tells us that there are between 0 and 22 empty seats.

5 a i x = 2, y = 0 Let’s extend our thinking

ii x = 0, y = 4 b i x = −1, y = 0

10 a (0, 9)

ii x = 0, y = 2

b T he height from which the soccer ball was kicked which is 9 m.

c i x = 1, y = 0 and x = 5, y = 0

c (3, 0)

ii x = 0, y = 5

d T he time at which the soccer ball hits the ground which is 3 seconds.

d i x = 0, y = 0 ii x = 0, y = 0

11 a Charlie

Let’s practice 6 a i 2

ii 2

iii 1

b i 1

ii 1

iii 1

c i 0

ii 0

iii 1

d i 1

ii 1

iii 1

7 a Minimum point: (4, −16)

b Maximum: (1, 4)

Minimum: (−1, 0)

b S ince the time taken for laying bricks only begins when the construction workers lay their first brick, both lines must pass through the point (0, 0) representing that they have laid 0 bricks after 0 minutes. 12 a T he maximum point is (2, 8). This means that at 2 minutes, Immanuel reached his top speed of 8 miles per hour. b T he domain, 0 < x < 18, tells us that Immanuel’s run went for 18 minutes. The range, 0 < y < 8, tells us that Immanuel reached a top speed of 8 miles per hour, and had a speed of zero at some points.

Answers mathspace.co

197

Topic 2 Assessment: Functions & Relations 1

SOL

Determine if each statement is always true, sometimes true, or never true. Justify your answer. a

A relations is a function.

A function is a relation.

The graph of a line is a function.

For a function f (x), there could be multiple outputs for f (−3).

Which of the following appears to be a function? A

1 −4 −3 −2 −1 −1

x 1

−4 −3 −2 −1 −1

−2

−3

−4

D 4

1 −4 −3 −2 −1 −1

x 1

−8 −6 −4 −2

x 2

−1

−2

−3

−4

For each relation: i

Find the y-value(s) when x = 1.

iii

Is it a function? Explain your answer.

x −2 −1 0 y −8 −1 0

1 1

2 8

Find the x-value(s) when y = 0.

b 5 4 3 2 1 −7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

198

−2

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

x 1 2 3

8 7 6 5 4 3 2 1

−4 −3 −2 −1 −1 −2

y −4

x 1

This set of ordered pairs represents a relation: {(−2, 3), (−1, −1), (0, 3), (1, 2), (1, 1), (2, −4)}

SOL

Plot the ordered pairs on a coordinate plane.

Which ordered pair would need to be removed from the set so that the remaining ordered pairs represent a function?

Which is NOT an element in the domain of this relation? {(−4, 7), (0, 6), (3, −1), (8, 0)} A

−1

−4

This set of points represents a relation between x and y. Find a value of k so that the relation does not represent a function. {(k, 5), (1, 3), (2, 6), (3, 5)}

For f (x) = 4 − 3x, evaluate each expression. Show your thinking. a

SOL

f (−5)

f (0)

If f (x) = (x − 4)2 − 2, what is f (9)? A

f (1) −9

For the graph of g(x), what is the value of: a

g(−4)

g(−2)

g(0)

g(4)

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

The value of a vintage car in dollars can be modeled by the function v(t) = 9t2 − 225t + 16000, where t is the number of years from year 2020. Find and interpret the value of v (5).

Topic 2 Assessment: Functions & Relations mathspace.co

199

For each of the graph: i

State the domain of the relation using inequality notation.

State the range of the relation using inequality notation.

2 −8 −6 −4 −2 −2

−6

−4

−8

−5

B 4

1 −4 −3 −2 −1 −1

−2

−3

−4

2 x 1

−4 −3 −2 −1 −1

−2

−4 −3 −2 −1 −1

1 4

−2

200

−3

Which graph has exactly one x-intercept and one y-intercept? A

SOL

−2

−4

SOL

−3 −2 −1 −1

x 2

1 −4 −3 −2 −1 −1

−3

−3 −4

All real numbers

All real numbers between −12 and 12

All real numbers less than or equal to 8.

All real numbers greater than or equal to 8.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

−2

−4

The graph shows a relation. Which best describes the range of this relation?

10 8 6 4 2 −10−8 −6 −4 −2 −2 −4 −6 −8 −10

x 2 4 6 8 10

Consider the functions shown on the graphs: i

State the coordinates of the x- and y-intercepts.

Determine if there is a minimum, maximum, or neither.

b 4

x −10

−5

−3 −2 −1 −1

1200

Based on the context, determine and interpret:

1000

The y-intercept The range

The domain

−4

The population growth of a sample of bacteria is shown in the graph, where t is measured in minutes. The initial population is 200.

−3

−15

x 1

−2

−10

y (no. of bacteria)

800 600 400 200

t 20

80 100 120

Performance task 16

Aicha is doing a fundraiser to support families in her neighborhood. She is offering subsidized tutoring. She creates this graph to show her pricing model based on the length of the tutoring session. 30

Dollars

25 20 15 10 5

Minutes 10 20 30 40 50 60 70 80 90 100 110 120

Is this a function or a relation? Explain your answer.

State the domain.

State the range.

Interpret the meaning of the domain for this context.

Find the slope for each segment of the function and explain its meaning based on the context.

Find the maximum cost for a tutoring session.

Explain why there is no x or y-intercepts in this scenario.

She decides to simplify her model and changes her price to P = 0.25m + 2, with the same domain. i

Determine which model has the lower price for 30 minutes of tutoring.

Determine which model has the lower price for 120 minutes of tutoring. Topic 2 Assessment: Functions & Relations mathspace.co

201

Answers

7 a f (x) = 4 − 3x

Topic 2 Assessment: Functions & Relations 1 a S ometimes true. Relations are only functions if their graphs pass the vertical line test or if the same input has different outputs. b A lways true. A function is a special type of relation, so all functions are relations. c S ometimes true. All lines are functions other than vertical lines, such as x = 2. d N ever true. There is exactly one output for all values in the domain of a function. A.F.2a

f (1) = 4 − 3(1)

Substitute x = 1

f (1) = 4 – 3

Evaluate the product

f (1) = 1

Evaluate the subtraction

b f (x) = 4 − 3x f (−5) = 4 − 3(−5) f (−5) = 4 + 15 f (−5) = 19

Start with the equation

c f (x) = 4 − 3x f (0) = 4 − 3(0) f (0) = 4 + 0 f (−5) = 4

Start with the equation

Substitute x = −5 Evaluate the product Evaluate the subtraction

Substitute x = 0 Evaluate the product Evaluate the subtraction

A.F.1g

2 B

8 C

A.F.2a

A.F.2g

3 a i y=1 ii x = 0 iii This is a function because every input has exactly one output.

9 a g(−4) = −4

b g(−2) = −3

c g(0) = −2

d g(4) = 0

A.F.1g 10 v (5) = 15 100 represents the value of the car after 5 years or in year 2025.

b i y = 2 and y = −2 ii x = 0 iii This is not a function because it fails the vertical line test. c i y=2 ii x = 2

A.F.2g 11 a i Domain: −4 ≤ x ≤ 7

ii Range: −8 ≤ y ≤ 9

b i Domain: −2 ≤ x ≤ 4

ii Range: −5 < y ≤ 3

A.F.1a, A.F.2b

iii This is a function. It passes the vertical line test and is a linear relationship which is a function. d i y = −4 and y = 0

12 B A.F.1a, A.F.2b 13 C

ii x = 1 iii F is not a function. An x-value maps to two different y-values, so it cannot be a function. A.F.1g, A.F.2a, A.F.2f 4 a 4

14 a i x-intercepts: (6, 0) and (0, 0) ii There is a maximum at the vertex (3, 9).

b i x-intercepts: (−2, 0) and (3, 0) y-intercept: (0, 3)

2 1 −4 −3 −2 −1 −1

A.F.2b y-intercept: (0, 0)

x 1

−2 −3 −4

b (1, 2) or (1, 1) A.F.2a

ii There is neither a minimum nor a maximum in this graph as it is a combination of linear segments. A.F.1a, A.F.2b 15 a y = 200, since this is the initial population of the bacteria colony when t = 0. b y ≥ 200, since the colony starts with 200 bacteria and increases exponentially, so 200 is the minimum. c t ≥ 0, since time cannot be negative, but is continuous and observations started when t = 0. A.F.2b

5 B A.F.2b 6 1 or 2 A.F.2a

202

Start with the equation

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Performance task 16 a T his is a function because each input (minutes) has only one output (dollars). b Domain: 15 ≤ x ≤ 120 c Range: 5 ≤ y < 25 and y = 30 d T he domain represents the possible minutes of tutoring session Aicha offers. e S egment 1: Slope = . The price increases by $1 for every 3 minutes. Segment 2: Slope = . The price increases by $2 for every 9 minutes. f

Segment 3: Slope = 0. The price remains constant. The maximum cost for a tutoring session is $30.

g T here are no x or y-intercepts because the graph does not intersect the x or y-axis. In this context, it means that there are no free tutoring sessions (y-intercept) and no tutoring sessions with zero minutes (x-intercept). h i F or 30 minutes of tutoring, the simpliﬁed model has a lower price (P = 0.25(30) + 2 = $9.50) compared to the original model ($15). ii For 120 minutes of tutoring, the original model has a lower price ($30) compared to the simpliﬁed model (P = 0.25(120) + 2 = $32). A.F.1a, A.F.1g, A.F.2a, MP1, MP2, MP4

Topic 2 Assessment: Functions & Relations mathspace.co

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3 Linear Functions Big ideas • A family of functions is defined by a unique set of characteristics shared by all functions that belong to that family. These characteristics give insight into the types of real-world situations that a function models. • There are many ways to represent a function (equation, table, graph, written description, etc.). The way a function is represented can affect what conclusions can be made.

Chapter outline 3.01 3.02 3.03 3.04 3.05 3.06

Slope (A.F.1) Transformations of linear functions (A.F.1) Slope-intercept form (A.F.1) Standard form (A.F.1) Point-slope form (A.F.1) Equations of parallel and perpendicular lines (A.F.1) Topic 3 Assessment

209 230 254 287 319 343 363

Linear equations can be used to model the fuel consumption of an airplane, which depends on factors such as speed, distance, and altitude.

3. Linear Functions Topic overview Foundational knowledge Evaluating standards proficiency The skills book contains questions matched to individual standards. It can be used to measure proficiency for each. Students should be proficient in these standards. 4.MG.4 — The student will identify, describe, and draw points, rays, line segments, angles, and lines, including intersecting, parallel, and perpendicular lines. 7.PFA.1 — The student will investigate and analyze proportional relationships between two quantities using verbal descriptions, tables, equations in y = mx form, and graphs, including problems in context. 8.MG.3 — The student will apply translations and reflections to polygons in the coordinate plane.

8.PFA.3 — The student will represent and solve problems, including those in context, by using linear functions and analyzing their key characteristics (the value of the y-intercept (b) and the coordinates of the ordered pairs in graphs will be limited to integers). A.EI.1 — The student will represent, solve, explain, and interpret the solution to multistep linear equations and inequalities in one variable and literal equations for a specified variable.

Big ideas and essential understanding A family of functions is defined by a unique set of characteristics shared by all functions that belong to that family. These characteristics give insight into the types of real-world situations that a function models. 3.01, 3.03 — Linear functions have a constant rate of change, often referred to as slope, and can be represented by the equation y = mx + b.

There are many ways to represent a function (equation, table, graph, written description, etc.). The way a function is represented can affect what conclusions can be made. 3.03, 3.04, 3.05 — Different representations of a function may highlight or hide different characteristics but they do not change the function itself.

3.02 — A relatively small number of transformations can be applied to the graph of a function to change what the function models without changing the family to which it belongs.

3.04, 3.05 — Changing the form of the equation of a function can highlight or hide different characteristics of the function but does not change the function itself.

3.06 — Parallel lines will have the same slope and never intersect. 3.06 — Perpendicular lines have slopes with opposite signs and they are reciprocals of one another. A vertical and horizontal line are perpendicular.

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Standards A.F.1diii — Write the equation of a linear function to model a linear relationship between two quantities, including those that can represent contextual situations. Writing the equation of a linear function will include the A.F.1a — Determine and identify the domain, range, zeros, following situations: iii) given the slope and a point on the slope, and intercepts of a linear function, presented line whose coordinates are integers; algebraically or graphically, including the interpretation of 3.03 Slope-intercept form these characteristics in contextual situations. 3.04 Standard form 3.01 Slope 3.05 Point-slope form 3.03 Slope-intercept form 3.06 Equations of parallel and perpendicular lines 3.04 Standard form A.F.1div — Write the equation of a linear function to 3.05 Point-slope form model a linear relationship between two quantities, 3.06 Equations of parallel and perpendicular lines including those that can represent contextual situations. A.F.1b — Investigate and explain how transformations to Writing the equation of a linear function will include the the parent function y = x affect the rate of change (slope) following situations: iv) vertical lines as x = a; and the y-intercept of a linear function. 3.04 Standard form 3.02 Transformations of linear functions A.F.1dv — Write the equation of a linear function to model a linear relationship between two quantities, including A.F.1c — Write equivalent algebraic forms of linear those that can represent contextual situations. Writing functions, including slope-intercept form, standard form, the equation of a linear function will include the following and point-slope form, and analyze and interpret the situations: v) horizontal lines as y = c. information revealed by each form. 3.04 Standard form 3.03 Slope-intercept form A.F.1 — The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.

3.04 Standard form 3.05 Point-slope form 3.06 Equations of parallel and perpendicular lines

A.F.1e — Write the equation of a line parallel or perpendicular to a given line through a given point. 3.06 Equations of parallel and perpendicular lines

A.F.1di — Write the equation of a linear function to model a linear relationship between two quantities, including those that can represent contextual situations. Writing the equation of a linear function will include the following situations: i) given the graph of a line; 3.03 Slope-intercept form 3.04 Standard form 3.05 Point-slope form 3.06 Equations of parallel and perpendicular lines

A.F.1f — Graph a linear function in two variables, with and without the use of technology, including those that can represent contextual situations. 3.02 Transformations of linear functions 3.03 Slope-intercept form 3.04 Standard form 3.05 Point-slope form 3.06 Equations of parallel and perpendicular lines

A.F.1g — For any value, x, in the domain of f, determine f (x), and determine x given any value f (x) in the range of f, given an algebraic or graphical representation of a linear function. 3.03 Slope-intercept form A.F.1h — Compare and contrast the characteristics of linear functions represented algebraically, graphically, in tables, and in contextual situations. 3.02 Transformations of linear functions 3.04 Standard form A.F.1h — Compare and contrast the characteristics of linear functions represented algebraically, graphically, in tables, and in contextual situations. 3.03 Slope-intercept form

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Future connections A.F.2 — The student will investigate, analyze, and compare characteristics of functions, including quadratic and exponential functions, and model quadratic and exponential relationships.

G.RLT.3 — The student will solve problems, including contextual problems, involving symmetry and transformation.

A2.F.1 — The student will investigate, analyze, and compare square root, cube root, rational, exponential, A.ST.1 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent and logarithmic function families, algebraically and graphically, using transformations. data; and analyze data and communicate results) with a focus on representing bivariate data in scatterplots A2.F.2 — The student will investigate and and determining the curve of best fit using linear and analyze characteristics of square root, cube root, quadratic functions. rational, polynomial, exponential, logarithmic, and G.RLT.2 — The student will analyze, prove, and justify the piecewise-defined functions algebraically and graphically. relationships of parallel lines cut by a transversal.

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Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

3.01 Slope Subtopic overview Lesson narrative In this lesson, students will learn to calculate slope from two points, a table, and a graph. By the end of the lesson, students will be able to calculate slope and understand its meaning in a real-world context.

Learning objective

3.01 Slope

Students: Page 104

After this lesson, you will be able to… • determine the slope of a line from two points or a graph.

Identify slope from a graph Recall the slope of a line is a. value that describes the line’s steepness.

Key vocabulary 

Slope slope The ratio of the vertical change (rise) to the horizontal change (run) between two points on a line

We can find the slope of a line by identifying the vertical and horizontal change or: Essential understanding

Linear functions have a constant rate of change, often referred to as slope, and can be represented by the equation y = mx + b. There are four types of slope: y y 5 5 4 4 3 3 This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning 2 2 standards. 1 1 x x Mathematical process goals 1 2 3 4 5 1 2 3 4 5 −5 −4 −3 −2 −1 −5 −4 −3 −2 −1 −1 −1 MPG1 — Mathematical Problem −2 Solving −2 −3goal by allowing students to explore and discover the −3 Teachers can incorporate this slope formula using an inquiry−4 −4 based activity. −5 −5

Standards

MPG3 — Mathematical Reasoning Positive Negative Teachers can incorporate this goal by having students use logical reasoning to analyze the relationship between the slope of a line and the steepness of the line on a graph. They can also encourage students to apply inductive y y 5 5 and deductive reasoning skills to justify the steps in calculating the slope of a line using two points with integer 4 4 coordinates. 3 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

Zero

x 1 2 3 4 5

2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

Undefined

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Prior connections 7.PFA.1 — The student will investigate and analyze proportional relationships between two quantities using verbal descriptions, tables, equations in y = mx form, and graphs, including problems in context.

A2.F.2 — The student will investigate and analyze characteristics of square root, cube root, rational, polynomial, exponential, logarithmic, and piecewisedefined functions algebraically and graphically.

Tools You may ﬁnd these tools helpful: • Graphing calculator

• Graph paper

Lesson supports The following supports may be useful for this lesson. More specific supports may appear throughout the lesson:

Concrete-Representational-Abstract (CRA) Targeted instructional strategies Concrete: Begin by engaging students with physical manipulatives to explore the concept of slope. Use items like ramps made from boards or cardboard to represent different inclines. Have students roll marbles or toy cars down these ramps to observe how the steepness affects speed. Provide coordinate grid mats and have students place physical objects, like counters or tiles, at specific points to form a line. Encourage them to measure the vertical change (rise) and horizontal change (run) between two points using rulers or measuring tapes. This hands-on experience helps students feel and see the rate of change physically.

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Representational: Next, transition to drawing representations of what they explored concretely. Have students sketch the ramps they used, labeling the rise and run for each. Use graph paper to plot the points from the concrete activity and draw lines connecting them. Encourage students to draw right triangles beneath the lines to visually represent the rise over run. Provide diagrams of coordinate planes with lines, and have students highlight and label the rise and run on each line. Using images of graphs, help students see how the steepness of a line relates to its slope. Abstract: Finally, introduce the symbolic representation of slope using the formula m =

. Explain how this

formula calculates the rate of change between two points on a line. Guide students through calculating slope from pairs of coordinates they plotted earlier. Practice finding the slope from tables by determining the change in y-values over the change in x-values. Work with the equation y = mx + b to show how the slope m represents the constant rate of change in a linear function. Connecting the Stages: Throughout the lesson, help students make connections between all three stages. When working with the slope formula, remind them of the rise and run they measured with the manipulatives. Ask them to compare their drawings to the calculations they perform, highlighting how the numbers in the formula correspond to the measurements in their sketches. Encourage discussions about how the physical experience of rolling objects down ramps relates to the abstract concept of slope.

Student lesson & teacher guide Identify slope from a graph Students are told the definition of slope and are shown graphs representing the different classifications of slope. They will then be shown that the slope of a line can be found from a graph by finding the ratio between the change in y-coordinates and the change in x-coordinates.

Students: Pages 104–105

3.01 Slope After this lesson, you will be able to… • determine the slope of a line from two points or a graph.

Identify slope from a graph Recall the slope of a line is a. value that describes the line’s steepness. Slope The ratio of the vertical change (rise) to the horizontal change (run) between two points on a line We can find the slope of a line by identifying the vertical and horizontal change or:

There are four types of slope: 5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

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211

The ratio of the vertical change (rise) to the horizontal change (run) between two points on a line We can find the slope of a line by identifying the vertical and horizontal change or:

There are four types of slope: 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

Zero

104

212

x 1 2 3 4 5

Negative

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

Positive 5 4 3 2 1

5 4 3 2 1

Mathspace Virginia SOL Algebra 1 mathspace.co

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

Undefined

Bridging rise over run to the slope formula Targeted instructional strategies For students to begin connecting different representations of the slope of a line, such as formula

and the slope

, provide the graph of a line with its points labeled and draw slope triangles to help students

determine the slope. y 8 (6, 7)

6 (3, 5)

4 2 (−3, 1)

(0, 3)

−4 −2 (−6, −1)

x 2

Ask students to choose two points on the graph and determine how operations could be used on the y-values to match the vertical change. Then ask them to do the same with the x-values. y

8 (⬚, 7)

(⬚, 5)

4 2 (⬚, 1) −4 −2 (⬚, −1)

4 (⬚, 3) x 2

Focus on y-values

2 (−3, ⬚)

−4 −2 (−6, ⬚)

(0, ⬚) 2

(3, ⬚)

(6, ⬚)

x 4

Focus on x-values

Ask students to predict how they could write a formula to help them find slope if they are only given two points on the line.

Reversing coordinates and rise over run Address student misconceptions When finding slope, students may incorrectly assume that slope is written

since x is written first in

(x, y). Alternatively, students may write coordinates as (y, x) before finding their slope. Many students remember slope as

as a way to remember that the vertical y-values are in the numerator

while the horizontal x-values are in the denominator.

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Examples Students: Pages 105–106

Apply the idea rise = 8 – 0 = 8 units run = 4 – 0 = 4 units

Subtract the y-coordinates of points A and B Evaluate Subtract the x-coordinates of points A and B Evaluate

From A, move 8 units up and 4 units to the right. b Express the direction of the movement in the previous question as a simpliﬁed ratio, comparing vertical movement to horizontal movement. Express the ratio in the form a : b.

Purpose Check that students can find the rise over run between two points on a graph. Create a strategy

Usewith the slope formula: m = , then convert to ratio. Reﬂecting students Ask students their answer would have changed if point A was above point B instead. Help them realize Apply how the idea thebe idea that theyApply would slope would be A negative. rise moving = 8 – 0 down vertically, Subtract so the the y-coordinates of points and B Substitute the value of the rise and run Emphasize that, for =this problem, counting 8 units Evaluateup corresponds to positive 8 and counting right corresponds to positive 4. The opposite bethe negative numbers. run = 4 – 0directions would Subtract x-coordinates of points A and B

Evaluate Evaluate Express as a ratio Students:From Page 1068 units up and 4 units A, move to the right. = 4 units

c Complete the directions that explain how to move from point A to point C. b Express the direction of the movement in the previous question as a simpliﬁed ratio, comparing vertical movement From A, move ⬚ units up and ⬚ units to the right. to horizontal movement. Express the ratio in the form a : b.

Create a strategy Create a strategy

We need to ﬁnd the difference in the y-values (i.e. rise) between points A and C, and the difference between their Use the slope formula: m = , then convert to ratio. x-values (i.e. run).

Apply the the idea idea Apply rise = 12 – 0

Substitute they-coordinates value of the rise and run Subtract the of points A and C

= 12 units

Evaluate Evaluate Subtract the x-coordinates of points A and C Express Evaluateas a ratio

run = 6 – 0 = 6 units

From A, move 12 units up and 6 units to the right. c Complete the directions that explain how to move from point A to point C.

214

A, the move ⬚ unitsofupthe and ⬚ units tointhe d From Express direction movement theright. previous question as a simpliﬁed ratio, comparing vertical movement Mathspace Virginia SOL Algebra 1 Teacher Edition to horizontal movement. Express the ratio in the form a : b. mathspace.co

Create a strategy

Create strategy We needato ﬁnd the difference in the y-values (i.e. rise) between points A and C, and the difference between their x-values (i.e. run). Use the slope formula: m =

, then convert to ratio.

= 4 units

Evaluate

From A, move 8 units up and 4 units to the right. b Express the direction of the movement in the previous question as a simpliﬁed ratio, comparing vertical movement Purpose to horizontal movement. Express the ratio in the form a : b.

Check that students can express the rise over run counted in part (a) as a ratio. Create a strategy

Apply the idea Reﬂecting students Usewith the slope formula: m = , then convert to ratio. rise = 8 – 0 the y-coordinates of points A and B Encourage students to describe the Subtract meaning of this ratio in their own words. What does the 2 represent? What = 8 units Evaluate does theApply 1 represent? the idea For students who struggle to explain it, provide them with a sentence frame to get started. run = 4 – 0 Subtract the x-coordinates of points A and B For example: “The ratio 2 : 1 means for each increase therise ⬚and (y-direction or y-values) by 2, the ⬚ (x-values) Substitute the value ofinthe run = 4 units Evaluate increase by 1.” From A, move 8 units up and 4 units to the right. Evaluate

Students: Page 106

Express as a ratio b Express the direction of the movement in the previous question as a simpliﬁed ratio, comparing vertical movement to horizontal movement. Express the ratio in the form a : b. c Complete the directions that explain how to move from point A to point C. From A, move ⬚ units up and ⬚ units to the right. Create a strategy Use the slope formula: m =

Create a strategy

, then convert to ratio.

We need to ﬁnd the difference in the y-values (i.e. rise) between points A and C, and the difference between their

Apply x-valuesthe (i.e.idea run).

Substitute the value of the rise and run

Apply the idea rise = 12 – 0 = 12 units run = 6 – 0

Evaluate Subtract the y-coordinates of points A and C Evaluateas a ratio Express Subtract the x-coordinates of points A and C

c Complete the that explain how to move from point A to point C. = 6directions units Evaluate From A, move ⬚ units up and ⬚ units to right. the right. From A, move 12 units up and 6 units to the

Create a strategy d Express the direction of the movement in the previous question as a simpliﬁed ratio, comparing vertical movement We to need to ﬁnd the difference in the the y-values (i.e. between horizontal movement. Express ratio in therise) form a : b. points A and C, and the difference between their

Purposex-values (i.e. run). Create a strategy Check that students can ﬁnd the rise over run between two points on a graph. Apply the idea

Use the slope formula: m =

, then convert to ratio.

Reﬂecting with students rise = 12 – 0 Subtract the y-coordinates of points A and C Ask students if there another way to solve this problem. Students could also count the vertical and = was 12 units Evaluate Apply the idea horizontal distances the points on the thex-coordinates graph. run =between 6–0 Subtract of points A and C = 6 units

Substitute the value of the rise and run Evaluate

Students:From Page 10612 units up and 6 units A, move to the right. Evaluate

Express as a ratio d Express the direction of the movement in the previous question as a simpliﬁed ratio, comparing vertical movement to horizontal movement. Express the ratio in the form a : b. 106

Mathspace Virginia SOL Algebra 1 mathspace.co

Create a strategy

Use the slope formula: m =

, then convert to ratio.

Apply the idea Substitute the value of the rise and run Evaluate Express as a ratio

106

Mathspace Virginia SOL Algebra 1 mathspace.co

Purpose Check that students can express the rise over run counted in part (c) as a ratio. Reﬂecting with students Ask student to explain why their answers to part (b) and part (d) are the same ratio. They should explain that the horizontal and vertical change between any two points will be the same for the entire line. 3.01 Slope mathspace.co

215

Students: Page 107

Example 2 Consider the graph shown: 10

8 6 4 2 −8 −6 −4 −2 −2 −4

x 2

−6 −8

Find the slope of the line represented by the graph.

Create a strategy To find the slope of a line represented by using the formula: Slope =

Apply the idea From point A(−2, 10) to point B(2, 2), move 8 units down and 4 units to the right. A

8 6 4 2

−8 −6 −4 −2 −2 −4

B x 2

−6 −8

The ratio of the rise to the run is

simplified into −2, the slope of the line.

The slope of the line is m = −2.

Idea summary Purpose The slope of a line is the ratio of the change in the y-coordinates (vertical change) to the change in the Ensure that students can find the slope of a line from a graph. x-coordinates (horizontal change). Expected mistakes Students might misread the axis labels and count by 1 instead of 2. For example, they might think point B is only two units to the right of point A. Refer them to the graph and encourage them to read the labels carefully. Reflecting with students Students can check their answer by picking two different points from the line and finding the new slope. If needed, prompt students to use the points (−1, 8) and (0, 6), which will also give us a slope of −2. 3.01 Slope mathspace.co

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107

Create a strategy To find the slope of a line represented by using the formula: Slope =

Apply the idea From point A(−2, 10) to point B(2, 2), move 8 units down and 4 units to the right. Compare and connect

English language learner support

use with Example 2

8 measure. An example might be 2 sets of staircases Give students a physical example of slope that they can 6 located in the school. Allow students to measure the height and depth of 2 or 3 steps from each staircase. 4

B After returning to class, call the height the “rise” and the 2 depth the “run.” Ask students which set of steps were x steepest. Help students compare the steepness −8 −6of −4each −2 staircase 2 4 6 by 8 asking them to discuss using the following −2 sentence frame: “The steeper the (explanation) the greater the (explanation).” −4

Note that the word “change” means different things in−6English. Be sure to explain that the word “change” when −8 talking about rate of change means “difference.” simplified into −2, the slope of the line.

The ratio of the rise to the run is

Students:The Page slope107 of the line is m = −2.

Idea summary The slope of a line is the ratio of the change in the y-coordinates (vertical change) to the change in the x-coordinates (horizontal change).

The slope formula Students will discover the slope formula by generalizing the method of counting the rise and run using a graph. They are provided with graphic visuals to support understanding of this generalization. 3.01 Slope mathspace.co

Students: Page 108

107

The slope formula Finding the ratio of the rise and run of the line works when it’s easy to see the graph, with clearly marked points on the line. We can extend this thinking to use the coordinates of two points and construct a general formula. The rise of the line is found with points that lie on a vertical line. These points share the same x-value. The distance between them is the difference in the y-values.

y (x2, y2)

rise = y2 − y1 y2 (x1, y1)

The run of the line is found with two points that lie on a horizontal line. These points share the same y-value. The distance between them is the difference in the x-values.

y (x2, y2)

run = x2 − x1 x1

(x1, y1) x2

We ﬁnd the slope by using the formula:

3.01 Slope mathspace.co m

slope

(x1, y1)

a point on the line

217

(x1, y1) x

We ﬁnd the slope by using the formula:

slope

(x1, y1)

a point on the line

(x2, y2)

a second point on the line

Example 3

Information gap

What is the slope of a line that passes through the points A(3, 5) and B(−2, 10)?

English language learner support

strategy Prepare Create a set ofa cards containing ordered pairs, graphs of lines, and slope calculations with missing elements. Begin Use thestudents slope formula: m = and distributing . Let point A = (x1, ytypes B = (xto y2). student in the pair. Student A by organizing into pairs different of cards 1) and point 2, each receives the Ordered Pairs cards, which each contain two ordered pairs, while Student B receives the Graph cards Apply the idea cards. The Graph cards show graphs of lines with two points marked but not labeled, and the and Slope Calculation Use the slope formula Slope Calculation cards display the slope formula with the ordered pairs plugged in but missing the final calculation.

Example cards could include:

Substitute x1 = 3, y1 = 5, x2 = −2, and y2 = 10

1. Ordered Pairs

Evaluate the subtraction

• Card 1: (2, 3) and (5, 7)

• Card 2: (−4, −1) and (−2, −3)

• Card 3: (3, −2) and (−1, −6)

• Card 2:

• Card 3:

• Card 2:

• Card 3:

Evaluate the division

2. Slope Calculation • Card 1: 108

3. Graphs

Mathspace Virginia SOL Algebra 1 mathspace.co

• Card 1: 9 8 7 6 5 4 3 2 1 −1−1

x 1 2 3 4 5 6 7 8 9

1 y x 1 −9 −8 −7 −6 −5 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7 −8 −9

1 y −2 −1−1 −2 −3 −4 −5 −6 −7 −8 −9

x 1

2 3 4 5 6 7

In the first task, Student A describes the ordered pairs on their card to Student B, who then identifies the corresponding graph. The students discuss and ensure the graph matches the ordered pairs. In the second task, using the matched graph and ordered pairs, Student B explains how to use the slope formula with the ordered pairs. Student A then helps complete the slope calculation by finding the final slope, with both students verifying the calculation together. After completing the tasks, pairs reflect on their communication process and how they clarified any misunderstandings, then share their findings and strategies with the class. For example, Student A might describe the ordered pairs (2, 3) and (5, 7), and Student B identifies the graph showing these points. Together, they would complete the slope calculation,

, discussing each step to ensure understanding.

This structured and interactive activity enhances students’ understanding of slope by requiring them to communicate mathematical ideas clearly, ask questions, and verify each other’s work. It fosters a collaborative learning environment where students practice both mathematical skills and language, ultimately building their confidence in using the slope formula and interpreting graphs.

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y (x2, y2)

Decompose the process for finding slope between two points

rise = y2 − y1

Student with disabilities support y2

Support students with developing computational thinking by decomposing finding the slope between two (x1, yand ) y1 points into smaller tasks then show 1 x students how the parts are connected. In small group instruction, work through a question finding the slope between two points. Throughout the solution, write out the steps you took. This can be turned into a model they can follow on future examples. The runcan of the line found with twothe points that lie of on finding a horizontal Create a graphic organizer of the steps that students use toisbreak down process the slope. y line. These points share the same y-value. The distance between Each of the following steps should(xhave a dedicated section with enough space for students to write their , y2) 2 them is the difference in the x-values. answers. run = x2 − x1

1. Label your two points as (x1, y1) and (x2, y2). x1 the 2. Subtract y2 − y1, to find (x , yrise. ) 1

3. Subtract x2 − x1, to find x2 the run. 4. Divide

5. Simplify as the a reduced fraction or integer. We ﬁnd slope by using the formula:

Examples Students: Pages 108–109

slope

(x1, y1)

a point on the line

(x2, y2)

a second point on the line

Example 3 What is the slope of a line that passes through the points A(3, 5) and B(−2, 10)?

Create a strategy Use the slope formula: m =

. Let point A = (x1, y1) and point B = (x2, y2).

Apply the idea Use the slope formula Substitute x1 = 3, y1 = 5, x2 = −2, and y2 = 10 Evaluate the subtraction Evaluate the division

Reﬂect and check By graphing the two points and sketching the line through them, we can see that 108 Mathspace Virginia SOL Algebra 1 it does in fact have a negative slope. mathspace.co

y 10 8 6 4 2 −2 −2

x 2

Example 4 Gasoline costs a certain amount per gallon. The table shows the cost of various amounts of gasoline in dollars: Number of gallons (x) Cost of gasoline ( y)

0 0

10 26.40

How much does gasoline cost per gallon?

20 52.80

30 79.20

40 105.60

3.01 Slope mathspace.co

219

Purpose Check that students can use the slope formula to find the slope of a line given two points. Reflecting with students Ask advanced learners whether it matters which point is labeled as (x1, y1) and which is labeled as (x2, y2). Then, ask them to explain why it does not matter how we label the points. They can use a graph to explain that the horizontal and vertical distances will be the same, regardless of how the points are labeled, or they could use an algebraic justification like the one shown:

Slope formula

Commutative property Factor out a −1 Divide −1 by −1

Applying the slope formula

use with Example 3

Address student misconceptions Students may think that slope is simply the ratio of the change in y-coordinates to the change in x-coordinates, andthe check and not Reﬂect consider order of the coordinates as they apply the formula. By graphing the two points and sketching the line through them, we can see that

y For example, calculating the slope between two points (x1, y1) and (x2, y2), students often think that it it does when in fact have a negative slope. 10 doesn’t matter which point is used as the starting point and which as the ending point. As a result, they might 8

miscalculate the slope as

instead of the correct formula

6, leading to a slope that has

the opposite sign of the correct answer.

4 2

Students may find it helpful to label their coordinates before plugging in values to the formula. −2 −2

x 8

Students: Page 109

Example 4 Gasoline costs a certain amount per gallon. The table shows the cost of various amounts of gasoline in dollars: Number of gallons (x) Cost of gasoline ( y)

0 0

10 26.40

20 52.80

30 79.20

40 105.60

How much does gasoline cost per gallon?

Create a strategy The cost per gallon of gasoline is the unit rate. This is the same as the ratio of the change in the cost of gasoline ( y) to the change in the number of gallons (x). We can ﬁnd the unit rate by using the slope formula, m =

, with any two input-output pairs in the table.

Apply the idea Write the slope formula Substitue two input-output pairs Evaluate the subtraction Evaluate the division

Reﬂect and check

220

The relationship between the cost of gasoline and the number of gallons purchased can be modeled by the equation y = 2.64x. If we use technology Mathspace Algebra 1 Teacher to graphVirginia this line,SOL we can see that all the Edition points in the table lie on the line. mathspace.co

110 100 90 80 70 60 50

y (40, 150.60) (30, 79.20) (20, 52.80)

Substitue two input-output pairs Evaluate the subtraction Evaluate the division

Reﬂect and check The relationship between the cost of gasoline and the number of gallons purchased can be modeled by the equation y = 2.64x. If we use technology to graph this line, we can see that all the points in the table lie on the line.

110 100 90 80 70 60 50 40 30 20 10

(40, 150.60) (30, 79.20) (20, 52.80) (10, 26.40) x 5 10 15 20 25 30 35 40 45 50

3.01 Slope mathspace.co

109

Purpose Check that students understand that a rate can be found by calculating the slope. Reﬂecting with students Ask various students to share with the class which two points they used in their calculations. Encourage them to explain why they chose their points. Point out that, regardless of which two points were used, everyone will get the same answer. However, it is usually easiest to use the smaller numbers in calculations.

Students: Page 110 Example 5 Mario wants to determine which of two slow-release pain medications is more rapidly absorbed by the body. For the liquid form, the amount of the medication in the bloodstream is presented in the graph shown.

A 35 30 25 20 15 10 5

The results for the capsule form are presented in the table below. Time (mins), t 4 7 10 13

t 1

2 3 4 5 6 7 8 9

Amount in blood (mgs), A 24.6 42.3 60 77.7

a At what rate, in milligrams per minute, is the liquid form absorbed?

Create a strategy Choose any two points that lie on the line, and use the formula for slope.

Apply the idea We use the points (0, 0) and (2, 8) that lie on the line. Write the formula for slope Substitute the values Evaluate

b At what rate, in milligrams per minute, is the capsule form absorbed?

Create a strategy

3.01 Slope mathspace.co

221

a At what rate, in milligrams per minute, is the liquid form absorbed?

15 10

Create a strategy

Choose any two points that lie on the line, and use the formula for slope.

The results for the capsule form are presented in the table below.

2 3 4 5 6 7 8 9

Apply the idea (mins), t (0, 0)Amount in blood We Time use the points and (2, 8) that lie(mgs), on theAline. 4 24.6 7 42.3 10 60 13 77.7

Write the formula for slope Substitute the values

Evaluate a At what rate, in milligrams per minute, is the liquid form absorbed? b At what rate, in milligrams per minute, is the capsule form absorbed? Create a strategy

PurposeChoose any two points that lie on the line, and use the formula for slope. Createemphasizes a strategy the concept of slope as rate of change in a real-world context. The student must This example Apply the idea Choose twoslope pointsof from table and use thethe slope formula. understand thatany the thethe line represents rate of absorption of the medication. We use the points (0, 0) and (2, 8) that lie on the line.

Apply thestudents idea Reﬂecting with Write the formula for slope We can use the points (4, 24.6) (7, 42.3) fromdirectly the table.corresponds to the rate of absorption. This can be a Point out that the slope of the lineand in the graph good opportunity to discuss the concept of slope as a rate of change in values various real-world contexts. Write the formula for slope Substitute the Evaluate Substitute the values

Students: Page 110

Evaluate b At what rate, in milligrams per minute, is the capsule form absorbed?

Create a strategy Choose any two points from the table and use the slope formula. 110 Mathspace Virginia SOL Algebra 1 mathspace.co

Apply the idea We can use the points (4, 24.6) and (7, 42.3) from the table. Write the formula for slope Substitute the values Evaluate

Purpose110 Mathspace Virginia SOL Algebra 1 This examplemathspace.co further develops the concept of slope as rate of change, and ensures that students understand that the concept of rate remains the same, regardless of how the data is presented. Reﬂecting with students Discuss how the method of finding the rate of change is similar for both the graph and the table. In both cases, we’re finding the slope between two points. The main difference is how those points are presented.

Students: Page 111 c In which form is the medication absorbed more rapidly?

Create a strategy Use the fact that the medication that is more quickly absorbed will have a higher rate.

Apply the idea 5.9 mg/min > 4 mg/min Comparing the two rates from part (a) and part (b), the capsule form is more quickly absorbed than the liquid form as it has a higher rate of absorption of 5.9 mg/min.

Idea summary 222

Mathspace We Virginia SOL Algebra 1 Teacher Edition ﬁnd the slope of a line by using the formula: mathspace.co

Purpose Check students’ ability to compare rates and determine which is larger. Expected mistakes Studentsc may expect lower number to represent a quicker rate because they are thinking about how it takes In which formthe is the medication absorbed more rapidly? less time to absorb. Remind students that a higher rate of absorption means the medication is absorbed more quickly because it’s a larger amount of medication being absorbed in the same timeframe. Create a strategy Use the fact that the medication that is more quickly absorbed will have a higher rate.

Reﬂecting with students Discuss Apply with the why it might be important for certain medications to be absorbed more quickly than thestudents idea 5.9 mg/min > 4 mg/min others. Comparing the two rates from part (a) and part (b), the capsule form is more quickly absorbed than the liquid form as

Students:it Page 111 rate of absorption of 5.9 mg/min. has a higher

Idea summary We ﬁnd the slope of a line by using the formula:

(x1, y1)

are the coordinates of the lower points

(x2, y2)

are the coordinates of the upper points

We can ﬁnd the slope in different ways depending on how it is represented: • • •

From a graph: Count the slope as the rise over the run, or ﬁnd two points and use the slope formula. Table: Find the difference in two consecutive outputs and divide by the difference in the corresponding inputs, or ﬁnd two input-output pairs and use the slope formula. Description: The slope is the rate given in the problem.

Practice What do you remember? Practice 1

Determine whether the following is a correct description of slope:

Students: Pages 111–115 a

the slope of each line as positive, negative, zero, or undefined: What do 2youClassify remember? a

14 12 10 10 a c 8 8 b 6 6 4 4 Classify the slope of each2line as positive, negative, zero, or undefined: 2 x 14 Determine whether the following is a correct description of slope: 12

−8 −6 −4 −2 2 4 y −2 14 −4 −6 12 −8

10 8 6 4 2

−8 −6 −4 −2 −2 −4 −6 −8

x 2

−8 −6 −4 −2 −2 −4 −6 −8

d x 2

14 12 10 8 6 4 2

−8 −6 −4 −2 −2 −4 −6 −8

3.01 Slope 4 mathspace.co 6 8

111

3.01 Slope mathspace.co

223

−8 −6 −4 −2 −2 −4 −6 −8

14 12 10 8 6 4 2

x 2

−8 −6 −4 −2 −2 −4 −6 −8

For each segments containing A and B: i

Find the rise going from A and B.

iii

Find the slope of the line.

A (−4, 0) and B (0, 2) y 5 4 3 2 B 1

A −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

224

14 12 10 8 6 4 2

x 2

−8 −6 −4 −2 −2 −4 −6 −8

14 12 10 8 6 4 2

−8 −6 −4 −2 −2 −4 −6 −8

14 12 10 8 6 4 2

x 2

−8 −6 −4 −2 −2 −4 −6 −8

14 12 10 8 6 4 2

Find the run going from A and B.

A (0, 4) and B (1, 0) y 5 A 4 3 2 1

x 1 2 3 4 5

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

x B 1 2 3 4 5

If we have two points and the slope formula m = (x2, y2)? Explain.

Consider the points A, B, and C on the coordinate plane: a

Describe how to move from point A to the following points: i

, does it matter which point is (x1, y1) and which point is

Point B

Point C

Express the direction of each movements as a simpliﬁed ratio, comparing vertical movement to horizontal movement. i

From point A to point B

From point A to point C

A 6 4

2 −6 −4 −2 −2

Determine the slope of the line.

Graph the y-intercept, and label it D.

2 4 6 8 10 12 14

−4

Let’s practice 6

Consider the straight line that passes through the points A, B, C, and D: 14 12 10 8 6 4 2

y C B

−8 −6 −4 −2 −2 −4 A −6 −8

Find the slope of the line using the points A and D.

Find the slope of the line using the points B and C.

What do you notice about the slopes? Will this be true for any two points?

Consider the points A and B on the straight line: A

6 4 2 −4 −3 −2 −1 B1 −2

x 2

−4 −6 −8

Describe how to move from point A to point B.

Express the direction of the movement in part (a) as a simpliﬁed ratio, comparing vertical movement to horizontal movement.

3.01 Slope mathspace.co

225

Identify the slope m for each line: a

−8 −6 −4 −2 −2 −4 −6 −8

Find the slope of the line that passes through the given points: a

(2, 0) and (0, 3)

(4, 7) and (1, 10)

(6, 4) and (3, 4)

(10, 5) and (15, −5)

(−8, 20) and (−4, 10)

(11, −17) and (3, −9)

(−15, −35) and (−30, −25)

(−2.25, −8) and (−1.75, 6.5)

(2.3, 4.8) and (5.6, 1.2)

Complete the table.

For each table of values: i

Find the slope.

x 1 2 3 4 5 y −3 −6 −9 −12

x −2 −1 0 1 2 y −10 −5 10

3 15

Two slopes are calculated. • Line 1 slope = 3 • Line 2 slope = 5 Which line is steeper? Explain why.

226

−8 −6 −4 −2 −2 −4 −6 −8

14 12 10 8 6 4 2

x 2

−8 −6 −4 −2 −2 −4 −6 −8

14 12 10 8 6 4 2

x 2

−8 −6 −4 −2 −2 −4 −6 −8

14 12 10 8 6 4 2

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

4 20

x y

−1

1 82

2 74

3 66

4 58

x y

−4 −9

−3

−2 −8

–1

0 −7

2 −6

Let’s extend our thinking 12

Pair up these points to create 3 lines, one with a positive slope, one with a negative slope and one with a slope of 0. A(3, 5), B(4, 6), C(−1, 8), D(7, 5), E(−3, 10), F(4, 12)

A ski resort has two ski runs as shown in the diagram: a

Find the slope of Run A, correct to two decimal places.

Find the slope of Run B, correct to two decimal places.

What is the slope of the steeper run?

Run A 21 m Run B

20 m 29 m

Consider the ramp shown in the diagram: a

Find the slope of this skateboard ramp if it rises 0.9 yd above the ground and runs 1 yd horizontally at the base.

The ramp can only be used as a ‘beginner’s ramp’ if for every 1 yd horizontal run, it has a rise of at most 0.5 yd. Can it be used as a ‘beginner’s ramp’?

Rise Run

Hiro believes that the slope of the line passing through the points (0, 0) and (−2, 5) and the slope of the line passing through the points (0, 0) and (2, −5) are the same. a

Plot the points (0, 0), (−2, 5), and (2, −5) on the coordinate plane.

Is Hiro correct? Explain how you know. y 7 6 5 A 4 3 B 2 1

Consider the points A and B on the straight line: a

Now, find y if the coordinates of C are (1, y) and C also lies on the same line.

Now, find x if the coordinates of C are (x, −19) and C also lies on the same line.

−5 −4 −3 −2 −1 −1 −2 −3

Bonnie recorded her savings (in dollars) over a few months in the table.

Months Savings

1 20

x 1 2 3 4 5

2 40

3 60

4 80

Plot the values in the coordinate plane where x corresponds to the months and y corresponds to the savings in dollars. Draw a line to connect the points from the origin.

Find the slope of the line.

Write an equation in the form y = mx representing the relationship between the number of months, x, and Bonnie’s monthly savings, y.

Using the equation in (c), ﬁnd the amount of Bonnie’s savings in one year.

The points A(26, m − 24), B(−1, m) and C(−10, 9) are plotted on the coordinate plane. Find m, given that A, B and C all lie on the same line.

Complete the coordinates using the points and slope provided. a c

(4, −3) and (1, ⬚), slope = −2

(5, 3) and (⬚, 63), slope = 4

b d

(5, 3) and (2, ⬚), slope = −4

(11, ⬚) and (−20, 16), slope =

3.01 Slope mathspace.co

227

Answers

b i −8 ii

3.01 Slope What do you remember? 1 a No

c No

2 a Positive slope

d No

d Positive slope

e Undefined slope

g Negative slope

h Undefined slope

Zero slope

3 a i 2

ii 4

iii

b i −4

ii 1

iii −4

4 No. Reversing the order of the points reverses the subtraction and will change the sign of both the numerator and the denominator, so when divided, the result will have the same sign either way. 5 a i From A, move 2 units down and 8 units to the right. ii From A, move 4 units down and 16 units to the right. b i −1 : 4

ii −1 : 4

2 4 6 8 10 12 14

Let’s practice 6 a 2

7 a From A, move 9 units down and 3 units to the right.

8 a m=3

b m = −4

9 a

b −1

c 0

d −2

−1

−1.09

−1

−10

−5

−4

−3

−2

−1

−9

−8.5

−8

−7.5

−7

−6.5

−6

13 a − 0.72

b −0.35

14 a 0.9

b No

−3

−6

−9

−12

−15

−18

c −0.72

6 y 5 4 3 2 1 −5 −4 −3 −2 −1−1

x 1 2 3 4 5

−2 −3 −4 −5 −6

b Y es, Hiro is correct. The ratio of change in y to change in x is the same between any two points on a line. The slope between the points (0, 0) and (2, −5) and the slope between (0, 0) and (−2, 5) is equal to

10 a i −3

228

−2

The slope of a line represents its steepness, and a larger slope means that the line increases more rapidly in the vertical direction as it moves horizontally. In this case, for every unit increase in the horizontal direction, Line 2 increases by 5 units vertically, while Line 1 increases by only 3 units. Therefore, Line 2 is steeper and increases faster than Line 1.

b −3 : 1

x y

11 Line 2 is steeper (increases faster) because it has a larger slope value than Line 1.

15 a

b 2

4 58

d i

c B oth slopes are the same. This will be true for any two points on a straight line, as the slope of a straight line is constant.

3 66

The negative slope needs to decrease from left to right, so B (4, 6) and C (−1, 8).

−4

2 74

The positive slope needs to increase from left to right, so the x and y-values of the first point need to be lower on the coordinate plane than the x and y-values for the second point. So, E (−3, 10) and F (4, 12).

2 −6 −4 −2 −2

1 82

12 The slope of 0 has to have y-values the same, this will have to be A (3, 5) and D (7, 5).

8 6

0 90

Let’s extend our thinking

b Zero slope

c Negative slope

or −0.25

−1 98

c i 5

b Yes

x y

16 a −1

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

b 7

17 a

100 90 80 70 60 50 40 30 20 10

Savings

b 20

Months 5

c y = 20x

d $240

b 15

c 20

18 m = 1 19 a 3

Answers mathspace.co

229

3.02 Transformations of linear functions Subtopic overview Lesson narrative In this lesson, students will examine transformations of the slope and y-intercept of a parent graph to create the form y = mx + b. Students will engage in an exploration where they analyze how the slope and y-intercept change the appearance of the graph of a function. By the end of the lesson, students should aim to be confident in identifying transformations from an equation or graph and creating graphs and equations using knowledge of transformations.

Learning objectives Students: Page 116

Key vocabulary 

function family



horizontal translation



parent function



vertical dilation



vertical reflection



vertical translation

 slope

Essential understanding A relatively small number of transformations can be applied to the graph of a function to change what the function models without changing the family to which it belongs.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG4 — Mathematical Connections To effectively incorporate mathematical connections into instruction, teachers can highlight the relationships between algebraic expressions and their graphical representations. They can start by exploring the linear parent function, y = x, and discussing how each transformation affects its graph. Teachers can use interactive graphing tools to show these changes in real-time, helping students visualize the impact of altering coefficients and constants in the equation. Additionally, connecting these transformations to real-world contexts, such as how changes in speed or direction affect motion, can make the abstract concepts more tangible.

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MPG5 — Mathematical Representations Teachers can encourage students to use different representations (e.g., equations, graphs, tables) to understand and explore the key concepts in the lesson. For example, when teaching transformations of linear functions, teachers can ask students to represent these transformations both algebraically and graphically. Similarly, when discussing the slope and y-intercept, teachers can ask students to represent these concepts in a real-world context, such as a table showing a company’s profit over time.

Content standards A.F.1 — The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.

A.F.1f — Graph a linear function in two variables, with and without the use of technology, including those that can represent contextual situations.

A.F.1b — Investigate and explain how transformations to the parent function y = x affect the rate of change (slope) and the y-intercept of a linear function.

A.F.1h — Compare and contrast the characteristics of linear functions represented algebraically, graphically, in tables, and in contextual situations.

Prior connections 8.MG.3 — The student will apply translations and reflections to polygons in the coordinate plane. 8.PFA.3 — The student will represent and solve problems, including those in context, by using linear functions and analyzing their key characteristics (the value of the y-intercept (b) and the coordinates of the ordered pairs in graphs will be limited to integers).

G.RLT.3 — The student will solve problems, including contextual problems, involving symmetry and transformation.

A2.F.1 — The student will investigate, analyze, and compare square root, cube root, rational, exponential, and logarithmic function families, algebraically and graphically, using transformations.

Engage Activity Starting salary options

60 mins

Students will investigate transforming linear functions by analyzing various salary options for Louis.

Understanding and skills

Will use

Will develop

Representing linear equations using function notation and understanding how to find slope from a graph or equation.

Identifying the effect on a graph or table of a given function of multiplying a linear function by a constant. Identifying the effect on a graph or table of a given function of adding a constant to a linear function.

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231

Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: Pencil, paper, graphing technology (recommended)

Support students with disabilities Support memory - use previously taught skills and concepts Provide the formula: f (x) = mx + b Where: • m is the slope • b is the y-intercept

Support for English language learners Three reads Have students read the task aloud. On the first read, ask students to describe the situation. Prompt: Students read the problem. Students think/write: Answer the question “What is the problem about?” Answers may look like: • Choosing a job and investigating the total money earned over at least 5 years for each annual pay option. Share: Students are called upon to discuss their answers with the class. On the second read, ask students to interpret the question. Prompt: Students read the problem. Students think/write: Answer the question “What does an answer look like?” Answers may look like: • Report with mathematical models and justification for Louis’ options. • How changing the starting bonus or fees affects the model. • How changing the annual salary affects the model. • How other factors could affect the model. On the third read, have students identify important information. Prompt: Students read the problem. Students think/write: Answer the question “What are the important pieces of information given in the question?” Answers may look like: • Annual salary • Signing bonus • Initial fees or payments

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Classroom guide Hook

Open questions

•

5 mins

Students compare two graphs of linear functions. What are the similarities and differences between these functions? 4

3 2 1 −4 −3 −2 −1 g(x)

−1

−2 −3

x 1

f (x)

−4

Slide 1 from Student Engage Activity

Implementation details Encourage students to share similarities and differences between the two linear functions; f (x) and g(x). Highlight student responses that compare the slope and y-intercept as these key features will help students in the activity when comparing the salary options and identifying the effects of transforming a linear function.

Launch After students read the instructions, begin a discussion on information that they think would be helpful to describing Louis’ salary options. Highlight responses such as different annual salary based on company or a signing bonus depending on the company. Before students form groups, let students know that they will be focusing on an annual salary based on a job of their choice for describing Louis’ options.

5 mins

Louis is thinking about his plans for after graduation. He is looking into the jobs he can get with different degrees or programs and what the starting salaries would be so that he can earn enough to help pay the school fees of his two younger sisters. What are Louis’ options? Slide 2 from Student Engage Activity

Important mathematical concepts: Linear equations Important contextual information: Salary, signing bonus, certification, bonus Suggested grouping: Form groups of 3 or 4 and assign roles

Continue when Students have read the Launch slide and understands the context of the problem.

Explore

Team roles

•

25 mins

Students will have successfully completed the activity when they have compared the linear models for each of the four options and explored and described how changing the starting signing bonus and annual salary affects the linear model. The graph, table, or equation will depend on the starting salary of the group’s chosen job.

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233

Anticipated strategies Table of values Students may use a table of values to compare each option. If students are choosing a table of values, make sure they are still considering the value of the slope and how the different options transformed it.

Write an equation Students may write an equation to compare each option. If f (x) = mx + b represents the total money earned in terms of years for Option 1, where m = annual salary and b = 0 since there is no signing bonus, then the other three options would be: Option 2: g(x) = 1.25 f (x)

Option 3: h(x) = f (x) + 10 000

Option 4: j(x) = f (x) − 5000

Encourage students to use multiple representations; graph, table, and equation when working on their report.

Graph Students may use a graph to compare each option. The graph will depend on the starting function. Here is a sample graph based on a salary of $20 000: y 1 00 000 75 000 f (x)

50 000 25 000

x 1

Misconceptions Misunderstanding y-axis and y-intercept What should the label be for the y-axis? How do you know? What does the y-intercept mean in context?

Misunderstanding slope in context What type of model can be used for each option? What affects the slope of the linear model?

Purposeful questions Ask the following questions to further student discussion and check student understanding throughout the task: • What is the starting annual salary for your group’s chosen job? How does this affect the options? • What are the pros and cons of each option? What makes you say that? • How does changing the starting salary affect the model? Can you show me your examples? • How does changing the signing bonus affect the model? Can you show me your examples? • Can you explain and describe the options using another method? How does this change or validate your predictions of how changing the annual salary/signing bonus affects the model? • What other factors could affect the annual salary and the model?

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Continue when Students have completed the following: mathematical models and justification for the overview of Louis’ options, a summary (with examples) of how changing the starting bonus or fees affects the model, a summary (with examples) of how changing the annual salary affects the model, and a summary of other factors that could affect the annual salary and its model.

Discuss

25 mins

Use a gallery walk for students to share their reports with the class leading to a whole class discussion. Consider sequencing the strategies presented from graph, to table, to equations.

Discussion guide Have students participate in a gallery walk and review each other’s reports for Louis’ options. Here are some questions to provide students to answer during gallery walk: • How does changing the starting signing bonus affect the model? Does this report change your thoughts on how this affects the model? Why or why not? • How does changing the annual salary affect the model? Does this report change your thoughts on how this affects the model? Why or why not? After the gallery walk, have students share their generalizations and examples. Sequence student responses so a graph, table, and equation are used as examples to describe the affect of changing the starting value ( y = f (x) + k) and annual salary ( y = af (x)) affects the linear model. If time, ask students what other factors could affect the annual salary and the model. Students may talk about raises and percent increases which could make the model nonlinear. These are good discussions to have as they are options in real life. However, the major take away is for students to realize that adding a constant to a linear function vertically translates the function and that multiplying a linear function by a constant changes the slope and vertically stretches or compresses a function. Transformations of linear functions will be focused on explicitly in the next lesson, but highlight student responses that get at this idea.

Tools You may ﬁnd these tools helpful: • Graphing calculator • Clear plastic sheets • Blank coordinate plane

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Student lesson & teacher guide Transformations of graphs In this section, students explore the affects of the slope and y-intercept on the graph of a linear function.

Students: Page 116

Understanding slope and intercepts Targeted instructional strategies To help students understand the concepts of slope and y-intercept, ﬁrst explain that the slope represents the rate of change in a linear equation, while the y-intercept is the point where the line crosses the y-axis. This can be shown visually: In the graph, the slope is 2, meaning that for every unit increase in x, y, increases by 2. The y-intercept is 3, meaning that when x = 0, y = 3. Once students understand these concepts, ask them to use the graph to predict how the graph and equation will change if: • The slope is cut in half • The line is shifted down 2 units • The line is reflected over the x-axis

y 6 5 4 3 y-intercept 2 1 −6 −5 −4 −3 −2−1−1

x 1 2 3 4 5 6

−2 −3 −4 −5 −6

Physically transform graphs Student with disabilities support Advise students to sketch the original graph on a clear plastic sheet and apply the transformations by moving or ﬂipping it with respect to a coordinate plane. This will allow students to test out multiple different transformations without having to draw the coordinate plane and function multiple times. This will also allow students to physically see how a transformation is applied and how it affects the graph, providing another avenue for understanding.

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Collect and display English language learner support As students are working, note how students describe the concepts of “translation,” “reflection,” “vertical stretch,” “vertical compression,” “horizontal stretch,” and “horizontal compression.” Collect the different ways that students find to understand these concepts and display them in a common place for the students to access. If students do not come up with alternative ways to word these concepts and are confused by them, suggest some of your own. For example: • Translation • Moves up, down, left, or right • Same rate of change, different y-intercept • Add to or subtract from the y-value • Reﬂection across x-axis • x-axis is a mirror • Rate of change is opposite sign

• y-coordinates change sign

• Vertical stretch • Pulling away from x-axis • Different rate of change, same x-intercept

• Steeper slope

• Vertical compression • Pushed towards x-axis • Different rate of change, same x-intercept

• Less steep slope

Take care to address any rewordings that contradict or are too similar to other concepts that the student will learn in the future, such as thinking that only one transformation can every happen to a function or that transformations are always vertical.

Using structure to identify dilations Address student misconceptions When ﬁrst learning transformations, students may struggle to identify the scale factor of a dilation or to identify whether a dilation represents a stretch or a shrink. For example, they may say factor of 2.

represents a stretch by a

Use technology to show students that a scale factor between 0 and 1 will decrease each of the y-values and make the slope less steep. A scale factor larger than 1 will increase the y-values and make the slope steeper.

Exploration Students: Page 116

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Suggested student grouping: Small groups In this exploration, students will manipulate the sliders for m (the slope) and b (the y-intercept) in the given Geogebra applet. They will observe the changes in the graph of a linear function as they adjust these two parameters. This exploration gives students a hands-on experience of how the slope and y-intercept affect the graph of a linear function. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What happens to the graph when the value of m changes? When the value of m changes, the slope of the graph changes. If m is positive, the graph slopes upwards as we move from left to right. If m is negative, the graph slopes downwards. The larger the absolute value of m, the steeper the graph. 2. What happens to the graph when the value of b changes? When the value of b changes, the y-intercept of the graph changes. This means the graph moves up or down along the y-axis. The value of b is the point where the graph crosses the y-axis. Purposeful questions • How would you describe the relationship between the slope and the steepness of the graph? • How does the y-intercept inﬂuence the position of the graph on the coordinate plane? Possible misunderstandings • Students may not fully grasp the concept of slope as a rate of change, leading them to not understand how the slope affects the steepness and direction (upward or downward) of the graph. • Students may not recognize that the slope changes from increasing to decreasing for negative values of m.

Applet creation extension Targeted instructional strategies To extend and develop technical skills, encourage students to recreate the applet in the student lesson. Students may deepen their understanding of variables by needing to define a variable with a slider or input box. Students may try to define the function for particular values, instead of in terms of the defined variables. Support them to recognize the difference between a parent and a particular function. Students are reminded of the of the definitions of slope and y-intercept and are introduced to the concept of parent functions and function families. They are shown examples of translations, dilations, and reflections with both graphs and equations. Students are told that the parent function f (x) = x can be transformed to graph and write equations.

Students: Pages 116–117

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A vertical translation, or shift, to the y-intercept of f (x) represents the transformed function f (x) + k.

In this example, the solid line represents the function f (x) = x.

The dashed line represents a shift 2 units down from f (x).

1 −4 −3 −2 −1

x 1

−1

The transformed function is represented by f (x) = x − 2.

−2 −3 −4

A horizontal translation, or shift, to the x-intercept of f (x) represents the transformed function f (x + k).

3 2 1 −4 −3 −2 −1

x 1

−1

The solid line represents the function f (x) = x, which is the same as above. Notice we can achieve the same transformed function by shifting the parent function 2 units right. The dashed line represents a shift 2 units right from f (x). The transformed function is represented by f (x − 2) = x.

−2 −3 −4

The slope of a line can be transformed by a vertical dilation, represented as kf (x) with k > 0. The line will either compress or stretch depending on the factor used on the slope, m.

3 2 1 −4 −3 −2 −1

x 1

−1

The black dashed line represents a vertical dilation (stretch) by a factor of 2 on f (x). The equation of the new line becomes f (x) = 2x. The blue dashed line represents a vertical dilation (compression) by a factor of . The equation of the new line becomes

−2

−3 −4

The dashed line represents another transformation on the parent function.

A vertical reflection occurs for kf (x) with k < 0.

2 1 −4 −3 −2 −1

−1

x 1

A change in the slope of f (x) = x from 1 to −1 represents a reﬂection of the line, which becomes f (x) = −x. This vertical reﬂection will transform f (x) = x into f (x) = −x.

−2 −3 −4

Transformations on the parent function f (x) = x can be used to graph and write equations.

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Examples Students: Page 118

Example 1 Determine what transformation has occured from the parent function f (x) = x. 4

3 2 1

Example 1

−4 −3 −2 −1

x 1

−1

Determine what transformation has occured from the parent −2 function f (x) = x. −3 y 4 −4 3 2

Create a strategy

Apply the idea x

First, determine whether the transformation −4 can−3 be−2 −1 Since the slope is negative, a vertical reflection has 1 2 3 4 achieved through a translation, dilation, or reflection. −1 occurred, which means it is in the form kf (x) with k < 0. −2 The line has also been stretched by a scale factor of 2, so −3 k = −2. −4 For our equation, f (x) = −2x.

Create a strategy

Apply the idea

Example 2

First, determine whether the transformation can be Since the slope is negative, a vertical reflection has PurposeConsider achieved the through a of translation, dilation, orfreflection. whicha means is in the kf (x)ofwith k < 0. graph the parent function (x) = x. Graph theoccurred, function after verticalitstretch ofform a factor 3, and

Show students to identify andWrite describe transformations of parent a vertical how translation of −2 units. the equation of the transformed line. The line has alsofunctions. been stretched by a scale factor of 2, so k = −2.

y 4 For our equation, f (x) = −2x.

Students: Pages 118–119

3 2

Example 2

1 −4 −3 −2 −1

x 1

Consider the graph of the parent function f (x) = x. Graph−1 the function after a vertical stretch of a factor of 3, and a vertical translation of −2 units. Write the equation of the −2transformed line. −3 y 4 −4 3 2

Create a strategy

A dilation will affect the slope of the line, and−4a translation will 1affect the y-intercept. −3 −2 −1 2 3 4 −1

−2 −3 −4

Create a strategy A dilation will affect the slope of the line, and a translation will affect the y-intercept. 118

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Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co 118

Mathspace Virginia SOL Algebra 1 mathspace.co

Apply the idea A scale factor of 3 for the vertical stretch means the original slope of 1 will be multiplied by 3. This creates the equation y = 3x, which is shown in the graph:

3 2 1 −4 −3 −2 −1

−1

−2 −3 −4

The second shift is 2 negative units, which means the y-intercept will change by −2, so the graph becomes:

3 2 1 −4 −3 −2 −1

−1

−2 −3 −4

The equation of the transformed line is y = 3x − 2.

Example 3 Purpose By ﬁrst identifying the slope and y-intercept, describe the transformations of the following lines from the parent line Illustrate how to graph and represent a function after applying a vertical stretch and a vertical translation. f (x) = x.

a mistakes g (x) = 5x Expected Students may not apply both the vertical stretch and the vertical shift correctly. It’s important to remember that Create a strategy the vertical stretch is applied ﬁrst, followed by the verticalApply shift.the idea The functions f (x) and g (x) have the same y-intercept, but slopes. This means a vertical dilation has Reﬂecting different with students occurred.

The slope is 5.

The y-intercept is 0.

This problem required us to use both a vertical stretch and shift. Discuss with (made students the by order f (x)a=vertical x has been vertically stretched steeper) • When the scale factor is greater than 1, a stretch a factor of 5. of these transformations and how it affected the ﬁnal result. Note that applying the translation ﬁrst, then the has occurred. dilation would in the equation y =0 3and (x 1,− 2) = 3x − 6. • Whenresult the scale factor is between a compression has occurred.

b g(x) = x − 2

12 10 8 6 4 2

Create a strategy 4 The functions f (x) and2g (x) have the same slope, but x

different y-intercepts. This means a vertical translation −8 −6 −4 −2 2 4 6 8 has occurred.

−2 −4

−6 −8

Translate down 2 ﬁrst

Apply the idea The slope is 1.

x −5−4−3−2 −1 1 2 3 4 5 6 7 −2 f (x) = x has been vertically translated (shifted) 2 units down. −4 −6 −8 3.02 Transformations of linear functions 119 −10 mathspace.co −12 The y-intercept is −2.

Then, dilate by a factor of 2

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241

−4 −3 −2 −1

Apply the idea A scale factor of 3 for the vertical stretch means the original slope of 1 will be multiplied by 3. This creates the equation y = 3x, which is shown in the graph:

−1

−2 −3 4 −4 3

Students:The Page 119of the transformed line is y = 3x − 2. equation

1 −4 −3 −2 −1

Example 3

−1

−2 −3

By ﬁrst identifying the slope and y-intercept, describe the transformations of the following lines from the parent line −4 f (x) = x. a g (x) = 5x The second shift is 2 negative units, which means the y-intercept will change by −2, so the graph becomes:

Create a strategy

Apply the idea

The functions f (x) and g (x) have the same y-intercept, but different slopes. This means a vertical dilation has occurred.

The slope is 5.

• When the scale factor is greater than 1, a stretch has occurred. • When the scale factor is between 0 and 1, a compression has occurred.

The y-intercept is 0.

2 1

f (x) = x has been vertically stretched (made steeper) by −4 −3 −2 −1 1 2 3 4 a factor of 5. −1

−2 −3 −4

The equation b g(x) = x − 2of the transformed line is y = 3x − 2.

PurposeCreate a strategy Apply the idea Test students’ understanding slopes andslope, y-intercepts and their is ability to identify how they transform a line. Example 3 f (x) and g (x)of The functions have the same but The slope 1. different y-intercepts. This means a vertical translation

The y-intercept is −2.

Create a strategy

Apply the idea

By mistakes ﬁrst identifying the slope and y-intercept, describe the transformations of the following lines from the parent line Expected has occurred. f (x) = x has been vertically translated (shifted) 2 units down. f (x) = x. might confuse the slope with the y-intercept, Some students leading to incorrect identification of a g (x) = 5x transformations. It is helpful to remind students of slope-intercept form, y = mx + b, and identify how the 3.02 Transformations of linear functions 119 transformation of f (x) corresponds to an operation on x. mathspace.co

Reflecting students Thewith functions f (x) and g (x) have the same y-intercept, The slope is 5. different slopes. This a vertical dilationand has y-intercept Discuss but with students how tomeans identify the slope in an equation and how they affect the graph of The y-intercept is 0. the line.occurred. Point out that a dilation is reflected by changes in the slope, and a translation reflected bybychanges in f (x) = x has been vertically stretchedis(made steeper) • When the scale factor is greater than 1, a stretch a factor of 5. the y-intercept.

has occurred. • When the scale factor is between 0 and 1, Students: Page 119 has occurred. a compression

b g(x) = x − 2

Create a strategy

Apply the idea

The functions f (x) and g (x) have the same slope, but different y-intercepts. This means a vertical translation has occurred.

The slope is 1. The y-intercept is −2. f (x) = x has been vertically translated (shifted) 2 units down.

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119

Purpose To ensure students understand how to identify the slope and y-intercept of a linear function and describe the transformations from the parent function f (x) = x. Expected mistakes Students might mistakenly think that the slope changes when the function is translated vertically. Students may also say the translation is 2 units up instead of down. Asking students to graph the parent function and transformed equation can help make the transformation more clear.

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Reflecting with students Challenge advanced learners to identify a second transformation that is equivalent to translating f (x) = x down 2 units. Students should say that the function could have been translated right 2 units instead. Discuss with advanced learners whether they think translating a function downward is always equivalent to translating the function to the right for linear functions. Would this be the case if the function had been reflected or dilated first? (No.) Is translating a linear function upwards equivalent to translating the function to the left (assuming no reflection or dilation occurred first)?

Students: Page 120 c

Create a strategy

Apply the idea

The functions f (x) and g (x) have different slopes and different y-intercepts. This means a vertical dilation and a translation have occurred. In addition, the new slope is negative, which means a reﬂection has occurred.

The slope is

The y-intercept is 4. f (x) = x has been reﬂected across the x-axis, compressed vertically (made less steep) by a factor of , and vertically translated 4 units up.

Example 4 Purpose To checkThe that students can identify multiple transformations from the parent function (x) =plan x. to set drama club is raising money for a ﬁeld trip to see a Broadway musical. To raise the money,f they up a face-painting stand during the high-school football game, and charge $4 per person. The function R (x) = 4x

Expected mistakes represents their revenue in dollars where x represents the number of faces painted. Studentsca might forget to mention that a negative slope represents a reflection. students put the two The club members spent $45 on face-painting supplies. Write the function P (x) that Ask represents their to proﬁt. slopes side-by-side to make the differences more clear. Create Create a a strategy strategy

Apply Apply the the idea idea

Reflecting with students The f (x) and g (x) have different slopes and The Profitfunctions is calculated by subtracting cost from the revenue. P (x) slope = 4x −is45 . y-intercepts. a vertical dilation and the orientation of the line and how the fraction changes Discuss different with students howThis themeans negative slope impacts The y-intercept is 4. a translation have occurred. In addition,will thethis newgraph slope isbe more steep or less steep than the parent graph the steepness of the line. For example, f (x) = x has been reflected across the x-axis, compressed negative, which means a reflection has occurred. f (x) = x?b Describe the transformation applied to R (x) to get P (x). strategy Students:Create Pagea 120

vertically (made less steep) by a factor of , and vertically translated 4 units up.

We have subtracted 45 from the revenue function which can be represented by P (x) = R (x) − 45. The two functions have the same slope, but their y-intercepts are different, meaning a translation has occurred.

Example 4

Apply the idea

Reﬂect and check

R (x) drama has been 45for units to get In context, this means revenue decreased The clubtranslated is raising down money a field tripPto(x). see a Broadway musical. To raisethat thethe money, theyisplan to set by theand costs of the$4 supplies, and that how weRget up a face-painting stand during the high-school football game, charge per person. Theisfunction (x) =the 4x profit function.of faces painted. represents their revenue in dollars where x represents the number a The club members spent $45 on face-painting supplies. Write the function P (x) that represents their profit. c The drama team realizes that they will need to paint 11 faces to break even at the current rate, so they decide to Create a strategy Apply the idea increase the cost per person to $8. The function N (x) = 8x represents their new revenue function. Profit is calculated by subtracting cost from the revenue. P (x) = 4x − 45 Describe the transformation from the original revenue function, R (x), to the new revenue function, N (x).

Create a strategy

b Describe the transformation applied to R (x) to get P (x). When comparing R (x) = 4x and N (x) = 8x, we can see the only difference is a change in the slope. This means a Purposedilation has occurred.

Create a strategy

Show students how transformations can be applied to a real-world context.

We have subtracted 45 from the revenue function which can be represented by P (x) = R (x) − 45. The two functions Apply the idea have the same slope, but their y-intercepts are different, meaning a translation has occurred. Expected Themistakes slope of N (x) is double the slope of R (x). This means R (x) has been vertically stretched by a factor of 2 to get (x). StudentsNmight not remember the difference between revenue and profit or how profit is calculated. Remind

Apply the idea

Reﬂect and check

them that the is the down money theytomake from face-painting andmeans profitthat is found by subtracting R (x) hasrevenue been translated 45 units get P (x). In context, this the revenue is decreased the by costs from the revenue. the costs of the supplies, and that is how we get the proﬁt function.

3.02 Transformations of linear functions Mathspace Virginia SOL Algebra 1 will need to paint 11 faces to break even at the current rate, so they decide to c120 The drama team realizes that they mathspace.co mathspace.co increase the cost per person to $8. The function N (x) = 8x represents their new revenue function. Describe the transformation from the original revenue function, R (x), to the new revenue function, N (x).

243

Create a strategy

Apply the idea

represents their revenue in dollars where x represents the number of faces painted. The functions f (x) and g (x) have different slopes and is . P (x) that represents their profit. a The club members spent $45 on face-painting supplies. The Writeslope the function different y-intercepts. This means a vertical dilation and The y-intercept is 4. a translation have occurred. In addition, the new slope is Create a which strategy idea reflected across the x-axis, compressed fApply (x) = x the has been negative, means a reflection has occurred. is calculated P (x) = 4x − 45 less steep) by a factor of , and vertically (made Students:Profit Page 120 by subtracting cost from the revenue. vertically translated 4 units up. b Describe the transformation applied to R (x) to get P (x).

Create a strategy Example 4 We have subtracted 45 from the revenue function which can be represented by P (x) = R (x) − 45. The two functions The clubslope, is raising for a field trip see a Broadway To raise the money, they plan to set havedrama the same but money their y-intercepts are to different, meaning amusical. translation has occurred. up a face-painting stand during the high-school football game, and charge $4 per person. The function R (x) = 4x represents faces painted. Apply the their idearevenue in dollars where x represents the number Reflectofand check a members spent $45 face-painting Write the function P (x)that thatthe represents profit. by R (x)The hasclub been translated down 45on units to get P (x).supplies. In context, this means revenuetheir is decreased the costs of the supplies, and that is how we get the profit function. Create a strategy Apply the idea P (x) = 4x − 45

Proﬁt is calculated by subtracting cost from the revenue.

c The drama team realizes that they will need to paint 11 faces to break even at the current rate, so they decide to increase the per personapplied to $8. The b Describe the cost transformation to Rfunction (x) to getNP(x) (x).= 8x represents their new revenue function. Purpose Describe the transformation from the original revenue function, R (x), to the new revenue function, N (x).

Students demonstrate that they can identify a transformation from an equation representing a contextual Create a strategy situation. Create a strategy We have subtracted 45 from the revenue function which can be represented by P (x) = R (x) − 45. The two functions Whenthe comparing R (x) but = 4xtheir andy-intercepts N (x) = 8x, we see themeaning only difference is a change in the slope. This means a have same slope, arecan different, a translation has occurred. Expected mistakes dilation has occurred.

StudentsApply may the switch translated down. Relate it back to the context, and ideathe function names, and say P(x) was Reﬂect and check ask if weR subtracted the costs from the revenue or from the profit. Apply the idea (x) has been translated down 45 units to get P (x). In context, this means that the revenue is decreased by The slope of N (x) is double the slope of R (x). This means R (x) been vertically stretched byisa how factor ofget 2 tothe getproﬁt thehas costs of the supplies, and that we function.

Students:NPages 120–121 (x).

c The drama team realizes that they will need to paint 11 faces to break even at the current rate, so they decide to increase the cost per person to $8. The function N (x) = 8x represents their new revenue function. Describe the transformation from the original revenue function, R (x), to the new revenue function, N (x). 120

Mathspace

Virginia SOL Algebra 1

mathspace.co Create a strategy

When comparing R (x) = 4x and N (x) = 8x, we can see the only difference is a change in the slope. This means a dilation has occurred.

Apply the idea The slope of N (x) is double the slope of R (x). This means R (x) has been vertically stretched by a factor of 2 to get N (x).

Reﬂect and check Using technology to compare the graphs of R (x) and N (x), we can see that N (x) does represent a vertical stretch, and its outputs are double the outputs of R (x). 120

Mathspace Virginia SOL Algebra 1 mathspace.co

12 8 4 −3 −2

−1

−4

x 1

−8 −12 −16

Example 5 Purpose To showUse students how tofrom identify transformations a function function that transformations the parent function f (x) = applied x to graphto the f (x) = 3xis− not 4. the parent function. Create a strategy 244

Mathspace Virginia Algebra 1 Teacher Edition First, identify the SOL transformations from the parent function. Then, use the transformations to graph the function. mathspace.co

Apply the idea 6

First, we can perform the vertical dilation by a scale factor of 3.

Reﬂect and check Expected mistakes technology compare R (x) and (x), the we can seefactor. that N (x) does verticalto stretch, StudentsUsing might identifytothe slopethe asgraphs 8 andofstate thatNas scale Use a represent table of avalues help and students outputs are double outputs of (x). scale factor is 2. see thatitsthe y-values havethe doubled, soRthe

x R(x) = 4x N(x) = 8x

0 0 0

1 4 8

2 8 16

3 12 24

4 16 32

12 8 4

Reflecting with students −3 −2 −1 1 2 3 −4 Ask students to identify a reasonable domain to represent the function. Encourage them to discuss their answers as a class. The domain for this should be the −8 set of whole numbers, since negative values and rational −12 or irrational values for the number of faces would not make sense in this context. −16

Students: Page 121

Example 5 Use transformations from the parent function f (x) = x to graph the function f (x) = 3x − 4.

Create a strategy First, identify the transformations from the parent function. Then, use the transformations to graph the function.

Apply the idea 6 5 4 3 2 1

y f (x) = x

The resulting function is f (x) = 3x. x 1 2 3 4 5 6

−6 −5 −4 −3 −2−1 −1 −2 −3 −4 −5 f (x) = 3x −6 6 5 4 3 2 1

First, we can perform the vertical dilation by a scale factor of 3. Every y-value will be 3 times the value of the y-value of the parent function.

Next, we can perform the vertical translation down 4 units. Every y-value will be 4 less than the y-value of the function y = 3x. The resulting function is f (x) = 3x − 4.

x −6 −5 −4 −3 −2−1 1 2 3 4 5 6 −1 −2 f (x) = 3x −3 −4 f (x) = 3x − 4 −5 −6

3.02 Transformations of linear functions Purpose mathspace.co Show students how to use transformations from the parent function to graph a given function.

121

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245

Understanding order of transformations

use with Example 5

Address student misconceptions Students may need support understanding that the order of transformations affects the final position of the resulting function. The order for transformations is similar to the order of operations since reflections and dilations use multiplication to affect a function and a translation uses addition or subtraction to affect a function. Highlight the following order for transformations: 1. Reflections and dilations 2. Translations

Students: Page 122

Idea summary The slope of a line is represented by

m (x1, y1) (x2, y2)

slope a point on the line a second point on the line

The parent function f (x) = x can be transformed to write new functions with changes to the slope and y-intercept. A vertical shift represented by f (x) + k will translate the graph of a total of k units up if k > 0 or k units down if k < 0. A horizontal shift represented by f (x + k) will translate the graph of a total of k units right if k > 0 or k units left if k < 0. A transformation of a vertical dilation kf (x) will stretch a graph’s slope if k > 1 or compress the graph if 0 < k < 1. A vertical reflection of k will change the sign of the slope.

Practice What do you remember?

Practice 1

Write the new equation of the function f (x) = x after a transformation of: a

a vertical shift 3 units down.

Students: Pages 122–127 c a vertical shift 2 units up.

a vertical dilation by a factor of .

a vertical dilation by a scale factor of 5.

graph of y = x is shown with a dashed line. Does the solid line show a vertical stretch or a vertical What do 2youThe remember? compression? a

Write the new equation of4the function f (x) = x after a transformation of: 4 3

a vertical shift 3 units down.

1 a vertical shift 2 units up.

−4 −3 −2 −1

x 1

−1

a vertical dilation by a factor of .

1 a vertical dilation by a scale factor of 5.

−4 −3 −2 −1

−1

−3

−3 −4

1 −4 −3 −2 −1

−2

−4

246

−2

x 1

−1 Mathspace Virginia SOL Algebra 1 Teacher Edition −2 mathspace.co

−4 −3 −2 −1

−1

−2

−3

−4

The graph of y = x is shown with a dashed line. Does the solid line show a vertical stretch or a vertical compression? a

1 −4 −3 −2 −1

x 1

−1

−4 −3 −2 −1

−2

−3

−4

3 2

1 2

−4 −3 −2 −1

−2

x 1

−1

x 1

−1

−2

−4 −3 −2 −1

x 1

−1

−2

−3

−4

Match the type of transformation that has been applied to the parent function f (x) to achieve each graph. Answers may be used more than once. i

Translation

Vertical Dilation

7 6 5 4 3 2 1

−7 −6 −5 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7

x 1 2 3 4 5 6 7

iii

Vertical Reflection 7 6 5 4 3 2 1 −7 −6 −5 −4 −3 −2 −1−1

x 1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

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247

7 6 5 4 3 2 1

−7 −6 −5 −4 −3 −2 −1−1

1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

7 6 5 4 3 2 1

1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

Match the type of transformation that has been applied to the parent function f (x) to achieve each function. Answers may be used more than once. i

Translation

Vertical Dilation

iii

Vertical Reflection

f (x) =

f (x) = x + 4

f (x) = 5x

f (x) = −x

For each type of transformation applied to y = x, explain whether the following key features change. If they change, explain how. • Slope • y-intercept • Domain a

Vertical dilation

Reﬂection across the x-axis

Determine which slope represents the steepest line: A

SOL

Vertical translation

−7

−1

Let f (x) = 2x and g(x) = 10x − 2. Compare the statements to compare the graph of g(x) to the graph of f (x). a b

The graph of g(x) is shifted ⬚ from the graph of f (x).

The graph of g(x) is ⬚ than the graph of f (x). A up

down

steeper

less steep

Let’s practice 8

The functions f (x) and g (x) = f (x) + k have been graphed on the coordinate plane. Determine the value of k for each graph. a

y 6

2 −8 −6 −4 −2 −2 f (x)

248

x 2

−4

−6 −4 −2 −2 f (x)

−4 −6

−6

−8

g(x) −8

−10

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x 2

6 g(x)

8 10

−10 −8 −6 −4 −2 −2 g(x) −4 f (x)

f (x) x

−8 −6 −4 −2 −2 g(x) −4

−6

−8

The functions f (x) and g (x) = kf (x) have been graphed on the coordinate plane. Determine the value of k for each graph. a

y 8

b 8

g(x)

4 f (x)

2 −8 −6 −4 −2 −2

x 6

−6

f (x)

−8

y g(x)

x 4

f (x)

−6

−8 −6 −4 −2 −2 g(x) −4

−8

−6

−8 −6 −4 −2 −2 f (x) −4

−4 −3 −2 −1 −2 g(x) −4

−4

−6 −8

−8

Describe the transformations of the given line from the parent function f (x) = x. a

1 −4 −3 −2 −1

−1

−2

x 1

−4 −3 −2 −1

−1

x 1

−2

−3

−4

3.02 Transformations of linear functions mathspace.co

249

1 −4 −3 −2 −1

−1

−2

−3

−4

−1

iii

−4 −3 −2 −1

b 4

−1

−4 −3 −2 −1

−2 −3

−4

−1

x 2

−1

−3

x 1

y = 3x + 1

−2

−4 −3 −2 −1

−1

−2

−3

−4

For each equation: i

Describe the transformations of the given line from the parent function f (x) = x.

Find the value of the y-intercept.

g (x) = x − 5

g (x) = x + 3

g(x) = −5 + 3x

g(x) = − x − 4

iii

Find the domain.

g(x) = 3 −

Find the range.

g (x) = 2x + 1

y = − 2x + 3

f (x) = −5x + 3

Use transformations from the parent function f (x) = x to graph each function. a

250

y = −x

x 1

−1

−2

−4 −3 −2 −1

Match each graph to the equation: i

x 1

f (x) = 3x

f (x) =

f (x) = −x − 4

Find the resulting equation when y = x undergoes these transformations. a

A dilation by a factor of 10

A dilation by a factor of

and a reﬂection across the x-axis

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

A vertical translation of 10 units up

A reﬂection about the x-axis and a vertical translation of 7 units down

A vertical compression by a factor of 0.75 and a shift of 2 units upwards

Explain the difference between the graphs of the equations y = 2x and y = −2x.

Let’s extend our thinking 16

Consider the graph of the function f (x) = x. Sketch the graph of the transformed function if:

the slope is tripled.

the slope is multiplied by −1.

the y-intercept is moved up 4 units.

the slope is halved, and the y-intercept is moved down 2 units.

−4 −3 −2 −1 −1

−2 f (x) = x

−3 −4

A personal tutor charges a transportation fee of $10 plus $20 per hour. The tutor’s fees can be modeled by f (x) = 20x + 10. For the beginning of the school year, the tutor is offering a reduced rate of $10 per hour, plus the transportation fee. The new rate can be modeled by r (x) = 10x + 10. Describe the transformation from the original fees, f (x), to the new fees, r (x).

A café owner models his daily coffee sales with the function C (x), where x is the number of cups of coffee sold. After a marketing campaign, the sales increase by a factor of 1.5. If he originally sold 100 cups in one day, how many cups will he sell in a day after the campaign?

Oreste draws the graph of g (x) = −f (x), as shown. Describe the error Oreste made.

3 2

g(x)

f (x)

−4 −3 −2 −1

x 1

−1

−2 −3 −4

Stavros has a bank account with $100. Every week, Stavros deposits an additional $50 into the account. A graph of Stavros’ bank balance reveals that the growth of his account is linear. Suppose that Stavros was instead depositing an additional $100 every week. Explain whether or not this could be represented by a vertical stretch of the current graph.

Balance 500 400 300 200 100 Weeks 1

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251

Answers

12 a i Vertical translation down 5 units ii −5

3.02 Transformations of linear functions

c Vertical stretch 3 a i. Translation c ii. Vertical Dilation 4 a ii. Vertical Dilation c ii. Vertical Dilation

b Vertical compression d Vertical stretch b iii. Vertical Reflection d ii. Vertical Dilation b i. Translation d iii. Vertical Reflection

5 a • The slope will not change. • The y-intercept will move up or down, depending on the direction of the translation.

• The domain will not change.

b • The slope will change based on the value of the scale factor. If the scale factor is greater than 1, the slope will be steeper. If the scale factor is greater than 0 but less than 1, the slope will be less steep.

• The y-intercept will not change.

• The domain will not change.

• The y-intercept will not change.

• The domain will not change.

iii All real x

ii 1

iii All real x

ii 5 f

ii −4

iii All real x

factor of

and vertically translated up 3 unit iii All real x

ii 3

iii All real x

ii 3 4 3 2 1

x 1

−3

9 a k=4

c k = −2

11 a Equation iv: y = 3x c Equation iii: y = −x

−3 −4

3 2

b Equation ii:

d Equation iv:

−4 −3 −2 −1 −1 −2 −3 −4

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

−4 −3 −2 −1 −1 −2

b V ertical stretch by a factor of 2 and a translation up 3 units.

d R eﬂection across the x-axis and a vertical translation up 2 units.

by a factor of .

c V ertical stretch by a factor of 4 and a translation down 1 unit.

4 3

10 a Reﬂection across the x-axis and a vertical compression

252

−4

d k=6

iv All real y

h i T he function has been vertically reflected, dilated by a scale factor of 2, and shifted up 3 units

b C

c k=5

iv All real y

g i Vertically reflected, vertically dilated by a scale

6 A

b k = −7

iv All real y

i V ertically reflected and vertically translated down 4 unit

−4 −3 −2 −1 −1 −2

8 a k = −4

iv All real y

e i V ertically dilated by a factor of 3 and vertically translated down 5 unit

13 a

Let’s practice

iv All real y

d i V ertically stretched by a factor of 2 and vertically translated up 1 unit

c • The slope will change to a negative value, so the line will decrease instead of increase.

iii All real x

ii 0

7 a B

iv All real y

compressed by a factor

d y = 5x

2 a Vertical compression

iv All real y

c i Reflected across the x-axis and vertically b

c y=x+2

iii All real x

ii 3

What do you remember? 1 a y=x−3

iii All real x

b i Vertically translation up 3 units

iv All real y

9 8 7 6 5 4 3 2 1

5 4 3 2 1

y=x+4

−9−8−7−6−5−4−3−2 −1 −1

−4 −3 −2 −1

1 2 3 4 5 6 7 8 9

−2 −3 −4 −5 −6 −7 −8 −9

−1

−2

2 1 −4 −3 −2 −1 −1

−2

14 a y = 10x

−3

c y = x + 10

d y = −x − 7

e y = 0.75x + 2 15 The equation y = 2x represents a line with a slope of 2 and a y-intercept at (0, 0). The equation y = −2x represents a line with a slope of −2 and a y-intercept at (0, 0). The difference between the lines is y = 2x is increasing from left to right while y = −2x is decreasing from left to right. Let’s extend our thinking 16 a

3 2

y = 3x

−4 −3 −2 −1 −1 −2

1 −4 y = x−2 2 −5

17 A vertical compression by a factor of vertical translation up 5 units.

followed by a

18 150 cups 19 Oreste reﬂected the graph of f (x) across the y-axis instead of the x-axis when drawing the graph of g (x). 20 Even if Stavros changed the amount he was depositing each week, his initial bank balance at Week 0 will not change. However, when the graph is transformed by a vertical stretch, the y-values of all the points would increase, including the initial amount. Since this is not the case for when Stavros changes the amount he deposits, this change cannot be represented by a vertical stretch of the current graph.

−3 −4

b y = −x

3 2 1

−4 −3 −2 −1 −1 −2

−3 −4

Answers mathspace.co

253

3.03 Slope-intercept form Subtopic overview Lesson narrative In this lesson, students will build on their understanding of linear relationships and formalize it into a consistent algebraic form: slope-intercept form. They will reason abstractly and quantitatively by writing equations in slopeintercept form from contexts and connecting the parts to quantities. Students will complete an exploration activity where they compare different representations of a function. By the end of the lesson, students should aim to be confident in writing the equation of a line in slope-intercept form, graphing lines from slope-intercept form, and being able to model real-world scenarios in slope-intercept form.

Learning objectives Students: Page 128

Key vocabulary 

slope



slope-intercept form



y-intercept

Essential understanding Different representations of a function may highlight or hide different characteristics but they do not change the function itself. Linear functions have a constant rate of change, often referred to as slope, and can be represented by the equation y = mx + b.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG 1 — Mathematical Problem Solving To incorporate mathematical problem-solving into instruction on the slope-intercept form, teachers can present students with real-world scenarios that require the application of y = mx + b. Teachers can begin by explaining the components of the slope-intercept form through relatable examples such as determining the cost of goods over time or calculating the trajectory of an object. By posing problems that require students to identify and interpret these components from graphs, tables, and equations, teachers can engage students in meaningful problem-solving activities. Encouraging students to create their own problems based on personal experiences can further enhance their understanding. 254

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MPG3 — Mathematical Reasoning

MPG4 — Mathematical Connections

Teachers can provide opportunities for students to apply recursive (inductive) reasoning by considering how to move from one point to the next using the slope. Students can be guided to derive the equation of a line from given data points or a graph, justifying each step in their calculations. Teachers can encourage students to analyze how changes in the slope or y-intercept affect the graph of the line, fostering a deeper understanding of linear relationships. By presenting problems that require evaluating the validity of different solution methods, students can strengthen their logical reasoning skills.

Teachers can facilitate activities that link the slopeintercept form to other areas of mathematics and realworld contexts. Students can be encouraged to connect linear equations to proportional relationships, functions, and patterns. By modeling real-world scenarios using slope-intercept form, such as calculating taxi fares or predicting earnings, students can see the practical applications of linear functions. Teachers can highlight the relationship between algebraic expressions and their graphical representations, reinforcing the idea of mathematics as an integrated field of study.

A.F.1di — Write the equation of a linear function to model a linear relationship between two quantities, A.F.1h — Compare and contrast the characteristics of including those that can represent contextual situations. linear functions represented algebraically, graphically, Writing the equation of a linear function will include the in tables, and in contextual situations. following situations: i) given the graph of a line;

A.EI.1 — The student will represent, solve, explain, and interpret the solution to multistep linear equations and inequalities in one variable and literal equations for a specified variable.

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A2.F.1 — The student will investigate, analyze, and compare square root, cube root, rational, exponential, and logarithmic function families, algebraically and graphically, using transformations.

A.ST.1 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on representing bivariate data in scatterplots and determining the curve of best fit using linear and quadratic functions.

Engage Activity The cost of living

60 mins

Students will investigate an individual’s earnings and expenses over time. They will make decisions on how many hours the person must work in order to set aside savings and afford various types of housing.

Understanding and skills

Will use Writing an equation in slope intercept form from a graph or table.

Will develop Writing a linear equation in slope intercept form from a written description in a real-world context. Graphing a linear function from an equation in slope intercept form. Interpreting the graph of a linear function in context.

Could extend Adjusting an equation in slope intercept form based on new information.

Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: None.

Support students with disabilities Support organization - solve multistep or complex problems Encourage students to read through the entire task and outline what’s required in each part. For this task students will need to: 1. Write three linear equations to model the amount of money their chosen person will earn if they work a different number of hours per week. 2. Explore an applet that models income and expenses and determine three levels of expenses based on their three income equations. 3. Write a report exploring two housing options and how the income and expenses ﬁt each option.

256

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In order to keep track of the details in each submission it may be beneﬁcial to print a checklist that clearly lists each submission and what is expected: • Three equations for income. • Three expense recommendations (one per income). • A report covering the income and expense considerations for two different housing options that includes: how much money the family needs to earn to afford the housing, what percent of their income the housing would be, how much their chosen person needs to work to also have $100 in savings each week. Assign one group number to be the time keeper for the activity to remind their group when to transition from one part of the work to the next. If desired, write a recommended timing outline on the board for the timekeepers to follow and check in with groups at regular intervals to see where time extensions are needed for the class as a whole.

Support for English language learners Three reads Have students read the task aloud. On the ﬁrst read, ask students to describe the situation. Prompt: Students read the problem. Students think/write: Answer the question “What is the problem about?” Answers may look like: • Providing advice to people who are looking to ﬁnd a new home. • Sharing information regarding how many hours are needed to work to afford the rent for different situations. Share: Students are called upon to discuss their answers with the class. On the second read, ask students to interpret the question. Prompt: Students read the problem. Students think/write: Answer the question “What could an answer look like?” Answers may look like: • Sharing moving advice with information about how much they need to work to afford the rent. • Report with percentage of income they would spend on rent in various situations. Engage the class in a discussion about the considerations when trying to move, invite students to share their experiences with moving, and create a list on the board documenting the information students want to know. On the third read, have students identify important information. Prompt: Students read the problem. Students think/write: Answer the question “What are the important pieces of information given in the question?” Answers may look like: • Savings • Hourly incomes • Additional information, such as profession or size of apartment, and how this plays a role in the task.

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Classroom guide Hook Students write observations about the discrepancies between rent prices and hourly wages for available jobs.

Notice and wonder

•

5 mins

What do you notice? What do you wonder? WANTED

Implementation details

ENGLISH TUTOR

A student may notice the hourly wage listed for a job, and wonder if they could afford to rent or buy a two bedroom house. Students may also wonder how much money you could earn per year with a given hourly wage, or how much they would need to earn in order to afford one of the apartments listed.

$10.2 - $12 Per Hour Contractor

NURSING ASSISTANT

Encourage students to try calculations to investigate their questions about affordability. Ask students what they think about the discrepancies and whether or not they see any issues in the gap between affordable housing and the salaries of available jobs.

FOR RENT $2.0 K 1100 sqft 4 Bedroom 1 Bathroom

$1.4 K 675 sqft

$36 Per Hour Full time (12 hr day)

2 Bedroom 1 Bathroom

FRY COOK

$1.2K 1100 sqft

$13.25 Per Hour Part-time

CLERK

1 Bedroom 1 Bathroom

$3.7K 1100 sqft

Slide 1 from Student Engage Activity

Launch

5 mins

Provide students time to read the information individually before forming groups. Important mathematical concepts: Linear equations, slope Important contextual information: Rent Suggested grouping: Form groups of 3 or 4 and assign numbers.

People are looking for a new home and have asked for your advice. Your group will be selecting one to investigate: • Danilo Santos just graduated high school and wants to move out of home. He earns $10.50 per hour (non-tipped) as a line cook. $270 in savings. • Lexie Smith wants to move to a larger apartment so her daughter can have her own room. Earns $5.95 per hour as a server, plus $9 per hour in tips. $500 in savings. • Elonso Ceja has a wife and two young children. He wants to move to the city so he spends less time commuting and more time with his family. Earns $21 per hour at a construction company. $430 in savings. Slide 2 from Student Engage Activity

Continue when Students have read the Launch slide and understand the context of the problem.

258

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Explore Students will begin the task by choosing a person to investigate with their group. That means that each groups answers will vary. As students are writing their reports, encourage students to consider factors when choosing a place to live and discuss the realistic/ unrealistic working hours versus rent prices of their chosen individual.

Numbered heads together

•

35 mins

Use the applet to investigate the problem. Expenses (incl. rent)

500

income per week $500 40 000

Anticipated strategies Write an equation Students can write the equation representing household earnings over time in slope-intercept form.

30 000

20 000

Students should be able to write an equation from a table or graph, but may struggle to write an equation 10 000 when given data in terms of real-life context. Ask students who are struggling to think about what 2275 a graph of a person’s income would look like. For 0 10 20 example: • Does the graph increase or decrease? Slide 5 from Student Engage Activity • What is the starting point of the graph? • Are there speciﬁc points that you can identify that will be on your graph?

Students can use this knowledge to help write equations to represent each line on the graph. Ask students to identify different parts of an equation in slope intercept form in terms of the situation. For example, the y-intercept represents how much money the person will start with. Students will produce three equations in slope-intercept form which show their total savings as a function of time.

Graph an equation Students can graph an equation given in slope intercept form. Students will graph the three equations using the applet, and use the graph to identify whether their chosen individual can afford an appropriate housing type and whether they will be able to save the desired $100 each month. For the desired housing type and savings of $100 per week, students must decide how many hours their person will work to meet those conditions.

Compare graphs Students can compare multiple graphs and draw inferences about how much their chosen person would need to earn to be able to afford different housing types.

Misconceptions Miscalculating the slope How do you calculate the slope of a line? How will the different hours worked per week change that?

Understanding graphical solutions Why do you think our graphs are overlapping? Why is this overlap point signiﬁcant? What does sharing a point in common mean?

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Purposeful questions When students are writing their recommendations, make sure to encourage meaningful information to further student discussion. Use the following questions to check for understanding and encourage critical thinking: • How do your housing choices affect the hours your chosen person will have to work? How do you know? • Are the required working hours realistic? What makes you say that? • Will the person be able to save $100 a week realistically? Why or why not?

Continue when Students have created three equations and graphed those equations on the applet. They have written an explanation of how much their chosen person needs to work in order to afford a speciﬁc housing type and a weekly savings of $100.

Discuss

15 mins

Have a class discussion to chat about student equations, graphs, and comparisons of graphs and equations in context. Consider making connections from the discussion to slope intercept form.

Discussion guide Begin by having at least one group present for each person; Danilo Santos, Lexie Smith, and Elonso Ceja. When presenting, students should be able to share their equations as well as the recommendations made for their person. Encourage groups to show how they determined how much their chosen person would be able to save, and how much they would have to work in order to afford their chosen housing type. In particular, emphasize the work done by groups who used different solving methods such as by solving the equations, identifying solutions on the graph, or by producing a table of values. Next, ask students about whether they think their solutions are realistic for their chosen person. For example, is it realistic for a parent to work 55 hours to afford a one bedroom apartment? Feel free to also address the common ﬁnancial recommendation which states that no more than 30% of a household’s income should go towards rent. By looking at the percentages of income that students have recommended the person spends, they can then access the reasonableness of this given the different circumstances. Take some time to address the different individual situations and see if students are able to propose solutions, whether for the individuals such as an additional adult taking on more employment, or societal solutions such as more affordable housing or higher pay wages.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 8 — 3.06 Slope-intercept form Algebra 1 — 3.01 Slope Algebra 1 — 3.02 Transformations of linear functions

Tools You may ﬁnd these tools helpful: • Graphing calculator • Graph paper 260

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Student lesson & teacher guide Slope-intercept form Students will be introduced to the slope-intercept form of a linear equation. They will be told the advantages of using slope-intercept form before engaging in an exploration about equivalent representations.

Students: Page 128

Anatomy of a linear function Targeted instructional strategies Have students annotate the equation f (x) = mx + b with as many connections to prior learning as they can. Also encourage students to draw connections to other facts about intercepts and rate of change. Consider providing them with one or two annotations to start with. For example, some possible annotations could be: Some possible facts could be: • Rate of change =

Function name

Input value

f (x) = mx + b

Output value

Slope

y-intercept

• Slope is the vertical dilation of the line y = x • y-intercept is the vertical translation of the line y = x • y-intercept is (0, b) or (0, f (0)) • x-intercept is (a, 0) where f (a) = 0

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Not checking for slope-intercept form before identifying values Address student misconceptions A common misconception is that the slope is always the first number after the equals sign for a linear equation. They may also not check to see if y has a coefficient before checking the coefficient of x. Encourage students to check that y is by itself on one side of the equation and to note that m is the coefficient of x, not the first number.

Compare and connect English language learner support Ask students to compare and connect the different representations of linear functions - table, graph, equation, and written context. Ask students which representation they find easiest to identify the slope from and how they are all connected. For example, these four representations could describe the same information: Verbal description: Annabelle is growing a tomato plant for the school garden. When she first planted it, the plant was 1 cm tall. She continues to track the growth and notices that it is growing 2 cm each week.

Graphically 9

8 7

Algebraically: y = 2x + 1

6 5 4 3

Numerically in a table

2 1

x y

x 1

2 3 4 5 6 7 8 9

0 1

1 3

2 5

3 7

4 9

5 11

Provide annotated diagrams and definitions Student with disabilities support Create a display of important prerequisite terms and vocabulary. Take time to review terms that students will use. Invite students to suggest language or diagrams to include that will support their understanding of linear equations in two variables, rate of change, slope and y-intercept. For example: The slope of a line:

• slope = (x2, y2)

• slope = x

rise (x1, y1)

262

run

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

• • Represents the rate of change • Ratio of independent variable : dependent variable

(0, b) x

The y-intercept of a line: • The point where the line crosses or touches the y-axis (dependent variable) • Occurs when x = 0 • Represents the initial value or ﬁxed amount

Exploration Students: Page 128

Suggested student grouping: Small groups Students look at the different representations for a linear function and identify any common features or differences that they see. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What do the table, description, graph, and equation representations have in common? Each representation has a constant rate of change (slope) of 1.5 between the independent variable (given as x or Time in hours) and the dependent variable (given as f (x) or Depth in inches). The dependent variable is always equal to 2 when the independent variable is 0, so each have a y-intercept value of 2.

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2. What is different about each representation? The table of values provides a discrete number of data points relating the two variables. The graph shows a continuous relationship between the two named variables, with some particular points highlighted. The description gives a summary of the relationship between two variables in a context. The function denotes a precise mathematical relationship between the two variables. Purposeful questions • What is a feature of one representation that is not present in another? • When would you use each representation? • How can we identify the rate of change or slope in each representation? • How can we identify the initial value or y-intercept in each representation? Possible misunderstandings • Students may look past the different features of the representations and only observe that all the representations correspond to the same linear relationship. Point out that some representations contain information that is not present in the other representations. After the exploration, students will be shown how to use various given information to write equations of lines in slopeintercept form. Students will be told that the graph of the equation represents points that satisfy the equation.

Students: Page 129 Given the information about a linear function, we can write the equation in slope-intercept form. 36 33 30 27 24 21 18 15 12 9 6 3

Consider this graph of a leaky faucet that drips at a rate of 4 cups per hour. It has already dripped 15 cups of water.

Let x represent the number of hours and y represent the number of cups of water that have leaked from the faucet. First, identify or solve for the slope, which represents the m in y = mx + b. In the graph, we can use a slope triangle to see the slope is . We can also see the rate of change in the context is 4. x 1

2 3 4 5 6 7 8 9

Next, identify or solve for the y-intercept, which represents the b in y = mx + b. In the graph, the y-intercept is at (0, 15), so b = 15. From the context, we can see the initial value of the leaked water is 15 cups. Lastly, rewrite y = mx + b, substituting the solved values of m and b. Since we have found m = 4 and b = 15, we can write the equation: y = 4x + 15 Slope-intercept form is especially helpful when we want to graph a linear function. The graph of a line represents the set of points that satisﬁes the equation of a line. 4

Consider the equation y = 2x − 1.

First, identify the value of b, and plot this as the y-intercept. Since the y-intercept is −1, plot a point at (0, −1).

3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3

Then, identify the value of m, and the value of the rise and the run. From the y-intercept, use the rise and run to create slope triangles to plot additional points. Each point on the graph is a solution of the equation, so every point satisﬁes y = 2x − 1.

−4

Example 1 264

5 4 3 2 1

Consider the following graph of a line: Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

−5 −4 −3 −2 −1

x 1 2 3 4 5

First, identify the value of b, and plot this as the y-intercept. Since the y-intercept is −1, plot a point at (0, −1).

3 2 1 −4 −3 −2 −1

Examples

−1

x 1

−2 −3

Students: Pages 129–130 −4 Example 1 5 4 3 2 1

Consider the following graph of a line:

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

a What is the slope of the line shown in the graph?

Create a strategy The slope of a line is equal to the vertical ‘rise’ divided by horizontal ‘run’. It is the ratio of the vertical change to the horizontal change.

Apply the idea

3.03 Slope-intercept form

129

Considering two points from the graph: (0, 1) and (5, 0), from (0, 1) requires moving 1 unit down and 5mathspace.co units to the right. The slope is

b What is the y-value of the y-intercept of the line shown in the graph?

PurposeCreate a strategy Apply the idea StudentsThe demonstrate that they can calculate the slope of a line from a graph. y-intercept is the point where the line intersects the

y 5 4 Expected mistakes Students may calculate the slope as positive rather than negative if they do not 3pay attention to the direction of 2 Apply the idea the line. 1 x to the right. Considering two points from the graph: (0, 1) and (5, 0), from (0, 1) requires moving 1 unit down and 5 units −5 −4 −3 −2 −1 1 2 3 4 5 slope130 is or . −1 Students:The Page −2 −3 −4 b What is the y-value of the y-intercept of the line shown in the graph? −5

y-axis.

Create a strategy

Apply the idea

The y-intercept is the point where the line intersects the y-axis.

Looking at the graph, the line intersects the y-axis at point (0, 1). Thus, the y-value ofy the y-intercept is y = 1. 5 4 3 2 1

c Write the equation of the line in slope-intercept form.

Create a strategy

−5 −4 −3 −2 −1 1 2 3 4 5 −1 Substitute the values of the slope and y-intercept in the slope-intercept form of the equation of a line. −2 −3 Apply the idea −4 −5 Start with the slope-intercept form

Substitute

and b Looking =1 at the graph, the line intersects the y-axis at point (0, 1). Thus, the y-value of the y-intercept is y = 1.

Reflect and check We can check our equation from the graph. c Write the equation of theusing line inpoints slope-intercept form. Since each point must be a solution to our equation, choose any point on the line. For this example, choose (5, 0). Create a strategy Substitute x = 5inand =0 Substitute the values of the slope and y-intercept theyslope-intercept form of the equation of a line.

Apply the idea

Simplify Start with the slope-intercept form

Since it is true that 0 = 0, this pointSubstitute satisfies our equation. and b = 1

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265

The y-intercept is the point where the line intersects the y-axis.

5 4 3 2 1

x −5 −4 −3 −2 −1 1 2 3 4 5 −1 −2 −3 Expected mistakes −4 Students might mistakenly identify the x-intercept. Students should label their intercepts on the graph with the −5

Purpose Students show that they can identify the y-intercept from a graph.

axis they intersect as a visual reminder before writing an answer.

Looking at the graph, the line intersects the y-axis at point (0, 1). Thus, the y-value of the y-intercept is y = 1.

Students: Page 130 c Write the equation of the line in slope-intercept form.

Create a strategy Substitute the values of the slope and y-intercept in the slope-intercept form of the equation of a line.

Apply the idea Start with the slope-intercept form Substitute

and b = 1

Reflect and check We can check our equation using points from the graph. Since each point must be a solution to our equation, choose any point on the line. For this example, choose (5, 0). Substitute x = 5 and y = 0 Simplify Since it is true that 0 = 0, this point satisfies our equation.

Purpose Students show that they can write an equation for a graphed line in slope-intercept form. Reﬂecting with students SOL Algebra 1 of writing the first term and how each form represents the same slope, Discuss 130 with Mathspace students Virginia the various ways mathspace.co for example: • • •

Students: Page 131 d Find the domain and range of the line.

Create a strategy The domain of a line in the graph represents all possible x-values that the line covers. The range of a line in the graph represents all possible y-values that the line covers.

Apply the idea

Reflect and check

Domain: −∞ < x < ∞

We can see from the graph that the line extends without any breaks or endpoints along the x and y-axis, confirming that the domain includes all real numbers.

Range: −∞ < y < ∞

The domain in interval notation is: (−∞, ∞)

266

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

The domain in set notation is: {x ∣ x∈} The range in interval notation is: (−∞, ∞)

d Find the domain and range of the line.

Create a strategy The domain of a line in the graph represents all possible x-values that the line covers. The range of a line in the graph represents all possible y-values that the line covers.

Apply the idea

Reflect and check

Domain: −∞ < x < ∞

We can see from the graph that the line extends without any breaks or endpoints along the x and y-axis, confirming that the domain includes all real numbers.

Range: −∞ < y < ∞

The domain in interval notation is: (−∞, ∞) The domain in set notation is: {x ∣ x∈} The range in interval notation is: (−∞, ∞) The range in set notation is: {y ∣ y∈}

PurposeExample 2 Show students how to in determine the domain Find the equation slope-intercept form for: and range from a graph of a line. a A line with a slope of

and a y-intercept of −3.

Approach unscaffolded problems

use with Example 1

Targeted instructional strategies Create a strategy d Find the domain and range of the line.

Substitute thethey values of thehave slope approached and y-intercept the in the slope-intercept form Ask students how would problem if it had just been:

Createthe a strategy Apply idea graph of a line: Consider the following

Write the equation of the line in slope-intercept form.

The domain of a line in the graph represents all possible x-values that the line covers. Start with the slope-intercept form

Encourage them to recognize that the parts of this

y The range of a line in the 5 graph represents all possible y-values that the line covers. Substitute and b = example −3 begin to decompose the problem into smaller

4 more manageable Reflect and check parts. From there, have them work 3 withcan a see partner to write a set steps for finding the Domain: −∞ < x < ∞ 2 We from the graph out that the lineof extends b A line with a slope of −3 and passes through the point (2,without 3). any breaks or endpoints along the x and y-axis, equation of a line from a graph by adding more detail, Range: −∞ < y < ∞ 1 x confirming that the domain includes all real numbers. including how they can check their answer. Create strategy 1 2 3 4 5 −5 a−4 −3 −2 −1 The domain in interval notation is: −1 Substitute the slope and point in the slope-intercept form, and solve for the y-intercept before (−∞, ∞) rewriting. −2 The domain in set notation is: −3 Apply the idea −4 {x ∣ x∈} y = mx + b Start with the slope-intercept form −5 The range in interval notation is:

Apply the idea

3 = −3(2) + b

3 = −6 + b 9=b y = −3x + 9 Students: Page 131

Substitute y = 3, m = −3, and x = 2 (−∞, ∞) Evaluate the multiplication The range in set notation is: Add 6 to both sides {y ∣ y∈} Rewrite using slope-intercept form

Example 2

3.03 Slope-intercept form mathspace.co

131

Find the equation in slope-intercept form for: a A line with a slope of

and a y-intercept of −3.

Create a strategy Substitute the values of the slope and y-intercept in the slope-intercept form

Apply the idea Start with the slope-intercept form Substitute

and b = −3

b A line with a slope of −3 and passes through the point (2, 3).

3.03 Slope-intercept form mathspace.co Substitute the slope and point in the slope-intercept form, and solve for the y-intercept before rewriting.

Create a strategy

Apply the idea

267

Example 2 PurposeFind the equation in slope-intercept form for: Check that theaequation line given the slope and y-intercept using the slope-intercept y-interceptof of a−3. a Astudents line with acan slopewrite of and form. Create a strategy

Reﬂecting with students

Substitute the values of the slope and y-intercept in the slope-intercept form

Ask students whether there is a difference between writing the equation as

and

. While

Apply the idea both equations represent the same line, the ﬁrst equation can be simpliﬁed, so the second equation is the Start with the slope-intercept form

preferred form.

Students: Page 131

Substitute

and b = −3

b A line with a slope of −3 and passes through the point (2, 3).

Create a strategy Substitute the slope and point in the slope-intercept form, and solve for the y-intercept before rewriting.

Apply the idea y = mx + b

Start with the slope-intercept form

3 = −3(2) + b

Substitute y = 3, m = −3, and x = 2

3 = −6 + b

Evaluate the multiplication

9=b

Add 6 to both sides

y = −3x + 9

Rewrite using slope-intercept form

3.03 Slope-intercept form

131

Purpose mathspace.co Check that students can write the equation of a line given a slope and a point other than the y-intercept.

Expected mistakes Students may make a mistake when substituting in the values from the point. They may also struggle with finding the equation without being told the y-intercept directly. Reflecting with students Have students reflect on whether they would get the same equation if they were asked for the equation for a line with the same slope and another point on this line. Remind them that the equation of a line represents all points on the line. If a point lies on the line, its coordinates will also make the equation true.

Students: Page 132 c A line that passes through the points (−4, 5) and (8, 8).

Create a strategy Substitute the given points into the slope formula to ﬁnd slope. Use the slope and one point to solve for the y-intercept before rewriting in slope-intercept form.

Apply the idea Start with the slope formula Substitute (x1, y1) = (5, −4) and (x2, y2) = (8, 8) Evaluate Simplify Choose (x1, y1) or (x2, y2) to substitute with m into y = mx + b

268

Evaluate the multiplication Mathspace Virginia SOL Algebra 1 Teacher Edition Subtract 2 from both sides mathspace.co Rewrite using slope-intercept form

c A line that passes through the points (−4, 5) and (8, 8).

Create a strategy Substitute the given points into the slope formula to ﬁnd slope. Use the slope and one point to solve for the y-intercept before rewriting in slope-intercept form.

Apply the idea Start with the slope formula Substitute (x1, y1) = (5, −4) and (x2, y2) = (8, 8) Evaluate Simplify Choose (x1, y1) or (x2, y2) to substitute with m into y = mx + b Evaluate the multiplication Subtract 2 from both sides Rewrite using slope-intercept form

Example 3 Purpose Check that can write equations the equation of a line inform. slope-intercept form given two points on the line. Writestudents each of the following in slope-intercept a −4y = 12 − 8x

Reﬂecting with students Connect to the previous lesson by asking students to identify the transformations of this line from the parent Create a strategy function y = x. Slope-intercept form is y = mx + b, where m is slope and b is the y-intercept.

Scaffolded in slope-intercept form Apply the ideachecklist for writing equations Reflect and check Student disabilities support Each termwith will be divided by −4 and simplified.

use with Example 2

Slope is the simplified ratio of

. By

Original equation To help students write equations of lines in slope-intercept formatwith increasingly information, looking the graph, we couldcomplex see a change of by provide a scaffolded set of steps foreach students how to use the given information Divide term byto−4use to determine counting down and left between points. This rate of to write the slope-intercept form of an equation. Simplify and reorder terms

1. Find The theslope-intercept slope, m. form of −4y = 12 − 8x is y = 2x − 3. • Given slope: Skip to step 2.

change matches the simplified ratio of , or a slope of 2.

• Given two points (x1, y1) and (x2, y2): Use slope formula • Given a graph: Choose two points on the graph, use 2.

y 5 4 3 and simplify. 2 1

and simplify. −2

−1

x 1

−1 −2 Find the y-intercept, b. −3 • Given the y-intercept: Skip to step 3. −4 −5 for b. • Given a point: Substitute values of m and (x, y) into y = mx + b and solve

• Given two points: Choose either (x1, y1) or (x2, y2) to use as (x, y) and follow same steps as above. • From a graph: If the line crosses the y-axis at an integer value, use this value for b. Mathspace Virginia SOL Algebra 1 3. Write132 equation as y = mx + b using your values of m and b. mathspace.co • Use the format y = ⬚x + ⬚ to help set up your equation • Check your equation for simplifying and formatting.

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269

Choose (x1, y1) or (x2, y2) to substitute with m into y = mx + b Evaluate the multiplication Subtract 2 from both sides

Students: Page 132

Rewrite using slope-intercept form

Example 3 Write each of the following equations in slope-intercept form. a −4y = 12 − 8x

Create a strategy Slope-intercept form is y = mx + b, where m is slope and b is the y-intercept.

Apply the idea

Reflect and check

Each term will be divided by −4 and simplified.

Slope is the simplified ratio of looking at the graph, we could see a change of

Divide each term by −4

counting down and left between points. This rate of

Simplify and reorder terms

change matches the simplified ratio of , or a slope of 2.

The slope-intercept form of −4y = 12 − 8x is y = 2x − 3.

5 4 3 2 1 −2

132

. By

Original equation

−1

x 1

−1 −2 −3 −4 −5

Mathspace Virginia SOL Algebra 1 mathspace.co

Purpose Show students how to rearrange a linear equation into slope-intercept form. Expected mistakes Students may miss that the original coefficient on the x and the coefficient on the y are both initially negative. Having students rearrange −4y = 12 − 8x to −4y = −8x + 12 before dividing can help clarify signs.

Students: Page 133 b 7x − 14y = −28

Create a strategy Isolate the y and rearrange terms to write in slope-intercept form, y = mx + b.

Apply the idea The x term will be moved first then divide by the coefficient of y. Original equation Subtract 7x from each side Write with x term before constant Divide each term by −14 Simplify The slope-intercept form of 7x − 14y = −28 is

Reflect and check Other forms of equations will create the same graphs as slope-intercept form but may highlight different information.

270

By creating a table of Algebra values and graphing, we see that 7x − 14y = −28 has the same slope and y-intercept as Mathspace Virginia SOL 1 Teacher Edition mathspace.co . 4

Isolate the y and rearrange terms to write in slope-intercept form, y = mx + b. Divide each term by −14

Apply the idea

Simplify The x term will be moved first then divide by the coefficient of y. Original The slope-intercept form of 7x − 14y = −28 is equation. Subtract 7x from each side

Reflect and check

Write with x term before constant Other forms of equations will create the same graphs as slope-intercept form but may highlight different information. Divide each term by −14 By creating a table of values and graphing, we see that 7x − 14y = −28 has the same slope and y-intercept as .

Simplify

The slope-intercept form of 7x − 14y = −28 is

Reflect and check

2 y-intercept

Other forms of equations will create the same graphs as slope-intercept form but may 1 highlight different information. x

By creating a table of values and graphing, we see that 7x − 14y = −28 has the same slope and y-intercept as −4 −3 −2 −1

−1

−2 −3 4 −4 3

2 y-intercept 1 −4 −3 −2 −1

−1

x 1

Purpose −2 Create a strategy Apply the idea Show students that equations in standard form can be rewritten in slope-intercept form. −3 Slope-intercept form is y = mx + b. Distribute any values outside parentheses and rearrange terms to isolate y.

We will first distribute −4 the then isolate the y to write the equation in slope-intercept form.

Students: Page 133

Original equation Distribute

and multiply

Subtract 2

Create a strategy

Apply the idea

Slope-intercept form is y = mx + b. Distribute any values outside parentheses and rearrange terms to isolate y.

We will first distribute the then isolate the y to write Simplify the equation in slope-intercept form. The slope-intercept form of

is Original equation Distribute

and multiply

3.03 Slope-intercept form Subtract 2 mathspace.co

133

Simplify The slope-intercept form of

3.03 Slope-intercept form

133

mathspace.co Purpose Show students how to use distributive property and rearrange terms to write equations in slope-intercept form.

Students: Page 134 Example 4 A bathtub has a clogged drain, so it needs to be pumped out. It currently contains 30 gallons of water. The table of values shows the linear relationship of the amount of water remaining in the tub, y, after x minutes. Time in minutes (x) Water remaining in gallons ( y)

0 30

1 28

2 26

3 24

a Determine the linear equation in slope-intercept form that represents this situation.

Create a strategy We can pick two points to calculate the rate of change for the slope. Then we can recognize that the y-intercept is given in the table of values. 3.03 Slope-intercept form

mathspace.co

Apply the idea Find the slope using the values (0, 30) and (1, 28):

271

A bathtub has a clogged drain, so it needs to be pumped out. It currently contains 30 gallons of water. The table of values shows the linear relationship of the amount of water remaining in the tub, y, after x minutes. Time in minutes (x) Water remaining in gallons ( y)

0 30

1 28

2 26

3 24

a Determine the linear equation in slope-intercept form that represents this situation.

Create a strategy We can pick two points to calculate the rate of change for the slope. Then we can recognize that the y-intercept is given in the table of values.

Example 4

Apply thehas idea A bathtub a clogged drain, so it needs to be pumped out. It currently contains 30 gallons of water. Find the slope usingshows the values (0, 30) and (1, 28):of the amount of water remaining in the tub, y, after x minutes. The table of values the linear relationship Time in minutes (x) Water remaining in gallons ( y)

Slope formula 0 1 2 3 30 28 26 24 Substitute (x , y ) = (1, 30) and (x , y ) = (1, 28) 1

Evaluate a Determine the linear equation in slope-intercept form that represents this situation. Notice that the initial value, or y-intercept is given in the table as (0, 30).

Createina the strategy Writing form y = mx + b the equation that represents this situation is y = −2x + 30. We can pick two points to calculate the rate of change for the slope. Then we can recognize that the y-intercept is given in the of values. Reﬂect andtable check If we had not noticed that the y-intercept was given, we could have substituted in any pair of values for x and y, and Apply theb.idea solved for Find the slope using the values (0, 30) and (1, 28): Slope formula b Draw the graph of this linear relationship with a clearly labeled scale. Only show the viable solutions. Substitute (x1, y1) = (1, 30) and (x2, y2) = (1, 28) PurposeCreate a strategy Check ifWe students can find thetime equation of we a line from table that represents a context. can’t have a negative (x ≥ Evaluate 0) and should enda the graph when the tub is empty ( y = 0). To plan our graph, we need to the ﬁndinitial whenvalue, the tub empty. is given in the table as (0, 30). Notice that or is y-intercept

Reflecting with students Writing in the form y = mx + b the equation that represents this situation is y = −2x + 30. Apply the ideathat if we had not noticed that the y-intercept was given, we could have substituted in any Make student aware To ﬁnd when empty: pair of values for x the andtub y,isand solved for b. Reﬂect and check = −2x +that 30 the y-intercept Originalwas equation If we had not ynoticed given, we could have substituted in any pair of values for x and y, and Substitute y = 0

Students:solved Pages for b.134–135 0 = −2x + 30 −30 = −2x

Subtraction property of equality

15 = x Division property of equality b Draw the graph of this linear relationship with a clearly labeled scale. Only show the viable solutions.

Create a strategy We can’t have a negative time (x ≥ 0) and we should end the graph when the tub is empty ( y = 0). To plan our graph, we need to ﬁnd when the tub is empty.

Apply the idea To ﬁnd when the tub is empty: 134

y = −2x + 30

−30 = −2x 15 = x

134

272

Original equation

Mathspace Virginia SOL Algebra 1 0 = −2x + 30 Substitute y = 0 mathspace.co

Subtraction property of equality Division property of equality

Mathspace Virginia SOL Algebra 1 mathspace.co

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

This tells us that we have the restriction that 0 ≤ x ≤ 15 and 0 ≤ y ≤ 30, which will help us choose the appropriate axes and scale. 30

Water in gallons

25 20 15 10 5 5

Time in minutes 10 15

We know that the bathtub begins with 30 gallons which is represented by the y-intercept at (0, 30) and that the slope of the linear equation is −2 which means that the tub loses 2 gallons of water every minute until there is no water in the tub at 15 minutes. 30

Water in gallons

25 20 15 10 5 5

Time in minutes 10 15

Reﬂect and check We can also graph the slope by using the idea of

. Since the slope is −2, we can write it as a fraction

and

identify that the change in y-values (or rise) is −2 and that the change in x-values (or run) is 1.

c Find the domain and range.

PurposeCreate a strategy Apply the idea Check that students can graph a line given constraints based on a real-world scenario. The domain of a function represents all possible x-values. Domain: 0 ≤ x ≤ 15 The range of a function represents all possible y-values. Notice that even though negative x-values could fit the Expected mistakes values that may fit the pattern but do not fit the pattern, it does not make sense with the context to have a StudentsConsider may have a hard time scaling the axes. Supportsnegative could amount includeofquestions such as “What is the biggest time. context. number of gallons and smallest number of gallons the graph needs to show on the graph?” Student could also Range: 0 ≤ y ≤ 30 be provided with these scaled axes: In the context, water is never added so it can never be above the starting amount of 30 gallons. Continuing the

Water in gallons pattern, the y-value will never drop below empty, or 0 gallons.

25 20 3.03 Slope-intercept form mathspace.co

135

10 5 5

Time in minutes 10 15

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273

Reﬂect and check We can also graph the slope by using the idea of

. Since the slope is −2, we can write it as a fraction

and

the change in y-values (or rise) is −2 and that the change in x-values (or run) is 1. Students:identify Pagethat 135

c Find the domain and range.

Create a strategy

Apply the idea

The domain of a function represents all possible x-values. Domain: 0 ≤ x ≤ 15 The range of a function represents all possible y-values. Consider values that may fit the pattern but do not fit the context.

Notice that even though negative x-values could fit the pattern, it does not make sense with the context to have a negative amount of time. Range: 0 ≤ y ≤ 30 In the context, water is never added so it can never be above the starting amount of 30 gallons. Continuing the pattern, the y-value will never drop below empty, or 0 gallons.

3.03 Slope-intercept form Purpose mathspace.co Show students how to find the domain and range of a function in a real-world context.

135

Students: Page 136

Purpose Check if students can apply their knowledge of lines, slope, and intercepts to answer real-world questions. 274

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Reflecting with students Ask students to compare the lines at different points in time. As them if they could design a tub that can empty even faster.

Advanced learners: Open questions for deeper insights

use with Example 4

Targeted instructional strategies Incorporate open-ended questions to challenge advanced learners and encourage deeper thinking about linear functions. For instance, ask students to create a real-world situation that can be modeled by a linear equation in slope-intercept form. Allow students to choose their own contexts based on their interests—such as economics, physics, or environmental science—and develop corresponding equations and graphs. By using open questions, you provide students with the opportunity to think critically, make connections, and extend their understanding of linear relationships beyond standard examples.

Students: Page 136

Purpose Teach students how to determine the y-intercept of a line from its equation or description. Expected mistakes Students may confuse the slope for the y-intercept. Make sure they understand that in the linear equation form y = mx + b, b represents the y-intercept. Reflecting with students Ask students why the coefficient of x (the slope) doesn’t affect the y-intercept. Discuss how changing the slope affects the steepness of the line, but not its position on the y-axis.

Students: Page 137

Idea summary The slope-intercept form of a line is:

y = mx + b m slope b

y-intercept

Slope-intercept form is useful when we know, or want to know the slope of the line and the y-intercept of the line.

Practice What do you remember?

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275

Practice Students: Pages 137–144

What do you remember? 1

What general equation represents slope-intercept form? What do m and b represent?

Consider the graph of each linear function. i

Determine the slope, m.

Determine the y-intercept, b.

b 4

1 −4 −3 −2 −1

−4 −3 −2 −1

−1

−2

−3

−4

x 1

In each of the equations, identify the value of m and b. a

y = 15x − 10

y = 6 + 8x

y = 7x

y=6−x

Determine whether each equation is in slope-intercept form. a

y = 2x − 1

y = 5x

4x + 3y = 0

y = 2y − 4

x = 2y

Find the equation in slope-intercept form for: a

A line with a slope of 2 and y-intercept of 5.

A line with a slope of

and y-intercept of 3.

A line with a slope of −2 and y-intercept of 0.

A line with a slope of

and y-intercept of −3.

3x − 2y = −4

Rewrite each equation in slope-intercept form: a

276

−1

x 1

y = 5 − 4x

y − 1 = 5 (x − 4)

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

y + 7 = −3x

Let’s practice 7

For each graph: i

What is the y-intercept?

1 −4 −3 −2 −1

−1

−4 −3 −2 −1

−2 −3

−4

6 5 4 3 2 1

x 1 2 3 4 5 6

x 1

−1

−3

6 5 4 3 2 1

−2

−6 −5 −4 −3 −2−1 −1 −2 −3 −4 −5 −6

What is the slope?

−6 −5 −4 −3 −2−1 −1 −2 −3 −4 −5 −6

x 1 2 3 4 5 6

For the line shown: y 5 4 3 2 1 −2

−1

x 1

−2

What are the values of the slope, m, and the y-intercept, b?

Write the equation of the line in slope-intercept form.

Find the value of y when x = 27.

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277

This graph shows the relationship between the time (x) in minutes and distance ( y) in kilometers of a highspeed train ride at a constant speed. y (kilometers) 5 4 3 2 1 1

x (minutes) 4 5

Find the equation of the line in slope-intercept form.

Find how far the train traveled, y, after 29 minutes.

Find the equation of the line in slope-intercept form. a

5 4

−2

1 −1

−1

x 1

−1

3 −2

−2

1 −2

−1

x 1

−1

= 3x − 2

−1

−1 −2

−2

−3 −3

−4

278

By rewriting the equation in the form y = mx + b: i

What is the slope, m?

What is the value of the y-intercept, b?

y−7=

9x − y − 8 = 0

3y = 12x − 15

y = 3 (4x − 3)

Consider the line passing through the point (7, 1) that has a slope of 4. a

Find the value of b, the y-intercept.

Use the slope and y-intercept to write the equation in slope-intercept form.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Find the equation in slope-intercept form for: a

A line with a slope of 8 and passes through the point (0, −4).

A line with a slope of −1 and passes through the point (0, 0).

A line with gradient

A line with gradient 8 and passes through the point (0, −4).

and passes through the point (0, 3).

For each line described: i

Find the slope of the line.

Find the equation of the line in slope-intercept form.

A line passes through the points (−3, 0) and (6, 6).

A line passes through the points (3, −3) and (5, −11).

A line passes through the points (−6, 7) and (−8, −4).

A line passes through the points (−7, 1) and (5, 2).

For each description: i

Find the y-value of the y-intercept.

Write the equation of the line in slope-intercept form.

iii

Find the x-value of the x-intercept of the line.

Sketch the graph of the line.

A line that has slope of

A line that has slope of −2 and passes through the point (3, −8).

and passes through the point (−10, 4).

For each of the following functions: i

Rewrite the equation in slope-intercept form using the function notation f (x).

Find the value of f (0).

iii

Graph the function, labeling each intercept.

y = 12 − 3x

y − (−5) = 5 (x − 2)

2y = −2x − 8

10x − 2y = 20

Sketch the graph of each line described. a

The line with a y-intercept of −2 and slope of −3.

The line with a y-intercept of 3 and slope of

The line with equation y = 2x + 5.

The line with equation

A cleaner charges an initial fee of $110 plus $20 per hour. a

Identify the slope and y-intercept of the scenario.

Write an equation to represent the total amount charged by the cleaner, y, as a function of the number of hours worked, x.

State the domain of the linear relationship.

A race car starts the race with 60 gallons of fuel. From there, it uses fuel at a rate of 1 gallon per minute. a

Complete the table of values. Number of minutes passed (x) Amount of fuel left in tank f (x)

Write the equation relating the number of minutes passed x and the amount of fuel left in the tank f (x).

Graph the function, including clearly labeled axes with an accurate scale.

Explain how the function would change if the car was repaired and now uses 0.9 gallons per minute.

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279

For each of the following pairs of linear functions, identify which function has the greater y-intercept: a • Function 1: The line with a slope of 4 that crosses the y-axis at (0, 6). • Function 2: y = x + 4 Function 1: b •

• Function 2:

x 2 4 6 y 2 −2 −6

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

• Function 1: x y

2 19

4 35

2 19

4 31

• Function 2: y = 4x + 6

6 51

Function 1: d • x y

• Function 2: y = 3x + 4

6 43

Function 1: e •

• Function 2:

x 2 4 6 y −5 −13 −21

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

1 2 3 4 5

x 1 2 3 4 5

• Function 1: The line with a slope of 7 that crosses the y-axis at (0, 5). • Function 2: y = (−m) x + 7

For each of the following pairs of linear functions, identify which function has the greater slope. a • b • Function 1: y = −3 + 3x Function 1: y = 1 + 8x • Function 2: • Function 2: x 0 1 2 y 1 6 11

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x y

0 −3

1 2

2 7

A carpenter charges an initial fee of $250 plus $35 per hour for labor. a

Write an equation to represent the total amount charged, y, by the carpenter as a function of the number of hours worked, x.

Interpret the slope of this linear function.

Interpret the y-intercept of this linear function.

Peta has budgeted $2500 to have a ramp built at the community center to increase accessibilty.

The carpenter works 8 hour days and estimated that the project would take one and a half days. He also priced the materials at $1800. Determine if Peta has budgeted enough for the ramp. Justify your answer.

Let’s extend our thinking 23

Which of the following linear relationships has the smaller corresponding x-value when y = 42? • Function 1: x y

0 2

1 10

2 18

• Function 2: y = 3x + 18 24

Two construction workers are competing to see who can lay the most bricks in one hour. The graph shows the number of bricks laid and the time, in minutes.

280

Use the given graph to compare the construction workers.

240

320

Bricks laid Charlie

200 160 120

Neville

80 40

Minutes 10 20 30 40 50 60 70 80

Edin and Marius are saving money for a trip. Edin saves $100 per month, and Marius saves $80 per month after starting with $60. Both start saving in January. a Write linear equations to represent the total amount ( y) saved by each friend after x months. b Interpret the meaning of the y-intercept of each equation. c Graph both equations on the same coordinate plane. Use an appropriate scale and label the axes. d Determine which friend will have saved more by June.

Some friends decide to go camping for the weekend. They cannot all ﬁt in one car so some of them catch a bus to the campground, which is 450 mi from home. Those in the car started driving at 8:00 am and arrived at the campground at 3:30 pm, driving at a constant speed. The bus also drives at a constant speed and takes the same route as the car. Its distance in miles, y, from home x hours after leaving is given by the equation y = 71x. a

Determine the speed of the car, in miles per hour.

Determine the speed of the bus, in miles per hour.

Determine which vehicle was traveling faster.

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281

For the three linear equations and their corresponding graphs: 10

y = x + 4, y = 2x + 4, y = 4x + 4 a

What do all of the equations have in common?

What do all of the graphs have in common?

What conclusion can be made about all lines that have the form y = mx + 4?

8 6 4 2

−8 −6 −4 −2 −2

−4

Look at the three lines: 8

• Equation 1: y = 2x + 4 • Equation 2: y = 2x + 8 • Equation 3: y = 2x − 4

y = 2x + 8

What do all the equations have in common?

What do all the graphs have in common?

Describe all lines that have the form y = 2x + b.

6 4

y y = 2x + 4

2 −8 −6 −4 −2 −2 −4

x 2

y = 2x − 4

−6 −8

Create a scenario where you would choose to use slope-intercept form, y = mx + b and explain what the slope and y-intercept mean in context.

The table shows the water level of a well that is being emptied at a constant rate with a pump where the time is measured in minutes and the water level in feet. Time (x) Water level ( y) a b c d

3 56.7

7 48.3

10 42

Write an equation to model the scenario. Interpret the key features of this linear function. Find the water level after 18 minutes. State the viable values of x for this scenario. Justify your answer.

This graph shows the relationship between the time (x) in seconds and distance ( y) in meters for a sprint that Jocel ran at a constant speed. a

Find the equation of the line in slope-intercept form.

Find how far Jocel ran, y, after 50 seconds.

10 9 8 7 6 5 4 3 2 1

y (meters)

x (seconds) 1 2 3 4 5 6 7 8 9 10

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A diver starts at the surface of the water and begins to descend below the surface at a constant rate. The table shows the depth below the surface of the diver over the ﬁrst 5 minutes. Number of minutes passed (x) (min) Depth of diver (y) (ft)

0 0

1 1.63

2 3.26

3 4.89

4 6.52

State the rate of change with units of ft/min.

Write an equation for the relationship between the number of minutes passed (x) and the depth below the surface (y) of the diver.

Determine the depth of the diver after 6 minutes.

Find how long the diver takes to reach 11.41 ft beneath the surface.

The diver has two air tanks, each of which last for 45 minutes. One is used for going down and one is used for returning to the surface. Determine an appropriate domain restriction for the function from part (b).

Assuming the diver comes back up at the same rate they go down, state the equation for the linear function which could describe the return to surface including the domain restriction. Sketch the full scenario as a graph.

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283

Answers

15 a i y = 12 iv

3.03 Slope-intercept form

20 12 8

1 y = mx + b, where m represents the slope, b represents the y-intercept. ii b = 4

3 a m = 15, b = −10

−8 −16

d m = −1, b = 6

e m = 8, b = 6

−20

, b = −1 b i y = −2

e No

4 8 12 16 20

−12

b m = 7, b = 0

b Yes

−20 −16 −12 −8 −4 −4

iv b = −2

iii m =

4 a Yes

What do you remember?

2 a i m=3

iii x = −15

c No

d Yes

ii y = −2x − 2

5 a y = 2x + 5

c y = −2x

6 a y = −4x + 5 c

b y = 5x − 19

−10 −8 −6 −4 −2 −2

d y = −3x − 7

−6

x 2 4 6 8 10

−4 −8 −10

Let’s practice 7 a i (0, 3)

ii m = 2

16 a i f (x) = −3x + 12

b i (0, 3)

ii m = 4

ii f (0) = 12

c i (0, 4)

ii m = −3

iii f (x) = 0 when x = 4

d i (0, 2)

ii m = −6

8 a m = −2, b = 3

b y = −2x + 3

f (x) (0, 12)

c y = −51

9 a y = 2x + 1

b y = 59 kilometers

10 a y = −2x + 5

b y = 3x + 1

c 11 a i m =

(4, 0) 5

ii b = 7

b i m=9

ii b = −8

c i m = 12

ii b = −9

d i m = 18

ii b = −12

ii f (0) = −15

e i m=4

ii b = −5

iii f (x) = 0 when x = 3

i m = −1

ii b = 4

12 a b = −27 13 y = 8x − 4

b i f (x) = 5x − 15

f (x) 5

b y = 4x − 27 b y = −x

d y = 8x − 4

−10

(3, 0) 5 10

−5 −5

14 a b i −4

ii y = −4x + 9

c i

d i

−10 −15 (0, −15)

284

iii x = −1

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

c i f (x) = −x − 4

ii f (0) = −4

iii f (x) = 0 when x = −4 iv

f (x)

x −1

5 (−4, 0) −10

5 (0, −4)

−5

−1

−5

−2

−10

b y = 20x + 110 c x≥0

18 a Slope: $20 per hour y-intercept: $110

d i f (x) = 5x − 10

ii f (0) = −10

iii f (x) = 0 when x = 2 iv

19 a

f (x) 10

Number of minutes passed (x)

Amount of fuel left in tank (y)

b f (x) = 60 − x

5 −10

(2, 0) 5

−5

Fuel (gallons) 60

−5

−10 (0, −10)

40 30 20

17 a

3 2 1 −2

−1

d T he rate of change is different, so the slope will change. This means the coeﬃcient of x would become 0.9 instead of 1, so the function would be y = 60 − 0.9x. 20 a Function 1

b Function 1

c Function 2

d Function 1

e Function 2

21 a Function 1

Function 2

b Function 2

22 a y = 35x + 250

b T he total amount charged increases by $35 for each additional hour of work.

2 1

x 1

−1

c T he minimum amount charged by the carpenter, which is the initial fee of $250. d T he total cost is Materials + Labor. It will cost $1800 + 250 + 35 ⋅ 12 = 2470, and 2470 < 2500, so she has budgeted enough for the ramp.

−2

Let’s extend our thinking

23 Function 1

4 3 2 1 −4 −3 −2

40 50 60

−3

−1

20 30

−2

Time (minutes) 10

−1

−1 −2

x 1

24 The number of bricks Neville can lay per minute is less than the number of bricks Charlie can lay per minute. Since the time taken for laying bricks only begins when the construction workers lay their ﬁrst brick, both lines must pass through the point (0, 0) representing that they have laid 0 bricks after 0 minutes.

Answers mathspace.co

285

25 a Edin’s savings: y = 100x

30 a y = −2.1x + 63

Marius’s savings: y = 80x + 60

b The y-intercept of Edin’s savings equation is 0. This means she didn’t start with any money before saving. The y-intercept of Marius’s savings equation is 60. This means he started with $60 before saving each month. c

Edin and Marius save money y, dollars 500 400 300 200

c 25.2 ft d T he lowest possible x-value is x = 0 because time can’t be negative. The largest possible x-value is x = 30 because we can’t have a negative water level, so we can solve the equation 0 = −2.1x + 63 to get x = 30 when the well is emptied. Once the well is emptied, the context is no longer valid. Time is always positive, so we get 0 ≤ x ≤ 30. 31 a y = 3x – 4

100

b y = 146 meters

32 a 1.63 ft/min

x, months since January −2 −1

b T he water level is falling at a constant rate of 2.1 feet per minute. The water level was 63 feet before any water was pumped out.

b y = 1.63x

1 2 3 4 5 6 7 8 9 10

c 9.78 ft d Edin 26 a 60 mi/h

d x=7 b 71 mi/h

c Bus

27 a The constant term of 4 is the same. b All the graphs intersect the y-axis at the point (0, 4). c The line will pass through (0, 4). 28 a The coefficient of x is the same. b All the graphs have the same gradient. c T he lines would be parallel on a coordinate plane because they have the same gradient but different y-intercepts. This means they will not intersect each other, because they are always the same distance apart. 29 A scenario where we are given the rate of change (slope) and the initial value (y-intercept). For example, if we are told how much a plumber charges as their ﬂat callout fee (y-intercept) and how much they charge per hour (slope).

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e T he domain will be x ∈ [0, 45] as after 45 minutes the diver will need to stop going deeper and start coming back to the surface. f

y = −1.63(x − 45) + 73.35 = −1.63x + 146.7 for x ∈ [45, 90] 90 80 70 60 50 40 30 20 10

Depth (ft)

Time (min) 10 20 30 40 50 60 70 80 90

3.04 Standard form Subtopic overview Lesson narrative In this lesson, students will expand their understanding of slope-intercept form and how it connects to equations represented in standard form. Students will make sense of writing and converting equations written in slope-intercept to standard form, and use standard form to find the x and y-intercepts. They will use the standard form and the intercepts to graph a line and describe other information about the graph. Students will also explore horizontal and vertical lines and their respective standard forms. By the end of the lesson, students should understand how to write and graph linear equations in standard form, recognizing key features such as y-intercepts and slopes.

Learning objectives Students: Page 145

Key vocabulary 

horizontal line



standard form (of a linear function)



vertical line

Essential understanding Different representations of a function may highlight or hide different characteristics but they do not change the function itself. Changing the form of the equation of a function can highlight or hide different characteristics of the function but does not change the function itself.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG3 — Mathematical Reasoning While teaching students to find the x and y-intercepts of a linear function given in standard form, teachers can ask them to use inductive reasoning to make conjectures about the intercepts. Similarly, when introducing the concept of rewriting a linear equation in different equivalent forms, teachers can encourage students to use deductive reasoning to determine which form might be most useful in different scenarios.

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MPG4 — Mathematical Connections

MPG5 — Mathematical Representations

Teachers can help students make connections between the slope-intercept form and standard form of a linear equation, highlighting how they represent the same function differently. Further, when discussing the interpretation of x and y-intercepts within the context of a problem, teachers can connect this concept to real-world situations, allowing students to see how mathematics is integrated with other disciplines.

Teachers can integrate this goal into their instruction by encouraging students to represent linear functions in various ways - algebraically in standard form, slopeintercept form, graphically, and in tables. Furthermore, when discussing the importance of x and y-intercepts within the context of a problem, teachers can encourage students to represent these intercepts in different forms, depending on the context and information needed.

Content standards A.F.1 — The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships. A.F.1a — Determine and identify the domain, range, zeros, slope, and intercepts of a linear function, presented algebraically or graphically, including the interpretation of these characteristics in contextual situations. A.F.1c — Write equivalent algebraic forms of linear functions, including slope-intercept form, standard form, and point-slope form, and analyze and interpret the information revealed by each form. A.F.1di — Write the equation of a linear function to model a linear relationship between two quantities, including those that can represent contextual situations. Writing the equation of a linear function will include the following situations: i) given the graph of a line; A.F.1dii — Write the equation of a linear function to model a linear relationship between two quantities, including those that can represent contextual situations. Writing the equation of a linear function will include the following situations: ii) given two points on the line whose coordinates are integers;

A.F.1diii — Write the equation of a linear function to model a linear relationship between two quantities, including those that can represent contextual situations. Writing the equation of a linear function will include the following situations: iii) given the slope and a point on the line whose coordinates are integers; A.F.1div — Write the equation of a linear function to model a linear relationship between two quantities, including those that can represent contextual situations. Writing the equation of a linear function will include the following situations: iv) vertical lines as x = a; A.F.1dv — Write the equation of a linear function to model a linear relationship between two quantities, including those that can represent contextual situations. Writing the equation of a linear function will include the following situations: v) horizontal lines as y = c. A.F.1f — Graph a linear function in two variables, with and without the use of technology, including those that can represent contextual situations. A.F.1h — Compare and contrast the characteristics of linear functions represented algebraically, graphically, in tables, and in contextual situations.

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A.EI.1 — The student will represent, solve, explain, and interpret the solution to multistep linear equations and inequalities in one variable and literal equations for a specified variable.

Engage Activity Purchasing baby food

60 mins

Students will work in groups to plan a shopping budget for baby food and formula.

Understanding and skills

Will use Writing and graphing linear equations in slope-intercept form.

Will develop Writing a linear equation in standard form from a written description of a real-world context. Graphing a linear function given an equation in standard form.

Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: Graph paper, paper, pencil

Support students with disabilities Support memory - use previously taught skills and concepts Provide the standard form for linear equations: Ax + By = C

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Answers may look like: • An explanation of how Alejandro should spend his money. • Equations, graphs, and calculations. On the third read, have students identify important information. Prompt: Students read the problem. Students think/write: Answer the question “What are the important pieces of information given in the question?” Answers may look like: • The cost of each type of item Alejandro must buy. • The amount of money Alejandro has to spend in total.

Classroom guide Hook

Which one doesn’t belong

•

5 mins

Students choose one of four linear equations.

Implementation details

Which one doesn’t belong? Select one option.

Two of the equations, y = 1.5 − 2x and y = 3 − 4x 4x + 2y = 3 3a + 2b = 8 A B are in slope-intercept form, students may pick out that one or both of the remaining equations y = 1.5 − 2x y = 3 − 4x C D do not belong because they aren’t in slope intercept form. Similarly, the remaining equations 3a + 2b = 8 and 4x + 2y = 3 are in standard Slide 1 from Student Engage Activity form (for which the formal vocabulary has not yet been introduced) so students may decide that the equations in slope-intercept form are the ones that do not belong. Encourage students to use the accurate vocabulary to describe characteristics of each equation. This could include terms such as coeﬃcient, constant, slope-intercept form, variables, and linear equation. Practicing the appropriate use of these terms and reviewing their deﬁnitions will help set up students for success in the proceeding activity.

Launch

5 mins

Provide students time to read the information individually before discussing context as a class and forming pairs. Some students may have younger siblings or family members and be very familiar with shopping for necessities and the cost of common baby goods. Some students may not know what the WIC program is or how it works, so be prepared to share some information with the class after reading the task so that all students understand how the voucher works. Important mathematical concepts: Linear equations, standard form Important contextual information: WIC program Suggested grouping: Form pairs

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Alejandro helps his foster parents run errands on the weekends. Usually they give him some cash and tell him which items he needs to buy. This weekend, Alejandro’s parents want him to buy some formula and baby food for his youngest brother Damian. Alejandro’s parents have given him the following information on the cost of baby food and infant formula: Baby Food (per 4 oz. jar) Infant Formula (per 13 fl. oz.)

Quick Mart $0.69 $18.99

Cloud 9 $0.50 $15.49

Cash n Carry $0.42 $19.99

Slide 2 from Student Engage Activity

Continue when Students have read the Launch and understand the context of the problem.

Explore

Think-pair-share

•

35 mins

Anticipated strategies Choose a store Students must ﬁrst choose which store they would select for Alejandro to purchase baby food from. Of the three options, some have higher prices for infant formula while some have higher prices for baby food, so it shouldn’t be immediately apparent that one store is cheaper than the others.

Create an equation Once students have selected a store to buy from, they must interpret the information given in the table and decide how to turn that information into an equation. In order to produce an equation, students will need to clearly deﬁne a set of variables. Based on the given information, students are most likely to produce an equation in standard form. Let x = number of jars of baby food and y = number of jars of baby food Quick Mart: 60 = 0.69x + 18.99y Cloud 9: 60 = 0.5x + 15.49y Cash n carry: 60 = 0.42x + 19.99y

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291

Create a graph In order to graph the equation they have written, students may try convert into slope intercept form, ﬁnd two points that solve the equation such as the intercepts, or create a table representing the different combinations Alejandro can purchase and graphing those values as ordered pairs. QuickMart

Cloud 9

Cash n Carry

x 20 40 60 80 100 120 140

Alejandro can only afford a maximum of three cans of formula at any of the stores. Encourage students to ﬁnd additional points by considering how many jars of baby food he could buy if he purchased 0, 1, 2, or 3 cans of formula. Students may have prior knowledge on the cost and quantity for baby food from younger siblings or relatives. Look for student calculations to determine how long formula lasts, and how many jars of baby food a baby might eat per day and facilitate sharing with the class.

Misconceptions Using an incorrect total What does it mean to have a WIC voucher? Including the WIC voucher, how much does Alejandro have to spend?

Identifying viable solutions - Although an equation can be produced to help solve for the number of jars of baby food and the number of cans of infant formula, the only solutions that make sense in context are integer solutions Does your answer make sense in terms of the situation? How do you know? Can you explain what (student solution as an ordered pair or list of quantities) represents in the situation? Is it possible to have a fractional amount of a jar of baby food/can of infant formula?

Purposeful questions Use the following questions to check for understanding and encourage critical thinking: • What inﬂuenced the store that you selected for Alejandro? • How did you create your equation? What do each of the variables represent? • How did you determine how many jars of baby food and cans of infant formula Alejandro should buy? Where does your solution lie on the graph? • How did you determine how many jars of baby food and cans of infant formula Alejandro should buy? Where does your solution lie on the graph? • Does this solution make sense? How do you know?

Continue when Students have chosen a store, written an equation, sketched a graph, and found possible purchase combinations. Most, if not all, students have written Alejandro’s note.

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Discuss

15 mins

Regroup students for a group discussion and presentation. Consider sequencing the strategies presented by starting with groups that used their equations or graphs to solve for the three viable solutions and then other methods, such as using a table or guess and check method.

Discussion guide For each of the stores, select one to two student groups to present. As students present, look for explanations on how they defined variables, wrote an equation, graphed the equation, and interpreted their solutions in context. Ask questions such as: • What do the coefficients/constants/variables in your equation represent in the situation? • How did you know that (given ordered pair) satisfies the shopping conditions? • How could you verify that the answer makes sense in context of the problem? If different groups chose different methods for finding their three viable solutions, capture the different methods at the end of presentations by facilitating a discussion. It is likely that most students used their equation or graph to directly solve for three viable solutions, but other groups may have relied on a table or guess and check method. Ask students which method they would choose if they were to redo the activity, and why.

Tools You may ﬁnd these tools helpful: • Graphing calculator • Graph paper

Student lesson & teacher guide Standard form Students are introduced to the components of standard form and are shown that slope-intercept form can be rewritten in standard form using familiar operations. Students then use standard form to graph equations in the coordinate plane.

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Students: Pages 145–146

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x-intercept: substitute y = 0 and solve:

2x − 3 (0) = 3

2x = 3

2 1 −4 −3 −2 −1

−1

x 1

y-intercept: substitute x = 0 and solve: 2 (0) − 3y = 3

−3y = 3

−2

Graph the intercepts, in this case line through the points.

−3 2x − 3y = 3 −4

and (0, −1), and draw the

We see that the graph still maintains the same slope and y-intercept as the original slope-intercept form, but standard form allows us to focus on other key features of the function.

Example 1

Understanding the features of standard form Consider the linear equation: Targeted instructional strategies The concept of standard form (Ax + By = C) can be abstract for students. To help them understand, present the a Determine whether the equation is written in standard form. If not, rewrite it in standard form. idea that standard form is a way to organize linear equations to easily compare and manipulate them. Create alines strategy Show students graphed in the coordinate plane written in standard y In order for the equation to bewith written in standard we must conﬁrmlook that the equation is written in the form form with the intercepts marked points. The form, equations should 4 Ax + By = C, that A, B and C are integers, and A and B are not both zero. similar except for the signs of the coefficients for y. 3

Ask students how the points marked on the graph connect to Apply to thedetermine idea 2 each equation. They should notice that the in values of form A, B,because and C Ahelp 2xthe + 6y Since in the equation, it’s not written standard mustyou be an integer and integers are set=of12 1 evaluatepositive the values of the whole intercepts. and negative numbers. To these convertequations, the equationask to standard form, can useand properties of equality and multiply each term by 2 to change Based on students towe identify articulate the rules −2 −1 1 2 3 4 5 6 7 the rational coeﬃcient of x. of standard form (A is a non-negative integer, B and C are integers, and −1 Original equation A and B are not both zero). Then, have them apply these rules to classify 2x − 6y = 12 −2 equations as being in standard form or not.

Multiplication property of equality

Provide guided practice on converting equations to standard form if they Evaluate the multiplication are not already in that form. A = 1, B = 8, and C = 32 now meet the requirements for standard form.

Use real-world problems, like the tour company scenario, to show the practical applications of standard form equations in solving complex problems. b Graph the equation using the standard form from part (a).

Create a strategy

Misconception: Slope-intercept form and standard form are unrelated To graph the equation, we will ﬁnd the x-intercept and y-intercept and then draw the line passing through these Address points. student misconceptions

Some students may believe that slope-intercept form and standard form are unrelated or serve different purposes. They may struggle to understand how these two forms can represent the same equation or line on a graph. To address this misconception, encourage students to convert the same equation from slope-intercept form to standard form and vice versa. Once they have both forms, they can graph each equation on the same graph. Mathspace Virginia SOL Algebra 1 forms can describe the same line. Reinforce the understanding that This will146 visually demonstrate that both mathspace.co different forms of equations are just different ways of representing the same information.

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Information gap English language learner support Pair students and provide one student with a card containing a linear equation in standard form, while the other receives the same equation in slope-intercept form and a corresponding graph card. Here are specific examples of cards that could be given for the activity: 1. Standard form: 2x + 3y = 6 • Slope-intercept form: • Graph card: A graph showing the line with a slope of

and a y-intercept of 2

2. Standard form: x − 4y = 8 • Slope-intercept form: • Graph card: A graph showing the line with a slope of

and a y-intercept of −2

3. Standard form: 3x + y = 9 • Slope-intercept form: y = −3x + 9 • Graph card: A graph showing the line with a slope of −3 and a y-intercept of 9 4. Standard form: 5x − 2y = 10 • Slope-intercept form: • Graph card: A graph showing the line with a slope of

and a y-intercept of −5

5. Standard form: x − 3y = −3 • Slope-intercept form: • Graph card: A graph showing the line with a slope of

and a y-intercept of 1

Students use these cards to describe their equations and graphs, facilitating discussion and collaboration to match the equations with the correct graphs. x-intercept: substitute y = 0 and solve:

Students must describe their 4 equations without showing the cards, using language related to intercepts, 2x − 3 (0) = 3 coefficients, and slopes. They then collaborate to determine if their cards match by discussing the conversion 3 =3 2 process and graphing features. Finally, they verify their match by plotting the2x equation and checking the y-intercept: substitute x = 0 and solve: 1 graph, reflecting on the language and steps used. This activity enhances mathematical communication and x 2 (0) − 3y = 3 understanding of and linear equations. −4 converting −3 −2 −1 1 2graphing 3 4 −1

−3y = 3

−2

Graph the intercepts, in this case line through the points.

−3

Examples 2x − 3y = 3 −4

and (0, −1), and draw the

see that the graph still maintains the same slope and y-intercept as the original slope-intercept form, but standard Students:WePage 146 form allows us to focus on other key features of the function.

Example 1 Consider the linear equation:

a Determine whether the equation is written in standard form. If not, rewrite it in standard form.

Create a strategy In order for the equation to be written in standard form, we must conﬁrm that the equation is written in the form Ax + By = C, that A, B and C are integers, and A and B are not both zero.

Apply the idea Since

in the equation, it’s not written in standard form because A must be an integer and integers are the set of

positive and negative whole numbers.

296

To convert the equation to standard form, we can use properties of equality and multiply each term by 2 to change Mathspace Virginia SOL Algebra the rational coeﬃcient of x. 1 Teacher Edition mathspace.co Original equation Multiplication property of equality

x-intercept: substitute y = 0 and a Determine whether the form. If not, rewrite it insolve: standard form. y equation is written in standard 4

2x − 3 (0) = 3

Create a strategy

2x = 3 2 In order for the equation to be written in standard form, we must conﬁrm that the equation is written in the form y-intercept: substitute x = 0 and solve: 1 Ax + By = C, that A, B and C are integers, x and A and B are not both zero. 2 (0) − 3y = 3 −4 −3 −2 −1 1 2 3 4 −1

Apply the idea Since

−3y = 3

−2

in the equation, it’s not written in standard because Ainmust an integer and thethe set of and (0,integers −1), and are draw Graphform the intercepts, this be case −3

positive and negative−4 whole numbers.

line through the points. To convert the equation to standard form, we can use properties of equality and multiply each term by 2 to change the rational coeﬃcient of x. We see that the graph still maintains the same slope and y-intercept as the original slope-intercept form, but standard form allows us to focus on other key features of the function. Original equation 2x − 3y = 3

Multiplication property of equality

Example 1

Evaluate the multiplication Consider the linear equation: A = 1, B = 8, and C = 32 now meet the requirements for standard form. a whether the equation is written standard form. If not, rewrite it in standard form. b Determine Graph the equation using the standard forminfrom part (a).

PurposeCreate a strategy Create a strategy Help students understand the requirements for an equation be inthat standard formisand how to convert it if it is In equation towill be find written standard form, we mustto confirm the the equation written in the form Toorder graphfor thethe equation, we the in x-intercept and y-intercept and then draw line passing through these not. Ax + By = C, that A, B and C are integers, and A and B are not both zero. points. Expected mistakes Apply the idea StudentsSince might forget multiply terms by the same number when to eliminate in theto equation, it’sall notthe written in standard form because A must be trying an integer and integersthe arefractional the set of coefficient. They only multiply the fraction, resulting in an incorrect equation. Encourage students to write positive andmight negative whole numbers. out the multiplication with each term separately ensure all terms areand multiplied by the To convert the equation to standard form, we canto use properties of equality multiply each termsame by 2 tovalue. change the rational coefficient of x.

Reﬂecting with students 146 Mathspace Virginia SOL AlgebraOriginal 1 equation It’s importantmathspace.co to note that not every equation can be immediately identified as being in standard form or not. Sometimes, we need to do a little work to convert it. In this case, it was multiplying every term by 2 to eliminate Multiplication property of equality the fractional coefficient. Evaluate the multiplication

Students:A Pages = 1, B = 8,146–147 and C = 32 now meet the requirements for standard form. b Graph the equation using the standard form from part (a).

Create a strategy To graph the equation, we will ﬁnd the x-intercept and y-intercept and then draw the line passing through these points.

Apply the idea To ﬁnd the x-intercept, set y = 0 and solve for x: x + 8(0) = 32

Substitute y = 0

x = 32

Simplify

So, the x-intercept is (32, 0). 146

Mathspace

Virginia SOL Algebra 1

To ﬁndmathspace.co the y-intercept, set x = 0 and solve for y: 0 + 8y = 32

Substitute x = 0

8y = 32

Simplify

y=4

Divide both sides by 8

So, the y-intercept is (0, 4). y 8 6 4 2 x 2

Reﬂect and check

10 12 14 16 18 20 22 24 26 28 30 32

3.04 Standard form mathspace.co

By using the intercepts, we accurately graphed the equation. The line passes through both intercepts (32, 0) and (0, 4).

297

So, the x-intercept is (32, 0). To ﬁnd the y-intercept, set x = 0 and solve for y: 0 + 8y = 32

Substitute x = 0

8y = 32

Simplify

y=4

Divide both sides by 8

So, the y-intercept is (0, 4). y 8 6 4

Apply the idea

To ﬁnd the x-intercept, set 2y = 0 and solve for x: x + 8(0) = 32

x = 32

Substitute y = 0 4

Simplify

10 12 14 16 18 20 22 24 26 28 30 32

So, the x-intercept is (32, 0). To ﬁnd the y-intercept, set x = 0 and solve for y:

Reﬂect and check

0 + 8y = 32 Substitute x = 0 By using the intercepts, we accurately graphed the equation. The line passes through both intercepts (32, 0) and (0, 4). 8y = 32 Simplify y=4

Divide both sides by 8

So, the y-intercept is (0, 4).

Example 2 y Purpose 8 Check students’ ability to graph a linear equation from A line with a slope of passes through the point (−8, 5).its standard form. 6

a mistakes Write the equation of the line in slope-intercept form. Expected 4 Students may struggle to identify the correct points where the line crosses the axes, or may not draw a straight Create a strategy line. Encourage students to label points and keep track of their points in a table ensure their points are following 2 Use the slope and the point to solve for the y-intercept and write in slope-intercept form. x the correct linear pattern.

Apply the idea

10 12 14 16 18 20 22 24 26 28 30 32

Reﬂecting with students Slope-intercept form Remind students that in the standard form, Ax + By = C, the x-intercept can be found by setting y = 0 and y =by 5, setting m = − , and −8 solving for y. solving for x, and y-intercept canSubstitute be found x = x0=and Reﬂect andthe check Evaluate the the multiplication By using the intercepts, we accurately graphed equation. The line passes through both intercepts (32, 0) and (0, 4).

Students: Page 147

Simplify Write in slope-intercept form

Example 2 A line with a slope of

passes through the point (−8, 5).

a Write the equation of the line in slope-intercept form.

Create a strategy Use the slope and the point to solve for the y-intercept and write in slope-intercept form.

3.04 Standard form mathspace.co

147

Apply the idea Slope-intercept form Substitute y = 5, m = − , and x = −8 Evaluate the multiplication Simplify Write in slope-intercept form

Purpose Check students’ ability to write an equation of a line in slope-intercept form given the slope and a point through which the line passes. 3.04 Standard form 147 mathspace.co

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Expected mistakes Students may confuse the y-value of the provided point for the y-intercept. Before substituting and solving, students should explicitly write out what each variable equals before solving for the y-intercept. Reﬂecting with students Challenge advanced learners by asking them to use the given information to write the equation in standard form directly, without first writing it in slope-intercept form, and generalize their solution method. The slope of the line in standard form is the negative ratio of the coefficients of x and y. The slope of

tells

us that the coefficient of x is 1, and the coefficient of y is 4. To find the value of C in Ax + By = C, we can simply substitute the given coordinates. x + 4y = (−8) + 4(5) = 12 The equation in standard form is x + 4y = 12.

Students: Page 148 b Convert this equation to standard form.

Create a strategy Rearrange the terms and multiply coeﬃcients as needed to ﬁt standard form.

Apply the idea Slope-intercept form from part a Multiply both sides by 4 Evaluate Move the x term to the other side of the equation. The standard form of the equation is x + 4y = 12. b Convert this equation to standard form.

strategy cCreate Solveafor the x-intercept and y-intercept of the standard form equation. Rearrange the terms and multiply coeﬃcients as needed to ﬁt standard form.

Purpose Create a strategy Check students’ Apply theability idea to convert an equation from slope-intercept form to standard form, understanding the Substitute 0 and solve to ﬁnd the x-intercept, (x, 0), and the y-intercept, (0, y). relationship between different formsSlope-intercept of linear equations. form from part a Apply the idea

Expected mistakes Multiply both sides by 4 Finding the x-intercept: Finding the y-intercept Students may distribute a −4 when converting to standard form, resulting in a negative A value when x + 4y = 12 Standard form equation from part b x + 4y = 12 Standard form equation from part b Evaluate rearranging. Some students may benefit from moving the x term before multiplying to ensure a positive A value. x + 4 (0) = 12

Substitute y = 0 Move the x term to the other (0)side + 4yof=the 12 equation. Substitute x = 0

x + 0 = 12formEvaluate the multiplication standard is x + 4y = 12. Students:The Page 148 of the equation x = 12 Simplify

4y = 12

Evaluate the addition

y=3

Divide both sides by 4

The x-intercept is (12, 0). The y-intercept is (0, 3). c Solve for the x-intercept and y-intercept of the standard form equation.

Create a strategy Substitute Example03and solve to ﬁnd the x-intercept, (x, 0), and the y-intercept, (0, y). Consider the line shown in the graph shown. Apply the idea Finding the x-intercept: x + 4y = 12

Standard form equation from part b

x + 4 (0) = 12

Substitute y = 0

x + 0 = 12 x = 12

Evaluate the multiplication Simplify

The x-intercept is (12, 0).

y 9 8 x + 4y = 12 Standard form7 equation from part b 6 (0) + 4y = 12 Substitute x = 0 5 4y = 12 Evaluate the addition 4 3 y=3 Divide both sides by 4 2 The y-intercept is (0, 3). 1

Finding the y-intercept

−1

Example 3

Write the equation of the line in standard form. Consider the line shown in the graph shown.

3.04 Standard form

y mathspace.co 9 Create a strategy 8 To write the equation of the line in standard form, we will ﬁrst write it in slope-intercept form y = mx + b, then convert 7 it to standard form Ax + By = C where A, B, and C are integers. 6

299

The standard form of the equation is x + 4y = 12.

c Solve for the x-intercept and y-intercept of the standard form equation.

PurposeCreate a strategy Substitute 0 andcan solve to ﬁnd the and x-intercept, they-intercepts y-intercept, (0,iny).a standard form equation. Check that students substitute solve (x, for0),x-and and Apply the idea Expected mistakes Finding the x-intercept: the y-intercept Students may forget to substitute 0 for y when solving forFinding x-intercept and vice versa. Ask students to graph their x + 4y = 12 Standard form equation from part b x + 4y = 12 in the Standard formsolution. equation from part b line and visually check that the intercepts graphed match the values original x + 4 (0) = 12

Substitute y = 0

(0) + 4y = 12

Substitute x = 0

Reﬂecting with x + 0 students = 12 Evaluate the multiplication 4y = 12 Evaluate the addition Ask students tox explain why we substitute 0 for y when finding the andsides vicebyversa. = 12 Simplify y = 3x-intercept, Divide both 4 The x-intercept is (12, 0).

The y-intercept is (0, 3).

Students: Pages 148–149 Example 3

Consider the line shown in the graph shown.

9 8 7 6 5 4 3 2 1 −1

−1

x 1

Write the equation of the line in standard form.

Create a strategy To write the equation of the line in standard form, we will ﬁrst write it in slope-intercept form y = mx + b, then convert it to standard form Ax + By = C where A, B, and C are integers.

Apply the idea 148

Mathspace

Virginia SOL Algebra 1

First, we ﬁnd the slope (m) of the line. mathspace.co Write the formula Substitute y2 = 4, y1 = 1, x2 = 0 and x1 = 2 Evaluate the subtraction Simplify The slope of the line is

Now, we use the slope-intercept form with the point (0, 4) to ﬁnd the y-intercept (b): Write the slope-intercept form Substitute

and b = 4

Now, we convert the slope-intercept form to standard form: Original slope-intercept form Multiply both sides by 2 Evaluate the multiplication Add 3x to both sides Evaluate and rearrange to standard form The equation of the line in standard form is 3x + 2y = 8.

Reﬂect and check To conﬁrm our equation, we can check the intercepts:

300

For the x-intercept, setAlgebra y = 0: 1 Teacher Edition Mathspace Virginia SOL mathspace.co Substitute y = 0 Evaluate the multiplication Divide both sides by 3

Evaluate the multiplication Add 3x to both sides Evaluate and rearrange to standard form The equation of the line in standard form is 3x + 2y = 8.

Reﬂect and check To conﬁrm our equation, we can check the intercepts: For the x-intercept, set y = 0: Substitute y = 0 Evaluate the multiplication Divide both sides by 3 Evaluate The x-intercept is

For the y-intercept, set x = 0: Substitute x = 0 Evaluate the multiplication Divide both sides by 2 Evaluate The y-intercept is (0, 4). These intercepts match the points given in the original problem, conﬁrming that the equation 3x + 2y = 8 is correct.

Purpose 3.04 Standard form mathspace.co To check students’ ability to determine the equation of a line in standard form from a graph.

149

Expected mistakes Students may mistakenly use the points to form the equation in slope-intercept form, y = mx + b, instead of standard form, Ax + By = C. Writing out the general standard form and aligning this form with their final answer may help students format their answers.

Students: Page 150 Example 4 A tour company travels to the Great Smoky Mountains National Park. They use a combination of buses and vans to get tourists to their destination. One bus can take 42 passengers, and one van can take 7 passengers. One day, they have 168 people register for the tour. a Write an equation in standard form that could be used to model the number of buses and vans they could use to transport all the people registered, assuming that each vehicle will be filled.

Create a strategy In words, we can start with the idea that: (Number of people on buses) + (Number of people on vans) = Total number of people and that: Number of people on buses = 42 ⋅ (Number of buses) and: Number of people on vans = 7 ⋅ (Number of vans) We will then need to deﬁne variables and write an equation using them.

Apply the idea Let b represent the number of buses used and let v represent the number of vans used. So the Number of people on buses = 42b and Number of people on vans = 7v which ﬁnally gives us the whole equation: 42b + 7v = 168

3.04 Standard form mathspace.co

301

Number of people on buses = 42 ⋅ (Number of buses) and: Number of people on vans = 7 ⋅ (Number of vans) We will then need to deﬁne variables and write an equation using them.

Example 4 Apply the idea Let b represent the number of buses used and let v represent the number of vans used. A tour company travels to the Great Smoky Mountains National Park. They use a combination of buses and vans to So getthe tourists to their destination. One bus can take 42 passengers, and one van can take 7 passengers. One day, they Number of people on buses = 42b have 168 people register for the tour. a andWrite an equation in standard form that could be used to model the number of buses and vans they could use to transport all the people registered, assuming that each vehicle will be filled. Number of people on vans = 7v

Create a strategy which ﬁnally gives us the whole equation: In words, we can start with the idea that:

42b + 7v = 168

(Number of people on buses) + (Number of people on vans) = Total number of people

Reﬂect and that:and check It is important to declare variables and it can be helpful to use variables that relate to the quantities in the scenario to Number of people on buses = 42 ⋅ (Number of buses) make sure we don’t mix them up. and: Number of people on vans = 7 ⋅ (Number of vans) b Graph the equation with an appropriate scale and labels. We will then need to deﬁne variables and write an equation using them.

Purpose Create a strategy This example checks if students can model a real-world problem using an equation in standard form. It Apply the idea In this case, there isn’t a clear independent and dependent variable, so we can put b on the horizontal axis and v on highlights the importance of understanding how represent the bvertical axis.the number Let represent of buses used and let to v represent thedifferent number ofquantities vans used. as variables in an equation. To determine an appropriate scale, we can ﬁrst ﬁnd the values of the intercepts as those can give an idea of the So the

Expected mistakes maximum values for each axis. Number ofofpeople onand buses = 42b Some students may forget to multiply the number buses vans by their respective capacities. Others may mixand up the variables and fail to assign them correctly to buses and vans. Students can write out what each variable represents as units for the values used to of help reduce errors. Number people on vans = 7v which ﬁnally gives us the whole equation: Reflecting with students + 7v = 168 of correctly defining and using variables in After solving this problem, it’s worth discussing the42b importance mathematical modeling. This can lead to a broader conversation about how mathematics can be used to solve real-world 150 problems. Mathspace Virginia SOL Algebra 1 Reﬂect and check mathspace.co

It is important to declare variables and it can be helpful to use variables that relate to the quantities in the scenario to Students:make Pages 150–151 sure we don’t mix them up.

b Graph the equation with an appropriate scale and labels.

Create a strategy In this case, there isn’t a clear independent and dependent variable, so we can put b on the horizontal axis and v on the vertical axis. To determine an appropriate scale, we can ﬁrst ﬁnd the values of the intercepts as those can give an idea of the maximum values for each axis.

150

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Mathspace Virginia SOL Algebra 1 mathspace.co

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Apply the idea Find the b-intercept:

Find the v-intercept:

42b + 7v = 168

Equation from part (a)

42b + 7v = 168

Equation from part (a)

42b + 7 (0) = 168

Substitute v = 0

42 (0) + 7v = 168

Substitute b = 0

42b = 168 b=4

Evaluate the product

7v = 168

Evaluate the product

Division property of equality

v = 24

Division property of equality

26 Number of vans (v) 24 22 20 18 16 14 12 10 8 6 4 2 Number of buses (b) 1

An appropriate scale would be going up by 1 along the b-axis to a maximum of 5 and going up by 2 or 4 along the v-axis to a maximum of 26. This will extend both of our axes just past where we need to plot the intercepts.

Reﬂect andidea check Apply the Notice it would not make sense to connect these dots with line. This is because the situation is discrete, Find thethat b-intercept: Findathe v-intercept: not continuous. 42b + 7v = 168 Equation from part (a) 42b + 7v = 168 Equation from part (a) If we were to include all points that fall on the line, this would mean we could use partial buses or vans to transport 42b + 7 (0) = 168 Substitute v = 0 42 (0) + 7v = 168 Substitute b = 0 people. While the vans or buses could only be partially filled, we would still require a whole number of buses and 42b = 168 Evaluate the product 7v = 168 Evaluate the product vans. b=4

Division property of equality

v = 24

Division property of equality

An appropriate scale be going up by 1 along the b-axis c Predict the number of vans that would be required if only 1 bus waswould available. 26 Number of vans (v) to a maximum of 5 and going up by 2 or 4 along the v-axis to a 24 maximum of 26. PurposeCreate22a strategy 20

extend of our justto past we need plot This example understand howbto appropriate scale forwhere graphing by the 18 We canhelps ﬁnd thestudents point along the b-axis where = This 1determine andwill then up an toboth the line andaxes across the v-axis to ﬁnd thetofinding 16 the intercepts. corresponding value for v, the number of vans. values of the intercepts, which give an idea of the maximum values for each axis. 14 12

10 idea Expected mistakes Apply the 8 Students might6choose an inappropriate scale that does not accurately the data to label Using the graph, we can seerepresent that the point (1, 18) liesor onforget the graph 26 4 Number of vans (v) the axes. Students can check the spacing between their entervals make theythey arewould evenneed and make of the equation. If only to 1 bus was sure available, 24 2 Number of buses (b) 22 18 vans. adjustments if20 needed. Encourage to write out what points from their graphs represent to see if the 1 2 3 4 students 5 18 chosen make sense. labels and values 16

Reﬂect14and check

Reflecting with 12 students Notice that it would not make sense to connect these dots with a line. This is because the situation is discrete, 10 Discuss not thecontinuous. importance of the intercepts in the context of this problem. What does each intercept represent in 8 6 terms ofIfthe number of buses andthat vans is it important to choose appropriate we were to include all points fall needed? on the line, Why this would mean we could use partialan buses or vans to scale? transport 4

people. While the vansNumber or buses could(b)only be partially filled, we would still require a whole number of buses and 2 of buses Students:vans. Pages 151–152 1 2 3 4 5 c Predict the number of vans that would be required if only 1 bus was available.

Create a strategy We can ﬁnd the point along the b-axis where b = 1 and then up to the line and across to the v-axis to ﬁnd the 3.04 Standard form corresponding value for v, the number of vans. mathspace.co

151

Apply the idea 26 Number of vans (v) 24 22 20 18 16 14 12 10 8 6 4 2 Number of buses (b) 1

Using the graph, we can see that the point (1, 18) lies on the graph of the equation. If only 1 bus was available, they would need 18 vans.

3.04 Standard form mathspace.co

303

c Predict the number of vans that would be required if only 1 bus was available.

Create a strategy We can ﬁnd the point along the b-axis where b = 1 and then up to the line and across to the v-axis to ﬁnd the corresponding value for v, the number of vans.

Apply the idea Using the graph, we can see that the point (1, 18) lies on the graph of the equation. If only 1 bus was available, they would need 18 vans.

26 Number of vans (v) 24 22 20 18 16 14 12 10 8 6 4 2 Number of buses (b) 1

Reﬂect and check We can check using the equation. 42b + 7v = 168

Equation from part (a)

42 (1) + 7v = 168

Substitute b = 1

42 + 7v = 168

3.04 Standard form mathspace.co

Evaluate the product

7v = 126

Subtraction property of equality

v = 18

Division property of equality

151

Idea summary Purpose The standard form of a line is: Show students how to use a graph to solve a real-world problem involving linear equations.

Ax + By = C

Reflecting with students A, B, C are integers Ask students to the practical implications of their answer. For example, are 18 vans an efficient To consider write the equation of the line in standard form, we will follow these steps: solution? What1. other factors might the tour company need to consider? Use the given information to write the equation in slope-intercept form, y = mx + b. 2. If m is a fraction, multiply each term in the equation by the denominator of m.

Decompose thex term problem 3. Move the to the y by side scaffolding of the equation. scaled graphs

use with Example 4

Student with disabilities Standard form is usefulsupport when we know, or want to know both intercepts of the line. Help students to develop the skills to identify and draw an appropriate scale for a graph by providing different levels of scaffolding.

Horizontal and vertical lines

This approach will encourage to useorcomputational thinking break down or decompose problems Recall that lines can also bestudents either horizontal vertical. These types of linestowill not look like slope-intercept form. that theyHowever, aren’t sure how to approach. they do follow another type of special pattern. Horizontal lines are the set of all points with a ﬁxed y-value. 1. Provide the axes already ylabeled and ask questions like: 4 They are parallel to the x-axis and have equations of the form y = a, • Why do you think this scale was chosen for the horizontal axis? vertical axis? 3 where a is a real number. Horizontal lines have a slope of zero. • How many values are 2 labeled on each axis? How many do you think is appropriate in general? Shown is the horizontal line y = 2.

1 −4 −3 −2 −1

this equation is in standard form with A = 0, since the in $ ( y) x ChargeNotice 1

−1

equation is equivalent to 0x + y = 2.

3 140 4

What happens to the equation y = mx + b if you substitute 0 for m? We get y = b.

120

−2 −3

100

−4

Vertical lines are the set of all points with a ﬁxed x-value. They are parallel to the y-axis and have equations of the form x = a, where a is a real number. Vertical lines have a slope that is undeﬁned.

Shown is the vertical line x = 1.

1 −4 −3 −2 −1

−1

Time in (x) form with B = 0. Notice this equation is hours in standard

x 1

−2

304

−3

Mathspace Virginia SOL −4 Algebra 1 Teacher Edition mathspace.co

152

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Virginia SOL Algebra 1

2. Provide students with a graph that has the axes partially labeled to build their confidence with identifying an appropriate scale. For example: 140

Charge in $ ( y)

Time in hours (x) 0

3. Provide an algorithm to help determine the labels on the coordinate plane such as: 1. Find the minimum and maximum x-values based on the context. Reﬂect check and maximum y-values using the model and the minimum and maximum x-values. 2. Find theand minimum We can check using the equation.

3. Calculate

42b + 7v = 168

rounding appropriately. 42 (1) + 7v = 168

Substitute b = 1

42 + 7v = 168

Students: Page 152

to get an idea of a reasonable amount to count by for each axis -

Equation from part (a) Evaluate the product

7v = 126

Subtraction property of equality

v = 18

Division property of equality

Idea summary The standard form of a line is:

Ax + By = C A, B, C

are integers

To write the equation of the line in standard form, we will follow these steps: 1. Use the given information to write the equation in slope-intercept form, y = mx + b. 2. If m is a fraction, multiply each term in the equation by the denominator of m. 3. Move the x term to the y side of the equation. Standard form is useful when we know, or want to know both intercepts of the line.

Horizontal and vertical lines Recall that lines can also be either horizontal or vertical. These types of lines will not look like slope-intercept form. However, they do follow another type of special pattern.

Horizontal and vertical lines y

Horizontal lines are the set of all points with a ﬁxed y-value.

4 Theytheir are parallel to the x-axis and have equationsThey of thewill formunderstand y = a, Students will learn about horizontal and vertical lines, properties, and their equations. that 3 where a is a real number. Horizontal lines have a slope of zero. horizontal lines have a set y-value and a slope of zero, and vertical lines have a set x-value and an undefined slope. 2 Shown is the horizontal line y = 2. They will also study the standard form of these lines’ equations. 1 x

−4 −3 −2 −1

−1

Notice this equation is in standard form with A = 0, since the equation is equivalent to 0x + y = 2.

What happens to the equation y = mx + b if you substitute 0 for m? We get y = b.

−2 −3 −4

3 2

Shown is the vertical line x = 1.

1 −4 −3 −2 −1

−1

−2 −3

x 1

Notice this equation is in standard form with B = 0.

3.04 Standard form mathspace.co

305

To write the equation of the line in standard form, we will follow these steps: 1. Use the given information to write the equation in slope-intercept form, y = mx + b. 2. If m is a fraction, multiply each term in the equation by the denominator of m. 3. Move the x term to the y side of the equation. Standard Students: Page 152 form is useful when we know, or want to know both intercepts of the line.

Horizontal lines are the set of all points with a ﬁxed y-value. They are parallel to the x-axis and have equations of the form y = a, where a is a real number. Horizontal lines have a slope of zero.

Shown is the horizontal line y = 2.

1 −4 −3 −2 −1

x 1

−1

Notice this equation is in standard form with A = 0, since the equation is equivalent to 0x + y = 2. What happens to the equation y = mx + b if you substitute 0 for m? We get y = b.

−2 −3 −4

Shown is the vertical line x = 1.

1 −4 −3 −2 −1

−1

x 1

Notice this equation is in standard form with B = 0.

−2 −3 −4

152

Mathspace Virginia SOL Algebra 1 mathspace.co

Annotated graph of horizontal and vertical lines Student with disabilities support Provide an annotated graph that illustrates the characteristics of horizontal and vertical lines, such as their equations, slopes, and how they appear on a graph. Have students fill in the chart during the lesson. An example graph is shown.

y 4 slope = undefined 3 x = −3 2 vertical 1 −4 −3 −2 −1 −1 −2 −3 −4

306

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x 3

slope = 0 y = −2 horizontal

Distinguishing between key features of horizontal and vertical lines Targeted instructional strategies Key characteristics of horizontal and vertical lines can seem similar, so having students connect the unique features to each graph is a useful exercise. Provide two tables that represent a horizontal and vertical line, respectively. x f (x)

−1 4

0 4

1 4

2 4

3 4

x g(x)

4 −1

4 0

4 1

4 2

4 3

Ask students to use the table to determine and describe: • the shape of the line • the slope of the line • the values that all points on each line share • how the values shared by all points on the table could shape the equation of the line Graph both tables of values and draw the lines through the points. Ask students to determine if the visual of each table changes or confirms their predictions from the tables before further instruction.

6 5 4 3 2 1 −3 −2 −1 −1

x 1 2 3 4 5 6 7

−2

Examples Students: Page 153 Example 5 Consider the line x = −8. a Plot the line on a coordinate plane.

Create a strategy

Apply the idea

We can use the fact that vertical lines have equations of the form x = a.

This will be a vertical line that crosses the x-axis at −8. y 4 2 x −8 −6 −4 −2

−2 −4

b Determine the domain and range of the line.

Create a strategy

Apply the idea

The domain of a vertical line is restricted to the single value of x at which the line is located.

Domain: {−8}

The range of a vertical line is all possible y-values since the line extends inﬁnitely in both the positive and

Range: (−∞, ∞)

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307

4 2 x

PurposeExample 5 Show students to xplot Consider how the line = −8.vertical lines on a coordinate plane.

−8 −6 −4 −2

−2 −4

a Plot the line on a coordinate plane.

Students: Page 153

Create a strategy

Apply the idea

b Determine domain and range the equations line. We can use thethe fact that vertical linesof have of the form x = a.

This will be a vertical line that crosses the x-axis at −8.

Create a strategy

Apply the idea

The domain of a vertical line is restricted to the single value of x at which the line is located. The range of a vertical line is all possible y-values since the line extends inﬁnitely in both the positive and negative y-directions.

y 4

Domain: {−8} Range: (−∞, ∞)

2 x

−8 −6 −4 −2

−2 −4

Example 6 PurposeWrite the equation of the line given. Show students how to determine the domain and range of a vertical line. a Determine the domain b and range of the line. y 4

Expected mistakes Create a strategy3 Apply the idea Students may confuse the 2 notation for the domain, which uses brackets for a single value, and the range, which The domain of a vertical line is restricted to the single Domain: {−8} uses parentheses to represent interval. Ask students to provide a verbal description what the single value in 1 line isan value of x at which the located.x Range: (−∞, ∞) the domain represents and the interval needed to represent all real numbers. −4 −3 −2 −1 1 2 3 4

The range of a vertical line is all possible y-values −1 since the line extends inﬁnitely in both the positive and Students:negative Page y-directions. 153 −2 −3

−4

Example 6 Applythe the idea of the line given. Write equation Horizontal lines have a ﬁxed y-value and have equations of the form y = a. The ﬁxed y-value for this line is −3, so the a y equation of the line 4is: y = −3 3 2 1 −4 −3 −2 −1 −1

x 1

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153

−2 −3 −4

Apply the idea Horizontal lines have a ﬁxed y-value and have equations of the form y = a. The ﬁxed y-value for this line is −3, so the equation of the line is: y = −3

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Purpose Show students how to write the equation of a horizontal line based on its graph. Reflecting with students The table of values for a horizontal line should reflect the equation written, in this case that all y-values equal 3. Finding points on the line and creating a table of values will confirm the correct equation of the line.

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Students: Page 154 b An undeﬁned slope through the point (2, −1).

Apply the idea Lines that have an undeﬁned slope are vertical lines which have equations with the form x = a. The x-value of the ordered pair (2, −1) is 2, so the equation can be written as: x=2

Idea summary

Purpose Horizontal lines are the set of all points with a ﬁxed y-value. They are parallel to the x-axis and have equations Show studentsofhow to write the equation a line with an undefined slope. the form y = a, where a is a realof number. Vertical lines are the set of all points with a ﬁxed x-value. They are parallel to the y-axis and have equations of

Reflecting with students the form x = a, where a is a real number. A vertical line’s slope is undefined since there is no horizontal change between points, so the denominator in the slope ratio is always 0. Drawing the vertical line on a coordinate plane and labeling points to count the vertical change shows this concept.

Practice

What do you remember?

3 1

State whether the each equation are written in standard form: 2 a

y = −x + 1

Write each of the following equations −4 in−3standard −2 −1 form. 1 a

SOL

3x + 2y = 6

−y + 4x = 8

3y = 8 − 4x

−1

−2

x 2

18x − 6y = 12

3y = 4x

Select the equation that is shown on the graph. −3 A 10x + 5y = −1 B −10x + 5y = −1 −4 C

−5x − 10y = −1

−x + 5y = −10

Three reads English language learner support

1 −4 −3 −2 −1 −1

x use with Example 6 2

To support finding equations of horizontal and vertical lines, students should use the first−2read to understand the −3 goal of the question, which involves finding the equation of the line. Students should write down any information −4 about equations of vertical and horizontal lines available. In the second read, students should focus on identifying the specific mathematical information in the problem, 4 For each equation, identify the slope, the x-intercept, and the y-intercept. such as the point or slope. They should write down this information with any key terms or variables that may be a 2x + y = −6 b 3x − 7y = −21 c 9x − y = 18 important for writing their answer. 5

State whether the graphs of the following lines are horizontal or vertical:

For the third read, students need to clarify how to representcand connect the information from the first two a y=2 b x = −4 y=3 d x = 17 reads. Discuss as a class,ensuring that all students understand the terms and processes involved. This approach h =x e y=0 f x=0 g y− =0 aids in breaking down the problem into manageable parts and reinforces comprehension through repetition and discussion. 6 The table shows some points on the line with equation y = 0. Does the line y = 0 represent the y-axis or y-axis?

154

x y

−6 0

−4 0

1 0

5 0

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b An undeﬁned slope through the point (2, −1).

Apply the idea Lines that have an undeﬁned slope are vertical lines which have equations with the form x = a. The x-value of the ordered pair (2, −1) is 2, so the equation can be written as:

Students: Page 154

x=2

Idea summary Horizontal lines are the set of all points with a ﬁxed y-value. They are parallel to the x-axis and have equations of the form y = a, where a is a real number. Vertical lines are the set of all points with a ﬁxed x-value. They are parallel to the y-axis and have equations of the form x = a, where a is a real number.

Practice What do you remember? Practice 1

State whether the each equation are written in standard form:

Students: Pages a 3x154–160 + 2y = 6 2

SOL

18x − 6y = 12

3y = 8 − 4x

3y = 4x

Select the equation that is shown on the graph.

C =−5x 3x + 2y 6 − 10y = −1

−x ++5y yD = −x 1 = −10

−y + 4x = 8

3y = 8 − 4x

10x + 5y = −1

−10x + 5y = −1

−5x − 10y = −1

−x + 5y = −10

2x + y = −6

3x − 7y = −21

−3 −4

2 1

9x − y = 18

State whether the graphs of the following lines are horizontal or vertical: a

y=2

x = −4

y=3

y=0

x=0

y−

The table shows some points on the line with equation y = 0. Does the line y = 0 represent the y-axis or y-axis?

1 2= 4x 3 4 3y

−2

For each equation, identify the slope, the x-intercept, and the y-intercept. a

18x − 6y = 12 x

−4 −3 −2 −1 d −1

Select the equation that is shown on the graph. A

Write each of the following equations in standard form. a

SOL

State whether the each written A 10x + 5y = −1 equationBare−10x + 5y =in−1standard form: a

Write each of the following equations in standard form.

−y + 4x = 8 What do youa remember?

y = −x + 1

−4 −3 −2 −1 −1

x = 17

x y

−6 0

−4 −4 1 0 0

x = 17

x y

−6 0

x 1

−2 −3 5 0

For each equation, identify the slope, the x-intercept, and the y-intercept. a

2x = −6 Virginia SOL Algebra b 3x 154 + y Mathspace 1 − 7y = −21

9x − y = 18

mathspace.co

State whether the graphs of the following lines are horizontal or vertical: a

y=2

x = −4

y=3

y=0

x=0

y−

The table shows some points on the line with equation y = 0. Does the line y = 0 represent the y-axis or y-axis?

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−4 0

1 0

5 0

Consider the sets of points in the following coordinate planes: i

State whether the set of points lies on a vertical or horizontal line.

Find the equation of the line that passes through the set of points.

−8 −6 −4 −2 −1

−8 −6 −4 −2 −2

−2

−4

−3

−6

−4

−8

Let’s practice SOL

Select the graph of the equation 3x + 4y = −2. A

7 6 5 4 3 2 1

−7 −6 −5 −4 −3 −2 −1−1

7 6 5 4 3 2 1

−7 −6 −5 −4 −3 −2 −1−1

1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

−7 −6 −5 −4 −3 −2 −1−1

1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

x 1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

For each of the following equations: i

Calculate the x-value of the x-intercept of the line.

iii

Draw the graph of the equation of the line on the coordinate plane.

3x − 5y = −15

5x − 3y = −15

Calculate y-value of the y-intercept of the line.

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Draw the graph of each of the following linear equations: a

2x = 6

6y = −3

3x − 2y = −12

Write the equation of the line.

For each of the following graphs: i

Find the slope.

1 −4 −3 −2 −1

−1

−2

−3

−4

−4 −3 −2 −1

−1

−2

−3

−4

For each of the following tables of values: i

Sketch the graph of the line that passes through the four points.

Find the equation of the line.

x 2 2 2 2 y −2 −1 0 1

x y

y = −2

−4 5

−2 5

0 5

2 5

Plot the following lines on a coordinate plane: a

−1

x 1

−1

−2

−4 −3 −2 −1

5x + 4y = 40

y=8

x=7

y=0

Consider the following line: a

State the y-value of the y-intercept.

Find the slope, m, of the line.

Find the equation of the line in the form y = mx + b.

Rewrite the equation of the line in standard form.

2 1 −4 −3 −2 −1

−1

−2 −3 −4

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x 1

For each graph, write the equation of the line in standard form. a

1 −4 −3 −2 −1

x 1

−1

−4 −3 −2 −1

x 1

−1

−2

−3

−4

Write the equation of the line in standard form. a

A straight line passes through the point (3, 5) with a slope of .

A straight line passes through the point (−2, 0) with a slope of 2.

A straight line passes through the point (−3, −1) with a slope of −1.

A straight line passes the point (−1, 2) and the point (3, 0).

Marvin went to the bulk food store to buy x lb of granulated sugar for $0.60/lb and y lb of powdered sugar for $0.90/lb. The total cost of all the sugar is $12. a

Select the graph of the linear function that relates the number of pounds of granulated sugar x and the number of pounds of powdered sugar y. A

y 18

6 3

x 3

D 21

6 x 6

y 21

Identify the restrictions on the independent variable.

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Salvina needs to buy a mix of red and green apples. She is going to spend $15 in total. Green apples cost $1.50 lb and red apples cost $3 lb. a

Write a linear equation in standard form to represent the scenario, deﬁning any variables.

Calculate and interpret the x-intercept of the linear equation from part (a).

Convert the equation from part (a) to slope-intercept form.

Graph the function, including clearly labeled axes with an accurate scale.

Apples

Green $1.50/lb

Red $3/lb

For each of the following pairs of linear functions, identify which function has the greater x-intercept: a • Function 1: The line with a slope of 3 and a y-intercept of −3. • Function 2: x − y = 3 • Function 2: b • Function 1: The line with a slope of −1 and a y-intercept of 4. x 0 1 y −6 −3 c

• Function 2: • Function 1: 5x − y = 15 x y

x y

0 4

1 2

0 3

1 5

• Function 2:

1 2

2 1

5 4 3 2 1 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5 6 7

• Function 1: • Function 2: 5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

314

2 4

2 0

e • Function 1: x y

0 6

• Function 2: 4x + y = 16

Function 1: d •

2 0

x y

x 1 2 3 4 5

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0 −8

1 −4

2 0

Which of the following linear relationships has an x-intercept? • Function 1: y = 3 • Function 2: 8

6 4 2 −8 −6 −4 −2 −2

x 2

−4 −6 −8

Let’s extend our thinking 21

The line ax + 4y = 12 passes through the point (8, −3). a

Solve for a.

Write the equation of the line in standard form.

Solve for the x-value of the x-intercept of the line.

The line kx + 3y = 15 passes through the point (13, −8). a

Find the value of k.

Solve for the y-value of the y-intercept of the line.

Given: • Line A: 5x + 3y + 5 = 0

• Line B: 7x + 6y − 3 = 0

Which line is steeper? Justify your answer. 24

Describe a scenario where using the standard form of a linear equation is more useful than the slope-intercept form.

Create a scenario where you would choose to use standard form, Ax + By = C, and explain what the intercepts mean in context.

Eﬃe is a entomologist and is currently studying mosquitoes and spiders. She knows that mosquitoes have six legs and spiders have eight legs. In her lab, she has a mix of mosquitoes and spiders. Between all the bugs, there is a total of 240 legs.

Create a model to represent this scenario. Deﬁne any variables and include appropriate labels in your model.

State and describe the possible number of mosquitoes.

Explain whether or not every point on the line represents a possible solution.

Nakisha is selling the baby outfits her daughter has outgrown and is buying second-hand toddler outfits. She is able to sell each baby outfit for $3 and she buys each toddler outfit for $5. In the end, she wants to make a profit of exactly $30. Profit = revenue − cost Revenue is the money earned Cost is the money spent a

Write an equation to represent the scenario, using x for the number of baby outfits sold, y for the number of toddler outfits bought, and the profit of exactly $30.

Calculate and interpret the x-intercept.

Nakisha has 25 baby outfits available to sell and needs 15 toddler outfits. Determine if she is able to make her desired profit of $30.

Using the additional information from part (c), state and interpret the restrictions on the viable values for x. 3.04 Standard form mathspace.co

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Answers

c i x=5 iii

3.04 Standard form

10 8 6 4 2

What do you remember? 1 a Yes

b No

c No

2 a 4x − y = 8

d Yes

−3−2 −1 −2 −4 −6 −8

b 4x − y = 8

c 6x + 2y = 1

d −4x + 3y = 0

3 B 4 a T he slope of the line is −2. The x-value of the x-intercept is −3. The y-value of the y-intercept is −6.

ii y = 10

10 a

3 1

c T he slope of the line is 9. The x-value of the x-intercept is 2. The y-value of the y-intercept is −18.

−4 −3 −2 −1 −1

2 3 4

−2

c Horizontal d Vertical

Vertical

is −7. The y-value of the y-intercept is 3.

e Horizontal f

x 1 2 3 4 5 6 7 8 9 10

b The slope of the line is . The x-value of the x-intercept

5 a Horizontal b Vertical

−3

g Horizontal h Vertical

−4

x-axis

7 a i The set of points lie on a horizontal line.

ii y = 3

b i The set of points lie on a vertical line.

ii x = −6

−4 −3 −2 −1 −1

Let’s practice

−2 −3

8 D 9 a i x = −5 iii

−4

ii y = 3 4

c 7

1 −7−6−5−4−3−2 −1 −1

1 2 3 4

−2

−3

b i x = −3

ii y = 5 6 5 4 3 2 1

−6 −5 −4 −3 −2 −1−1 −2 −3 −4 −5 −6

316

−6 −5 −4 −3 −2 −1 −1

−4

iii

10 8 6 x

1 2 3

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

2 −2

x 2

ii y = 3

11 a i 0

b i Undefined

ii x = 4

c i Undefined

ii x = −1

d i 0

ii y = −2

−3 −2 −1

ii x = 2

12 a i y 4 3 x 1

2 3 4

14 a y = 2

−2 −3 −4

ii y = 5

b i y 8 7 6 5 4 3 2 1

−6 −5 −4 −3 −2 −1 −1 −2

b m = −3

c y = −3x + 2

d 3x + y = 2

15 a x + 2y = 2

b 3x − y = 4

16 a 2x − 3y = −9

b 2x − y = −4

c x + y = −4

d x + 2y = 3

17 a B b The minimum value for the independent variable, x is 0 . So, 0 ≤ x ≤ 20.

and the maximum is x 1 2 3 4

18 a 3x + 6y = 30 where x represents green apples and y represents red apples b x = 10, so she could buy 10 green apples and no red apples for $15. c

y 12

Red apples

x 1

2 3 4

2 1

Green apples 2

19 a Function 2

b Function 1

c Function 2

d Function 2

e Function 2

−2 −1 −1

1 2 3 4 5 6 7 8 9

20 Function 2

21 a a = 3 4

−3 −4

b x=5

c y=5

23 Line A. Writing both equations in slope-intercept form:

−2

b 3x + 4y = 12

22 a k = 3

−2 −1 −1

Function 2

Let’s extend our thinking

−2

−3

−4 −3 −2 −1 −1

−1

−4

−4 −3 −2 −1

x 1

−2

13 a

x 1

2 3 4 5 6

A is

, we can see the slope of line , which has a larger magnitude than

, the

slope of line B. 24 A scenario where we are creating a mixture or different portions of two substances and know the total amount. For example, if we are buying a mix of wheat and rye flour

Answers mathspace.co

317

at the store and they have different prices and know that we need a total of 16 cups of flour. 25 A scenario where we are given the unit rates for two items that we are creating a mixture of to get a certain total. For example, if we are told the cost of almonds per pound (A), the cost of peanuts per pound (B), and are told the total amount to be spent (C). 26 a E quation model: 6x + 8y = 240 where x represents mosquitos and y represents spiders Graph:

Spiders

35 30 25 20 15 10 5

Mosquitoes 5 10 15 20 25 30 35 40 45 50

b There are between 0 and 40 mosquitoes. c W e can only have whole numbers for the number of mosquitoes and spiders, so no, every point on the line does not represent a solution, for example there are no valid points between (0, 30) and (4, 27).

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27 a 3x − 5y = 30 b The x-intercept is x = 10 which means if she sells 10 baby outfits, she will buy 0 toddler outfits. c I f we substitute y = 15 into our equation, we find that x = 35. This means that in order to buy 15 toddler outfits and make a profit of $30, she would need to sell 35 baby outfits, but she only has 25, this is not enough. It is not possible to make her desired profit. d T he viable values for x are between 10 and 25, inclusive. If she was to sell less than 10 baby outfits she would not make her desired profit, so she must sell at least 10 baby outfits and she only has 25 baby outfits, so that is the maximum value.

3.05 Point-slope form Subtopic overview Lesson narrative In this lesson, students will extend their thinking to include writing the equation of a line multiple different ways with the same form using point-slope form. They will connect that while the point used from the line may change, the slope will be consistent. Students will connect that the point-slope form of an equation will simplify to the same slope-intercept or standard form regardless of the point on the line used. Students will be able to use the graph of a line to write multiple representations of an equation in point-slope form. By the end of the lesson, students should aim to be confident in being able to determine equations of lines in point-slope form, and connect multiple representations of the same line.

Learning objectives Students: Page 161

Key vocabulary 

coordinates



ordered pair



point-slope form

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG2 — Mathematical Communication Teachers can foster Mathematical Communication by encouraging students to explain their reasoning when choosing between point-slope form, slope-intercept form, and standard form for representing linear functions. They can also be asked to justify their reasoning when deciding which form is most beneficial based on the context or the characteristics they are being asked to find.

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MPG4 — Mathematical Connections

MPG5 — Mathematical Representations

The teacher can help students make Mathematical Connections by emphasizing the connection between slope-intercept form and the point-slope form introduced in this lesson. Teachers can also help students make connections between the slope formula and point-slope form, and between the different ways that linear functions can be represented algebraically.

Teachers can facilitate Mathematical Representations by teaching students to graph a linear function given in point-slope form, and then asking them to convert the equation to slope-intercept form, identify the slope and y-intercept, and plot these values on the graph. Additionally, teachers can provide opportunities for students to compare and contrast the characteristics of linear functions represented in various forms algebraically in point-slope form, slope-intercept form, standard form, graphically, in tables, and in contextual situations.

A.F.1dii — Write the equation of a linear function to model a linear relationship between two quantities, including those that can represent contextual situations. Writing the equation of a linear function will include the following situations: ii) given two points on the line whose coordinates are integers; A.F.1diii — Write the equation of a linear function to model a linear relationship between two quantities, including those that can represent contextual situations. Writing the equation of a linear function will include the following situations: iii) given the slope and a point on the line whose coordinates are integers; A.F.1f — Graph a linear function in two variables, with and without the use of technology, including those that can represent contextual situations.

A.EI.1 — The student will represent, solve, explain, and interpret the solution to multistep linear equations and inequalities in one variable and literal equations for a specified variable.

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Engage Activity The slope of the group

60 mins

Students will use the heights of each person in their group to determine where each person needs to stand in the classroom so that the top of their heads form a straight line. Students will use their understanding of linear equations to create the line.

Understanding and skills

Will use Finding slope between two points. Understanding that two points create a line.

Will develop Writing a linear equation given two points.

Could extend Writing a linear equation using point-slope form.

Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: Tape measure, yardstick, rulers, or other measuring tools, paper, pencil, graphing technology (optional) • Form groups in advance of the class period to ensure that no two students in a single group are the same height.

Support students with disabilities Support memory - use previously taught skills and concepts Provide the formula for slope:

Where: • (x1, y1 ) is one point on the line • (x2, y2 ) is a different point on the line

Support for English language learners Critique, correct, and clarify While students are discussing the calculations for slope and writing the equations of the lines, display the following incorrect calculation and reasoning: The difference between our heights are 2 in and we are standing 12 in apart so the next person needs to be 2 in shorter and stand 12 in apart. Ask students to identify the error, critique the reasoning, and write a short explanation of how the students’ reasoning could be improved.

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Classroom guide Hook

Co-craft questions

Students create questions about the image of characters standing from tallest to shortest.

•

5 mins

What mathematical questions can you ask about this image?

Slide 1 from Student Engage Activity

Implementation details Students may wonder about the heights of the individuals in the picture, how far apart they are standing from one another, or the equation of the line that can be seen by the top of their heads forming a straight line. Encourage students to think about the slope of the line, or whether it is indeed a straight line.

Launch

5 mins

Let students know that they will be getting up and moving today in class in order to determine where each group member needs to stand in the classroom so that the top of their heads form a straight line, similar to the picture in the hook. Let students create groups and make sure that each group has the necessary materials to complete task. It may be helpful to have starting point labeled on ground and a measuring tape or yard stick on the ground as well as a measuring tape or yard stick for students to use to determine heights.

You will be working with your group to figure out how to arrange yourselves in the classroom so that the top of your heads form a straight line.

Suggested grouping: Groups of 3 or 4 and assign roles Slide 2 from Student Engage Activity

Continue when Students have read the Launch and understand the context of the problem.

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Explore

Team roles

•

35 mins

Students could create a table of values, equation, or graph. Students may also add a photo of them standing in the classroom creating a line.

Anticipated strategies Create a table Students can create a table of values that include the heights of each person in their group. Students could start generating a table as shown here: Horizontal distance from table, feet 10 ? ? ?

Height, inches 68 66 62 59

Note that students may use a different location as their origin. In order to ﬁnd the horizontal distance from the table for each group to stand, groups need to determine a slope that would allow each person’s height to be reached in the horizontal distance provided in the classroom. Students could use guess and check or other methods to determine the horizontal position. Solve for horizontal position

Using definition of slope Students can use the definition of slope to solve for the horizontal position (this can be used to determine the horizontal position for each person to stand to complete x-values in the table of values shown above). Possible example: Let where the groups’ table intersects with the ground be the origin. If students choose a slope of 2, students could use the definition of slope to find the horizontal distance for each group member to stand. The difference in y-values between 66 and 68 is −2. The second group member must be one foot away from the first group member. Algebraically:

So,

= 2 and using substitution

= x − 10 and −1 + 10 = x. Which gives us 9 = x. Similar process could be used to determine other horizontal distances for group members. Horizontal distance from table, feet 10 9 7 5.5

Height, inches 68 66 62 59

Solve for horizontal position

Using slope-intercept form Students use slope-intercept form of linear equation and group members’ height to solve for horizontal position. Students could choose a y-value for the y-intercept that is less than or equal to the shortest person in the group and choose a slope.

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Using above information, if students chose a slope of 2 and wanted the tallest person of 68 inches to stand 10 feet away from the table, the linear equation in slope-intercept form would be y = 2x + 48 Students could then plug in known heights for each person and then solve for x. Example: If group member was 62 inches tall, students could substitute 62 for y and solve for x. 62 = 2x + 48 and 14 = 2x, so 7 = x The group member with a height of 62 inches needs to stand 7 feet away from table.

Create a graph Students create a ﬁxed horizontal distance for a person in their group and choose a slope that would ensure all heights of people in group would be met in distance available in classroom. Students test this by drawing a graph and highlighting points on function and explaining x- and y-values of points in context (horizontal distance and height of group member). This could be done with graphing technology.

Misconceptions Misinterpreting units for height, such as not interpretting 5′8″ as 68 inches : What units is your group currently using for height? Are the heights of each group member correctly recorded in the table? How can we check?

Choosing a slope that does not include each person’s height based on horizontal distance students chose Have you tested these positions yet based on your slope? What would the line look like?

Purposeful questions Use the following questions to check for understanding and encourage critical thinking: • What information do you know about your line so far? • What information is needed in order to create a line, using your heads as ‘points’? • Why did your group choose this value for the slope of the line? • Why do group members only have one place they can stand once two people in your group are in position? • Does this solution make sense? How do you know?

Continue when Students have a visual representation of a line created by the top of the heads of each group member as well as a description of the process of determining where each group member should stand in classroom.

Discuss

15 mins

Begin with group presentations. After group presentations, consider introducing the point-slope formula and asking students to review the task and see how this equation could help determine where each person needs to stand.

Discussion guide Invite each group to show their visual representation of their line as well as the process for determining where each group member needed to stand to create line. If possible, start with the visual models and move to algebraic methods. If time allows, feel free to manipulate the slope formula and get the difference between the y-values isolated on one side, introducing the point-slope formula. This form of a linear equation is helpful when we know a point on the line (that is usually not the y-intercept) and the slope of the line. Ask students to review the task in their groups and see how this equation could help determine where each person needs to stand in the classroom so the top of group members’ heads form a straight line. Note that point-slope formula will be formally introduced in the next lesson.

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Tools You may ﬁnd these tools helpful: • Graphing calculator • Graph paper

Student lesson & teacher guide Point-slope form Students will learn how to use the point-slope form to find the equation of a line when given a point on the line and the slope. They will understand how to represent the coordinates in an ordered pair, calculate the slope from two points or via a given rate of change in a scenario. Students will then learn how to convert the point-slope form of an equation into slope-intercept form.

Students: Pages 161–163

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325

The same function could also be represented as

3 (−2, 2)

2 1

since the slope remains constant at line (−2, 2).

−4 −3 −2 −1 −1

but uses the point on the

−2 −3 −4

We can also ﬁnd the equation in point-slope form when given the coordinates of two points on the line by following these steps: 1. Choose which ordered pair will represent (x1, y1) and the other (x2, y2).

9 y 8 7 6 5 4 3 2 1 −1 −2 −3 −4 −5 −6 −7 −8 −9

x1, y1 x 1

326

x2, y2

9 y 8 7 6 5 4 3 2 1 −1 −2 −3 −4 −5 −6 −7 −8 −9

2. Solve for slope using

(3, 5) x 1

(7, −2)

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

−2 −3 −4 −5 −6 −7 −8 −9

x2, y2

9 y 8 7 6 5 4 3 2 1

2. Solve for slope using

(3, 5) x 1

−1 −2 −3 −4 −5 −6 −7 −8 −9

(7, −2)

3. Rewrite y − y1 = m(x − x1), substituting m and (x1, y1).

9 y 8 7 6 5 4 3 2 1

162

(3, 5)

1 2 3 4 5 6 7 8 9 −1 (7, −2) −2 −3 −4 −5 −6 −7 −8 Mathspace Virginia SOL Algebra 1 −9 mathspace.co

We are also able to ﬁnd the slope-intercept form of an equation written in point-slope form by solving the equation for y.

Example 1

Using point-slope Given the point (4, −1) andform the slope m = 2, write the equation of the line in point-slope form. Targeted instructional strategies Create amay strategy Some students beneﬁt from explicit steps for writing an equation of a line using point-slope form.

To write the equation of the line in point-slope form, we use the formula: y − y1 = m(x − x1) where (x1, y1) is a point on

1. Identify a point theslope. line (x1, y1). the line and mon is the

2. Determine the slope of the line, m. Apply the 3. Substitute theidea point and the slope into the point-slope form equation y − y1 = m(x − x1). y − (−1) = 2(x − 4)

Substitute x1 = 4, y1 = −1, and m = 2

4. Simplify the equation if, necessary. y + 1 = 2(x − 4)

Simplify

ProvideThe a graph of a with different point-slope equation of line the line in point-slope form is y + 1 = 2(x − 4). form equations at labeled points to use as a guide to demonstrate using the steps. An example is shown:

Example 2 By following these steps, students can write the equation The graph of a linear function is shown. Write the equation of the line in of any line given a point and the slope. This process can point-slope form for each of the given points on the line. also be reversed to ﬁnd the slope and a point on the line from an equation in point-slope form.

7 6 5 4 3 2 1

6 5 4 23 3 2 1

−4−3−2 −1 1 −1 −2 −6−5−4−3−2 −1 −1 −3 −2 −4 −3 −5 −4

x 4 5 6 7 8 9 10 11 x 1 2 3 4 5 6

−5 −6

3.05 Point-slope form mathspace.co

3.05 Point-slope form

163

327

Stronger and clearer each time English language learner support In order to practice writing the equation of a line in point-slope form, students will be given their own line and be given the first line of their written statement: “We have a graph of a linear function. We see two points on the line: (⬚, ⬚) and (⬚, ⬚). We can use the point-slope form to write the equation of the line.” Students should write out the next steps of the problem in their own words, gradually refining and improving their language with each iteration. Similarly, in the second problem, the teacher can say, “We have another line that passes through the points (⬚, ⬚) and (⬚, ⬚). We need to write the equation for this line in point-slope form. Then, we need to figure out if the point (⬚, ⬚) lies on this line.” Again, students should be asked to paraphrase this in their own words, with each iteration becoming clearer and clearer. Repeat this process until the student’s explanation is accurate and clear. This strategy can also be used in group work, where students take turns explaining their solutions and giving feedback to each other. This helps students to learn from each other and to practice using mathematical language, improving their understanding and ability to communicate about the math problems.

Provide an annotated diagram Student with disabilities support To assist students who struggle to remember the meaning of each of the variables in point-slope form, consider providing students with a resource sheet with a worked example and explained equation. For example, y 12 9 6

Stay as is, represent any point on the line

3 −12 −9 −6 −3 −3

y − y1 = m (x − x1)

y − 7 = 3(x − 1) y − 1 = 3(x + 1) x 3 6 9 12

−6 y + 8 = 3(x + 4) −9

y-coordinate slope or x-coordinate of the given rate of of the given change point point

−12

Misunderstanding of point-slope form Address student misconceptions A common misconception students may have is that the point used in the point-slope form must be the y-intercept. It’s important to emphasize that any point on the line can be used in the point-slope form. Teachers can demonstrate this by using multiple points on the same line to derive the same equation in point-slope form.

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Examples

x 1

−1 −2 −3 −4 −5 −6 −7 −8 −9

8 9 (7, −2)

Students: Page 163

We are also able to ﬁnd the slope-intercept form of an equation written in point-slope form by solving the equation for y.

Example 1 Given the point (4, −1) and the slope m = 2, write the equation of the line in point-slope form.

Create a strategy To write the equation of the line in point-slope form, we use the formula: y − y1 = m(x − x1) where (x1, y1) is a point on the line and m is the slope.

Apply the idea y − (−1) = 2(x − 4) 9 y y + 1 = 2(x − 4)

Substitute x1 = 4, y1 = −1, and m = 2 3. Rewrite y − y1 = m(x − x1), substituting m and (x1, y1).

Simplify

8 The equation of the line in point-slope form is y + 1 = 2(x − 4).

7 6 5 4 3 2 Example 1 2 Purpose

(3, 5)

1 2 3 4 formulate 5 6 7 8 equation 9 −1 Check whether of aofline form given a point and the slope. The graphstudents of a linear can function is shown.the Write the equation the in linepoint-slope in y (7, −2)

−2

6 5 −4 Advanced learners: Derive the point-slope form of a linear equation 4 use with Example 1 −5 3 −6 Targeted −7 instructional strategies 2 −8 1 x −9 understanding of the point-slope form of a linear equation, challenge advanced learners To build a deeper to −6−5−4−3−2 −1 1 2 3 4 5 6 −1 −3 form for each of the given points on the line. point-slope

derive itWe forare themselves. Help them get started by explaining that m represents the slope, the (x, y) represents any also able to ﬁnd the slope-intercept form of an equation written in point-slope form by solving equation for y. −2 point on the line in general, and (x1, y1) represents a specific point on the line. −3 −4

−5= Example 1 guide them to find the slope. Once they find the slope of the line, m If students are stuck, −6 need to multiply both sides of the equation by x − x1.

, they simply

Given the point (4, −1) and the slope m = 2, write the equation of the line in point-slope form.

After deriving the equation, ask them to describe situations where this form would be more useful than Create aor strategy slope-intercept standard form. Some examples are: To write equationof ofathe linewhen in point-slope form,slope we use the aformula: y − ythe • Finding the the equation line given the and point on line.− x1) where (x1, y1) is a point on 1 = m(x the lineaand is the slope. • Graphing linemwhen only the slope and a point other than the y-intercept are known. • Finding the equation of a line when given a graph that does not show the y-intercept, or where the Apply the idea y-intercept does not have integer coordinates. y − (−1) = 2(x − 4) y + 1 = 2(x − 4)

Substitute x1 = 4, y1 = −1, and m = 2

Simplify

3.05 Point-slope form mathspace.co

163

The equation of the line in point-slope form is y + 1 = 2(x − 4).

Students: Pages 163–164

Example 2 The graph of a linear function is shown. Write the equation of the line in point-slope form for each of the given points on the line.

6 5 4 3 2 1 −6−5−4−3−2 −1 −1

x 1 2 3 4 5 6

−2 −3 −4 −5 −6

3.05 Point-slope form mathspace.co

329

a (−6, 1)

Apply the idea From the graph, we can choose any two points to ﬁnd the slope. We can either count the . the slope formula

from the graph or use

Start with the slope formula Substitute (−6, 1) = (x1, y1) and (6, −3) = (x2, y2) Evaluate the subtraction Simplify Find the equation of the line in point-slope form: Start with the point-slope form a (−6, 1)

Substitute y1 = 1, m =

, and x1 = −6

Apply the idea b (3, −2) From the graph, we can choose any two points to ﬁnd the slope. We can either count the . the slope theformula idea PurposeApply

from the graph or use

Check that write the ofand a line given a graph and a point. The students slope from can the previous partequation was will in usepoint-slope (3, −2) as (x1, yform 1). Start with the slope formula Given y − y1 = m(x − x1), the resulting equation will be

Expected mistakes Substitute (−6, 1) = (x , y1) and (6, −3) = (x2, y2) Students may forget to flip the sign of the x-value. Ask1 students to distribute and write their equation in slopeintercept form and graph to see if it matches the original graph and contains the needed point. Evaluate the subtraction Example 3 Reflecting with students Simplify linestudents passes through the two points (−3, 7) and −3). for any point on the line, and the only thing that changes Point outA to that the slope remains the(2, same Find the equation of the line in point-slope form: a Write theform equation of the line in point-slope form. in the point-slope of the equation is the coordinates of the specific point. Start with the point-slope form

strategy Students:Create Pagea 164

, and x1 = −6 Substitute y1 = 1, m = We will ﬁrst ﬁnd the slope of the line using the two points. Then we will pick one of the points to substitute into the point-slope equation. b (3, −2)

Apply the idea Apply the idea

Find the slope of the line: The slope from the previous part was and will use (3, −2) as (x1, y1). Start with the slope formula Given y − y1 = m(x − x1), the resulting equation will be Substitute (x1, y1) = (−3, 7) and (x2, y2) = (2, −3) Evaluate the subtraction

Example 3 Simplify Purpose A line passes through the two points and Find the equation of the line in point-slope Ensure students can write the equation(−3, of 7)aform: line(2,in−3). point-slope form given a specific point on the line. a Write the line inStart point-slope − xthe with theform. point-slope formula y −equation y1 = m (x of 1)

Expected mistakes y − 7 = −2 (x − (−3)) Substitute y1 = 7, m = −2, and x1 = −3 a strategy StudentsCreate may forget to include the negative sign when substituting the y-coordinate into the equation. Students y − 7 = −2 (x + 3) Simplify We their will ﬁrst ﬁnd the slope of the line using pick equation one of the points the balance can check equation by substituting (3,the −2)two forpoints. the xThen and we y inwilltheir to seetoifsubstitute the twointo sides point-slope equation. when evaluated. 164

Mathspace

Virginia SOL Algebra 1

mathspace.co Apply thestudents idea Reflecting with the slope offor thestudents line: Use thisFind opportunity to connect multiple representations of the same line. Show them how these Start with theor slope formula form. This can help students see the consistency in equations simplify to the same slope-intercept standard the slope and the flexibility in the point selection.

Substitute (x1, y1) = (−3, 7) and (x2, y2) = (2, −3) Evaluate the subtraction

330

Simplify Mathspace Algebra Teacher Edition Find theVirginia equationSOL of the line in1point-slope form: mathspace.co Start with the point-slope formula y − y1 = m (x − x1) y − 7 = −2 (x − (−3))

Substitute y1 = 7, m = −2, and x1 = −3

y − 7 = −2 (x + 3)

Simplify

Compare and connect

use with Example 2

a (−6, 1)language learner support English

ChooseApply two different the idea points on the same line, for example, (2, 3) and (5, 7). Have students write the point-slope form equation using both points, waiting to solve for slope. This would look like From the graph, we can choose any two points to ﬁnd the slope. We can either count the from the graph or use • For (2, 3) : y 3 = m(x − 2). the slope − formula • For (5, 7) : y − 7 = m(x − 5) Start with the slope formula

Students compare the two equations, discussing how both represent the same line despite different forms. Highlight the flexibility of the point-slope form(−6, and can(6,adapt to, different points on the line. Ask students −3) = (x Substitute 1) =how (x1, y1it) and 2 y2) to discuss whether the slope will be different for the two equations or if m represents the same value in both. Evaluatedifferent the subtraction Students reflect on why and how choosing points changes the appearance but not the essence of the equation. Simplify

Sentence frames could include: Find the equation of the line in point-slope form: • “Using point (2, 3), the equation is ⬚ because ⬚.” Start with the point-slope form • “Using point (5, 7), the equation is ⬚ because ⬚.” , and x1 = −6 Substitute y1 = 1, m = • “Both equations represent the same line because”

Students can then solve for slope using the slope formula and compare their equations to confirm their b (3, −2) predictions in the discussion. By comparing and contrasting different representations, students develop a deeper understanding Apply the idea of the point-slope form’s versatility and application. The slope from the previous part was

and will use (3, −2) as (x1, y1).

Given y − y1 = m(x − x1), the resulting equation will be

Students: Pages 164–165 Example 3

A line passes through the two points (−3, 7) and (2, −3). a Write the equation of the line in point-slope form.

Create a strategy We will ﬁrst ﬁnd the slope of the line using the two points. Then we will pick one of the points to substitute into the point-slope equation.

Apply the idea Find the slope of the line: Start with the slope formula Substitute (x1, y1) = (−3, 7) and (x2, y2) = (2, −3) Evaluate the subtraction Simplify Find the equation of the line in point-slope form: y − y1 = m (x − x1)

Start with the point-slope formula

y − 7 = −2 (x − (−3))

Substitute y1 = 7, m = −2, and x1 = −3

y − 7 = −2 (x + 3)

Simplify

Reﬂect and check 164

Mathspace

Virginia SOL Algebra 1

We can conﬁrm we have correctly written this in point-slope form: y − y1 = m (x − x1), as we can see the coordinates mathspace.co (−3, 7), and a slope of −2 are represented correctly. This is easier to see in the previous line: y − 7 = −2 (x − (−3))

b Determine whether the ordered pair (−10, 21) lies on the same line as (−3, 7) and (2, −3).

Create a strategy We can substitute the ordered pair into the equation we wrote in part (a) and determine whether the statement is true. If so, we can conﬁrm whether the point lies on the same line as (−3, 7) and (2, −3). 3.05 Point-slope form mathspace.co Apply the idea We have y − 7 = −2(x + 3)

Point-slope form from part (a)

331

Purpose Check if students can find the slope of a line given two points and write the equation of the line in point-slope form. Expected mistakes Students may need a reminder on how to find the slope from two points. Remind students that slope is the change in y-values over the change in x-values, or display the slope formula. If needed, students can also graph their points on a coordinate plane and count the rise over run. Reflecting with students Ask students to think about the advantages of using the point-slope form of a line. Discuss how this form Reﬂect check emphasizes theand relationship between the slope of the line and a specific point on the line. We can conﬁrm we have correctly written this in point-slope form: y − y1 = m (x − x1), as we can see the coordinates

7), and165 a slope of −2 are represented correctly. This is easier to see in the previous line: y − 7 = −2 (x − (−3)) Students:(−3, Page b Determine whether the ordered pair (−10, 21) lies on the same line as (−3, 7) and (2, −3).

Apply the idea We have y − 7 = −2(x + 3)

Point-slope form from part (a)

(21) − 7 = −2((−10) + 3)

Substitute y = 21 and x = −10

14 = −2(−7) Reﬂect and check

Evaluate the subtraction and addition

14 correctly written Evaluate the We can confirm14 we= have this in multiplication point-slope form: y − y1 = m (x − x1), as we can see the coordinates Since equation is true, the ordered pair (−10, 21) satisfies equation andline: as ayresult is on same line (−3, 7),the andresulting a slope of −2 are represented correctly. This is easier to seethe in the previous − 7 = −2 (x −the (−3)) as (−3, 7) and (2, −3). b Determine whether the ordered pair (−10, 21) lies on the same line as (−3, 7) and (2, −3). Reflect and check Any ordered pair that satisfies the equation will be on the same line as (−3, 7) and (2, −3).

Create a strategy

We can substitute the ordered pair into the equation we wrote in part (a) and determine whether the statement is true. If so, we can conﬁrm whether the point lies on the same line as (−3, 7) and (2, −3).

Example 4 PurposeApply the idea Check ifWe can determine if awork point liestheon a line with the equation given point-slope form. Astudent carpenter charges for a day’s using given equation, where y is the cost andinx is the number of have hours worked:

y − 7 = −2(x + 3) Point-slope form from part (a) Expected mistakes y −and 125x == 50 (x − 2) (21) − 7 = −2((−10) + 3) Substitute y = 21 Students may forget to distribute the slope to all terms in−10 the parentheses when using point-slope form. This can 14meaning = −2(−7)of each number Evaluateinthe a incorrect Interpret theequation thesubtraction equation. and addition lead to an of the line. 14 = 14 Evaluate the multiplication Remind students that the slope applies to more than just x and should be distributed accordingly when writing Create Since thea resulting is true, the ordered pair (−10, 21) satisfies the equation and as a result is on the same line the equation of astrategy line inequation point-slope form. as (−3, 7) and (2, −3). Recall the point-slope form: Encourage students to substitute given points from the problem into another form of the linear equation to (x − point-slope x1) y − y1 = mthe check their work and ensure they have correctly applied form. Reflect and check Any ordered pair that satisfies the equation will be on the same line as (−3, 7) and (2, −3).

Students:Apply Page the165 idea

The slope is represented by the number 50. In this context this is the rate the carpenter charges for one hour of work. So, the carpenter charges $50 for each hour of work.

Example 4 slope formula, we also know the point (2, 125) ﬁts our context. This means after 2 hours of work, the Using the point total cost is $125. A carpenter charges for a day’s work using the given equation, where y is the cost and x is the number of hours worked: y − 125 = 50 (x − 2) a Interpret the meaning of each number in the equation.

Create a strategy 332

Recall the point-slope form: 1 Teacher Edition Mathspace Virginia SOL Algebra mathspace.co

Apply the idea

y − y1 = m (x − x1)

3.05 Point-slope form mathspace.co

165

Example 4 A carpenter charges for a day’s work using the given equation, where y is the cost and x is the number of hours worked: y − 125 = 50 (x − 2) a Interpret the meaning of each number in the equation.

Create a strategy Recall the point-slope form: y − y1 = m (x − x1)

Apply the idea The slope is represented by the number 50. In this context this is the rate the carpenter charges for one hour of work. So, the carpenter charges $50 for each hour of work. Using the point slope formula, we also know the point (2, 125) ﬁts our context. This means after 2 hours of work, the total cost is $125.

Purpose Show students how to interpret and understand the real-world meaning of numbers in an equation.

Students: Page 166

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165

b Draw the graph of the linear equation from the point-slope form. Clearly label the axes with labels, units, and an appropriate scale.

Create a strategy We need two points to plot a line. We can read from the equation that one point on the line is (2, 125). To get another point, we can use the slope from the given point or substitute in an x-value in the domain and solve for y. Since the x-axis will be the number of hours worked and generally an 8-hour work day is reasonable, showing the graph for 0 ≤ x ≤ 8 would be a good scale.

Apply the idea Before we graph, we label our axes. To do that, we need to know what our maximum and minimum values should be. We can use the slope or the equation to determine the y-value when x = 8. We can ﬁnd that the y-value when x = 8 is y = 425. 450

Charge in $ ( y)

450

400

Charge in $ ( y)

400

350

300

250

200

150

100

Hours (x) 1

Hours (x)

A reasonable scale for the x-axis is from 0 to 8, going up by 1. A reasonable scale for the y-axis is from 0 to 450, going up by 50, with one tick between each label at the 25s.

The given point is (2, 125). Since the slope is 50, we can go up 50 units and right 1 unit from our given point to plot another point on the line.

c Predict the charge for 6 hours of work using the graph.

PurposeCreate a strategy Apply the idea StudentsWewill apply their point-slope knowledge to create a graph. can go to x = 6 on the x-axis and then go up to the line and across to the y-axis to make our prediction.

450

Charge in $ ( y)

400 350 300 250 200 150

3.05 Point-slope form mathspace.co

100 50

Hours (x)

333

We need two points to plot a line. We can read from the equation that one point on the line is (2, 125). To get another point, we can use the slope from the given point or substitute in an x-value in the domain and solve for y. Since the x-axis will be the number of hours worked and generally an 8-hour work day is reasonable, showing the graph for 0 ≤ x ≤ 8 would be a good scale.

Apply the idea Expected mistakes Before graph, we label ourlabeling axes. To do that, we need to axes. know what our axes maximum andbe minimum values should be. Students may we have trouble with and scaling the These could provided if needed.

We can use the slope or the equation to determine the y-value when x = 8. We can ﬁnd that the y-value when x = 8 is Charge in $ (y) y = 425.

450

Charge in $ ( y)

400

450

400

350

300

350

300

250

300

250

400

250

200

150

100 50 Hours (x)

100 50 1

Charge in $ ( y)

100 50

Hours (x)

Hours (x) 2

A reasonable scale for the x-axis is from 0 to 8,

The given point is (2, 125).

going up by 50, with one tick between each label

point on the line.

Discuss why a reasonable scale for the x-axis from 0 to 8 going up by 1 and why a reasonable scale for the going up by 1. Since the slope is 50, we can go up 50 units and y-axis from A0reasonable to 450 going up by 50 with one tick between each at our thegiven 25s.point to plot another right 1 label unit from scale for the y-axis is from 0 to 450,

Students: Page at the 166 25s.

c Predict the charge for 6 hours of work using the graph.

Create a strategy

Apply the idea

We can go to x = 6 on the x-axis and then go up to the line and across to the y-axis to make our prediction.

450

Charge in $ ( y)

400 350 300 250 200 150 100 50

Hours (x) 1

The charge for 6 hours of work will be $325.

Purpose 166 Mathspace Virginia SOL Algebra 1 Check if students can use a graph to solve real-world problems. mathspace.co Reflecting with students Ask students if they could use different points on the line to find the same answer. Ask students to provide context to the points they choose and if the points chosen are reasonable.

Students: Page 167 d Give an example of a non-viable solution if the carpenter only uses this model for a maximum of 10 hours per day. Explain your answer.

Create a strategy Since the carpenter only uses this model between 0 and 10 hours, any solution outside of this interval will not be viable.

Apply the idea One possible non-viable solution would be (11, 575) which would represent working 11 hours and getting paid $575, but the model only applies to a maximum of 10 hours, so we don’t actually know what would happen for 11 hours.

334

Reﬂect Virginia and check Mathspace SOL Algebra 1 Teacher Edition mathspace.co Any solution with x < 0 or x > 10 would be non-viable.

Explain your answer.

Create a strategy Since the carpenter only uses this model between 0 and 10 hours, any solution outside of this interval will not be viable.

Apply idea d Givethe an example of a non-viable solution if the carpenter only uses this model for a maximum of 10 hours per day. OneExplain possible non-viable your answer. solution would be (11, 575) which would represent working 11 hours and getting paid $575, but the model only applies to a maximum of 10 hours, so we don’t actually know what would happen for 11 hours.

Create a strategy Reﬂect and check

Since the carpenter only uses this model between 0 and 10 hours, any solution outside of this interval Any solution with x < 0 or x > 10 would be non-viable. will not be viable.

Apply the idea One possible non-viable solution would be (11, 575) which would represent working 11 hours and getting paid $575, model summary only applies to a maximum of 10 hours, so we don’t actually know what would happen for 11 hours. Purposebut the Idea The point-slope form of a line is: to domain and range with real-world models. Check if students understand restrictions

Reﬂect and check

y − y1 = m (x − x1) solution with x < 0 or x > 10 would be non-viable. Students:Any Page 167

Idea summary

m x1 y1

slope x-coordinate of a point on the line y-coordinate of the same point

Point-slope form is useful when we know or want to know the slope of the line and a point on the line. The point-slope form of a line is:

Practice What do you remember?

y − y1 = m (x − x1) m x1 y1

slope x-coordinate of a point on the line y-coordinate of the same point

Point-slope form is useful when we know or want to know the slope of the line and a point on the line. State the point-slope form of a linear equation.

Consider a straight line with slope 1 going through point A(2, 1). What is the slope between A and any other

point on the line? Practice 3

For each of the following equations, state the form they are written in:

= −4xremember? +5 Whata doy you

Practice b

y − 1 = 5(x − 4)

cState5xthe − 3y =8 point-slope form of a linear equation. d y = −2 Students: 2Pages 167–172 Consider a straight line with slope 1 going through point A(2, 1). What is the slope between A and any other 4 point True or The line with the equation y − 5 = 4(x + 6) contains the point (−5, 6). onfalse? the line? 1

For each of the following equations, state the form they are written in:

What do youa remember? y = −4x + 5 b

y − 1 = 5(x − 4)

y = −2

State the cpoint-slope 5x − 3y = 8form of a linear equation.

Consider a straight line with slope 1 going through point A(2, 1). What is the slope between A and any other 4 True or false? The line with the equation y − 5 = 4(x + 6) contains the point (−5, 6). point on the line?

For each of the following equations, state the form they are written in:

3.05 Point-slope form mathspace.co

167

form True or false? The line with the equation y − 5 = 4(x + 6) contains the point (−5, 6).3.05 Point-slope mathspace.co

167

y = −4x + 5

y − 1 = 5(x − 4)

5x − 3y = 8

y = −2

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335

Which graph correctly represents the function y + 3 = −2(x − 1)? A

1 −4 −3 −2 −1 −1

x 1

−3

−3 −4

−2

−4

−4 −3 −2 −1 −1

−2

x 1

−4 −3 −2 −1 −1

−2

−3

−4

Select the linear equation that could represent the following graph. A

y + 3 = 2(x + 3)

y − 7 = 2(x − 2)

y − 7 = 2(x − 5)

y − 4 = 2(x − 5)

4 2 −8 −6 −4 −2 −2 −4 −6 −8

Let’s Practice 7

336

Using the point-slope formula, ﬁnd the equation of the following lines: a

A line passes through (−5, 9) and has a slope of 2.

A line passes through (−2, −1) and has a slope of .

A line passes through (8, 2) and has a slope of −3.

A line passing through the point (7, 1) has a slope of

A line passing through the point (4, 0) has a slope of −5.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

x 2

For each of the following lines: i

Find the slope, m, of the line.

Write the equation of the line in point-slope form.

A line passes through the two points (7, 6) and (9, 12).

A line passes through the two points (4, −8) and (2, 0).

For each of the following table of values: i

Find the slope, m, of the line represented in the table.

Write the equation of the line in point-slope form.

x 1 2 3 4 y −3 2 7 12

x y

1 9

2 7

3 5

4 3

Write the equation of each line in point-slope form, using the point shown on the graph. a

5 4

3 2 1 1

−2 −3

5 4 3 2 1

−7 −6 −5 −4 −3 −2 −1 −1

5 4

1 −3 −2 −1 −1 −2 −3

x 1

−7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7

−5 −4 −3 −2 −1 −1

7 6 5 4 3 2 1

1 2 3 4 5 6 7

x 1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7 −8 −9 −10

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337

A linear function is defined by the equation a

Select the graph that could represent the equation. A y

5 x

−5

x −5

5 −5

C y

5 x

−5

x −5

5 −5

b 12

Identify the range of the function.

y + 8 = 3(x + 2)

y + 1 = −2(x + 1)

2x + 3y = 12

y − 2 = 3 (x − 7)

y + 7 = −3(x + 2)

y−7=

(x − 5)

y − y1 = m(x − x1 )

A race car uses fuel at the rate of 0.9 gallons per minute. After running for 12 minutes, the car has 48 gallons of fuel left in the tank. a

338

Rewrite the following linear equations in standard form: a

Identify the domain of the function.

Rewrite the following linear equations in slope-intercept form: a

−5

Sketch the graph of the lines on the coordinate plane given the following equations: a

Select the linear equation that could describe the relationship between the number of minutes x the car is running and the number of gallons of fuel y left in the tank. A y + 12 = −0.9 (x + 48)

y + 48 = 0.9 (x + 12)

C y − 12 = 0.9 (x − 48)

y − 48 = −0.9 (x − 12)

Graph the function chosen in part (a). Explain how the y-intercept relates to the context.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

A plumber charges a fixed amount for a call out fee plus $30 per hour of work. He charged a total of $80 for 2 hours of work. He cannot work for more than 8 hours in any given day. a

Select the graph of the linear function that relates the number of hours of work x and the total charge y for the day. A y

20 x 1

C y

20 x 1

Identify the domain of the function.

Let’s extend our thinking 17

Create a scenario where you would choose to use each of the following forms and explain why you would use that form over the others: a

Point-slope form, (y − y1) = m(x − x1)

Standard form, Ax + By = C

Slope-intercept form, y = mx + b

Sally receives an order to make key chains for an upcoming festival. She is told to make at least 50 and at most 100 key chains, and will be paid $8 for each key chain she makes. Sally makes 80 key chains and is paid $665 altogether. a

Let x represent the number of key chains Sally makes and y represent the amount she is paid. Write an equation in point-slope form relating x and y.

Convert the equation from part (a) to slope-intercept form.

Interpret the y-intercept in the context of the problem.

Explain whether or not we can use the equation in part (a) to predict the amount paid for 10 key chains.

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339

Paige is planning an event and the brochure for the caterer at the venue shows the following formulas for the meal costs. Point-slope form: y − 500 = 12.5(x − 10) Slope-intercept form: y = 12.5x + 375 Compare and contrast the information presented by each equation.

Create a scenario where you would choose to use point-slope form, y − y1 = m(x − x1), and explain what the point and slope mean in context.

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Answers

3.05 Point-slope form What do you remember? 1 For slope m and point (x1, y1), point-slope form: (y − y1) = m(x − x1)

b Point-slope form

c Standard form

−7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7

2 1 3 a Slope-intercept form

7 6 5 4 3 2 1

d Standard form

1 2 3 4 5 6 7

4 False c

5 B

6 C

2 1

Let’s practice 7 a y − 9 = 2(x + 5)

c y −2 = −3(x − 8)

−4 −3 −2 −1 −1

−2 −3 −4

e y = −5(x − 4) 8 a i 3

ii y − 6 = 3(x − 7)

b i −4

ii y + 8 = −4(x − 4)

9 a i 5

6 5 4 3 2 1

ii Possible answers:

y + 3 = 5(x − 1)

−6 −5 −4 −3 −2 −1−1 −2 −3 −4 −5 −6

y − 2 = 5(x − 2) y − 7 = 5(x − 3) b i −2

ii Possible answers:

y − 9 = −2(x − 1)

x 1 2 3 4

y − 7 = −2(x − 2) y − 5 = −2(x − 3) b

10 a c y − 1 = −2(x − 2)

d y − 2 = 6(x − 2) b Domain: x ∈ (−∞, ∞)

11 a A c Range: y ∈ (−∞, ∞)

13 a

b y = 3x − 19

14 a 4x + 3y = 9

b 3x + y = −13

c 2x + 3y = 31 15 a D b 80

12 a

8 y 7 6 5 4 3 2 1 −8−7−6−5−4−3−2 −1 −1

−2 −3 −4 −5 −6 −7 −8

d mx − y = (mx1 − y1)

70 60 50 40 30 x

1 2 3 4 5 6 7 8

x 1

2 3 4 5 6 7 8 9

The y-intercept is at (0, 58.8) and represents the number of gallons of fuel in the tank before the car starts running. 16 a D

b Domain: x ∈ [0, 8]

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Let’s extend our thinking 17 a A scenario where we are given a rate of change (slope) and both variables at a particular moment (point). For example, if we are told how much someone deposits into their savings account per week and the amount in their bank account after a certain number of weeks. b A scenario where we are given the rate of change (slope) and the initial value (y-intercept). For example, if we are told how much a plumber charges as their flat callout fee (y-intercept) and how much they charge per hour (slope). c A scenario where we are given the unit rates for two items that we are creating a mixture of to get a certain total. For example, if we are told the cost of almonds per pound (A), the cost of peanuts per pound (B), and are told the total amount to be spent (C).

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18 a (y − 665) = 8(x − 80) b y = 8x + 25 c Sally is initially paid $25 for the order. d T his equation is only valid for 50 ≤ x ≤ 100, since 10 is outside of this interval, we cannot predict the amount paid for 10 key chains. 19 In point-slope form, it is easy to see the slope, or cost per meal, as well as the total cost for 10 meals. In slope-intercept form, it is easy to see the slope, or cost per meal, as well as the y-intercept, or cost to hire the caterer before ordering meals. 20 A scenario where we are given a rate of change (slope) and both variables at a particular moment (point). For example, if we are told how much someone deposits into their savings account per week and the amount in their bank account after a certain number of weeks.

3.06 Equations of parallel and perpendicular lines Subtopic overview Lesson narrative In this lesson on equations of parallel and perpendicular lines, students will explore the properties and relationships of these lines. Through interactive exploration, they will learn that parallel lines have the same slope and never intersect, while perpendicular lines have slopes that are negative reciprocals of each other. Students will practice writing equations for lines that are parallel or perpendicular to given lines and passing through specific points. By the end of the lesson, students should be proficient in identifying and constructing equations for parallel and perpendicular lines both graphically and algebraically.

Learning objective Students: Page 173

Key vocabulary 

negative (opposite) reciprocal



parallel



perpendicular

Essential understanding Parallel lines will have the same slope and never intersect. Perpendicular lines have slopes with opposite signs and they are reciprocals of one another. A vertical and horizontal line are perpendicular.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG 1 — Mathematical Problem Solving

MPG4 — Mathematical Connections

To encourage mathematical problem solving, teachers can present real-world scenarios, such as urban planning or art design, where identifying and working with parallel and perpendicular lines are crucial. Problem-solving tasks can include challenges like finding the equations of lines given specific conditions, determining if lines are parallel or perpendicular based on their slopes, and solving geometric problems involving these types of lines.

Teachers can help students make connections between the concepts of slope, parallel and perpendicular lines, and different forms of linear equations. They can also connect these mathematical concepts to real-world contexts by providing examples and problems that involve parallel and perpendicular lines in real-world situations.

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MPG5 — Mathematical Representations Teachers can integrate this goal by providing opportunities for students to represent parallel and perpendicular lines in various ways - algebraically (through equations), graphically (through graphs), and in contextual situations (through word problems or real-world scenarios). They can also encourage students to make connections between these different representations and to discuss the advantages and disadvantages of each form depending on the context and information needed.

A.F.1dii — Write the equation of a linear function to model a linear relationship between two quantities, including those that can represent contextual situations. Writing the equation of a linear function will include the following situations: ii) given two points on the line whose coordinates are integers; A.F.1diii — Write the equation of a linear function to model a linear relationship between two quantities, including those that can represent contextual situations. Writing the equation of a linear function will include the following situations: iii) given the slope and a point on the line whose coordinates are integers; A.F.1e — Write the equation of a line parallel or perpendicular to a given line through a given point. A.F.1f — Graph a linear function in two variables, with and without the use of technology, including those that can represent contextual situations.

Prior connections 4.MG.4 — The student will identify, describe, and draw points, rays, line segments, angles, and lines, including intersecting, parallel, and perpendicular lines.

A.EI.1 — The student will represent, solve, explain, and interpret the solution to multistep linear equations and inequalities in one variable and literal equations for a specified variable.

Future connections G.RLT.2 — The student will analyze, prove, and justify the relationships of parallel lines cut by a transversal. A2.F.2 — The student will investigate and analyze characteristics of square root, cube root, rational, polynomial, exponential, logarithmic, and piecewise-defined functions algebraically and graphically.

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Engage Activity Making it parallel

60 mins

Students will explore the connection between parallel lines and their equations in a guess-and-check applet.

Understanding and skills

Will use Understanding that parallel lines are lines that do not intersect.

Will develop Understanding that parallel lines have equal slopes.

Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: None.

Support students with disabilities Support language - write explanations of mathematical thinking Provide students with sentence starters to help write down their thoughts. For the ﬁrst applet of the Engage: • It was challenging to ⬚ because ⬚ • I knew the lines would be parallel because ⬚ • I noticed that parallel lines ⬚ For the second applet of the Engage: • When the grid was visible I tried ⬚ instead of ⬚ • The grid was easier/harder to use because ⬚ • I know that two lines are parallel when ⬚

Support for English language learners Stronger and clearer each time Students are prompted to discuss their observations with their partner after exploring the applet. Provide more support for students to guide this discussion: 1. Explain to yourself a method for making sure the lines are parallel before you check them with the applet. 2. Convince your partner that your method will ensure the lines are parallel before you check them with the applet. 3. Write a convincing argument to convince the class that the lines are parallel before checking with the applet. Encourage students to adjust their explanation after each round to incorporate their partner’s feedback.

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Classroom guide Hook Students compare the similarities and differences between two parallel lines.

Implementation details

Open questions

5 mins

What are the similarities and differences between the two lines shown?

Record student thoughts on the board as conjectures to revisit at the end of the lesson. Encourage students to compare the type of functions as well as characteristics of the linear functions, such as slope and y-intercepts. It may be helpful to use the student responses as an opportunity to review and practice appropriate vocabulary around linear functions, graphing on the coordinate plane, and parallel lines.

•

3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

Slide 1 from Student Engage Activity

Launch Give students several minutes to investigate the applet before forming pairs. Important mathematical concepts: Parallel lines, slope, intersection, coordinate plane

5 mins

Consider the following applet. Check

Suggested grouping: Form pairs

Continue when Students have explored the applet and understand how the applet functions.

Slide 2 from Student Engage Activity

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Explore

Think-pair-share

•

35 mins

This is a discovery-based activity. Students who have successfully completed the task will have ideas about the conditions for parallel lines and will be ready to share those ideas with the class.

Anticipated strategies Guess and check The goal of the first applet is to have students guess the proper placement for the lines to be parallel and to check their guess until they are successful. They can continue this process into the second applet, but encourage them to be more efficient in their guessing. Identify specific points When working in the second applet, students will now be able to make the lines parallel by identifying points that make the slopes equal.

Misconceptions Thinking the slopes are close enough to be parallel when they are not quite equal How can we be sure these lines are parallel? What do we look for? What would happen if we zoomed out on these lines?

Purposeful questions While pairs are interacting with the first applet: • What strategies make it easier to make the lines parallel? What would help you find a parallel line on the first check? • Between you and your partner, who is more accurate with their parallel lines? Why do you think that is? • What characteristics do you think make lines parallel? While pairs are interacting with the second applet: • Will guess and check still help you find the parallel line? Explain. • Is it easier to make the lines parallel now? Why or why not? • How can we use the grid to ensure our lines are parallel?

Continue when Students have successfully made parallel lines on each applet and completed the reﬂection questions.

Discuss

15 mins

Have a class discussion, invite students to share their strategies for creating parallel lines on a gridless plane and when the cartesian plane was provided. Consider making connections from the discussion to the slopes of parallel lines.

Discussion guide Begin by having students share the challenges with creating parallel lines on a gridless plane. Encourage students to share the maximum number of tries they needed to make parallel lines. Then, move students into discussing their strategies that they felt worked for making the lines parallel. When students are sharing their strategies, try to start with observations based on the distance between the lines or maintaining “equal space” between lines and end with students who discuss making the slopes equal. Next, invite students to share how their strategies changed or adapted once the grid was provided. If possible, end the discussion with pairs who identiﬁed two points, found the slope between those points, then matched the slope with their line. 3.06 Equations of parallel and perpendicular lines mathspace.co

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Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Algebra 1 — 3.01 Slope Algebra 1 — 3.03 Slope-intercept form Algebra 1 — 3.04 Standard form Algebra 1 — 3.05 Point-slope form

Tools You may ﬁnd these tools helpful: • Graphing calculator • Graph paper

Student lesson & teacher guide Parallel and perpendicular lines Students learn how slopes of parallel and perpendicular lines relate, and are shown examples of graphs of equations of each. They learn that perpendicular slopes are negative reciprocals, and see an example of finding the equation of a line perpendicular to a given equation through a particular y-intercept.

Students: Page 173

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Identifying properties of parallel and perpendicular slopes Targeted instructional strategies As a way to introduce parallel and perpendicular slopes, provide a graph with three equations and their slopeintercept form labeled. Two of the lines should be parallel and the third should be perpendicular to both. An example graph is shown: 4

3 2 1 −4 −3 −2 −1 −1

Tell students that f (x) and g(x) are parallel, and ask students to look at the equations and discuss what similarities they see in their equations. Students should notice that the equations have the same slope.

x 1

−2 −3 −4

Tell students that h(x) is perpendicular to both f (x) and g(x), and have them discuss how the slopes relate to one another. They should notice that the numerator and denominator are flipped and signs are opposites. This serves as a visual introduction to relating slopes and equations of parallel and perpendicular lines before the exploration and lesson.

Three reads English language learner support To help students interpret and solve problems with equations of parallel and perpendicular lines, provide students with different problems involving parallel and perpendicular lines. 1. First Read: Comprehension Begin by having students read a problem involving parallel and perpendicular lines, focusing on understanding the general context. For example, a problem might state: “Determine if the lines given by the equations y = 2x + 3 and

are parallel, perpendicular, or neither.” Ask students to describe what

the problem is asking them to focus on, such as the slopes, writing new equations, or graphing. 2. Second Read: Identifying information In the second read, students should identify and underline key mathematical information. For instance, students should underline the equations y = 2x + 3 and

. They should note the slopes m = 2 and

and intercepts b = 3 and b = 4. Encourage students to highlight terms like “slope” and “intercept” to focus on the essential parts of the equations. 3. Third Read: Clarifying the task In the final read, students should clarify what the problem is asking them to do. They should focus on the task: determining if the lines are parallel, perpendicular, or neither. Have them rephrase the task in their own words, such as “I need to compare the slopes to see if they are the same for parallel lines or negative reciprocals for perpendicular lines.” Encourage students to think about what makes lines parallel or perpendicular based on their slopes. By guiding students through these three reads, they will better understand the context, identify critical information, and clarify the task, leading to a deeper comprehension and accurate problem-solving.

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Visual reminders of parallel and perpendicular slopes Student with disabilities support Provide an annotated graph of parallel and perpendicular lines and their equations to reinforce and remind students of key characteristics of each. An example is shown. 4 3

y = 2x + 3

3 2

2 1 −4 −3 −2 −1 −1 −2

x 1

y = 2x − 4

1 −4 −3 −2 −1 −1 −2

−3

−4

Parallel lines = same slope

x 1

y = 2x − 3

Perpendicular lines = opposite reciprocal slopes

The signs of perpendicular slopes Address student misconceptions When checking if two lines are perpendicular, students will often forget that the slopes of the two lines must be opposite reciprocals, not just reciprocals. A common mistake will be similar to stating that two lines are perpendicular because their slopes are 3 and . The missing negative sign for one of the slopes is evident in the fact that the product is 1 instead of −1. Ask students to check the product of the slopes to make sure that the lines are perpendicular, or even to sketch the two lines for a visual check.

Exploration Students: Page 173

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Suggested student grouping: In pairs In this activity, students will be exploring the relationship between the slopes of two lines and their orientations to each other. They will manipulate the positions of two lines on a graph and observe changes in their slopes when the lines become parallel or perpendicular. Students will also experiment with the y-intercepts to understand their role in the position and orientation of the lines. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Move the red line until you get the message “a and b are parallel lines”. What do you notice about the slopes when the two lines are parallel? When the two lines are parallel, their slopes are the same. 2. Can you create two different parallel lines with the same y-intercept? No, if two lines have the same slope and the same y-intercept, they are the same line. 3. What do you notice about the slopes when the two lines are perpendicular? Move the red line until you get the message “a and b are perpendicular lines”. When the two lines are perpendicular, the slopes are negative reciprocals of each other. 4. Can you create two distinct perpendicular lines with the same y-intercept? Yes, you can have two distinct perpendicular lines with the same y-intercept. The lines will intersect at the y-intercept. Purposeful questions • How can we determine if slopes are parallel if we don’t have a graph? • What does it mean for two lines to be coinciding? • How can we check that two slopes are perpendicular without a graph? Possible misunderstandings • Students may assume that the y-intercept of a line helps determine whether lines are parallel or perpendicular since it affects whether lines are parallel or coinciding. Encourage students to classify lines as parallel or perpendicular based on slope first, and if parallel, to go back and double check to determine if the lines coincide.

Advanced learners: Proving vertical and horizontal lines are perpendicular Targeted instructional strategies The lesson introduces the fact that perpendicular lines have slopes that are negative reciprocals, but this rule cannot be used as justification when one of the lines has an undefined slope (vertical line) and the other has a zero slope (horizontal line). Ask students to consider why this rule cannot be used for this special case of perpendicular lines. Make students aware that an undefined slope is not a value, so we cannot use it in calculations. Encourage advanced learners to justify why vertical and horizontal lines are perpendicular without using that rule. Students will be challenged to think beyond slope and consider the definition of perpendicular lines as lines that intersect to form right angles.

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Students: Pages 173–174

7 6 5 4 3 2 1

Perpendicular lines have slopes with opposite signs and they are reciprocals of one another.

y y = 3x − 2

These are called negative reciprocals (or opposite reciprocals). x

−7 −6 −5 −4 −3 −2 −1−1

1 2 3 4 5 6 7

−2 y −3 7 −4 6 −5 5 −6 4 −7 3

y = 3x − 2

The product of slopes of perpendicular lines is −1 unless one of the lines has an undeﬁned slope. The lines y = 3x − 2 and y =

x + 4 are perpendicular.

Perpendicular thewith same y-intercept. Perpendicular lines lines may havehave slopes opposite signs and they are reciprocals of one another. These are called negative reciprocals (or opposite reciprocals).

The product of slopes of perpendicular lines is −1 unless one 2 Consider a line perpendicular to y = 3x − 2 with We the slope must be the negative 1 of y-intercept. the lines has anknow undeﬁned slope. x the same −7 which −6 −5 −4 1 the 2 3y-intercept 4 5 6 7 must be the reciprocal is−3 m−2 = −1−1 and = −2. Thesame, lines ywhich = 3x −is2band y=

x + 4 are perpendicular.

−2 lines Therefore, the equation of a perpendicular line with the Perpendicular same y-intercept is may y = have x − the 2. same y-intercept. −3 −4 −5 Example 1 −6 −7

The line AB passes through the points (−2, 9) and (3, −21).

Examples Consider a line perpendicular to y = 3x − 2 with the same y-intercept. We know the slope must be the negative a Write the equation of the line. which is m = Students:reciprocal Page 174

and the y-intercept must be the same, which is b = −2.

Create a strategy

Therefore, the equation of a perpendicular line with the same y-intercept is y = x − 2. To find the equation, we need to know the slope and the y-intercept. We will find the slope using m = will find the y-intercept using y = mx + b.

, and we

Example 1

Apply The linethe ABidea passes through the points (−2, 9) and (3, −21). a Write the equation of the line.

Create a strategy

Slope formula Substitute (x1, y1) and (x2, y2)

To find the equation, we need to know the slope and the y-intercept. We will find the slope using m = Evaluate the subtraction will find the y-intercept using y = mx + b. Evaluate the division

, and we

The slope the line is m = −6. Now, we will use y = mx + b with the slope we found and one of the points. We can Apply theofidea use either point because either will result in the same answer. Slope formula y = mx + b Slope-intercept form of a linear equation 9 = −6(−2) + b −3 = b

Substitute (x m1,=y−6 and (x , y ) Substitute 1) and (x2,1 y21) Evaluate the multiplication Evaluate theproperty subtraction Subtraction of equality

b = −3

Reﬂexive property of equality Evaluate the division

9 = 12 + b

This meansofthe at Now, (0, −3). The slope they-intercept line is m =is−6. we will use y = mx + b with the slope we found and one of the points. We can use either point because either will in the same answer. Substituting m = −6 and b = −3 into result slope-intercept form of a linear equation, we ﬁnd the equation of the line to be

352

y = −6x − 3. y = mx + b Slope-intercept form of a linear equation Mathspace Virginia SOL Algebra 1 Teacher Edition Substitute m = −6 and (x1, y1) mathspace.co 9 = −6(−2) + b Reﬂect and9 check = 12 + b Evaluate the multiplication The equation of the line in standard form is 6x + y = −3. −3 = b Subtraction property of equality

Slope formula Substitute (x1, y1) and (x2, y2) Evaluate the subtraction Evaluate the division The slope of the line is m = −6. Now, we will use y = mx + b with the slope we found and one of the points. We can use either point because either will result in the same answer. y = mx + b

Slope-intercept form of a linear equation

9 = −6(−2) + b

Substitute m = −6 and (x1, y1)

9 = 12 + b

Evaluate the multiplication

−3 = b

Subtraction property of equality

b = −3

Reﬂexive property of equality

This means the y-intercept is at (0, −3). Substituting m = −6 and b = −3 into slope-intercept form of a linear equation, we ﬁnd the equation of the line to be y = −6x − 3.

Reﬂect and check The equation of the line in standard form is 6x + y = −3.

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Purpose Students demonstrate that they can write the equation of a line given two points. Reﬂecting with students Ask students to check their work by graphing the line and verifying that it passes through the given points.

Students: Page 175 b Find the equation of the line that passes through (1, 5) and is parallel to the line AB.

Create a strategy Since this line is parallel to the line AB, we know that it will have the same slope as to ﬁnd the y-intercept of the parallel line.

which was −6. We only need

Apply the idea Just like we did in the previous part, we will substitute the slope, and the x- and y-values of the point into y = mx + b. y = mx + b

Slope-intercept form of a linear equation

5 = −6(1) + b

Substitute m = −6 and the point (1, 5)

5 = −6 + b

Evaluate the multiplication

11 = b

Addition property of equality

b = 11

Reﬂexive property of equality

The equation of the parallel line is y = −6x + 11.

Reﬂect and check Using technology to graph the lines, we can see that they are parallel, and they pass through the speciﬁed points from parts (a) and (b).

y 10 5 x −3 −2 −1 −5

−10 −15 −20

Example 2 Purpose Considerhow the line − 3y −6. Show students to 4x find the= equation of a line that passes through a specific point and is parallel to a given line. a Find the equation of the line that is perpendicular to the given line and has the same y-intercept. 3.06 Equations of parallel and perpendicular lines mathspace.co Create a strategy For the new line to be perpendicular to the given line, its slope must be the opposite reciprocal of the slope of the given line.

353

−5 −10 −15 −20

Students: Pages 175–176

Example 2 Consider the line 4x − 3y = −6. a Find the equation of the line that is perpendicular to the given line and has the same y-intercept.

Create a strategy For the new line to be perpendicular to the given line, its slope must be the opposite reciprocal of the slope of the given line. To ﬁnd the y-intercept, we can either substitute x = 0 or rearrange to slope-intercept form.

Apply the idea Rearranging the equation of the given line to slope-intercept form gives us: Given equation Subtract 4x from both sides Divide both sides by −3 The slope of the given line is

3.06 Equations of parallel and perpendicular lines mathspace.co

175

Purpose Show students how to find the equation of a line that is perpendicular to a given line and passes through a given point. Expected mistakes Students might not know how to start the equation, especially since the given line is in standard form. Start by asking them what we know about lines that are perpendicular, then ask them how we can find the slope of the given line. Reflecting with students Ask students to consider if this is the only line that would be perpendicular to the given line. Point out that the instructions said the new line needed to have the same y-intercept of the given line, which is why we came up with this specific line. But any line with a slope of

354

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

would be perpendicular to the given line.

Students: Page 176

Purpose Students demonstrate that they can convert an equation in slope-intercept form into standard form. Expected mistakes Students might leave the coefficient of x as a rational number. Remind them that standard form requires all coefficients to be integers. Reflecting with students Ask students to compare the given equation with the new equation in standard form. Have them share the things they notice with a partner, then with the whole class.

Students: Page 176

Purpose Challenge students to apply their understanding of linear functions to find the x-intercepts, or zeros, of the function.

3.06 Equations of parallel and perpendicular lines mathspace.co

355

Step-by-step checklist method of solving

use with Example 2

Student with disabilities support Incorporate algorithmic thinking by guiding students to develop a clear, step-by-step procedure for writing the equation of a line that is parallel or perpendicular to a given line and passes through a specific point. Encourage them to identify each step needed, sequence them logically, and apply the algorithm consistently to various problems. Some students may benefit from being given the procedure and being shown how to follow an algorithm. 1. Find the slope of the given line. 2. Determine the slope of the new line, based on whether it is parallel or perpendicular to the given line. 3. Write the equation of the new line in point-slope form, using the given point that the new line passes through. 4. Rearrange the equation to be in standard form. Some students may benefit from having accompanying graphs or examples alongside the checklist for additional support.

Students: Page 177 Example 3 A mirror is placed along the x-axis. A laser beam is projected along the line y = −x + 4 which reﬂects off the mirror.

6 5 4 3 2 1 −1

x 1

−1

a A normal is a line which is perpendicular to the surface of the mirror at the point of reﬂection. Find the equation of the normal.

Create a strategy

Apply the idea

We can do a quick sketch of the normal to help:

Since the mirror is a horizontal line, the normal must be a vertical line if it is to be perpendicular. This means it will be of the form x = a.

Since it goes through the point where the laser hits the mirror, (4, 0), the equation of the normal will be x = 4.

6 Normal

5 4

Laser

3 2 1 −1

−1

x 1

2 3 4 5 6 7 Surface of the mirror

b The angles that the laser and its reflection make with the normal will be congruent. If the angle between the laser beam and the normal is 45°, find the equation of the path of the reflection.

Purpose Show students to find the equation of a vertical line that is perpendicular to a horizontal line. Create ahow strategy

Since the angles are congruent, know that the angle formed between the normal and the reﬂection will also be 45°. We can label this on our diagram: 7

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Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

6 5 4 3

Normal

a A normal is a line which is perpendicular to the surface of the mirror at the point of reﬂection. Find the equation of the normal.

Create a strategy

Apply the idea

Wemistakes can do a quick sketch of the normal to help: Expected

Since the mirror is a horizontal line, the normal must be a vertical line if it is to be perpendicular. This means it will Students might misread ythe question and think they needbetooffind the equation of the dashed line. Ask them to the form x = a. 7

reread the question and determine which line representsSince the surface of the mirror (the x-axis). Then, check for it goes through the point where the laser hits the 6 understanding by asking them to draw the normal on themirror, graph. Normal (4, 0), the equation of the normal will be x = 4. 5 4

Reflecting with students Laser 3 Share with students that normal lines are used in physics when working with the law of reflection and centripetal 2 force (the force acting 1on a object moving in a circular motion). They are also used in calculus when working x with tangent lines. −1

−1

2 3 4 5 6 7 Surface of the mirror

Students: Pages 177–178

b The angles that the laser and its reflection make with the normal will be congruent. If the angle between the laser beam and the normal is 45°, find the equation of the path of the reflection.

Create a strategy Since the angles are congruent, know that the angle formed between the normal and the reﬂection will also be 45°. We can label this on our diagram: 7

6 Normal

5 4 3 2

45° 45°

1 −1

−1

x 1

3.06 Equations of parallel and perpendicular lines mathspace.co

177

3.06 Equations of parallel and perpendicular lines mathspace.co

357

Purpose Show students how to find a perpendicular line in a contextual situation. Expected mistakes Students might assume that the reflection would be perpendicular to the laser without understanding the rest of the information given in the problem. Point out that not all reflections form 90° angles. Ask students to explain how they knew the reflection would be perpendicular to the laser to asses their understanding of the given information.

Students: Page 178

Practice Students: Pages 178–181

What do you remember? 1

Explain what you know about parallel and perpendicular lines.

Using the given graph, determine if:

a and b are perpendicular

b and e are perpendicular

a and d are parallel

b and c are parallel

y e a

−4 −3 −2 −1 −1

−3

Given the graphs of the lines:

• y=x+5 • y=x+2 • y=x−3

Determine whether the following statements are true or false:

358

They all have the same slope.

They all have positive y-intercepts.

They all have positive x-intercepts.

They are all parallel to the line y = x.

They all have slope equal to 1.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

d x 4 b

−2 −4

x −5 y=x+5 y=x+2 −5 y=x−3

State whether each of the following lines are parallel, perpendicular, or neither to the line a

3x − 4y = −12

8x + 6y = 6

6x − 2y = 4

Let’s practice 5

Determine which, if any, of the lines are parallel. A

y = 3x − 4

y = −3x + 1

3x + y = 4

2x − 3y = 4

Determine which, if any, of the lines are perpendicular. A

y = 2x − 4

y = −3x + 1

3x + 2y = 0

2x + 4y = 4

Find the equations of the lines that are: a

Parallel to x = −2, and 4 units away from the line x = −2.

Parallel to y = −4, and 2 units away from the line y = −4.

Perpendicular to y = 4, and 3 units away from the line x = 2.

Perpendicular to x = 0, and 1 unit away from the line y = 1.

Find the equation of the line, in slope-intercept form, that is: a

Parallel to the line y = 8x − 3 and cuts the y-axis at 5.

Parallel to the line y = −2x + 9 and passes through the point (−3, 1).

Parallel to the line y =

x − 1 and passes through the point (0, 0).

Parallel to the line y =

and passes through the point (3, 6).

Perpendicular to y =

+ 7, and goes through the point (0, 6).

Perpendicular to the line y = −2x and passes through the point (2, 1).

Perpendicular to the line y =

Perpendicular to the line y = 5x + 1 and intercepts the y-axis at y = 2.

x − 2 and passes through the point (−2, 4).

Find the equation of the line, in standard form, that is: a

Parallel to the line 3x + 6y = −2 and intercepts the y-axis at y = 3.

Parallel to the line 4x − 2y = 0 and intercepts the y-axis at y = −1.

Perpendicular to the line x − 2y = 3 and intercepts the y-axis at y = 1.

Perpendicular to the line 4x + y = −2 and intercepts the y-axis at y = 0.

3.06 Equations of parallel and perpendicular lines mathspace.co

359

For each graph: i

Write an equation for the line parallel to line l that goes through point P.

Write an equation for the line perpendicular to line l that goes through point P.

y 7 6 5 4 3 2 1

−7 −6 −5 −4 −3 −2 −1−1

7 6 5 4 3 2 1

−7 −6 −5 −4 −3 −2 −1−1

1 2 3 4 5 6 7

−2 −3 −4 P −5 −6 −7

7 6 5 4 3 2 1

P x 1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

Consider the equation f (x) =

−7 −6 −5 −4 −3 −2 −1−1 P

x 1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

+ 2.

Graph a line parallel to f (x) that passes through the point (4, −3).

Graph a line perpendicular to f (x) that passes through the point (1, −2).

Consider the line x = 0. a

Determine the equation of a line that is perpendicular to x = 0.

Determine if there is more than one possible answer for part (a). Explain.

Describe and correct the error in writing an equation of the line that passes through (4, 2) and is parallel to the line y =

360

−2 −3 −4 −5 −6 −7

−7 −6 −5 −4 −3 −2 −1−1

x + 2.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Two different phone companies charge customers an initial activation fee plus a monthly fee. The graph shows the total amounts paid by two customers over a 5-month period. The lines are parallel. a

Did one of the customers pay a greater activation fee? Explain.

Did one of the customers pay a greater monthly fee? Explain.

250 225 200 175 150 125 100 75 50 25

B A

x 1

Let’s extend our thinking 15

The segment joining point A (1, 10) and B (4, 9) is parallel to the segment joining point C (−2, 5) and D (−11, y). a

Find the slope of

Find the value of y.

Consider the points A (5, 6) and B (−13, −22). a

Find the slope of

Find the midpoint of

Write the equation of the perpendicular bisector of

. . . Express your answer in standard form.

Laurelei wants to create a concrete path around her pool as seen in the ﬁgure. Each unit on the ﬁgure is 1 ft. The equations of the lines that mark the edge of the pool are as follows: • a : x = −3 • b:y=x+5 • c:y=3 • d : y = −x + 5 • e:x=3

5 y r q b p

s d e

−5

t x 5

Determine the equations of the lines which would mark the outside edges of the path: p, q, r, s and t. −5

Write the equations of three lines that will form a rectangle with one side represented by y = 3x + 5.

3.06 Equations of parallel and perpendicular lines mathspace.co

361

Answers

7 6 5 4 3 2 1

3.06 Equations of parallel and perpendicular lines What do you remember?

−7−6−5−4−3−2 −1 −1 −2 −3 −4 −5 −6 −7

1 Parallel lines always have the same slope. They will never intersect. Perpendicular lines have negative reciprocal slopes. They will always intersect exactly once and form a 90° angle where they intersect. 2 a No

b Yes

c No

d Yes

3 a True

b False

c False

d True

e True 4 a Neither

b Parallel

c Perpendicular

d Neither

Let’s practice

x 1 2 3 4 5 6 7

12 a A ny horizontal line is perpendicular to x = 0. For example, y = 0. b The line x = 0 is vertical, so any horizontal line is perpendicular to it. Such a line is of the form y = a, a is a real number. 13 These steps describe how to find a perpendicular line that goes through the point (4, 2). To find a parallel line with a slope of , use the following steps:

5 A and B D and E 6 A and F C and D 7 a x = 2 or x = −6

b y = −2 or y = −6

c x = −1 or x = 3

d y = 0 or y = 2

8 a y = 8x + 5

b y = −2x − 5

g y = −4x − 4

14 a Y es, customer B paid a $75 activation fee, while customer A paid a $50 activation fee. b N o, since both lines are parallel, both customers paid the same rate. Let’s extend our thinking

9 a x + 2y = 6

b 2x − y = 1

c 2x + y = +1

d x − 4y = 0

15 a

b y=8

10 a i y = x + 2

ii y = −x + 6

16 a

b (−4, −8)

b i y = −2x − 4

c i y=3

ii x = 2

d i

11 a

7 6 5 4 3 2 1 −7−6−5−4−3−2 −1 −1 −2 −3 −4 −5 −6 −7

362

c 9x + 14y = −148 17 The equations for the lines that mark the outside edges of the path are as follows: • p : x = −4

• q : y = x + 6 • r : y = 4 • s : y = −x + 6 x 1 2 3 4 5 6 7

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

• t : x = 4 18 There must be one line that is parallel with the given line that will have a slope of 3. There must also be two lines perpendicular to the given line that will have negative reciprocal slopes of −

and different y-intercepts.

For example, y = 3x + 7,

, and

Topic 3 Assessment: Linear functions 1

For each of the line on the coordinate plane, determine whether the slope is positive, negative, zero, or undeﬁned. a

b 4

4 3

−4 −3 −2 −1 −1

−2

−3

−4

−2

−3

−2

SOL

−4 −3 −2 −1 −1

−4

Select the graph that matches the equation 4x − 3y = 6. a

y 7 6 5 4 3 2 1

−7 −6 −5 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7

x 1 2 3 4 5 6 7

7 6 5 4 3 2 1 −7 −6 −5 −4 −3 −2 −1−1

x 1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

Topic 3 Assessment: Linear functions mathspace.co

363

7 6 5 4 3 2 1

−7 −6 −5 −4 −3 −2 −1−1

1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

7 6 5 4 3 2 1

1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

Consider the straight line that passes through the points A, B, and C:

Find the slope of the line using the points A and B.

Explain how you can ﬁnd the slope between A and C, without calculating it.

Find the coordinates of the y-intercept.

Find the zeros of the function.

Write the equation of the line in slope-intercept form.

y C B

1 −4 −3 −2 −1 A

x 1

−1

−2 −3 −4

Find the equation of each line: y

1 −4 −3 −2 −1

−1

x 1

−4 −3 −2 −1

−3

−4

Find the values of the range of f (x) =

Consider the table of values of a linear function: x

364

−5 −11

−2 −5

−1

x 1

−2

f (x)

0 −1

3 5

− 1 when the domain is {−3, 4, 8}.

5 9

Sketch the graph of the function.

Find the coordinates of the y-intercept.

Find the slope of the line.

Find the equation of the linear function.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

A band plans to record a demo at a local studio. The cost of renting a studio is $250 plus $50 per hour. The maximum amount of time the studio can be rented is 100 hours. Let y represent the cost in dollars of renting the studios for x hours. a

A line is graphed to represent the scenario. Determine the: i

b c 8

slope of the line.

y-intercept of the line.

Write an equation for the line. Determine the domain and range of the function.

Consider the equation −2x + 3y = 24, assuming that y is a function of x. a

Rewrite the equation using the function notation f (x).

Sketch the graph of the function.

A gym offers aerobics classes. The monthly cost in dollars, y, of taking x classes can be modeled by the linear equation y = 15x + 40. Describe what the y-intercept means in the context.

Consider the line passing through the point (−3, −5) that has a slope of −2. a

Write the equation of the line in the forms: i

b SOL

Standard form

Slope-intercept form

iii

Find the coordinates of the x-intercept. Which form did you use to ﬁnd it? Explain why.

Let f (x) = 2x. The graph of g(x) is shown. a

SOL

Point-slope form

9 y 8 7 6 5 4 3 2 g(x) 1

The slope of g(x) is ⬚ the slope of f (x).

The y-intercept is g(x) is ⬚ the y-intercept of f (x).

twice

one half

two less than

two more than

−4 −3 −2 −1−1 −2 −3 −4

x 2

Plot each point on the coordinate plane. A line will extend through the two points. Let f (x) = 2x. The graph of g(x) is: • Shifted down three units from the graph of f (x) = 2x • Half as steep as f (x) Plot two points to create a line that represents g(x). The coordinates of the points must be integers. 8

6 4 2 −8 −6 −4 −2 −2

x 2

−4 −6 −8

Topic 3 Assessment: Linear functions mathspace.co

365

Consider the functions represented by: Function A: x y

0 2

1 5

2 8

3 11

Function B: y

4 14 5

Determine: a

The function with the greater slope.

The function with the smaller value of y-intercept.

The function with the larger value x-intercept.

x −5

5 y = 2x + 1 −5

Draw the graph of each of the linear equations: b

a 15

4x = −12

Describe how each function has been transformed from the parent function y = x. a

3x + 6y = 8

y = 2x

y=x−5

y=x+3

y = 0.3x

Determine whether each line is parallel, perpendicular, or neither to the line

Explain your reasoning. b

a 17

SOL

Consider the equation of a line

20x − 8y = 12

Find the equation of a line that is parallel to the line and goes through the point (0, 4).

Find the equation of a line that is perpendicular to the line and goes through the point (0, 4).

Which equation represents the line that passes through the points (3, −1) and (−1, 5)? A

SOL

2x + 5y = −7

Which equation best represents line m? A

y = −3x + 3

y = −3x

y = 3x + 3

y = 3x − 6

D 9 y 8 7 6 5 4 3 m 2 1 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8 −9

366

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

x 1

SOL

The graph of line k is shown.

9 y 8 7 6 5 4 3 2 1

Which number is closest in value to the slope of line k? A B C

−2

−1

−4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7 −8 −9

x 1 2 3 4 5 6 7 8 9

Performance task 21

Felix, Graciana, and Stella are best friends trying to raise money to go to music camp this summer. The total cost for the group is $500. They have each researched jobs they can do around their neighborhood and have presented their ﬁndings in different ways. Felix thinks they should mow lawns, Graciana wants to wash cars, and Stella thinks walking dogs is the best option. • Mowing lawns • Walking dogs Lawns mowed 2 4 6 • Washing cars f (x) = 5x − 1

Money earned $18 $38 $58

Dollars 40 30 20 10 Dogs 1

2 3 4 5 6 7 8 9

Can the relationship for each type of job be represented by a linear function? Explain.

Which job will help the friends reach their goal the fastest, assuming each type of job takes the same amount of time to do? How much work will it take? Explain why the option you chose requires less work than the other two options.

Stella realizes her graph does not accurately show how the money will be earned from walking dogs. Explain why the graph is not accurate and create a new graph to ﬁx the mistake.

Topic 3 Assessment: Linear functions mathspace.co

367

Answers

8 a b

Topic 3 Assessment: Linear functions 1 a Positive slope

8 f (x) 7 6 5 4 3 2 1

b Zero slope

c Negative slope

d Undeﬁned slope

A.F.1a 2 C A.F.1f

−12 −10 −8 −6 −4 −2 −1 −2

3 a b I t will be the same as the slope between A and B, as the slope between any 2 points on the same line is the

A.F.1c, A.F.1f 9 This means that there is a ﬂat fee of $40 per month, even if no classes are taken.

same, so it will also be . d −2

c (0, 1)

A.F.1a

A.F.1a, A.F.1di 4 a y=2

10 a i y + 5 = −2 (x + 3)

b x = −2

ii 2x + y = −11

iii y = −2x − 11

A.F.1di, A.F.1div, A.F.1dv

. Any form with a valid explanation.

For example, I used standard form so that when the y-term is set to 0, we don’t need to rerrange and just need to divide both sides by 2.

5 Range: A.F.1a, A.F.1g 6 a 5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

b (0, −1)

A.F.1a, A.F.1c, A.F.1diii

11 a A

b D

A.F.1b x

1 2 3 4 5

6 4 2 −8 −6 −4 −2 −2

d y = 2x − 1

c 2

−4

A.F.1a, A.F.1dii, A.F.1f 7 a i 50

x 2

−6 −8

ii 250

b y = 50x + 250

A.F.1b, A.F.1f

c Domain: [0, 100]

Range: [250, 5250]

A.F.1a, A.F.1diii

13 a Function A A.F.1a, A.F.1h 14 a 4 3 2 1 −2 −1 −1 −2 −3 −4 −5 −6

368

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

b Function B

x 1 2 3 4 5 6 7 8

c Function A

b 4

Performance task

21 a Y es. Each function has a constant slope. For each uniform increase in the x-values, the y-values also increase by a uniform amount.

3 2 1

−4 −3 −2 −1 −1

b M owing lawns will help the friends reach their goal the fastest.

−2

They only need to mow 51 lawns to get $500 while they would need to wash 101 cars or walk 100 dogs to make the same amount of money.

−3 −4

c 4

Mowing lawns requires less work because they can make $10 per lawn mowed, but they could only make $5 per car or $5 per dog for the other two options.

c S tella’s graph is incorrect because it is continuous, however the domain for the number of dogs is discrete because you can only have non-negative integer values of dogs. The correct graph would be:

2 1 −4 −3 −2 −1

−1

x 1

−2

Dollars

−3

−4

40 30

A.F.1f

15 a Steeper, vertical dilation by a factor of 2 b Translated 5 units downwards

10 Dogs

c Translated 3 units upwards

1 2 3 4 5 6 7 8 9 10

d Less steep, vertical dilation by a factor of 0.2 A.F.1b

A.F.1a, A.F.1h, MP1, MP2, MP5

16 a N either since their slopes are not the same and also do not have a product of −1. b Parallel since the slope of 2x + 5y = −7 is the same slope to the line

which has

c Perpendicular since the product of the slopes of is −1.

20x − 8y = 12 and

The slopes of 20x − 8y = 12 and and

are

, respectively.

d N either since their slopes are not the same and also do not have a product of −1. A.F.1e 17 a

A.F.1e 18 D A.F.1dii 19 A A.F.1di 20 C A.F.1a

Topic 3 Assessment: Linear functions mathspace.co

369

Systems of Equations & 4 Inequalities Big ideas • A standard algorithm can be followed to solve a wide range of equations. This algorithm is reliable and useful in a variety of situations, but there is often a more efficient method that can be used based on the structure of the equation. • A solution set is the collection of all values that make an equation or inequality true.

Chapter outline 4.01 4.02 4.03 4.04 4.05

Write and graph linear systems (A.EI.2) Substitution method (A.EI.2) Elimination method (A.EI.2) Two variable linear inequalities (A.EI.2) Systems of linear inequalities (A.EI.2) Topic 4 Assessment

374 401 425 447 475 501

Paths crossing in a park is similar to the solution of a system of linear equations. Each path represents an equation, and the crossing point is the solution.

4. Systems of Equations & Inequalities Topic overview Foundational knowledge Evaluating standards proficiency The skills book contains questions matched to individual standards. It can be used to measure proficiency for each. Students should be proficient in these standards. 8.PFA.5 — The student will write and solve multistep linear inequalities in one variable, including problems in context that require the solution of a multistep linear inequality in one variable. A.EI.1 — The student will represent, solve, explain, and interpret the solution to multistep linear equations and inequalities in one variable and literal equations for a specified variable.

A.EO.1 — The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables. A.EO.2 — The student will perform operations on and factor polynomial expressions in one variable.

Big ideas and essential understanding A standard algorithm can be followed to solve a wide range of equations. This algorithm is reliable and useful in a variety of situations, but there is often a more efficient method that can be used based on the structure of the equation. 4.01 — Systems of linear equations can be solved using many different methods. Graphing is an efficient method when the equations are provided in slope-intercept form and the system has integer solutions. 4.02 — Systems of linear equations can be solved using many different methods. Substitution is an efficient method when at least one of the equations has an isolated variable. 4.03 — Systems of linear equations can be solved using many different methods. Elimination is an efficient method when the coefficients are not equal to one.

372

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

A solution set is the collection of all values that make an equation or inequality true. 4.04 — The value(s) of the variables that make an equation or inequality true make up its solution set. A two variable linear inequality has an infinite number of solutions so its solution set can be represented by a half plane. 4.05 — The value(s) of the variables that make every equation or inequality in a system true make up its solution set. A system of two variable linear inequalities can have an infinite number of solutions so its solution set can be represented by an intersection of half planes (should that intersection exist).

Standards A.EI.2 — The student will represent, solve, explain, and interpret the solution to a system of two linear equations, a linear inequality in two variables, or a system of two linear inequalities in two variables. A.EI.2a — Create a system of two linear equations in two variables to represent a contextual situation. 4.01 Write and graph linear systems 4.02 Substitution method 4.03 Elimination method A.EI.2b — Apply the properties of real numbers and/ or properties of equality to solve a system of two linear equations in two variables, algebraically and graphically. 4.01 Write and graph linear systems 4.02 Substitution method 4.03 Elimination method A.EI.2c — Determine whether a system of two linear equations has one solution, no solution, or an infinite number of solutions. 4.01 Write and graph linear systems 4.02 Substitution method 4.03 Elimination method

A.EI.2d — Create a linear inequality in two variables to represent a contextual situation. 4.04 Two variable linear inequalities A.EI.2e — Represent the solution of a linear inequality in two variables graphically on a coordinate plane. 4.04 Two variable linear inequalities A.EI.2f — Create a system of two linear inequalities in two variables to represent a contextual situation. 4.05 Systems of linear inequalities A.EI.2g — Represent the solution set of a system of two linear inequalities in two variables, graphically on a coordinate plane. 4.05 Systems of linear inequalities A.EI.2h — Verify possible solution(s) to a system of two linear equations, a linear inequality in two variable, or a system of two linear inequalities algebraically, graphically, and with technology to justify the reasonableness of the answer(s). Explain the solution method and interpret solutions for problems given in context. 4.01 Write and graph linear systems 4.02 Substitution method 4.03 Elimination method 4.04 Two variable linear inequalities 4.05 Systems of linear inequalities

Future connections A2.EI.1 — The student will represent, solve, and interpret the solution to absolute value equations and inequalities in one variable.

A2.EI.3 — The student will solve a system of equations in two variables containing a quadratic expression.

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4.01 Write and graph linear systems Subtopic overview Lesson narrative In this lesson, students will examine the graph of linear systems to determine whether there will be zero, one, or infinite number of solutions. Then, students will work with contextual situations involving systems of linear equations, where they will define variables, create models (using equations, tables and/or graphs), and interpret the meaning and viability of solutions in contextual situations. This lesson provides opportunities for students to attend to precision by solving accurately and using appropriate labels and scales on graphs. By the end of this lesson, students should be able to create and interpret models of systems of linear equations in a context, as well as justify that their solutions fit contextual constraints.

Learning objectives Students: Page 184

Key vocabulary 

intersection



non-viable solution



solution (to a system of equations)



system of equations



viable solution

Essential understanding Systems of linear equations can be solved using many different methods. Graphing is an efficient method when the equations are provided in slope-intercept form and the system has integer solutions.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG3 — Mathematical Reasoning Teachers can incorporate this goal by having students use the slope and x-intercept to graph the system of equations in order to determine how many solutions exist (one solution, no solution, infinite solutions). 374

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MPG4 — Mathematical Connections

MPG5 — Mathematical Representations

This can be achieved by having students draw connections between the concept of a system of two linear equations and real-world scenarios. For instance, teachers can connect to A.EO.1a by guiding students to create two linear equations that represent a given real-world situation, or connect to A.F.1f by reviewing the graphing of a single linear equation using slope and x-intercept.

Teachers can incorporate this goal by asking students to represent the systems of two linear equations graphically, and then analyze the graphs to identify the number of solutions. For instance, students can create tables of values and determine when the input and output values are the same, or graph both equations on the same coordinate plane and discuss the significance of the point of intersection.

Content standards A.EI.2 — The student will represent, solve, explain, and interpret the solution to a system of two linear equations, a linear inequality in two variables, or a system of two linear inequalities in two variables. A.EI.2a — Create a system of two linear equations in two variables to represent a contextual situation. A.EI.2b — Apply the properties of real numbers and/ or properties of equality to solve a system of two linear equations in two variables, algebraically and graphically.

A.EI.2c — Determine whether a system of two linear equations has one solution, no solution, or an infinite number of solutions. A.EI.2h — Verify possible solution(s) to a system of two linear equations, a linear inequality in two variable, or a system of two linear inequalities algebraically, graphically, and with technology to justify the reasonableness of the answer(s). Explain the solution method and interpret solutions for problems given in context.

Prior connections A.EO.1 — The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

Future connections A2.EI.3 — The student will solve a system of equations in two variables containing a quadratic expression.

Engage Activity Recipes on a budget

60 mins

Students will work with a partner to determine the servings of a recipe that can be made within a set budget.

Understanding and skills

Will use Graphing the relationship between two real-world quantities.

Will develop Recognizing the points on a line as the solutions that satisfy the relationship. Understanding that the solutions to a system of equations is the intersection point of the two equations.

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Could extend Determining the cost for n servings of a recipe using the prices from local grocers.

Preparation and materials • Materials: Graph paper or blank grids for graphing, downloadable USDA food budget table (optional) • Download and print copies of the students graphic organizer from the student Launch slide.

Support students with disabilities Support visual-spatial - create and interpret visual representations Provide handouts of the USDA food budget so they can reference costs of various ingredients. Provide graph paper so that students can graph the number of batches of their recipe versus the quantity of a given ingredient.

Support for English language learners Compare and connect with discussion supports Ask students to reﬂect on the differences between using a table and a graph to represent how much of each of their two ingredients they could buy with the given budget. Consider sharing the following sentence prompts: • I can determine how many servings of my recipe I can make with $15 using a table or a graph because they have... in common. • I would choose to use a graph/table because... • The table is helpful for determining how much of each ingredient I could buy with my budget because...

Classroom guide Hook

Notice and wonder

•

5 mins

Consider the snippet of the USDA Cost of Food at Home data from December 2020. What do you notice? What do you wonder? Male 14-18 years Female 14-18 years

Thrifty plan $41.40 $39.50

Low-cost plan $58.10 $49.20

Moderate plan $73.60 $58.90

Liberal plan $85.10 $72.90

Slide 1 from Student Engage Activity

Students are reading the table of values provided by the United States Department of Agriculture (USDA) regarding the weekly and monthly grocery costs by gender and age for different budget levels. The screen shows two rows: male teenager and female teenager for one week, but the full table has been provided as an optional material to be printed.

Implementation details The hook provides students an opportunity to discuss food costs and budgets which can often be a very different experience for students of different socioeconomic backgrounds. Allow students space to share as much or as little of their personal experience as they are comfortable with.

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Launch

5 mins

Today you will be choosing a recipe to explore with your partner: Arroz con Pollo

Saag Paneer

Spaghetti marinara

Egg drop soup

Ingredients 4 oz chicken 3 oz Spanish rice

Ingredients 4 oz saag simmer sauce B 3 oz paneer cheese

Ingredients 4 oz spaghetti noodles 3 oz marinara sauce

Ingredients 4 cups of chicken broth D 3 eggs

Slide 2 from Student Engage Activity

Have recipe cards prepared to share with students based on the recipe selected. Provide students with graphing paper or unlabeled graphs to work on. Recommended: Collect grocery ads or provide a way for students to research grocery prices at their local stores as part of the extension of this activity. Important mathematical concepts: Unit conversions involving ounces, cups, and pounds Important contextual information: Budget, groceries Suggested grouping: Form pairs

Continue when Students have had the opportunity to read and discuss the food budgets provided by the USDA

Explore

Think-pair-share

•

30 mins

You’ve been given $15 to create as many servings of the recipe as possible. Meat, eggs, and dairy chicken thighs 16oz paneer cheese 12oz eggs 18ct Grains and starches rice spaghetti noodles

Price per $2.19 $6.00 $1.88

Pantry marinara sauce 14 oz saag simmer sauce 10oz chicken broth 32oz

Price per container $1.48 $3.50 $1.99

Price per pound $1.00 $2.42

Slide 5 from Student Engage Activity

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Anticipated strategies Ingredients Strategy All of the recipes use a ratio of 4 : 3. The ordered pairs for the graph will be (4, 3), (8, 6), and (16, 12) or the reverse, depending on the order of the ingredients. 18

Second Ingredient

First Ingredient 2 4 6 8 10 12 14 16 18

Budget Strategy With the prices provided and a budget of $15 you could buy: • 143 eggs • 109 oz of chicken thighs • 141 oz of marinara • 68 oz of Spanish cooked rice • 42 oz of palak gravy • 99 oz of spaghetti noodles • 30 cups of chicken broth • 30 oz of paneer Arroz con pollo: 135

Saag paneer:

120

105

15 30 45 60 75 90 105120135

Spaghetti marinara: 135

Egg drop soup:

135

120

105

x 15 30 45 60 75 90 105120135

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5 10 15 20 25 30 35 40 45

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x 15 30 45 60 75 90 105120135

• Arroz con pollo: The lines intersect close to (36, 48) which means we can make 12 servings. • Saag paneer: The lines intersect close to (15, 20) which means we can make 5 servings. • Spaghetti marinara: The lines intersect close to (48, 64) which means we can make 16 servings. • Egg drop soup: The lines intersect close to (19, 26) which means we can only make 6 servings.

Misconceptions Choosing a scale What are the key values we want our graphs to display? What’s a factor we can count by to include these values within 10 units on our graph? Interpreting the meaning of the line What do the points on your graph represent? What would other points represent? Understanding units How do we ﬁnd the price per ounce if we know the price per pound?

Purposeful questions As students work through this problem they will need to balance the reality of purchasing ingredients with the simpliﬁcation caused in a mathematical model. Use prompts to encourage students to explain their strategy, the assumptions they had to make, and the edge cases of their model so that they are fully exploring the context: • What strategy did you use? • Does this seem like a reasonable answer? What assumptions did you make? • How much of just one ingredient could you buy? Is this represented on your model? Where?

Continue when All students have estimated the point of intersection between the line for the ingredient ratios and the line for the budget.

Discuss

15 mins

Begin with small group discussions to compare solutions for the same recipe, then select one of each recipe to share and discuss as a class. Make connections between the different representations chosen and how graphs can be used to solve for intersections.

Discussion guide Allow students to see the results from recipes they did not choose. Start by having partners with the same recipes huddle and compare solutions. Then, coordinate a short gallery walk or elect a presenter from each recipe group to share their ﬁndings and summarize their struggles. Students may recognize that you can’t go to the store and buy just 1 egg or just 1 ounce of spaghetti. Highlight the differnces in solutions for students who focused on purchasing packages of ingredients instead of the prices per ounce or unit rates for the ingredients. • What is a ‘solution’ to the different parts of this activity? What do these ‘solutions’ represent? • How was your recipe different from other recipes? • Do you think these recipes are affordable? Explain. Extension: Have groups research the prices in their local stores and create a model based on the quantities of ingredients that can actually be purchased, the budget restriction, and the recipe ratios.

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Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lesson: Grade 8 — 3.05 Characteristics of linear functions

Tools You may ﬁnd these tools helpful: • Graphing calculator • Graph paper • Ruler • Clear plastic sheets

• Blank coordinate plane • Frayer model graphic organizer • Spreadsheet application

Student lesson & teacher guide Write and graph linear systems Students learn what deﬁnes a system of equations and its solutions, then go through the different types of solutions to a system of equations and how they relate to the graphical representation of the system.

Students: Pages 184–185

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The number of people signing up to join the soccer team can be represented by the equation p(x) = 3x + 11 where x represents the number of days the signup has been open.

y 25 20

The number of people signing up to join the lacrosse team next season can be represented by the equation r(x) = 7 + 3x.

r(x)

15 p(x) 10 5

x 1

When two lines are parallel and distinct, they have no points of intersection. The corresponding system of equations has no solutions, which means the teams will never have the same number of players.

The amount of money Fernando makes at his construction job is shown by the equation F(x) = 10 + 2x and the amount of money Heather makes at McDonalds is represented by the equation H(x) = 2(x + 5) where x represents the number of hours they work.

y 25 20

When two lines are identical, they intersect at every point. The corresponding system of equations has infinitely many solutions. This means that Fernando and Heather make the same amount of money at any point in time.

H(x)

15 10 F(x) 5

x 1

The number of solutions can be determined by looking at the equations without graphing. • One solution: the lines have different slopes so they intersect at one point • No solution: the lines have the same slope and different y-intercepts so they are parallel and will never intersect • Infinitely many solutions: the lines have the same slope and y-intercept so they are the same line A solution to a system of equations in a given context is said to be viable if the solution makes sense in the context and non-viable if it does not make sense within the context, even if it would otherwise be algebraically valid.

Example 1 Graph each system of equations and state the solution.

Create a strategy

4.01 Write and graph linear systems mathspace.co

We can convert the second equation in the system to slope-intercept form and graph both equations using the y-intercept and slope.

381

Concrete-Representational-Abstract (CRA) Aproach Targeted instructional strategies Concrete: Engage students with a hands-on activity to model systems of linear equations. Create a large coordinate grid on your classroom floor using tape or chalk. Provide students with lengths of string or rope to represent linear equations. Have them physically place the strings on the grid to model equations by positioning the string according to the slope and y-intercept. As they lay down the strings, students can observe where the lines intersect, are parallel, or coincide, representing one, zero, or infinitely many solutions. Representational: Transition students to drawing the systems of equations on graph paper. Teach them how to plot points using tables of values and draw the lines accurately, emphasizing appropriate scales and labels on the axes. Encourage them to represent the same equations they modeled with strings, now on paper. This helps students see how the physical activity relates to graphical representations, reinforcing their understanding of solutions as points of intersection on the graph. Abstract: In upcoming lessons students will learn to solve algebraically using methods such as substitution or elimination, focusing on manipulating the equations using numbers and symbols. For now you can focus on students connecting the equations to the graphs to identify the number of solutions the system will have to help them understand how the abstract equations relate back to the graphs and physical models.

Always, sometimes, never English language learner support Have students consider which of the following statements are always, sometimes, or never true to check their understanding of the concepts. • Systems of equations where the slopes are the same will have no solution. (answer: sometimes - depends on whether the y-intercept is the same or not) • Systems of equations where the slopes are different will have exactly one solution. (answer: always) • Systems of equations where the y-intercept are the same will have no solutions (answer: never). • A solution to a system of equations is viable. (sometimes - it depends on the context)

Identifying solutions from a graph Address student misconceptions Students may think that the solution to a system of equations must be easily readable off the graph to exist. Two common ways this can come up is if the solution does not occur on the grid lines or if it occurs out of the view of the graph the student drew. Challenge these misconceptions by graphing with technology and changing the view. 4

1 −4 −3 −2 −1 −1

x 1

382

1 −4 −3 −2 −1 −1

−2

−3

−4

Point of intersection (solution) is not easily readable

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x 1

Point of intersection (solution) is not visible on the graph

x 1

y 25

Assist20graphing

When two lines are identical, they intersect at every point. The corresponding system of equations has infinitely many solutions. Thisprinted means that Fernando axes. and Heather makeallowing the same amount of to use Provide students with larger grid paper or provide prelabeled Consider students 10 F(x) at anySince point in time. a digital tool like GeoGebra or Desmos to graphmoney the lines. accuracy is important when identifying the

Student support H(x) 15 with disabilities

solution from a graph, if technology is not available, consider scribing or pairing students to have one student x explain and the other 1 to2draw. 3 4 5 The number of solutions can be determined by looking at the equations without graphing. • One solution: the lines have different slopes so they intersect at one point • No solution: the lines have the same slope and different y-intercepts so they are parallel and will never intersect • Infinitely many solutions: the lines have the same slope and y-intercept so they are the same line

Examples

solution to185–186 a system of equations in a given context is said to be viable if the solution makes sense in the context Students:A Pages and non-viable if it does not make sense within the context, even if it would otherwise be algebraically valid.

Example 1 Graph each system of equations and state the solution.

Create a strategy We can convert the second equation in the system to slope-intercept form and graph both equations using the y-intercept and slope.

Apply the idea Second equation Divide equation by 4 Evaluate the division

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185

Purpose Show students how to solve a system of equations that has infinite solutions by graphing. Reﬂecting with students Ask students if they need to graph the lines in order to find the solutions to this system. Let students know that if the two equations are equivalent, then the system will always have an infinite number of solutions.

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Students: Pages 186–187

Purpose Show students how to ﬁnd the solutions to a system of equations with no solutions by graphing.

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Expected mistakes Students may wonder if the point of intersection is simply not visible on the graph and assume that this system has one solution. Encourage students to graph enough points to get a full picture of each line. Reﬂecting with students Ask students if they can think of another way to identify when a system of equations has no solutions. Remind students that a system has no solutions if the two lines are parallel but not equivalent. How can they check if the lines are parallel? How can they make sure that the lines are not equivalent?

Advanced learners: Determining no solution from standard form

use with Example 1

Targeted instructional strategies When solving a system of linear equations, a situation may arise where there is no solution. This happens when the two lines represented by the equations are parallel to each other and never intersect. As an extension opportunity for advanced learners, encourage students to identify a relationship between the coefficients and constants in a system of equations that has no solution. One explanation is that the coefficients of the x and y terms in both equations will be proportional, and the constants will not be. For instance, in the given problem, the coefficients of x and y in the first equation are half of those in the second equation, but the constant in the first equation is not half of that in the second. Therefore, the lines will be parallel and there will be no solution to the system.

Students: Page 187

Purpose Show students how to interpret a context into a system of linear equations. Expected mistakes Students may be confused by the equation presented in the given information and try to solve it immediately. Remind students to read the question carefully and consider what each side of the equation represents. Reﬂecting with students Point out that the two equations in the system contain expressions from either side of the given information. Ask students what this means in the context of the problem.

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Students: Pages 187–188

Time (weeks) 8 9 10

Car Mileage (Rodica) 200(8) + 9000 = 10 600 200(9) + 9000 = 10 800 200(10) + 9000 = 11 000

Car Mileage (Yuwei) 500(8) + 6000 = 10 000 500(9) + 6000 = 10 500 500(10) + 6000 = 11 000

When we get to 10 weeks in the table we can see that the mileage for both cars is 11 000. So we know the mileage will be the same after 10 weeks. c Graph the system of equations. Choose an appropriate scale for the axes.

PurposeCreate a strategy Show students we can system of equations a same tablemileage of values. Based onhow the solution in solve part (b),awe know that the cars will using have the of 11 000 miles after 10 weeks. This will help us decide on an appropriate scale. Car Mileage (Yuwei) ReﬂectingTime with(weeks) students Car Mileage (Rodica) 8 200(8) 9000 = 10 600 500(8) + 6000 10 000 Apply ifthe idea Ask students they can think of a+ different method for ﬁnding the=solution, that isn’t graphing. Have students + 9000 = 10we 800 500(9) + 6000 = 10y-values 500 Used mileage gives an Since the9number of weeks the x-value, by 1 along the solve the equation given in200(9) theisinstructions tocan seecount that equating the of the twocar equations 10 200(10) + 9000 = 11 000 at 10 weeks. 500(10)Since + 6000 = 11 000 x-axis to slightly beyond the point of intersection the Mileage equation that has the solution for the system of equations. 11 000

mileage is the y-value, we can count by 500 along the y-axis to slightly When we get to 10ofweeks in the table we can see that the mileage for both cars 10 is 000 11 000. So we know the mileage beyond the point intersection at 11 000 miles. 9000 Students:willPage be the188 same after 10 weeks.

8000 7000 6000 c Graph the system of equations. Choose an appropriate scale for the axes. 5000 4000 Create a strategy 3000 2000 Based on the solution in part (b), we know that the cars will have the same mileage of 11 000 miles after 10 weeks. 1000 This will help us decide on an appropriate scale. Time (weeks) −1

1 2 3 4 5 6 7 8 9 10 11

Apply the idea Used car mileage Since the number of weeks is the x-value, we can count by 1 along the d Identify the solution system of equations intrepret it inthe the context of the problem. x-axis to slightly beyondfor thethe point of intersection at and 10 weeks. Since Mileage 11 000 mileage is the y-value, we can count by 500 along the y-axis to slightly 10 000 Create a strategy beyond the point of intersection at 11 000 miles. 9000 We can use the graph from part (c) to find the point of intersection and use the labels on the axes to interpret it. 8000

Apply the idea The point of intersection from the graph is (10, 11 000): The x-axis measures time so 10 represents 10 weeks. The y-axis measures mileage so 11 000 represents 11 000 miles. The solution (10, 11 000) means that after 10 weeks both cars will have the same mileage of 11 000 miles.

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7000 6000 5000 Used car mileage 4000 Mileage 113000 000 102000 000 1000 9000 Time (weeks)

8000 1 2 3 4 5 6 7 8 9 10 11 −1 7000 Mathspace Virginia SOL Algebra 1 Teacher Edition 6000 mathspace.co d Identify the solution for the system of equations and intrepret it in the context 5000 of the problem. 4000 3000 Create a strategy 2000

c Graph the system of equations. Choose an appropriate scale for the axes.

Create a strategy Based on the solution in part (b), we know that the cars will have the same mileage of 11 000 miles after 10 weeks. ThisTime will help us decide on anMileage appropriate scale. (weeks) Car (Rodica) Car Mileage (Yuwei) 200(8) + 9000 = 10 600 500(8) + 6000 = 10 000 200(9) + 9000 = 10 800 500(9) + 6000 = 10 500 Used car mileage Since the10number of weeks is the x-value, can count500(10) by 1 along the = 11 000 200(10) + 9000 = 11we 000 + 6000 x-axis to slightly beyond the point of intersection at 10 weeks. Since the Mileage mileage theto y-value, we in can count the the y-axis to slightly When weisget 10 weeks the tableby we500 canalong see that mileage for both cars is11 000 11 000. So we know the mileage 10 000 beyond thesame pointafter of intersection will be the 10 weeks. at 11 000 miles. 8

Apply the 9 idea

9000 8000 7000 c Graph the system of equations. Choose an appropriate scale for the axes. 6000 5000 Create a strategy 4000 3000 Based on the solution in part (b), we know that the cars will have the same mileage of 11 000 miles after 10 weeks. 2000 This will help us decide on an appropriate scale. 1000 Time (weeks)

Apply the idea

−1

1 2 3 4 5 6 7 8 9 10 11

Used car mileage Since the number of weeks is the x-value, we can count by 1 along the x-axis to slightly beyond the point of intersection at 10 weeks. Since the Mileage d Identify the solution for the system of equations and intrepret it in the context 11of000 the problem. mileage is the y-value, we can count by 500 along the y-axis to slightly 10 000 beyond the point of intersection at 11 000 miles. 9000

PurposeCreate a strategy 8000 We can use theto graph from part to find the of intersection and useby thegraphing. labels on the axes to interpret it. Show students how conﬁrm the (c) solution to point a system of equations 7000 6000

Apply thestudents idea 5000 Reﬂecting with 4000 The point of intersection the graph is (10,cars 11 000): Ask students if they think thefrom mileage of the will intersect again. We can see Used from car themileage graph that the two 3000 Mileage The x-axis measures time so 10 represents 10 weeks. The y-axis measures lines will not meet again. 112000 000 mileage so 11 000 represents 11 000 miles.

101000 000 Time (weeks) 9000 1 2 3 4 5 6 7 8 9 10 11 −1 8000 7000 6000 d Identify the solution for the system of equations and intrepret it in the context of the problem. 5000 4000 Create a strategy 3000 2000on the axes to interpret it. We can use the graph from part (c) to find the point of intersection and use the labels 1000 Time (weeks)

solution (10, 11 000) means that after 10 weeks both cars will have the Students:The Page 188 same mileage of 11 000 miles.

Apply the idea

−1

1 2 3 4 5 6 7 8 9 10 11

Used car mileage

The point of intersection from the graph is (10, 11 000): The x-axis measures time so 10 represents 10 weeks. The y-axis measures 188 Mathspace SOL Algebra mileage so 11 000Virginia represents 11 0001 miles. mathspace.co

The solution (10, 11 000) means that after 10 weeks both cars will have the same mileage of 11 000 miles.

11 000 10 000 9000 8000 7000 6000 5000 4000 3000 2000 1000 −1

188

Mileage

Time (weeks) 1 2 3 4 5 6 7 8 9 10 11

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Purpose To show students how to interpret the solution of a system of equations in a real-world context.

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Use technology to reduce computation errors

use with Example 2

Student with disabilities support For students who face difficulties with computation or using small buttons on a calculator, encourage them to use a spreadsheet for part (b). For example, starting with these formulas and dragging down. 1 2 3 4

A Time (weeks) 0 = A2 + 1 ⋮

B Car Mileage (Rodica) =200 ∗ A2 + 9000 =200 ∗ A3 + 9000 ⋮

C Car Mileage (Yuwei) = 500 ∗ A2 + 6000 = 500 ∗ A3 + 6000 ⋮

Which results in: 1 2 3 4 5 6 7 8 9 10 11 12

A Time (weeks) 0 1 2 3 4 5 6 7 8 9 10

B Car Mileage (Rodica) 9000 9200 9400 9600 9800 10 000 10 200 10 400 10 600 10 800 11 000

C Car Mileage (Yuwei) 6000 6500 7000 7500 8000 8500 9000 9500 10 000 10 500 11 000

Students: Page 189

Example 3 Bixia is saving up her quarters and dimes in a jar. She has a total of $24.50 in 125 coins. a Write a system of equations that models this situation.

Create a strategy We’ll need to write a system of equations with the information provided in the problem, and we’re only given two numbers. The two totals give us information about each equation: one of the equations utilizes the units of a dollar amount throughout, and the other equation must include the number of coins as its units. We should deﬁne the unknown variables. Since we don’t know how many of each type of coin Bixia has, we can say that x = the number of quarters and y = the number of dimes.

Apply the idea

Reﬂect and check We can conﬁrm that the system of equations makes sense by checking the units of each equation. Since the total of x + y = 125 is the total number of coins, it should make sense that the number of quarters added to the number of dimes is equal to the total number of coins, which is what is being represented in this equation.

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The total of 0.25x + 0.10y = 24.50 represents the fact that Bixia has $24.50. The expression 0.25x represents the value of 1 quarter multiplied by how many quarters are in the jar. This term is the dollar amount of all the quarters combined. The expression 0.10x represents the value of 1 dime multiplied by how many dimes are in the jar. This term is the dollar amount of all the dimes combined. When these two expressions are added together, the result is the total value of all the coins, or $24.50. Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co b Graph the system of equations. Use appropriate axes, labels, and scales.

that x = the number of quarters and y = the number of dimes.

Apply the idea

Example 3 Bixia is saving up her quarters and dimes in a jar. She has a total of $24.50 in 125 coins.

Reﬂect and check

a Write a system of equations that models this situation. We can conﬁrm that the system of equations makes sense by checking the units of each equation. Since the total of x + y = 125 is the total number of coins, it should make sense that the number of quarters added to Create a strategy the number of dimes is equal to the total number of coins, which is what is being represented in this equation. We’ll need to write a system of equations with the information provided in the problem, and we’re only given two numbers. The total of 0.25x + 0.10y = 24.50 represents the fact that Bixia has $24.50. The expression 0.25x represents the The give us information equation: equations utilizes units of amount valuetwo of 1totals quarter multiplied by howabout manyeach quarters are inone the of jar.the This term is the dollarthe amount of aalldollar the quarters throughout, andexpression the other equation must include the number of multiplied coins as itsbyunits. combined. The 0.10x represents the value of 1 dime how many dimes are in the jar. This term We should theofunknown variables. SinceWhen we don’t know many of each type of coin Bixia weiscan is the dollardeﬁne amount all the dimes combined. these twohow expressions are added together, thehas, result thesay total that = the number valuex of all the coins,oforquarters $24.50.and y = the number of dimes.

Apply the idea b Graph the system of equations. Use appropriate axes, labels, and scales.

Purpose Create a strategy Show students how to write a system of equations from a situation involving the sum of two variable terms and Based the check equation, x represents the number of quarters because we know that a quarter is $0.25, and y Reﬂectonand their known totals. represents the number of dimes because we know that a dime is $0.10. We can calculate the intercepts of both We can conﬁrm that the system of equations makes sense by checking the units of each equation. equations and use those to determine the maximum values on our x- and y-axes. Since the total of x + y = 125 is the total number of coins, it should make sense that the number of quarters added to Expected mistakes the number of dimes is equal to the total number of coins, which is what is being represented in this equation.

StudentsApply may the notidea recognize that there is enough information to form two equations and only note that The total of 0.25x + 0.10y = 24.50 represents the fact that Bixia has $24.50. The expression 0.25x represents the 125 coins = $24.50 which not an equation First, we’ll calculate theisx-intercept of x + y = we 125: can use to solve the problem. Remind students that we have value of 1 quarter multiplied by how many quarters are in the jar. This term is the dollar amount of all the quarters two variables; the number of quarters and the number of dimes. + y = 125 0.10x represents Firstthe equation combined. Thexexpression value of 1 dime multiplied by how many dimes are in the jar. This term is the dollar amount all the dimes combined. When x + (0) =of125 Substitute y =these 0 two expressions are added together, the result is the total

Students:value Pages 189–190 of all the coins, or $24.50. x = 125

Evaluate the addition

Now, we’ll calculate the y-intercept of x + y = 125: b Graph the system equations. Use appropriate axes, labels, and scales. x + y = of 125 First equation (0) + y = 125

Substitute x = 0

Create a strategy

y = 125 Evaluate the addition Based on the equation, x represents the number of quarters because we know that a quarter is $0.25, and y represents the number of dimes because we know that a dime is $0.10. We can calculate the intercepts of both equations and use those to determine the maximum values on our x- and y-axes.

Apply the idea First, we’ll calculate the x-intercept of x + y = 125: x + y = 125

First equation

x + (0) = 125

Substitute y = 0

x = 125

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189

4.01 Write and graph linear systems mathspace.co

189

Evaluate the addition

Now, we’ll calculate the y-intercept of x + y = 125: x + y = 125

First equation

(0) + y = 125

Substitute x = 0

y = 125

Evaluate the addition

The x-intercept of x + y = 125 is (125, 0) and the y-intercept is (0, 125). Next, we’ll calculate the x-intercept of 0.25x + 0.10y = 24.50: 0.25x + 0.10y = 24.50

Second equation

0.25x + 0.10(0) = 24.50

Substitute y = 0

0.25x = 24.50

Evaluate the multiplication

x = 98

Divide both sides by 0.25

Finally, we’ll calculate the y-intercept of 0.25x + 0.10y = 24.50: 0.25x + 0.10y = 24.50

Second equation

0.25(0) + 0.10y = 24.50

Substitute x = 0

0.10y = 24.50

Evaluate the multiplication

y = 245

Divide both sides by 0.10

The x-intercept of 0.25x + 0.10y = 24.50 is (98, 0) and the y-intercept is (0, 245). Since the x-intercepts have a maximum value of 125, we can draw an x-axis up to 130 and count by 10. Since the y-intercepts have a maximum value of 245, we can draw a y-axis up to 250 and count by 50.

Bixia’s coin jar Number of dimes 200 150 100

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Finally, we’ll calculate the y-intercept of 0.25x + 0.10y = 24.50: 0.25x + 0.10y = 24.50

Second equation

0.25(0) + 0.10y = 24.50

Substitute x = 0

0.10y = 24.50 Evaluate the multiplication The x-intercept of x + y = 125 is (125, 0) and the y-intercept is (0, 125). y = 245 Divide both sides by 0.10 Next, we’ll calculate the x-intercept of 0.25x + 0.10y = 24.50: The x-intercept of 0.25x + 0.10y = 24.50 is (98, 0) and the y-intercept is (0, 245). 0.25x + 0.10y = 24.50 Second equation Since the x-intercepts have a maximum value of 125, we can draw an x-axis 0.25x + 0.10(0) = 24.50 Substitute y =a0maximum value of up to 130 and count by 10. Since the y-intercepts have 200 0.25x 245, we can draw a y-axis up =to24.50 250 and Evaluate count by the 50. multiplication x = 98

Divide both sides by 0.25

Finally, we’ll calculate the y-intercept of 0.25x + 0.10y = 24.50: 0.25x + 0.10y = 24.50

Second equation

Bixia’s coin jar Number of dimes

150 100

0.25(0) + 0.10y = 24.50 Substitute x = 0 The x-intercept of x + y = 125 is (125, 0) and the y-intercept is (0, 125). 0.10y = 24.50 Evaluate the multiplication Next, we’ll calculate the x-intercept of 0.25x + 0.10y = 24.50: y = 245 Divide both sides by 0.10 0.25x + 0.10y = 24.50 Second The x-intercept of 0.25x + 0.10y = 24.50 is (98, 0) equation and the y-intercept is (0, 245).

0.25x + have 0.10(0)a =maximum 24.50 value Substitute = 0can draw an x-axis Since the x-intercepts of 125,y we = 24.50 the multiplication 130 andthe count by 0.25x 10. the y-intercepts have a maximum value of cup to Interpret solution toSince the system of Evaluate equations. 245, we can draw a y-axis up 250 and Divide count by 50.sides by 0.25 x =to98 both

50 Number of quarters 10

Bixia’s coin jar

110

Number of dimes 200

a strategy Apply the idea Finally, we’ll calculate the y-intercept of 0.25x + 0.10y = 24.50: PurposeCreate 150 We students can use theto graph to determine the solution to the Challenge solve the system of equations using graph.to the system of equations is (80, 45). Since 0.25x + 0.10y = 24.50 Second equation Theasolution system and the axes labels to interpret the meaning of 0.25(0) + 0.10y = 24.50 Substitute x = 0

x is the number of quarters, we know that Bixia had 100

the point intersection. Since the solution to the system 80 quarters, and since y is the number of dimes, we know Reﬂecting with of students = 24.50 Evaluatetothe multiplication is not clear on the graph,0.10y we can use technology graph that Bixia had 45 dimes in the jar. Ask students how many solutions there will be for this system of equations.50 Ask them how they know this. the system of equations.

y = 245

Divide both sides by 0.10

x-intercept Students:The Page 190 of 0.25x + 0.10y = 24.50 is (98, 0) and the y-intercept is (0, 245).

Number Bixia’s coin jar of quarters 10 30 50 70 Number of dimes

Since the x-intercepts have a maximum value of 125, we can draw an x-axis Example 4 count by 10. Since the y-intercepts have a maximum value of up to 130 and 200 we can the draw a y-axistoup 250 and by 50. c245,Interpret solution thetosystem of count equations. Tyson is saving money in order to purchase a new smartphone for $800 when the latest model is released. 150 He currently has $350 saved up and is able to put away $100 each month.

110

Create a strategy

Apply the idea

a a system of equations to represent the to situation. We Write can use the graph to determine the solution the system and the axes labels to interpret the meaning of Create the pointaofstrategy intersection. Since the solution to the system is not clear on theof graph, we can use to graph To write a system equations, we willtechnology need to deﬁne the system of equations. some variables. Let’s choose y to represent an amount

The solution to the system of equations is (80, 45). Since 100 x is the number of quarters, we know that Bixia had Apply the idea 80 quarters, and since y is the number of dimes, we know 50 that Bixia hadvariables, 45 dimesthe in the jar. of money Tyson has Using these amount

of money (in dollars), and x to represent the number of months that have passed.

quarters saved over time can be represented by y =Number 350 +of100x. 10 can30be represented 50 70 90 by 110 The price of the smartphone y = 800.

Example c Interpret4the solution to the system of equations. Purpose Mathspace VirginiainSOL Algebra 1 190 Tyson is asaving money order to purchase a new smartphone for $800 when the latest model is released. Create strategy Apply the idea Show students how to interpret the solution to a system of equations in context. mathspace.co

He currently has $350 saved up and is able to put away $100 each month. We can use the graph to determine the solution to the The solution to the system of equations is (80, 45). Since a Write astudents system of labels equations to represent the situation. Reﬂecting with system and the axes to interpret the meaning of x is the number of quarters, we know that Bixia had the point of intersection. the solution the system the 80 quarters, andxsince the number of dimes,found we know Encourage students to checkSince the solution bytosubstituting values for andyyisinto the equations in part (a). Create a strategy thehad idea is not clear on the graph, we can use technology to graph Apply that Bixia 45 dimes in the jar. the system of equations. write a 190 system of equations, we will need to deﬁne Using these variables, the amount of money Tyson has Students:To Page some variables. Let’s choose y to represent an amount of money (in dollars), and x to represent the number of months that 4 have passed. Example

saved over time can be represented by y = 350 + 100x. The price of the smartphone can be represented by y = 800.

Tyson is saving money in order to purchase a new smartphone for $800 when the latest model is released. He currently has $350 up and1 is able to put away $100 each month. Mathspace Virginiasaved SOL Algebra 190 mathspace.co a Write a system of equations to represent the situation.

Create a strategy

Apply the idea

To write a system of equations, we will need to deﬁne some variables. Let’s choose y to represent an amount of money (in dollars), and x to represent the number of months that have passed.

Using these variables, the amount of money Tyson has saved over time can be represented by y = 350 + 100x. The price of the smartphone can be represented by y = 800.

190

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Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Purpose Check that students can write a system of equations for a given situation. Expected mistakes Students may be confused that one of the equations does not involve x as a variable. Remind students that the equations in a system only need to include one of the variables at least.

Students: Page 191 b Sketch the two lines representing these equations on the coordinate plane.

Create a strategy

Apply the idea

All of the values involved in the question are multiples Tyson’s cell phone savings of $50, so we can use this for the scale of the y-axis. Amount of money 900 Also, both x and y only make sense for positive values in this context, so we only need to think about the ﬁrst 800 quadrant. 700 b Sketch the two lines representing these equations on the coordinate plane. 600

Create a strategy All of the values involved in the question are multiples of $50, so we can use this for the scale of the y-axis. Also, both x and y only make sense for positive values in this context, so we only need to think about the ﬁrst quadrant.

500

Apply the idea

400

Tyson’s cell phone savings 300 200 Amount of money 900 100 Time (months) 800 1 2 3 4 5 6 7 8 −1 700 600 500

c If the new phone is to be released in 5 months’ time, determine if Tyson 400will be able to afford it on release.

Purpose Apply the idea Check that students can graph a system of equations.

The point of intersection on the graph occurs at (4.5, 800), meaning that in 4.5 months, Tyson will have Students:saved Page 191Therefore Tyson will have saved enough $800. money before the phone is released.

300

Reﬂect and 200 check 100 equation of Tyson’s savings is linear, in While the model Time (months) reality, he probably puts money away 1 2 3 4 5 6once 7 per 8 month or −1 once per week, depending on how often he gets paid.

So although the point of intersection is at x = 4.5 months, Tyson might not actually reach $800 in savings until the c If the new phone is to be released in 5 months’ time, determine if Tyson will be able to afford it on release. end of the 5th month.

Apply the idea

Reﬂect and check

The point of intersection on the graph occurs at While the model equation of Tyson’s savings is linear, in (4.5, 800), meaning that in 4.5 months, Tyson will have reality, he probably puts money away once per month or Example 5 saved $800. Therefore Tyson will have saved enough once per week, depending on how often he gets paid. money before phone is released. Gordiano madethe two trips to a flower shop to purchase rosesSo and sunflowers. On his trip, he purchased roses although the point of first intersection is at x = 4.54 months, and 4 sunflowers and paid $12. The following day, GordianoTyson went back theactually flower reach shop and purchased roses mighttonot $800 in savings12until the and 8 sunflowers for $16. Without graphing, determine the number of solutions. end of the 5th month.

Example 5 PurposeCreate a strategy Gordiano made two trips to a flower shop to purchase roses and first trip,tohesolve purchased 4 roses Check that interpret the solution for a system of sunflowers. equationsOn inhis context practical problems. First,students define thecan variables. and 4 sunflowers and paid $12. The following day, Gordiano went back to the flower shop and purchased 12 roses Based on the problem, we see that each trip to the flower shop is represented in its own equation. Gordiano buys

and with 8 sunflowers for $16. Without graphing, determine the number of solutions. Reflecting students 4 roses, then 12, so the expressions 4x and 12x indicate the cost for the roses. Gordiano buys 4 sunflowers, then Ask students if theysocan of and a way to check will have saved enough without checking the 8 sunflowers, the think terms 4y 8y indicate thewhether cost of theTyson sunflowers. point of Let intersection. Have students consider which line has the greater y-value at 5 months. Point out that we x = the cost per rose and let y = the cost per sunflower. can check Tyson has the enough money saved tothe buy the laptop at any pointform in time by checking Nowwhether that we have defined variables, we can convert equations to slope-intercept and determine the which line has Create the greater y-value at that time. number of solutions. Lines that have the same slope will always have one solution. Lines that have the same slope a strategy with different y-intercepts will have zero solutions, and parallel lines with the same y-intercept will have infinite First, define the variables. solutions. Based on the problem, we see that each trip to the flower shop is represented in its own equation. Gordiano buys 4 roses, then 12, so the expressions 4x and 12x indicate the cost for the roses. Gordiano buys 4 sunflowers, then 8 sunflowers, so the terms 4y and 8y indicate the cost of the sunflowers. Let x = the cost per rose and let y = the cost per sunflower.

4.01 Write and graph linear systems mathspace.co

191

4.01form Write and graph linear Now that we have deﬁned the variables, we can convert the equations to slope-intercept and determine the systems mathspace.co number of solutions. Lines that have the same slope will always have one solution. Lines that have the same slope with different y-intercepts will have zero solutions, and parallel lines with the same y-intercept will have infinite solutions.

391

The point of intersection on the graph occurs at (4.5, 800), meaning that in 4.5 months, Tyson will have saved $800. Therefore Tyson will have saved enough money before the phone is released.

Students: Pages 191–192

While the model equation of Tyson’s savings is linear, in reality, he probably puts money away once per month or once per week, depending on how often he gets paid. So although the point of intersection is at x = 4.5 months, Tyson might not actually reach $800 in savings until the end of the 5th month.

Example 5 Gordiano made two trips to a flower shop to purchase roses and sunflowers. On his first trip, he purchased 4 roses and 4 sunflowers and paid $12. The following day, Gordiano went back to the flower shop and purchased 12 roses and 8 sunflowers for $16. Without graphing, determine the number of solutions.

Create a strategy First, define the variables. Based on the problem, we see that each trip to the flower shop is represented in its own equation. Gordiano buys 4 roses, then 12, so the expressions 4x and 12x indicate the cost for the roses. Gordiano buys 4 sunflowers, then 8 sunflowers, so the terms 4y and 8y indicate the cost of the sunflowers. Let x = the cost per rose and let y = the cost per sunflower. Now that we have defined the variables, we can convert the equations to slope-intercept form and determine the number of solutions. Lines that have the same slope will always have one solution. Lines that have the same slope with different y-intercepts will have zero solutions, and parallel lines with the same y-intercept will have infinite solutions.

4.01 Write and graph linear systems mathspace.co

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Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

191

Purpose Challenge students to graph and solve a system of equations representing a real-world scenario, interpreting the solution in context. Expected mistakes Students may not recognize that a negative value for x for the solution is not viable in context. Reﬂecting with students Ask students why the solution to the system of equations is non-viable. Ask students what this solution might tell us about the real-world scenario. Perhaps the prices of the ﬂowers are not linear, or perhaps there was a sale on one of the days.

Students: Page 193

Idea summary The solution to a system of linear equations is the ordered pair of the point of intersection of the lines. Systems of equations may have one solution, no solutions, or infinitely many solutions. Graphing a system can help to determine the number of solutions a system will have: • • •

One solution: the lines have different slopes so they intersect at one point No solution: the lines have the same slope and different y-intercepts so they are parallel and will never intersect Infinitely many solutions: the lines have the same slope and y-intercept so they are the same line

Practice What do you remember? 1

For each graph, select the solution to the system of equations. a

A (−1, −7)

(−6, −7)

C (−7, −6)

(1, −6)

y 4 2

E No solution

x −8

−6

−4

−2

−2 −4

4.01 Write and graph linear systems −6 mathspace.co −8

393

Practice Students: Pages 193–197

What do you remember? 1

For each graph, select the solution to the system of equations. a

A (−1, −7)

(−6, −7)

C (−7, −6)

(1, −6)

y 4 2

E No solution

x −8

−6

−4

−2

−2 −4 −6 −8

A (−3, 0)

C (3, 5)

(−3, 5) 8

Inﬁnitely many solutions

E No solution

4 2

−8 −6 −4 −2 −2

−4 −6 −8

State the number of solutions for the systems of equations: a

y 6 5 4 3 2 1

−6−5−4−3−2 −1 −1

x 1 2 3 4 5 6

−2 −3 −4 −5 −6

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5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

6 5 4 3 2 1

3 2 1

−6−5−4−3−2 −1 −1

−2 −3 −4

Solve each system of equations graphed: y

4 2

10 12

−5 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7 −8

9 y

8 7 6 5 4 3 2 1

−3 −2 −1−1 −2 −3 −4 −5 −6 −7 −8 −9

8 y 7 6 5 4 3 2 1

−4 −2 −2 −4 −6 −8 −10 −12 −14 −16 −18

−4 −3 −2 −1 −1

1 2 3 4 5 6

−2 −3 −4 −5 −6

x 1

2 3 4 5

9 y 8 7 6 5 4 3 2 1

−9−8 −7−6−5−4−3−2 −1−1 −2 −3 −4 −5 −6 −7 −8 −9

1 2 3 4 5 6 7 8 9

x 1 2 3 4

Consider the equations y = 2x and y = 28 − 2x which have the following tables of values: y = 2x x

y = 28 − 2x

x 3 y 22

4 20

5 18

6 16

7 14

State the values for x and y which satisfy both equations. 5

Consider the equations x + y = 3 and y = x − 5 which have the following tables of values: x + y = 3 y = x − 5 x y

4 −1

5 −2

6 −3

x y

4 −1

5 0

6 1

State the values for x and y which satisfy both equations.

4.01 Write and graph linear systems mathspace.co

395

Consider the equations y = 4x − 11 and y = −2x + 13 which have this table of values:

x 2 3 4 5 6

State the values for x and y which satisfy both equations.

y = 4x − 11 −3 1 5 9 13

y = −2x + 13 9 7 5 3 1

Let’s practice 7

Use the graph to solve the systems of equations: 5 4 3 2 1 −2

−1

a 8

−1 −2 −3 −4 −5

x −2y = 4 2

y = −8 + 2x

Sketch the two lines representing the equations on the coordinate plane.

Solve the system of equations using the graph.

Consider the system of equations:

Sketch the two lines representing these equations on the coordinate plane.

State how many solutions this system of equations has.

Consider the equations y = −2x and y = −12 − 4x. Represent each function using a table of values.

Find the solution to the system of equations.

Solve the system of equations.

(−4, 8)

Consider the equations y = 3x and y = −35 − 4x. a

y=x−2

For each system of equations:

a 11

Represent each function using a table of values.

Consider the system of equations:

Select the solution to this system of equations.

396

(0, 0)

Inﬁnitely many solutions

(−4, 0)

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

No solution

Consider the system of equations:

Select the solution to this system of equations.

SOL

(0, −5)

No solution

(0, −8)

(1, −5)

The system of linear equations is graphed as shown.

9 8 7 6 5 4 3 2 1

What is the solution to this system of equations?

(0, −3)

(1, −4)

(2, −2)

(3, 0)

−9−8−7−6−5−4−3−2−1 −1 −2 −3 −4 −5 −6 −7 −8 −9

For each equation: i

Rewrite the equation as a system of equations.

Create a graph to represent each equation.

iii

Solve the system of equations.

x − 2 = −3x – 6

(−8, −5)

x 1 2 3 4 5 6 7 8 9

−5x − 22 = 7x + 38

Sarah is determining the dimensions of a rectangular frame. The length, l, of the frame is 5 cm more than the width, w. The perimeter of the frame is equal to 50 cm. Sarah came up with the following equations to solve for the length and width of the frame:

Determine the correct dimensions of the rectangular frame.

Interpret the solution in terms of the context.

A pair of linear equations has no solutions. One of the equations is y = −3x − 2. Determine if the following can be the other equation: a

y = −3x + 2

A system of linear equations has infinitely many solutions. One of the equations is y = 4x − 5. Determine if the following can be the other equation: a

y = −3x − 3

y = 8x − 10

y = 4x + 5

3y = 12x − 15

2y = 8x − 10

The sum of two mystery numbers is 4. The difference of the two numbers is −2. a

Write a system of equations to represent the situation.

Find the solution to the sytem of equations.

Rochelle and Mohamad are sister and brother. Rochelle’s age is 11 more than 4 times the age of Mohamad. The sum of their ages is 21. a

Write a system of equations to represent the situation.

Determine how old Rochelle and Mohamad are.

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397

Kang and Elvia are working on an assignment together. They have broken down the work into 9 parts of the same size. Kang works at 2 times the speed of Elvia but has 9 pieces of work to do for another subject. a

Write a system of equations to represent the situation.

Solve the system of equations and interpret the result in terms of the context.

Is it possible to have a pair of linear equations with exactly two solutions? Explain why or why not.

Consider the system of equations:

Without graphing, is there a solution? Explain your answer.

Let’s extend our thinking 24

Write a scenario to represent the system of equations and its solution. Explain what the solution to the system means in terms of the scenario.

18 16 14 12 10 8 6 4 2

Two equations, y1 and y2 represent the growth of two different house plants over time. Use the graph of y1 and y2 to support the claim that the two plants will never reach the same height on the same day.

x 2 4 6 8 10 12 14 16 18 20 22 24

25 20 15 10 5 −40 −30 −20 −10 −5 −10 y1 −15 −20 y2 −25

x 10 20 30

Describe a situation where it would not be reasonable to solve a system of equations by graphing.

Homer plans to start taking an aerobics class. Nonmembers pay $4 per class. Members pay a one-time $8 sign-up fee but only have to pay $2 per class. Should Homer join as a member or take classes as a nonmember? Create and analyze a model, then use it to justify your response.

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Answers

d i

y 6 5 4 3 2 y = −2x + 1 1

4.01 Write and graph linear systems What do you remember? 1 a B

b E

2 a 1

b Infinitely many

c 0

−6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6

d 1

3 a (4, −6)

c (6, −3)

b (−1, 3)

4 x = 7, y = 14

d (−4.5, -2)

y 6 5 4 5y −20 = 15x 3 2 1

6 x = 4, y = 5 Let’s practice b (0, −2)

7 a (6, 4) 8 a i

y = −x − 2

b Infinitely many solutions

2 3 4 y = 2x − 5

10 a Example answer: y = -2x

-8

-7

-6

-5

-4

y = -12 − 4x

ii (1, −3)

y = −4x − 4

1 2 3 4 5 6

−4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8 −9 −10

b i

−6 −5 −4 −3 −2 −1 −1 −2 y = 3x + 4 −3 −4 −5 −6

c (4, 0)

4 y 3 2 1

y = −x

ii (1, -1) 9 a

5 x = 4, y = -1

1 2 3 4 5 6

7 6 5 4 3 2 1

-8

-7

-6

-5

-4

b (−6, 12)

y = −4x + 5 x

−7−6−5−4−3−2 −1−1 −2 −3 −4 −5 −6 −7

1 2 3 4 5 6 7

11 a y = 3x y = -35 − 4x x x -5 -3 -1 -5 -3 y y -15 -9 -3 -15 -23

-1 -31

b (−5, −15) 12 D

ii No solution c i

6 5 4 3 2 1 −7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7

ii (-1, -3)

13 C

y y = 2x − 1

14 C 15 a i

x 1 2 3 4 5 6 y=x−2

iii (−1, −3)

ii y 5 4 3 2 1

−5 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 y = x − 2 −7 −8 −9

x 1 2 3 4 5

y = −3x − 6

Answers mathspace.co

399

Let’s extend our thinking

b i ii 7 y

iii (−5, 3)

6 5 4 y = −5x − 22 3 2 1

−7 −6 −5 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 y = 7x + 38 −7

The solution to the system (14, 5) means that after 14 days, Maria and Arjun have the same amount of rice remaining.

x 1

25 The point of intersection on the graph has negative x- and y-values. In terms of the height of a plant over time, only positive values make sense. Since the two lines do not intersect in the first quadrant, they will never reach the same height on the same day.

16 a l = 15 cm, w = 10 cm

b Yes

17 a Yes

b No

c No

d Yes

18 a No

b No

c Yes

d Yes

19 a Equation 1: y + x = 4

Equation 2: y − x = -2

b x = 3, y = 1 20 a Let x = Mohamad’s age and y = Rochelle’s age

Equation 1: y + x = 21

Equation 2: y = 4x + 11

b Rochelle is 19 years old and Mohamad is 2 years old. 21 a Let x = the number of parts Elvia completes and y = the number of parts. Kang completes.

Equation 1: x + y = 9

Equation 2: y = 2x - 9

b K ang did 3 parts of the assignment and Elvia did 6 parts. 22 No, it’s not possible. Two lines can either intersect at a single point (one solution), coincide resulting in infinitely many solutions, or be parallel (no solution). They can’t intersect at exactly two points. 23 No, the lines have the same slope, so they are parallel and will never intersect.

400

24 Answers will vary. For example: Maria has 15 lbs of rice and cooks 4 lbs of rice every 3 days. Arjun has 12 lbs of rice and cooks 1 lbs of rice every 2 days.

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26 If you need an exact solution to the system and the solution contains irrational numbers or fractions not easily readable on the scale of your graph, then solving the system by graphing is not feasible. 27 Let x = the number of classes taken and y = the total cost. We can use y = 4x to model the total cost for nonmembers and y = 2x + 8 to model the total cost for members. We can use the graph of the equations to analyze the models. 22 Total cost (Dollars) 20 18 16 14 12 10 y = 4x 8 6 4 y = 2x + 8 2 of classes taken Number −10 −8 −6 −4 −2−2

2 4 6 8 10

−4

Homer should pay the nonmember fee if he is only planning on taking 4 or less classes and if he decide he will take more than 4 classes, he should pay for the membership. The cost of the classes for being a nonmember will be cheaper than the member cost until the cost is the same, at $16 for 4 classes. After that, the nonmember cost will become more expensive so it makes sense for him to pay for a membership if he plans on taking 5 or more aerobics classes.

4.02 Substitution method Subtopic overview Lesson narrative In this lesson, students will solve systems of equations algebraically using the substitution method. Students should recognize the limitations of accuracy when solving systems graphically, compared to algebraic methods. Students will work with contextual situations involving systems of linear equations, where they will define variables, create and solve systems of equations, and interpret the meaning and viability of solutions in contextual situations.

Learning objectives Students: Page 198

Key vocabulary 

solution (to a system of equations)



substitution method

Essential understanding Systems of linear equations can be solved using many different methods. Substitution is an efficient method when at least one of the equations has an isolated variable.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving

MPG3 — Mathematical Reasoning

Teachers can engage students in problem solving by posing real-world situations that require creating and solving systems of equations. For example, teachers can present a scenario in which two different companies offer different pricing structures for a service, and students must create a system of equations to determine under what conditions each company’s offer is better. Teachers can then guide students through the process of using the substitution method to solve the system of equations and interpret the solution in the context of the problem.

Teachers can promote mathematical reasoning by guiding students to make connections between their current understanding of linear equations and the substitution method. Teachers can also encourage students to reason about the number of solutions (one, none, infinite) a system of equations might have, based on the results of the substitution method.

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MPG4 — Mathematical Connections Teachers can facilitate the making of mathematical connections by relating the concept of systems of equations to real-world situations. They can also help students see the connection between the graphical representation of a system of equations and its algebraic solution. For instance, teachers can discuss how the point of intersection of two lines in a graph corresponds to the solution of the system of equations.

Prior connections A.EO.1 — The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

A.EI.1 — The student will represent, solve, explain, and interpret the solution to multistep linear equations and inequalities in one variable and literal equations for a specified variable.

Future connections A2.EI.3 — The student will solve a system of equations in two variables containing a quadratic expression.

Engage Activity Concert ticket prices

60 mins

Students will determine ticket prices for a local concert given a set of parameters such as venue cost, discount presales, and proﬁt goals. Although it is not guaranteed students will know to solve by substitution on their own, the system lends itself to substitution which can be connected to the formal process of substitution.

Understanding and skills

Will use Solving linear systems of equations by graphing.

Will develop Writing a system of two-variable linear equations to represent constraints in a real-world context and interpret solutions.

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Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: Paper, pencil, graph paper, graphing technology

Support students with disabilities Support organization - solve multistep or complex problems Chunk the task into smaller parts. For example, spend 2 minutes picking a venue, 3 minutes determining ticket prices, 2 minutes determining how much money the band should make, 12 minutes determining how many of each ticket type they should sell to meet the proﬁt goals decided by the group, and the rest of the time writing up the formal mathematical model. You may have group members volunteer to be the time keeper for the activity or you can keep track of time for the whole class and remind them when they should be transition from one part of the work to the next. If you use slides or other visual displays during class time you can visibly display the chunked sections of time for students.

Support for English language learners Collect and display As pairs are working, listen for and collect vocabulary, phrases, and methods students use for writing equations and graphing their solutions. Consider grouping language for each part of the process (writing equations and solving equations), and be sure to include different methods for solving. Continue to update collected student language throughout the entire activity. Remind students to borrow language from the display as needed. Some terms and phrases may include: variables, coeﬃcients, intercepts, solutions, coordinates, linear, equation, equal, and values.

Classroom guide Hook Students compare the similarities and differences between two systems of equations.

Open questions

5 mins

What are the similarities and differences between the two systems of equations? 3 x + 10 8

Implementation details

Both systems are represented graphically since students are already familiar with solving a system by identifying the point of intersection of two lines.

Ask students to make a list of observations about each system and then compare the two lists. Students can break down their observations into those about the graphs and those about the equations, and if they are feeling conﬁdent in their understanding of the Hook they can also list observations about the connection between each graph and its corresponding equations.

•

y = 50 − 2 x

5 x + 8 y = 320

10 0 10 20 30 40 50 x

3 x + 10 8

0 10 20 30 40 50 x

Slide 1 from Student Engage Activity

A key observation that ties into the motivation behind solving systems algebraically is that one of the systems has an integer solution and thus can be easily solved by graphing, while the other system can be estimated, but is more diﬃcult to solve for exact values. Amplify student observations around this difference when possible.

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Launch Ask students to share about a concert or performance event they have attended or want to attend. Ask groups to discuss how much tickets cost for the event, and how ticket prices at different venues can vary. If any students have ever planned a concert or show, ask them to share about their experience and what work they had to do to ensure people could show up on the day of the event. Important mathematical concepts: System of equations, solution to a system of equations, and point of intersection. Important contextual information: Price, presale tickets, door tickets, deposit, discount, and venue.

5 mins

Felipe’s band, Diamonds Under Pressure, are renting a venue for their first show. These are the details for three venues in their area: Slim’s: $885 for three hours, maximum occupancy of 325 people

Seifert Center: Outdoor venue, $300 for 3.5 hours, maximum occupancy of 100 people

Caffiene Den: $285 for three hours, maximum occupancy of 60 people, only ages 16 and up

Slide 4 from Student Engage Activity

Suggested grouping: Form groups of 3 or 4 and assign roles.

Explore

Team roles

•

25 mins

Anticipated strategies Create a graph Students may wish to solve their system by graphing, since they are most familiar with this method. This can be done by hand or with graphing technology. Answers will vary depending on how much students chose to sell tickets for and which venue they selected. The following solution is for Slim’s if presale tickets cost $15 and door tickets cost $20: p = number of presale tickets, d = number of door tickets Solution: (11.8, 41.15) The band should sell about 12 presale tickets and 42 door tickets in order to make a proﬁt of $115 and cover the deposit on the venue.

48 d 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2

p 5 10 15 20 25 30 35 40 45 50 55 60 65

Solve algebraically The system lends itself to substitution method as one equation has an isolated quantity. Students may recognize that they can solve and then substitute this expression into the other equation, or may work algebraically to set the equations equal to each other before solving.

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Answers will vary depending on how much students chose to sell tickets for and which venue they selected. The following solution is for Slim’s if presale tickets cost $15 and door tickets cost $20: p = number of presale tickets, d = number of door tickets 15p = 177 15p + 20d = 1000 Solution: (11.8, 41.15) The band should sell about 12 presale tickets and 42 door tickets in order to make a proﬁt of $115 and cover the deposit on the venue. Guess and check Once students have determined their ticket prices, they may use a strategic guess and check method and test different combinations of ticket prices until they ﬁnd one that meets the criteria.

Misconceptions Interpreting the given information. What are the two unknown quantities we want to solve for? What requirements do we have for those quantities? How can we represent those requirements in a system of equations?

Purposeful questions Use the following questions to check for understanding and encourage critical thinking: • Which venue did your group choose to rent? • What do we want to determine in order to help the band? • How did you determine the number of each type of ticket? • What other mathematical model can you use to justify your decisions?

Continue when Students have created a system of equations with a solution that can be interpreted in the context of ticket sales. Students should also be able to show their method for solving, whether that be by graphing, algebraic manipulation, or some other method.

Discuss

25 mins

Have a whole class discussion. Consider sequencing responses by grouping visual representations (graphs and tables) together and algebraic representations (substitution method or guess and check) together and inviting students to make connections between the two strategies as well as discussing the pros and cons for each.

Discussion guide Once students have solved and presented their ﬁndings, you can begin to place emphasis on the various ways that groups chose to solve for the number of presale and door tickets. For groups who solved graphically, ask about how they created their graph and how they identiﬁed the solution from the graph. For groups that solved using an algebraic method, ask them to show their work and see if they can generalize their method. It is likely that the groups who solved algebraically either did so by guess and check or substitution, in which case you can emphasize the appropriate vocabulary and naming of the method they used if the students don’t already know what it is called.

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Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Algebra 1 — 1.04 Multistep equations Algebra 1 — 4.01 Write and graph linear systems

Tools You may ﬁnd these tools helpful: • Graphing calculator • Grid paper

Student lesson & teacher guide Substitution method Students will start by engaging in an exploration to build conceptual understanding of the substitution process.

Students: Page 198

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Develop algorithmic thinking with flowcharts Targeted instructional strategies Encourage students to think algorithmically by having them work in pairs to create a flowchart that could be used to solve any system of linear equation using substitution. Have one student work on a system of equations problem and have the other student be the documenter, asking question about the other student’s thinking and process, while taking notes. Provide the students with four different problems and have them switch roles with each problem: • One with a variable isolated and that has a unique solution, like: • One that needs rearranging and has a unique solution, like • One that needs rearranging and has no solution, like: • One that needs rearranging and has infinite solutions, like Then have the pair of students work together to create a flowchart. For students, who need some support, provide a starting point. For example: Start Input Equation 1 and Equation 2 Does one equation have an isolated variable? Yes Substitute the expression of the isolated variable into the other equation Is there only one variable remaining?

Does one equation have a variable with coefficient of 1? Yes Isolate the variable

Choose one of the equations to isolate one of the variables. Consider minimizing the number of steps or use of fractions. Isolate the variable.

There was an error. Go back to the original equations

Yes

Systems with infinite or no solutions Address student misconceptions Students may forget that a system of equations can have either inﬁnitely many or no solutions and instead think that they have done something wrong when reaching an equation for either of these scenarios. For example, a student getting the equation x + 2 = x − 4 after substituting may think they have substituted incorrectly because they cannot get any solutions for x. Remind students that a system of equations does not need to have only one solution. Justify this by connecting equations to graphs and showing some relevant examples (such as in the ‘collect and display’ support above).

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Collect and display English language learner support As students are working, note how students describe the concept of “substitution” and how they relate this to “no solutions” and “infinitely many solutions.” Collect the different ways that students find to understand these concepts and display them in a common place for the students to access. If students do not come up with alternative ways to word these concepts and are confused by them, suggest some of your own. For example: • Substitution • Replace variable with equivalent expression • Rearrange for y, then combine equations • Rearrange for x, then combine equations • No solutions • Same x-terms, different constants • Eliminates all the variables, but not all the constants • Simplifies to a not true equation • Parallel lines

3 2 1

−4 −3 −2 −1 −1

−2 −3 −4

• Inﬁnitely many solutions • Same x-terms and same constants • Simpliﬁes to 0 = 0 • Equations are equivalent • Coinciding lines

3 2 1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

Take care to address any rewordings that contradict or are too similar to other concepts that the student will learn in the future, such as ensuring students know that this only applies to systems of linear equations and nonlinear systems can intersect in other ways- like two solutions. For example, students are likely to rearrange equations to eliminate y in the substitution step, but this will not always be the most eﬃcient method.

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Visualizing the substitution method Student with disabilities support Help students who have diﬃculty manipulating algebraic expressions to understand how the substitution method works by providing a visual representation of how it works. For example, 3x − y = 7

y = 3x − 7

2x + 3y − 1 = 0 2x + 3(3x − 7) − 1 = 0 2x + 9x − 21 − 1 = 0 11x = 22 x=2

y = 3(2) − 7 y = −1

Highlight how we showed that y = 3x − 7 from Equation 1, which then lets us ‘substitute’ the expression into Equation 2 to replace y. Similarly, solving for x = 2 allows us to replace x with 2 to then solve for y in the ﬁnal steps.

Exploration Students: Page 198

Suggested student grouping: Small groups Students are challenged to discover the concept of substituting expressions from one equation into another as a means of solving a system. The initial problem requires students to recognize that the values for x and y are shared by both equations, and the follow-up problem challenges students to extend the idea of substitution beyond constants to include algebraic expressions. By the end of the exploration, students should be exposed to the idea that they can substitute the solution of one equation into the other, even when the solution is an algebraic expression.

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Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. How could you solve this without graphing? This system of equations can be solved by substituting the solution y = 9 (from the second equation) into the ﬁrst equation, by replacing y with 9 to get x + 3 (9) = 6 which is a single-variable equation. Solving the equation gives us the solution x = −21. Combining this with the solution for y from the second equation, we have the ﬁnal solution for the system of equations: x = −21, y = 9 2. Would that method still work if the second equation was y = 9 − 2x? This method would still work, except instead we are replacing y with 9 − 2x, which gives us the singlevariable equation x + 3 (9 − 2x) = 6 which can be solved by distributing the multiplication to expand the parentheses, collecting the like terms, and then isolating x. This gives the solution

, which we then substitute into either equation to ﬁnd the

solution for y (in the same way that we solved the previous problem). Purposeful questions • We have the solution y = 9 in the second equation. How does this relate to the value of y in the ﬁrst equation? • Using the information given, can we form an equation that only has one variable? • What happens if we substitute y = 9 − 2x into the ﬁrst equation? Possible misunderstandings • Students may have the misconception that they can only substitute constant values into an equation and thus think that the method cannot be applied in the second question. • Students may not substitute the value of y as a singular expression and thus forget to contain the whole expression inside a pair of parentheses.

Advanced learners: Introduction to algebraic proofs and proof by contradiction Targeted instructional strategies Introduce advanced learners to the concept of proving mathematical statements to give them a deeper understanding of the three types of algebraic solutions that are possible when solving a system algebraically. Teach them that proofs are a way of using definitions and proven facts to justify a assumption or hypothesis. When solving a system of equations, we can consider the problem to say, “Prove that there is a single solution, or xy-coordinate pair, that makes both equations true, simultaneously.” Then, we would begin our proof by saying, “Assume there is a single solution to the system of equations”, and proceed to solve the system algebraically until we find a single value for x and y, thus proving the assumption. For parallel lines or identical lines, we do not find a single value for x and y. Instead, we get an equation that is not true (meaning there is no solution to the system) or an equation that is true for all values of x and y (meaning there are infinitely many solutions to the system). These cases contradict the original assumption that there was a single solution, which is why we do not end up with a single value for x and y. Explaining this information can give students a deeper understanding of solutions that result in a = b, where a and b are different numbers or a = a, where a is a number. It can also prepare them for future work with algebraic and geometric proofs.

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After the exploration, students are guided through the procedure of substitution and the types of solutions they encounter are connected to the graphical solutions they have found previously. It is important that students understand the underlying concept of solving a system of equations algebraically- that we want to ﬁnd values for the variables that make all the equations in the system true. To ﬁnd these values, we must assume that the value of y is the same in both equations, which is what lets us substitute the equality from one equation into the other. If students do not understand where this assumption comes from, remind them that the solution(s) to a system of equations can be represented by their intersection on the coordinate plane. In other words, the solution(s) is exactly where both equations are true for the same pair of x- and y-values.

Students: Pages 198–199

The following steps can be used to solve a system of equations using substitution: Steps

Example x − y = 5, 2x + 3y = 6 x−y=5 x=5+y 2(5 + y) + 3y = 6 10 + 2y + 3y = 6

Given system 1. Isolate a variable in one of the equations 2. Substitute the resulting expression into the other equation

10 + 5y = 6 3. Solve the equation for the variable

5y = −4

4. Substitute the value into one of the original equations 5. Solve for the remaining variable 6. Write the solution as an ordered pair

Recall that a solution to a system of equations is the set of ordered pairs that make all equations in the system true and that a system of linear equations can have three types of solutions: y

1 −4 −3 −2 −1

−1

x 1

−2

−4 −3 −2 −1

−1

x 1

−2

−4 −3 −2 −1

−1

−3

−4

A system of linear equations with no solution.

Solving algebraically will result in an Solving algebraically will result in equations of the form x = a and y = b, equation of the form a = b, where a and b are different numbers. where a and b could be the same number.

−2

−4

A system of linear equations with one solution.

x 1

A system of linear equations with inﬁnitely many solutions. Solving algebraically will result in equations of the form a = a, where a is a number.

4.02 Substitution When the solution to a system of equations does not consist of integer values it is diﬃcult to determine the exact method mathspace.co solution by graphing, so solving algebraically using a method like substitution is necessary.

411

4. Substitute the value into one of the original equations 5. Solve for the remaining variable 6. Write the solution as an ordered pair

Recall that a solution to a system of equations is the set of ordered pairs that make all equations in the system true and that a system of linear equations can have three types of solutions: y

1 −4 −3 −2 −1

−1

x 1

−4 −3 −2 −1

−1

−2

x −4 −3 −2 −1

−1

−3

−4

A system of linear equations with no solution.

−2

−4

A system of linear equations with one solution.

x 1

A system of linear equations with inﬁnitely many solutions. Solving algebraically will result in equations of the form a = a, where a is a number.

When the solution to a system of equations does not consist of integer values it is diﬃcult to determine the exact solution by graphing, so solving algebraically using a method like substitution is necessary.

Examples The following support may be useful for the examples in this section.

Labeling equations Targeted instructional strategies The standard notation for a system of equations is by a large left curly bracket around all the equations. Substitution method 199 However, some students may not recognize this as indicating multiple equations. 4.02 Make sure to introduce mathspace.co students to the standard notation before they begin independent practice, so they are familiar with it. Encourage students to label the equations so they can keep track of and reference them when giving reasons for their work. This can also be very useful when teaching students a method for solving, as has been done for the examples. For example, the standard notation:

is equivalent to:

412

3x − y = 7

2x + 3y − 1 = 0

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Students: Page 200 Example 1 Solve the following systems of equations using the substitution method. a

Create a strategy We ﬁrst want to number our equations to make them easier to work with. 1

y = x + 11

y = 3x + 19

Since both equations already have y isolated, we can start by substituting equation 1 into equation 2 to eliminate y from the equation. We can then solve for x, and substitute this value back into one of the equations to solve for y.

Apply the idea Now we will use x = −4 to solve for y:

First we will solve for x: y = 3x + 19 x + 11 = 3x + 19

Equation 2

y = x + 11

Equation 1

Substitute y = x + 11

y = −4 + 11

Substitute x = −4

y=7

Evaluate the addition

11 = 2x + 19

Subtract x from both sides

−8 = 2x

Subtract 19 from both sides

−4 = x

Divide both sides by 2

So the solution to the system of equations is x = −4, y = 7 and written as (−4, 7).

Reflect and check Substitute the value of x and y back into both of the equations from the original system. Evaluate to check if both equations are true. Equation 1: y = x + 11

Write the first equation

7 = −4 + 11

Substitute x = −4, y = 7

7=7

Evaluate the addition

This is a true statement so the ordered pair is a solution to the first equation. Equation 2: y = 3x + 19

Write the second equation

7 = 3(−4) + 19 Substitute x = −4, y = 7 7 = −12 + 19

Evaluate the multiplication

7=7

Evaluate the addition

This is also a true statement so the ordered pair is a solution to the second equation. Since the ordered pair is a solution to both equations it is a solution to the system of equations.

Purpose Students demonstrate that they can substitute values from one equation into another. Both equations are given with y being isolated to direct students towards the most straightforward substitution. Reﬂecting with students Point out to students that we substituted Equation 1 into Equation 2. Ask if there would be a difference if we had 200 Mathspace Virginia SOL Algebra 1 substituted Equation 2 into Equation 1 instead. Make students aware that, while this question may not have a mathspace.co difference, there are systems where substituting one equation into the other will be more eﬃcient.

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Students: Page 201

Purpose Test students’ understanding of substitution, requiring them to use parentheses to appropriately substitute the value for x as a whole expression. Check that students can identify which variable will be easier to substitute, rather than defaulting to isolating y. Expected mistakes Students may not use parentheses when substituting the expression for x into the first equation. Help them see how that would lead to an incorrect solution graphically. Reflecting with students Discuss with students why solving for y would not be as efficient for this system. Point out that they would get the same answer, but it would require more steps.

Students: Page 201

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Purpose Remind students of the different possible types of solutions to a system of equations, with this example having no solutions. Reﬂecting with students Connect this system to students’ understanding of solving systems by graphing from the previous lesson. Graphically, this represents two parallel lines that never intersect. In other words, there are no points that would make both equations true, so there is no solution.

Checking solutions with back-substitution

use with Example 1

Targeted instructional strategies Remind students that after ﬁnding the solution to a system of equations, it is important to check their work. One effective method for doing this is back-substitution. This involves substituting the values of the variables back into the original equations to ensure they hold true. In the case of example 1b, where x = −14, y = −11, substitute these values back into the original equations: 2x − 3y = 5 2(−14) − 3(−11) = 5

Original equation 1 Substitute x = −14, y = −11

−28 + 33 = 5

Simplify

5=5

Simplify

As we can see, the value we computed for equation 1 holds true. We would then do the same for equation 2. If both equations hold true, then our solution is correct. Alternatively, students could check their work by graphing both equations and ﬁnding the point of intersection.

Students: Page 202 Example 2 The length of a rectangle is 3 inches less than twice its width. If the perimeter of the rectangle is 48 inches, ﬁnd the length.

Create a strategy For this real-world situation, we should start by deﬁning the variables, then build a system of equations to model the situation. The length of the rectangle in inches is unknown, so we can give it the variable l. Since the length and width make up the perimeter of the rectangle and the width in inches is also unknown, we can give it the variable w.

Apply the idea Based on the ﬁrst sentence in the problem, we know that the length is equal to 3 inches less than twice its width. The equation that models this relationship is l = 2w − 3. We know that the perimeter of a rectangle is equal to the sum of twice the length and twice the width, so the 4.02 Substitution method equation that models this relationship is 2l + 2w = 48. mathspace.co We can write our system of equations as:

415

Apply the idea Based on the ﬁrst sentence in the problem, we know that the length is equal to 3 inches less than twice its width. The equation that models this relationship is l = 2w − 3. We know that the perimeter of a rectangle is equal to the sum of twice the length and twice the width, so the equation that models this relationship is 2l + 2w = 48. We can write our system of equations as:

Solve for w:

Solve for l: 2l + 2w = 48

Second equation

l = 2w – 3

2(2w − 3) + 2w = 48

Substitute l = 2w – 3

l = 2(9) – 3 Substitute w = 9

4w − 6 + 2w = 48

Distributive property

l = 15

6w − 6 = 48

Combine like terms

6w = 54

Add 6 to both sides

w=9

Divide both sides by 6

First equation Evaluate the multiplication and subtraction

The length of the rectangle is 15 inches.

Reflect and check To confirm that our solution is correct, we need to check both the criteria provided in the problem. First, we verify that the length is 3 inches less than twice the width. We found the length to be 15 inches and the width to be 9 inches. “Twice the width” is 2 ⋅ 9 = 18 and “3 inches less” than that is 18 − 3 = 15. So this criteria is satisfied. Next, we verify that the perimeter is 48 inches. Remember perimeter is found by adding all of the sides (both lengths and both widths for a rectangle). 9 + 9 + 15 + 15 = 48 so this criteria is also satisfied.

Purpose This example demonstrates how we can model a problem with a system of equations, which can then be solved to determine the solution to the problem. Reﬂecting with students For contextual questions like this, students should also check their solutions for reasonableness. Are both the length and width positive values? Is the length greater than the width? This is an important skill for interpreting algebraic solutions to real-world problems.

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Example 3 A theater club at a high school charges a student rate and an adult rate to attend the spring musical. The cost for a student ticket is $3 and the cost for an adult ticket is $7. If 200 people attended the show and the theater club at the school raised $700, determine how many students and how many adults attended.

Create a strategy A system of equations will help us solve for the unknown variables, which are the number of students, x, and the number of adults, y. Based on the total amounts provided in the problem, we will write two equations in standard form to model the situation and solve the system to ﬁnd the solution.

Apply the idea A system of equations that models the situation follows:

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To solve the system using the substitution method, at least one equation must have an isolated variable. We can Mathspace Virginia SOL Algebra 1 Teacher Edition easily isolate either x or y in the ﬁrst equation because they do not have coeﬃcients. mathspace.co x + y = 200 First equation x = 200 – y

Subtract y from both sides

Apply the idea A system of equations that models the situation follows:

To solve the system using the substitution method, at least one equation must have an isolated variable. We can easily isolate either x or y in the ﬁrst equation because they do not have coeﬃcients. x + y = 200 x = 200 – y

First equation Subtract y from both sides

We will substitute the expression 200 − y for x in the second equation to solve for y. 3x + 7y = 700 3(200 − y) + 7y = 700

Example 3 600 − 3y + 7y = 700

Second equation Substitute x = 200 − y Distributive property

4y = 100 Combine like terms A theater club at a high school charges a studentboth ratesides and an rate to attend the spring musical. The cost for a y = 25 Divide by adult 4 student ticket is $3 and the cost for an adult ticket is $7. If 200 people attended the show and the theater club at the Now weraised will solve fordetermine x using y how = 25.many students and how many adults attended. school $700, x + y = 200

First equation

Create a strategy x + 25 = 200

Substitute y = 25 A system of equations will help us solve for the unknown which are the number of students, x, and the x = 175 Subtract 25 fromvariables, both sides number of adults, y. So the solution to the system of equations is x = 175, y = 25 and written as (175, 25). Based on the context, this means Based the totalattended amountsthe provided the problem, will attended. write two equations in standard form to model the that 175on students spring in musical and 25 we adults situation and solve the system to ﬁnd the solution.

Apply the idea

Idea summary of equations that models the situation follows: PurposeA system systems of equations using therequires substitution method leads to one three solutions: This example Solving presents a common set-up that students to solve a of system of equations. •

One solution: When solving algebraically an equation of the form x = a or y = b is reached

• No solutions: When solving algebraically an equation of the form a = b is reached Expected mistakes To solve• the systemmany usingsolutions: the substitution method,algebraically at least one an equation must have an isolated variable. We can Infinitely When solving equation of the a = a is reached Students might not understand how to model the situation using a system ofform equations. Help them see that two easily isolate either x or y in the first equation because they do not have coefficients. equations are needed: one for the total number of people and one for the total amount of money raised. x + y = 200 First equation Students may not interpret the solution in the context of the problem. Highlight the importance of clearly defining x = 200 – y Subtract y from both sides their variables at the beginning of the problem, which can help them with interpreting the solutions in context. We will substitute the expression 200 − y for x in the second equation to solve for y. 3x + 7y = 700

Choosing ‘nice’ 3(200 − variables y) + 7y = 700

Second equation Substitute x = 200 − y

Targeted instructional strategies 600 − 3y + 7y = 700 Distributive property

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4y = into 100 a system Combine terms Converting a contextual problem of like equations can be confusing for students that are not y = 25 Divide both sides by 4 comfortable with this degree of abstraction. To alleviate the confusion, it can be helpful to select variables that will solve for x using = 25. studentsNow canwe easily identify, suchy as using s for the number of students and a for the number of adults. x + yquestion, = 200 First students equation to choose variables that they ﬁnd suitable when writing Unless speciﬁed by the encourage 200 Substitute y = 25 real-world problemsx +as25a =system of equations. x = 175

Subtract 25 from both sides

So the solution to the system of equations is x = 175, y = 25 and written as (175, 25). Based on the context, this means

Students:that Page 203 175 students attended the spring musical and 25 adults attended.

Idea summary Solving systems of equations using the substitution method leads to one of three solutions: • • •

One solution: When solving algebraically an equation of the form x = a or y = b is reached No solutions: When solving algebraically an equation of the form a = b is reached Inﬁnitely many solutions: When solving algebraically an equation of the form a = a is reached

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Practice Students: Pages 204–207

What do you remember? 1

Select the solution to this system of equations. a

(−1, −5)

(1, 5)

(−5, −1)

(5, 1)

(−8, −2)

(−2, 2)

(−2, −8)

(2, −2)

A 2

State whether these statements are true or false. a

The method of substitution involves isolating one variable in one equation and substituting this expression into the other equation.

The substitution method can be used to solve any system of linear equations.

Once you solve one of the equations for one variable, you must substitute it back into the first equation.

Solving a system by substitution will always result in the same solution, regardless of which variable you solve for first.

A system of linear equations can have exactly two solutions, no solutions, or infinitely many solutions.

Solve the following systems of equations by substitution: a

Identify the number of solutions for each system of equations: a

A group of students were solving the equations:

Tomi substituted x from the first equation into the second.

Carla substituted y from the second equation into the first.

Regie substituted y from the first equation into the second.

Junard substituted x from the second equation into the first.

Write the equation that each student would end up with and comment on the efficiency of solving each. 6

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If you solve a system of linear equations by substitution and find that the result is a false statement such as 0 = 5, what does that tell you about the equations? Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

What does it mean if you solve a system of equations by substitution and get a true statement, such as 0 = 0 or 5 = 5?

Let’s practice 8

Solve the following systems of equations by substitution: a

Solve the equation 3x − 7y = 4 for x.

Substitute your expression for x from part (a) into −12x + 28y = −16 and solve for y.

Solve the system of equations using the substitution method:

Verify each solution by graphing.

For each system of equations: a

Find the solution to the system.

Check the solution algebraically.

For this system of linear equations:

iii

For the system of equations:

Explain the steps you would take to solve the system of equations and why you would do it in that way. 13

Do the equations y = 3x + 5 and y = 3(x + 5) have infinitely many solutions? Explain your answer.

A car rental agency offers two types of cars for rent: compact cars and SUVs. Today, the agency rented out a total of 50 cars, generating $3000 in revenue. The rental fee for a compact car is $40 per day, and the rental fee for an SUV is $60 per day. a

Write a system of equations to represent the context.

How many compact cars and how many SUVs were rented today?

Judy and Jorge both walk from their houses to the bus stop every morning. Judy walks 1.1 mi further, and together they walk 3.1 mi. Find the distance that Jorge walks.

At a bookstore, Stella purchased a novel and a magazine for a total cost of $59. The price of the novel, n, is $10 less than twice the price of the magazine, m. This situation can be represented by the following system of equations:

Find the solution to the system of equations.

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The total cost of 5 rulers and 3 books is $15.30. If the cost of a ruler is a and a book is $2.70 more expensive, find the cost of one ruler.

A rectangular garden bed has a perimeter of 13.8 m. The length of the garden bed is 2.5 m longer than the width. a

Find the width of the garden bed.

Find the length of the garden bed.

Rustam bought some fresh produce from the farmers’ market. He picked up 4 oranges and 5 plums. The total cost of Rustam’s fruit was $13.33. Valentina went to the same shop and bought 1 orange and 1 plum. The total cost of Valentina’s fruit was $2.89. a

Write a system of equations where f represents the price of an orange and g represents the price of a plum.

Solve for the prices of oranges and plums at the farmers’ market.

A man is ﬁve times as old as his son. Four years ago the man was thirteen times as old as his son. a

Write a system of equations where x represents the age of the man and y represents the age of his son.

Solve for x and y.

Determine whether the solution makes sense in terms of the context. Explain your answer.

The number of new jobs created in Miami varies greatly each year. The number of jobs created in 2013 was 440 000 less than double the number of jobs created in 2004. This is equivalent to an increase of 10 000 jobs created from 2004 to 2013. a

Write a system of equations to represent the context.

Find the solution to the system of equations.

Determine whether the solution is viable in terms of the context. Explain your answer.

Valentina has $2000 to invest, and wants to split it up between two accounts: Account A earns 8% annual interest, while Account B earns 9% annual interest. Her target is to earn $177 total interest from the two accounts in one year. a

Write a system of equations to represent the context.

Solve the system of equations.

State if you would make the same investment as Valentina. Explain your answer.

The function f (x) = 0.47x + 8.9 represents the US annual bottled water consumption (in billions of gallons) and the function g(x) = −0.17x + 14.2 represents the US annual soda consumption (in billions of gallons). For both functions, x is the number of years since 2009. a

Determine the year in which the bottled water and soda consumption in the US is the same.

Determine whether the solution is viable in terms of the context. Explain your answer.

Let’s extend our thinking 24

A pizza shop offers two types of pizzas: Margherita and Pepperoni. Today, the shop sold a total of 100 pizzas, generating $2000 in revenue. The price for a Margherita pizza is $20, and the price for a Pepperoni pizza is also $20. How many Margherita pizzas and how many Pepperoni pizzas were sold today?

At Soul Food Express you can buy two orders of oxtail stew and a slice of sweet potato pie for $38.49, or you can get ﬁve orders of oxtail stew and three slices of sweet potato pie for $99.22. Construct a system of equations to model the situation and use it to determine the cost of 4 oxtail stews and 2 slices of sweet potato pie.

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Edilene has to read a book for class and only has 8 days before the test. So far, Edilene has read 90 pages. She is planning on reading 15 pages each day between now and the test. The book is 215 pages long. a

Construct a system of equations to model the situation.

Construct models to represent the system of equations and determine whether she will ﬁnish her book before the test.

Create a scenario for each system of equations where the solution is viable in terms of the scenario. Explain your answer. a

Shufang spent $55.25 to purchase 9 flowers. He bought two different types of flowers. Rhododendron cost $6.45 each and chrysanthemums cost $5.75 each. Construct a system of equations to model the scenario and use the model to determine how many of each type of flower Shufang purchased.

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Answers

10 a i (7, 4) ii

c m = 7, n = −2

d p = 9, q = −2

14 13 12 11 10 9 8 7 6 5 4 3 2 1

e x = −3, y = −4

y = −5, z = 3

−4−3−2 −1−1

4.02 Substitution method What do you remember? 1 A 2 a True

b D b True

c False

d True

e False 3 a x = 1, y = 2

4 a No solution.

b a = 5, b = 9

Infinitely many solutions.

5 Answer varies: Tomi will get y = 2(6 − y) − 3, which simplifies to y = 12 − 2y − 3, then finally 3y = 9. This approach is efficient as it results in a simple equation where we can solve for y directly.

(7, 4)

x 1 2 3 4 5 6 7 8 9 10 11 12 13 14

−2 −3 −4

b Infinitely many solutions.

c Infinitely many solutions. d No solution. e No solution.

b i (−6, 5) ii

b Answer varies: Carla will get x + 2x − 3 = 6, which simplifies to 3x − 3 = 6, then finally 3x = 9. This approach is also efficient because it results in a simple equation where we can solve for x directly.

(−6, 5)

c Answer varies: Regie will get 6 − x = 2x − 3, which simplifies to 3x = 9. This approach is efficient because it results in a simple equation where we can solve for x directly.

−14−13−12−11−10−9 −8 −7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4

d Answer varies: Junard will get

6 If the process of substitution leads to a false statement, it indicates that the system of equations has no solutions. In other words, the lines represented by the equations do not intersect at any point. 7 If you get a true statement, it indicates that the system of linear equations has infinitely many solutions. This could mean that the equations represent the same line. Let’s practice 8 a a = 11, b = 6 or (11, 6) b m = −3, n = 5 or (−3, 5) c x = −8, y = −3 or (−8, −3)

11 i

x 1 2 3 4

a x = 6, y = −5 or (6, −5) 5x + 4y = 10

Write the first equation

5(6) + 4(−5) = 10

Substitute x = 6, y = −5

30 − 20 = 10

10 = 10

3x + 7y = −17

3(6) + 7(−5) = −17

18 − 35 = −17

−17 = −17

b Infinitely many solutions

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Evaluate the multiplication Evaluate the subtraction Write the second equation Substitute x = 6, y = −5 Evaluate the multiplication Evaluate the subtraction

ii a x = 2, y = 3 or (2, 3) 4x + 3y = 17

Write the first equation

4(2) + 3(3) = 17

Substitute x = 2, y = 3

8 + 9 = 17

17 = 17

9x − 4y = 6

9(2) − 4(3) = 6

d p = 6, q = 4 or (6, 4)

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which simplifies to

y + 3 + 2y = 12, then finally 3y = 9. This approach is slightly less efficient because the equation involves a fractional coefficient, which can be a bit harder to work with. Once the fraction is handled the resulting equation is also simple.

9 a

14 13 12 11 10 9 8 7 6 5 4 3 2 1

18 − 12 = 6

6=6

Evaluate the multiplication Evaluate the addition Write the second equation Substitute x = 2, y = 3 Evaluate the multiplication Evaluate the subtraction

iii a f = 0.4, g = 1.2 or (0.4, 1.2) b

−7f + 2g = −0.4

Write the first equation

−7(0.4) + 2(1.2) = −0.4

Substitute f = 0.4, g = 1.2

−2.8 + 2.4 = −0.4

Evaluate the multiplication

−0.4 = −0.4

−2.1f + g = 0.36

Evaluate the addition Write the second equation

−2.1(0.4) + 1.2 = 0.36

Substitute f = 0.4, g = 1.2

−0.84 + 1.2 = 0.36

Evaluate the multiplication

0.36 = 0.36

Evaluate the addition

16 a m = 23 and n = 36 or (23, 36) b The magazine costs $23 and the novel costs $36. 17 a = $0.90 18 a 2.2 m

b 4.7 m

19 a b f = $1.12, g = $1.77 20 a b x = 30, y = 6

iv a b

Write the first equation

Substitute c = , d =

Evaluate the multiplication

Evaluate the addition

Write the second equation

Substitute c = , d =

Evaluate the multiplication

Evaluate the addition

12 Example explanation: In this case, it may be easier to solve for y in the first equation and then substitute it into the second equation. This would leave us with an equation in terms of x, so we could then solve for x. We would then substitute the value of x and solve for y using either equation. Finally, we could substitute the values of x and y into both original equations to check our answer. This would avoid fractions and unnecessary rearranging. 13 No. If we substitute one equation into the other, we do not end up with a true statement: 3 (x + 5) = 3x + 5 3x + 15 = 3x + 5 15 ≠ 5

c Y es. It is possible for a man to be 30 years old and have a son who is 6 years old. Four years ago, their ages were 26 years old and 2 years old. These are valid values for their ages. 21 a b x = 4 50 000, y = 4 60 000 c Y es, both values make sense in the context. That is, it is possible to have created 450 000 and 460 000 jobs, respectively. 22 a b x = $300, y = $1700 c V alentina’s choice will not yield the highest return. It would be better to invest all $2000 in Account B since it earns more interest than Account A. 23 a 2017 b Y es. It is possible that in 2017 people consumed over 12 billion gallons each of bottled water and soda. Let’s extend our thinking 24 Since both types of pizzas have the same price, there are an infinite number of combinations of Margherita pizzas and Pepperoni pizzas that could have been sold to total 100 pizzas and $2000 in revenue. For example, all 100 pizzas could have been Margherita pizzas, or all 100 could have been Pepperoni pizzas, or any combination in between. 25 Let x = the cost of an order of oxtail stew and y = the cost of a slice of sweet potato pie.

Since this is always false, the two equations would not have any solutions. 14 a b The agency rented out 0 compact cars and 50 SUVs.

Since oxtail stew costs $16.25 and sweet potato pie costs $5.99, an order of four oxtail stews and two slides of sweet potato pie costs $76.98.

15 1 mi

Answers mathspace.co

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26 a Let x = the number of days and y = the number of pages.

225 210 195 180 165 150 135 120 105 90 75 60 45 30 15

b A ny answer where negative integers are valid for both the independent and dependent variable. For example, a situation where x is the number of days before or after a specific date, and y is the outside temperature in degrees Fahrenheit.

y y = 215

y = 15x + 90

x 1

27 a A ny context where positive fractions are valid solutions. For example, the weights of items (such as fruit).

2 3 4 5 6 7 8 9 10

Based on the graph, Edilene will not finish her book before the test.

c A ny answer where the independent variable can be a negative integer and the dependent variable can be a positive integer. For example the daily balance in two different bank accounts based on different savings and spending habits. 28 Let R = the total number of rhododendron and C = the total number of chrysanthemums.

Shufang purchased 5 rhododendron and 4 chrysanthemums.

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4.03 Elimination method Subtopic overview Lesson narrative In this lesson, students will develop and use another strategy for solving systems of linear equations in two variables: the elimination method. Students will construct and solve models of systems of equations based on real-world scenarios. Mathematical reasoning is required to discover that multiplying an equation by a constant does not change it’s graph and the sum of two equations will produce an equation whose graph intersects at the same point as the solution to the original system of equations, and students will be able to use this understanding to identify equivalent systems of equations and justify why the elimination method works to solve a system of equations. Students will be able to analyze the structure of a system of equations and choose the most efficient method of solving and be able to explain the reasoning for their choice. By the end of the lesson, students will be able to create systems of linear equations to solve and interpret contextual problems.

Learning objectives

4.03 Elimination method

Students: Page 208

After this lesson, you will be able to… • find solutions to a system of equations algebraically using the elimination method. • interpret solutions to a system of equations in a real-world context. • determine the number of solutions to a system of equations. • create systems of equations to represent constraints from a real-world context.

The elimination method The elimination method is an algebraic method for solving systems of equations where like terms are aligned. Key vocabulary 

In this system, the equations are aligned: elimination method

3x + 4y = 12 x + y = 11

and in this system, the equations are not aligned: Essential understanding 3x = 12 − 4y Systems of linear equations can be solved using many different methods. Elimination is an efficient method when the x + y = 11 coefficients are not equal to one. Elimination method A method of solving a system of equations by adding or subtracting the equations until only one Standards

variable remains This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals

Exploration

MPG4 — Mathematical Connections Begin bycreate graphing the system of equations and identifyingbythe solution. the student’s prior knowledge of Teachers can mathematical connections for students connecting + 2y = 10 method. They can also help students see the solving linear equations and polynomial operations to the3xelimination connections between algebra and geometry by discussingx the of the solution in the context of graphs of + 2ymeaning =8 linearNext, equations. perform each operation using the original equations and consider the result. What do you notice? Record your observations. 1.

Divide the second equation by 2 and regraph the system. What do you notice?

Multiply the ﬁrst equation by 2 and regraph the system. What do you notice?

Add the equations together and regraph the system. What do you notice?

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MPG5 — Mathematical Representations Teachers can show students how to represent systems of linear equations using a variety of forms of linear equations, such as slope-intercept, point-slope, and standard form. They can also guide students in using different representations such as graphs or algebraic equations to solve real-world problems using the elimination method. For example, teachers can demonstrate how to use a graphing calculator or online graphing tools to verify the solutions found using the elimination method. Additionally, teachers can emphasize the importance of interpreting the solutions in the context of the problem, thus making the connection between the mathematical representation and its real-world meaning.

Prior connections A.EO.2 — The student will perform operations on and factor polynomial expressions in one variable.

A.EI.1 — The student will represent, solve, explain, and interpret the solution to multistep linear equations and inequalities in one variable and literal equations for a specified variable.

Future connections A2.EI.3 — The student will solve a system of equations in two variables containing a quadratic expression.

Tools You may ﬁnd these tools helpful: • Graphing calculator • Ruler • Blank coordinate plane 426

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Student lesson & teacher guide The elimination method Students learn the elimination method for solving a system of equations and also how to recognize when two equations have like terms aligned. Students then explore how different operations can affect a system of equations and be used to solve it.

Students: Page 208

4.03 Elimination method After this lesson, you will be able to… • find solutions to a system of equations algebraically using the elimination method. • interpret solutions to a system of equations in a real-world context. • determine the number of solutions to a system of equations. • create systems of equations to represent constraints from a real-world context.

The elimination method The elimination method is an algebraic method for solving systems of equations where like terms are aligned. In this system, the equations are aligned: 3x + 4y = 12 x + y = 11 and in this system, the equations are not aligned: 3x = 12 − 4y x + y = 11 Elimination method A method of solving a system of equations by adding or subtracting the equations until only one variable remains

Exploration Misaligning equations Begin by graphing the system of equations and identifying the solution. Address student misconceptions

3x + 2y = 10

x +which 2y = 8 can lead to incorrect combining of the Students may misalign terms between the two equations each operation using the original equations and consider the result. What do you notice? Record equations.Next, For perform example, if the two equations are x + 4y = 7 and 2x = 4y − 9 students may misalign them as your observations.

x + 4y = 7 1.

Divide the second equation by 2 and regraph the system. What do you notice?

2x =the 4yﬁrst − 9equation by 2 and regraph the system. What do you notice? Multiply

3. Add the equations and regraph the system.from What the do you notice? which can tempt students to try together and subtract one equation other without any rearranging. 4.

Multiply the ﬁrst equation by −1 and add the equations together. Graph the new equation with the

Multiply the second equation by 3 and add the equations together. Graph the new equation with the system. What do you notice?

Multiply the second equation by −3 and add the equations together. Graph the new equation with the system. What do you notice?

Remind students to always align equations using the = sign, so that they can clearly match the left- and system. What do you notice? right-hand sides of both equations.

What happens to the solution to a system of linear equations if one equation is multiplied by a number and then added to the other equation?

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427

Explicitly teaching which elimination approach is appropriate Targeted instructional strategies Demonstrate for students the different operations that can be applied when using the elimination method and explain why each method is appropriate for the situation. • When both equations have a variable with the same coeﬃcient with the same sign, we can subtract one equation from the other. For example: 1 3x + y = 4 2 x + y = 11 Subtracting equation 2 from equation 1 gives (3x + y) − (x + y) = 4 − 11 ⟶ 2x = −7 or

• When both equations have a variable with the same coeﬃcient with opposite signs, we can add the equations together. For example: 1

3x − y = 4

x + y = 11

Adding the equations together gives (3x − y) + (x + y) = 4 + 11 ⟶ 4x = 15 or

• When the coeﬃcient of a variable in one equation is a multiple of the coeﬃcient of that variable in the other equation, we can scale one equation and then combine the equations. For example: 1

3x − y = 4

x + 2y = 11

Multiplying equation 1 by 2 gives 3x − y = 4 ⟶ 6x − 2y = 8 which now has the same coeﬃcient for y as equation 2. • When neither equation has any coeﬃcients of a variable as a multiple of the other, we scale both equations using a lowest common multiple. For example: 1

2x + 5y = 7

3x − 4y = 10

Multiplying equation 1 by 3 gives 2x + 5y = 7 ⟶ 6x + 15y = 21 and multiplying equation 2 by 2 gives 3x − 4y = 10 ⟶ 6x − 8y = 20 so both equations now have the same coeﬃcient for x.

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Collect and display English language learner support

4.03 Elimination method

As students are working, note how students describe the concept of “elimination” and how it relates to “adding and subtracting equations” and to “coefficients”. Collect the different ways that students find to understand these concepts and display them in a common place for the students to access. If students do not come withyou alternative After this up lesson, will be ableways to… to word these concepts and are confused by them, suggest some of your own. For example: • find solutions to a system of equations algebraically using the elimination method. • Elimination • interpret solutions to a system of equations in a real-world context. • determine the number of solutions to a system of equations. • Want the same or opposite coefficients • create systems of equations to represent constraints from a real-world context. • Combining equations • Get rid of a variable • Adding/subtracting equations The elimination method • Add/subtract the left-hand sides together and the right-hand sides together The elimination method is an algebraic method for solving systems of equations where like terms are aligned. • Combine the equations In this system, the equations are aligned: • Combine like terms 3x + 4y = 12

Take care to clear up any misconceptions that students may form while coming up with their own deﬁnitions. x + y = 11 For example, adding and subtracting equations is only equivalent to adding or subtracting coeﬃcients if both and in this system, the equations are not aligned: equations have their variable termed aligned. 3x = 12 − 4y

x + y = 11

Exploration Elimination method A method of solving a system of equations by adding or subtracting the equations until only one

Students: Page 208 variable remains

Exploration Begin by graphing the system of equations and identifying the solution. 3x + 2y = 10 x + 2y = 8 Next, perform each operation using the original equations and consider the result. What do you notice? Record your observations. 1.

Divide the second equation by 2 and regraph the system. What do you notice?

Multiply the ﬁrst equation by 2 and regraph the system. What do you notice?

Add the equations together and regraph the system. What do you notice?

Multiply the ﬁrst equation by −1 and add the equations together. Graph the new equation with the system. What do you notice?

Multiply the second equation by 3 and add the equations together. Graph the new equation with the system. What do you notice?

Multiply the second equation by −3 and add the equations together. Graph the new equation with the system. What do you notice?

What happens to the solution to a system of linear equations if one equation is multiplied by a number and then added to the other equation?

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Suggested student grouping: In pairs Students are presented with a system of linear equations and instructed to perform various operations with them to investigate how multiplying equations by a constant doesn’t affect the solution of the system and that adding the equations of a system together can help find the solution if the coefficients for a variable cancel each other out. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Divide the second equation by 2 and regraph the system. What do you notice? After dividing the second equation by 2 and regraphing the system, we can see that the second equation did not change graphically. The only thing that changed was the algebraic representation of the second equation. Dividing the entire equation by a constant doesn’t change its graph. 2. Multiply the first equation by 2 and regraph the system. What do you notice? After multiplying the first equation by 2 and regraphing the system, we can see that the first equation did not change graphically. The only thing that changed was the algebraic representation of the first equation. Multiplying the entire equation by a constant doesn’t change its graph. 3. Add the equations together and regraph the system. What do you notice? Adding the equations together resulted in the equation 4x + 4y = 18 The graph of this equation intersects at the same point as the original two equations. 4. Multiply the first equation by −1 and add the equations together. Graph the new equation with the system. What do you notice? Multiplying the first equation by −1 and adding the equations together resulted in the equation −2x = −2 which can be simplified to x = 1. This is a vertical line that passes through the point of intersection of the original system of equations. We now know the value of x that solves the system of equations. 5. Multiply the second equation by 3 and add the equations together. Graph the new equation with the system. What do you notice? Multiplying the second equation by 3 and adding the equations together resulted in the equation 6x + 8x = 34 This equation intersects with the solution to the original system of equations, but does not help us know the value of either x or y. 6. Multiply the second equation by −3 and add the equations together. Graph the new equation with the system. What do you notice? Multiplying the second equation by −3 and adding the equations together resulted in the equation −4y = −14 which has a distinct solution of

. The graph of this equation is a horizontal line that passes through the

intersection of the original two graphs at . We now know the value of y that solves the system of equations. Purposeful questions • Does adding the equations together change the solution to the system of equations? • What is required for the resulting sum equation to give part of the solution to the system of equations? • Does multiplying or dividing an equation in a system by a constant change the solution? Possible misunderstandings • Students may not realize that the different operations are meant to be applied to the original system of equations and instead apply the operations sequentially.

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Advanced learners: Justifying efficient methods for solving various systems Targeted instructional strategies Encourage advanced learners to critically analyze when graphing, substitution, or elimination is the most effective method for solving a system of linear equations. Facilitate discussions where students compare the efficiency of each method based on different types of systems they encounter. Have them consider factors such as the simplicity or complexity of the coefficients, the ease of isolating variables, and the practicality of graphing with precise accuracy. By justifying their choices and explaining their reasoning, students will deepen their understanding of the methods and develop strategic problem-solving skills.

Students: Page 209

Examples Students: Pages 209–210

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Purpose Show students how to prove that two different systems of equations are equivalent by graphing and comparing them. Expected mistakes Students may think that the two systems are not equivalent because the equations have different numbers in them. Remind students that we can have equivalent equations when the entire equation is multiplied or divided by a constant.

Students: Pages 210–211

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Purpose Show students that two systems of equations may not be equivalent, but will still produce the same solution, when some combination of equivalent equations are added together.

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Expected mistakes Students may think that because the two systems are not the same, they can’t have the same solution. Remind students that adding equivalent versions of the original equations together creates an equation whose line will intersect at the same point as the original system of equations.

Students: Pages 211–212

Purpose Show students how to solve a system of equations algebraically using the elimination method. Expected mistakes Students may think that they need to perform the elimination method twice, once for each variable. Remind students that they can solve for the second variable by substituting into either of the original equations.

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Students: Pages 212–213

Purpose Check that students can apply the elimination method to solve a system of equations. Expected mistakes Students may not apply the operations to both sides of the equation when multiplying an equation by a constant. This would result in equations that are not equivalent.

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Students: Pages 213–214

Purpose This example is designed to teach students that in some cases, both equations in a system must be multiplied by a factor before proceeding with the elimination method. This is necessary when the coefficients of the variable to be eliminated are not exact multiples of each other. Reﬂecting with students After solving this problem, discuss with students why it was necessary to multiply both equations in this case, and what would happen if they didn’t. You can also discuss the importance of keeping track of all operations performed on the equations to avoid mistakes, especially in more complex systems of equations.

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Students: Page 214

Purpose Check that students can write a system of equations to represent a real-world scenario. Expected mistakes Students may incorrectly assign the variables in the equation representing the difference in scores which will result in their solutions being the wrong way around. It may help students to use more explicit variables like H for “higher” and L for “lower” scorer, or simply use the words “higher” and “lower” in place of variables. Reﬂecting with students Ask students if they can match every aspect of the system of equations to the information given about the scenario.

Students: Page 214

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Purpose Check that students can solve a system of equations using the elimination method. Reﬂecting with students Some students may have subtracted one equation from the other when applying the elimination method. Ask students if they would expect this approach to give a different result, or the same result. let students know that any addition or subtraction of the two equations will have the same solutions as the original system.

Students: Page 215 c Does the solution make sense in terms of the context? Explain your answer.

Apply the idea Yes. Assuming that the tests were out of 100, then 72 and 56 are both valid test scores to obtain.

PurposeExample 4 Show students how to interpret the solutions in context and evaluate their validity. Determine the number of solutions to each system without solving.

Reﬂecting with students a c Does the solution make sense in terms of the context? Explain your answer. Ask students to come up with contextual constraints that would make the solution not make sense in terms of the context. Apply idea Createthe a strategy

Yes. Assuming that the tests were out of 100, then 72 and 56 are both valid test scores to obtain. different representations of the same line.

that if one equation in the system is a multiple of the other, the system has infinite solutions because they are Students:Recall Page 215 If the coefficients of the x and y terms are multiples but the constant is not a multiple, the system has no solutions because they represent parallel lines and will never cross.

Example 4 Lastly, if the x and y terms are not multiples then the lines intersect and have one solution. Determine the number of solutions to each system without solving.

Apply the idea

Since neither the entire equation nor the coefficients of the x and y terms are multiples, these lines intersect exactly a once. Therefore, this system has one solution.

Create a strategy

Recall that if one equation in the system is a multiple of the other, the system has infinite solutions because they are different representations of the same line. b If the coefficients of the x and y terms are multiples but the constant is not a multiple, the system has no solutions because they represent parallel lines and will never cross.

Create a strategy Lastly, if the x and y terms are not multiples then the lines intersect and have one solution. Recall that if one equation in the system is a multiple of the other, the system has infinite solutions because they are Apply idea differentthe representations of the same line. Since neither the entire equation nor the coefficients of the x and y terms are multiples, these lines intersect exactly Apply once. the idea We can manipulate thehas second equation by dividing each term by 2: Therefore, this system one solution. Original equation b

Divide each term by 2

Purpose This is exaclty the same as the firstthe equation. Therefore, both equations represent the same line, there aretheir infinite a strategy To checkCreate if students can determine number of solutions to a system of equations byand analyzing solutions. Recall that if one equation in the system is a multiple of the other, the system has infinite solutions because they are structure. different representations of the same line. c Apply the idea We can manipulate the second equation by dividing each term by 2:

Apply the idea

Original equation

In this system, the coefficients of the x and y terms are multiples but the constant is not a multiple. Divide each term by 2 Therefore, the system has no solutions because they represent parallel lines and will never cross.

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This is exaclty the same as the1first equation. Therefore, both equations represent the same line, and there are infinite 4.03 Elimination method 215 Mathspace Virginia SOL Algebra Teacher Edition mathspace.co solutions. mathspace.co

Lastly, if the x and y terms are not multiples then the lines intersect and have one solution. a

Apply the idea Since neither the entire equation nor the coefficients of the x and y terms are multiples, these lines intersect exactly Create a strategy once. that215 if one equation in the system is a multiple of the other, the system has infinite solutions because they are Students:Recall Page Therefore, this system has one solution. different representations of the same line. If the coefficients of the x and y terms are multiples but the constant is not a multiple, the system has no solutions because they represent parallel lines and will never cross. b Lastly, if the x and y terms are not multiples then the lines intersect and have one solution.

Apply idea Createthe a strategy Since theequation entire equation nor the of the x and terms has are infinite multiples, thesebecause lines intersect exactly Recall neither that if one in the system is acoefficients multiple of the other, the ysystem solutions they are once. different representations of the same line. Therefore, this system has one solution.

Apply the idea We can manipulate the second equation by dividing each term by 2: b

Original equation Divide each term by 2

Create a strategy Recall that if one equation in the system is a multiple of the other, the system has infinite solutions because they are This is exaclty the same of asthe thesame first equation. Therefore, both equations represent the same line, and there are infinite different representations line. solutions.

Apply the idea We can manipulate the second equation by dividing each term by 2: c Original equation

Purpose Divide each term by 2 To showApply students how simplifying equations can help determine the number of solutions to a system of the idea equations. In this system, the coefficients of the x and y terms are multiples but the constant is not a multiple. Therefore, the system hasas nothe solutions becauseTherefore, they represent linesrepresent and will never cross.line, and there are infinite This is exaclty the same first equation. bothparallel equations the same

Students:solutions. Pages 215–216

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Apply the idea In this system, the coefficients of the x and y terms are multiples but the constant is not a multiple. Therefore, the system has no solutions because they represent parallel lines and will never cross. 4.03 Elimination method mathspace.co

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Purpose To show students how to determine that a system of equations has no solutions by recognizing that the equations represent parallel lines.

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Students: Page 216

Practice Students: Pages 216–219

What do you remember? 1

The method of elimination involves: A

Substituting one variable into another equation

Graphing the equations

Adding or subtracting equations to eliminate a variable

Isolating a variable

Consider the following system of equations:

Describe how to use multiplication to eliminate the variable y.

Now, combine the two equations using the method you explained.

Select the solution to each system of equations:

(1, 2)

(−3, 4)

(−3, 1)

(5, 9)

(2, 1)

(4, −3)

(1, −3)

(9, 5)

Solve each system of equations using the elimination method: a

Let’s practice 5

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Consider the following system of equations:

What value can we multiply each equation by in order to eliminate the variable y?

Solve for x and y.

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For the system of linear equations:

What number should the second equation be multiplied by to eliminate y?

Eliminate y and find the value of x that satisfies both equations.

Substitute the x-value to find the value of y that satisfies both equations.

Solve each system of equations using the elimination method: a

Consider each system of equations: i

Solve the system of equations.

Verify each solution algebraically.

Consider the following systems of equations: i

Use the elimination method to solve each pair of linear equations:

Verify each solution by graphing.

a 10

Consider this system of equations:

Explain how to eliminate the variable y.

Explain why eliminating y would help solve the system of equations.

Consider each pair of systems of equations and state whether they are equivalent. a

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The train in an amusement park has 15 cabs that can hold a total of 56 people. Some cabs hold 4 people while some hold 6 people. Let x be the number of four-passenger cabs and y be the number of six-passenger cabs. This situation can be represented by the following system of equations:

Select the correct number of four-passenger cabs and six-passenger cabs. A x = −2, y = 17

b 13

x = 12, y = 3

x = 17, y = −2

When comparing her test results, Judy noticed that the sum of her Geography test score and Math test score was 137, and that their difference was 29. Judy scored higher on her Geography test than her Math test. a

Write a system of equations where x represents Judy’s Geography score and y represents her Math score.

Solve for Judy’s Geography score using the elimination method.

Now, solve for Judy’s Math score.

12 pens and 5 rulers cost $70 while 3 pens and 25 rulers cost $65. a

Write a system of equations where x represents the price of a pen and y represents the cost of a ruler.

Solve for the price of a pen using the elimination method.

Now, solve for the price of a ruler.

Fred spent $55.25 to purchase 9 ﬂowers. He bought rhododendrons which cost $6.45 each and chrysanthemums which cost $5.75 each. a

If R is the number of rhododendrons and C is the number of chrysanthemums that Fred bought, construct two equations describing the total number of ﬂowers bought and the total amount spent in dollars.

Solve for the number of rhododendrons and chrysanthemums that Fred purchased.

For each system of equations, explain which strategy you would use and why: b

For each system of equations: i

State whether the substitution method or the elimination method would be more efficient for solving.

Explain why you chose the method in part (i).

x = 3, y = 12

Determine whether the solution is viable in terms of the context.

a 17

Identify the number of solutions for each system of equations: a

Let’s extend our thinking 19

Describe a situation where it would be more efficient to use the elimination method rather than the substitution method to solve a system of equations and explain why.

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Consider this system of equations:

Explain how to rewrite the system of equations so that it has integer coeﬃcients. 21

Ricardo solved the following system of equations used to model wildlife populations in a wildlife sanctuary, where r represents the number of rhinos and h represents the number of hippopotami.

Explain why the solution to the system of equations is not viable.

Explain how you might interpret the solution to the system of equations to make sense in terms of the context.

Elias solved a nonlinear system of equations using elimination. His work is shown. Aria chose to solve by graphing, and she found two solutions. a

Explain the mistake Elias made that caused him to only find one solution.

Find the second solution.

Original equations

Multiply equation 2 by 3

Add the equations, solve for y

Substitute y, solve for x

The solution is (1, −3). 23

Without solving, how can you determine the number of solutions to the system of equations?

Kwabena bought some fresh produce. He picked up 2 oranges, and 3 bananas. The cost of the Kwabena’s shopping was $18.30. Amra also went to the same shop and bought 5 oranges and 7 bananas. The cost of the Amra’s shopping was $44.03.

Construct a system of equations to model the scenario.

Explain two different ways to approach ﬁnding the cost per orange and the cost per banana.

At a donut shop, two types of donuts are available: classic donuts and premium donuts. The donut shop offers two deals for these donuts: Deal A with 3 classic donuts and 5 premium donuts for $6; Deal B with 6 classic donuts and 10 premium donuts for $12. a

Write the systems of equation where c represents the price of one classic donut and p represent the price of one premium donut.

Solve for the prices of the classic and premium donuts using the elimination method.

Omeida’s piggy bank contains 70 nickels and dimes with a total value of $3.85. a

Construct a system of equations to model the scenario and use it to determine how many of each coin Omeida has.

Suppose the coins in the piggy bank were only dimes and quarters. Revise the model in part (a) for this change. Explain how this changes the solution. 4.03 Elimination method mathspace.co

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Answers

9 a i

4.03 Elimination method

7 6 5 4 3 2 1

What do you remember? 1 C 2 a M ultiply the second equation by 7 then combine the two equations using addition.

b j = 2, k = 8 or (2, 8)

c d = 8, e = 4 or (8, 4)

d m = 7, n = 4 or (7, 4) b i (2, 8)

Let’s practice 5 a M ultiply the first equation by 5 and the second equation by 7.

(2, 8)

8 6

b x = −3

7 a x = 6, y = 7 or (6, 7)

b s = 9, t = 6 or (9, 6)

c p = 0, q = 5 or (0, 5)

d x = 6, w = −3 or (6, −3)

−8 −6 −4 −2 −2

e x = −5, y = −3 or (−5, −3)

x = −7, y = −9 or (−7, −9)

−4

g x = 40, y = −2 or (40, −2)

h x = −8, y = −2 or (−8, −2)

8 a i x = −7.6, y = 3.4 or (−7.6, 3.4) ii

b x = −1, y = −3 6 a 7

1 2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

b D

4 a a = −9, b = 6 or (−9, 6)

−7−6−5−4−3−2 −1 −1

b 17x = −125 3 a B

0.5x + 3y = 14

Write the first equation

0.5(7.6) + 3(3.4) = 14

Substitute x = 7.6, y = 3.4

3.8 + 10.2 = 14

Evaluate the multiplication

14 = 14

6x − 4y = 32

c i (2, −9) ii

Write the second equation

6(7.6) − 4(3.4) = 32

Substitute x = 7.6, y = 3.4

45.6 − 13.6 = 32

Evaluate the multiplication

32 = 32

Evaluate

b i p = −38, q = 10.2 or (−38, 10.2) ii

p + 4q = 2.8

−38 + 4(10.2) = 2.8

Substitute p = −38, q = 10.2

−38 + 40.8 = 2.8

Evaluate the multiplication

2.8 = 2.8

Write the first equation

Evaluate

4 y 3 2 1 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7 −8 −9 −10 −11 −12 −13 −14

Evaluate

x 1

(2, −9)

d i (8, 3) ii

4 3

y (8, 3)

0.1p + q = 6.4

Write the second equation

0.1(−38) + 10.2 = 6.4

Substitute p = −38, q = 10.2

−3.8 + 10.2 = 6.4

Evaluate the multiplication

−8−6−4−2 −1

6.4 = 6.4

Evaluate

x 2 4 6 8

x 2 4 6 8 10 12 14

−2 −3 −4

10 a E xample answer: Multiply the first equation by 70 and multiply the second equation by 1 800. Then add the equations. b An equation with one unknown variable can be solved.

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Let’s extend our thinking

11 a Equivalent

b Not equivalent

c Equivalent

d Not equivalent

19 Answer may look like:

b No

If the coefficients of one variable are the same or opposite in both equations, it’s often easier to use the elimination method because you can add or subtract the equations to eliminate that variable right away. With substitution, you would first have to solve one equation for one variable, which could be more work.

12 a D 13 a

b Judy scored 83 on Geography. c Judy scored 54 on Math. 14 a

b x = $5

c y = $2 15 a b F red purchased 5 rhododendrons and 4 chrysanthemums. 16 a A nswers may vary. For example, one could use technology to find the accurate intersection point. Alternatively, one could use the substitution method as the first equation is already in the form y = mx + c. b A nswers may vary. For example, one can simply read the solution from the graph. Alternatively, one could use the substitution method as both equations are already in the form y = mx + c. c A nswers may vary. For example, one could use the substitution method as Equation 1 is already in the form y = mx + c. Alternatively, one could use the elimination method by multiplying Equation 1 by 2 and then adding the two equations to eliminate the y variable. d A nswers may vary. For example, one could use the elimination method by multiplying Equation 2 by 3 and then adding the two equations to eliminate the y variable. Alternatively, one could use the substitution method by solving Equation 2 for x or y and then substituting it into Equation 1. 17 a i Substitution ii An equation is already solved for a variable. b i Elimination ii One variable already has the same coefficient. c i Elimination ii The coefficients of the variables are multiples of each other. d i Substitution ii One equation is already solved for a variable.

20 Multiply Equation 1 by 10 and multiply Equation 2 by 100. 21 a T he solution to the system is r = 4.8, h = 4.2 is not feasible because you cannot have part of a hippo or a rhinocerous. b T he solution results in 4.8 rhinos and 4.2 hippopotami. Depending on the circumstances, you might be able to round to 5 rhinos and 4 hippopotami in order to make sense in context. 22 a W hen Elias substituted y to solve for x, he only found one value for x because he forgot that −1 could also be a solution to x2 = 1. b (−1, −3) 23 If you add the two equations, both variables would be eliminated, resulting in a true statement 0 = 0. This means there are infinitely many solutions. Each original equation represents a line, and these lines coincide (i.e., are the same line), hence there are infinitely many solutions. 24 a Let x = the cost per orange and let y = the cost per banana.

b O ne way to use the elimination method in finding the cost per orange and banana would be multiplying the first equation by -5 and the second equation by 2. That way, the x-terms would be eliminated after adding the equations and we would begin solving for the cost per banana. Another way to use the elimination method in finding the cost per orange and banana would be multiplying the first equation by 7 and the second equation by -3. That way, the y-terms would be eliminated after adding the equations and we would begin solving for the cost per orange. The system could be converted to slope-intercept form for graphing as well, and the point of intersection would be where we find the cost per orange and the cost per banana.

18 a No solution b An infinite number of solutions c No solution d An infinite number of solutions

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26 a Let x = the number of nickels and let y = the number of dimes.

25 a b B y multiplying the first equation by 2, we get 6c + 10p = 12, which is identical to the second equation. This implies that there are an infinite number of solutions for the system. Therefore, we cannot determine the exact prices of the classic and premium donuts.

There are 63 nickels and 7 dimes. b Let x = the number of dimes and let y = the number of quarters.

There would be 91 dimes and -21 quarters. This is not a possible solution to the system because Omeida could not have a negative number of coins, so there couldn’t be 70 dimes and quarters in the piggy bank.

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4.04 Two variable linear inequalities Subtopic overview Lesson narrative In this lesson, students will combine their understanding of one-variable linear inequalities and two-variable linear equations to understand linear inequalities in two variables. This lesson focuses on creating, graphing, and solving linear inequalities. Students use problem-solving skills to develop an understanding that a two-variable linear inequality is represented as a region of the coordinate plane bounded by a line, and the solution is an infinite set of ordered pairs within a specified region. Then, they will shift to contextual problems where the problem context may impose further constraints on the solutions, creating viable and non-viable solutions. By the end of this lesson, students should be comfortable with identifying the key features of a linear inequality from both an algebraic and graphical representation, as well as understanding what the features mean mathematically and in context. A solid understanding of what the solution region of an inequality represents will also be useful moving forward when considering systems of linear inequalities, and when learning about quadratic inequalities.

Learning objectives Students: Page 220

Key vocabulary 

boundary line



linear inequality (in two variables)



non-viable solution



viable solution

Essential understanding The value(s) of the variables that make an equation or inequality true make up its solution set. A two variable linear inequality has an infinite number of solutions so its solution set can be represented by a half plane.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving Teachers can incorporate problem-solving into the lesson by having students apply their understanding of solving one-variable linear inequalities to solving linear inequalities in two variables.This includes understanding the implications of inequality symbols on the graph, such as whether the boundary line should be solid or dashed. Students should also recognize that the solutions to the inequality are represented by the shaded region on the graph, and understand the meaning of these solutions in the context of the problem. 4.04 Two variable linear inequalities mathspace.co

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MPG4 — Mathematical Connections

MPG5 — Mathematical Representations

Teachers can help students make connections by relating the concept of linear inequalities in two variables to real-world scenarios, and to their prior knowledge of graphing linear functions. This includes understanding how the inequality symbol influences the graph of the inequality, and what the solutions to the inequality represent in real-world contexts.

Teachers can integrate this goal into their instruction by having students represent linear inequalities using various forms, such as algebraic expressions, graphs, and tables. Students should be guided in interpreting specific points on the graph in the context of the problem, understanding that these points represent potential solutions to the inequality. The solution should communicate both the meaning of the shaded region and the significance of solid or dashed lines on the graph.

A.EI.2h — Verify possible solution(s) to a system of two linear equations, a linear inequality in two variable, or a system of two linear inequalities algebraically, graphically, and with technology to justify the reasonableness of the answer(s). Explain the solution method and interpret solutions for problems given in context.

A.EI.2e — Represent the solution of a linear inequality in two variables graphically on a coordinate plane.

Prior connections 8.PFA.5 — The student will write and solve multistep linear inequalities in one variable, including problems in context that require the solution of a multistep linear inequality in one variable.

A.EO.1 — The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

Future connections A2.EI.1 — The student will represent, solve, and interpret the solution to absolute value equations and inequalities in one variable.

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Engage Activity Homecoming dance ticket sales

60 mins

Students will work in groups to determine a way to present the multiple solutions of a linear inequality in two variables within The context of selling tickets to the Homecoming dance.

Understanding and skills

Will develop Writing solutions to a two-variable linear inequality from a written description within a real-world context. Interpreting solutions to a two-variable linear inequality from a written description within a real-world context. Displaying the solution space of a linear inequality.

Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Groups will need a way to create a presentation (digital or physical) to share • Materials: Pencil, paper, and calculators for working out arithmetic are recommended, but not required

Support students with disabilities Support memory - carry out algorithms Provide students a worked example of how to graph inequalities in the plane and test for viable solutions.

Support for English language learners Three reads Have students read the task aloud. On the ﬁrst read, ask students to describe the situation. Prompt: Students read the problem Students think/write: Answer the question “What is the problem about?” Answers may look like: • Selling tickets to a dance • Finding the number of singles and couples tickets to sell without exceeding 500 attendees • Finding feasible solutions for the different combinations of singles and couples tickets that can be sold for a school dance Share: Students are called upon to discuss their answers with the class. On the second read, ask students to interpret the question. Prompt: Students read the problem Students think/write: Answer the question “What does an answer look like?” Answers may look like: • The number of singles and couples tickets you can sell • Combinations of singles and couples tickets that add to 500 or less attendees • A shaded region or all solutions to an inequality in two variables On the third read, have students identify important information. Prompt: Students read the problem Students think/write: Answer the question “What are the important pieces of information given in the question?” Answers may look like: • The total number of attendees allowed is 500 • A singles ticket is good for one person and a couples ticket is good for two people

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Classroom guide Hook Students write observations about the ﬂyer for a Homecoming dance that includes date, times, and ticket prices.

Notice and wonder

•

5 mins

What do you notice? What do you wonder?

A Night Beyond the Stars Implementation details Encourage students to discuss the ticket options, prices, and timeline for buying tickets. Students may wonder if the prices have changed since 2011 as well as how many tickets are sold for the dance, how many tickets were sold, what types of tickets were sold, and how much money is made off of ticket sales.

Homecoming 2011 Saturday, September 24th 7:30 pm - 10:30 pm Tickets : $15 single / $25 couple : $20 each at door On sale : Sept. 16th - Sept. 23rd

Slide 1 from Student Engage Activity

Launch

5 mins

Maiko is a council member on her school’s student council. This year they are in charge of organizing the Homecoming dance. The committee decides to offer both couples and singles tickets, but they can only allow a maximum of 500 attendees. Slide 2 from Student Engage Activity

Give students time to process individually what ticket combinations meet or exceed the 500 student maximum so that each student enters their group ready to participate in the group task. Suggested grouping: Form groups of 3 or 4 and assign roles

Continue when Students have read the Launch and understand the context of the problem.

Explore

Team roles

•

35 mins

Use prompts to help groups progress from limited displays of a subset of possible solutions to representing all possible solutions in a graph and/or an inequality. Students are unlikely to reach a perfect solution on their own. Focus on ways to improve their presentation by encouraging the Perspective to point out what is missing. Avoid presenting a perfect solution to any groups as they work.

Anticipated strategies Create a table Students create a table to track and represent number of singles tickets, number of couples tickets, and total attendees.

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Students may focus on edge cases or a small subset of the sample space: singles 0 500 0 100 100

couples 0 0 250 200 100

capacity 0 500 500 500 300

When presented as a visual such as a table, or designed sample space, not all cases are possible to represent. Encourage groups to move from listed or tabular displays to graphical or algebraic models. Create an equation or inequality Students write an inequality that models the restriction (this may need prompting to develop from an equation to an inequality). Let x represent the singles tickets sold and let y represent the couples tickets sold. Then x + 2y ≤ 500,

x + y ≤ 250, y ≤

x + 250 are all examples of inequalities that model the number of students

at the dance. Create a graph Students plot points representing the number of singles and couples tickets sold. A graphical representation of the above inequality:

Misconceptions Focus on meeting the maximum What does it mean to have a maximum capacity of 500? Is it possible that student council doesn’t ﬁll up the dance?

y 450 400 350 300 250 200 150

Missing that a couples ticket represents two attendees

100

Use edge cases to probe. What is the largest amount of couples tickets that could be sold? What is the largest amount of singles tickets that could be sold? How many students will go to the dance if student council sells 100 couples tickets and 100 singles tickets?

x 50 100 150 200 250 300 350 400 450

Purposeful questions Use the following questions to check for understanding and encourage critical thinking: • How many different combinations are there? What are some ways we can record or display these combinations? • What are the best case scenarios for ticket sales to this dance? Is there more than one “best”? • Is it possible for the dance not to meet the maximum occupancy? How do we show that? • Motivate edge cases: what is the least number of tickets that can be sold, what is the most? What if the student council only sold couples tickets?

Continue when All groups have created a visual display of potential cases and at least considered the following (but should consider more): only couples tickets, only singles tickets, at least one combination of tickets equalling 500 students, at least one combination of tickets equalling less than 500 students.

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Discuss

15 mins

Begin with group presentations. After group presentations, consider making a direct connection with writing an algebraic inequality and allow students to work with more restrictive conditions in the context.

Discussion guide Call on each group to share their presentation. Encourage the observing groups to add to or improve their sample space after each presentation if the group had different combinations. After presentations, students can then explore this graphical applet: https://www.geogebra.org/m/ubhvqg9u. This applet displays the solutions and nonsolutions to the homecoming dance problem. Extension: If no group presented an algebraic inequality, ask students to write the algebraic inequality that models the graph shown in the applet. Give students more restrictive conditions to explore: • How does the problem change if 75% of tickets sold have to be couples tickets? • How does the problem change if the maximum number of singles tickets that can be sold is 20 more than the number of couples tickets? • How can the student council model their ticket sales revenue if couples tickets cost $25 and singles tickets cost $15?

Tools You may ﬁnd these tools helpful: • Graphing calculator • Ruler • Blank coordinate plane

Student lesson & teacher guide Two variable linear inequalities Students learn the deﬁnition for a two variable linear inequality and then explore the relationship between the graph of a two variable linear inequality and the features of the inequality.

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Students: Page 220

Concrete-Representational-Abstract (CRA) Support Targeted instructional strategies Concrete: Begin by drawing a large coordinate plane on the floor using tape or chalk. Engage students by having them use their bodies to represent points on the plane. Introduce a linear inequality, such as y > x + 2, and ask students to stand at coordinates they believe satisfy the inequality. Encourage each student to explain why they chose their position. For example, a student might stand at (3, 6) and share their reasoning. This physical activity allows students to explore which regions of the plane represent solutions to the inequality by moving themselves to different points. Next, using string have some students who are not standing on the graph lay down the string to represent the line y = x + 2 and discuss where all of the students are standing in relation to the line. Students should notice that they are all above the line and no student is on the line. Representational: Transition to the representational stage by having students draw the coordinate plane on graph paper. Instruct them to graph the boundary line of the inequality y = x + 2. Explain that they should use a dashed line because the inequality does not include the boundary (it’s “greater than,” not “greater than or equal to”). Teach them how to test a point, like (0, 0), to determine which side of the line to shade. Have them shade the region where the inequality holds true. Encourage them to label their graphs with the inequality and any key points. This visual representation helps students connect their physical movements on the floor to a drawn graph. Abstract: Move on to the abstract stage by focusing on the algebraic representation of the inequality. Explain how to write inequalities in slope-intercept form and interpret the symbols. Teach students how to substitute values into the inequality to check if a point is a solution. Discuss how the inequality symbol determines the type of boundary line and the solution region. Present real-world problems and have students create inequalities to model constraints. Encourage them to solve the inequalities algebraically without relying on graphs or physical models. This helps students work with abstract symbols and understand the underlying mathematics.

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Misinterpreting Real-World Constraints Address student misconceptions Students may struggle to correctly translate real-world situations into two-variable inequalities. They might misassign variables or use incorrect inequality symbols, leading to mathematical representations that do not accurately reflect the given context. This can result in solutions that are not viable for the problem at hand. To address this misconception, encourage students to begin by clearly defining each variable and specifying what it represents in the context. Have them write statements such as “Let x represent the number of hours spent on activity A, and y represent the number of hours spent on activity B.” Guide them through identifying key words and phrases like “at least,” “no more than,” “minimum,” and “maximum,” and discuss how these translate to the appropriate inequality symbols (>, <, ≥, ≤). Use real-world examples and have students practice by writing inequalities from given scenarios, first paraphrasing the constraints in their own words. Visual aids like charts or diagrams can help organize information and clarify relationships between variables. Additionally, model the thought process by working through examples step-by-step, explaining how you determine which inequality symbol to use based on the context. This approach helps students connect the language of the problem to its mathematical representation.

Assist graphing Student with disabilities support Provide students with templates or technology to help with graphing linear inequalities. A template could include a partially completed graph where they need to determine some key features like where to shade and whether the line is dashed or solid. A graphing calculator or graphing software could be used to check answers or to aid in the physical graphing. For example, students could graph by moving the points and adjusting the toggles: https://www.geogebra.org/m/ftyhkaww.

Exploration Students: Page 220

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Suggested student grouping: Individual Students investigate the relationship between the inequality symbol of a two variable linear inequality and the graphical representation of the inequality, and also explore the conditions for whether a point satisfies the inequality or not. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What do you notice about the label when the point is in the shaded vs. the unshaded region? Why do you think that happens? When the point is in the shaded region, the label is ‘true’. When the point is in the unshaded region, the label is ‘false’. This happens because the shaded region represents the points on the coordinate plane that correspond to ordered pairs which make the inequality true. 2. What do you notice about the label when the point is on the boundary line? Why do you think that happens? When the point is on the boundary line, the label is ‘true’ for the inclusive inequality symbols (≤ and ≥) and false for the non-inclusive inequality symbols (< and >). This happens because the ordered pairs represented by the boundary line only make the inequality true when the inequality symbol is inclusive. 3. How does the inequality symbol affect the graph and the label on the point? Why do you think that happens? When the inequality symbol changes, the shaded region representing the solution set for the inequality changes to match it, which consequently affects the label on the point depending on whether it is inside or outside the shaded region. Purposeful questions • Is there any relationship between the inequality symbol and the region that is shaded? • Is there any relationship between the inequality symbol and whether the boundary line is solid or dashed? • How does the label on the point relate to its position? Possible misunderstandings • Students may try to position the point on the boundary line and toggle between inclusive and non-inclusive inequality symbols without the label on the point changing. Let students know that placing the point exactly on the boundary line is very difficult, so it is hard to demonstrate this directly. Following the exploration, students learn the definition for the boundary line and are shown examples which demonstrate how the boundary line and shaded region relate to the inequality.

Students: Page 220–221

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Examples Students: Page 222 Example 1 Consider the inequality y ≤ −2x + 5 a Graph the boundary line y = −2x + 5.

Create a strategy We can graph the equation of the boundary line using the y-intercept and slope.

Apply the idea 5

y = −2x + 5

3 2 1 −5 −4 −3 −2 −1 −1

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−2 −3 −4 −5

x 1

2 3 4 5

y ≤ −2x + 5 a Graph the boundary line y = −2x + 5.

Create a strategy We can graph the equation of the boundary line using the y-intercept and slope.

Apply the idea

Example 1

y = −2x + 5

Consider the inequality

y ≤ −2x + 5 a Graph the boundary line y = −2x + 5.

−5 −4 −3 −2 −1 −1

2 3 4 5

−2

Create a strategy

−3 We can graph the equation of the boundary line using the y-intercept and slope. −4

−5

Apply the idea

4 b Determine whether the point (4, 2) is a solution to the inequality y ≤ −2x + 5. y = −2x + 5

2 Purpose Create a strategy Show students how to identify and graph the boundary 1line for the inequality. x

We will substitute the ordered pair into the inequality to determine whether the statement is true. −5 −4 −3 −2 −1

2 3 4 5

−1 Reﬂecting with students Apply the idea −2 Ask students if they can relate the boundary line to the−3inequality. Point out that the boundary line is the linear y ≤ −2x + 5 inequality equation generated by replacing theGiven inequality symbol−4with an equal symbol.

2 ≤ −2(4) + 5

2 ≤ −3 Students: Page 222

Substitute (4, 2)

−5

Evaluate the multiplication and addition

Since 2 is not less than or equal to −3 the statement is false and the point (4, 2) is not a solution to the inequality y ≤ −2x + 5. b Determine whether the point (4, 2) is a solution to the inequality y ≤ −2x + 5.

a strategy cCreate Determine whether the point (0, 0) is a solution to the inequality y ≤ −2x + 5. We will substitute the ordered pair into the inequality to determine whether the statement is true. Create a strategy Apply the idea

We will substitute the ordered pair into the inequality to determine whether the statement is true. y ≤ −2x + 5 Given inequality

Apply the2idea ≤ −2(4) + 5

Substitute (4, 2) y2 ≤≤ −2x Given inequality −3 + 5 Evaluate the multiplication and addition 0 ≤ −2(0) + 5 Substitute (0, 0) is false and the point (4, 2) is not a solution to the inequality Since 2 is not less than or equal to −3 the statement 0≤5 y ≤ −2x + 5.

Evaluate the multiplication and addition

Since the statement is true, the point (0, 0) is a solution to the inequality y ≤ −2x + 5. c Determine whether the point (0, 0) is a solution to the inequality y ≤ −2x + 5. 222

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Virginia SOL Algebra 1

Purpose mathspace.co Create a strategy Show students how to check algebraically whether a point is a solution to an inequality. We will substitute the ordered pair into the inequality to determine whether the statement is true.

Expected mistakes Apply the idea Students might mistake the inequality sign for an equality sign. It’s important to remind them that an inequality y ≤ −2x + 5 Given inequality includes values that are less than or equal to −2x + 5, not just equal to. 0 ≤ −2(0) + 5

0≤5 Reﬂecting with students

Substitute (0, 0)

Evaluate the multiplication and addition

Since the true, the point (0, 0) why is a solution to the(4, inequality + 5. to the inequality. This will help Ask students to statement explain inis their own words the point 2) is noty ≤a−2x solution reinforce their understanding of inequalities. 222

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y ≤ −2x + 5

Given inequality

2 ≤ −2(4) + 5

Substitute (4, 2)

2 ≤ −3

Evaluate the multiplication and addition

Since 2 is not less than or equal to −3 the statement is false and the point (4, 2) is not a solution to the inequality ≤ −2x + 5. Students:y Page 222

c Determine whether the point (0, 0) is a solution to the inequality y ≤ −2x + 5.

Create a strategy We will substitute the ordered pair into the inequality to determine whether the statement is true.

Apply the idea y ≤ −2x + 5

Given inequality

0 ≤ −2(0) + 5

Substitute (0, 0)

0≤5

Evaluate the multiplication and addition

Since the statement is true, the point (0, 0) is a solution to the inequality y ≤ −2x + 5.

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Purpose Check that students can determine whether a point is a solution to an inequality or not. Reﬂecting with students Ask students what it means for a point to be a solution to the inequality. Challenge students to ﬁnd two more points that are solutions to the inequality.

Students: Page 223

Purpose Show students how to determine the region for the solution to the inequality by checking points. Expected mistakes Students may incorrectly shade the side of the graph that does not include the test point. Remind students that the test point is a solution to the inequality, so the side of the graph that includes it is the solution set. 458

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Reﬂecting with students Ask students to relate the “or equal to” condition in the inequality to the correct feature of the graph, to reinforce their understanding of why the line of the graph remains solid.

Students: Pages 223–224

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Purpose Show students how to graph a two variable linear inequality. Reflecting with students Encourage advanced learners to investigate how altering the coefficients and constants in the inequality affects its graph and solution set. Invite them to create their own inequalities by selecting different values for the coefficients and constant, and then graph each one. Ask students to observe how changes in these values influence the slope and position of the boundary line, as well as the direction of the shading.

Students: Pages 224–225

Purpose Check that students can determine whether an ordered pair is in the solution set of the inequality or not.

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Reﬂecting with students Encourage students to check the point using their graph. Ask guiding questions such as: “Is it inside the shaded region, or outside?”

Decompose to write an algorithm

use with Example 2

Targeted instructional strategies Introduce algorithmic thinking by guiding students to develop a step-by-step process for graphing linear inequalities in two variables. Encourage them to break down the procedure into sequential steps. Have students apply their algorithm to various inequalities to test its effectiveness and refine it as needed. Ensure they test their process on cases like horizontal and vertical lines. 1. Rearrange to a commonly used form, such as y = mx + b or x = a, but with the correct inequality. 2. Plot the boundary line • Use a dashed line for < or >

• Use a solid line for ≤ or ≥

3. Choose a test point not on the line, such as (0, 0) and substitute the test point into the original inequality. 4. Shade the correct region. • If the test point satisfies the inequality, shade the region containing the test point. • If not, shade the opposite side of the boundary line.

Students: Pages 225–226

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Purpose Show students how to interpret the graph of a shaded region as an inequality. Expected mistakes Students may incorrectly determine the inequality sign based on the direction of the shaded region. Remind them that for a solid boundary line, the inequality is either ≤ or ≥, and the direction of the inequality is determined by whether the shaded region is above (≥) or below (≤) the boundary line. 462

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Reﬂecting with students Encourage students to conﬁrm their solution using a test point. Discuss with students why choosing a point that is on the boundary line is not a good test point. Point out that the origin, (0, 0), is often a good choice, unless the line crosses the origin.

Students: Page 227 Example 4 A pick-up truck has a maximum weight capacity of 3000 pounds. Each box of oranges weighs 8 pounds and each box of grapefruits weighs 12 pounds. a Write an inequality to represent the number of boxes of oranges and grapefruit that can be in the truck.

Create a strategy

Apply the idea

Our unknown values are the number of boxes of oranges and the number of boxes of grapefruits. Let x represent the number of boxes of oranges in the truck. Let y represent the number of boxes of grapefruit in the truck.

The weight of all the orange boxes is the product of the weight of one box, 8, and the number of boxes, x.

Create a strategy

Apply the idea

Weight of orange boxes: 8x pounds

The weight of all the grapefruit boxes is the product of We should consider the structure of the inequality. The the weight of one box, 12, and the number of boxes, y. Example 4 capacity is 3000 pounds, so we know maximum weight Weight of grapefruit boxes: 12y pounds that the total weight of the boxes ≤ 3000. From there, The total the sumweighs of the weights ofand orange wepick-up can use the has deﬁned variables and numbers the pounds. A truck a maximum weight capacityfrom of 3000 Eachweight box ofisoranges 8 pounds each boxes and grapefruit boxes. problem to build the other12 part of the inequality. box of grapefruits weighs pounds. Total weight: 8x + 12y pounds a Write an inequality to represent the number of boxes of oranges and grapefruit that can be in the truck. The truck can carry at most 3000 pounds, so we get our inequality: 8x + 12y ≤ 3000 Our unknown values are the number of boxes of oranges The weight of all the orange boxes is the product of the and the number of boxes of grapefruits. Let x represent weight of one box, 8, and the number of boxes, x. the number of boxes of oranges in the truck. Let y Weight oftoorange 8xnumbers pounds of orange and b Create a graph of the region containing the points corresponding all the boxes: different represent the number of boxes of grapefruit in the truck. grapefruit boxes that can be loaded into the truck. The weight of all the grapefruit boxes is the product of the weight of one box, 12, and the number of boxes, y. PurposeWe should consider the structure of the inequality. The maximum weight capacity is 3000 pounds, so we know Create ahow strategy of grapefruitlinear boxes:inequality. 12y pounds Show students to interpret a contextual problem intoWeight a two-variable that the total weight of the boxes ≤ 3000. From there, We will graph the region representing 8x + 12y ≤ 3000. The total weight is the sum of the weights of orange we can use the deﬁned variables and numbers from the Expected mistakes boxes and grapefruit boxes. We need to identify our boundary line, then graph it. Then, we need to decide which side of the boundary line to shade. problem to build the other part of the inequality. Students may not recognize that the scenario can be modeled by a single and instead write one Total weight: 8x + 12yinequality, pounds

Apply the idea inequality for each variable, 8x ≤ 3000 and 12y ≤ 3000, which is not accurate interpretation. The truck canan carry at most 3000 pounds, so we get our The boundary line is 8x + 12y = 3000.

inequality:

Students:This Pages is a line227–228 in standard form, so we can graph it by ﬁnding the intercepts.

8x + 12y ≤ 3000

Find the x-intercept by setting y = 0 and solving: 8x + 12y = 3000 Given equation b Create a graph of the region containing the points corresponding to all the different numbers of orange and 8x − 12(0) = 3000 Substitute y=0 grapefruit boxes that can be loaded into the truck. 8x = 3000 Evaluate the multiplication

Create a strategy

x = 375

Divide both sides by 8

We will graph the region representing 8x + 12y ≤ 3000. We need to identify our boundary line, then graph it. Then, we need to decide which side of the boundary line to shade.

Apply the idea The boundary line is 8x + 12y = 3000. This is a line in standard form, so we can graph it by ﬁnding the intercepts. Find the x-intercept by setting y = 0 and solving: 8x + 12y = 3000

Given equation

8x − 12(0) = 3000

Substitute y = 0

8x = 3000

Evaluate the multiplication

x = 375

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grapefruit boxes that can be loaded into the truck.

Create a strategy We will graph the region representing 8x + 12y ≤ 3000. We need to identify our boundary line, then graph it. Then, we need to decide which side of the boundary line to shade.

Apply the idea The boundary line is 8x + 12y = 3000. This is a line in standard form, so we can graph it by ﬁnding the intercepts. Find the x-intercept by setting y = 0 and solving: 8x + 12y = 3000

Given equation

8x − 12(0) = 3000

Substitute y = 0

8x = 3000

Evaluate the multiplication

x = 375

Divide both sides by 8

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Purpose Check that students can graph the solution set for a two-variable linear inequality.

Students: Page 228

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Purpose Show students how to determine whether a solution is viable or non-viable in context. Reﬂecting with students Ask students what they think are reasonable domain and range constraints for the inequality. It’s important to note that negative solutions, while mathematically correct, may not make sense in the context of a problem. For example, we can’t have negative boxes of fruits.

Three reads

use with Example 4

English language learner support Advise students to read through the instructions a few times, focusing on gathering different information each time in order to build up their understanding of what the question is asking. On the first read, students should aim to identify the scenario presented in the question. Ask students “What do you think is happening in this question?” or “Can you explain what this question is about?” On the second read, students should aim to interpret the problem and the mathematics vocabulary by answering questions like “What is the question asking you to find?” and “What information should be included in the answer?” On the third read, students should look for important information in the instructions. In this question, the important information includes: the maximum weight capacity of the truck, the weight of each box of oranges, and the weight of each box of grapefruits. Finally, students can put the information together to answer the question. They can be prompted by questions like “Would the orange and grapefruit boxes have the same weight or different weights?” or “What variables could you use to represent the different weights?”

Students: Page 229

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Practice Students: Pages 229–235

What do you remember? 1

State the equation of the boundary line for each inequality. a

y ≥ 4x − 3

y ≤ − x + 1

For each inequality, state whether they are represented using a solid or dashed line on a graph: a

y ≤ 4x − 4

y < 3x − 5

y ≥ 2x − 3

y>5

For each shaded region, determine whether the following points lie in the shaded region: i

(2, 4)

(−3, 2)

iii

(−1, −6)

(3, −3)

6 5 4 3 2 1

−3 −2 −1 −1 −2 −3 −4 −5 −6

466

x > 12

6 5 4 3 2 1

−7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6

1 2 3 4 5 6 7

x 1

2 3

For each inequality: i

Determine if the origin (0, 0) satisﬁes the inequality.

Graph the region that satisﬁes the inequality.

x>3

y ≤ −2

x ≥ −1

Are these statements true or false? a

If a point lies in the shaded region of the graph of an inequality, it is not a solution to the inequality.

The point (0, 0) lies in the solution region of the inequality x + y > 0.

If an inequality has < or > symbol, the graph of the inequality will have a dashed line.

In the context of budgeting, the inequality 10m + 20s ≤ 100 could represent spending on movies (m) and snacks (s) with a total budget not exceeding $100.

The inequalities 3x − 2y > 6 and 3x − 2y ≥ 6 will have exactly the same graph.

Changing 2x + 3y ≤ 20 to 2x + 3y ≤ 30 will shift the line upwards on the graph.

The inequality x ≥ 0 represents all points to the right of the y-axis, including the y-axis itself.

When graphing an inequality, we shade the region that contains the solutions to the inequality.

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Select the graph that represents the inequality 2x − 7y > 14. A

x −5

5 −5

x −5

5 −5

Select the graph that represents the inequality y ≤ −3x + 4. A

x −5

5 −5

x −5

5 −5

−5

5 −5

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Select the inequality that describes the following region: A

y 8

4 2

−8 −6 −4 −2 −2

2 3 4 5

−4 −6 −8

Select the inequality that describes the following region: A

8 6 4 2

−8 −6 −4 −2 −2 −4 −6 −8

Let’s practice 10

The line y = x − 5 is shown.

Graph the solution to y > x − 5.

Graph the solution to y ≤ x − 5.

4 3 2 1 −5 −4 −3 −2 −1 −1

−2 −3 −4 −5

For each inequality: i

Determine whether the point (2, 3) satisfies the inequality or not.

Graph the region that satisﬁes the inequality.

y ≤ 3x + 5

y ≥ 2x + 4

y < 3x + 2

Does the point satisfy the inequality? a

x − y > 10 i

b c

(−2, −9)

(15, 20)

iii

(16, 7)

(10, −6)

(−2, −3)

iii

(0, 0)

(2, 7)

(−2, 4)

iii

(−4, 30)

(2, −13)

y > 3x i

(12, 20)

y > −8x − 3 i

468

(6, −50)

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For each inequality: i

State the coordinates of the x- and y-intercepts of the boundary line.

Graph the region that satisﬁes the inequality.

3x + 2y < 12

5x + 2y ≥ 10

Write the inequality that describes the following shaded regions: a

y 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

5 4 3 2 1

1 2 3 4 5

5 4 3 2 1

1 2 3 4 5

5 4 3 2

1 2 3 4 5

−4 −3 −2 −1 −1

x 1 2 3 4 5 6

−2 −3 −4

Applicants for a particular job are asked to sit a numeracy test and verbal reasoning test. Successful applicants must obtain a minimum combined score of 43 for both tests. Write an inequality to represent the situation. Define all variables.

The two entertainment options for the school holidays are movies and ice-skating. • Movies, m, cost $12 each • Ice-skating, s, costs $21 per session The inequality 12m + 21s ≤ 150 represents the total amount of money you can spend. What does the point (0, 0) on this graph signify?

The point (0, 0) signifies that there is no money spent on movies or ice-skating sessions.

The point (0, 0) signifies that you spent all your budget on movies and ice-skating sessions.

The point (0, 0) signifies that you attended only movies.

The point (0, 0) signifies that you attended only ice-skating sessions.

Throughout university, Luigi works as a mentor, getting paid $10 per hour, and as a barista getting paid $13 per hour. The number of hours he works in each job can vary from week to week, and he needs to be able to at least cover his weekly expenses of $260. Write an inequality to represent the context. Deﬁne all variables.

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Lisa is being careful with her spending so she can later purchase a car. She has allocated no more than $450 each month for both travel expenses and eating out. On average, each time she eats out costs $18, and each travel journey costs $5. Write an inequality to represent the context. Deﬁne all variables.

In the new basketball season, Mario is looking to beat his personal best of scoring 102 points in total for the whole season (this does not include points from fouls). Write an inequality to represent the context. Deﬁne all variables.

SOL

an the inequality x + y > 10 have integer solutions where both x and y are less than 3? Explain your C reasoning.

an the inequality x − y > 10 have integer solutions where both x and y are less than 3? Explain your C reasoning.

Emily works as a barista. • She earns $10 per hour when she works during the day. • She earns $15 per hour when she works at night. • She wants to earn at least $675 per week. Which graph best represents this situation? A

Emily’s Weekly Earnings

Emily’s Weekly Earnings Number of hours worked at night

Number of hours worked at night 70

10 10

Number of hours worked during the day

20 30 40 50 60 70

Emily’s Weekly Earnings

Emily’s Weekly Earnings Number of hours worked at night

Number of hours worked at night 70

Number of hours worked during the day 10

470

20 30 40 50 60 70

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Number of hours worked during the day 20 30 40 50 60 70

Number of hours worked during the day 10

20 30 40 50 60 70

Let’s extend our thinking 22

Oprah is thinking about how to use her last arcade tokens. A game of Table Tennis costs 2 tokens and a game of Frog Days costs 5 tokens. She has a total of 30 tokens left. Describe the different ways Oprah could spend her tokens. Use a mathematical model to justify your response.

Fiona has set aside $13.20 in her shopping budget for fruits or vegetables this week. The following table lists the prices for different fruits and vegetables at her supermarket. Fruit Apples Bananas Grapes

Price $1.05 each $0.25 each $3.59 per bag

Vegetable Broccoli Cucumbers Snap peas

Price $1.57 per head $0.50 each $4.98 per 8 oz bag

Choose two produce items for Fiona to buy and create a model to help determine the different combinations she can afford within her budget. Then, make a recommendation for how much of each item you think she should purchase. Explain your reasoning.

Explain how you would change your model if Fiona’s goal was to purchase as much produce as possible.

A book seller makes a proﬁt of $9 − $10.50 on every book sold online and $5 − $5.75 on every book sold in store. The company wants to make a proﬁt of at least $270 a day selling books through online and in-store sales. a

Construct an inequality and graph to model the scenario. Describe the domain and range.

Given the context, determine whether the ordered pair (−5, 100) is a valid solution to the inequality. Justify your answer.

Sean is thinking about how to use his remaining spending money for snacks. A pack of dried fruit costs $6 and a bag of mixed nuts costs $4. He has $48 remaining. a

Construct an inequality and graph to model the scenario. Describe the domain and range.

Supposing that Sean does not need to spend all of it, describe how many different ways Sean can spend his remaining money on some combination of dried fruit and mixed nuts.

Monica works part-time as a tutor and also as a sales associate. The number of hours she works in each role varies from week to week, and she needs to earn at least a certain amount weekly to cover her expenses. The graph of the inequality relating the number of hours she works as a tutor, x, and as a sales associate, y, is shown.

45 40 35 30 25 20

Write the rule for the inequality.

Find Monica’s hourly rate of pay for each job.

Find the amount of money she needs weekly to cover her expenses.

x 5 10 15 20 25 30 35 40 45

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Answers

Let’s practice 10 a

4.04 Two variable linear inequalities

5 4 3 2 1

What do you remember? 1 a y = 4x − 3

c y = −x + 1 b Dashed

c Solid

d Dashed

3 a i Yes

ii No

iii No

iv No

b i Yes

ii Yes

iii Yes

iv No b

4 a i No ii

5 4 3 2 1

x 1 2 3 4 5

5 4 3 2 1

−6 −5−4 −3 −2 −1 −1 −2 −3 −4 −5 −6

x 1 2 3 4 5

ii y

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

e False

c True

d True

True

g True

h True

6 C 7 B 8 C 9 B

472

6 5 4 3 2 1 −4 −3 −2 −1 −1 −2 −3 −4

1 2 3 4 5

b False f

x 1 2 3 4 5 6

b i Yes

c i Yes 5 4 3 2 1

x 1 2 3 4 5

6 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

11 a i Yes ii

1 2 3 4 5

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

b i No

5 a False

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

d x = 12

2 a Solid

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x 1 2 3 4 5 6

c i No

17 10x + 13y ≥ 260

6 5 4 3 2 1 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4

18 19 x

20 a N o, the inequality x + y > 10 cannot have integer solutions where both x and y are less than 3. If x and y were both 3, the sum would be 6, which is less than 10.

1 2 3 4

b Yes, for example x = 2 and y = −9. 21 A

d i Yes ii

6 5 4 3 2 1 −6−5−4 −3 −2 −1 −1 −2 −3 −4 −5 −6

Let’s extend our thinking 22 Oprah could play 58 different combinations of Table Tennis and Frog Days. If she only played Frog Days she could play 6 more games in total and if she only played Table Tennis she could play 15 more games in total. If she wanted to use all of her tokens and play both games, she could play Table Tennis 5 times and Frog Days 4 times or she could play Table Tennis 10 times and Frog Days 2 times.

x 1 2 3 4 5 6

12 a i No

ii No

iii No

iv Yes

b i No

ii Yes

iii No

iv Yes

c i Yes

ii No

iii Yes

iv Yes

13 a i (4, 0), (0, 6) ii

8 7 6 5 4 3 2 1 −4 −3−2 −1 −1 −2 −3 −4

number of plays for Frog Days

10 8 6 4

x 1 2 3 4 5 6 7 8

2 number of plays for Table Tennis 3 6 9 12 15 18

b i (2, 0), (0, 5) ii

8 7 6 5 4 3 2 1 −4 −3 −2 −1 −1 −2 −3 −4

Using a mathematical model, let x = the number of times Oprah plays Table Tennis and y = the number of times she plays Frog Days. Then, the number of games Oprah can play can be modeled by 2x + 5y ≤ 30 or the shaded region in the graph:

x 1 2 3 4 5 6 7 8

23 a T here are several combinations of produce that would work for this problem. Suppose that Fiona were to go with apples and bananas. Let x = the number of apples and let y = the number of bananas purchased. A model that could be used for the purchase might be a linear inequality 1.05x + 0.25y ≤ 13.20. Any combination of apples and bananas that costs less than or equal to $13.20 will work. A suggestion might be 7 apples and 7 bananas, one for each day of the week. The cost for this would be $1.05(7) + $0.25(7) = $9.10, which is within her budget. b The cheapest option is bananas which are $0.25 each

14 a y ≥ 3x − 3 c y ≤ −2 15 x + y ≥ 43 16 A

bananas. This would

b y < −2x − 4

so Fiona can afford

d y > −3x + 6

mean Fiona would be eating about 7.5 bananas per day. If Fiona is purchasing a combination of two produce items, we want to change the model to include the two cheapest produce items and make

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sure the amounts purchased are as close to $13.20 as possible. It appears that bananas and cucumbers are the cheapest produce, so Fiona will be able to purchase the most food if she buys some combination of those two items. A graph would help us find the maximum amounts of produce to purchase. Let x represent the number of bananas and y represent the number of cucumbers Fiona purchases. The new inequality would be 0.25x + 0.50y ≤ 13.20. A graph that models this is shown: cucumbers

4 x 4

26 a y ≥ 30 − 0.75x or 30x + 40y ≥ 1200

5 10

b S he earns $30 per hour as a tutor and $40 per hour as a sales associate.

bananas 50

Any combination of produce closest to the boundary line would be within Fiona’s budget, and use the money she has. Some possible combinations could include: cucumbers

cost

number of items

$13.00

24 a Let x = the number of books sold online, and y = the number of books sold in store on one day. If the profits were based on the least amount the book seller makes on the books, an inequality may be 9x + 5y ≥ 270. number of books sold in store

number of books sold online 10 15 20 25 30 35 40

b T he ordered pair (−5, 100) is not a valid solution since the book seller cannot sell a negative number of books.

474

8 6

b T here are 61 different ways, one for each integer coordinate inside the solution set of the inequality.

60 55 50 45 40 35 30 25 20 15 10 5

12 10

bananas

y 14

25 a Let x = the number of packs of dried fruit, and y = the number of bags of mixed nuts that Sean buys. 6x + 4y ≤ 48

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c She needs $1200 to cover expenses.

4.05 Systems of linear inequalities Subtopic overview Lesson narrative In this lesson, students will solve systems of inequalities in two variables in both algebraic and real-world contexts. Students will recognize that the solution is not a single point, but a region of the coordinate plane. This lesson will build on students’ abilities to graph a single linear inequality in two variables as a half-plane and their knowledge that the solution will be the intersection or overlap of the regions. Both procedural and application problems are presented as worked examples to ensure a breadth of understanding. Students should be confident with graphing the solution set given the equations and creating the system of equations given a context.

Learning objectives Students: Page 236

Key vocabulary 

non-viable solution



system of inequalities



viable solution

Essential understanding The value(s) of the variables that make every equation or inequality in a system true make up its solution set. A system of two variable linear inequalities can have an infinite number of solutions so its solution set can be represented by an intersection of half planes (should that intersection exist).

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving Teachers can incorporate problem-solving by having students apply their understanding of solving one-variable linear inequalities to solving systems of linear inequalities in two variables. This includes understanding that the solution to the system is represented by the overlap of the shaded regions. Furthermore, students should recognize that for a point to be included in the solution set, it must lie within the overlapping region or on solid boundary lines.

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MPG2 — Mathematical Communication

MPG4 — Mathematical Connections

Teachers can foster mathematical communication by encouraging students to explain their understanding of the graph of a system of inequalities, including the significance of overlapping shaded regions and solid boundary lines. Students should also be able to communicate the meaning of the overlap in the context of the problem and be able to justify why points on solid lines are included in the solution set.

Teachers can facilitate activities that link systems of inequalities to students’ prior knowledge and other mathematical concepts. Students can be guided to connect their understanding of graphing single inequalities to graphing systems, recognizing how overlapping regions represent the solution set. Teachers can encourage students to apply this concept to real-world contexts, such as modeling constraints in economics or engineering, highlighting the interdisciplinary applications of systems of inequalities. By drawing parallels between algebraic methods and graphical representations, students can see the interconnectedness of different areas within mathematics.

A.EI.2g — Represent the solution set of a system of two linear inequalities in two variables, graphically on a coordinate plane.

Prior connections A.EO.1 — The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

A.EI.1 — The student will represent, solve, explain, and interpret the solution to multistep linear equations and inequalities in one variable and literal equations for a specified variable.

Future connections A2.EI.1 — The student will represent, solve, and interpret the solution to absolute value equations and inequalities in one variable.

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Engage Activity Student council parade

60 mins

Students will work in groups to plan the ﬂoats for a parade given size constraints and a restricted budget.

Understanding and skills

Will use Writing numerical inequalities.

Will develop Testing ordered pairs to the constraints of an inequality. Graphing the solution space for a system of parallel inequalities. Restricting a solution space for a system of inequalities by adding a new constraint. Interpreting the solutions to a system of linear inequalities within a speciﬁc context.

Could extend Revising the solution space for a system of linear inequalities based on new conditions.

Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: Paper, pencil, calculators, graphing technology

Support students with disabilities Support organization - solve multistep or complex problems Chunk the task into smaller parts: for example, spend 5 minutes each on solving for the maximum amount of ﬂoats, the least amount of empty space, the longest parade, and the least amount spent. You may have group members volunteer to be the time keeper for the activity or you can keep track of time for the whole class and remind them when they should be transition from one part of the work to the next. If you use slides or other visual displays during class time you can visibly display the chunked sections of time for students.

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Prompt: Students read the problem. Students think/write: Answer the question “What does an answer look like?” Answers may look like: • Different parade arrangements with their associated costs. • Solutions to a system of linear inequalities with justification about their feasibility. On the third read, have students identify important information. Prompt: Students read the problem. Students think/write: Answer the question “What are the important pieces of information given in the question?” Answers may look like: • The length of each ﬂoat. • The budget of the senior class. • The cost of each ﬂoat given its size.

Classroom guide Hook Students create questions about a picture of a parade.

Implementation details

Co-craft questions

Slide 1 from Student Engage Activity

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

5 mins

What mathematical questions could we ask about this image?

Contextual questions may arise from the image and will provide an opportunity to discuss parades and introduce vocabulary students may or may not be familiar with. Amplify mathematical questions regarding the parade such as the cost, number of people, timing, length of the parade or number of ﬂoats.

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•

Launch The Launch presents students with the length constraints of the parade. Give students think time to consider how this information may inﬂuence the length of the parade. Important mathematical concepts: Inequalities

5 mins

In this task, you’ll be helping design a school parade. It will be organized with the following parameters: • Small floats are 15ft. • Large floats are 30ft. • There must be 10ft in between floats. Class of 2022 Football Stars!!

Important contextual information: Parades, ﬂoats Suggested grouping: Form groups of 3 or 4 and assign numbers.

10-ft space

30 ft

10-ft space

15 ft

Slide 2 from Student Engage Activity

Explore

Numbered heads together

•

35 mins

In the ﬁrst stage of the Explore, students are working with the constraints on the length of the ﬂoats and the overall length of the parade.

Anticipated strategies Write an inequality Students create a compound inequality with an expression representing the lengths of all small and large floats. This inequality should look similar to: 150 ≤ 15x + 10y < 250 where x represents the number of large floats and y represents the number of small floats in the parade. Make a table Students list the possible combinations of floats and their total parade lengths in a table of values. Plot points Students think of individual combinations that work and plot them as points on a graph to create the solution plane.

Misconceptions Space between floats Which floats in the parade require space? Will the number of empty spaces be equal to the number of floats in the parade? How do you know? Inclusive or exclusive If Maiko could make the parade exactly 150 feet long would that fit the requirements? What about exactly 250 feet long? What does “at least” mean? What does “less than” mean? Encourage students who are struggling to find examples to test edge cases: What is the least and most amount of large floats that Maiko could use to fit in the length restrictions? What about the least and most amount of small floats? Students have experience drawing and shading linear inequalities. Ask them how we model situations where more than one solution is valid and how we can apply that to this situation. In the second stage of the Explore, students are given information about the budget constraints and costs to operate each float. 4.05 Systems of linear inequalities mathspace.co

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Anticipated strategies Write an inequality Students create a compound inequality with an expression representing the cost of the parade and given budget. This inequality should look similar to: 600x + 250y ≤ 2500 where x represents the number of large ﬂoats and y represents the number of small ﬂoats in the parade. Make a table Students use their previous list of combinations and test them within the new constraint. Plot points Students use their previous graph and graph the new linear inequality (either on the same grid or a new one) to look for overlap.

Misconceptions Taking length restrictions into consideration This combination ﬁts within the budget, but what is its length? Does it ﬁt the length restrictions? How can we check both restrictions at the same time?

Purposeful questions Use the following questions to check for understanding and encourage critical thinking: • What are the variables in this problem? What stays constant? • Is there a pattern? Have we ever solved a problem like this before? • Is there another way to (draw, explain, say) that? • How did you organize the information?

Continue when Students have identified multiple combinations of large and small floats that fit both the length and budget restrictions.

Discuss

15 mins

Begin with group presentations. Consider connecting student representations with the graph presented by graphing software. Use student algebraic equations if they exist, or work together as a class to model this system of linear inequalities. Ask students to compare and connect their solutions to the graph.

Discussion guide The discussion will progress in two main parts: the length conditions of the parade and the budget restrictions of the parade. While groups develop ideas for how to calculate the total length of the parade, look for discussion on the space between floats. Encourage groups to sketch a few samples with different combinations of float sizes and calculate the length of the parade and the spaces in between. If groups are struggling with solutions encourage them to test the edge cases. If groups are listing out every possible combination encourage them to model the solution space in other ways (specifically, a graph but try to let this be the group’s idea). Ideally some groups in the previous work developed either algebraic inequalities or a graph to model the solution space of the length constraints. If not all groups made it to this point, encourage a share out and discussion of what different group’s work looks like. For developing the budget restriction, encourage groups to try modeling with an inequality and on a graph the budget restrictions. Again, guide students to updating a graphical model of how the solution space has been reduced now that there is a finite budget.

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Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lesson: Algebra 1 — 4.04 Two variable linear inequalities

Tools You may ﬁnd these tools helpful: • Graphing calculator • Graph paper • Blank coordinate plane • Ruler

Student lesson & teacher guide Systems of linear inequalities Students begin by exploring an example system of linear inequalities to investigate the requirements for a point to be a solution.

Students: Page 236

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Concrete-Representational-Abstract (CRA) Approach Targeted instructional strategies Concrete: Begin by providing each student with a large coordinate plane printed on paper and placed on their table. Introduce a system of linear inequalities, such as y > x + 2 and y ≤ −2x + 6. Give students sheets of tissue paper in different colors—one color for each inequality. Instruct them to graph the boundary line of the first inequality on the coordinate plane. Then, have them cut and tape a piece of the first color of tissue paper over the region that represents the solution set for that inequality. Repeat the process with the second inequality using the second color of tissue paper. When both pieces are taped onto the graph, students will see where the colors overlap, representing the solution set for the system of inequalities. This hands-on activity helps students physically visualize how the solution sets interact. Have students take this even further by providing pipe cleaners to tape along the solid boundary line to emphasize that solutions can be found on the solid boundary of the solution set but not on the other boundary. Representational: Transition to the representational stage by having students draw the inequalities on graph paper. Instruct them to accurately graph the boundary lines for each inequality. Teach them to use a dashed line for inequalities like y > x + 2 and a solid line for inequalities like y ≤ −2x + 6. Instead of tissue paper, have them use colored pencils or highlighters to lightly shade the solution region of each inequality, using the same colors as the tissue paper. The overlapping shaded area will visually represent the solution set to the system. This step connects their physical activity to a drawn representation. Abstract: Teach students how to use the algebraic inequalities to graph the boundary line and then use test points to determine the solution set.

Collect and display English language learner support As students are working, note how students describe the different aspects of the graph for the solution set of a system of inequalities, like “regions”, “boundary lines” and “test points”. Collect the different ways that students ﬁnd to understand these concepts and display them in a common place for the students to access. If students do not come up with alternative ways to word these concepts, suggest some of your own. For example: • Regions • Two linear inequalities make four regions • Possible solution sets • Correct region is where both inequalities are true • Shade the correct region • Boundary lines • Linear equations • Change the inequality symbol to = • Separates the regions • Edges of the regions • Solid if ≤ or ≥, dashed if < or > • Test points • Not on the boundary lines • Check which region is the solution • Used to check the inequalities Take care to address any rewordings that contradict or are too similar to other concepts that the student will learn in the future, such as ensuring students know that only one of the regions is the solution set, so the terms are not interchangeable.

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Support graphing with templates and/or technology Student with disabilities support Provide students with templates or technology to help with graphing linear inequalities. A template could include a partially completed graph where they need to determine some key features like where to shade and whether the line is dashed or solid. A graphing calculator or graphing software could be used to check answers or to aid in the physical graphing.

Clearly indicate the solution set Address student misconceptions While doing the work to solve a system of inequalities, it can be helpful for students to shade the solution set to each individual inequality and then have the region with overlapping shading be the solution to the system. This can sometimes appear as if the student has shaded the union of two sets, rather than their intersection. Encourage students to do just a rough sketch when shading both and then draw another more precise graph which just shades the solution set. 4

1 −4 −3 −2 −1 −1

x 1

−2

1 −4 −3 −2 −1 −1

x 1

−2

−3

−4

Three inequalities: x > −1, y > x − 3 and 2x + 5y < 13

Solution set to the system

Exploration Students: Page 236

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Suggested student grouping: Individual Students are presented with an example system of linear inequalities where both regions are graphed onto the coordinate plane. Students are instructed to move around a manipulable point to determine a relationship between the position of the point with respect to the shaded regions and whether or not the point is labeled as a solution or not. Students can investigate different examples by clicking the refresh symbol in the top-right-hand corner of the applet. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Drag the point to a spot where the label says “Solution”. What do you notice about this region? The label for the point says “Solution” when the point is contained in the region where the shaded regions for each linear inequality overlap. In other words, the point is labeled as a solution when inside the region of intersection of both linear inequalities. 2. Drag the point all the way along the dashed boundary line. What happens to the label and why do you think that is? When dragging the point along the dashed boundary line, the label remains as “Not a solution”. This happens because the dashed boundary line is not part of the solution set for that linear inequality, so it is not included in the solution set for the system. 3. Drag the point all the way along the solid boundary line. What happens to the label and why do you think that is? When dragging the point along the solid boundary line, the label changes from “Not a solution” to “Solution” when the boundary line intersects with the shaded region for the other linear inequality. This happens because the solid boundary line is included in the solution set for its linear inequality, so it is included in the solution for the system when it overlaps with the solution set for the other linear inequality. 4. How could you verify your assumptions algebraically? These assumptions could be veriﬁed algebraically by substituting points into the linear inequalities and evaluating whether or not they are true. By testing points on the boundary line we can determine general relationships that verify the assumptions made. Purposeful questions • How do the boundary lines relate to the inequalities shown? • What do the different shaded regions on the coordinate plane represent? • What does it mean for a point to be labeled as a “Solution”? Possible misunderstandings • Some of the examples involve one linear inequality having a solution set that is a subset of the solution set for the other linear inequality, or perhaps the individual solution sets will have no overlap (at least as is visible on the graph). Students may be very confused about what these situations represent, since they are not conventional systems of linear inequalities. For students where this is a blocker, instruct them to refresh the graph to get an example they are more comfortable with, and let them know that these unconventional examples will be covered later in the lesson.

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Method for graphing the solution set Targeted instructional strategies Students can sketch a graph of the solution set to a system of linear inequalities by following the steps: 1. Sketch the boundary line for each inequality. If the inequality uses ≤ or ≥ then the line is solid, otherwise it is dashed. 2. Determine which side of each boundary line satisﬁes each inequality. This can be done using test points or by observing the inequality symbol when a variable is isolated. 3. Shade the region of the graph which satisﬁes both inequalities. Following the exploration, students are given an explanation for how to represent and interpret systems of linear inequalities graphically, and are presented with some examples (both conventional and unconventional) which demonstrate the different types of systems that students should be familiar with.

Students: Pages 236–237

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Examples Students: Page 238

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Purpose Show students how to interpret a system of linear inequalities from a graph. Reﬂecting with students Ask students if they can think of another way to test the direction for the inequality signs. Let students know that they can choose any point in the shaded region, substitute it into the equation for both boundary lines, and then choose the inequality signs which would make the statements true.

Students: Pages 238–239

Purpose Show students how to check whether a point is in the solution set for a system or not. Expected mistakes Students may state that (−2, −4) is part of the solution set since it lies on the boundary line. Remind students that a point on a dashed boundary line will not be part of the solution. Reinforce this concept by showing students algebraically why the point (4, −4) is not a solution to the system.

Students: Pages 239–240

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Purpose Check that students can graph a system of linear inequalities. Reﬂecting with students Ask students if they can think of other methods for determining which region of the coordinate plane should be shaded. Possible methods include using the inequality signs to determine whether regions are above or below the boundary lines, or testing one point in each of the four distinct regions separated by the boundary lines.

Students: Page 240

Purpose Check that students can determine whether or not a point is a solution to the system. 488

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Expected mistakes Students might forget that they need to test the point in both inequalities in the system. If they only test it in one and it works, they might incorrectly conclude that it is a solution to the system.

Students: Page 240

Purpose Check that students can represent a scenario as a system of linear inequalities. Expected mistakes Students may choose the wrong direction for their inequality signs. Remind students that a ‘minimum’ is the lowest possible value, so the variables must be greater than or equal to that amount.

Students: Page 240

Purpose Check that students can graph a system of linear inequalities. Expected mistakes If students are used to shading “above” or “below” the line, they may struggle to shade the region representing the solution to the inequality x ≥ 14. Guide students to look at the values on the x-axis to help them determine the direction in which the x-values are larger than 14.

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Reflecting with students Encourage advanced learners to determine the possible ranges of successful test scores when given specific scores within the constraints. For example: • If a successful applicant scored 10 on the verbal reasoning test, what range must their quantitative reasoning score have fallen in? • If an applicant scores 14 on the quantitative reasoning test, was must they get on the verbal reasoning test? • If an applicant’s verbal reasoning score is between 15 and 19, what must they score on the quantitative reasoning test to be successful?

Students: Page 241 c Suppose the maximum of the verbal reasoning test was a score of 50. Is the solution (15, 56) a viable solution in the context?

Apply the idea No. This would mean that the score for the quantitative reasoning test was 15 and the verbal reasoning test 56. By viewing the graph, we can see that this point technically satisﬁes both inequalities, but we now know that the maximum possible score on the verbal reasoning test is a 50.

d Update the system of inequalities that models the new information about the tests.

Purpose Create a strategy Check that students can recognize when a solution is not viable.

cWe Suppose the maximum of the of verbal reasoning test was score 50. of Is the 56) a viable in now know that the maximum the verbal reasoning testa was a of score 50, solution and that (15, y represents thesolution score on context? the the verbal reasoning test.

Reﬂecting with students Ask students they can establish a domain and range for the scenario. What additional information do they Apply ifthe idea need in No. order to have upper and lower bounds for xbeand y?a score Since thewould scoremean for thethat verbal reasoning test cannot above 50, y≤ 50 may be included in This the score for the quantitative reasoning test of was 15the andinequality the verbal reasoning test 56. theviewing system the of inequalities modeling thethis relationship: By graph, we can see that point technically satisﬁes both inequalities, but we now know that the

Students:maximum Page 241 possible score on the verbal reasoning test is a 50.

d Update the system of inequalities that models the new information about the tests.

Create a strategy We nowIdea know summary that the maximum of the verbal reasoning test was a score of 50, and that y represents the score on the verbal Thereasoning solution totest. a system of inequalities lies in the region where the solutions of more than one linear inequality overlaps. Since solutions to systems of inequalities can have many solutions, we use a graph to

Apply show the idea the solution set. Since the score for the verbal reasoning test cannot be above a score of 50, the inequality y ≤ 50 may be included in the system of inequalities modeling the relationship:

Idea summary Purpose The solution to a system of inequalities lies in the region where the solutions of more than one linear Check that students can modify their modeled solution to reﬂect can new domain and range restrictions. inequality overlaps. Since solutions to systems of inequalities have many solutions, we use a graph to show the solution set.

Reﬂecting with students Students may also include some lower bounds for their inequalities, such as:

Have students justify their choice of lower bounds using reasons from the context.

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241

Apply the idea Since the score for the verbal reasoning test cannot be above a score of 50, the inequality y ≤ 50 may be included in the system of inequalities modeling the relationship:

Students: Page 241

Idea summary The solution to a system of inequalities lies in the region where the solutions of more than one linear inequality overlaps. Since solutions to systems of inequalities can have many solutions, we use a graph to show the solution set.

Practice Students: Pages 242–249

What do you remember? 1

Select the graph of the solution set of: A

10 8

4 2

−10 −8 −6 −4 −2 −2 −4 −6 −8 −10

10 8

4 2

−10 −8 −6 −4 −2 −2 −4 −6 −8 −10

x 2 4 6 8 10

Systems of −10 −8 −6 4.05 −4 −2 2 linear 4 inequalities 6 8 10 −2 mathspace.co −4 −6 −8 −10

2 4 6 8 10

−10 −8 −6 −4 −2 −2 −4 −6 −8 −10

241

x 2 4 6 8 10

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Select the graph of the solution set of: A

4 2

−10 −8 −6 −4 −2 −2 −4 −6 −8 −10

10 8

4 2

2 4 6 8 10

−10 −8 −6 −4 −2 −2 −4 −6 −8 −10

2 4 6 8 10

Select the graph of the solution set of: A

10 8

4 2

−10 −8 −6 −4 −2 −2 −4 −6 −8 −10

492

x 2 4 6 8 10

−10 −8 −6 −4 −2 −2 −4 −6 −8 −10

10 8

−10 −8 −6 −4 −2 −2 −4 −6 −8 −10

10 8

x 2 4 6 8 10

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−10 −8 −6 −4 −2 −2 −4 −6 −8 −10

x 2 4 6 8 10

30 24

−10 −8 −6 −4 −2 −6

18 6

−10 −8 −6 −4 −2 −6

2 4 6 8 10

−12

−18

−24

−30

2 4 6 8 10

Select the graph of the solution set of: A

y 10 8 6 4 2

−10 −8 −6 −4 −2 −2 −4

B 10 8 6 4 2

10 8

4 2

−10 −8 −6 −4 −2 −2 −4 −6 −8 − 10

x 2 4 6 8 10

−10 −8 −6 −4 −2 −2 −4 −6 −8 − 10

2 4 6 8 10

−6 −8 − 10

−10 −8 −6 −4 −2 −2 −4 −6 −8 − 10

2 4 6 8 10

x 2 4 6 8 10

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Write the system of inequalities that is represented on each graph: a

12 10 8 6 4 2

−10 −8 −6 −4 −2 −2 −4 −6 −8

5 4 3 2 1

−5 −4 −3 −2 −1 −1

2 4 6 8 10

2 3 4 5

−3 −4 −5

2 1

3 2 1

x 1

−7 −6 −5 −4 −3 −2 −1 −1

2 3

−2 −3

2 3 4 5

−2

−4 −5

−3 −4

−6 −7

−5

x 1

−2

−5 −4 −3 −2 −1 −1

The graph shows the region for the system of inequalities:

(−3, 2)

(2, 5)

7 6 5 4 3 2 1

(−1, −6)

(3, 2)

−3 −2 −1−1

(4, 4)

Determine whether the points are located in the region.

x 1

2 3 4 5 6 7

−2 −3 −4 −5 −6 −7

Let’s practice 7

Which ordered pair satisfies the system of inequalities? 2x − y > −6 2x + 2y < 0 A

494

(2, 6)

(−3, 6)

(1, −3)

(−6, 1)

Is the statement true or false? a

Solutions to an inequality fall only in the shaded region and not on the boundary lines.

The solution set to a linear inequality is a region of the graph and can include an infinite number of points.

All the points on the boundary line of a linear inequality are included in the solution if the inequality is “less than or equal to” or “greater than or equal to”.

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Graph each system of inequalities on the coordinate plane: a

Consider the graph of a

8 6 4 2

If the system was changed to

graph the new system of inequalities on a coordinate plane. b

Is (−6, −8) a solution to the system in part (a)?

If the system was changed to

Is (0, 6) a solution to the system in part (a)?

2 4 6 8 10

12 10 8 6 4 2

graph the new system of inequalities on the coordinate plane. b

−10 −8 −6 −4 −2 −2 −4 −6 −8 −10 −12

Consider the graph of a

−12−10 −8 −6 −4 −2 −2 −4 −6 −8

x 2 4 6 8

Consider the following graphs: Graph 1 10 8 6 4 2 −6 −4 −2 −2 −4 −6 −8 −10

Graph 2

x 2

8 10 12

10 8 6 4 2 −6 −4 −2 −2 −4 −6 −8 −10

x 2

8 10 12

Graph 1 shows the solution set to the system:

If Graph 2 shows the solution set when one of the inequalities in the system is changed, state the new system of inequalities.

Is (3, 0) a solution to either system? Explain how you know. 4.05 Systems of linear inequalities mathspace.co

495

For each system of inequalities: i

Find the x-coordinate of the point at which the boundary lines intersect.

Find the y-coordinate of the point of intersection.

iii

Graph the solution set.

x ≤ 5 and y < 3

y ≤ 3x − 4 and y > −4x + 1

3x + y > 5 and 3x + y < 7

y < 4x − 2 and y < 3x − 3

Sheila sells popcorn to raise money during summer break. The popcorn comes in two ﬂavors, classic butter which costs $2 each and cheese which costs $3 each. Sheila needs to sell at least $50 worth of popcorn with at least 8 of the cheese popcorn. Let b be the number of butter popcorn and c be the number of cheddar popcorn. This situation can be represented by the following system of equations:

Select all the viable solutions.

b = 6, c = 13

b = 18, c = 6

b = 10, c = 10

b = 11, c = 9

b = 13, c = 8

Applicants for a particular university are asked to sit a numeracy test and verbal reasoning test. Successful applicants must obtain a minimum score of 17 on the numeracy test and a minimum combined score of 37 for both tests. Let x and y represent an applicant’s score on the numeracy and verbal reasoning test respectively. a

It’s not possible to get a negative score therefore y ≥ 0 is one inequality. Write two more inequalities from the information in terms of x and y.

Graph the solution to the system of inequalities.

Which points represent scores that would make the applicant successful? i

(14, 16)

(33, 18)

iii

(20, 13)

(17, 22)

If an applicant obtains a score of 24 in the numeracy test, find the minimum integer score they need to obtain in the verbal reasoning test to be successful.

Yreka Bakery makes two types of cookies: plain and iced. They have enough oven space to bake 15 dozen cookies each day. Each dozen iced cookies requires 0.8 pounds of icing. Yreka Bakery can make no more than 32 pounds of icing per day. If x is the number of dozens of iced cookies and y is the number of dozens of plain cookies, write a system of inequalities to represent the scenario.

Throughout university, Jimmy works as a barista, getting paid $10 per hour, and as a mentor getting paid $13 per hour. The number of hours he works in each job can vary from week to week, but he never works more than 27 hours in total each week, and he needs to be able to at least cover his weekly expenses of $260. If b represents the number of hours worked as a barista and m represents the number of hours worked as a mentor, write a system of inequalities to represent the scenario.

Let’s extend our thinking 18

496

Without graphing, determine whether each system of inequalities has no solutions or infinitely many solutions. Explain your answer. a

y > 4 and y < 4

x ≤ −2 and x ≥ −2

9x − y < 7 and 9x − y > 7

2x − y ≤ 7 and 2x − y ≥ 7

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Is it possible for an ordered pair to satisfy one part of the linear inequality pair but not the other? Explain with a diagram what this would look like.

David’s Pizza makes two types of pizzas: Vegan and Meat Lover’s. Based on recent sales they know they will sell at least twice as many Meat Lover’s Pizzas as they do Vegan Pizzas. David’s Pizza can use 42 pounds of vegan cheese each day, and each Vegan Pizza requires 0.6 pounds of vegan cheese.

If v is the number of Vegan Pizzas and m is the number of Meat Lover’s Pizzas, write a system of inequalities to represent the scenario.

Is (28, 70) a viable solution in terms of the scenario? Explain.

Is (25.25, 71) a viable solution in terms of the scenario? Explain.

In a team CrossFit competition, each team is to be made up of no more than 10 people. For a particular challenge, each team member must complete the course and the team’s total time in completing the course must be under 20 minutes. Women take on average 2.4 minutes and men take on average 1.5 minutes to complete the course. a

If w is the number of women and m be the number of men on the team, write a system of inequalities to represent the scenario.

Graph the system of inequalities.

Give an example of one viable and one non-viable solution to the number of men and women on the team. Explain your answer for both.

Throughout university, Tom works as a mentor, getting paid $15 per hour, and as a ﬁtness instructor getting paid $18 per hour. The number of hours he works in each job can vary from week to week, but he never works more than 29 hours in total each week, and he needs to be able to at least cover his weekly expenses of $270. a

If x represents the number of hours Tom works as a mentor and y represents the number of hours he works as a ﬁtness instructor, construct a system of inequalities to represent the scenario.

Graph the system of inequalities.

If Tom works 6 hours as a mentor in one week, what is the minimum number of hours he can work as a ﬁtness instructor so that he can cover his expenses?

Is it possible for him to work the same number of hours in both jobs and still be able to cover his expenses?

A bicycle manufacturer employs a mechanic and a painter to construct two types of bikes: racing bikes and mountain bikes. The time each worker spends on each bike is shown (assume both workers can work on the same bike at the same time). Let x represent the number of racing bikes built and y represent the number of mountain bikes built. Each worker can work at most 40 hours in a week. Mechanic

Painter

Racing Bike

5 hours

10 hours

Mountain Bike

8 hours

4 hours

Write an inequality relating x and y to the total time spent working by the mechanic.

Write an inequality relating x and y to the total time spent working by the painter.

4.05 Systems of linear inequalities mathspace.co

497

Answers

8 6

4.05 Systems of linear inequalities

4 2

What do you remember?

−8 −6 −4 −2 −2

1 C

2 4 6 8

−4

2 C

−6 −8

3 B 4 B

10 a y c x < 1, y ≤ 3

5 a

8 6 4 2

d x < 1 and y < 1 and y ≥ − x − 5 6 a No

b Yes

c No

b True

c True

−10 −8 −6 −4 −2 −2 −4 −6 −8 −10 −12

d No

e Yes Let’s practice

b Yes

2 4 6 8 10

7 C 8 a False 9 a

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

11 a y 12 10 8 6 4 2

−12−10 −8 −6 −4 −2 −2 −4 −6 −8

1 2 3 4 5

b No

x 2 4 6 8

12 a b

5 4 3 2 1

−3 −2 −1 −1 −2 −3 −4 −5

6 5 4 3 2 1 −3−2−1 −1 −2 −3 −4 −5 −6

498

b N o, since both systems include at least one strict inequality, their point of intersection is not a valid solution. 13 a i 5

1 2 3 4 5 6 7

iii

x 1 2 3 4 5 6 7 8 9

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

ii 3 6 5 4 3 2 1

−4−3−2−1 −1 −2 −3 −4 −5 −6

x 1 2 3 4 5 6 7 8

ii −1.857

b i 0.714 iii

6 5 4 3 2 1 −6−5−4−3−2−1 −1 −2 −3 −4 −5 −6

17 x

Let’s extend our thinking

1 2 3 4 5 6

18 a N o solutions. The regions are on opposite sides of the line y = 4. b I nfinitely many solutions. The regions have the infinite set of points on the line x = −2 in common. c N o solutions. The regions are on opposite sides of the boundary line.

c i No point of intersection ii No point of intersection iii

7 6 5 4 3 2 1

d I nfinitely many solutions. The regions have the infinite set of points on the boundary line in common.

19 Yes. In the diagram, the dashed lines represent the two inequalities y = x + 2 and y = 3x + 2. The darker shaded region is the intersection of the regions that satisfy the inequalities y < x + 2 and y < 3x + 2. There are points like (−2, −2) that solves y < x + 2 but not y < 3x + 2.

−4 −3 −2 −1 −1 −2 −3 −4

d i −1

2 3 4

5 4 3 2 1

ii −6

iii

4 3 2 1 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7 −8 −9

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1

2 3 4

b Y es, since 28 vegan pizzas and 70 meat lovers pizzas would satisfy the constraints, and pizzas can only be made in whole numbers. c N o. While (25.25, 71) satisfies the constraints, you cannot cook a fraction of a pizza.

15 a x ≥ 17, x + y ≥ 37 36 32 28 24 20 16 12 8 4

21 a b

x 4 8 12 16 20 24 28 32 36

c i No d 13

x 1 2 3 4 5

20 a

14 A, B, D

ii Yes

iii No

iv Yes

14 y 13 12 11 10 9 8 7 6 5 4 3 2 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Answers mathspace.co

499

c V iable solution: Any point which satisfies the constraints and is made up of positive integers, since the variables represent the number of men and women. For example, (3, 4). Nonviable solution: Any point which includes at least one non integer value, since you cannot have part of a person as a team member. For example, (3.1, 4). 22 a b

y 30 25 20 15 10 5 x 5

c 10

d Yes

23 a 0 ≤ 5x + 8y ≤ 40 b 0 ≤ 10x + 4y ≤ 40

500

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Topic 4 Assessment: Systems of Equations & Inequalities 1

State whether the point (−3, 4) is a solution to each system of equations: a

Estimate the solution of the system of equations: 5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1 2 3 4 5

Solve these systems of equations: a

Consider this system of equations: a

Explain how to eliminate the variable y.

Explain why eliminating y would help solve the system of equations.

The owner of an amusement part wants to purchase a theme park train that has 12 cabs. Some cabs hold 3 people while some hold 6 people. The owner plans to have the train carry 78 people. Let x be the number of three-passenger cabs and y be the number of six-passenger cabs. This situation can be represented by the following system of equations:

Graph and solve the system of equations.

Determine whether the solution is viable in terms of the context.

A group of 5 adults and 2 children paid $78 for the movie tickets while a group of 9 adults and 4 children paid $144. Find the cost of a movie ticket for an adult.

Robert went to the school supplies store. He bought 3 pens and 6 notebooks. He paid $14.58. Jennifer bought 2 pens and 5 notebooks at the same store. She paid $11.72. Let p represent the price of a pen and n represent the price of a notebook. a

Write a system of equations.

Solve for p, the price of a pen, and for n, the price of a notebook.

Topic 4 Assessment: Systems of Equations & Inequalities mathspace.co

501

Determine whether each system of two linear equations has one solution, no solution, or an inﬁnite number of solutions. a

3x + 6 = 1 b y = −2x – 11 x − y = −1 4x + 2y = −22

2a + b = 23 d b = −2a + 35

5x − 3y = 12 x − 0.6y = −4

The graph of y = −4x + 2 is shown.

A second line, y = mx + b, intersects this line at the one point (0, 2).

What value of m is not possible? Explain your answer.

4 3 2 1 −2

−1

x 1

−2

SOL

Elle began graphing the system of inequalities shown:

7 4

To complete the graph, Elle must shade the region which represents the solution set to the system of inequalities. What region of the graph needs to be shaded?

5 3

4 3

−9−8−7−6−5−4−3−2 −1 −1

1 2 3 4 5 6 7 8 9

−2 −3 −4

SOL

Using the inequalities shown, create a system of two inequalities that could be represented by this graph.

≥

≤

9 8 7 6 5 4 3 2 1 −9−8−7−6−5−4−3−2 −1−1 −2 −3 −4 −5 −6 −7 −8 −9

502

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

x 1 2 3 4 5 6 7 8 9

For each system of inequalities on the coordinate plane: i

Sketch each of the inequalities separately on the coordinate plane.

Sketch the solution set of the system of linear inequalities.

a 13

Consider the system of inequalities:

State whether each point is a solution to the system: a SOL

(2, 0)

(6, 4)

(−7, −6)

What is the solution to this system of equations?

A 16

For which system of inequalities is (−2, 3) a solution? A

SOL

(3, −5)

(1, 4)

(2, 3)

(3, 2)

(4, 1)

A novelist makes a proﬁt of $7.5 on every copy sold online and $2 on every copy sold in store. The novelist wants to make a proﬁt of at least $220 a day selling copies through online and in-store sales. a

Construct an inequality and graph to model the scenario. Deﬁne the variables used.

Given the context, determine whether the ordered pair (−5, 130) is a valid solution to the inequality. Justify your answer.

Emma makes a blend of coffee which uses a $2 per pound grade of coffee beans and a $5 per pound grade of coffee beans. She uses at least twice as many $5 per pound beans than $2 per pound beans. The blend of coffee beans costs at most $38. Let x be the number of pounds of $2 per pound beans and y be the number of pounds of $5 per pound beans.

SOL

Write a system of inequalities to represent the scenario.

Is (2, 6) a viable solution in terms of the scenario? Explain.

Is (4, 6.5) a viable solution in terms of the scenario? Explain.

Xyla can spend no more than $30 to buy almonds and walnuts. She will pay $5 per pound for almonds and $10 per pound for walnuts. Which graph best represents the number of pounds of almonds and the number of pounds of walnuts Xyla can buy? Snacks Snacks A B 9

Walnuts (lb)

Almonds (lb) 1

2 3 4 5 6 7 8 9

Almonds (lb) 1

2 3 4 5 6 7 8 9

Topic 4 Assessment: Systems of Equations & Inequalities mathspace.co

503

Snacks 9

Snacks

Walnuts (lb)

Almonds (lb) 1

2 3 4 5 6 7 8 9

Walnuts (lb)

Almonds (lb) 1

2 3 4 5 6 7 8 9

Performance task 19

Rosa is volunteering at the local preschool for their annual scavenger hunt where four groups of students will follow clues to search in the nearby park for their group’s hidden trophy. Rosa has been tasked with placing the trophy for each group. She has mapped out the route that each group will follow, but she only has time to make one straight pass through to drop off the trophies. These are the routes that each group of students will take. Group A: 2x + 10 = y Group B: x − 5y = −52 Group C: y = −x + 2 Group D: 3y − x = −8 Each group starts on the west side of the park at x = −10, they move across the park from west to east until they reach the other side at x = 10

504

Draw the graph of each group’s path and determine the equation of the path Rosa should follow to drop off the trophies and identify the coordinates where she should drop each trophy. Remember Rosa’s path must be a straight line.

Explain how you decided on the best path for Rosa and the locations for the trophy drops. Is there another option for the path Rosa might take and where she could drop the trophies?

Based on your first choice for where to hide the trophies, which group do you think might find their trophy first? Explain the factors that might contribute to who finds their trophy first.

Ahmad works for a popular concert venue and is planning their ﬁrst event of the summer. He needs to hire a media company to take pictures and video of the event to be used in future marketing campaigns. The company charges $12 per photo and $5 for each minute of video. Ahmad has a budget of $2500 for media. a

Write an inequality to represent these constraints.

What are the additional constraints on this scenario that are not modeled by the inequality? Explain why they exist in this context and write additional inequalities to represent them.

If Ahmad requests 2 hours of video, what is the maximum number of photos his budget will allow? Explain.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Answers

10 A A.EI.2g

Topic 4 Assessment: Systems of Equations & Inequalities 1 a No

b Yes

c Yes

d No A.EI.2g

A.EI.2h 2 (1.5, −4)

12 a i 4

A.EI.2b

3 a x = 3, y = 26

y≤2

c x = −4, y = 28

d p = −1, q = 3

−4 −3 −2 −1 −1

A.EI.2b

−2

4 a M ultiply the ﬁrst equation by 70 and multiply the second equation by 2100. Then add the equations. b T he result after using elimination is a linear equation with one unknown variable which can be solved using inverse operations.

−3 −4

A.EI.2h

5 a x = −2, y = 14

−4 −3 −2 −1 −1

−2

−3

−4

x y ≥ 3x − 4

4 2 −8 −6 −4 −2 −2

x 2

b No A.EI.2b, A.EI.2h

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

6 A movie ticket for the adults costs $12. A.EI.2a, A.EI.2b, A.EI.2h

1 2 3 4 5

b p = $0.86, n = $2.00

7 a A.EI.2a, A.EI.2b

b i 4

8 a One solution b Inﬁnite number of solutions

3 2

c No solutions

d No solutions

−4 −3 −2 −1 −1

A.EI.2c 9 The value of m ≠ −4. Since the lines intersect at the one point (0, 2), we know that b = 2. If m = −4 and b = 2, then the system would have an inﬁnite number of solutions, however we want a single solution or point of intersection, so m ≠ −4.

−2

x 1

3x + 7y < −14

−3 −4

A.EI.2c

Topic 4 Assessment: Systems of Equations & Inequalities mathspace.co

505

17 a

b Y es, since 2 pounds of $2 per pound beans and 6 pounds of $5 per pound beans would satisfy the constraints and both of the numbers of pounds are nonnegative real values.

2 1

−4 −3 −2 −1 −1

4x − 5y ≥ 10

−2

c No. The point (4, 6.5) does not satisfy either inequality. If we substitute x = 4 and y = 6.5, we get 2(4) + 5(6.5) = 40.5 > 38. We can also see that the amount of $5 per pound beans is not at least twice the amount of $2 per pound beans.

−3 −4

4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6

A.EI.2f, A.EI.2h 18 B x

A.EI.2e

1 2 3 4 5

Performance task 19 a

A.EI.2e, A.EI.2g B

13 a No

b Yes

c No

d Yes

A.EI.2h 14 B, C

A.EI.2b

−9−8−7−6−5−4−3−2 −1 −1 −2 −3 −4 D −5 −6

15 C A.EI.2b 16 a Let x = the number of copies sold online, and y = the number of copies sold in store on one day. If the proﬁts were based on the least amount the book seller makes on the books, an inequality may be 7.5x + 2y ≥ 220. 110

Number of copies sold in store

100 90 80 70 60 50 40 30 20 10

Number of copies sold online 5 10 15 20 25 30 35 40 45 50 55

b T he ordered pair (−5, 130) is not a valid solution since the novelist cannot sell a negative number of copies. A.EI.2d, A.EI.2h

506

14 y 13 12 11 10 9 8 7 6 5 4 3 2 1

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

1 2 3 4 5 6 7 8 9

The equation for Rosa’s path will vary. It must be a straight line that intersects each of the other lines. Sample equation:

b Answers will vary. Sample answer: I started at the point where the paths of groups B and C will cross because I thought it would save Rosa time to drop two trophies off at the same spot. Then I drew a line from that point that intersected with the other two lines. I counted the slope and then used the point-slope formula to write the equation of the line before converting it into slope-intercept form. There are many other paths Rosa could take that would intersect with all four lines. Such as the vertical path along x = −6. c Answers will vary. Sample answer. It depends where each group starts from. If they all start from the left side of the graph at x = −10 then group B would probably find their trophy first with group C finding theirs shortly after. If they started from the right, group D would find theirs first. But either way group A would find theirs somewhere in

the middle. But this all depends on how quickly each group moves and how fast they figure out their scavenger hunt clues. So, it is possible that a group whose trophy is the farthest away could find it first. A.EI.2a, A.EI.2b, A.EI.2h, MP1, MP2, MP3 20 a 12p + 5v ≤ $2500, where p represents the number of photos taken and v represents the length of the video footage in minutes. b Additional constraints: p ≥ 0 : Ahmad cannot request a negative number of photos. v ≥ 0 : Ahmad cannot request a negative length of video. p and v should be whole numbers since partial photos or video minutes are impractical. c 2 hours of video is equivalent to 120 minutes of video. Substituting 120 into the inequality for v and solving for p gives p ≤ 158.33, so the maximum number of photos his budget will allow is 158 because you cannot take a third of a photo so we must round down. A.EI.2d, A.EI.2f, A.EI.2g, A.EI.2h, MP4, MP5

Topic 4 Assessment: Systems of Equations & Inequalities mathspace.co

507

Exponents, Radicals, & 5 Exponential Functions Big ideas • Changing the form of an expression or equation can reveal information that was previously unknown. • Expressions are the building blocks of algebra. They can be used to represent and interpret real-world situations. • The properties of real numbers can be applied to many types of expressions. • A family of functions is defined by a unique set of characteristics shared by all functions that belong to that family. These characteristics give insight into the types of real-world situations that a function models.

Chapter outline 5.01 5.02 5.03 5.04 5.05 5.06 5.07 5.08 5.09

Product rule (A.EO.3) Power rule (A.EO.3) Quotient rule (A.EO.3) Zero and negative exponents (A.EO.3) Rational exponents (A.EO.4) Simplify radicals (A.EO.4) Operations with numerical radicals (A.EO.4) Characteristics of exponential functions (A.F.2) Graphs of exponential functions (A.F.2) Topic 5 Assessment

513 525 538 552 569 589 600 614 630 653

The arrangement of seeds in a pinecone follows a pattern that can be represented using exponents and radicals.

5. Exponents, Radicals, & Exponential Functions Topic overview Foundational knowledge Evaluating standards proficiency The skills book contains questions matched to individual standards. It can be used to measure proficiency for each. Students should be proficient in these standards. 6.NS.3 — The student will recognize and represent patterns with whole number exponents and perfect squares. 7.CE.1 — The student will estimate, solve, and justify solutions to multistep contextual problems involving operations with rational numbers. 7.NS.1 — The student will investigate and describe the concept of exponents for powers of ten and compare and order numbers greater than zero written in scientific notation.

7.NS.3 — The student will recognize and describe the relationship between square roots and perfect squares. 8.NS.1 — The student will compare and order real numbers and determine the relationships between real numbers. 8.PFA.1 — The student will represent, simplify, and generate equivalent algebraic expressions in one variable. A.F.1 — The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.

Big ideas and essential understanding Changing the form of an expression or equation can reveal information that was previously unknown. 5.01 — The law for multiplying exponential expressions with the same base allows us to simplify complex expressions and uncover underlying patterns that inform more efficient methods for computation and analysis. 5.02 — The law for raising an exponential expression to another exponent allows us to simplify complex expressions and uncover underlying patterns that inform more efficient methods for computation and analysis. 5.03 — The law for dividing exponential expressions with the same base allows us to simplify complex expressions and uncover underlying patterns that inform more efficient methods for computation and analysis.

510

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

5.04 — Understanding the underlying patterns for zero and negative exponents can inform more efficient methods for computation and analysis. 5.06 — The special relationship between square roots and perfect squares and cube roots and perfect cubes allows us to reveal the simplest form of a radical expression. Expressions are the building blocks of algebra. They can be used to represent and interpret real-world situations. 5.05 — The laws of integer exponents also apply to rational exponents and can be useful when simplifying expressions.

The properties of real numbers can be applied to many types of expressions. 5.07 — Operations can be applied to radical expressions in much the same way that they can be applied to real numbers. In fact, many radical expressions are real numbers.

A family of functions is defined by a unique set of characteristics shared by all functions that belong to that family. These characteristics give insight into the types of real-world situations that a function models. 5.08, 5.09 — Exponential functions have a constant growth factor, often referred to as the common ratio, and can be represented by the equation y = abx. 5.08 — Exponential functions are characterized by a constant growth factor. This determines what types of real world situations exponential functions can model.

Standards A.EO.3 — The student will derive and apply the laws of exponents. A.EO.3a — Derive the laws of exponents through explorations of patterns, to include products, quotients, and powers of bases. 5.01 Product rule 5.02 Power rule 5.03 Quotient rule 5.04 Zero and negative exponents A.EO.3b — Simplify multivariable expressions and ratios of monomial expressions in which the exponents are integers, using the laws of exponents. 5.01 Product rule 5.02 Power rule 5.03 Quotient rule 5.04 Zero and negative exponents

A.EO.4d — Generate equivalent numerical expressions and justify their equivalency for radicals using rational exponents, limited to rational exponents of

and

. 5.05 Rational exponents A.F.2 — The student will investigate, analyze, and compare characteristics of functions, including quadratic and exponential functions, and model quadratic and exponential relationships.

A.F.2e — Given an equation or graph of an exponential function in the form y = abx (where b is limited to a natural number), interpret key characteristics, including y-intercepts and domain and range; interpret key characteristics as related to contextual situations, where applicable. A.EO.4 — The student will simplify and determine 5.08 Characteristics of exponential functions equivalent radical expressions involving square roots of 5.09 Graphs of exponential functions whole numbers and cube roots of integers. A.F.2f — Graph an exponential function, f (x), in two A.EO.4a — Simplify and determine equivalent radical variables using a variety of strategies, including expressions involving the square root of a whole number transformations f (x) + k and kf (x), where k is limited to in simplest form. rational values. 5.06 Simplify radicals 5.09 Graphs of exponential functions A.EO.4b — Simplify and determine equivalent radical expressions involving the cube root of an integer. 5.06 Simplify radicals A.EO.4c — Add, subtract, and multiply radicals, limited to numeric square and cube root expressions. 5.07 Operations with numerical radicals

A.F.2g — For any value, x, in the domain of f, determine f (x) of a quadratic or exponential function. Determine x given any value f (x) in the range of f of a quadratic function. Explain the meaning of x and f (x) in context. 5.08 Characteristics of exponential functions

5. Exponents, Radicals, & Exponential Functions mathspace.co

511

Future connections A.EI.3 — The student will represent, solve, and interpret the solution to a quadratic equation in one variable. A2.EO.2 — The student will perform operations on and simplify radical expressions.

A2.EO.3 — The student will perform operations on polynomial expressions in two or more variables and factor polynomial expressions in one and two variables. A2.F.2 — The student will investigate and analyze characteristics of square root, cube root, rational, polynomial, exponential, logarithmic, and piecewisedefined functions algebraically and graphically.

512

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

5.01 Product rule Subtopic overview Lesson narrative In this lesson, students will determine the product of powers. Students will engage in an interactive exploration, where they discover the relationship between exponential and expanded forms of two numbers in exponential form being multiplied together. Students will recognize that multiplying terms with like bases means to add the powers. By the end of this lesson, students will be able to find the product of powers expression.

Learning objectives

5.01 Product rule

Students: Page 252

After this lesson, you will be able to… • derive the product of powers from patterns. • use the product of powers law to simplify algebraic expressions.

Product rule

Key vocabulary 

Interactive exploration

exponential form

Explore online to answer the questions



product rule

mathspace.co

Essential understanding Usefor the interactiveexponential exploration in 5.01 to answer questions. The law multiplying expressions withthese the same base allows us to simplify complex expressions and 3 uncover1.underlying inform more efficient methods computation and analysis. When a patterns is writtenthat in expanded form, how many a’s arefor being multiplied? 2.

When a3 ⋅ a2 is written in expanded form, how many a’s are being multiplied?

By counting the number of a’s being multiplied in expanded form, what is a3 ⋅ a2 in exponential form?

Standards 4. By counting the number of a’s being multiplied in expanded form, what is a4 ⋅ a5 in exponential form? 5. Isaddresses there a faster to multiply terms with Mathematics the same base? This subtopic theway following Virginia 2023 Standards of Learning standards.

Mathematical process goals

Teachers canthe incorporate by guiding students through exploration of exponent patterns by Consider expressionthis a5 ⋅goal a3. Notice that the terms share like the bases. demonstrating how the Product Law is derived from multiplying two exponential expressions with the same base by Let’s think about what this would look like if we distributed the expression: breaking it down into repeated addition of exponents. Students can articulate their understanding by presenting their a5 ⋅ a3 = (a ⋅ a ⋅ a ⋅ a ⋅ a)(a ⋅ a ⋅ a) findings to the class, creating a written summary, or engaging in group discussions where they justify and explain = a ⋅ a ⋅ a ⋅ and a ⋅ aderivations, ⋅a⋅a⋅a a5 ⋅ a3processes their reasoning. Students can share their thought using mathematical language, notation, 5 3 8 and symbols to clearly express their ideas. a ⋅ a = a We can see that there are eight a’s being multiplied together, and notice that 8 is the sum of the powers in the original expression. We can write this in exponential form as a8, where a is the base and 8 is the power. So, in our example above, a5 ⋅ a3 = a5 + 3 = a8

5.01 Product rule

mathspace.co We can avoid having to write each expression in expanded form by using the product rule. It states that when multiplying two powers with the same base, we add the powers.

For any base number a, and any numbers m and n as powers, am ⋅ an = am + n

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MPG3 — Mathematical Reasoning

MPG4 — Mathematical Connections

Teachers can support this goal by facilitating activities where students have to use problem-solving skills to understand and apply the Product Law. For example, students could be tasked to justify why the order in which multiplication is applied does not affect the result or to determine whether a given mathematical statement involving the Product Law is valid or not.

Teachers can incorporate this goal by helping students to see how the concept of exponents and the Product Law connect with other mathematical concepts they’ve learned, such as the commutative property, order of operations, and the distributive property. They should also connect these concepts to different areas of mathematics and other disciplines. For example, teachers can highlight the use of the Product Law in scientific notation, which is widely used in science.

Content standards A.EO.3 — The student will derive and apply the laws of exponents. A.EO.3a — Derive the laws of exponents through explorations of patterns, to include products, quotients, and powers of bases.

A.EO.3b — Simplify multivariable expressions and ratios of monomial expressions in which the exponents are integers, using the laws of exponents.

Prior connections 6.NS.3 — The student will recognize and represent patterns with whole number exponents and perfect squares.

7.NS.1 — The student will investigate and describe the concept of exponents for powers of ten and compare and order numbers greater than zero written in scientific notation.

Future connections A2.EO.3 — The student will perform operations on polynomial expressions in two or more variables and factor polynomial expressions in one and two variables.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 6 — 3.03 Introduction to exponents Grade 7 — 1.05 Scientific notation

Tools You may ﬁnd these tools helpful: • Paper • Straightedge

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Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Lesson supports The following supports may be useful for this lesson. More speciﬁc supports may appear throughout the lesson:

Develop pattern analysis skills Targeted instructional strategies As students write out expressions in expanded form to find a generalized form for simplifying products, encourage them to organize their findings in a table and to cover a variety of examples including special cases. Encourage them to observe and analyze the data to recognize patterns between the exponents and the resulting product. Through this pattern analysis, guide students to conclude that when multiplying powers with the same base, the exponents are added. This will build their pattern analysis skills that will be used throughout this and other topics.

Stronger and clearer each time English language learner support Place students in pairs and give them an example like 45 ⋅ 43. Have students write down how they would simplify this expression. Give students time to think about how they will explain what they did to solve the problem verbally. Then, without looking at what they wrote down, let students explain to their partner their answer and what strategy they used to solve it. Encourage partners to ask clarifying questions such as, “How do you know the answer is correct?” or “What other method could you use to verify this answer?” Have students read what they wrote and revise using newly learned words/information as it applies. This will provide students with an opportunity to use terms they learned in the lesson like “base”, “exponent”, “expanded form”, or “product rule”.

Graphic organizer for exponent laws Student with disabilities support Students will create a graphic organizer with three columns, using paper and a straightedge, to organize the exponent laws. In the ﬁrst column, students will list the name of the law. In the second column, students will write the law algebraically. In the third column, students will write a worked example of the law. Name of law Product rule

Exponent law am ⋅ an = am + n

Example x2 ⋅ x5 = x7

Students can add to and reference this graphic organizer throughout the unit.

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Students may incorrectly multiply the exponents Address student misconceptions Students may incorrectly multiply the exponents when simplifying an expression. For example, they might show the following work: a5 ⋅ a2 = a10 Remind students exponents represent the number of times that the base is being multiplied to itself. Encourage them to write expressions in expanded form rather than applying the rule directly. This may need to be reiterated and shown multiple times throughout the lesson as students develop their understanding of the rule.

Student lesson & teacher guide Product rule Students will start the lesson by engaging in an exploration.

Exploration Students: Page 252

5.01 Product rule After this lesson, you will be able to… • derive the product of powers from patterns. • use the product of powers law to simplify algebraic expressions.

Product rule Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 5.01 to answer these questions. 1.

When a3 is written in expanded form, how many a’s are being multiplied?

When a3 ⋅ a2 is written in expanded form, how many a’s are being multiplied?

By counting the number of a’s being multiplied in expanded form, what is a3 ⋅ a2 in exponential form?

By counting the number of a’s being multiplied in expanded form, what is a4 ⋅ a5 in exponential form?

Is there a faster way to multiply terms with the same base?

When multiplying a number by itself repeatedly, we are able to use exponent notation to write the expression more simply. Here, we are going to look at a rule that allows us to simplify products of expressions with exponents. Consider the expression a5 ⋅ a3. Notice that the terms share like bases. Let’s think about what this would look like if we distributed the expression: a5 ⋅ a3 = (a ⋅ a ⋅ a ⋅ a ⋅ a)(a ⋅ a ⋅ a) a 5 ⋅ a3 = a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a

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Mathspace Virginia SOL Algebra 1 Teacher Edition a 5 ⋅ a 3 = a8 mathspace.co We can see that there are eight a’s being multiplied together, and notice that 8 is the sum of the powers in the original expression. We can write this in exponential form as a8, where a is the base and 8 is the power.

Suggested student grouping: In pairs This exploration aims to build an understanding of the power rule. Students will use a GeoGebra applet to compare a product of two exponential expressions with the same base with its expanded form. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. When a3 is written in expanded form, how many a’s are being multiplied? There are three a’s being multiplied together. 2. When a3 ⋅ a2 is written in expanded form, how many a’s are being multiplied? There are five a’s being multiplied together. 3. By counting the number of a’s being multiplied in expanded form, what is a3 ⋅ a2 in exponential form? The exponential form of a3 ⋅ a2 is a5. 4. By counting the number of a’s being multiplied in expanded form, what is a a4 ⋅ a5 in exponential form? The exponential form of a4 ⋅ a5 is a9.

5.01 Product rule

5. Is there a faster way to multiply terms with the same base? Yes, a faster way to multiply terms with the same base is to add the exponents together. Purposeful questions After this lesson, you will be able to… • What does an exponent represent? • derive the product of multiplication. powers from patterns. • Exponents represent repeated If we multiply a number 3 times (a3) and multiply it two more • use the product of powers law to simplify algebraic expressions. 2 times (a ), how many times have we multiplied the number altogether? • What is the relationship between the two original exponents and the number of a’s in expanded form? • How can we simplify the expression without expanding it?

Product rule

Possible misunderstandings

Interactive • When looking at numericalexploration bases, students might want to multiply the base by the exponent. Remind them Explore online to answer Explore online to answer the questions that exponents tell us the numberthe of questions times a base is multiplied by itself, and relate this back to the expanded form of the expression. mathspace.co • When focusing on the exponents, students might not realize that the base is the same in all the expressions. Point out this rule only worksinwhen bases the same. Usethat the interactive exploration 5.01 to the answer theseare questions. 1.

When a33 is written in expanded form, how many a’s are being multiplied?

By counting the number of a’s being multiplied in expanded form, what is a44 ⋅ a55 in exponential form?

Is there a faster way to multiply terms with the same base?

3 2 the product rule. They will discover that when multiplying powers with the same base, Students will be2.introduced 2 is written in expanded form, how many a’s are being multiplied? When a3 ⋅ ato 3 2 the exponents 3. can By becounting added the to simplify expression. number ofthe a’s being multiplied in expanded form, what is a3 ⋅ a2 in exponential form?

Students: Pages 252–253 5. Is there a faster way to multiply terms with the same base?

Let’s think about what this would look like if we distributed the expression: 5 3 a5 ⋅ a3 = (a ⋅ a ⋅ a ⋅ a ⋅ a)(a ⋅ a ⋅ a)

a55 ⋅ a33 = a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a 5 3 8 a 5 ⋅ a 3 = a8

We can see that there are eight a’s being multiplied together, and notice that 8 is the sum of the powers in the original expression. 8 We can write this in exponential form as a8, where a is the base and 8 is the power.

So, in our example above, a55 ⋅ a33 = a55 ++ 33 8 = a8 We can avoid having to write each expression in expanded form by using the product rule. It states that when multiplying two powers with the same base, we add the powers. m m+ +n n For any base number a, and any numbers m and n as powers, am ⋅ ann = am

That is, when multiplying terms with a common base: • Keep the same base • Find the sum of the exponents

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a5 ⋅ a3 = (a ⋅ a ⋅ a ⋅ a ⋅ a)(a ⋅ a ⋅ a) a 5 ⋅ a3 = a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a a 5 ⋅ a 3 = a8 We can see that there are eight a’s being multiplied together, and notice that 8 is the sum of the powers in the original expression. We can write this in exponential form as a8, where a is the base and 8 is the power. So, in our example above, a5 ⋅ a3 = a5 + 3 = a8 We can avoid having to write each expression in expanded form by using the product rule. It states that when multiplying two powers with the same base, we add the powers. For any base number a, and any numbers m and n as powers, am ⋅ an = am + n That is, when multiplying terms with a common base: • Keep the same base • Find the sum of the exponents When using the product rule for exponents, the coefficients are handled separately from the exponents. Let’s take a look at an example. 3a2 ⋅ 4a5 • Multiply the numerical coefficients: 3 ⋅ 4 = 12 252 Mathspace Virginia SOL Algebra 1 • Apply the product rule to the exponents: a2 ⋅ a5 = a2 + 5 = a7 mathspace.co The final result is: 3a2 ⋅ 4a5 = 12a7 When using the product rule for exponents, the coefficients are handled separately from the exponents. Let’s take a look at an example.

Example 1

3a2 ⋅ 4a5 • Multiply the numerical coefficients: 3 ⋅ 4 =2 12 ⬚ Fill in the blank to make the equation true: b ⋅2b 5= b2 +23+ 5 • Apply the product rule to the exponents: a ⋅ a = a = a7

Examples

The final result is:

strategy Students:Create Pagea 253

We can use the exponent law: am ⋅ an = am + n

3a2 ⋅ 4a5 = 12a7

Apply the idea Example 1 Since we know that we can add powers when two common bases are being multiplied together, we know that the +3 Fill in the the equation true: added b2 ⋅ b⬚to = b22on blank boxblank must to bemake the power that is being the right hand side of the equation. Therefore, we know that 3 must go into the blank box to make the equation true.

Create a strategy We can use the exponent law: am ⋅ an = am + n

b2 ⋅ b3 = b2 + 3

Apply the idea

Example 2

Since we know that we can add powers when two common bases are being multiplied together, we know that the blank box must be the power that is being added to 2 on the right hand side of the equation. Therefore, we know that Simplify m2 ⋅ m7 + r3 ⋅ r2, giving your answer in exponential form. 3 must go into the blank box to make the equation true. b2 ⋅ b3 = b2 + 3

Create a strategy We can use the exponent law: am ⋅ an = am + n

Apply the idea Example 2 Purpose m2 ⋅ m7 + r3 ⋅ r2 = m2 + 7 + r3 + 2 Add the powers of the bases m and r 2 Simplify understand m2 ⋅ m7 + r3 ⋅ rand , giving your5answer form.specifically the product rule. Help students lawsinofexponential exponents, Simplify the powers =apply m9 + rthe Create strategy Reﬂecting witha students m We cancan use the law: ⋅ anwhole = am + nclass or for advanced learners by asking students if the base can This problem be exponent extended fora the Example 3 represent any number or if it can only represent positive integers for the rule to hold true. For example, students Apply the idea can substitute values such as2b = 2 +or b3=+ 2−2 to verify that b2 ⋅ b3 = b5 for all real values of b. 2 7 3 7 Simplify: m ⋅m +r ⋅r =m

a 35 ⋅ 39

= m9 + r5

Add the powers of the bases m and r

Simplify the powers

Create a strategy Use the product rule to simplify the expression.

Example 3

Apply the idea Simplify:

a 35 ⋅ 39

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35 ⋅ 39 = 35+9 14

Apply the product rule Simplify

Create Virginia a strategy Mathspace SOL Algebra 1 Teacher Edition mathspace.co Use the product rule to simplify the expression. 5.01 Product rule mathspace.co

Apply the idea 5

5+9

253

• Apply product rule to the exponents: a2 ⋅ a5 = a2 + 5 = a7 Apply thethe idea Sincefinal we result know is: that we can add powers when two common bases are being multiplied together, we know that the The blank box must be the power that is being added to 22on the right hand side of the equation. Therefore, we know that 3a ⋅ 4a5 = 12a7 3 must go into the blank box to make the equation true. b2 ⋅ b3 = b2 + 3

1 Students:Example Page 253

Fill in the blank to make the equation true: b2 ⋅ b⬚ = b2 + 3

Example 2

Create a strategy

Simplify m2 ⋅the m7exponent + r3 ⋅ r2, giving your We can use law: am ⋅ ananswer = am + nin exponential form.

Create a strategy Apply the idea

m m+n We can the that exponent law: ⋅ an = awhen Since weuse know we can adda powers two common bases are being multiplied together, we know that the blank box must be the power that is being added to 2 on the right hand side of the equation. Therefore, we know that Apply idea 3 must the go into the blank box to make the equation true. +3 m2 ⋅ m7 + r3 ⋅ r2 = m2 + 7 + r3 + 2 of the bases m and r b2 Add ⋅ b3 =the b2 powers

= m9 + r5

Simplify the powers

Example 2 3 PurposeExample Simplify m2 ⋅ m7 + r3 ⋅ r2, giving your answer in exponential form. StudentsSimplify: demonstrate that the product rule only applies to products with the same bases. Create a9 strategy 5

a 3 ⋅3 Expected Wemistakes can use the exponent law: am ⋅ an = am + n Students might think the expression is not fully simplified because of the addition sign between the terms. Create a strategy the that idea the bases and exponents are different, so these are not like terms. Compare it to a more Remind Apply students Use the product to simplify 2the 2 rule 7 expression. +2 ⋅ m7as + rx32⋅ r+2 x, =m + r3them Add powers of the m and r familiar example, m such to +help see that thetheexpression is bases fully simplified. = m9 + r5

Apply the idea

Students: Page 253 35 ⋅ 39 = 35+9

Simplify the powers

Apply the product rule

= 314

Simplify

Example 3 Simplify: a 35 ⋅ 39

5.01 Product rule mathspace.co

253

5.01 Product rule

253

Create a strategy Use the product rule to simplify the expression.

Apply the idea 35 ⋅ 39 = 35+9

Apply the product rule

Simplify

Purpose mathspace.co Reinforce understanding of the product rule of exponents and its application in simplifying expressions.

Students: Page 254 b c7 ⋅ c6

Create a strategy Use the product rule to simplify the expression.

Apply the idea c7 ⋅ c6 = c7+6

Apply the product rule

= c13

Simplify

c 5d5 ⋅ 3d3

Create a strategy Multiply the coeﬃcients, then use the product rule with the exponents.

Apply the idea 5d5 ⋅ 3d3 = 5 ⋅ 3 ⋅ d5 ⋅ d3 5+3

= 15 ⋅ d

Group the coeﬃcients and variables Multiply the coeﬃcients and apply the product rule

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519

b c7 ⋅ c6

Create a strategy Use the product rule to simplify the expression.

Purpose theinidea Further Apply practice applying the product rule of exponents to simplify expressions. 7

b c ⋅c

c7 ⋅ c6 = c7+6

Apply the product rule

Students: Page 254 = c13

Simplify

Create a strategy 5 3d3 cUse5d the⋅product rule to simplify the expression.

Createthe a strategy Apply idea Multiply the product rule with the exponents. c7coeﬃcients, ⋅ c6 = c7+6 then use the Apply the product rule

Apply the idea

= c13

Simplify

5d5 ⋅ 3d3 = 5 ⋅ 3 ⋅ d5 ⋅ d3 c 5d5 ⋅ 3d3 = 15 ⋅ d5+3 8

Create a strategy

= 15d

Group the coeﬃcients and variables Multiply the coeﬃcients and apply the product rule Simplify

Multiply the coeﬃcients, then use the product rule with the exponents. d −3m2n5 ⋅ 7m3n

Apply the idea

Purpose 5 Create a 5d strategy ⋅ 3d3 = 5 ⋅ 3 ⋅ d5 ⋅ d3 Group the coefficients and variables Check that students can multiply coefficients and apply the product rule with exponents to simplify expressions. 5+3 Multiply the coefficients, simplify the exponents. with one base at a time. Multiplyrule the to coefficients and apply the Work product rule = 15 ⋅then d use the product

Students: Page 254

Apply the idea

= 15d8

Simplify

−3m2n5 ⋅ 7m3n = −3 ⋅ 7 ⋅ m2 ⋅ m3 ⋅ n5 ⋅ n d −3m n ⋅ 7m3n = −21 ⋅ m2+3 ⋅ n5+1 2 5

5 6

Create a strategy

= −21m n

Group the coeﬃcients and variables Multiply the coeﬃcients and apply the product rule Simplify

Multiply the coeﬃcients, then use the product rule to simplify the exponents. Work with one base at a time.

Example 4

Apply the idea 2 5 3 3 −3m n the ⋅ 7m n = −3in⋅ scientific 7 ⋅ m2 ⋅ mnotation. ⋅ n5 ⋅ n Group the coefficients and variables Multiply and write answer 2+3 5+1 5 13 ⋅n Multiply and apply the product rule = −21 ⋅ m ) (6.04 the × 10coefficients ) (2.7 × 10

= −21m5n6

Simplify

Create a strategy Multiply coeﬃcients and use product rule for exponents.

Example 4 Purpose Apply idea Multiplythe and write the answer scientiﬁc notation. Ensure students understand howinto multiply coefficients, add exponents with the same base, and simplify (2.7 × 105) (6.04 × 1013) = 2.7 × 105 × 6.04 × 10135 Rewrite parentheses as multiplication (2.7 × 10 ) (6.04 × 1013) expressions with multiple variables. 5 13 = 2.7 × 6.04 × 10 × 10

Commutative property of multiplication

= 16.308 × 105+13

Product rule of exponents

Expected mistakes Create a strategy Simplify = 16.308 × 1018of adding them. StudentsMultiply mightcoeﬃcients multiply the exponents instead Alternatively, they might try to add the and use product rule for exponents. 19 Rewrite in scientific notation help them see that the 1.6308By × 10 coefficients rather than multiplying =them. grouping the coefficients and variables, coefficients are while the exponents of common bases are added. Apply themultiplied idea 254

(2.7 × 105) (6.04 × 1013) = 2.7 × 105 × 6.04 × 1013

Mathspace Virginia SOL Algebra 1 = 2.7 × 6.04 × 105 × 1013 mathspace.co

Rewrite parentheses as multiplication

Commutative property of multiplication Combining the commutative property with the product rule use with Example 3 5+13 = 16.308 × 10

Targeted instructional strategies

= 16.308 × 10

Product rule of exponents

Simplify

When simplifying expressions with =coefficients variables, encourage in scientific notationstudents to organize their 1.6308 × 1019 and multipleRewrite work by grouping the coefficients together and grouping the common bases together. Explicitly connect this regrouping to their prior knowledge of the commutative property, a ⋅ b = b ⋅ a. 254

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Create a strategy Multiply the coeﬃcients, then use the product rule with the exponents.

Apply the idea 5d5 ⋅ 3d3 = 5 ⋅ 3 ⋅ d5 ⋅ d3

Group the coeﬃcients and variables

3 5 5+3 For example, (−3m2n5)(7m be written as coeﬃcients −3 ⋅ m2 ⋅ nand ⋅7 ⋅ m3the⋅ n. By the Multiply the apply product rulecommutative property, we can = 15n) ⋅ dcould 8 change the order of multiplication without affecting the final product. Simplify = 15d

(−3m2n5) (7m3n) = −3 ⋅ m2 ⋅ n5 ⋅ 7 ⋅ m3 ⋅ n

d −3m2n5 ⋅ 7m3n

Write with multiplication symbols

= (−3 ⋅ 7) ⋅ (m ⋅ m ) ⋅ (n ⋅ n) 5

Group coefficients and common bases

= (−21) ⋅ (m ) ⋅ (n ) Create a strategy

Evaluate the multiplication

5 6

= −21m n

Multiply the coeﬃcients, then use the product rule to simplify the exponents. Work with one base at a time.

By organizing their work in this way, students can prevent simple arithmetic errors or mistakes in applying the product Apply rule. the idea −3m2n5 ⋅ 7m3n = −3 ⋅ 7 ⋅ m2 ⋅ m3 ⋅ n5 ⋅ n 2+3

= −21 ⋅ m

Students: Page 254

5+1

⋅n

Group the coeﬃcients and variables Multiply the coeﬃcients and apply the product rule

= −21m5n6

Simplify

Example 4 Multiply and write the answer in scientiﬁc notation. (2.7 × 105) (6.04 × 1013)

Create a strategy Multiply coeﬃcients and use product rule for exponents.

Apply the idea (2.7 × 105) (6.04 × 1013) = 2.7 × 105 × 6.04 × 1013

254

Rewrite parentheses as multiplication

= 2.7 × 6.04 × 105 × 1013

Commutative property of multiplication

= 16.308 × 105+13

Product rule of exponents

= 16.308 × 1018

Simplify

= 1.6308 × 1019

Rewrite in scientific notation

Mathspace Virginia SOL Algebra 1 mathspace.co

Purpose This example helps students understand the process of multiplying numbers in scientific notation and the use of the product rule for exponents. Expected mistakes Students may forget that numbers written in scientific notation only contain the ones place (not the tens place), thus not completing the final step in the worked solution. Remind them that 16.308 × 1018 is not written in proper scientific notation and provide support as needed.

Students: Page 255

Idea summary For any base number a, and any numbers m and n as powers, am ⋅ an = am + n When multiplying terms with like bases, we add the powers.

Practice What do you remember? 1

When multiplying two powers with the same base, do we add, subtract, multiply, or divide the powers?

Is each expression equivalent to 6p8? a

6p2 ⋅ 6p4

6p4 ⋅ p2

p2 ⋅ p2 ⋅ p2 ⋅ 6p2

3⋅3⋅3⋅3⋅y⋅y⋅y

Write each expression in exponential form: a

a⋅a⋅b⋅b

6p4 ⋅ p4

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Practice Students: Pages 255–256

What do you remember? 1

When multiplying two powers with the same base, do we add, subtract, multiply, or divide the powers?

Is each expression equivalent to 6p8? a

6p2 ⋅ 6p4

6p4 ⋅ p2

p2 ⋅ p2 ⋅ p2 ⋅ 6p2

6p4 ⋅ p4

Write each expression in exponential form: a

a ⋅ a ⋅ b ⋅ b

3⋅3⋅3⋅3⋅y⋅y⋅y

3 ⋅ u ⋅ u ⋅ u ⋅ 5 ⋅ v ⋅ v ⋅ v

−3 ⋅ c ⋅ c ⋅ b ⋅ 5⋅ c ⋅ c ⋅ a ⋅ b

(3p) ⋅ (3p) ⋅ (3p)

g ⋅ x13 ⋅ ⬚ = g2 ⋅ x21

x9y8 ⋅ ⬚ = x11y15

Let’s practice 4

Write the term that would make each statement true. a e

a3 ⋅ ⬚ = a6

x16y6 ⋅ ⬚ = x22y13

Simplify each expression in exponential form. a

22 ⋅ 23

39 ⋅ 310

y2 ⋅ y6

x4 ⋅ 10x3

4y5 ⋅ 3y2

4y3 ⋅ 6y

8y9 ⋅ 5y7

8y4 ⋅ 8y3

9m2 ⋅ 6m2

7y3 ⋅ 5y4

u2 ⋅ u6 ⋅ u3

6y7 ⋅ 6y5 ⋅ 6y3

m 4y ⋅ 6y ⋅ 3y 6

q 2 ⋅ q3

s9 ⋅ s10

y5 ⋅ y2

p3 ⋅ p

b ⋅b

6y2 ⋅ y5

u2 ⋅ u6 ⋅ u3

4y ⋅ 5y ⋅ 6y

3y7 ⋅ 5y8 ⋅ 2y

d2 ⋅ d6

x4 ⋅ x3

m ⋅m ⋅m

w9 ⋅ w7

a4 ⋅ a3

f ⋅f ⋅f

n8 ⋅ n6 ⋅ n3

am ⋅ a n

a3 ⋅ −2a8

5z5 ⋅ 3z8 ⋅ z9

6y7 ⋅ 6y5 ⋅ 6y3

4y ⋅ 6y ⋅ 3y

3y6 ⋅ 4y

2w4 ⋅ w10 ⋅ w7

4 2

3 4

5g h ⋅ 8g h

6bc ⋅ 8b a ⋅ 0

8m2n4 ⋅ 7m6n

t5 ⋅ t12 + y10 ⋅ y4

k7 ⋅ k8 + w13 ⋅ w3

43y2 ⋅ ⬚ = 415y15

Simplify each expression in exponential form. p2 ⋅ p3 + w9 ⋅ w3

a ⋅ a2 + c

Write the term that would make each statement true. a e

Simplify each expression in exponential form.

a 9

8y ⋅ 9y

i 8

Simplify each expression in exponential form.

i 7

a3 ⋅ ⬚ = a4

9h5 ⋅ ⬚ = 27h7

-5b1 ⋅ 1 ⋅ ⬚ = 35b1 ⋅ 2

Clifton and Duncan are discussing how to simplify the expression x7 ⋅ x4. Clifton thinks that x7 ⋅ x4 = x28, because they think that when multiplying expressions, you also multiply the exponents. Duncan argues that x7 ⋅ x4 = x11, as they believe the exponents should be added when multiplying expressions with the same base. Who is correct, Clifton or Duncan? Explain your reasoning.

Explain why the expression a4 ⋅ b4 cannot be simplified using the product rule.

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Let’s extend our thinking 12

Multiply the expressions. Write your answer using scientific notation. a

(2 × 103) (4 × 104)

(2 × 104) (6 × 107)

(4.6 × 106) (6.37 × 103)

What quantity multiplied by itself results in x100?

The length of the base of a triangle is 3c4 and the length of its perpendicular height is 6c8. What is the expression for the area of the triangle?

Fill in the missing number using positive integers: a

3mc⬚ ⋅ ⬚bm⬚ ⋅ 2b⬚c3 = −6b4c5m7

If 2x ⋅ 2 y = 210, x ≥ y. a

How many positive integer values of x and y satisfy the inequality?

Why can we not list the possible solutions if we allow negative integers?

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Answers

10 Duncan is correct. One possible explanation: We can assign x a value, and then show which student’s equation is true.

5.01 Product rule

Let x = 2. What do you remember?

For Clifton’s equation: (2)7 ⋅ (2)4 = (2)28 → 128 ⋅ 16 ≠ 268 435 456

1 Add 2 a No

b No

2 2

c Yes

4 3

3 a a b

For Duncan’s equation:

d Yes 3 3

4 2

b 3 y

c 15u v

d −15c b a

b a1

c g ⋅ x8

d x2y7

b 319

c y8

d 10x7

g 40y16

h 64y7

e (3p)3 Let’s practice 4 a a3 e x6y7 5 a 25 e 12y7 i

54m

m 72y 6 a q

e y

24y4

35y

n 72y

b s

7 a 6y

e u11 i

72y17

k u

o 120y c d

l 14

p 30y16 d x

g w

h a

m11

k f 15

b 12y

216y15

c a

m+n

11 The product of powers law requires the bases to be the same. Since the bases in the expression are different, we cannot simplify it using the product of powers law. Let’s extend our thinking b 1.2 × 1012 10

c 2.9302 × 10

n17 11

2w21

g 15z22

h 216y15

40g7h6

k 0

56m8n5

8 a p5 + w12

b a3 + c

c t17 + y14

d k15 + w16

9 a 3h2

b −14

d 412 y13

Another way to explain that Duncan is correct is by stating that when multiplying x7 and x4, we are essentially combining the number of times x is multiplied by itself. x7 represents x multiplied by itself 7 times, and x4 represents x multiplied by itself 4 times. Therefore, in total, x is being multiplied by itself 11 times, leading to x11.

12 a 8 × 107

d −2a

(2)7 ⋅ (2)4 = (2)11 → 128 ⋅ 16 = 2048

13 x50 14 9c12 square units 15 a 3mc2 · −1 bm6 · 2b3c3 = −6b4c5m7 b 16 a 5, x = 9, y = 1, x = 8, y = 2, x = 7, y = 3, x = 6, y = 4, x = 5, y = 5. b T here would be an infinite number of possible solutions, x = 11, y = −1, x = 12, y = −2 etc.

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5.02 Power rule Subtopic overview Lesson narrative In this lesson, students will determine the power of a power. Students will explore how a power of a power rule is similar to the product of powers property. Students will engage in an interactive exploration, where they discover the relationship between exponential and expanded form of a number in exponential form being raised to another power. By the end of this lesson, students will be able to simplify expressions that contain the power of a power.

Learning objectives

5.02 Power rule

Students: Page 257

After this lesson, you will be able to… • derive the power of power law from patterns. • use the power of a power law to simplify algebraic expressions.

Power rule

Key vocabulary Interactive exploration 

power of aExplore productonline rule to answer the questions



power rule

mathspace.co

Essential understanding Use the interactive exploration in 5.02 to answer these questions. The law for raising an exponential expression to another exponent allows us to simplify complex expressions and 1. After clicking “Show expanded form”, how many groups of a2’s are there? uncover underlying patterns that inform more efficient methods for computation and analysis. 2. Click “Show answer”. What do you notice about the exponent in the simplified form? 3.

Adjust the sliders to change the base’s exponent and the exponent outside the parentheses. What happens to the simplified form (am)n?

Standards

This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards. The power rule states that for any base number, a, and any numbers m and n as power,

Mathematical process goals

(am)n = am ⋅ n

That— is,Mathematical when simplifying a term with a power that itself has aMPG4 power:— Mathematical Connections MPG3 Reasoning • Keep the same base Teachers can incorporate this goal by facilitating activities Teachers can incorporate this goal by helping students • Multiply the exponents connect the Power of a Power Law to other mathematical that require students to apply logical reasoning to2 3 2 2 2 (a ) = (a ) (a ) (a ) concepts they’ve learned, such as base and exponent understand and use the Power of a Power Law. This 23 ) = (a ⋅ a) (a ⋅ a) (a ⋅ a) notation, the order of operations, and the concept of could involve proving the Power of a Power Law(athrough 2 3 if scientific repeated multiplication of exponents, or determining (a ) = a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ anotation. Teachers can show students how the Power of a Power Law is used across different areas conclusions made using the Power of a Power Law are (a2)3 = a6 of mathematics and in other disciplines, like physics or valid. Encourage them to use number sense to reason When usingof therepresentations. power rule for exponents, the coefficient is handled separately the exponents. Let’s take aused. look engineering, wherefrom powers of 10 are commonly from a variety at an example. 4

(3x2)4

• Raise the coefficient to the power: 3 = 81 • Multiply the exponents, applying the power rule: (x2)4 = x2 ⋅ 4 = x8 Therefore, (3x2)4 simplifies to: 8

81x

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MPG5 — Mathematical Representations Teachers can incorporate this goal by asking students to represent their understanding of the Power of a Power Law using a variety of methods. For example, they could use visual representations to demonstrate why the Power of a Power Law works, or use symbolic representations to simplify expressions involving powers of powers. They should also be able to apply the Power of a Power Law to expressions with variables as bases, connecting this to the mathematical skills they’ve previously acquired.

Content standards A.EO.3 — The student will derive and apply the laws of exponents.

A.EO.3b — Simplify multivariable expressions and ratios of monomial expressions in which the exponents are integers, using the laws of exponents.

A.EO.3a — Derive the laws of exponents through explorations of patterns, to include products, quotients, and powers of bases.

Prior connections 6.NS.3 — The student will recognize and represent patterns with whole number exponents and perfect squares.

7.NS.1 — The student will investigate and describe the concept of exponents for powers of ten and compare and order numbers greater than zero written in scientific notation.

Future connections A2.EO.3 — The student will perform operations on polynomial expressions in two or more variables and factor polynomial expressions in one and two variables.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lesson: Algebra 1 — 5.01 Product rule

Lesson supports The following supports may be useful for this lesson. More specific supports may appear throughout the lesson:

Compare and connect English language learner support Present students with the following three expressions: (x4)3 x4 ⋅ x3 (x4y)3 Individually, give students a few minutes to simplify each expression. Encourage them to show their mathematical thinking by writing out each step or by justifying each step with a rule or property.

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In groups of two, ask students to compare their strategies and discuss how the strategies are the same or different. Consider providing discussion prompts such as “Which strategy do you believe is the most efficient in determining a solution?” and “How are the stategies for simplifying each expression the same or different?” Listen for and amplify observations of advantages and disadvantages to different approaches, such as applying an exponent law directly versus expanding the expressions each time. This will help students connect different approaches and determine the effectiveness of each one through partner and whole-class discussions.

Use an example to highlight the differences between exponent laws Targeted instructional strategies Students often confuse the product rule, power rule, and power of a product rule. To highlight their differences, provide a worked example that requires all three laws and instruct students to justify each step using an exponent law. For example, (3x2)3(x5)2 = (3x2)3(x5 ⋅ 2) 2 3 10

= (3x ) (x ) = (33 ⋅ x2 ⋅ 3) (x10) 6

= (27x ) (x ) = 27x6 + 10 16

= 27x

⬚

In pairs, give students time to discuss the patterns they see. Invite pairs to explain the differences between the rules to the rest of the class. Example responses are shown: • The power rule is used when only one base is inside the parentheses. • The power of a product rule is the same as the power rule, but we use it when there is more than one base inside the parentheses. • We use the product rule when multiplying two expressions that have the same base. • We multiply exponents when using the power rule and power of a product rule. • We add exponents when using the product rule.

Comparison chart for exponent laws Student with disabilities support Students may struggle with the differences between the exponent laws, specifically the product rule, power rule, power of a product rule. To help students recognize similarities and differences in laws, use a comparison chart like the one shown. Product rule a m ⋅ a n = am + n Example:

Power rule (am)n = amn Example:

Power of a product rule (ambn)p = ampbnp Example:

When to use:

Explicitly point out that the product law is only used when we can see both equivalent bases like in the case of x3 ⋅ x5.

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We typically use the power rule when the exponent is outside of a grouping symbol such as (x3)5. The power of a product rule is used when there are different bases inside the grouping symbol, such as (x3 ⋅ y2)5. This new exponent would be appplied to every base in the group. An example of a completed organizer is shown: Product rule a m ⋅ a n = am + n Example: x3 ⋅ x5 = x8 When to use: We can see both bases being multiplied and bases are the same.

Power rule (am)n = amn Example: (x3)5 = x15 When to use: We can see only one base and it is inside grouping symbols.

x3⋅ x5

( x 3 )5

Power of a product rule (ambn)p = ampbnp Example: (2x3)5 = 25x15 When to use: We can see more than one base inside grouping symbols. ( 2 x 3 )5

Student lesson & teacher guide Power rule Exploration Students: Page 257

5.02 Power rule After this lesson, you will be able to… • derive the power of power law from patterns. • use the power of a power law to simplify algebraic expressions.

Power rule Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 5.02 to answer these questions. 1.

After clicking “Show expanded form”, how many groups of a2’s are there?

Click “Show answer”. What do you notice about the exponent in the simplified form?

Adjust the sliders to change the base’s exponent and the exponent outside the parentheses. What happens to the simplified form (am)n?

The power rule states that for any base number, a, and any numbers m and n as power, (am)n = am ⋅ n

528

That is, when simplifying a term with a power that itself has a power: • KeepVirginia the same base Mathspace SOL Algebra 1 Teacher Edition • Multiply the exponents mathspace.co (a2)3 = (a2) (a2) (a2) (a2)3 = (a ⋅ a) (a ⋅ a) (a ⋅ a)

Suggested student grouping: Small groups Students will use a GeoGebra applet to explore how changes in a base’s exponent and the exponent outside the parentheses affect the simplified form of (am)n. Students will use sliders to adjust these values and answer questions based on their observations. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. After clicking “Show expanded form”, how many groups of a2’s are there? There are 4 groups of a2. 2. Click “Show answer”. What do you notice about the exponent in the simplified form? The exponent in the simplified form is 8 which is the product of the base’s exponent and the exponent outside the parentheses.

5.02 Power rule

3. Adjust the sliders to change the base’s exponent and the exponent outside the parentheses. What happens to the simplified form (am)n? The exponent in the simplified form changes. It always equals the product of the base’s exponent and the exponent outside thelesson, parentheses. After this you will be able to… • derive the power of power law from patterns.

Purposeful questions • use the power of a power law to simplify algebraic expressions.

• In the first expression, what does the exponent of 4 represent? • What patterns do you notice as only n changes? as only m changes? rule to formulate a rule to help in simplifying calculations? • Can Power you generalize

Possible misunderstandings Interactive exploration Explore onlinethe to answer the questions • Students might misread first question and, instead of stating the groups of a2, state the total number 2 of a’s. Point out that “groups of a ” means we should could the number of a2’s in the first line, or count the mathspace.co parentheses (grouping symbols) in the second line.

Use the interactive exploration in 5.02 to answer these questions.

Students are introduced to the“Show power rule. They thatofwhen raising 1. After clicking expanded form”,will howdiscover many groups a2’s are there? an exponent to another power, they can multiply the2.exponents together to simplify the expression. Click “Show answer”. What do you notice about the exponent in the simplified form? 3.

Adjust the sliders to change the base’s exponent and the exponent outside the parentheses.

Students: Page What 257happens to the simplified form (am)n?

The power rule states that for any base number, a, and any numbers m and n as power, (am)n = am ⋅ n That is, when simplifying a term with a power that itself has a power: • Keep the same base • Multiply the exponents (a2)3 = (a2) (a2) (a2) (a2)3 = (a ⋅ a) (a ⋅ a) (a ⋅ a) (a2)3 = a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a (a2)3 = a6 When using the power rule for exponents, the coefficient is handled separately from the exponents. Let’s take a look at an example. 4

(3x2)4

• Raise the coefficient to the power: 3 = 81 • Multiply the exponents, applying the power rule: (x2)4 = x2 ⋅ 4 = x8 Therefore, (3x2)4 simplifies to: 81x8

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Examples Students: Page 258 Example 1 Simplify (a5)3.

Create a strategy Use the power rule.

Apply the idea (a5)3 = a5 ⋅ 3 =a

Multiply the powers of the variable Evaluate the multiplication

Idea summary

Purpose For any base number a, and any numbers m and n as power, Check if students can use the power rule correctly. mn m⋅n (a ) = a

That is, when simplifying a term with a power that itself has a power: Expected mistakes • add Keepthe theexponents same base like they would do with the product of two exponential expressions with the Students might Example 1 • Multiply the exponents same base. Have students expand the expression to see why the exponents should be multiplied. Simplify (a5)3.

Reflecting with students Power of a product rule understanding of the power rule for advanced learners or for all Use open-ended questions that extend Create a strategy mn 15 5n 15 students. For Use theexample, power rule.ask: “Find all possible integer values of m and n such that (a ) = a .” or “If (a ) = a , Exploration what could n be, and why?” Alternatively, ask students to create their own examples and challenge their Apply idea classmates to the solve them. Consider the following expressions: (a5)3 = a5 ⋅ 3

Students: Page 258= a15

Idea summary

Multiply the powers of the variable ⋅ 3)2 and 22 ⋅ 32 Evaluate the (2 multiplication (4 ⋅ 5)3 and 43 ⋅ 53 (7 ⋅ 2)4 and 74 ⋅ 24 (3 ⋅ 6)5 and 35 ⋅ 65

Evaluate each expression record your results in the table: For any base number and a, and any numbers m and n as power, ⋅n (am )n = amResult Expression 2 itself has a power: That is, when simplifying a term with a power that (2 ⋅ 3)

• •

Keep the same base Multiply the exponents

Power of a product rule Exploration

22 ⋅ 32 (4 ⋅ 5)3 43 ⋅ 53 (7 ⋅ 2)4 74 ⋅ 24 (3 ⋅ 6)5 35 ⋅ 65

1. What you notice in the results of the expressions? Consider the patterns followingdo expressions: 2.

258

Can you form a rule or law based on your observations? (2 ⋅ 3)2 and 22 ⋅ 32 (4 ⋅ 5)3 and 43 ⋅ 53 (7 ⋅ 2)4 and 74 ⋅ 24 (3 ⋅ 6)5 and 35 ⋅ 65

Mathspace Virginia SOL Algebra 1 mathspace.co

Evaluate each expression and record your results in the table:

530

Expression (2 ⋅ 3)2 22 ⋅ 32 (4 ⋅ 5)3 Mathspace Virginia SOL Algebra 1 Teacher Edition 43 ⋅ 53 mathspace.co (7 ⋅ 2)4 74 ⋅ 24 (3 ⋅ 6)5

Result

= a15

Evaluate the multiplication

Idea summary

Power of aForproduct any base numberrule a, and any numbers m and n as power, (am )n = am ⋅ n

Exploration That is, when simplifying a term with a power that itself has a power: •

Keep the same base

Students: Page • 258 Multiply the exponents

Power of a product rule Exploration Consider the following expressions: (2 ⋅ 3)2 (4 ⋅ 5)3 (7 ⋅ 2)4 (3 ⋅ 6)5

and and and and

22 ⋅ 32 43 ⋅ 53 74 ⋅ 24 35 ⋅ 65

Evaluate each expression and record your results in the table: Expression (2 ⋅ 3)2 22 ⋅ 32 (4 ⋅ 5)3 43 ⋅ 53 (7 ⋅ 2)4 74 ⋅ 24 (3 ⋅ 6)5 35 ⋅ 65

258

Result

What patterns do you notice in the results of the expressions?

Can you form a rule or law based on your observations?

Mathspace

Virginia SOL Algebra 1

Suggested student grouping: In pairs mathspace.co Students will evaluate a series of mathematical expressions and record their results in a table. The expressions are designed to encourage students to recognize and think about the properties of exponents, specifically the power of a product rule. Here is the completed table: Expression (2 ⋅ 3)2 22 ⋅ 32 (4 ⋅ 5)3 43 ⋅ 53 (7 ⋅ 2)4 74 ⋅ 24 (3 ⋅ 6)5 35 ⋅ 65

Result 36 36 8000 8000 2401 2401 7776 7776

Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here.

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1. What patterns do you notice in the results of the expressions? I noticed that the ﬁrst and second expressions in each pair always give the same result. It’s like the power gets distributed to both numbers in the first expression. 2. Can you form a rule or law based on your observations? Yes, it seems like when you have a product of numbers raised to a power, you can raise each number to that power individually. Purposeful questions • What happens to the product when it’s raised to a power? How does this compare to raising each factor to that power separately? • Can you expand the pairs of expressions to show that they are equivalent? For instance, can you expand (2 ⋅ 3)2 and 22 ⋅ 32 and conﬁrm there are the same number of each base? • Can you apply the rule you came up with to an expression such as (42 ⋅ 3)5? Possible misunderstandings • When formulating a rule, students may not clarify that it only works for multiplication or a product of two numbers. Encourage them to be more specific with their rule by asking if, as stated, their rule would work on an expression such as (2 + 3)2. Highlight the importance of including this distinction in their rule.

Applying the power of a product rule incorrectly Address student misconceptions Students may incorrectly apply the power of a product rule and only raise the closest base to the exponent instead of multiplying the exponent of every base by the new power, especially if one of the bases is numerical (a coefficient). It may be helpful to show students the connection between the power of a product and the power rule by writing an expression in expanded form. (am ⋅ bn)p = (am)p ⋅ (bn)p m⋅p

n⋅p

⋅b

Power of a product rule Power rule

In addition, highlight the fact that a coefficient is a numerical base to the power of 1, so the power is also applied to it. (5ambn)p = 5p ⋅ amp ⋅ bnp Students will be introduced the the power of a product rule. They will discover that when an entire expression is raised to a power, each base in the expression will get raised to that power.

Students: Page 259 For the product of any numbers a and b in the base, and for any number n in the power, (ab)n = anbn The power of a product rule states that a product raised to a power is equivalent to the product of the two factors each raised to the same power. (ab)4 = (a4) (b4) (ab)4 = (a ⋅ a ⋅ a ⋅ a) (b ⋅ b ⋅ b ⋅ b) (ab)4 = a4b4

Example 2 Simplify (a9 ⋅ b3)4

Create a strategy 532

n n Mathspace Virginia SOL(ab) Algebra Edition We can use the rule: = an1bTeacher mathspace.co

Apply the idea (a9 ⋅ b3)4 = (a9)4 (b3)4

Start with the power of a product rule

For the product of any numbers a and b in the base, and for any number n in the power, (ab)n = anbn The power of a product rule states that a product raised to a power is equivalent to the product of the two factors each raised to the same power.

Examples

(ab)4 = (a4) (b4) (ab)4 = (a ⋅ a ⋅ a ⋅ a) (b ⋅ b ⋅ b ⋅ b)

Students: Page 259

(ab)4 = a4b4

Example 2 Simplify (a9 ⋅ b3)4

Create a strategy For the product of any numbers a and b in the base, and for any number n in the power, We can use the rule: (ab)n = anbn

(ab)n = anbn

The power of a product rule states that a product raised to a power is equivalent to the product of the two factors Apply the idea each raised to the same power. 4 (a9 ⋅ b3)4 = (a9)4 (b3)4 Start (ab) with4 the = (apower ) (b4) of a product rule = a9 ⋅ 4 ⋅ b3 ⋅ 4 = a36b12

Multiply the powers (ab)4 = (a ⋅ a ⋅ a ⋅ a) (b ⋅ b ⋅ b ⋅ b) Evaluate4the powers (ab) = a4b4

Example 2 Example 3 PurposeSimplify (a9 ⋅ b3)4 22 Check ifSimplify students how to use the power of a product rule. (−2xunderstand ) . Create a strategy

Reflecting witha students Create strategy We can use the rule: (ab)n = anbn Discuss Use withthe students the importance of the multiplication symbol. More specifically, make students aware that power rule. this ruleApply only applies when the exponential expressions inside the parentheses are being multiplied. the idea The power of the a product rule cannot be applied to expressions such as (a + b)2 or (a − b)2. Inform students that Apply idea (a9 ⋅ b3)4 = (a9)4 (b3)4 Start with the power of a product rule 2 2 in 2⋅2 they will explore(−2x this2)further the next topic (Topic 6 - Polynomials). = =(−2) Multiply Multiply the the powers powers of the variable a9 ⋅ 4x⋅ b3 ⋅ 4

Students: Page 259

= =4xa436b12

Evaluate and coefficient Evaluate the the multiplication powers

Idea summary

Example 3

The power of a product rule states that for the product of any numbers a and b in the base, and for any Simplifynumber (−2x2)2.n in the power, (ab)n = anbn

CreateInaother strategy words, a product raised to a power is equivalent to the product of the two factors each raised to the power. Use thesame power rule.

Apply the idea (−2x2)2 = (−2)2 x2 ⋅ 2 = 4x4

Multiply the powers of the variable Evaluate the multiplication and coefficient

Idea summary

5.02 Power rule

259

mathspace.co Purpose The power of a product rule states that for the product of any numbers a and b in the base, and for any Check if students understand how to use the power of a product rule when the expression has a coefficient. number n in the power, n

n n

(ab) = a b Expected mistakes In other words, a product raised to a power is equivalent to the product of the two factors each raised to the Students may multiply the −2 by the exponent instead of raising it to the exponent. Remind students that −2 is same power. also a base which can be rewritten as (−2)1.

Reﬂecting with students Encourage students to think about the placement of the negative sign. Does −(2x2)2 have the same solution as (−2x2)2? Why or why not? If students need help justifying their answers, remind them that exponents represent repeated multiplication, so the expression can be expanded. 5.02 Power rule mathspace.co

259

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Create a strategy Use the power rule.

Apply the idea (−2x2)2 = (−2)2 x2 ⋅ 2

Multiply the powers of the variable

Students: Page 259 = 4x4

Evaluate the multiplication and coefficient

Idea summary The power of a product rule states that for the product of any numbers a and b in the base, and for any number n in the power, (ab)n = anbn In other words, a product raised to a power is equivalent to the product of the two factors each raised to the same power.

Practice

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Students: Pages 260–261

259

What do you remember? 1

When simplifying a term with a power that itself has a power, do we add, subtract, multiply, or divide the exponents?

Consider (r2)4. a

Is each expression equivalent to (r2)4? i

r2 ⋅ r4 4

iii (r ⋅ r) v b

(r ⋅ r) ⋅ (r ⋅ r ⋅ r ⋅ r)

(r ⋅ r) ⋅ (r ⋅ r) ⋅ (r ⋅ r) ⋅ (r ⋅ r)

(r2)4 = r2 ⋅ 4

r2 ⋅ r2 ⋅ r2 ⋅ r2

Is each statement correct? i

(r2)4 = r2 + 4

Fill in the box to complete the rule: (r2)4 = r⬚

Let’s practice 3

Consider the following terms which form a pattern: (a2)1, (a2)2, (a2)3, (a2)4, … What would the ﬁfth term be? Simplify your answer in exponential form.

Consider the following terms which form a pattern: (x2)3, (x2)6, (x2)9, (x2)12, … What would the eighth term be? Simplify your answer in exponential form.

534

Simplify each expressions: a

(−x3)4

(4y4)3

(−2x2)2

(−3p2)5

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

(8y6)2

d h

(2r2s)4

Simplify each expression in exponential form: a

(c9)2

y8 ⋅ (3y)5

(u6)3 ⋅

6(r6)5 ⋅ 4(r4)7

(x9y)7 ⋅ (xy2)4

m ( p10)6 ⋅ ( p4)3

((x3)4)5

( f 8)6

(w3)4

(w9v6)4

(k5)3 ⋅ (k2)8

10(r7)5 ⋅ (2rs)4

( p5)9 ⋅ (q6)10

Simplify each expression in exponential form: a

9y9 ⋅ 8(−y)8 ⋅ 7y7

(−5y6) ⋅ (−3y7) ⋅ (−4y7)

(−9b4) ⋅ 4b4

2v2w ⋅ (−5u2v3) ⋅ 3u3w4

(−8q4) ⋅ p2 ⋅ (−7q4) ⋅ p3

(−7y5z) ⋅ (−3yx3) ⋅ (−6y2) ⋅ (−4y2)

Complete each statement: a

( j 2)5

(abc)⬚ = a3b3c3

(0.1a⬚)2 ⋅ (10b4)⬚ = 0.1a6b4

(⬚m⬚n⬚)3 = 8m6n9

(4m2)⬚ ⋅ (⬚b⬚)2 = 256m6b4

[(2a⬚)3]2 = 64a12

In a laboratory experiment, a type of algae quadruples in size every day. A sample starts with a size of 10 units and can be modeled by the expression 10 ⋅ (22)5, where 22 represents the daily quadrupling growth factor and 5 represents the number of days. Simplify the expression to calculate the final size of the algae after 5 days. State the operations or rules applied in each step of your work.

Consider the expression shown: (x6 ⋅ x2)3 ⋅ (x2 ⋅ x5)4 Simplify the expression two different ways: using either the product of powers law or the power to a power law first. Show your work and justify each step.

Consider (33)2. a

Expand and simplify the expression (33)2 in exponential form.

Explain how the simplified expression demonstrates the power rule.

Consider the expressions simplified by two students: Student 1:

Student 2:

2 2 2

(3 + 4) = 3 + 4

(3 ⋅ 4)2 = 32 ⋅ 42

= 9 + 16

= 9 ⋅ 16

= 25

= 144

Describe the difference between the two workings and determine if the students’ solutions are correct. Explain your reasoning.

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Let’s extend our thinking 13

Dante and Daniella are trying to simplify the expression n ⋅ (r7)3 ⋅ n5 ⋅ (r3)6. • Dante wrote 1

n ⋅ (r7)3 ⋅ n5 ⋅ (r3)6 = n6 ⋅ r21 ⋅ r18 = n6 ⋅ r39

2 • Daniella wrote: 1

n ⋅ (r7)3 ⋅ n5 ⋅ (r3)6 = n6 ⋅ r10 ⋅ r9

= n6 ⋅ r19

Whose work is correct? Explain. 14

Write (16 p)4 in the form ab, where a is a prime number.

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Answers

10 Starting with the product of powers law: (x6 ⋅ x2)3 ⋅ (x2 ⋅ x5)4 = (x6+2)3 ⋅ (x2+5)4

5.02 Power rule What do you remember? 1 Multiply 2 a i No

ii No

iii Yes

iv Yes

v Yes b i No

ii Yes

c (r2)4 = r8

= (x8)3 ⋅ (x7)4

= x8 ⋅ 3 ⋅ x7 ⋅ 4

= x24 ⋅ x28

= x24 + 28

Product of powers law

= x52

Combine the exponents

5 a x12

b 64y12

e 4x4

6 a j

−243p10

c 64y12

d 9y12

g 16u20v20

h 16r8s4 12

c f

d w

b c

e 243y13

u20

g w36v24

h k31

24r58

x67y15

k 160r39s4

n x

b −4y15

d 3y14

c −60y

e −36b8

f 5 5

21y6zx3

h −12y11

g −30v w u i

p45q60

7 a 504y24

5 8

56p q

8 a (abc)3 = a3b3c3 b (2m2n3)3 = 8m6n9 c d (4m2)3 ⋅ (2b2)2 = 256m6b4 e (0.1a3)2 ⋅ (10b4)1 = 0.1a6b4 f

Multiply the exponents for each term

law:

m p

Power to a power law

(x6 ⋅ x2)3 ⋅ (x2 ⋅ x5)4 = x6 ⋅ 3 ⋅ x2 ⋅ 3 ⋅ x2 ⋅ 4 ⋅ x5 ⋅ 4 Power to a power

= x18 ⋅ x6 ⋅ x8 ⋅ x20

Multiply the exponents for each term

= x18+6+8+20

Product of powers law:

= x52

Combine the exponents

4 x48

Simpliﬁcation of exponents within parentheses

Starting with the power to a power law:

Let’s practice 3 a

Product of powers law

[(2a2)3]2 = 64a12

9 10 ⋅ (22)5 = 10 ⋅ 22⋅5

Power to a power law: Multiply the exponents

= 10 ⋅ 210

Simplify the exponent to 10

= 10 ⋅ 1 024

Evaluate the exponent

= 10 240

Final result representing the size of the algae after ﬁve days

11 a Expanding (33)2 results in 33 ⋅ 33, which further expands to 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3, simplifying to 36. b T he expression (33)2 represents 33 being multiplied by itself. This results in 3 being used in multiplication 3 times in each of the 2 groups, totaling 3 · 2 = 6 multiplications of 3, and therefore demonstrating the “power to a power” law for (33)2 = 36. 12 Explanations may vary. One possible answer: Student 1 made a mistake by attempting to use the product to a power law on a sum of two numbers, while Student 2 correctly applies the product to a power law to a product. If we use a calculator to evaluate the expressions, only Student 2’s solution is correct. Let’s extend our thinking 13 Dante’s working is correct. When simplifying a term with a power that itself has a power the base is kept the same and the exponents are multiplied. Daniella added the powers. 14 216p

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5.03 Quotient rule Subtopic overview Lesson narrative In this lesson, students will determine quotients using the division property of exponents. Students will recognize that the division property allows for subtraction of exponents when the bases are the same and that the coefficients follow standard division rules. Students will complete two exploration activities that investigate the connection between expanded form and exponential form. By the end of the lesson, students will be able to simplify problems using the division property of exponents.

Learning objectives

5.03 Quotient rule

Students: Page 262

After this lesson, you will be able to… • derive the quotient of powers law from patterns. • use the quotient of powers law to simplify algebraic expressions.

Quotient rule

KeyExploration vocabulary 

 quotient rule power of a quotient rule Expand each of the expressions for the following values:

Expression

Values

Essential understanding

Substitute values

Expanded form

Simplified exponential form

m = 5, n = 3 The law for dividing exponential expressions with the same base allows us to simplify complex expressions and uncover underlying patterns that inform s = 4, t = 2 more efficient methods for computation and analysis.

Standards

x = 6, y = 1

1. What do you the notice about the relationship between the exponents simplified form and the original This subtopic addresses following Virginia 2023 Mathematics Standardsinofthe Learning standards. exponents?

Mathematical process 2. What happens to goals the base when you divide two expressions with the same base? MPG4 — Mathematical Connections MPG1 — Mathematical Problem Solving 6 2 If we wanted to simplify the expression a ÷ a , we could write as: To itincorporate this goal, teachers can connect the Teachers can integrate this goal into their instruction Quotient Law to previous lessons on the Product and by providing a range of problems that require students Power of a Power laws, as well as to concepts from other to apply the Product and Power of a Power laws and disciplines such as science (e.g., scientific notation in the Quotient Law. Teachers should include problems physics or chemistry). They can also relate the Quotient that model real-world situations and require students We can see that common factors are divided out of the expanded expression, leaving a4. Law to real-world contexts and problem situations, to strategize and determine acceptable solutions. 6 2 students to see the relevance and applicability of ForConsider example,the they could ask students to calculate the . expression a ÷ a , which can be written as helping growth of a bacterial population over time, using the laws the mathematical concepts they are learning. Let’s think about what this would look like if we expanded the expression: of exponents.

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MPG5 — Mathematical Representations Teachers can incorporate this goal by asking students to represent and describe mathematical ideas in relation to the Quotient Law using a variety of methods. For instance, they could ask students to visually represent the process of dividing exponents using the Quotient Law on a number line or using algebra tiles. They could also ask students to create symbolic representations of the law and to interpret representations in different forms, such as simplifying expressions involving powers of 10 using the Quotient Law.

Content standards A.EO.3 — The student will derive and apply the laws of exponents.

A.EO.3b — Simplify multivariable expressions and ratios of monomial expressions in which the exponents are integers, using the laws of exponents.

A.EO.3a — Derive the laws of exponents through explorations of patterns, to include products, quotients, and powers of bases.

Prior connections 6.NS.3 — The student will recognize and represent patterns with whole number exponents and perfect squares.

7.NS.1 — The student will investigate and describe the concept of exponents for powers of ten and compare and order numbers greater than zero written in scientific notation.

Future connections A2.EO.3 — The student will perform operations on polynomial expressions in two or more variables and factor polynomial expressions in one and two variables.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Algebra 1 — 5.01 Product rule Algebra 1 — 5.02 Power rule

Lesson supports The following support may be useful for this lesson. More specific supports may appear throughout the lesson:

Choose unambiguous variables Student with disabilities support When using multiple variables in a question, make sure that they are visually distinct. Combinations to avoid include: • b and d • m and n • p and q • u, v, and w

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Variables that look like numbers or symbols should also be avoided whenever possible. This will also depend on handwriting, but the most common examples are: • o and 0 • l and 1 • z and 2 • s and 5 • g and 9 • t and +

Student lesson & teacher guide Quotient rule Exploration Students: Page 262

5.03 Quotient rule After this lesson, you will be able to… • derive the quotient of powers law from patterns. • use the quotient of powers law to simplify algebraic expressions.

Quotient rule Exploration Expand each of the expressions for the following values: Expression

Values

Substitute values

Expanded form

Simplified exponential form

m = 5, n = 3 s = 4, t = 2 x = 6, y = 1 1.

What do you notice about the relationship between the exponents in the simplified form and the original exponents?

What happens to the base when you divide two expressions with the same base?

If we wanted to simplify the expression a6 ÷ a2, we could write it as:

We can see that common factors are divided out of the expanded expression, leaving a4. Consider the expression a6 ÷ a2, which can be written as

Let’s think about what this would look like if we expanded the expression:

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Suggested student grouping: In pairs Sudents will explore the quotient rule by using concrete examples to generalize the pattern. They will begin by completing the table shown, then they will answer questions that encourage them to observe the relationship between the original and simplified exponents. Expression

Values

Substitute values

Expanded form

Simpliﬁed exponential form

m = 5, n = 3

s = 4, t = 2

x = 6, y = 1

Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What do you notice about the relationship between the exponents in the simpliﬁed form and the original exponents? The exponent in the simpliﬁed form is the difference between the original exponents. 2. What happens to the base when you divide two expressions with the same base? The base remains the same. We only subtract the exponents. Purposeful questions • Why do you think the rule of subtracting exponents when dividing expressions with the same base works? • How can you confirm that the simplified form you came up with is equal to the original expression? Possible misunderstandings • Students may think that the base also gets subtracted or divided when dividing expressions with the same base. It’s important to clarify that only the exponents are subtracted according to the quotient rule for exponents.

Critique, correct and clarify English language learner support Present students with an incorrectly simplified expression, such as

Place students in pairs and asks them to discuss this statement with their partner. Provide guiding questions such as, “Are there any reasoning errors?” or “Can you justify this answer?” As students discuss with their partner, encourage them to correct the statement and discuss their reasoning together. Finally, give students a few minutes to rewrite the statement and to clarify their explanations. Invite students to share their critiques and explanations with the class.

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5.03 Quotient rule After this lesson, you will be able to… • derive the quotient of powers law from patterns. • use thequotient quotient of powers law to simplify algebraic expressions. Applying the rule incorrectly

Address student misconceptions

rule If bases Quotient are numerical, students may incorrectly apply the quotient rule and divide the bases of the exponents. For example, Exploration

Expand each of the expressions for the following values:

For expressions that contain a combination of coefficients and variables, students may subtract coefficients or Expression Substitute values Expanded form Simplified exponential form divide exponents. For example, Values m = 5, n = 3 s = 4, t = 2 x = 6, y = 1

Students will discover that when dividing powers with like bases, the exponents can be subtracted to simplify the expression. 1.

What do you notice about the relationship between the exponents in the simplified form and the original exponents?

Students: Pages 262–263 2.

What happens to the base when you divide two expressions with the same base?

If we wanted to simplify the expression a6 ÷ a2, we could write it as:

We can see that common factors are divided out of the expanded expression, leaving a4. Consider the expression a6 ÷ a2, which can be written as

Let’s think about what this would look like if we expanded the expression:

We can avoid having to write each expression in expanded form by using the quotient rule:

where a is any base number, and m and n are powers. That is, when dividing terms with a common base: • Keep the same base • Find the difference in the power. Of course, we can also write this law in the form am ÷ an = am − n 262

Mathspace

Virginia SOL Algebra 1

Whenmathspace.co using the quotient rule for exponents, the coefficients are handled separately from the exponents. Let’s take a look at an example.

• Divide the numerical coefficients: 12 ÷ 2 = 6 • Apply the quotient rule to the exponents: The final result is:

Example 1 Simplify the following expressions:

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a Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Create a strategy

Expand and simplify the expressions in the quotient.

• Divide the numerical coefficients: 12 ÷ 2 = 6 • Apply the quotient rule to the exponents:

Examples The final result is: Students: Page 263 Example 1

Simplify the following expressions: a

Create a strategy Expand and simplify the expressions in the quotient. Of course, we can also write this law in the form am ÷ an = am − n When using the quotient rule for exponents, the coefficients are handled separately from the exponents. Let’s take a Apply the idea look at an example. Expand the expressions • Divide the numerical coefficients: 12 ÷ 2 = 6

Divide the like terms

• Apply the quotient rule to the exponents:

Simplify the expression

11 So, finalsimplifies The result is: to z .

Reflect and check We can instead use the rule of exponents which states that when we divide terms with the same base, we subtract the exponents.

Example 1

Subtract the exponents Simplify the following expressions: Simplify the exponent

Create a strategy b

PurposeExpand and simplify the expressions in the quotient. a strategy StudentsCreate simplify an algebraic expression using the quotient of powers rule. Apply the idea

We can simplify the fractions and use the rule of exponents which states that when we divide terms with the same

Reflecting with students base, we subtract the exponents. Expand the expressions Extend this example for the whole class or for advanced learners by asking students how the answer would change if the original expression was

Divide the like terms

. If students seem stuck, encourage them to group the coefficients

together and group the expressions with z together. So,

Simplify the expression

simplifies to z11.

5.03 Quotient rule mathspace.co

Reflect and check

263

Point outWethat coefficients different which bases, so they are not included applying thewequotient canthe instead use the ruleare of exponents states that when we divide termswhen with the same base, subtract rule to the exponents.

. Therefore, this expression would simplify to

Subtract the exponents

Students: Pages 263–264

Simplify the exponent

Create a strategy We can simplify the fractions and use the rule of exponents which states that when we divide terms with the same base, we subtract the exponents.

5.03 Quotient rule mathspace.co

263

543

Apply the idea Rewrite the quotient as a product of quotients Divide the coefficients and apply the quotient of powers law Simplify the exponents 5 2

simplifies to 3m n .

So,

Idea summary

Purpose The quotient of powers law for exponents: Show students how to simplify more complex algebraic expressions involving both coefficients and variables with exponents. Expected mistakes a is any base number Students might subtract the coefficients rather than dividing them. If they did not group the coefficients together m is a power and group the other bases separately, as shown above, encourage them to do so. This should help them see n is a power that the coefficients are still divided, while the exponents of the other terms are subtracted. That is, when dividing terms with a common base: •

Keep the same base

Provide additional examples of problems with multiple bases • Find the difference in the power.

use with Example 1

Targeted instructional strategies Students may benefit from seeing more examples of simplifying expressions with the quotient rule, specifically examples where of the a quotient rule cannot Power quotient rule be applied. For example, ask students if the quotient rule can be applied to the expression

Exploration

. Ask students to justify their answers.

Use this example to highlight that, similar to the product rule, the quotient rule can only be applied to Consider the mathematical expression expressions with the same base. Other examples that can be used for discussion are shown: Apply the idea

Rewrite the quotient as a product of quotients 4 expressions: Complete the table below by2writing the expanded forms of the 3 expressions and then simplifying your Expression

Divide and apply the quotient of powers law Expanded form the coefficients Simplify as the quotient of two powers Simplify the exponents

Students:So,Page 264 simplifies to 3m n . 5 2

Idea summary The quotient of powers law for exponents:

1. 2.

a is any base numberbetween the original expression and the expanded form? Can you identify any patterns or relationships m is a power Can you write a rule based on your observations? n is a power That is, when dividing terms with a common base:

264

• Keep the same base Mathspace Virginia SOL Algebra 1 mathspace.co • Find the difference in the power.

Power of a quotient rule Exploration Consider the mathematical expression

544

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co Complete the table below by writing the expanded forms of the expressions and then simplifying your expressions: Expression

Expanded form

Simplify as the quotient of two powers

The quotient of powers law for exponents:

is any base number

is a power

Power of a quotient rule m is a power

That is, when dividing terms with a common base:

Exploration • Keep the same base •

Find the difference in the power.

Students: Page 264

Power of a quotient rule Exploration Consider the mathematical expression

Complete the table below by writing the expanded forms of the expressions and then simplifying your expressions: Expression

264

Expanded form

Simplify as the quotient of two powers

Can you identify any patterns or relationships between the original expression and the expanded form?

Can you write a rule based on your observations?

Mathspace Virginia SOL Algebra 1 mathspace.co

Suggested student grouping: In pairs In this exploration, students will be working with mathematical expressions of the form

. Students will then

look for patterns or relationships between the original expression and its expanded form, and try to formulate a rule based on their observations. Expression

Expanded form

Simplify as the quotient of two powers

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545

Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Can you identify any patterns or relationships between the original expression and the expanded form? Yes, I can see that the numerator and the denominator of the fraction are each raised to the original exponent when the fraction is expanded. 2. Can you write a rule based on your observations? Yes, the rule is that when you have a fraction raised to an exponent, you can apply the exponent to both the numerator and the denominator separately. This rule can be written as

Purposeful questions • What does an exponent represent? • How would the expressions change if the base was a whole number instead of a fraction? Possible misunderstandings • Students may have difficulty writing the expanded form. Remind them that the exponent represents how many times the fraction is multiplied by itself. Students will apply the previously seen power of a power rule to the newly learned quotient rule for exponents.

Students: Page 265 When the power is applied to the entire quotient, we use the power of a power law to write an expression for the power of a quotient rule as:

Consider the expression When the power is applied to the entire quotient, we use the power of a power law to write an expression for the power of a quotient rule as: This can be expanded as . Keep in mind we cannot simplify this any further because a and b are different bases. Consider the expression

Example 2

Examples

This canthe be expanded as Simplify following expressions:

. Keep in mind we cannot simplify this any further because

Students:a Page 265 and b are different bases. a

Example 2 Create a strategy Simplify the following expressions: The power of a quotient rule of exponents states that

. Apply this rule to the given expression.

Apply the idea Create a strategy

Apply the power of a quotient rule

The power of a quotient rule of exponents states that

. Apply this rule to the given expression.

Apply the power rule

Apply the idea

Simplify the exponents Apply the power of a quotient rule

So, the simplified form of

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Apply Edition the power rule Mathspace Virginia SOL Algebra 1 Teacher mathspace.co

Reflect and check

Simplify the exponents Notice that the expression cannot be simplified further, as the bases in the numerator and denominator are not the same and the numerical coefficients are fully simplified.

Create a strategy The power of a quotient rule of exponents states that

. Apply this rule to the given expression.

Apply the idea Apply the power of a quotient rule Apply the power rule Simplify the exponents

So, the simplified form of

Reflect and check Notice that the expression cannot be simplified further, as the bases in the numerator and denominator are not the same and the numerical coefficients are fully simplified.

Purpose Students apply the power of a quotient rule and the power of a power rule to simplify an algebraic expression. Expected mistakes Students might forget to apply the power to the coefficients in both the numerator and the denominator when using the power of a quotient rule. Remind students that coefficients are numerical bases, so an expression such as 2x6 actually has 2 bases. We must apply the power to all bases in the expression. 5.03 Quotient rule mathspace.co

Students: Page 266

265

Create a strategy We can apply the power of a quotient rule and the quotient rule to simplify the expression.

Apply the idea Apply the power of a quotient rule

Apply the quotient rule Evaluate the subtraction Therefore,

simplifies to a12b9.

Idea summary

Purpose When we raise a fraction to a power, we can use the power of a quotient rule: Check students’ ability to apply multiple rules of exponents (power of a product, power of a power, and quotient of powers) to simplify expressions.

Practice What do you remember? 1

5.03the Quotient rule When dividing two exponential terms with the same base, which operation should be used to simplify mathspace.co exponents? A

Addition

Subtraction

Rewrite each expression using a single exponent.

Multiplication

Division

547

Reflecting with students Encourage students to share different strategies they used to simplify these expressions, and highlight ones that are bdifferent from the one shown. If students did not use alternative methods, encourage them to simplify the problem by applying the quotient rule first. Create a strategy

Quotient We can apply the power of a quotient rule andrule the quotient rule to simplify the expression. Apply the idea

Evaluate the subtraction

Power ofthe a product Apply power of rule a quotient rule

Evaluate the multiplication

quotientto rule Point out that the result is the same. UseApply this the example remind students that, as long as the rules are applied Evaluate the subtraction correctly, they can be applied in any order.

Students:Therefore, Page 266

simplifies to a12b9.

Idea summary When we raise a fraction to a power, we can use the power of a quotient rule:

Practice What do you remember? Practice 1

When dividing two exponential terms with the same base, which operation should be used to simplify the

Students: Pages 266–268 exponents? A 2

Addition

Subtraction

Multiplication

Division

Rewrite each expression using a single exponent.

What do you remember? a

Addition 4

Subtraction

C c

Multiplication

Division

Simplify each expression in exponential form. f8÷f6

y8 ÷ y4

Complete each statement: a

x⬚ ÷ x3 = x7 266

548

Complete each statement:

⬚ Rewrite each a xexpression ÷ x3 = x7 using a bsingle exponent.

a 4

When3dividing two exponential terms with the same base, which operation should be used to simplify the Simplify each expression in exponential form. 8 exponents? b y8 ÷ y4 c n5 ÷ n3 d w6 ÷ w2 a f ÷f6 A

Mathspace Virginia SOL Algebra 1 mathspace.co

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

n5 ÷ n3

w6 ÷ w2

Let’s practice 5

Rewrite each expression using a single exponent. a

a10 ÷ a5

m5 ÷ m2

j 11 ÷ j 4

(240u32) ÷ (8u9) ÷ (5u12) f

15x18 ÷ 15x8 ÷ 15x5

Simplify: a

9r8 ÷ (4r2)

Simplify:

12r30 ÷ 12r8 ÷ 12r7

(m12)9 ÷ (m4)2

A student simplifies the expression 115 ÷ 1112 and gets 117 as the result. Explain the student’s mistake.

Explain why each statement is incorrect, then give the correct solution: a

When simplifying the expression

, Stein’s first line of work was a3 ⋅ a2

Describe what Stein may have done to get this line of work and determine its correctness. 12

Explain why m5 ÷ z3 is not equal to

Let’s extend our thinking 13

Two students were asked to simplify the expression

James simplified the expression as follows:

Neil simplified the expression as follows:

Which student is correct? Explain.

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Find the value of a and b in the equation:

Simplify: a

Complete the work by stating the exponent rule(s) used at each step to simplify the expression:

⬚ and ⬚

Simplify the exponents inside each group

⬚

⬚ and ⬚

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Simplify the exponents

Answers

11 Stein’s work is incorrect. Explanations may vary. One possible explanation: Stein may have applied the quotient rule for each term in the numerator separately, leading to subtract the exponent in the denominator from each exponent in the numerator. One way to determine if his line of work is correct is by expanding each expression and then simplifying them to show their equivalence. So

5.03 Quotient rule What do you remember? 1 B 2 a x4

b m5

c d9

d p8

b y

c n

4 a 10

b 5

c 11

3 a f

d w

a ⋅ a ⋅ a ⋅ a ⋅ a = a5, showing that Stein’s line of work is not equivalent to the expression and therefore is incorrect. 12 The expression m5 ÷ z3 indicates division of two different

Let’s practice

bases each raised to a power, whereas

5 a f7

b g3

c h3

d i2

6 a u11

c 3m7

d k13

e a5

g j7

k 9j2k3

b b18

c 125y6

d x24y−12

e a

−6

30 −15

h 27u v

means

raising the fraction of the bases to a power. The quotient rule applies to expressions with the same base, and since m and z are different bases, this law cannot be used to simplify

7 a

= a7 while a3 ⋅ a2 =

into

Let’s extend our thinking 13 Neil is correct. James is incorrect since he divided the exponents instead of subtracting them. 14 a = 3, b = 6

8 a n20r20

b 10x2y11z15

d m

e 6u11

g 2p

h 9v y j

2 11

k t v w

2 14 11

a b c

9 The student’s solution is incorrect because they subtracted the numerator from the denominator. According to the quotient rule, the exponent in the denominator must be subtracted from the exponent in the numerator. The expression should be 115 ÷ 1112 = 115−12 = 11−7 10 a When dividing terms with the same base, subtract the exponents, so

b 343a−21

y7 5 11

15 a 3a+6

100

, not t. The solution is 6.

Power of a product rule and power of a quotient rule

Simplify the exponents inside each group

Quotient rle

Product to a power rule and power of a power law Quotient rule and product of powers rule Simplify the exponents

b W hen raising a power to a power, we multiply the exponents rather than add them which is done here. We will have a different answer when we divide like bases. The solution is x2y22.

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5.04 Zero and negative exponents Subtopic overview Lesson narrative In this lesson, students will apply their knowledge of the division property and power property to simplify expressions with negative exponents. Students will complete two exporation activities to help them use the zero property to simplify expressions with an exponent of zero and recognize that an exponent of zero for a numeric or algebraic expression will result in one. By the end of this lesson, students will be able to simplify exponential expressions with negative exponents and apply the zero property of exponents.

Learning objectives

5.04 Zero and negative exponents

Students: Page 269

After this lesson, you will be able to… • simplify algebraic expressions using the zero rule for exponents. • simplify algebraic expressions with negative exponents.

Zero and negative exponents Nowvocabulary let’s look at the quotient of powers rule when m and n are equal. Key 

negative exponent rule

 reciprocal Interactive exploration



zero rule

Explore online to answer the questions

mathspace.co Essential understanding Understanding the underlying patterns for zero and negative exponents can inform more efficient methods for Use the interactive exploration in 5.04 to answer these questions. computation and analysis. 1. Use the applet to complete the table of values for the number of sections in the paper created based on the number of folds we make:

Standards Number of folds (exponent)

Number of the sections created This subtopic addresses following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical 2. As theprocess number of goals folds decreases, what is the pattern in the number of sections created? 3. Moving from right to left, complete the table of values using the pattern you found above: MPG2 — Mathematical Communication −2 To achieve the4goal of teachers can encourage students to articulate their thinking 4−1Mathematical 40 41 Communication, 42 43 and reasoning using the vocabulary and 16 symbolic 64 notation of exponents. Teachers can foster classroom discussions around the reasoning behind the rules of exponents and the implications of negative exponents in scientific notation. These discussions can help students clarify their thinking and deepen their understanding.

The zero rule states that a0 = 1 The zero rule tells us any non-zero base raised to the power of zero is equal to 1. For example: Substitute 0 = 3 − 3 Quotient rule 552

Mathspace Virginia SOL Algebra 1 Teacher Edition Write in expanded form mathspace.co

Divide out common factors

MPG4 — M athematical Connections Teachers can facilitate activities that link the properties of exponents to other areas of mathematics and real-world contexts. Students can be guided to connect negative exponents with the concept of reciprocals and division, reinforcing their understanding of inverse operations. Teachers can encourage discussions on how the zero exponent property relates to patterns in exponential sequences, highlighting the continuity of mathematical principles. Connections to scientific notation can be made by demonstrating how negative exponents are used to represent very small numbers in science and engineering.

Content standards A.EO.3 — The student will derive and apply the laws of A.EO.3b — Simplify multivariable expressions exponents. and ratios of monomial expressions in which the exponents are integers, using the laws of exponents. A.EO.3a — Derive the laws of exponents through explorations of patterns, to include products, quotients, and powers of bases.

Prior connections 6.NS.3 — The student will recognize and represent patterns with whole number exponents and perfect squares.

7.NS.1 — The student will investigate and describe the concept of exponents for powers of ten and compare and order numbers greater than zero written in scientific notation.

Future connections A2.EO.3 — The student will perform operations on polynomial expressions in two or more variables and factor polynomial expressions in one and two variables.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 7 — 1.05 Powers of 10 with negative exponents Algebra 1 — 5.03 Quotient rule

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Student lesson & teacher guide Zero and negative exponents Students begin the lesson by engaging in an exploration.

Students: Page 269

5.04 Zero and negative exponents 5.04 Zero and negative 5.04 Zero and negative exponents exponents After this lesson, you will be able to… After this algebraic lesson, you will be able to…the zero rule for exponents. • simplify expressions using this algebraic lesson, you will be able to… •After simplify expressions using the zeroexponents. rule for exponents. with negative usingnegative the zeroexponents. rule for exponents. • simplify algebraic expressions with • simplify algebraic expressions with negative exponents.

Zero and negative exponents Zero and negative exponents Now let’s look at the quotient of powers rule when m and n are equal. Zero and negative exponents Now let’s look at the quotient of powers rule when m and n are equal. Now let’s look at the quotient of powers rule when m and n are equal.

Interactive exploration Interactive Explore online to exploration answer the questions Interactive Explore to exploration answer the questions Connecting toonline prior knowledge of negative and zero exponents Targetedmathspace.co instructional strategies

Explore online to answer the questions

mathspace.co

Use experience themathspace.co interactivewith exploration in 5.04 to zero answer these questions. Prior student negative and exponents includes powers of ten and scientific notation. Use the interactive exploration in 5.04 to answer these questions. Connecting 1.to this knowledge could help of students rules for negative andcreated zero exponents. Useprior the applet to complete the table values forestablish the number of sections in the paper based on Use the interactive exploration in 5.04 to answer these questions. 1. finish Usenumber the applet to filled complete tablesuch of values for the of sections in the paper created based on the of folds we make: Help students a partially outthe table as the onenumber below: 1.

Usenumber the applet to complete the table of values for the number of sections in the paper created based on the of folds we make:

0 Power of tentheNumber Expanded form Standard number folds(exponent) we make: ofoffolds 2 21 form 22 23 24 0 1 2 3 3 Number of folds (exponent) 2 2 2 2 24 Number 10 10 ⋅ of 10sections ⋅ 10 created 1000 0 1 2 3 Number of folds (exponent) 2 2 2 2 24 Number of sections created 102 10 ⋅ 10 100 of sections 2. AsNumber the number of foldscreated decreases, what is the pattern in the number of sections created? 101 2. Moving As the number of folds decreases, is the the the number of sections created? 3. from right to left, complete what the table ofpattern values in using pattern you found above: 2. the number of folds decreases, is the the the number of sections created? 100 3. As Moving from right to left, complete what the table ofpattern values in using pattern you found above: 3. Moving 4−2 from 4−1 right 40to left, 41 complete 42 the 43 table of values using the pattern you found above: 10-1 2 4−2 4−1 40 41 4 43 16 64 2 10-2 4−2 4−1 40 41 4 43 16 64

10-3

The zero rule states that a = 1 0

The zero rule states thatnegative a =1 Discuss The withzero students make theisvalue negative, but focused instead on moving rule tellshow us any non-zero exponents base raised todid thenot power of zero equal to 1. The zero rule states that a0 = 1 zerothe rulenumerator tells us any non-zero base raised to the power of zero is equal to 1. the baseThe from to the denominator. Also discuss how the expanded form was divided by 10 each For example: The zero rule tells us any non-zero base raised to the power 1 of zero is equal to 1. example: time theFor power decreased by 1. A power of 0 meant 10 would be divided by 10, giving 1. Substitute 0 = 3 − 3 For example:

Discussion supports

Substitute 0 = 3 − 3 Quotient Substituterule 0=3−3 Quotient rule Quotient rule Write in expanded form

Write in expanded form English language learner support

Write expanded Divideinout commonform factors

Provide sentence starters to help students discuss Divide out commonconcepts factors and problem-solving strategies about zero and Divide out common factors negative exponents: • “The exponent is negative, so I will …” • “When the exponent is zero, the result is … because …” • “I can simplify an expression with a negative exponent by …” • “Comparing the expressions with both positive and negative exponents, I noticed …” Encourage students to use terms such as “numerator”, “denominator”, “reciprocal”, and “quotient rule” in their explanations.

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269

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269

Use color-coding with various examples Student with disabilities support To help students apply the negative exponent rule, provide multiple examples with varying bases. Give them these steps to follow for each example: 1. Write the expression as a fraction with 1 in the denominator. 2. If a base has a positive exponent, draw a red box around it. These expressions will stay in the numerator. 3. If a base has a negative exponent, draw an arrow pointing to the denominator. We will move these expressions to the denominator and make the exponent positive. 4. If nothing is left in the numerator, write a 1 in the numerator. A few examples using this method are shown:

Negative exponents are reciprocals not negatives Address student misconceptions Students often think that a negative exponent makes the base negative. For example, they might incorrectly think that 3−2 equals −9. In reality, a negative exponent means that the reciprocal of the base is raised to the

5.04 Zero and negative exponents

corresponding positive power. So, 3−2 actually equals

This may need to be reiterated and shown multiple times throughout the lesson as students develop their After this lesson, you will be able to… understanding of the rule. • simplify algebraic expressions using the zero rule for exponents. • simplify algebraic expressions with negative exponents.

Exploration Students:Zero Pageand 269negative exponents Now let’s look at the quotient of powers rule when m and n are equal.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 5.04 to answer these questions. 1.

Use the applet to complete the table of values for the number of sections in the paper created based on the number of folds we make: Number of folds (exponent) Number of sections created

As the number of folds decreases, what is the pattern in the number of sections created?

Moving from right to left, complete the table of values using the pattern you found above: 4−2

4−1

42 16

43 64

The zero rule states that a0 = 1 The zero rule tells us any non-zero base raised to the power of zero is equal to 1. For example: Substitute 0 = 3 − 3

5.04 Zero and negative exponents mathspace.co

555

Suggested student grouping: Individual Students will engage in a hands-on exploration using either a digital slider tool or a physical sheet of paper. They will be asked to fold the paper in half, each time unfolding it to observe the number of sections created. They will examine how the number of folds relates to the power of 2, and question what it means when 2 is raised to the power of zero. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here.

5.04 Zero and negative exponents

1. Use the applet to complete the table of values for the number of sections in the paper created based on the number of folds we make: The number of sections created should be: 1, 2, 4, 8, 16 2. As the number ofthis folds decreases, is the pattern in the number of sections created? After lesson, you will bewhat able to… As the numbers of folds decreases, theusing number of sections created is halved, or divided by 2. • simplify algebraic expressions the zero rule for exponents. simplify algebraic expressions with negative exponents. 3. Moving from• right to left, complete the table of values using the pattern you found above:

The numbers that will complete the table are:

, , 1, 4

Zero and negative exponents Purposeful questions let’s look the quotient powers rule when m andton the are equal. • WhyNow do you thinkatany numberof(except zero) raised power of zero results in one? Can you explain this using the concept of folding the paper? Interactive exploration • What happens to the number of sections created when you increase the number of folds? Now what Explore online to answer the questions happens when we go in the opposite direction, and decrease the number of folds?

mathspace.co Possible misunderstandings • When working backwards to understand what 40 or 4−1 might equal, students may fail to understand why Use the interactive exploration in 5.04 to answer these questions. to divide by 4 each time as the power gets smaller. They may incorrectly think that we should subtract Use theby applet to complete the table of values the number sections in the paper created based on 4 instead1.of divide 4. This misunderstanding mayforarise from aoflack of understanding of the relationship the number of folds we make: between division and subtraction. It’s important to stress that when you decrease a power, it equates to 3 4 dividing the previous the base, not represent repeated multiplication. Number result of foldsby (exponent) 20 subtracting 21 22 it,2since 2powers Number of sections created

Students will be2.introduced to the zerodecreases, power and exponent rules.ofThey willcreated? learn that any base raised to a As the number of folds whatnegative is the pattern in the number sections power of 0 equals 1. They will discover this concept by exploring problems that involve using the quotient property 3. Moving from right to left, complete the table of values using the pattern you found above: when the bases of the powers are equivalent. 4−2

4−1

Students: Pages 269–270

42 16

43 64

The zero rule states that a0 = 1 The zero rule tells us any non-zero base raised to the power of zero is equal to 1. For example: Substitute 0 = 3 − 3 Quotient rule Write in expanded form Divide out common factors

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269

The negative exponent rule states:

The negative exponent rule tells us any non-zero base raised to a negative exponent is equal to 1 divided by the same base raised to the opposite positive exponent. Consider the following: Write in expanded form The negative exponent rule states: Group common factors Divide common factors The negative exponent rule tells us any non-zero base raised to a negative exponent is equal to 1 divided by the Multiplicative identity same base raised to the opposite positive exponent. Consider the following: Write in expanded form Write in exponential form Group common factors

And also:

Quotient rule factors Divide common Evaluate the subtraction Multiplicative identity .

Therefore,

Write in exponential form

We use this rule to write negative exponents as positive exponents or positive exponents as negative exponents. Andcan also: Quotient rule

Example 1

Evaluate the subtraction Simplify x5 ÷ x5 by first writing the expression in expanded form.

Examples

Therefore,

strategy Students:Create Pagea 270

We can this rule to negative exponents as positive exponents as negative exponents. Write theuse expression aswrite fraction and write it in expanded formexponents to cancel or outpositive the common factors.

Apply the idea Example 1 Write the expression as fraction Simplify x5 ÷ x5 by first writing the expression in expanded form. Write in expanded form

Create a strategy

Write the expression as fraction and write it inout expanded form to cancel out the common factors. Divide common factors Simplify

Apply the idea

Write the expression as fraction

Reflect and check

, we could have written x5 ÷ x5 as = x5 − 5 = x0. Now we can see that x0 = 1. Write in expanded form This should be no surprise; the initial expression asks us “What do we get when we divide x5 by itself?”, to which the answer is simply 1, since anything divided by itself is equivalent to 1. Divide out common factors m

m−n

Recall that a ÷ a = a

Simplify

Reflect and check Recall that am ÷ an = am − n, we could have written x5 ÷ x5 as = x5 − 5 = x0. Now we can see that x0 = 1. This should be no surprise; the initial expression asks us “What do we get when we divide x5 by itself?”, to which the answer is simply 1,Virginia sinceSOL anything by itself is equivalent to 1. Algebradivided 1 270 Mathspace mathspace.co

Purpose Students will simplify by expanding before cancelling similar factors, thus showing that a base raised to a power of 0 is 1. 270

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557

Expected mistakes Students may think that the value should be 0 since all of the pairs will “cancel out”. Remind them that “canceling” refers to dividing a number by itself, which results in 1. Reﬂecting with students If students struggle to understand why the result is 1, replace x with a quantity. Then, group each division or “cancellation” to emphasize the result is 1, as shown:

Reminding students that any value divided by itself is 1 reminds them that the phrase “canceling factors” actually refers to dividing a value by itself.

Students: Page 271 Example 2 Simplify 9p0.

Create a strategy

Apply the idea

We can use the zero rule: a0 = 1

Since the base of p has a power of 0, the whole expression is equal to 9 ⋅ 1. So by simplifying this, we have: 9p0 = 9 ⋅ 1 = 9

Example 2 PurposeExample03 Simplify 9p .

Check students can apply the zero rule to simplify expressions with a coeﬃcient. Write the following with a negative exponent:

Create a strategy

Apply the idea

Expected mistakes We can use the zero rule: a0 = 1 Since the base of p has a power of 0, the whole Students may say the entire expression is equal to 1. Remind students that a coeﬃcient is a different base that is expression is equal to 9 ⋅ 1. So by simplifying this, 1 0 multiplied to the variable: 9 ⋅ p . we have: Create a strategy We can use the product rule, am ⋅ an = am + n, and the negative exponent rule,

Students: Page 271

9p0 =. 9 ⋅ 1 = 9

Apply the idea Each fraction3is to the power of 1 so we can start by applying the product rule. Example Apply the product rule Write the following with a negative exponent: Evaluate the addition Negative exponent rule

Create a strategy

rule the negative exponent rule, We can use the product rule, a ⋅ a = Power am + n, and m

Therefore,

Apply the idea

expressed with a negative exponent is g−4.

Each fraction is to the power of 1 so we can start by applying the product rule.

Example 4

Apply the product rule

Evaluate the addition Express the following with positive exponents. 5 ⋅ x−2 ⋅ a6 3a2 ⋅ x−4 ⋅ rule Negative exponent

558

Power rule Mathspace Virginia SOL Algebra 1 Teacher Edition Create a strategy mathspace.co −4 Therefore, expressed n We can use product of powers rule:with am ⋅ aanegative = am + n exponent is g .

Create a strategy We can use the product rule, am ⋅ an = am + n, and the negative exponent rule,

Apply the idea Each fraction is to the power of 1 so we can start by applying the product rule. Apply the product rule Evaluate the addition Negative exponent rule Power rule Therefore,

expressed with a negative exponent is g−4.

PurposeExample 4 Make students aware that, not only can expressions with negative exponents be rewritten with positive Express the following with positive exponents. exponents, but the reverse is also true. 2 −4 −2 6 3a ⋅ x

⋅5⋅x

⋅a

Expected mistakes a strategy StudentsCreate may forget to ﬁnd the reciprocal of

, or keep the exponent positive instead of making it negative.

We can use product of powers rule: am ⋅ an = am + n

Reﬂecting with students Help students see that this is an application of the converse of the negative exponent rule, example, a = g and n = 4:

. In this

Zero and negative exponents 271 While it is not common to leave answers with negative exponents, this type5.04 of example can be useful when mathspace.co simplifying more complex expressions.

Provide additional examples for the negative exponent rule

use with Example 3

Targeted instructional strategies Students may benefit from seeing multiple examples where they must apply the negative exponent rule and its converse. Examples can include the following: 1 2 3 4 5 As students gain confidence with expressions with one variable, show examples that include multiple variables or that require applying the product or quotient rules first.

5.04 Zero and negative exponents mathspace.co

559

Negative exponent rule Power rule Therefore,

expressed with a negative exponent is g−4.

Students: Pages 271–272

Example 4 Express the following with positive exponents. 3a2 ⋅ x−4 ⋅ 5 ⋅ x−2 ⋅ a6

Create a strategy We can use product of powers rule: am ⋅ an = am + n

Apply the idea Multiply the coefficients Commutative property Product rule Evaluate the addition 5.04 Zero and negative exponents Negative exponent rule

271

mathspace.co

Multiply

Idea summary

Purpose For any numeric or algebraic expression a, the zero rule tells us that Check students’ understanding of the application of the product of powers rule and the correct interpretation of a0 = 1 negative exponents. The negative exponent rule states: Expected mistakes Students may multiply exponents instead of add. They may also combine the exponents of non-like terms together, or perform the fraction multiplication incorrectly. Apply Raisetheaidea fraction to a negative exponent

Multiply the coefficients Reﬂecting withuse students We can this same understanding to raise a fraction to a negative exponent. Commutative property StudentsLet’s may wonder if they can rewrite the terms with positive exponents before multiplying. If some students first review what a reciprocal is. used this strategy, encourage them to share their results with the class. If no one tried this method, give Product rule . The reciprocal of is . The reciprocal of 91 is . The reciprocal of is students a few minutes to solve the problem in this way. Evaluate the addition So, toshould find thebe reciprocal, you expression, need to invert or flip the fraction. have an integer, you put that integer as the Their result the same showing we canIf you write terms with positive exponents before Negative exponent rule denominator, and 1 as the numerator. performing operations. Multiply

Exploration Students: Page 272 Consider the expression

. We want to write this without negative exponents.

Idea summary

Apply the exponent to the terms inside the parentheses For any numeric or algebraic expression a, the zero rule tells us that Rewrite as a division 0 a =1 Apply the negative exponent rule The negative exponent rule states: Multiply by the reciprocal Simplify 1.

Complete the working above by filling in the blanks.

2. a Complete the statement: In general, exponent Raise fraction to a negative We can use this same understanding to raise a fraction to a negative exponent. Let’s first review what a reciprocal is. When raising a fraction to any negative exponent, we find the reciprocal of the fraction in the parentheses, then apply . The reciprocal of is . The reciprocal of 91 is . The reciprocal of is . the power of a quotient law: So, to find the reciprocal, you need to invert or flip the fraction. If you have an integer, you put that integer as the denominator, and 1 as the numerator.

560

Mathspace Virginia SOL Algebra 1 Teacher Edition 272 Mathspace Virginia SOL Algebra 1 mathspace.co Exploration mathspace.co Consider the expression

. We want to write this without negative exponents.

Let’s first review what a reciprocal is. The reciprocal of

. The reciprocal of

is . The reciprocal of 91 is Multiply

So, to find the reciprocal, you need to invert or flip the fraction. If you have an integer, you put that integer as the denominator, and 1 as the numerator.

Raise a Exploration fraction to a negative exponent Idea summary For anythat numeric or algebraic a, the zero rule tells us that Students will discover when raising aexpression fraction to a negative power, both the numerator and denominator are 0 = 1 a raised to thatConsider power, the andexpression the negative. exponent rule this willwithout still apply. We want to write negative exponents. The negative exponent rule states:

Students: Page 272

Apply the exponent to the terms inside the parentheses Rewrite as a division Apply the negative exponent rule

Raise a fraction to a negative Multiply by exponent the reciprocal

We can use this same understanding Simplify to raise a fraction to a negative exponent. Let’s 1. first review whatthe a reciprocal is. by filling in the blanks. Complete working above . The reciprocal of is . The reciprocal of 91 is The reciprocal of isthe statement: 2. Complete In general,

So, to find the reciprocal, you need to invert or flip the fraction. If you have an integer, you put that integer as the denominator, and 1 as the numerator. When raising a fraction to any negative exponent, we find the reciprocal of the fraction in the parentheses, then apply theExploration power of a quotient law: Consider the expression 272

. . We want to write this without negative exponents.

Mathspace Virginia SOL Algebra 1 Apply the exponent to the terms inside the parentheses mathspace.co

Graphic organizer of exponent laws

Rewrite as a division

Student with disabilities support

Apply the negative exponent rule

If students have not already done so, have them create a graphic organizer of the exponent laws. Alternatively, Multiply by the reciprocal provide a complete graphic organizer for them to use and reference throughout the remainder of the year. Simplify An example 1.organizer is the shown: Complete working above by filling in the blanks. of law 2. Name Complete the statement: In general,Exponent law

Example Product rule a ⋅a =a x ⋅ x = x7 mn mn Power rule (a ) = a (x3)4 = x12 n raising a fraction the reciprocal Power When of a product rule to any negative exponent, (ab)n = we anbfind (2x)3 of = the 8x3fraction in the parentheses, then apply m

the power of a quotient law:

Quotient rule

m+n

Power 272 of a quotient Mathspace rule Virginia SOL Algebra 1 mathspace.co

Zero rule

a0 = 1

x0 = 1

Negative rule Fraction with negative exponent

5.04 Zero and negative exponents mathspace.co

561

Raise a fraction to a negative exponent We can use this same understanding to raise a fraction to a negative exponent. Let’s first review what a reciprocal is. Exploration The reciprocal of

. The reciprocal of

is . The reciprocal of 91 is

Students:So,Page 272 to find the reciprocal, you need to invert or flip the fraction. If you have an integer, you put that integer as the denominator, and 1 as the numerator.

Exploration Consider the expression

. We want to write this without negative exponents. Apply the exponent to the terms inside the parentheses Rewrite as a division Apply the negative exponent rule Multiply by the reciprocal Simplify

Complete the working above by filling in the blanks.

Complete the statement: In general,

When raising a fraction to any negative exponent, we find the reciprocal of the fraction in the parentheses, then apply the power of a quotient law:

Suggested student grouping: Individual Students will apply their previously learned power laws with the aim to generalize a rule for applying a negative power to a quotient. 272

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Ideal student responses

These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Complete the working above by ﬁlling in the blanks.

2. Complete the statement: In general,

=⬚

Purposeful questions • Which exponent law is used in the first step? • If we apply the negative exponent first, how can we use fraction division to get the same result? • How would you explain this rule in your own words? Possible misunderstandings • Students may not realize that “Multiply by the reciprocal” refers to the division of the fractions in the previous step. • Students may only apply the exponent to the numerator and not the denominator. Remind them of the power of a quotient law from the previous lesson.

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Advanced learners: Algebraic derivation of the exponent laws Targeted instructional strategies Extend the power law for advanced learners by investigating how the laws of exponents extend to zero and negative exponents. Begin by asking students to consider expressions like both the expanded form and the power rule of exponents. Power rule

and simplify them using

Expanded form

Then, have them generalize their findings by describing what happens when a non-zero base is raised to the zero power and when a non-zero base is raised to a negative power. Having students justify and generalize these patterns algebraically helps them construct a deeper conceptual understanding of the zero and negative exponent properties and gain familiarity with working with abstract or generalized proofs.

Examples Students: Page 273 Example 5 Express

using positive exponents.

Create a strategy

Apply the idea

We can use 5 the exponential rule: Example Express

Rewrite using the reciprocal

using positive exponents.

Example 6 a strategy Apply the idea PurposeCreate StudentsWedemonstrate that they can raise a quotient of powers to a negative exponent. can use the exponential rule: Write

with positive exponents.

Rewrite using the reciprocal

Students: Page 273

Create a strategy First rewrite 6 with a positive exponent and then apply the quotient of powers rule. Example

Apply the idea Write

with positive exponents. Apply the negative exponent rule

Create a strategy

Apply the quotient of powers rule First rewrite with a positive exponent and then apply the quotient of powers rule.

Apply the idea

Apply the power rule

Reflect and check

Apply the negative exponent rule

An alternative method for solving this problem is applying the quotient of powers rule first, then the power rule, finally the negative exponent rule: Apply the quotient of powers rule 5.04 Zero and negative exponents

mathspace.co Apply the quotient of powers rule Apply the power rule

563

Write

with positive exponents.

Create a strategy First rewrite with a positive exponent and then apply the quotient of powers rule.

Apply the idea Apply the negative exponent rule Apply the quotient of powers rule Apply the power rule

Reflect and check An alternative method for solving this problem is applying the quotient of powers rule first, then the power rule, finally the negative exponent rule: Apply the quotient of powers rule Apply the power rule Write as a division Negative exponent rule Multiply by the reciprocal Multiply

Purpose Use multiple laws of exponents to simplify an expression with powers. 5.04 Zero and negative exponents mathspace.co

273

Expected mistakes Students might forget to apply the power to the coefficients in the numerator and denominator. If this happens, encourage them to review the power of a product rule from previous lessons.

Students: Page 274

Purpose Use multiple laws of exponents to simplify an expression with powers.

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Expected mistakes Students may subtract the exponents in the wrong order, or put the c on the wrong side of the fraction bar. They may also forget to apply the negative power rule to the d. Reﬂecting with students If we apply the negative power rule to the d in the first step, we can see how expanding the powers will give the same result.

Negative exponent rule

Expand powers Cancel similar pairs of factors Simplify

Avoiding negatives in the quotient rule

use with Example 7

Address student misconceptions When simplifying

, students may attempt to avoid getting a negative exponent by subtracting c4 − c3. Use the

expanded form of the expression to show students that, after canceling like terms, the remaining c would be in the denominator. This means the result from the subtraction would be −1, not 1. Generalize by reminding students that the exponent in the result is always obtained by subtracting the exponent of the divisor from the exponent of the dividend.

Students: Page 274

Practice Students: Pages 274–276

What do you remember? 1

True or false: a

a-4 = -a4

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565

Rewrite each expression using a single exponent. a

Fill in the missing value to complete the pattern. m5 = 1 ⋅ m ⋅ m ⋅ m ⋅ m ⋅ m m4 = 1 ⋅ m ⋅ m ⋅ m ⋅ m m3 = 1 ⋅ m ⋅ m ⋅ m m2 = 1 ⋅ m ⋅ m m1 = 1 ⋅ m

m0 = ⬚

Evaluate: a

( j )0

(k ⋅ t)0

s-6

v-12

y-2x3

a-5b-4c4

Let’s practice 5

Express the following using only positive exponents: a

t-2

e 6

Express the following using only negative exponents: a

r-3

Determine the missing exponent so that following equation is always true. (a3b-5)⬚ =a-15b25

Fill in the missing exponent to make the following equations true. a

566

g12 ÷ g12 = g⬚

j16 ÷ j16 = j ⬚

Rewrite with positive exponents: a

a-9

p-2

3x-4

8p-3

2x-8y3

m 3a4b-5c-7

7x-9

p-2q3

m-5n-4p4

4-2k-3l7

Simplify: a

(w0 + s)2

(x7)0 + x - x0

18a0

11a0

(a0)79

9 ⋅ (15x6)0

(3m4)3 ⋅ mq0

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(6a)0

Simplify using the appropriate exponent rule(s). Final answer should contain only positive exponents. a

(4x4y3)3(2x− 4y5)−2

(3x10y7)4(5x4y−5)−3

What is the value of x0? Explain why this remains the same for any value of x. What is 04 equal to?

Explain the difference between 2x0 and (2x)0.

Explain why 0−4 is undefined.

Let’s extend our thinking 15

Write

Celeste and Donovan are writing an equivalent expression for

at least three different ways.

Celeste writes down

as a single term.

-10

and Donovan writes down x

Who is correct and why? 17

Nigel simplified the expression as follows: 1 2 3 4 His teacher noticed that there was an error in his solution. Identify the error and explain how to correct it.

A student was writing 5a-1 without negative indices and wrote and write the correct answer.

When asked to find an expression that is equivalent to -20, a student responded 1. Is this answer correct? Explain why or why not.

. Explain why their working is incorrect,

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Answers

13 a 0 b

5.04 Zero and negative exponents What do you remember? 1 a True

b False

c False

d True

2 a x3

b y16

c r6

d f 16

b 1

c 1

d 1

3 1 4 a 1 Let’s practice 5 a e h2

• •

b 10p−8q−5

• a-2b2

8 a 0

b 0

c 11

d 7

9 a

b an

10 a 4

b x

c d5

d p8

17 There was an error in step 1 of the solution. Nigel forgot to distribute the power of the exponent in the denominator leading to an incorrect ﬁnal answer. To correct it, distribute the power of the exponent in the denominator, so that the denominator becomes 3644610

g 11

h 1

k 27m13

h 48x9y14

b 4p

For example,

we can show that their answers

are equivalent.

e 18

16 Both Celeste and Donovan are correct. Using the negative exponent rule x−n =

5y5

12 1. For any value of x, we can rewrite x0 as a fraction with the same numerator and denominator.

568

The expression (2x)0 represents the entire quantity 2x raised to the power of 0. When a non-zero base is raised to the power of 0, the result is always 1. This means that (2x)0 simpliﬁes to 1.

15 There are many different representations, such as:

7 −5

11 a

14 The expression 2x0 represents the number 2 times x raised to the power of 0. Any non-zero number raised to the power of 0 is equal to 1. This means that 2x0 simpliﬁes to 2.

Let’s extend our thinking

6 a u−5

and dividing by zero is impossible.

since 3 − 3 = 0.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

18 The −1 exponent only applies to the term directly in front of it. In this case it only applies to the a, not the 5. The correct work is 5a−1 = . 19 The correct value is −1 because the negation applies to the base, not to the exponent.

5.05 Rational exponents Subtopic overview Lesson narrative In this lesson, students will build on their prior knowledge of integer exponents and use this understanding to construct arguments for conversion rules from radicals to exponents. Students will complete an exploration activity that connects previously learned exponent rules and rational multiplication to rational exponents. They will use their rules and repeated reasoning to translate expressions from radical form to exponential form and vice versa. By the end of this lesson, students should be able to translate fluently between radical and exponential forms using properties of exponents and an understanding of how rational exponents are defined.

Learning objectives Students: Page 277

Key vocabulary 

index



radical



radicand

 rational exponent

Essential understanding The laws of integer exponents also apply to rational exponents and can be useful when simplifying expressions.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving Teachers can incorporate this goal into their lesson by emphasizing the problem-solving aspect of understanding rational exponents. They can provide a variety of practice problems that require students to apply their understanding of rational exponents, and encourage students to draw upon their prior knowledge of square roots and cube roots to solve these problems. Teachers can also provide real-world scenarios where students need to apply their knowledge of rational exponents, thus allowing them to become better problem solvers.

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MPG4 — Mathematical Connections

MPG5 — Mathematical Representations

Teachers can incorporate this goal into their lesson by continually relating the new concept of rational exponents back to the prior knowledge of square roots and cube roots. They can also show how this concept connects to the Power of a Power Law and other mathematical rules or laws. Teachers can also highlight connections between this concept and real-world situations or applications.

Teachers can integrate this goal into their lesson by showing different ways to represent the concept of rational exponents, such as through symbolic notation, diagrams, or examples. They can also have students create their representations and explain their thinking. The teacher can also demonstrate how to generate equivalent numerical expressions using rational exponents and how to justify their equivalency.

Content standards A.EO.4 — The student will simplify and determine equivalent radical expressions involving square roots of whole numbers and cube roots of integers.

A.EO.4d — Generate equivalent numerical expressions and justify their equivalency for radicals using rational exponents, limited to rational exponents of

and

Prior connections 7.NS.3 — The student will recognize and describe the relationship between square roots and perfect squares.

7.CE.1 — The student will estimate, solve, and justify solutions to multistep contextual problems involving operations with rational numbers.

A.EO.3 — The student will derive and apply the laws of exponents.

Future connections A2.EO.2 — The student will perform operations on and simplify radical expressions.

Engage Activity Exponents exploration

45 mins

Students will be using the exponent mutiplication rule to come to conclusions about negative and fractional powers.

Understanding and skills

Will use Using laws of integer exponents to rewrite numerical expressions.

Will develop Understanding the meaning of negative exponents. Understanding the meaning of fractional exponents. Using and understanding the exponent multiplication rule.

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Could extend Generalizing the fractional and negative exponent rules.

Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: None

Support students with disabilities Support conceptual processing - solve abstract problems Set up the investigation so that students move from the concrete to the abstract: Remind students of previously learned exponent rules and of how to solve the equation ⬚ + ⬚ = 1. Then extend to 16⬚ ⋅ 16⬚ = 161.

Support for English language learners Three reads Have students read the task aloud. On the ﬁrst read, ask students to describe the situation. Prompt: Students read the problem. Students think/write: Answer the question “What is the problem about?” Answers may look like: • The exponents multiplication rule. • Investigating exponents. • Adding exponents together. Share: Students are called upon to discuss their answers with the class. On the second read, ask students to interpret the question. Prompt: Students read the problem. Students think/write: Answer the question “What does an answer look like?” Answers may look like: • Numbers that make the given equation true. • An investigation about exponents. On the third read, have students identify important information. Prompt: Students read the problem. Students think/write: Answer the question “What are the important pieces of information given in the question?” Answers may look like: • The exponents multiplication rule: xa ⋅ xb = xa + b • 16⬚ ⋅ 16⬚ = 161 • The multiplication rule is used to add exponents together.

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Classroom guide Hook

Open questions

•

5 mins

Consider the following equation, where a and b represent two numbers: a+b=1 Come up with as many number combinations as you can for a and b which make this equation true. Slide 1 from Student Engage Activity

Students ﬁnd values for a and b that add up to 1.

Implementation details The hook will prompt students to start thinking about what numbers have a sum of 1. Students will come up with a myriad of solutions; however, ensure that all students are aware that there are solutions which include negative numbers, fractions, and zero.

Launch

5 mins

On the left side of the given equation we have two terms with the same base (16) being multiplied together:

Slide 2 from Student Engage Activity

Introduce students to the exponent multiplication rule and ensure that all students understand the basic functionality of the rule. At this point, you may want to show students a short proof on why the multiplication rule works. Important mathematical concepts: Exponents, base, multiplication rule, positive numbers, negative numbers, fractions, true equation. Suggested grouping: Form pairs

Continue when All students have a basic understanding of the exponent multiplication rule.

Explore

Think-pair-share

•

25 mins

Anticipated strategies Create true equations Students will use the exponent multiplication rule to come up with four solutions, in the form of exponents, to the equation 16⬚ ⋅ 16⬚ = 161. Solutions will consist of positive numbers, negative numbers, zero, and fractions. Some examples of solutions are as follows: 1 and 0, 4 and −3, −7 and 8,

and .

Evaluate fractional and negative exponents Students will use reverse engineering to determine the meaning of fractional and negative exponent expressions by using a set of exponents from their true equations and speculating. Students should be reminded of the exponent multiplication rule, xa ⋅ xb = xa + b, but encouraged to not use it in their reasoning. Students may speculate the meaning of fractional or negative indices by reverse engineering their equation. 572

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Misconceptions Confusing the power and the root in fractional exponents. How else can we represent a fraction?

Not understanding the exponent multiplication rule. What operation do we use to combine the exponents? How does the exponent multiplication rule work? Why do you think it works?

Purposeful questions information About The Questions • What numbers add together to equal one? And thus, what exponents can we use to make 161? • Can we use negative numbers? Fractions? What about zero? • What is the answer to 161? How will that help us ﬁnd the values of 16⬚?

• Can you ﬁnd a pattern for 16⬚ when you use the exponents: −3, −2, −1, 0, 1, 2, 3? And what about: , , ?

Continue when All students have found pairs of exponents which make the equation true, as well as determined what their pair of numbers mean as exponents. Some students have been able to fully convince the class of what their exponents mean. Few students have been able to consolidate their knowledge to write exponent rules for fractional and negative exponents.

Discuss

25 mins

Use a group discussion for students to share their work with the class. Consider sequencing the strategies presented from negative to fractional exponents.

Discussion guide In the discussion, we want to give students ample time to share their ideas and solutions. Each pair of students will likely have a different set of numbers, so allow each group to present their solutions to the class and show what their exponents actually mean. Students will be explaining what their set of exponents means, ie. what a fractional or negative exponent means to then evaluate 16 to their chosen exponents. When asking students to explain what 16⬚ actually means, we want them to explain what 16 to their set of exponents means. Students will need to start by determining what 161 equals, to then think about what their exponents could possibly mean. We want students to work through various theories before being explicitly told what the fractional and negative powers mean. You may walk around the room and assist students in their thinking process. Encourage questions and discussion around what these exponents could mean, and if students need a push in the right direction, you can create a list on the board of exponent patterns. Extension: Have students formalize rewriting a numerical radical as an expression involving a rational exponent, and vice versa. Students can use their numerical solutions, coupled with the variables provided in the problem, to formalize the laws. For negative exponents, we’re looking for:

For fractional exponents, we’re looking for:

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Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 7 — 1.08 Square roots and perfect squares Algebra 1 — 5.02 Power rule Algebra 1 — 5.03 Quotient rule Algebra 1 — 5.04 Zero and negative exponents

Tools You may ﬁnd this tool helpful: • Scientiﬁc calculator

Student lesson & teacher guide Rational exponents Students will learn about rational exponents and their laws, and how they can be used to rewrite radicals. They will examine the relationship between exponents and radicals through a series of mathematical statements provided in the exploration activity.

Students: Page 277

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Critique, correct and clarify English language learner support Before students share their responses, display the following incorrect statement: “I know that because we have to multiply 4 by one half.”

equals 2

Invite students to identify the error, critique the reasoning and write a correct explanation. Listen for and amplify the language students use to describe the correct process for simplifying expressions with fractional indices.

Create visuals to support mathematical language Student with disabilities support Provide this vocabulary list, then have students fill in the blanks. Include this visual on a word wall or in notes. • Rational exponent • Radical form • Radicand • Exponential form • Base • Index • Radical

Make explicit connection between radicals and rational exponents Address student misconceptions Students may mistakenly believe that a rational exponent means the base is to be divided, rather than understanding it as a representation of a radical. This misconception may stem from an overgeneralization of the concept that exponents represent repeated multiplication. To address this misconception, emphasize the connection between roots and exponents, illustrating the concept with numerous examples. Explain that the cube root of a.

is equivalent to the square root of a, and

is equivalent to

Also, remind students of the product law and how it applies to rational exponents. For example, demonstrate that

equals

which simpliﬁes to a1.

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Exploration Students: Page 277

Suggested student grouping: Small groups In this exploration, students will apply the laws of exponents to fill in missing parts of some mathematical statements. Through this process, they will develop an understanding of the relationship between rational exponents and radicals. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Which exponent law is being used? The law being applied is the power rule. 2. What values would make each statement true? The missing values would be , , and

respectively.

3. Generalize the relationships by completing these statements:

and

and Purposeful questions • What is the value of x in 2x = 1? How can this help us find the missing value in the first and second statement? • What is the difference between the first and last terms of the second statement? Based on this, we can conclude that is the same as what? • Both terms in the statement 81 = (8⬚)3 have a base of 8. That means we need to find the value that makes the exponents the same. In other words, what can we multiply to 3 so that the result is 1? Possible misunderstandings • Students might struggle to see that the power rule is being applied in each statement, and therefore not realize they need to multiply the exponents. Consider rewriting the statements to use the product rule instead:

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Students: Pages 277–278

In a similar way, we can look at the value

. Using the product law:

From the definition of a cube root:

Once again, by comparison we see that

These results can be generalized to:

and

Notice that the index of the radical becomes the denominator of the rational exponent. When there is no index shown, it is a square root. Rational exponent

a3 =

Index

Radical

Base

Radicand

Exponential form

Radical form

We can use these rules for rewriting radicals along with the laws of exponents to simplify expressions involving radicals and rational exponents. Product rule Quotient rule Power rule Power of a product Power of a quotient Identity exponent Zero rule Negative exponent rule Rational exponent

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1 3

a =

Radical

Base

Radicand

Exponential form

Radical form

Advanced learners: Connect integer exponent laws to rational exponents Targeted instructional strategies 278

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mathspace.co Help advanced learners discover that the laws of exponents are identical for integer and rational exponents, so they do not actually need to learn anything new here, just apply previously learned concepts.

This can be aided by presenting students with the laws of exponents for integer and rational exponents side by side, and asking students to identify similarities and differences. Alternatively, present students with the left side of each rational exponent rule and have them complete the right side. Product rule

a m ⋅ a n = am + n

Power rule

(am)n = amn

Power of a product

(ab)m = am ⋅ bm

Quotient rule

Power of a quotient Zero exponent

a0 = 1

Negative exponent Guide students to the realization that the only difference is the replacement of integer exponents (m and n) with rational exponents

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Provide a table of perfect squares and perfect cubes Student with disabilities support To support students to recognize perfect squares and perfect cubes, create a table like this to have available. Perfect squares (1)2 = 1 (2)2 = 4 (3)2 = 9 (4)2 = 16 (5)2 = 25 (6)2 = 36 (7)2 = 49 (8)2 = 64 (9)2 = 81 (10)2 = 100

Perfect cubes (1) = 1 (2)3 = 8 (3)3 = 27 (4)3 = 64 (5)3 = 125 (6)3 = 216 (7)3 = 343 (8)3 = 512 (9)3 = 729 (10)3 = 1000 3

Examples Students: Page 279 Example 1 For a Rewrite in radical form.

Create a strategy Use the rule

Apply the idea

Reflect and check We can also write this as

, but writing the root without an index is usual for a square root.

b Evaluate

Purpose Create a can strategy Check students rewrite an expression with a fractional exponent of We now know that

Expected mistakes

as a radical expression.

is the square root of 36.

StudentsApply may the covert ideathis to

if the connection between radical and exponential forms is not solidiﬁed. It

could also stem from misunderstanding Fromthe partnotation. (a) Using 36 = 62 Simplify

Reflect and check We could also evaluate this using exponents. Using 36 = 62 Power rule Simplify

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Example 1 Reflect and check For

can also write this as Students:WePage 279

, but writing the root without an index is usual for a square root.

a Rewrite in radical form.

Create a strategy b Evaluate Use the rule

Create a strategy Apply idea We nowthe know that

. is the square root of 36.

Apply the idea Reflect and check

From part (a)

We can also write this as

Using 36 = 62 , but writing the root without an index is usual for a square root. Simplify

Reflect and check

b Evaluate We could also evaluate this using exponents.

Create a strategy

Using 36 = 62

We now know that

is the square root of 36. Power rule

Apply the idea

Simplify From part (a) Using 36 = 62

PurposeExample 2 Simplify Check students evaluate an expression with a fractional exponent. Write incan exponential form. Reflect and check

Reﬂecting with students We coulda also evaluate this using exponents. Create strategy Apply the idea Ask students which form they would rather use to evaluate the expression (exponential form or radical form) We can write this expression in exponential form2using the Using 36 =used 6 and to explain their reasoning. If all students the same method, discuss both methods as a class before fact: asking which they prefer. Power rule

Students: Page 279

Simplify

Example 2 Write

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in exponential form.

Create a strategy

Apply the idea

We can write this expression in exponential form using the fact:

Purpose Check students can convert a radical expression to exponential form.

Reﬂecting with students This expression cannot be evaluated as 6 is not a perfect cube and it contains no perfect cube factors. If it was evaluated on a calculator, it would be a non-ending, non-repeating decimal, meaning it is best to leave it in this exact form.

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Students: Page 280 Example 3 Evaluate without a calculator. a

Create a strategy We can rewrite −216 with an exponent of 3 since it is a perfect cube, and then use the power rule to simplify.

Apply the idea Rewrite −216 using its factors

Example 3

Rewrite in exponential form

Evaluate without a calculator.

Use the power rule Simplify

Reflect and check Create a strategy

We may also notice that . We can rewrite −216 with an exponent of 3 since it is a perfect cube, and then use the power rule to simplify.

Apply the idea Rewrite −216 using its factors

Purpose Rewrite in exponential form Create a can strategy Check students evaluate an expression with a fractional exponent without a calculator. We can use the power of a power law, Use thenthe convert torule radical form to evaluate. power

Reﬂecting with students Simplify the idea the process of evaluating fractional exponents and the different ways this can be done Discuss Apply with students Reflect andFor check and represented. example, an alternative way would be rewriting −216 in terms of its prime factors. Power rule

We may also notice that

Students: Page 280

. Rational exponent law Rewrite 8 using its factors

b Rewrite in exponential form

Create a strategy

Evaluate the cube root We can use the power of a power law, then convert to radical form to evaluate. Evaluate the power

Apply the idea Reflect and check Power rule way and got If we had used the power of a power law the other , we would not have been able to evaluate as easily. This is because 85 = 32 768 and it is not easy to identify the cube root of 32 768 without a calculator. Rational exponent law Rewrite 8 using its factors Rewrite in exponential form 280

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Evaluate the cube root Evaluate the power

Reflect and check If we had used the power of a power law the other way and got , we would not have been able to evaluate as easily. This is because 85 = 32 768 and it is not easy to identify the cube root of 32 768 without a calculator.

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Rewrite 8 using its factors Rewrite in exponential form Evaluate the cube root Evaluate the power

Purpose To show students that the laws of exponents can be used in reverse to break an exponent into two factors, which allows for evaluation Virginia or SOLsimplification. Algebra 1 280 Mathspace mathspace.co

Expected mistakes Upon initial inspection, students may not recognize that we can break this into two exponents. Show students that the laws of exponents are a biconditional statement, so can be used to simplify or break apart expressions. For example: If (a3)5, then we can simplify to a15. If a15, then (a3)5 is also true.

Students: Page 281

Create a strategy Use the laws of exponents to simplify, then use rational exponents to simplify.

Apply the idea Use the power rule and quotient laws

Simplify exponents using operations on fractions Product of a power law Simplify the exponent Using 33 = 27 Apply the power rule

Reflect and check All of the exponent laws can be used for rational exponents, not just integer exponents.

Idea summary Purpose In addition to the laws of exponents we have seen, we can simplify rational exponents by converting to and Provide an opportunity advanced learners to simplify an expression involving rational exponents using from radicalfor form. exponents laws. Expected mistakes and Students may not realize that the laws of exponents can be applied to rational exponents as well as integer exponents. Some students may benefit from a review of operations with fractions before doing this example. We can simplify expressions involving rational exponents using the laws of exponents.

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What do you remember? 1

Use the properties of exponents to define a rational exponent that would make the statement true:

Prime factorization for perfect cubes

use with Example 3

Targeted instructional strategies Explicitly teach or review the algorithm of how to decompose into prime factors. Break down the prime factorization for a perfect cube, and explain how this can be translated to an expression with rational exponents. For example, the prime factorization of 216 from part (a) can be found as follows:

216 2

108 2

54 27

Create a strategy Use the laws of exponents to simplify, then use rational exponents to simplify.

Apply the idea

This shows us that 216 can be rewritten as 23 ⋅ 33. Since the radicand is negative, we can use (−2)3 ⋅ 33, as the Use the power rule and quotient laws product of these factors is −216. Now, the exponent laws can be used to simplify the expression. Simplify exponents using operations on fractions

Rewrite −216 using its prime factors

Product of a power law

Apply the product rule

Simplify the exponents

Simplify the exponent 3

Using 3 = 27 Evaluate

Apply the power rule

Reflect and check

Students:AllPage 281 of the exponent laws can be used for rational exponents, not just integer exponents.

Idea summary In addition to the laws of exponents we have seen, we can simplify rational exponents by converting to and from radical form.

and

We can simplify expressions involving rational exponents using the laws of exponents.

Practice What do you remember? 1

Use the properties of exponents to define a rational exponent that would make the statement true: a

Explain the similarities between a rational exponent and a radical expression.

Simplify each expression: a

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Practice Students: Pages 281–283

What do you remember? 1

Use the properties of exponents to define a rational exponent that would make the statement true: i

Explain the similarities between a rational exponent and a radical expression.

Simplify each expression: a

Use the laws of exponents to fully simplify each expression. If possible, evaluate the expression. Otherwise, write in exponential form. a

Select all of the expressions that are equivalent to A

: C

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Let’s practice 6

Write each expression in an equivalent radical form. a

Write each expression in an equivalent exponential form. a

Use rational exponents to justify that each of the following equations is true: a

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For each expression: i

Write in radical form.

For each expression: i

Write in the form

a 11

Write in the form

Evaluate.

Blair has attempted to fully simplify an expression with positive variables in reduced radical form and showed his work: Rewrite the rational exponent as a radical

Evaluate.

For each expression: i

Evaluate.

Take the square root of 64

Evaluate the cube

Identify where Blair has made errors and explain the errors.

Fully simplify the expression, showing the correct steps of work.

Identify and correct the error in the work:

Let’s extend our thinking 14

Simplify: a

Aliyah is trying to simplify the expression using the laws of exponents.

Write the expression 52 as a radical in two different ways, one using a square root and one using a cube root. Explain your steps.

. She claims that

. Is Aliyah correct? Justify your answer

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Without calculating, which is greater:

A cube has sides of length

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? Explain your reasoning.

cm. Find the volume of the cube.

Answers 10 a i

b i

c i

ii 32

d i

ii 125

5.05 Rational exponents What do you remember? 1 a i

b A rational exponent and a radical expression represent the same quanitites in different forms. The radicand is the same as the base of the power. The denominator of the rational exponent is the index of the radical. 2 a 8

b −6

c 5

d 4

3 a 4

b 49

g 81

e 16

11 a i

b i

c i

ii 25

d i

ii 16

4 B, D, F 5 A, E Let’s practice 6 a

7 a

12 a B lair has made an error in Steps 1. He has incorrectly converted between radical and exponential forms. When we express rational exponents as radicals, the denominator represents the index of the radical, but he made it a power of a square root. b Rewrite the rational exponent as a radical

Write the radicand as a power of 3

Take the cube root

This ia an alternative strategy, he could have used as well:

8 a b c

Rewrite the radical as an exponent

Rewrite the exponents to have a common denominator

Simplify the multiplication by adding exponents

13 The error is 64 ≠ 28 since 64 = 26 . So, the work should be:

d 9 a i

ii 9

b i

ii 3

c i

ii −2

d i

ii 4

e i

ii 11

ii −10

g i

ii −6

h i

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Let’s extend our thinking 14 a 2

b 1

represents the cube root of 64. The square 64, while root of a number greater than 1 is always greater than the cube root of the same number.

15 Aliyah is correct. To simplify the expression, we can write it in exponential form: . Since 6 divided by 3 equals 2, the expression simpliﬁes to 112. 16 The expression 52 can be written as a square root as It can be written as a cube root as . This is because squaring and taking the square root (or cubing and taking the cube root) are inverse operations.

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is greater because it represents the square root of

18 25 cm3 .

5.06 Simplify radicals Subtopic overview Lesson narrative In this lesson, students will learn to simplify radical expressions, focusing on square and cube roots. The exploration involves students using prime factorization to break down numbers under the radical sign, identifying perfect square and cube factors. They will simplify expressions by factoring out these perfect squares or cubes. Students will also engage in problems that require them to apply properties of radicals. By the end of the lesson, students should be proficient in simplifying various radical expressions involving both square and cube roots.

Learning objectives

5.06 Simplify radicals

Students: Page 284

After this lesson, you will be able to… • simplify square roots of whole numbers. • simplify cube roots of integers.

Simplify square and cube roots Radical expressions have many parts as shown in the following diagram: Key vocabulary 

index



Parts of a Radical perfect cube



perfect square



radical



radicand Index



simplified radical form

Radical symbol

Essential understanding

Radicand

The special relationship between square roots and perfect Radical squares and cube roots and perfect cubes allows us to reveal the simplest form of a radical expression. Index The number on a radical symbol that indicates Standards which type of root it represents. For instance, the

Radicand The value or expression inside the radical symbol.

Perfect square of Learning standards. index onaddresses a cube root is following 3. The index on a square This subtopic the Virginia 2023 Mathematics Standards root is usually not written, but would be 2. A number that is the result of multiplying two of the Mathematical process goals same integer. Radical MPG3 — cube Mathematical Reasoning MPG1 A —mathematical Mathematical Problem Solving Perfect expression that uses a root, such Teachers can incorporate goal into the lesson Teachers incorporate this goalroot into the lesson by as acan square root , or nth A number that is the resultthis of multiplying three of by students to justify why a square root is in simplest providing and modeling real-world problems that involve asking the same integer together. the concept of simplifying radicals. For example, teachers form when the radicand has no perfect square factors other than one, and a cube root is in simplest form when can provide a problem where students have to find the Radical written in simpliﬁed factored anyfactors further.other For square the radicand radicandcannot has nobe perfect cube than one. side lengthexpressions of a squareare given its area, and theradical area isform a if the roots,square. this means there will are no remaining of the radicand that are squares and for cube roots Students willperfect need to use logical reasoning to this make and perfect Students need to applyfactors the process means theresquare are noroots remaining factors the radicand that are perfect cubes. If an expression test this mathematical statement.is in simplified radical of simplifying to solve this of problem. form, and there is still a number left in the radicand, the result will be irrational. We can use the following facts to simplify radical expressions, for a, b ≥ 0 and m, n positive integers, 5.06 Simplify radicals mathspace.co

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MPG5 — Mathematical Representations Teachers can incorporate this goal into the lesson by asking students to represent their process of simplifying radicals using different methods, such as through visual representations or written explanations. For example, when teaching the process of simplifying square roots, teachers can ask students to create a tree diagram to illustrate the prime factorization of the radicand.

Content standards A.EO.4 — The student will simplify and determine equivalent radical expressions involving square roots of whole numbers and cube roots of integers.

A.EO.4b — Simplify and determine equivalent radical expressions involving the cube root of an integer.

A.EO.4a — Simplify and determine equivalent radical expressions involving the square root of a whole number in simplest form.

Prior connections 7.NS.3 — The student will recognize and describe the relationship between square roots and perfect squares. 7.CE.1 — The student will estimate, solve, and justify solutions to multistep contextual problems involving operations with rational numbers.

8.NS.1 — The student will compare and order real numbers and determine the relationships between real numbers. A.EO.3 — The student will derive and apply the laws of exponents.

Future connections A.EI.3 — The student will represent, solve, and interpret the solution to a quadratic equation in one variable.

A2.EO.2 — The student will perform operations on and simplify radical expressions.

Tools You may find this tool helpful: • Step-by-step graphic organizer

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Student lesson & teacher guide Simplify square and cube roots Students will learn about the parts of a radical and the properties that can be used to simplify them.

Students: Page 284

5.06 Simplify radicals After this lesson, you will be able to… • simplify square roots of whole numbers. • simplify cube roots of integers.

Simplify square and cube roots Radical expressions have many parts as shown in the following diagram: Parts of a Radical Radical symbol

Index

Radicand

Radical

Index

Radicand

The number on a radical symbol that indicates which type of root it represents. For instance, the index on a cube root is 3. The index on a square root is usually not written, but would be 2.

The value or expression inside the radical symbol. Perfect square A number that is the result of multiplying two of the same integer.

Radical A mathematical expression that uses a root, such as a square root , or nth root

Perfect cube A number that is the result of multiplying three of the same integer together.

Radical expressions are written in simpliﬁed radical form if the radicand cannot be factored any further. For square roots, this means there are no remaining factors of the radicand that are perfect squares and for cube roots this means there are no remaining factors of the radicand that are perfect cubes. If an expression is in simplified radical form, and there is still a number left in the radicand, the result will be irrational. We can use the following facts to simplify radical expressions, for a, b ≥ 0 and m, n positive integers,

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Create a set of steps for simplifying radical expressions Targeted instructional strategies Have students develop algorithmic thinking by creating a set of steps which can be used for simplifying radical expressions. Remind students to include a step for verifying their answer. In particular, students can use a Step-by-Step graphic organizer like the one shown. Procedure: Steps Step 1: Step 2: Step 3: Step 4: Step 5:

Details

If students are having difficulties coming up with a set of steps to follow, offer the following steps: 1. Identify the prime factorization of the radicand. 2. Circle pairs of common factors (for square roots) or groups of three common factors (for cube roots). 3. Write one factor from each circle outside the radical. 4. Evaluate the product of factors outside the radical. 5. Evaluate the product of factors inside the radical. Once students have written their steps, have them consider questions like: • How can we check that our answer is correct? • Could these steps be done in a different order? • Can we think of a different set of steps that could be used? • Do we need any additional steps for questions that have different values? • How can we make our mathematical language more precise?

Compare and connect English language learner support Present students with the methods of simplifying radical expressions from the lesson:

Prime factorization method

Perfect square method

Individually, give students a few minutes to compare the two methods. Then in pairs, students can discuss how the strategies are the same or different. Consider providing discussion prompts such as “Which strategy is easier for you to understand?” and “How are the stategies for simplifying each expression the same or different?”. Listen for and amplify comments about what might make one approach more efficient or easier to understand. This will help students connect different approaches and determine the effectiveness of each one through partner and whole-class discussions.

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Provide a resource sheet of perfect squares and cubes Student with disabilities support Provide students with a table or resource sheet of the first 20 perfect squares and square roots. An example of a table showing the first 5 perfect squares and square roots is shown below. Perfect squares

On the same sheet, include the first 10 perfect cubes and cube roots of positive values. An example of a table showing the first 5 perfect cubes and cube roots is shown below.

Square roots

Perfect cubes

(1) = 1

(2) = 4

(2)3 = 8

(3)2 = 9

(3)3 = 27

(4)2 = 16

(4)3 = 64

(5)2 = 25

(5)3 = 125

Square cubes

Cube roots and negatives Address student misconceptions Students may remember from their work with square roots that the radicand of a square root must be non-negative, and incorrectly extend this to all types of radical expressions. Challenge this misconception by asking students to evaluate (−2)3 and therefore evaluate

Exploration Students: Page 285

Exploration We can use prime factors to help us split a number into a product of a perfect square and a remainder. If we wanted to simplify

, we want to see if 60 has any factors which are perfect squares. Using the factor tree, we can see that 60 = 22 ⋅ 3 ⋅ 5, so the perfect square factor is 22 = 4, and the remainder is 15.

How can you identify if a number has a perfect square factor after drawing a factor tree?

Is it possible to get two different factor trees for the same number?

Is it possible to get two different prime factorizations?

Can every radical be simplified?

How would you rewrite 180 as a product of a perfect square and a remainder?

What would happen if we didn’t use the largest perfect square?

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Perfect square method This is the quickest method for simplifying a radical:

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Suggested student grouping: Small groups Students will investigate how to use prime factors to split a number into a product of a perfect square and a remainder. The goal is to better understand the simplification of radicals. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. How can you identify if a number has a perfect square factor after drawing a factor tree? After drawing a factor tree, a number has a perfect square factor if any of the prime factors repeat twice. The pair (or pairs) of the same prime number is a perfect square factor. 2. Is it possible to get two different factor trees for the same number? Yes, you can have different factor trees for the same number depending on which factors you choose to break down ﬁrst. However, the prime factorization will always be the same regardless of how the tree is drawn. 3. Is it possible to get two different prime factorizations? No. Every integer greater than one has a unique prime factorization. 4. Can every radical be simpliﬁed? No, because some radicals do not contain perfect square factors, such as

5. How would you rewrite 180 as a product of a perfect square and a remainder? Exploration Using prime factorization, 180 can be written as 22 ⋅ 32 ⋅ 5. Therefore, 180 can be rewritten as the product of the perfect square 36 (22 ⋅ 32) and the remainder 5. We can use prime factors to help us split a number into a product of a perfect square and a remainder. 6. What would happen if we didn’t largest perfect square? If we wanted to simplify , we use wantthe to see if 60 has any factors which are perfect squares. If we didn’t use the largest perfect square when simplifying a radical, equivalent Using the factor tree, we canwe seewould that 60still = 22get ⋅ 3 ⋅ an 5, so the 60 2 perfect square factor is 2 = 4, and the remainder is 15. expression, but it wouldn’t be in its simplest form. Purposeful questions2

• What is the significance of identifying perfect square factors in a radicand? 3 10 • What steps do you follow when simplifying radicals using prime factorization? • How would this process change if the expression was a cube root? 2

Possible misunderstandings 1.

How can you identify if a number has a perfect square factor after drawing a factor tree?

Is it possible to get two different prime factorizations?

Can every radical be simplified?

• Students may misunderstand and think that different factor trees may lead to different prime factorizations. 2. Is it possible to get two different factor trees for the same number? To address this, hightlight the difference between factors and prime factors.

Students: Page How 285would you rewrite 180 as a product of a perfect square and a remainder? 5. 6.

What would happen if we didn’t use the largest perfect square?

Prime factorization method We can use a few steps to help us simplify any radical: 1. Find prime factorization of radicand 2.Group factors in groups equal to index of radical expression 3. Use multiplication property of radicals 4. Simplify any rational factors Perfect square method This is the quickest method for simplifying a radical: 1. Find largest perfect square (or cube) factor of radicand 2. Use multiplication property of radicals 3. Use the properties of radicals to factor 4. Simplify any perfect square (or cube) factors

Example 1 594

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Create a strategy

How can you identify if a number has a perfect square factor after drawing a factor tree?

Is it possible to get two different factor trees for the same number?

Is it possible to get two different prime factorizations?

Can every radical be simplified?

How would you rewrite 180 as a product of a perfect square and a remainder?

What would happen if we didn’t use the largest perfect square?

Use factor trees to visualize prime factorization Targeted instructional strategies

Students may prefer writing the prime factorization of a number using factor trees because it is a visually appealing way to organize their work. Help them relate their factor tree back to a radical expression, so they Prime factorization method can continue the simplification process. We can use a few steps to help us simplify any radical: 1. Find prime factorization of radicand

2.Group factors in groups equal to index of radical expression

3. Use multiplication property of radicals

4. Simplify any rational factors

2 Perfect square method

6 2

This is the quickest method for simplifying a radical: 1. Find largest perfect square (or cube) factor of radicand

Examples

2. Use multiplication property of radicals

Students: Pages 285–286

4. Simplify any perfect square (or cube) factors

3. Use the properties of radicals to factor

Example 1 Simplify

Create a strategy Split 12 into its prime factors and use

Apply the idea Find prime factorization of 12 Rewrite using perfect square Product of radicals property

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285

Square root of a perfect square

Example 2 Purpose Simplify how to simplify radicals by using prime factorization and the property of radicals. Show students Create strategy Reflecting witha students Split −64learners into factors. the negative sign. Ask advanced to Notice consider whether the square root of a positive number always results in a smaller Apply the idea number. To encourage discussion, remind them to consider all types of positive numbers, not only whole Find prime factorization of 12 Apply the idea and check numbers. The square root of a number less than 1 is not aReflect smaller number. For example, the square root of

is , which is larger than .

Students: Page 286

Negative radicands can have rational cube roots. Can the FactorRewrite radicand using perfect square same be said for square roots? Rewrite using perfect cube Product of radicals property Cube root of perfect Square root of cube a perfect square

Example 2 3 Example Simplify the expression Simplify

Create a a strategy strategy Create To simplify thefactors. expression, wethe can first simplify Split −64 into Notice negative sign. the square root and then multiply the result by the coefficient outside the root.

Apply the idea Apply the idea

Reflect and check Negative radicands can have rational cube roots. Can the Factor radicand Factor 18 into 9 ⋅ 2 same be said for square roots? 5.06 Simplify radicals Rewrite using perfect cube mathspace.co Use the property of square roots that Cube root of perfect cube Find the square root of 9, which is 3 Multiply the coefficients together

595

Example 2 Simplify

Create a strategy Split −64 into factors. Notice the negative sign.

Apply the idea

Reflect and check Factor radicand Rewrite using perfect cube

Negative radicands can have rational cube roots. Can the same be said for square roots?

Cube root of perfect cube

Example 3 Purpose Apply idea Simplifythe the expression Show students how to simplify cube roots, especially when dealing with negative numbers. Find prime factorization of 12

Reﬂecting witha students Create strategy Rewrite using perfect square Discuss To with students the difference between rootsroot and rootsthe when with negative simplify the expression, we can first simplify cube the square andsquare then multiply resultdealing by the coefficient outside theSquare root. Product ofare radicals property (imaginary numbers are introduced in Algebra 2), numbers. roots of negative numbers undefined while cube roots of negative numbers result inofnegative, real numbers. This is a key concept in understanding Square root a perfect square Apply the idea different types of roots. Factor 18 into 9 ⋅ 2

Use the property of square roots that Example 2 learners: Generalizing Advanced radical simplification to higher-order roots Find the square root of 9, which is 3 Targeted instructional strategies Simplify

use with Examples 1 and 2

Multiply the coefficients together

Encourage advanced learners to extend the method of simplifying square roots and cube roots to higher-order Therefore, the simplified formExamples of is 1 and . 2, prompt students to explore how radicals like a strategy radicals.Create After working through or . Split −64 into factors. Notice the negative sign.

This allows them tocheck discover patterns and generalize the simplification process for any radical expression. Reflect and Providing opportunities for students to generalize justify their methods helpsboth them connect the lesson to Let’s check our solution by substituting back into theand original expression and evaluating sides. Apply the idea Reflect and check broader mathematical conceptsFactor andSubstitute enhances critical thinking skills. with its decimal approximation 4.24have rational cube roots. Can the Negative radicands can radicand same be said for square roots? Evaluate Rewrite using perfect cube Substitute Cube root of perfectwith cubeits decimal approximation 1.41

Students: Page 286

Evaluate The values on both sides are approximately equal, confirming that our solution is correct. The slight difference is due to the rounding Example 3 off of the square roots to two decimal places. Simplify the expression 286

Mathspace

Virginia SOL Algebra 1

mathspace.co Create a strategy

To simplify the expression, we can first simplify the square root and then multiply the result by the coefficient outside the root.

Apply the idea Factor 18 into 9 ⋅ 2 Use the property of square roots that Find the square root of 9, which is 3 Multiply the coefficients together Therefore, the simplified form of

Reflect and check Let’s check our solution by substituting back into the original expression and evaluating both sides. Substitute

with its decimal approximation 4.24

Evaluate Substitute

with its decimal approximation 1.41

Evaluate The values on both sides are approximately equal, confirming that our solution is correct. The slight difference is due to the rounding off of the square roots to two decimal places.

286

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Purpose This example shows students how to simplify a square root expression by factoring and then using the property of square roots. Expected mistakes Students might forget to multiply the 3 from to the 3 that was already in front of the radical, or they may try to add the 3’s. Remind them that, when we simplify the perfect square factors from the radicand, we must multiply the result to any values that were already in front of the radicand.

Students: Page 287

Practice Students: Pages 287–288

What do you remember? 1

Find the largest square number that divides exactly into 75.

True or False:

Consider the expression a

Complete the following statement: ‘An expression in the form this fact.’

b 4

Consider the expression a

b 5

Simplify the expression

can be simpliﬁed to ⬚. We can rewrite 49 as the perfect square, ⬚2, to use

Complete the following statement: then it can be simpliﬁed to . If one of the factors of 108 is a ‘If an expression can be written as cube number then it can be simpliﬁed as above. 108 does have a cube factor, ⬚, which can be written as the cube ⬚3.’

Simplify the expression

Complete the second statement by the example on the first statement: a

Statement 1: Statement 2:

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597

Determine if each expression is written in simplest form: b

Let’s practice 7

SOL

Simplify:

Express each expression in simplest form: a

What is

in simplest form? B

A SOL

−

What is the value of

in simplest radical form?

Let’s extend our thinking 12

Is each expression rational or irrational? a

Find the value of x in the equation

The volume of a cube is 3456 cubic inches. What is the length of each of its sides?

In general, if we take the cube root of x, where x is some negative integer with a rational cube root, what can we say about the resulting expression?

Use your knowledge of simplifying radicals to simplify these variable expressions. a

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Answers

8 a −1

b −20

9 a

1 25

2 False

b Rational

c Irrational

d Irrational

Rational

g Rational

h Rational

e 3

5.06 Simplify radicals What do you remember?

can be simpliﬁed to a. 3 a ‘ An expression in the form We can rewrite 49 as the perfect square, 72, to use this fact.’

11 A Let’s extend our thinking

b 4 a ‘ If an expression can be written as then it can be simpliﬁed to . If one of the factors of 108 is a cube number then it can be simpliﬁed as above. 108 does have a cube factor, 27, which can be written as the cube 33.’ b 5 a

12 a Irrational e Irrational

b Yes

c No

d Yes

b 27

c 3

h 735

13 x = 10 14

inches

15 If x is a negative integer with a rational cube root and we take the cube root of it, the cube root of x must be negative. 16 a x2

b 6 a Yes

10 B

d y4

Let’s practice 7 a 5 e

Answers mathspace.co

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5.07 Operations with numerical radicals Subtopic overview Lesson narrative In this lesson, students will build on their knowledge of simplifying expressions and square roots by learning how to add, subtract, multiply, and divide radical expressions. By the end of this lesson, students will be able to perform operations with numerical radicals.

5.07 Operations with numerical Learning objectives radicals Students: Page 289

After this lesson, you will be able to… • add and subtract radicals. • multiply and divide radicals.

Multiplication of radicals

Key vocabulary We have seen that we can simplify radicals using the property 

If we wanted to multiply two radicals together, we could combine them back in the same way. That is radicand

Unlike adding and subtracting radicals, these do not have to be like terms before multiplying. Essential understanding

If we want to find the product of radicals that have a coefficient, we can use the properties of multiplication to Operations applied to radical expressions in much the same way that they can be applied to real numbers. rearrangecan thebe expression. In fact, many radical expressions are real numbers. Say we have , which we can rewrite as . Using the commutative property, we can also write . We can simplify this by multiplying the numbers together in one group and the this expression as radicals together in another. This becomes

Standards

This subtopic addresses the following Virginiathe 2023 Mathematics Standards Learning standards. Although simplifying radicals before starting question is not necessary withofmultiplication and division like it is with addition and subtraction, it can still help us get the job done quicker sometimes.

Mathematical process goals MPG3 — Mathematical Reasoning Example 1

The teacher can incorporate this goal into the lesson by challenging students to justify their steps when simplifying, Simplify the expression adding, subtracting, and multiplying radicals. Students can be asked to explain why only “like radicals” can be combined oranswer why theinproduct of two radicals Give your the simplest radical form.with the same index can be found by multiplying the radicands. This encourages the application of logical reasoning and validation of conclusions.

Create a strategy Multiply the radicals together.

Apply the idea Rearrange the terms Perform the multiplication can simplifyVirginia our answer since 175 = 25 ⋅Edition 7, where 25 is a perfect square. SOL Algebra 1 Teacher 600WeMathspace mathspace.co

Write the radical as a product of its factors Evaluate

MPG4 — Mathematical Connections

MPG5 — Mathematical Representations

Teachers can help students make connections between prior knowledge and the current lessons. For instance, when introducing the concept of adding and subtracting radicals, the teacher can connect this to the previous knowledge of combining like terms in algebraic expressions. When discussing the difference between rational and irrational numbers, the teacher can relate it to the concept of operations with radicals. They can also relate mathematics to other subjects, such as science, by showing how these concepts are used in real-world contexts.

The teacher can incorporate this goal into the lesson by encouraging students to represent and describe the process of simplifying, adding, subtracting and multiplying radicals using different methods. Teachers can use physical, visual, symbolic, verbal, or contextual representations. For example, students can visually represent the process of multiplying radicals on a number line or through geometric models. Encouraging students to use different representations can deepen their understanding and enhance their communication skills.

Content standards A.EO.4 — The student will simplify and determine equivalent radical expressions involving square roots of whole numbers and cube roots of integers.

A.EO.4c — Add, subtract, and multiply radicals, limited to numeric square and cube root expressions.

8.PFA.1 — The student will represent, simplify, and generate equivalent algebraic expressions in one variable. A.EO.3 — The student will derive and apply the laws of exponents.

Future connections A2.EO.2 — The student will perform operations on and simplify radical expressions.

Tools You may ﬁnd this tool helpful: • Highlighters

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Lesson supports The following supports may be useful for this lesson. More speciﬁc supports may appear throughout the lesson:

Concrete-Representational-Abstract (CRA) Approach Targeted instructional strategies Concrete: Begin by engaging students with physical manipulatives to explore radicals and their operations. Use square tiles or blocks to create squares that represent perfect squares. For example, have students build a square using 16 tiles to represent the number 16. Show them that the length of one side (4 tiles) is the square root of 16. To demonstrate adding radicals, ask students to build two squares, such as one with 9 tiles and another with 16 tiles, and then combine the tiles to see how the areas add up. This hands-on experience helps students visualize radicals and understand how to add and subtract them by manipulating actual objects. Representational: Transition to the representational stage by having students draw pictures of the squares they built with tiles. Encourage them to sketch squares on graph paper, labeling the side lengths (square roots) and the areas (the numbers under the radicals). For example, they can draw a square with a side length of 5 units to represent . To illustrate adding radicals, have them draw two squares side by side and show how the side lengths and areas relate when combined. Use drawings of rectangles to represent multiplying radicals, where the side lengths are radicals, and the area represents the product under a new radical. These visual representations help students connect their concrete experiences to mathematical concepts on paper. Abstract: Move on to the abstract stage by introducing the mathematical symbols and rules for operating with radicals. Teach students how to add and subtract radicals by combining like terms, such as simplifying ,and how to divide radicals to 3 + 4 = 7. Show them how to multiply radicals using the property by simplifying expressions like

. Provide practice problems that involve simplifying and performing

operations with radicals, and guide students in writing out each step using proper notation. Remind them to think back to the squares and drawings they used earlier to make sense of the abstract symbols. Connecting the stages: Help students make connections between all three stages by encouraging them to relate the abstract symbols back to their drawings and the physical manipulatives. Ask guiding questions like: • “How does combining these two radicals on paper relate to the squares you drew?” • “Can you imagine how multiplying these radicals is like creating a larger rectangle from your square drawings?” By linking the abstract expressions to the representational drawings and the concrete manipulatives, students can monitor their thinking and choose the representation that helps them understand best. This integrated approach reinforces their learning and builds a deeper understanding of operations with radicals.

Add 1’s to distinguish between coefficients and radicands Student with disabilities support When multipling, adding and subtracting radicals, it is important that students are able to distinguish between radicands and constants or coefficients. Confusing these values can cause them to incorrectly multiply or combine the wrong values. For example, when evaluating , students might multiply and 4 to get . When introducing operations with radicals, encourage students to check that all terms include a coefficient and a radical expression. If a term is missing one, they can add a coefficient of 1 or a radical expression of . For example, •

•

Then when explaining how to multiply, add or subtract two radicals, students can create a list of steps that clearly identify which values are multiplied or combined for each type of expression.

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Student lesson & teacher guide Multiplication of radicals Students learn how to multiply radicals. The students will explore simplifying before and after multiplying radicals.

Students: Page 289

5.07 Operations with numerical radicals After this lesson, you will be able to…

5.07 Operations with numerical radicals • add and subtract radicals. • multiply and divide radicals.

Multiplication of radicals We have seen can simplify Afterthat thiswe lesson, you willradicals be ableusing to… the property

If we wanted to and multiply two radicals • add subtract radicals.together, we could combine them back in the same way. That is • multiply and divide radicals. Unlike adding and subtracting radicals, these do not have to be like terms before multiplying. If we want to find the product of radicals that have a coefficient, we can use the properties of multiplication to Multiplication of radicals rearrange the expression. We have seen that we can simplify radicals using . Say we have , which we can rewrite as the property . Using the commutative property, we can also write . We can simplify by combine multiplying theback numbers in one this If weexpression wanted to as multiply two radicals together, wethis could them in thetogether same way. Thatgroup is and the radicals together in another. This becomes Unlike adding and subtracting radicals, these do not have to be like terms before multiplying. Although simplifying radicals before starting the question is not necessary with multiplication and division like it is If weaddition want to find product of radicals that us have coefficient, can use the properties of multiplication to with and the subtraction, it can still help getathe job donewe quicker sometimes. rearrange the expression. Say we have

, which we can rewrite as . Using the commutative property, we can also write . We can simplify this by multiplying the numbers together in one group and the radicals together in another. This becomes Simplify the expression

Example 1 this expression as

Examples Give your answer in the simplest radical form.

radicals before starting the question is not necessary with multiplication and division like it is Students:Although Page simplifying 289 with addition and subtraction, it can still help us get the job done quicker sometimes. Create a strategy Multiply the radicals together.

Example 1 Apply the idea Simplify the expression

Rearrange the terms Give your answer in the simplest radical form. Perform the multiplication We can simplify our answer since 175 = 25 ⋅ 7, where 25 is a perfect square. Create a strategy Multiply the radicals together.

Apply the idea

Write the radical as a product of its factors Evaluate Simplify Rearrange the terms Perform the multiplication

We can simplify our answer since 175 = 25 ⋅ 7, where 25 is a perfect square. Write the radical as a product of its factors Evaluate Simplify

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Simplify the expression Give your answer in the simplest radical form.

Create a strategy Multiply the radicals together.

Apply the idea Rearrange the terms Perform the multiplication We can simplify our answer since 175 = 25 ⋅ 7, where 25 is a perfect square. Write the radical as a product of its factors Evaluate Simplify

Purpose Check that students are able to multiply the radicals together.

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Expected mistakes can be simplified further. Remind them to always check for Students may not realize that or check whether perfect square factors before finalizing their answer.

Students: Page 290

Example 2 Simplify the expression Give your answer in the simplest radical form.

Create a strategy Use the distributive property to multiply and then simplify the result.

Apply the idea Distributive property Evaluate the multiplication Evaluate the perfect square

Idea summary

Purpose In general, we found that: Check students can apply the distributive property and simplify expressions involving radicals. Expected mistakes Students might multiply and 3 when distributing the radical to the first term. Point out that is a radical expression, but 3 is and a constant. Addition subtraction of radicals This about may help them To help We students their work, encourage them to rewrite 3 as but. what looked organize at simplifying radicals which related to multiplication and division, addition and see that 3 is a subtraction? Let’snot explore and to subtraction of radicals. coefficient, so it does get addition multiplied the radicand of . Let’s look at

Reflecting with students 16 and 9 are perfect squares, so we can simplify to 4 + 3 = 7. can be simplified and wonder if it is possible to rewrite it as before applying Students may notice that Let’s look at whether is the same as the distributive property. Encourage them to solve the problem using both methods. Then ask questions like, “Did you get the same result?” and “Which method do you think is more efficient?” So, we can see that, in general,

, and similarly

So, how can we add and subtract radicals? Well unfortunately, if the radicands, a and b, are different values, we . But if they were the same number, then just like can not simplify the expression collecting like terms in algebra. We can summarize this as:

604

Sometimes we are asked to add and subtract radicals that have different radicands (arguments). In this case, we can Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co try to simplify one of the radicals so that we have the same radicands. When adding and subtracting radicals, they must have the same radicand before we can simplify. Be sure to simplify all radicals ﬁrst.

Apply the idea

Example 2

Distributive property

Simplify the expression

Students:Give Page your 290 answer in the simplest radical form.

Evaluate the multiplication Evaluate the perfect square

Create a strategy Use the distributive property to multiply and then simplify the result.

Idea summary

general, Apply In the idea we found that: Distributive property Evaluate the multiplication Evaluate the perfect square

Addition and subtraction of radicals

We looked at simplifying radicals which related to multiplication and division, but what about addition and Addition andLet’ssubtraction of radicals subtraction? explore addition and subtraction of radicals.

Idea summary

Let’s look at addition . and subtraction of radicals. Students will explore In general, we found that: 16 and 9 are perfect squares, so we can simplify

Students:Let’s Page look 290 at whether

to 4 + 3 = 7.

is the same as

So, we can see that, in general,

, and similarly Addition and subtraction of radicals

So, how can we add and subtract radicals? Well unfortunately, if the radicands, a and b, are different values, we We looked at simplifying radicals which related to multiplication and division, but what about addition and . But if they were the same number, then just like can not simplify the expression subtraction? Let’s explore addition and subtraction of radicals. collecting like terms in algebra. We can summarize this as: Let’s look at . 16 and 9 are perfect squares, so we can simplify

to 4 + 3 = 7.

Let’s look at whether is the same as Sometimes we are asked to add and subtract radicals that have different radicands (arguments). In this case, we can try to simplify one of the radicals so that we have the same radicands. So, weadding can see that, in general,radicals, they must, have and similarly When and subtracting the same radicand before. we can simplify. Be sure to simplify all radicals So, how canﬁrst. we add and subtract radicals? Well unfortunately, if the radicands, a and b, are different values, we . But if they were the same number, then just like can not simplify the expression collecting like terms in algebra. We can summarize this as:

Sometimes we are asked to add and subtract radicals that have different radicands (arguments). In this case, we can try to simplify one of the radicals so that we have the same radicands. When adding and subtracting radicals, they must have the same radicand before we can simplify. Be sure to simplify all radicals ﬁrst.

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Adding and subtracting radicals Address student misconceptions Students may add or subtract the radicals before having a common radicand. Discuss with students the importance of having the same radicand first. A connection to combining like terms may help students 290 Mathspace Virginia SOL Algebra 1 understand the concept. mathspace.co For example, replace the radicals in one of the examples below with variables. Then, combine like terms and resubstitute the radicals at the end. Ask students to explain whether the variables changed or whether two different variables can be combined.

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Create a flowchart for simplifying radical expressions Targeted instructional strategies When learning operations with radicals, encourage students to use Computational Thinking to develop a list of steps that they can use to simplify various expressions. Use multiple examples to help them refine their list and encourage them to use specific, mathematical terminology. Since the steps for multiplying are different from the steps for adding and subtracting, students can create a type of flowchart, like the one shown. Which operation are we using?

Multiply

Add/Subtract

1. Simplify each radical if possible 2. Multiply coefficients and multiply radicands 3. Simplify further if possible

1. Simplify each radical if possible 2. Group like radicals 3. Add coefficients of the like radicals

Multiply + Add/Subtract 1. Simplify each radical if possible 2. Follow steps for multiplying 3. Follow steps for add/subtract

Compare and connect English language learner support Provide students with the following tables which compare adding and subtracting radicals to adding and subtracting terms in an algebraic expression. Table 1 Radical expression

Algebraic expression 5t − 3t

Expression

Combine like terms Table 2 Radical expression

Algebraic expression a − b + 4a

Expression

5a − b

Combine like terms Table 3 Radical expression Expression Combine like terms

Algebraic expression 4x − y + 4y + 2x 6x − 3b

In small groups, students should prepare responses to the questions “How are the expressions different? What are like terms? How is combining like terms the same for each expression?” Listen for and amplify observations of how the approaches are similar, such as adding/subtracting coefficients while radicals/variables are unchanged. Invite groups to share their responses to the question prompts. This will help students connect operations with radicals and combining like terms in algebraic expressions through partner and whole-class discussion.

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Examples Students: Page 291 Example 3 Simplify:

Create a strategy Simplify each radical before subtracting.

Apply the idea We can find that 24 = 8 ⋅ 3 where 8 is a perfect cube. Write the radicals as a products of their factors Evaluate Evaluate

Example 4 Purpose Ensure that able to simplify radicals by determining their factors before subtracting. Fully students simplify theare expression . Expected mistakes Create a strategy StudentAdd maylike subtract and come to a ﬁnal answer of 2 instead of the radicals. that is being subtracted. the 1 in front of the

. If students get 2, remind the students of

Apply the idea

Advanced Example 3 learners: Reverse the problem

Group like radicals

Simplify:

Evaluate

Targeted instructional strategies

use with Example 3

to simplify, provide them with the simplified result, Instead of presenting students with the expression which isCreate . Challenge students to come up with different combinations of radical expressions involving cube a strategy roots that would simplify to this result. Encourage them to devise original problems where the simplification Simplify each radical before subtracting. Example leads back to . 5 Apply the ideastudents This reversal pushes to think more deeply about the properties of radicals and how they can Consider the rectangle shown. We can that 24 = them 8 ⋅ 3 where 8 is a perfect cube. outcome. It fosters creative problem-solving and allows manipulate andfind combine to achieve a specific students to apply their understanding inWrite a new engaging way. the and radicals as a products 8 cmof their factors Evaluate

Students: Page 291

Evaluate

5 50 cm

a Find the exact perimeter of the rectangle. Give your answer in the form

, where a and b are integers.

Example 4 Create a strategy Fully simplify the expression Add the four side lengths of the rectangle.

Create a strategy Apply the idea Add like radicals.

Add the lengths and widths

Apply the idea

Perform multiplication Group like radicals 5.07 Operations with numerical radicals mathspace.co

Evaluate

291

Example 5 Consider the rectangle shown.

5.07 Operations with numerical radicals mathspace.co 8 cm

607

Example 3 PurposeSimplify: Ensure that students are able to add like radicals. Create a strategy

Reﬂecting witheach students Simplify radical before subtracting. . Highlight Ask the students to explain in their own words why the problem cannot be simplified past Applysuch the idea observations as “the radicals are not like radicals” or “we cannot subtract the terms because they are not alike”. If students struggle to24 explain their reasoning, We can find that = 8 ⋅ 3 where 8 is a perfect ask cube.them why we cannot subtract the terms in the expression 48a − 3b. This may help them explain that the terms are different, and therefore, cannot be combined. Write the radicals as a products of their factors Evaluate

Color-coding with like radicals

use with Example 4

Evaluate

Student with disabilities support

Students have already learned how to add and subtract algebraic terms by combing like terms. Adding and subtracting radical expressions can be done in the same way. Example 4 If students have difficulty differentiating between terms like and , use a highlighter to indicate the simplify the expression . differentFully types of terms in the expression. More specifically, have students highlight identical radicands with the same color. Create a strategy Add like radicals.

This should help students identify the like radicals, which can help them group like terms and combine the Apply ideaterms. coefficients ofthe those Group like radicals Evaluate

Students: Pages 291–292 Example 5 Consider the rectangle shown.

8 cm

5 50 cm

a Find the exact perimeter of the rectangle. Give your answer in the form

, where a and b are integers.

Create a strategy Add the four side lengths of the rectangle.

Apply the idea Add the lengths and widths Perform multiplication We can split up both

and

into two factors, with one being a perfect square. 5.07 Operations with numerical radicals mathspace.co

Write the radical as a product of its factors Evaluate

and

Evalute the product of the coeﬃcients Evaluate

b Find the exact area of the rectangle.

Purpose Create a strategy Check that students are able to find the perimeter of a shape. Use the formula: Area of rectangle = Length ⋅ Width

Apply the idea 608

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Substitute Length

and Width

Write the radical as a product of its factors Evaluate

and

Evaluate the product of the coeﬃcients

291

Expected mistakes Students may add the radicals together without getting a common radicand first. Point out that a common radicand is needed to combine, similar to combining like terms with the same variable. Reflecting withsplit students We can up both and into two factors, with one being a perfect square. Discuss with students the importance of including the units in their solution. Because the perimeter represents Write the radical as a product of its factors the length around the outside of the rectangle, the units help us understand the exact length. If the units were Evaluate and not listed, we would state the perimeter as units. Evalute the product of the coeﬃcients

Students: Page 292

Evaluate

b Find the exact area of the rectangle.

Create a strategy Use the formula: Area of rectangle = Length ⋅ Width

Apply We canthe splitidea up both

and

into two factors, with one being a perfect square. Substitute Length and Width Write the radical as a product of its factors Write the radical as a product of its factors Evaluate and Evaluate and Evalute the product of the coeﬃcients Evaluate the product of the coeﬃcients Evaluate Evaluate the product of the radicals Evaluate

b Find the exact area of the rectangle.

Create a strategy

Idea summary

formula: Area of rectangle = Length ⋅ Width PurposeUse theWhen adding and subtracting radicals, they must have the same radicand before we can simplify. Be sure to Check that students areradicals able to find the area of a shape. simplify all first.

Apply the idea

Substitute Length and Width Reﬂecting with students Write theofradical as a product of individually its factors StudentsPractice may choose to multiply the radicands first instead simplifying them before multiplying. This results in a very large radicand, thus making it more difficultand when simplifying the radical. Encourage Evaluate students to discuss which method they find simpler orEvaluate more efficient. the product of the coeﬃcients What do you remember? Evaluate the product of the radicals

Students: 1Page 292

Evaluate Express the following as a single radical expression: a

Determine if the two radicals could be written in terms of the same radical: aIdea summary and

and

When adding and subtracting radicals, they must have the same radicand before we can simplify. Be sure to Multiply simplify:first. simplifyand all radicals a b c d e

Practice i

What do you remember?

Practice 1

Express the following as a single radical expression:

a Students: Pages 292–295 292

and What do youa remember? 3

Mathspace Virginia SOL Algebra 1 Determine if the two radicals could be written in terms of the same radical: mathspace.co

and

Multiply and simplify:

Express the a following as a single bradical expression: a

e i

f j

l 5.07 Operations with numerical radicals mathspace.co

609

Determine if the two radicals could be written in terms of the same radical: and

a 3

and

Multiply and simplify: a

Consider the equation a

b 5

and answer the following questions.

Evaluate each of the following to one decimal place: i

iii

Does

Is the equation

In general does

true or false?

Let’s practice 6

Simplify: a

Simplify completely: a

Complete the equation to create a true statement. Each option may be used more than once. 2

610

Simplify completely: a

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

and

Find the exact area of the rectangle.

7 ft

35 ft

Find the area of the trapezoid in simpliﬁed radical form.

Find the perimeter of the triangle in simpliﬁed radical form.

Consider the rectangle: a

Find the exact perimeter of the rectangle. Give your answer in the form where a and b are integers.

Find the exact area of the rectangle.

Iain is participating in a race around the track shown in the diagram, which has dimensions in miles. Find the length of one lap around this race track.

Bob is about to paint the perimeter of a rectangular handball court, which has the dimensions shown in the diagram.

Find the total perimeter he has to paint.

The body surface area of a person in square meters can be modeled by

, where A is the surface

area, h is the height of the person in inches, and w is the weight of the person in pounds. Use the model to ﬁnd the surface area of a person who is 75 inches tall and weighs 172 pounds.

5.07 Operations with numerical radicals mathspace.co

611

Let’s extend our thinking 17

The surface area of a human body can be approximated by the formula

where A is the surface area of the body, h is the height of the person in cm, and w is the weight of the person in kg. Find the surface area of a person who is 164 cm tall and weighs 63 kg. 18

Uma solved for the height of a triangle whose area is

and whose base measures

1 2 3

4 5 Did she make an error? 19

Aviva and Jillian are traveling from the shore at Point A to their home at Point C. Jillian wants to swim home and travels directly from A to C. Aviva wants to avoid the water and decides to run home via point B. a

How far does Jillian have to swim to get home?

How far does Aviva have to run to get home?

How much further did Aviva travel than Jillian?

Give two different pairs of values for k and m which make the following equation true:

Identify and correct the error in the following work:

Sasha is simplifying the expression Explain how you know.

612

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

and says that the answer is

. Is she correct?

Answers

Let’s extend our thinking

5.07 Operations with numerical radicals

What do you remember?

18 She didn’t make an error.

1 a

19 a

2 a Yes

b Yes

c No

d Yes

3 a 45

b 5

ii 4.5

iii 5.2

4 a i 2.6

b 52 yd

c 20 Any two pairs of values where either k or m (or both) are zero. For example: k = 0, m = 4 k = 2, m = 0 21 When adding two radicals you do not combine the radicands. Instead, you can solve by

b No c No 5 False Let’s practice

22 Sasha is correct. Her work would have looked like

6 a

7 a 6

g 8

8 9 a

h 0

10 11

square units

units b 36 cm2

13 a 14 15 16

= 2.03 m2

Answers mathspace.co

613

5.08 Characteristics of exponential functions Subtopic overview Lesson narrative In this lesson, students will investigate the characteristics of exponential functions. They will explore how exponential functions grow by constant factors, using examples like y = 5x and y = 2x. Students will complete an exploration that includes creating tables of values, comparing different exponential functions, and identifying key features such as y-intercepts and asymptotes. Students will also analyze graphs and contextual situations modeled by exponential functions. By the end of the lesson, students should understand how to identify and describe the key characteristics of exponential functions.

5.08 Characteristics of exponential Learning objectives functions Students: Page 296

After this lesson, you will be able to… • identify the key features of an exponential function. • interpret key features of an exponential function in context.

Characteristics of exponential functions

Key vocabulary Exponential relationships include any relations where the outputs increase by a constant factor or decrease by a 

constant factor for consistent changes in x.  constant factor  exponential function  exponential relationship asymptote An exponential relationship can be modeled by a function with the independent variable in the exponent, known as an exponential function:

Essential understanding

f (x) = abx

a growth Leading Exponential functions have a constant factor,coefficient often referred to as the common ratio, and can be represented x b Base where b > 0, b ≠ by 1 a constant growth factor. This determines what by the equation y = ab . Exponential functions are characterized x Independent variable types of real world situations exponential functions can model. This determines what types of real world situations Dependent variable exponential functions can model. f (x)

Exploration

Standards Consider the following equations with a = 1: This subtopic y = 5x addresses the following Virginia 2023 Mathematics Standards of Learning standards. x −2 process −1 0 goals 1 Mathematical

y 1 5 25 MPG1 — Mathematical Problem Solving

Teachers can integrate this goal by presenting students with problems that require them to apply the concepts of y = 2x both linear and exponential functions. For example, students can be asked to solve problems that involve population x decay, −2 or financial −1 0problems 1 2 growth or involving compound interest. Students should be encouraged to use their understanding of concepts such as the y, domain, and range to find solutions. y 1 2 4 614

For each of the functions, think about the following questions: Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co 1. What happens to y as x increases?

Compare y = 5x and y = 2x. How are they similar? How are they different?

What is the y-intercept for each of the functions? How does this relate to the value of a?

MPG2 — Mathematical Communication

MPG3 — Mathematical Reasoning

Teachers can encourage students to explain their reasoning and solutions in a clear and precise manner, using the correct mathematical language and notation. For instance, when comparing and contrasting the characteristics of exponential and linear functions, students should be asked to articulate their observations and conclusions in a clear and coherent manner.

Teachers can apply this goal by asking students to justify their steps when solving problems or when determining the characteristics of exponential functions. For example, students should be able to explain why the y-intercept, domain, and range are useful in understanding and solving problems involving these functions.

MPG5 — Mathematical Representations Teachers can incorporate mathematical representations into their instruction of exponential functions by using visual aids like graphs to illustrate how these functions grow. Creating tables of values helps students see the numerical patterns and the constant multiplicative rate of change. Graphing software or graph paper can be used to plot exponential functions, highlighting key features such as y-intercepts and asymptotes. Additionally, teachers can present real-world scenarios modeled by exponential functions to contextualize the concepts and engage students with practical applications.

Prior connections A.F.1 — The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Algebra 1 — 2.03 Evaluating functions Algebra 1 — 2.04 Characteristics of functions

5.08 Characteristics of exponential functions mathspace.co

615

Tools You may ﬁnd this tool helpful: • Graphing calculator

Student lesson & teacher guide Characteristics of exponential functions Students are introduced to the exponential function in the form f (x) = abx before they take part in an exploration to investigate common key features of exponential functions.

Students: Page 296

5.08 Characteristics of exponential functions After this lesson, you will be able to… • identify the key features of an exponential function. • interpret key features of an exponential function in context.

Characteristics of exponential functions Exponential relationships include any relations where the outputs increase by a constant factor or decrease by a constant factor for consistent changes in x. An exponential relationship can be modeled by a function with the independent variable in the exponent, known as an exponential function:

f (x) = abx a b x f (x)

Leading coefficient Base where b > 0, b ≠ 1 Independent variable Dependent variable

Exploration Consider thekey following equations a = 1: Identifying features ofwith exponential functions x

y = 5 instructional strategies Targeted x

−2

−1

Ask students to generalize their understanding of the key features of an exponential function by considering the y 1 5 25 similarities between various functions and how they determine key features. Prompt student y = 2x investigation with questions like: • What is the bx when x value −2 of −1 0 x =1 0? 2 • Are there any values of x that make an exponential function undefined? y 1 2 4 • How does the asymptote relate to the range of an exponential function? • How can the asymptote of following an exponential Forwe eachdetermine of the functions, think about the questions:function?

616

What happens to y as x increases?

Compare y = 5x and y = 2x. How are they similar? How are they different?

What is the y-intercept for each of the functions? How does this relate to the value of a?

Does either function1have an x-intercept? Mathspace4. Virginia SOL Algebra Teacher Edition mathspace.co 5. Create a table of values for y = 1x. Does it have the same properties as y = 5x and y = 2x?

We can determine whether a function is exponential by dividing consecutive function values to see if they have a

Compare and connect English language learner support Present two representations of the same exponential function, such as a labeled graph and a table like the ones shown: 4

• f (x) = 3x

f (x) = 3x

x −3

2 1 −4 −3 −2 −1 −1

−2

x 1

−1

−2

−3

−4

Ask students to answer the following question: Which representation best shows the key characteristics of the function? Which characteristics appear in both representations? Listen for and amplify observations of equivalent characteristics, such as identifying the y-intercept on a graph and matching it to (0, 1) in a table, or that the range will never be negative. This will help students connect different representations and determine the benefits of different displays using whole-class discussion. Providing students with multiple examples to compare and connect can help students familiarize themselves with the key features of the equation as well. For example, the equation y = 2(3)x has the graph: 6

5 4 3 2 1 −3

−2

−1

x 1

−2

Students may make connections such as: • The y-value of the y-intercept is the same as the initial value in the equation. • The exponential is above the x-axis when the initial value is positive. • The exponential is increasing when the common factor is greater than 1.

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617

Labeled graphs of exponential functions Student with disabilities support Provide students with an example graph of an exponential function to draw, highlight, and label with key terms such as domain, range, asymptote, and y-intercept. These features can be color-coded and annotated to provide a reference for future problems.

Domain 7 6 5 4 3 2 1

Range

−4 −3 −2 −1 −1

5.08 Characteristics of exponential functions −2 −3

y-intercept

1 2 3 4 asymptote

Asymptotes and behavior

After this lesson, you will be able to…

Address student misconceptions

• identify the key features of an exponential function.

• interpret key features of an exponential in context. Looking at a graph, students may assume that the function exponential graph actually reaches the asymptote. They must be reminded that visually, the graph looks like it reaches the x-axis, but it never actually touches or crosses it.

To better understand this concept, use the tracing tool on a graphing calculator to show that, as x decreases, Characteristics of exponential functions the y-values get very very small, but they are never 0. Alternatively, students can substitute negative x-values, Exponential relationships include any relations where the outputs increase by a constant factor or decrease by a such as x = −100 or x = −1000, into a function. constant factor for consistent changes in x.

This willAn reinforce their understanding of why the range is y > 0 rather than y ≥ 0. exponential relationship can be modeled by a function with the independent variable in the exponent, known as an exponential function:

f (x) = abx

Exploration

a b x f (x)

Students: Page 296

Leading coefficient Base where b > 0, b ≠ 1 Independent variable Dependent variable

Exploration Consider the following equations with a = 1: y = 5x x

−2

−1

y = 2x x y

−2

−1

For each of the functions, think about the following questions: 1.

What happens to y as x increases?

Compare y = 5x and y = 2x. How are they similar? How are they different?

What is the y-intercept for each of the functions? How does this relate to the value of a?

Does either function have an x-intercept?

Create a table of values for y = 1x. Does it have the same properties as y = 5x and y = 2x?

We can determine whether a function is exponential by dividing consecutive function values to see if they have a constant factor.

618

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5.08 Characteristics of exponential functions

Suggested student grouping: Individual Students examine two tables of values for functions of the form y = bx, for different values of b. Students will look for patterns in how the outputs change as x increases, determine any common trends in key characteristics such as intercepts, and determine if a third function follows a similar pattern. Ideal student responses After thismay lesson, youfrom will beother able to… These ideal responses differ correct student responses. Less formal responses can be • identify theprecise key features of an exponential function. connected with the more mathematical language presented here. • interpret key features of an exponential function in context.

1. What happens to y as x increases? For both functions, the value of y increases at a faster and faster rate as x increases. 2. Compare y = 5x and y = 2x. How are they similar? How are they different? Characteristics of exponential functions Both tables approach 0 as x decreases, while the values of y increase as x increase. However, the values of Exponential relationships include any relations where the outputsxincrease by a constant factor or decrease by a y forconstant y = (5)xfactor increase more rapidly than the values of y = 2 . for consistent changes in x. 3. WhatAnisexponential the y-intercept for can each the functions? How this relate to the value of a? known as relationship be of modeled by a function with does the independent variable in the exponent, The an y-intercept offunction: both y = (5)x and y = (2)x is (0, 1). In both functions, the value of a is 1. exponential

4. Does either function have an x-intercept? f (x) = abx According to the tables provided, neither function has an x-intercept. a

Leading coefficient

x x it have 5. Create a table of values for y =b1x. DoesBase wherethe b > same 0, b ≠ 1 properties as y = 5 and y = 2 ? variabledoes not show the same rate of change, but does In the table for y = (1)x, all outputx values Independent are 1. This graph f (x) Dependent variable have the same y-intercept.

Exploration Purposeful questions • If the tables continued in each function, what would you notice about the values of each function? Consider the following equations with a = 1: • Are the values changing at a constant rate? y = 5x • Can the values of the function be negative? x

−2

−1

Possible misunderstandings y 1

• Students may believe that, if the table continued, the values would eventually reach 0. They may also 2x every function must have both an x-intercept and a y-intercept. assumey =that • Students xmay −2 think that = 1x is1an exponential function since the variable is in the exponent. Make students −1 y 0 2 aware that exponential functions are defined by their rate of change. Since the rate of change for y = 1x is y 1 2 4 constant, it is not exponential. For each of the functions, think about the following questions: 1. What happens are to y as x increases?to identifying an exponential function in a table by looking for a After the exploration, students introduced Compare y =then 5x and y = 2x. How are they similar? How areofthey different? functions, including domain, constant factor.2.Students are shown common characteristics exponential 3. What is the y-intercept for each of the functions? How does this relate to the value of a? intercepts, and asymptotes. 4.

Does either function have an x-intercept?

5. Create a table of values for y = 1x. Does it have the same properties as y = 5x and y = 2x? Students: Pages 296–297

We can determine whether a function is exponential by dividing consecutive function values to see if they have a constant factor. The base, or constant factor, is the number being multiplied repeatedly. It tells us how quickly the output values are growing or shrinking. We can find the base by dividing a term by the previous term, as shown below: 296

Mathspace Virginia SOL Algebra 1 mathspace.co

f (x)

×3 Constant factor:

×3

9 3 =3 =3 1 3

27 =3 9

In this example, we see that the function is growing exponentially. A function grows exponentially when it increases by a constant factor. Exponential functions change at a faster rate than linear functions. In the table below, we are adding 3 to each term, 5.08 Characteristics of exponential functions but the terms do not grow as quickly. mathspace.co x

619

f (x)

9 3 =3 =3 1 3

27 =3 9

×3 Constant factor:

×3

f (x)

All exponential functions of the form y = abx have these features in common: • The domain is all real values of x. • The range is y > 0. • For a context, the domain and range may be further restricted. • The y-intercept is at (0, a). • The common factor is b. • There is a horizontal asymptote at y = 0.

8 7 6 5 4

y = 3x

3 2 1 −4 −3 −2 −1

x 1

Asymptote

A line that a curve or graph approaches as it heads toward positive or negative infinity.

An exponential function can get infinitely close to an asymptote, but it can never cross it. This means that an exponential function of this form will not have an x-intercept.

Examples Students: Page 298 Example 1 5.08 Characteristics of exponential functions mathspace.co

Consider the table of values for the function y = 3(2)x. x

−3

−2

−1

3072

297

a Describe the behavior of the function as x increases.

Create a strategy We want to identify if the values of y are increasing or decreasing as x increases.

Apply the idea

Reflect and check

As x increases, the function increases at a faster and faster rate.

We can see that the equation has a constant factor that is greater than 1. This is why the function is increasing.

b Determine the y-intercept of the function.

Purpose Create a strategy Students demonstrate they can determine the behavior of the function.

The y-intercept occurs when x = 0. We can read these coordinates from the table.

620

Mathspace Virginia SOL Algebra 1 Teacher Edition Apply the idea mathspace.co (0, 3)

Reflect and check We can see that the equation has a leading coefficient of 3. This is the value of the y-intercept, and the result of substituting x = 0 into the equation.

Expected mistakes Example 1 Students may say the function increases without realizing that the function is increasing at a variable rate (not a x constantConsider rate). the table of values for the function y = 3(2) . x Help students see outputs rate by10asking them to ﬁnd the difference in the −3 that −2 the −1 0 are1increasing 2 3at a faster 4 5 outputs. Show them that the differences at larger y 3 6 are12increasing 24 48 96 and 3072larger intervals, showing them that the function increases at a faster and faster rate. a Describe the behavior of the function as x increases.

Reflecting with students Invite students share the answers they formulated with the class. Discuss how describing the behavior as Create atostrategy “increasing at an to simply statingasthe function is “increasing”. We want to increasing identify if the rate” valuesisofpreferred y are increasing or decreasing x increases. Make students aware that we could be even more specific with exponential functions. For example, students Example 1 the idea Reflect and check may sayApply the outputs are doubling or increasing by a factor of 2 since the constant factor of the function is 2. As x increases, the function increases at a faster and Consider the table of values for the function y = 3(2)x. 2

We can see that the equation has a constant factor that is greater than 1. This is why the function is increasing. 4 5 10

y 3 6 12 b Determine the y-intercept of the function.

rate.298 Students:faster Page x

−3

−2

−1

3072

a Describe the behavior of the function as x increases.

Create a strategy

The y-intercept occurs when x = 0. We can read these coordinates from the table. Create a strategy We want to identify if the values of y are increasing or decreasing as x increases.

Apply the idea

Reflect and check

(0, 3) the idea Apply

We can see the equation has a leading coefficient Reflect andthat check of 3. This is the value of the y-intercept, and the result of We can see that the equation has a constant factor that is substituting x = 0 into the equation. greater than 1. This is why the function is increasing.

As x increases, the function increases at a faster and faster rate. c State the domain of the function. b Determine the y-intercept of the function.

Purpose Create a strategy a strategy StudentsCreate demonstrate they can use a table to ﬁnd the y-intercept of an exponential function.

The domain is the complete set of possible values for x. For exponential functions, the graph extends indefinitely in The y-intercept occurs when x = 0. We can read these coordinates from the table. both horizontal directions.

Reﬂecting with students Apply to theexplain idea why a in any exponential functionReflect check Ask students of theand form y = abx is always the value of the Apply the idea Reflect and check (0, 3)Students should use the zero power law to justify We can see that the equation has a leading coefficient y-intercept. their reasoning. All real values of x. All exponential equations of the form y = abx have a

of 3. This is the value of the y-intercept, and the result of domain of all real x. substituting x = 0 into the equation.

Students: Page 298 d State the range of the function. c State the domain of the function.

Create a strategy Create a strategy

The range is the complete set of possible values for y. We can see the graph extends indefinitely up towards the left, The domain is the complete set of possible values for x. For exponential functions, the graph extends indefinitely in but it approaches an asymptote at y = 0 towards the right. both horizontal directions.

Apply the idea Apply the idea

y>0 All real values of x.

Reflect and check Reflect and check

All exponential equations of the form y = abxx have a All exponential equations of the form y = ab have a range of y > 0 for positive values of a. domain of all real x.

d State the range of the function.

298

Mathspace

Virginia SOL Algebra 1

mathspace.co Purpose Create a strategy Students will demonstrate they recall that the domain of all exponential functions is the same, but may be The range is the complete set of possible values for y. We can see the graph extends indefinitely up towards the left, represented different ways. but it approaches an asymptote at y = 0 towards the right.

Expected mistakes Apply the idea Reflect and check Students may confuse domain and range, and state the domain as x > 0. Point out that there xare negative y>0 All exponential equations of the form y = ab have a x-values in table, and those inputs lead to real outputs. range of y > 0 for positive values of a.

298

Mathspace Virginia SOL Algebra 1 mathspace.co

5.08 Characteristics of exponential functions mathspace.co

621

The domain is the complete set of possible values for x. For exponential functions, the graph extends indefinitely in both horizontal directions.

Apply the idea

Reflect and check

All real values of x.

All exponential equations of the form y = abx have a domain of all real x.

Students: Page 298 d State the range of the function.

Create a strategy The range is the complete set of possible values for y. We can see the graph extends indefinitely up towards the left, but it approaches an asymptote at y = 0 towards the right.

Apply the idea

Reflect and check

y>0

All exponential equations of the form y = abx have a range of y > 0 for positive values of a.

298

Mathspace

Virginia SOL Algebra 1

Purpose mathspace.co Students demonstrate that they can use the asymptote and end behavior to find the range. Reflecting with students Encourage students to find the y-values for various negative values of x to help them see that the outputs will never reach 0 and never be negative. Ask students to use a calculator to evaluate the function at large, negative x-values, such as x = −100 or x = −1000, and help them see that the value is extremely small, but still greater than 0.

Students: Page 299

Example 2 A population of bacteria can be modeled with the equation p(t) = 100(2)t, where p(t) is the population after t days. This graph shows the population over time.

2000 1800 1600 1400 1200 1000 800 600 400 200

t 1 2 3 4 5 6 7 8 9

a Identify and interpret the p-intercept.

Create a strategy The p-intercept is the vertical intercept and will occur when t = 0. We can read this off the graph or use the equation.

Apply the idea

Reflect and check

Using the equation: p(t) = 100(2)t

Start with the equation

p(0) = 100(2)0

Let t = 0

= 100(1)

Use the zero power law

= 100

Evaluate the product

Reading from the graph is not as precise as using the equation, but is still an important approach.

This means that initially there were 100 bacteria.

b Estimate and interpret when p(t) = 1600.

Purpose Create a strategy Students demonstrates that they can determine meaning of key characteristics in context. We can create a table of values using the equation or read it off the graph.

622

Mathspace Virginia SOL Algebra 1 Teacher Edition Apply the idea mathspace.co 2000 1800

Starting from p(t) = 1600, we can go across to the curve and then down to find the t-value for when p(t) = 1600.

a Identify and interpret the p-intercept.

Create a strategy The p-intercept is the vertical intercept and will occur when t = 0. We can read this off the graph or use the equation. Expected mistakes Students may struggle to interpret the p-intercept in context. Guide students to interpret the meaning of each Apply the idea Reflect and check variable ﬁrst (t and p), then have them apply this to the p-intercept. Using the equation:

Reading from the graph is not as precise as using the equation, but is still an important approach.

Start with the equation p(t)students = 100(2) Reflecting with 0 Let = 0 the class how they found the p-intercept. Highlight the different responses, p(0) = 100(2) Encourage students to discusstwith Use the zerograph power law = 100(1) such as finding the intercept from the and finding it from the equation. = 100

Evaluate the product

means 299–300 that initially there were 100 bacteria. Students:This Pages

b Estimate and interpret when p(t) = 1600.

Create a strategy We can create a table of values using the equation or read it off the graph.

Apply the idea 2000 1800 1600 1400 1200 1000 800 600 400 200

Starting from p(t) = 1600, we can go across to the curve and then down to find the t-value for when p(t) = 1600.

t 1 2 3 4 5 6 7 8 9

This means that after 4 days, the population reaches 1600.

Reflect and check We can confirm this by creating a table of values: t p(t)

0 100

1 200

2 400

3 800

5.08 Characteristics of exponential functions mathspace.co

299

4 1600

c Identify and interpret the domain and range in this context.

Purpose Create a strategy Show students how to graphically ﬁnd an input when given the value of an output.

We can read the domain and range from the graph, or think about what is possible given the context.

Expected mistakes Reflectthe and check Apply idea Students might think 1600 represents the independent variable value and try to input it into the equation for t. We can confirm this by creating table of values: Since t represent time, it cannot abe negative. This means the domain is x ≥ 0. Additionally, they may struggle to read the graph to ﬁnd the t-value at p = 1600. Thet initial population from 0 1 is 100, and 2 it is growing 3 4 there, so the range is y ≥ 100.

p(t) 100 Students: Page 300

200

400

800

1600

Reflect and check Without the context, the domain and range of y = 100(2)x would be different.

c Identify and interpret the domain and range in this context.

Create a strategy

Idea summary

We can read the domain and range from the graph, or think about what is possible given the context. The base of the exponent is the constant factor, or the number being multiplied repeatedly. We can find it by dividing one output by the previous output.

Apply the idea

All exponential functions of the form y = abx have the following features in common: Since t represent time, it cannot be negative. This means the domain is x ≥ 0. • The domain is −∞ < x < ∞. The initial population is 100, and it is growing from there, so the range is y ≥ 100. • The range is y > 0. • If there is a context, the domain and range may be different based on the realistic constraints. Reflect• and Thecheck y-intercept is at (0, a). Without• theThere context, domainasymptote and rangeatofyy==0.100(2)x would be different. is a the horizontal

Practice Idea summary

5.08 Characteristics of exponential functions mathspace.co

The base of the exponent is the constant factor, or the number being multiplied repeatedly. We can find it by output by the previous output. What dividing do youone remember?

623

Create a strategy We can read the domain and range from the graph, or think about what is possible given the context.

Apply the idea Since t represent time, it cannot be negative. This means the domain is x ≥ 0. The initial population is 100, and it is growing from there, so the range is y ≥ 100.

Reflect and check Without the context, the domain and range of y = 100(2)x would be different.

Idea summary

Purpose The base of the exponent is the constant factor, or the number being multiplied repeatedly. We can find it by Identify the meaning of key characteristics exponential functions in context. dividing one output by the previousof output. All exponential functions of the form y = abx have the following features in common:

Expected mistakes • The domain is −∞ < x < ∞. Students may •state domain Thethe range is y > 0.and range of a general exponential function (all real x and y > 0) rather than interpreting the domain range in context. them when based working in real-world situations, there are • If there is a context, the domainRemind and range may that be different on the realistic constraints. • The y-intercept is at (0, a). often constraints on the variables. There is a horizontal asymptote at y = 0.

•

Reflecting with students and Ask the Reflect students tocheck use abstraction to explain the limitations of the given model based on the context. We can confirm ifthis byrange creatingtruly a table of values: Practice Have them consider the does go up to infinity and to justify their answer. t

p(t) 200 400 800 1600 Advanced learners: Encourage exploration of parameter variations What do100 you remember?

use with Example 2

Targeted instructional strategies Would an exponential function generate the values shown in each table? To deepen advanced learners’ c Identify and interpret the understanding, domain and range encourage in this context.students to explore how changing the parameters a b t x x 1 2 3 4 5 1Invite 2 them 3 to4investigate 5 6 what happens in the exponential function p(t) = 100(2) affects the6 population model. f (x) 5 25 125 625 3125 15 625 f (x) 4 9 11.5 15 13.5 11 when the initialapopulation Create strategy or the growth factor is altered. Have students predict the outcomes and then graph the newWe functions them original model. can read to thecompare domain and rangewith fromthe the graph, or think about what is possible given the context. 1

What type of function would generate values as shown in the table?

Additionally, prompt them to connect these mathematical models to other real-world scenarios, such as Apply the x idea 1 2 3 4 5 6 analyzing investment growth with various interest rates. This exploration not only enhances their conceptual Since t frepresent cannot7.6 be negative. (x) 2 time, 3 it4.8 11.8 18This means the domain is x ≥ 0. understanding but also allows them to see the broader applications of exponential functions. The initial population is 100, and it is growing from there, so the range is y ≥ 100. A Approximately exponential B Exactly exponential C and Notcheck exponential Reflect

Students:Without Pagethe300 context, the domain and range of y = 100(2)x would be different. 300

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Idea summary

The base of the exponent is the constant factor, or the number being multiplied repeatedly. We can find it by dividing one output by the previous output. All exponential functions of the form y = abx have the following features in common: The domain is −∞ < x < ∞. The range is y > 0. If there is a context, the domain and range may be different based on the realistic constraints. The y-intercept is at (0, a). There is a horizontal asymptote at y = 0.

• • • • •

Practice What do you remember? 1

Would an exponential function generate the values shown in each table? a

x f (x)

2 25

3 125

4 625

5 3125

6 15 625

x f (x)

1 4

2 9

3 11.5

What type of function would generate values as shown in the table? x f (x)

624

1 5

1 2

2 3

3 4.8

4 7.6

5 11.8

6 18

Mathspace A Virginia SOL Algebra 1 Teacher Edition Approximately exponential mathspace.co C Not exponential

Exactly exponential

4 15

5 13.5

6 11

Practice Students: Pages 300–304

What do you remember? 1

Would an exponential function generate the values shown in each table? x 1 2 3 4 f (x) 5 25 125 625

6 15 625

1 4

f (x)

2 9

3 11.5

4 15

5 13.5

y = 2(7)x

6 11

What type of function would generate values as shown in the table? x

1 2

f (x)

5 3125

2 3

3 4.8

4 7.6

5 11.8

Approximately exponential

Not exponential

6 18 B

Exactly exponential

The function n(x) = 40(3)x represents the number of cells after x hours. a

State which characteristic of the graph is represented by 40.

State what 40 represents in this context.

State how the number of cells changes from one hour to the next.

Let’s practice 4

Which one of the following exponential functions rises most steeply? A

y = 2(2)x

For f (x) = 5x, find f (3).

Consider the function y = −(2x). a

y = 2(5)x

y = 2(3)x

Complete the table of values: x y

−5

−4

−3

−2

−1

Can the value of y ever be zero or positive? Explain your answer.

Describe the behavior of the function as x increases.

State the domain of the function.

State the range of the function.

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625

For each function: i

Identify the equation of the horizontal asymptote.

Find the y-intercept.

iii

Find one other point on the curve.

1 −4 −3 −2 −1

8 7

−2

−3

−4

−5

−6

x 1

y −4 −3 −2 −1 −1

−4 −3 −2 −1

x 1

−7 −8

The graph of g(x) = 2x is shown: a

Find the value of g(3).

Find the value of x when g(x) = 32.

36 32 28 24 20 16 12 8 4

−3 −2 −1

Consider the graph of the functions y = 3x and y = 10x: State the coordinates of the point of intersection of the two curves.

Describe what happens to the values of y for each function as x gets increasingly larger.

Describe what other features these functions have in common.

9 y = 10x

−6

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y = 3x

−5 −4 −3 −2 −1 −3

626

x 1 2 3 4 5

The local seagull population is changing according to the exponential function f (t) = 1000(2)t where t is the number of years that have passed. a

Describe what is happening to the population each year.

Describe what 1000 represents in context.

Find f (3) and describe what it represents in context.

Consider the graphs of the two exponential functions R and S: a

y 18 16 14 12 10 8 6 R 4 2

One of the graphs is of y = 4x and the other graph is of y = 6x. Identify which is the graph of y = 6x. Explain your answer.

For x < 0, determine the relationship between y = 6x and y = 4x. Explain your reasoning.

−3 −2

x 1

For each pair of functions, select the function that increases more rapidly for x > 0: a

−1

y = 4x and y = 5x

y = 2x and y = 3(2)x

Do either of the functions y = 9x or y = −(9x) have x-intercepts? Explain your answer.

Let’s extend our thinking 14

The number of layers, y, resulting from a rectangular piece of paper being folded in half x times, is shown in the graph. a

Interpret the meaning of the y-intercept in this context.

Describe what happens to the thickness with each fold.

If a rectangular piece of paper is folded 10 times, find the resulting number of layers.

If a rectangular piece of paper of thickness 0.02 mm is folded 11 times, find the total resulting thickness.

10 9 8 7 6 5 4 3 2 1

x 1

Find the missing coordinate in each ordered pair so that the pair is a solution of y = −3x: a

(5, ⬚)

(−1, ⬚)

(⬚, −81)

The graph of f (x) = 3x is shown. Find the length of the line segment PQ.

y P

x −5 −4 −3 −2 −1

Q1 2 3 4 5

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A lottery winner has two options: 1. Getting $1 000 000 today 2. Getting $1 today, $2 tomorrow, $4 the next day, and so on with the amount of money each day doubling for a month. The second option can be modeled by A(d) = 2d, where d is the number of days after today (today is d = 0). a

Calculate A(0) and interpret its meaning in the context.

Calculate A(18) and interpret its meaning in the context.

1000000

Using the graph, determine on which day the lottery winner would get more than $1 000 000 using the second option.

900000

Determine which option would give the larger prize. Explain.

600000

A ($)

800000 700000

A (d) = 2d

500000 400000 300000 200000 100000

d 2 4 6 8 10 12 14 16 18 2022242628

Winston and Rasiah want to organize a lunch time dominos tournament. They want a knockout tournament, where the winner of each round progresses to the next round until there are only two players left. The diagram shows the draw for the final, semi-final and quarter final rounds. Semi-Final

Quarter-Final Final

Complete the table of values for the total number of players, p, that the competition can accommodate given a number of rounds, r. Number of rounds (r) Number of players ( p)

628

Winner

Winston and Rasiah want to make sure that each round of the tournament has every spot filled. Find the values of p for which a tournament can be formed.

Over two weeks, they can fit in 8 rounds of play. Determine how many players can they accept into the tournament.

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Answers

11 a Graph R is the graph of y = 6x as it passes through the point (1, 6) and as the base is 6, it will be increasing by a factor of 6 which will increase more quickly than the function y = 4x.

5.08 Characteristics of exponential functions What do you remember? 1 a Yes

b F or x < 0, the graph of y = 6x is below the graph of y = 4x. This is because negative values of x result in fractional function values, and for any negative value of x, 6x will result in a smaller value than 4x.

b No

2 A 3 a The y-intercept

b The initial number of cells

c It triples

12 a y = 5x

b y = 3(2)x

Let’s practice

13 No, y = 9x is always greater than 0, and y = −(9x) is always less than 0.

4 D

Let’s extend our thinking

5 f (3) = 125

14 a The y-intercept is equal to the intial value. There is 1 layer of paper when no folds have been made.

6 a

−5

−4 −3 −2 −1 0

b The thickness of the paper doubles with each fold

−1 −2 −4 −8 −16 −32

c 1024 layers

b N o. For a > 0 the expression ax is greater than zero for all real x. So, − (ax) will be less than zero.

d 40.96 mm

15 a −243

b −3

d 4

c As x increases, the function decreases at a faster and faster rate.

16 81 units

d All real x.

17 a A (0) = 1. This represents how much money the winner would get today if they chose option 2.

e y<0 7 a i y=0

ii (0, 1)

iii Answers may vary, for example: (2, 4) b i y=0

ii (0, 0.5)

iii Answers may vary, for example: (1, 2) c i y=0

ii (0, 2)

iii Answers may vary, for example: (1, 6) d i y=0

ii (0, −1)

iii Answers may vary, for example: (1, −8) 8 a g(3) = 8

b x=5

9 a (0, 1) b As x increases both functions increase towards inﬁnity. c B oth functions have the same domain of all real x and the same range y > 0. They both have the same asymptote y = 0.

b A (18) = 262 144. This represents that on the 18th day after it started, the winner would get $261 144 if they chose option 2. c A fter 20 days the lottery winner would get more than $1 000 000. d T he second option would end up with drastically more than the ﬁrst option. On just the 20th day, the second option earns more, so adding all 30 days it would be way better to go with the second option. 18 a

Number of rounds (r)

Number of players ( p)

b Any power of 2 such as 2, 4, 8, 16, 32, 64 … c 256 players

10 a The population doubles each year b 1000 is the initial population of seagulls. c f (3) = 3200. This says that after 3 years, there will be 3200 seagulls.

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5.09 Graphs of exponential functions Subtopic overview Lesson narrative Students will explore and develop ideas of exponential change, creating and analyzing mathematical models to make sense of situations. They will explore key features of exponential functions and explain connections between the equations, tables, and graphs, recognizing that the base is the number being multiplied each time and identifying relationships between the coefficient and constant factor values. By the end of the lesson, students should be able to examine the key components of exponential equations, describe key characteristics, and graph exponential functions.

Learning objectives Students: Page 305

Key vocabulary 

exponential function

Essential understanding Exponential functions have a constant growth factor, often referred to as the common ratio, and can be represented by the equation y = abx.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving Teachers can integrate this goal by posing problems that involve the graphing of linear and exponential functions, and encouraging students to utilize various strategies such as creating a table of values, plotting points, and identifying key features to solve them. Real-world contexts such as population growth, compound interest, and radioactive decay could be used to construct problems, enhancing students’ problem-solving skills.

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MPG2 — Mathematical Communication

MPG4 — Mathematical Connections

Teachers can facilitate mathematical communication by encouraging students to express their reasoning and thought process as they graph functions and identify their key characteristics. Students could also be asked to discuss and justify the differences between the graphs of linear and exponential functions, thus enhancing their mathematical vocabulary and precision.

Teachers can encourage students to relate the concepts of linear and exponential functions to their prior knowledge of functions (A.F.1 standard) and to other areas of mathematics. They can also help students connect the concept of transformations to their understanding of other mathematical concepts, and to real-world contexts such as population growth and compound interest.

A.F.2f — Graph an exponential function, f (x), in two variables using a variety of strategies, including transformations f (x) + k and kf (x), where k is limited to rational values.

Prior connections A.F.1 — The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.

Engage Activity Career planning

60 mins

Students will investigate career opportunities and create a model showing how much money Leroy, Rosaria, and Alejandro earned in the ﬁrst ten years for their career choices.

Understanding and skills

Will develop Graphing exponential functions. Identifying key features of exponential functions. Comparing key features of exponential functions, represented in the same or different ways. Identifying or applying transformations to exponential functions from an equation, table, or graph.

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Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: Graphing technology (recommended)

Support students with disabilities Support organization - making tables and graphs To support development of organizational skills, check in with students as they are completing tables, graphs, or recording the money earned each year in some other way for each career. Ask the represent in each group to share the values and any patterns they may have noticed. Ask students questions to check and further their understanding, such as “how did you ﬁnd the total money earned each year?” or “how can we generalize this pattern?”. This will give students an opportunity to ﬁx any errors that may impact the outcome of each graph.

Support for English language learners Three reads Have students read the task aloud. On the ﬁrst read, ask students to describe the situation. Prompt: Students read the problem. Students think/write: Answer the question “What is the problem about?” Answers may look like: • Students graduating and choosing a career. • The amount people earn based on their career paths and annual salary increases. • Building exponential functions to represent salary and earnings over time. Share: Students are called upon to discuss their answers with the class. On the second read, ask students to interpret the question. Prompt: Students read the problem. Students think/write: Answer the question “What does an answer look like?” Answers may look like: • A graph of three exponential functions. • A visual that shows how fast each person’s earnings grow over ten years. On the third read, have students identify important information. Prompt: Students read the problem. Students think/write: Answer the question “What are the important pieces of information given in the question?” Answers may look like: • The salary for each career. • The percent increase for each salary. • The tuition cost and number of years of schooling required for each career.

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Classroom guide Hook Students compare the graphs of three different exponential functions.

Open questions

•

5 mins

What are the similarities and differences between f (x), g(x), and h(x)?

h(x)

3 2

f (x) g(x) −4 −3 −2 −1

1 −1

x 1

−2 −3 −4

Slide 1 from Student Engage Activity

Implementation details Encourage students to compare key features of f (x), g(x), and h(x). If students write equations of the exponential functions, invite students to share their equations in their descriptions of similarities and differences.

Launch

5 mins

Leroy, Rosaria, and Klara are researching career opportunities for the future: • Leroy has decided to enter the workforce. He found a job as a carpenter where he makes $29 100 per year. Leroy’s income increases at a rate of 2% each year. • Rosaria is interested in fashion design and is interested in a job that would have an annual salary of $44 500 after graduation. The four year program costs $13 950 per year. Rosaria’s income would increase at a rate of 4% each year. • Klara is looking into a career in sports management. She is interested in pursuing an associate’s degree that costs $2500 for each of the two years. She found a job with an annual salary of $34 000. Klara’s income would increase at a rate of 2.5% each year. Slide 2 from Student Engage Activity

Have students complete the three reads routine as needed to help discuss the information presented in the Launch. Invite students to highlight key information given in the Launch. Lead a discussion about salaries and discuss yearly salary, yearly percent increase, and other factors that impact the money earned each year. Important mathematical concepts: Exponential functions Important contextual information: Salaries, yearly percent increase Suggested grouping: Form groups of 3 or 4 and assign roles

Continue when Students have read the Launch and understand the context of the problem.

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Explore

Team roles

•

35 mins

Encourage students to create a model showing how much money Leroy, Rosaria, and Alejandro earned in the ﬁrst ten years, as well as to use their data to justify the path they would choose.

Anticipated strategies Write an equation Students may write an equation for each career path. Let x represent the time in years. Leroy: L(x) = 29 100(1.02)x Rosaria: R(x) = 44 500(1.04)x−4, note that this exponential function is horizontally shifted to the right 4 units due to the four year program. Klara: K(x) = 3400(1.025)x−2, note that this exponential function is horizontally shifted to the right 2 units due to the associate’s degree.

Create a table Students may create a table that shows the yearly salary for at least the ﬁrst ten years. Another table students may choose to create is the total money saved (taking into account cost of programs or other factors) to explain their choice of career paths. For example, the beginning of a table showing Klara’s income could look like this:

Years of employment 1 2 3 4 5

Annual salary, $ 34 000 34 850 35 721 36 614 37 530

Note that the salaries are rounded to the nearest integer, but students may have decimal values here. Encourage students to think about what the values represent so students can make adjustments, such as round to two decimal places or round to the nearest integer. Student work could also look like this:

Years after graduation 0 1 2 3 4

Total money earned, $ 0 −2500 −5000 −5000 + 34 000 = 29 000 63 850

Note that this is assuming that Klara did not have any other jobs during pursuing an associate’s degree, not taking into account taxes, and multiple other factors. Encourage students to think about assumptions and other factors that would affect their model when comparing the career paths.

Create a graph

Students may create a graph that represents each career path. 60 000

R(x) K(x)

L(x)

40 000

20 000 x 5 634

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Misconceptions Starting all student’s careers at the same time When would Leroy/Rosaria/Klara’s job start? How does this affect the model? Can you show me where this is represented in your model?

Purposeful questions Use the following questions to check for understanding and encourage critical thinking: • What is your model so far? What information is important for your model? • How do your models compare so far? What makes you say that? • Based on your data, which path would you choose? • Could you represent your model in another way? • What other factors could affect your data? • How much is each person earning at the start (year zero)? Year 1? Year n? • How did you account for tuition costs in your model? • Which pathway would you choose and why?

Continue when Students have created a model showing how much money each student earns in the ﬁrst ten years, and justiﬁed which path they would choose out of the three.

Discuss

15 mins

Begin with a gallery walk followed by a whole class discussion. Consider sequencing the strategies presented from creating a table, to creating a graph, to writing an equation.

Discussion guide Have students complete a gallery walk to see the various models students created as well as justification for choosing a specific path. As students complete the gallery walk, display the following questions to have students discuss: • What are the similarities and differences between your group’s models and other groups? • After viewing other groups’ work, would your choice stay the same? Why or why not? After the gallery walk, invite students to share their responses. Sequence student responses so every type of model is highlighted, such as writing an equation, creating a table, and creating a graph. Next, ask students how their choice has evolved or stayed the same during the gallery walk. Students may share reasons that take into account outside factors that would affect the total money earned each year and this is encouraged. To continue the discussion, ask students who would earn the most money after 2, 4, or 10 years. There is no “best” career choice, so the discussion should encourage comparing the exponential models and what the values mean in context. As an extension you may wish to give students the following prompts: • Have students research a job of their choice and create a similar model showing how much money they could earn in the first ten years with a certain yearly percent increase, such as 2%. • Ask students how long it takes for Leroy, Rosaria, and Klara to reach a yearly salary of ⬚, such as $48 000?

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Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lesson: Algebra 1 — 5.08 Characteristics of exponential functions

Tools You may ﬁnd these tools helpful: • Graphing calculator

• Graph paper

• Spreadsheet application

Student lesson & teacher guide Graphs of exponential functions Students are introduced to the components of the graph of an exponential function f (x) = abx + k, including values such as the y-intercept, constant factor, and vertical shift. Students then engage in an exploration examining how changing the base and exponent of an exponential function affect the graph’s characteristics.

Students: Page 305

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Characteristics of exponential functions with translations Address student misconceptions Students may make assumptions based on the exponential form y = a(b)x which do not generalize to the exponential form y = a(b)x + k. For example, they may think the asymptote of an exponential function will always be at the x-axis or the y-intercept will always be (0, a). Have students consider which characteristics a vertical translation will affect and how they will be affected. Encourage them to make new generalizations for exponential functions that have been translated. For exponential functions of the form y = a(b)x + k: • The y-intercept is (0, a + k) • The horizontal asymptote is y = k • If a > 0, the range is (k, ∞) • If a < 0, the range is (−∞, k)

Always, sometimes, never English language learner support Have students consider which of the following statements are always, sometimes, or never true to check their understanding of the vocabulary and key features. • The asymptote for an exponential function will be a horizontal line (answer: always) • The graph of an exponential function will have an x-intercept (answer: sometimes - depends on the values of a and k) • The range of an exponential function can be (−∞, ∞) (answer: never) • The graph of an exponential function will have a y-intercept (answer: always - unless there is a domain constraint for a context) • An exponential function of the form y = bx − 1 has an x-intercept at the origina (answer: always)

Concrete-Representational-Abstract (CRA) Approach Targeted instructional strategies Concrete: Engage students with physical manipulatives to explore exponential growth and decay. Use objects like coins, counters, or blocks to model situations where quantities double or halved. For example, start with one coin and show how it doubles each time: 1 coin becomes 2, then 4, then 8, and so on. Allow students to build these sequences themselves, physically adding the coins to see how the amount increases exponentially. This hands-on activity helps students understand the concept of a constant multiplier and how exponential functions model rapid change. Representational: Transition to the representational stage by having students create tables and graphs based on their manipulative activities. Ask them to record the number of coins at each step in a table, noting how the quantity changes. Then, have them plot these values on a graph with the step number on the x-axis and the quantity on the y-axis. Encourage them to draw the curve that connects the points, showing the exponential growth or decay. Visual representations like tables and graphs help students see patterns and relationships in the data. Abstract: Move on to the abstract stage by introducing the mathematical equations for exponential functions. Teach students how to write an equation in the form y = abx, where a is the starting amount and b is the base or common ratio. Show them how the equations relate to the tables and graphs they created. For example, if they started with one coin and it doubled each time, the equation would be y = 1 × 2x. Provide problems where they use the equation to calculate values for different x or they observed.

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Connecting the stages: Help students make connections between all three stages by discussing how the physical manipulatives relate to the graphs and equations. Ask guiding questions like: • “How does the doubling of coins show up in our equation?” • “What does the shape of the graph tell us about how the quantity changes?” Encourage students to refer back to their manipulatives and drawings when working with abstract equations. By linking the concrete activities, visual representations, and symbolic equations, students can deepen their understanding of exponential functions and choose the representation that helps them think through problems best.

Exploration Students: Page 305

Suggested student grouping: Individual Students will use a GeoGebra applet to explore what happens to an exponential graph that is vertically stretched or translated. They will use sliders to change the variables a and k in the equation y = a(2)x + k. The goal is to see how these changes affect the horizontal asymptote, the shape of the graph, and the y-intercept. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Which variables(s) change the horizontal asymptote? k shifts the horizontal asymptote up or down. 2. Which variables(s) change the shape? a will stretch or compress the graph vertically, thus altering its shape. 3. Which variable(s) change the y-intercept? Both a and k affect the y-intercept, causing it to shift up or down along the y-axis.

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4. Set the k slider to k = 0. Describe the graph when a is positive. Describe the graph when a is negative. When a is positive, the graph increases indefinitely toward positive infinity. All of the y-values are positive. When a is negative, the graph decreases indefinitely toward negative infinity. All of the y-values are negative. 5. Set the a slider to a = 1. Describe what happens when k is positive. Describe what happens when k is negative. When k is positive, the graph has the same shape as the original graph, but it shifts up k units. When k is negative, the graph has the same shape as the original graph, but it shifts down k units. Purposeful questions • When a is positive (or negative), how would you describe the behavior of the graph? What about the outputs? • When you adjust k, does the graph increase in the same way as the original? What feature of the graph changes with k? • What happens to the graph when a is zero? How does this affect the equation? • Can you predict the effects on the y-intercept if both a and k are changed simultaneously? Possible misunderstandings • Students might not realize that the value of a stretches or compresses the graph vertically. To emphasize this, let k = 0 and focus only on the positive values of a. Identify specific y-values to show students that they have changed by a factor of a.

Using transformations to graph an exponential function Targeted instructional strategies Students may benefit from explicit instruction on how to use transformations to graph exponential functions. Show students how to identify the transformations of the function, then discuss how those transformations will affect the function. First, explain that we can write the general form of a transformed exponential function as: f (x) = a ⋅ bx + k Then, explain that we always begin by graphing the parent function, y = bx. We can do this by creating a table of values. For example, a table of values for the parent function f (x) = 3x is shown. −2

−1

f (x)

Next, we can identify the transformations. Since a and k represent vertical transformations, they will only affect the y-values. • a describes the vertical dilation. This will change each y-value in the table by a factor of a, meaning we multiply each y-value by a. For g(x) = 2(3)x: x

−2

−1

f (x)

g(x)

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• A negative value of a represents a reflection across the x-axis. This will change the sign of each y-value in the table. For h(x) = −2(3)x: −2

−1

g(x)

h(x)

−2

−6

−18

−54

• k describes the vertical translation. If k is positive, we add k to each y-value. If k is negative, we subtract k from each y-value. For j(x) = 3x − 4: −2

−1

f (x)

j(x)

−3

−1

Finally, students can plot the two functions in each table to visualize and verify the transformations. Students are then introduced to the key characteristics of both equations and graphs of exponential functions, including the range, y-intercept, and rate of change.

Students: Pages 305–306

For k > 0, the graph will shift up k units. For k < 0, the graph will shift down k units. y

6 f (x) = 2x

4 2 −4 −3 −2 −1

(0, a) 1 2

4 (0, a + 4) 2

x 3

Parent Function f (x) = 2x

f (x) = 2x + 4

−4 −3 −2 −1

Vertical shift 4 units up f (x) = 2x + 4

The key features of an exponential function can be found in both the equation and the graph. Given f (x) = abx, the leading coefficient, a, represents the y-intercept and is plotted as the point (0, a). It also determines the range of the function. • When a > 0, the range is y > 0 • When a < 0, the range is y < 0. The value of a also affects the rate of change of the function. If two functions have the same value for b, a larger value for a will have a greater rate of change. 8

7 6

640

Mathspace Virginia SOL Algebra 5 1 Teacher Edition mathspace.co 4 a > 0, b > 1

y −2

−1

x 1

−2 −3 a < 0, b > 1 −4

−5

−6

y −2

−1

x 1

−2

−3 a < 0, b > 1

5 a > 0, b > 1

−4

−5

−6

1 −2

−1

−7

−1

−8

y > 0, Increases at an increasing rate

y < 0, Decreases at an increasing rate

The absolute value of a, tells us whether the graph’s height will be made taller or shorter. If ∣a∣ > 1, then every y-coordinate of the function is multiplied by a factor a that is greater than 1. The points on the graph move further away from the x-axis, increasing the steepness of the graph. This is called a vertical stretch. If 0 < ∣a∣ < 1, then every y-coordinate of the function is multiplied by a factor a that is between 0 and 1. The points on the graph move closer to the x-axis, decreasing the steepness of the graph. This is called a vertical compression.

Examples Students: Page 307 Example 1 Virginia SOL Algebray1 = 2.5(4)x. 306 Mathspace Consider the exponential function mathspace.co

a Draw the graph of the function.

Create a strategy We can identify both the y-intercept and constant factor from the equation since it is of the form y = abx. Using these two key features, we can plot other points to the left and right of the y-intercept and connect them with a smooth curve.

Apply the idea The function has a constant factor of 4 since that is the base of the exponent, and a y-intercept at (0, 2.5) as that is the coefficient. We can start by plotting the y-intercept and choosing an appropriate scale. We can see that the function will always be positive, so we don’t need any negative y-values.

y 40 35

We know that the constant factor is 4. If we want to go from −2 to 2 on the x-axis, we need to go up to at least 2.5(4)2 = 40 on the y-axis.

30 25 20

This would be an appropriate scale as it doesn’t have too many labels or tick marks and is easy to read.

15 10 5 −2

−1

Either using a table of values or using the common ratio, we can plot another three points to get a good shape for the graph. Then we can connect the points with a smooth curve.

y 40 35 30 25 20

5.09 Graphs of exponential functions mathspace.co

15 10 5

641

y-axis. 20

This would be an appropriate scale as it doesn’t have too many labels or tick marks and is easy to read.

15 10 5 −2

−1

Either using a table of values or using the common ratio, we can plot another three points to get a good shape for the graph. Then we can connect the points with a smooth curve.

y 40 35 30 25 20 15 10 5 −2

−1

x 1

Purpose Students demonstrate they can use the key features of an exponential equation to graph the function.

Students: Page 308

5.09 Graphs of exponential functions mathspace.co

642

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

307

Purpose Students demonstrate they can use appropriate technology to verify graphs done by hand.

Students: Page 308

Purpose Students demonstrate that the leading coeﬃcient plays a key role in the rate of change of the graph of an exponential function. Expected mistakes Students might not be able to explain how the leading coefficient affects the graph. They may also struggle to picture y = 4x, making it hard for them to compare the two graphs. Encourage students to substitute a few values of x into both equations and have them compare the y-values. Ask questions like, “Which function’s outputs are larger/smaller?” and “If this function’s outputs are larger, how will it compare to the graph of the other function? Reflecting with students Extend this example for advanced learners or for the entire class by asking students how varying the leading coefficient a in y = a(4)x changes the graph. Encourage them to consider values of a that are less than 1, greater than 1, or negative, and predict the resulting transformations. Ask questions such as: • How is has the graph of y = −2.5(4)x been transformed from y = 4x ? • How has the graph of y = 0.5(4)x been transformed from y = 4x ? • Order the graphs of y = 2.5(4)x, y = 4x, y = 0.5(4)x from least steep to most steep. • Will the graph of 0.25(4)x or y = −2(4)x be more steep? Prompt students to justify their reasoning and, if necessary, sketch the graphs to visualize these changes.

Spreadsheets, code, or tools to reduce computational work

use with Example 1

Student with disabilities support Show students how to use formulas in spreadsheets or other tools to create tables of values. For example, 1. Open a spreadsheet program, such as GeoGebra, GoogleSheets, Excel, Desmos, etc. 2. Fill in headings A 1 2 3 4 5 6 7

B f(x)=2.5*4^x

C g(x)=4^x

5.09 Graphs of exponential functions mathspace.co

643

3. Fill in the domain by typing the first value and then in the cell below using a formula and then dragging down. Emphasize that an equals sign must preceed the formula. 1 2 3 4 5 6 7

A x -3 =A2+1

B f(x)=2.5*4^x

C g(x)=4^x

4. Fill in the function values by typing the formula and then dragging down. A 1 2 3 4 5 6 7

x -3 -2 -1 0 1 2

B f(x)=2.5*4^x =2.5*4^A2

C g(x)=4^x =4^A2

5. Use the table of values to sketch the function. A 1 2 3 4 5 6 7

x -3 -2 -1 0 1 2

B f(x)=2.5(4^x) 0.0390625 0.15625 0.625 2.5 10 40

C g(x)=4^x 0.015625 0.0625 0.25 1 4 16

Students: Page 309 Example 2 In 2010, Bob counted 3 rabbits in his backyard. He noticed they double every week. Let f (x) describe this scenario, and use it to answer the questions below. a Find the y-intercept.

Create a strategy Imagine time in weeks as x, the independent variable. The y-intercept occurs when x = 0 at week 0 of counting.

Apply the idea y=3 The y-value of the y-intercept is the starting number of rabbits which is 3. The y-intercept is (0, 3).

b Write an equation describing this scenario.

Purpose Create a strategy Apply the idea Check the students’ understanding of the concept of the y-intercept in the context of a real-life problem and Since the rabbits double every months, we can start by We know our y-intercept, or a = 3. Our constant factor how to determine it constant from a given identifying the factor. scenario. b = 2. We can write our equation as f (x) = 3(2)x.

644

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co c Graph the function.

Create a strategy

Apply the idea

Example 2 In 2010, Bob counted 3 rabbits in his backyard. He noticed they double every week. Let f (x) describe this scenario, and use it to answer the questions below. a Find the y-intercept.

Expected mistakes StudentsCreate may misunderstand the scenario and think that Apply the y-intercept a strategy the idea is the number of rabbits after one week, not at the start. Emphasize how the y-intercept represents the initial value (or the value when the independent Imagine time in weeks as x, the independent variable. y=3 variableThe is zero) in the context of the problem. y-intercept occurs when x = 0 at week 0 of counting.

Students: Page 309

The y-value of the y-intercept is the starting number of rabbits which is 3. The y-intercept is (0, 3).

Example b Write an 2 equation describing this scenario. In 2010, Bob counted 3 rabbits in his backyard. He noticed they double every week. Let f (x) describe this scenario, Create a strategy Apply the idea and use it to answer the questions below. Since the rabbits double every months, we can start by We know our y-intercept, or a = 3. Our constant factor a Find thethe y-intercept. identifying constant factor. b = 2.

Create a strategy Imagine time in weeks as x, the independent variable. The y-intercept occurs when x = 0 at week 0 of counting. c Graph the function.

We can write our equation as f (x) = 3(2)x.

Apply the idea

y=3 The y-value of the y-intercept is the starting number of

Purpose rabbits which is 3. The y-intercept is (0, 3). Create a strategy Apply the idea Checks if students can formulate an exponential function from a real-life scenario. Draw the curve using the equation from part (b).

24 b mistakes Write an equation describing this scenario. Expected 21 Students might formulate a linear equation due to misunderstanding the doubling nature of the problem. Create a strategy Apply the idea 18 Help students realize that “doubling” something means we multiply the previous value by 2, which correlates to 15 Since the rabbits double every months, we can start by We know our y-intercept, or a = 3. Our constant factor exponential growth. Linear growth is when a constant value is added to previous outputs. y = 3(2)x

identifying the constant factor.

Students: Page 309

b = 2.

9 We can write our equation as f (x) = 3(2)x. 6 3

c Graph the function.

Create a strategy

y 24

Example 3

Consider the function y = −4x − 1:

a Find the y-intercept.

Substitute x = 0 into the function.

Apply the idea

Draw the curve using the equation from part (b).

Create a strategy

−1

y = 3(2)x

Apply the idea 9 6

y = −4x − 1

Write the function

= −4 − 1

Substitutexx = 0

0 3

= −2−1

3 Evaluate 4 5

PurposeExample 3 5.09 Graphs of exponential functions 309 mathspace.co Ensures that students can accurately plot the graph of an exponential function and link it to the real-life x Consider the function y = −4 − 1: scenario. a Find the y-intercept.

Reﬂecting with students Create a strategy Apply the idea Discuss with students which method(s) they used to graph the functions. Remind students that, to graph any Substitute x = 0 into the function. y = −4x − 1 Write the function function, we can substitute values of x into the equation to create a table of values, then plot the points to graph Substitute x = 0 = −40 − 1 the function. However, identifying the key features of a function is usually more efficient. = −2

Evaluate

5.09 Graphs of exponential functions mathspace.co

309

5.09 Graphs of exponential functions mathspace.co

645

9 6 3

−1

Students: Page 309

Example 3 Consider the function y = −4x − 1: a Find the y-intercept.

Create a strategy

Apply the idea

Substitute x = 0 into the function.

y = −4x − 1

Write the function

= −40 − 1

Substitute x = 0

= −2

Evaluate

5.09 Graphs of exponential functions Purpose mathspace.co Students will demonstrate they can ﬁnd the y-intercept of an exponential function algebraically.

309

Reﬂecting with students Remind students that the y-intercept is the point where a graph intersects the y-axis, which happens when x = 0. This is a fundamental concept in understanding and graphing any type of function, not just exponential functions.

Students: Page 310 b Graph the function as a transformation of the function f (x) = −4x.

Create a strategy

Apply the idea x

Start by graphing the function f (x) = −4 .

Draw the curve passing through each plotted point.

y −5 −4 −3 −2 −1 −2 −4 −6

x 1

−5 −4 −3 −2 −1 y = −(4x) −1 −2

x 1

−4 y = −(4x)

−6 −8

−8

−10

−12

−14

−16

This curve of the function y = −(4x) − 1 must be shifted down 1 unit from the function f (x) = −(4x).

Idea summary

Purpose We can use the y-intercept, the constant factor, b, and the vertical shift, k. to graph an exponential function in x Check students vertical shifts to key graph transformations of exponential functions. thecan formuse y = ab + k and identify features: •

When a > 0 and b > 1, the function is increasing at an increasing rate.

Reﬂecting with students • When a < 0 and b > 1, the function is decreasing at an increasing rate. Students may •benefit from a the deeper k > 0, will shift graphdiscussion k units up. of graphing with transformations. Begin by pointing out that the • k < function 0, will shiftwill the graph units down. parent exponential be in kthe form y = bx. In this case, the parent function is y = 4x. We can begin by graphing the parent function using known points or by creating a table of values. Next, ask students to identify which two transformations have occurred. In this example, a = −1 which indicates Practice a reflection across the x-axis. Students can graph this transformation by making the y-values of the parent function negative.

What do you remember? 646

1 Match eachSOL graph to one1 Teacher of the equations: Mathspace Virginia Algebra Edition mathspace.co i y = 2x ii y = −2x a 8

iii

b 8

y = −2(2)x

y = −(4 )

−6

−6 −8

−8

−10

−12

−14

−16can graph this transformation Finally, the constant value of k = −1 indicates a translation of 1 unit down. Students by subtracting 1 from the y-values of their reflected function.

This curve of the function y = −(4x) − 1 must be shifted down 1 unit from the function f (x) = −(4x).

Students: Page 310

Idea summary We can use the y-intercept, the constant factor, b, and the vertical shift, k. to graph an exponential function in the form y = abx + k and identify key features: • • • •

When a > 0 and b > 1, the function is increasing at an increasing rate. When a < 0 and b > 1, the function is decreasing at an increasing rate. k > 0, will shift the graph k units up. k < 0, will shift the graph k units down.

Practice What do you remember?

Practice 1

Match each graph to one of the equations: x

i y 310–313 =2 Students: Pages

y = −2x

a 8

iii b

What do you remember?6

Match each graph to one 2of the equations: x −4 −3 −2 −1 1 2 3 4x i y = 2x ii y = −2 −2

iii

−4 −3 −2 −1 −2

−6

y −6 8

310

−8

Mathspace Virginia SOL Algebra 1 mathspace.co

−4 −3 −2 −1 −2

x 1

y = −2(2)x

−6

−8

x 2

−4 −3 −2 −1 −2

−4

−4 −3 −2 −1 −2

−4

y = −2(2)x

2 −4 −3 −2 −1 −2

−4

−6

−8

5.09 Graphs of exponential functions mathspace.co

647

For f (x) = 3x, will the transformation change the equation of the horizontal asymptote? a

A vertical dilation by a factor of 5

A translation down 2 units

A refection across the x-axis

A vertical dilation by a factor of

Describe how the graph of y = 2x is transformed to get each graph: a

y = 2x + 4

y = 2x − 6

y = 5(2)x

Let’s practice 4

Draw the graphs of the functions y = 2x, y = 3x and y = 5x by hand or using technology, then answer the following questions: a

State whether the following statements are true for all of the functions: All of the curves have a maximum value.

All of the curves pass through the point (1, 2).

iii All of the curves have the same y-intercept.

None of the curves cross the x-axis.

State the y-intercept of each curve.

Describe what happens to the values of y as x gets increasingly larger.

Consider the function y = 4(2x). a

Find the y-value of the y-intercept of the curve.

Can the function values ever be negative?

State an appropriate scale for the axes to graph y = 4(2x). Justify your choice.

Graph y = 4(2x) by hand or using technology.

List the domain and range for the function.

Consider the functions f (x) = 4(2)x and g(x) = 2(4)x. a

Graph the two functions using technology.

10 8 6 4 2

Complete the table of values for y = −5 . x y

648

Compare the domain and range of f (x) and g(x).

Consider the given graph of y = 5x. a

−2

−1

Graph y = 5x and y = −5x on the same coordinate plane.

Compare the domain and range of y = 5x and y = −5x.

Describe a transformation of the graph of f (x) = 5x that would obtain g(x) = −5x.

−3 −2

−1

−2 −4 −6 −8 −10

y = 5x x 1

Gabby starts by saving 6 pennies in her piggy bank. She decides to triple the amount she saves every week. a

Write an equation to match the scenario.

Graph the function.

Use your function to determine how much money Gaby will have after 10 weeks.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

The graph of f (x) = 3x is shown.

5 4 3 2 1

Describe a transformation of the graph of f (x) = 3 that would obtain g(x) = 3x − 5.

Sketch the graph of g(x) = 3x − 5 on the same set of axes as f (x) = 3x.

f (x) = 3x

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

The graph of f (x) = 8x is shown. a b

Describe transformation of the graph of f (x) = 8 that would obtain g(x) = 0.5(8)x. x

1 2 3 4 5

6 x

Sketch the graph of g(x) = 0.5(8) on the same set of axes as f (x) = 8 .

f (x) = 8x

−4 −3 −2 −1 −2

−4 −6 −8

Consider a graph of y = 5x:

y x

Describe a transformation of the graph of y = 5 that would obtain the graph of y = 5x + 1.

Sketch the graph of y = 5x + 1.

25 20 15 10 5 x −3 −2

Of the two functions y = 4x and

−1

, which is increasing more rapidly for x > 0?

5.09 Graphs of exponential functions mathspace.co

649

The graphs of A, B, and y = 6x are shown. a b

What transformation has been applied to y = 6 to get the graph of A? What transformation has been applied to y = 6 to get the graph of B?

10 9 8 7

y = 6x

6 5 4 3

2 1

x 1

In a laboratory, the number of bacteria in a petri dish is recorded, and the bacteria are found to double each hour. a

Complete the table of values. Number of hours passed (x) Number of bacteria ( y)

0 1

At this rate, how many bacteria will be present in the petri dish after 15 hours?

Using the table, graph the number of bacteria over time.

Interpret the meaning of the y-intercept in this context.

Let’s extend our thinking 15

Consider the original graph y = 3x. The function values of the graph are multiplied by 2 to form a new graph. a

For each point on the original graph, find the point on the new graph. Point on original graph Point on new graph

(−1, ⬚)

(0, 1)

(1, 3)

(2, 9)

(0, ⬚)

(1, ⬚)

(2, ⬚)

State the equation of the new graph.

Graph both functions on the same coordinate plane and compare them.

The function y = 3 (2x) is shown. a

Describe a transformation of the graph of y = 3(2x) that would obtain y = −3(2x).

Sketch the graph of y = −3(2x) on the same set of axes as y = 3(2x).

The number of bacteria over time is to be modeled by an exponential function, with x representing time and y representing the number of bacteria. Which function should be used? Explain.

When the only transformation applied to the function f (x) = bx is a dilation by a factor of a, what are the coordinates of the y-intercept? Explain your reasoning.

650

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Answers

e Domain: All real numbers, Range: y > 0 6 a

5.09 Graphs of exponential functions

7 6

What do you remember?

5 4

b y = −2x

1 a x

c y=2 2 a No

d y = −2(2) b Yes

f (x)

c No

b The graph is translated down 6 units.

d The graph is dilated by a factor of 5. Let’s practice iii Yes

iv Yes

• Domain: All real values of x

• Range: All positive real values of y

7 a

−2

−1

10 8 6 4 2

b No

The y-intercept of this function is at (0, 4). Moving to the right, for every increase in x by 1, the y-values increase by a factor of 2. We could show x-values in the interval −3 ≤ x ≤ 3. Since the function values are all positive, an appropriate minimum and maximum y-values would be −2 ≤ y ≤ 4(2)(2)(2) = 32. Having 34 tick marks would be way too many, something closer to 10 is a more appropriate number, so we could go up by 4 on the y-axis. The axes could look like this: y

3 2

d 32

x 1

−1

−5

−25

x 1

y 5x

d A reﬂection about the x-axis 8 a f (x) = 6(3x) b

10 8 6 4 2 3 2

c B oth have domain: All real x. The function y = 5x has range: y > 0 and y = −5x has the opposite range: y < 0.

y 5x

2 4 6 8 10

20 16

2 3 4

c A ny appropriate scale that allows for the visibility of the key features of the graph, with suﬃcient justiﬁcation. For example:

3 2

x 1

c As x gets larger, the function values all increase exponentially.

g(x)

−4 −3 −2 −1 −1

b Shared y-intercept of (0, 1)

5 a y=4

b E ven though the graphs of the functions are different, they both have the same domain and range:

c The graph is dilated by a factor of .

ii No

3 1

d No

3 a The graph is translated up 4 units.

4 a i No

2 4 6 8 10

y 6 3x x 1

c Gabby will have $3542.94 in her account.

28 24 20 16 12 8 4 3 2

x 1

Answers mathspace.co

651

9 a A translation down 5 units b

5 4 3 2 1

Let’s extend our thinking

15 a

f (x) = 3x x 1 2 3 4 5

5 4 3 2 1 1 2 3 4 g(x) = 3x 5 5

Point on original graph

(0, 1) (1, 3) (2, 9)

Point on new graph

(0, 2) (1, 6) (2, 18)

b y = 2 · 3x c

y 8

10 a A vertical dilation by a factor of 0.5. b 8

y 3x

6 4

y 2 3x

2 x

5 4

3 2 f (x) = 8

3 2

16 a A reﬂection across the x-axis

g(x) = 0.5(8)x

11 a Vertical translation up 1 unit b

20 15 10 x 1

13 a Vertical dilation by a factor of b Vertical dilation by a factor of 2 Number of hours passed (x)

Number of bacteria (y)

b 32 768 bacteria c

20 18 16 14 12 10 8 6 4 2

y 2x

x 1

d The initial number of bacteria, in this case 1.

652

y 3 (2x)

x 1 2 3 4 5

y 3 (2x)

17 The coordinates of the y-intercept are (0, a). When a dilation is applied to f (x) = bx, the equation becomes f (x) = a (b)x. If we substitute x = 0 into the transformed equation, we will get y = a.

12 y = 4x

14 a

c y = 3 (2x). The number of bacteria can’t be negative and all values of y = −3 (2x) are negative, so it must be y = 3 (2x).

5 1

5 4 3 2 1 5 4 3 2 1 1 2 3 4 5

3 2

For all values of x, the graph of y = 2 ⋅ 3x is above 3x.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Topic 5 Assessment: Exponents, Radicals, & Exponential Functions 1

Without using any exponent rules, show how to simplify each expression to have a single positive exponent. a

a 5 ⋅ a2

(a3)2

Describe the rule of exponents that is being derived. Write out the rule, including the restriction on the values of a. Same numerator and denominator simpliﬁes to 1

Simplify: a

x3 ⋅ 6x2

−2y7 ⋅ 2y2 ⋅ −2y6

(−2x3)4

Find the value of a and b in the equation:

Fill in each box. Evaluate if possible.

4y10 ⋅ 5y−4

For which value of x does A

(5mp)2 ⋅ mp−2

(4a)−3

Simplify fully: a

6m3n7 ⋅ 9m6n

Simplify each radical fully. a

Simplify, giving your answers with positive exponents: a

SOL

Quotient rule

128

Which is equivalent to A

simplify to B

256

? C

512

1024

in simplest form? B

Topic 5 Assessment: Exponents, Radicals, & Exponential Functions mathspace.co

653

Find the perimeter of the triangle in simpliﬁed radical form. 27

Write with a rational exponent: b

Write in radical form.

a 14

SOL

For each expression:

24x12

Which of the following is equivalent to

A 17

Evaluate.

Which represents this expression in simplest form?

A SOL

Consider the function f (x) = 7x.

Key Features

Complete the table by identifying the key features of the function.

654

Growth or decay Domain Range y-intercept

For each of the following functions: i

Find the y-value of the y-intercept.

State the domain.

iii

State the range.

Sketch the graph of the function.

y = −3(4x)

y = 5(2)x

For each function, describe the transformation from f (x) = 2x, then sketch the graph on the same axes as f (x) = 2x. a

c10d5

g(x) = 3 ⋅ 2x

h(x) = 2x + 3

r(x) = 2x − 5

The local rabbit population is changing according to the exponential function f (t) = 400 (2)t where t is the number of years that have passed. a

Classify the relationship as either growth or decay.

Describe what is happening to the population each year.

Describe what 400 represents in context.

Find f (3) and explain what it represents in context.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

The local rat population triples every year. The graph of the population r (number of rats) 4500 is shown. 4000

Identify the r-intercept and explain what it represents in context.

Identify the range of the graph.

3500

Estimate R(2).

3000

Describe the effect on the population if it was doubling each year instead.

R(t)

2500 2000 1500 1000 500

t (number of years) 0.5

1.5

2.5

Performance task 22

Car commercials on television are always touting “great deals” like “ 0 down at signing” and “0% APR for the first year”. But what does that really mean for how much you would actually have to pay for a new car? Investigate this by finding a car you would like to buy online and identifying the: • Sale price • Cost of any additional features you would like to add • Cost of any additional fees (including sales tax) a

What is the total cost of the car you would like to buy?

After explaining depreciation to your friend, they decide they will wait and save money until they can afford to buy the car without a loan. You think this might be a good idea too, but then you consider inflation. The average rate of inflation in the U.S. is around 3% per year. That is, the average price of goods increases by approximately 3% per year.

If your car followed this inflation rate, how much would it cost after 5 years? 10 years? Is it worth it to wait? Explain.

Topic 5 Assessment: Exponents, Radicals, & Exponential Functions mathspace.co

655

Answers

9 a

Topic 5 Assessment: Exponents, Radicals, & Exponential Functions 1 a a5 ⋅ a2 = (a ⋅ a ⋅ a ⋅ a ⋅ a) ⋅ (a ⋅ a) Write each power

A.EO.4c 10 A A.EO.4b

in expanded form

= a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a

= a7 Write in exponential form

Simplify expanded form

11 C A.EO.4b

Write each power in expanded form

units

A.EO.4c

Simplify fractions

Simplify expanded form

Write in exponential form

c (a ) = a ⋅ a

13 a

Expand the square

14 a i

ii 10

b i

ii 4

c i

expanded form

=a⋅a⋅a⋅a⋅a⋅a

Simplify expanded form

= a6

Write in exponential form

d i d

A.EO.4d

= (a ⋅ a ⋅ a) ⋅ (a ⋅ a ⋅ a) Write each power in

ii 4

Write each power in expanded form

A.EO.4d

Simplify fraction

Simplify expanded form

15 D A.EO.3b 16 A

A.EO.3a

A.EO.3b 0

2 The zero power property: a = 1, where a ≠ 0.

A.EO.3a 3 a 6x

e 25y8

b 8y f

9 8

c 54m n

Key Features Growth or decay

Growth

Domain

All real numbers

Range

y>0

y-intercept

(0, 1)

d u

16x12

A.EO.3b

A.F.2e

4 a = 5, b = 4

18 a i y = −3

A.EO.3b 5 a

ii All real values of x

iii y < 0 iv

y 4 2

A.EO.3b 6 a 20y6

b 25m3

−2

A.EO.3b 7 a 9

b −5

−4

A.EO.4a, A.EO.4b 8 a

A.EO.4c

656

−10 −8 −6 −4 −2

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b i y=5

ii All real values of x

d Vertical translation 5 units downwards

iii y > 0

45 40 35 30 25 20 15 10 5

2 1 −4 −3 −2 −1 −1

y y = 2x x 1

2 3 4

−2 −3

y = 2x − 5

−4 −5 x

−3 −2 −1

2 3 4 5

A.F.2e, A.F.2f

A.F.2f 20 a Growth

19 a Vertical dilation by a factor of 3 (stretch)

b The population doubles each year

c 400 is the initial population of rabbits.

y = 3 ⋅ 2x

d f (3) = 3200. This says that after 3 years there will be 3200 rabbits.

5 4

A.F.2e, A.F.2g

3 2 y = 2x

1 −4 −3 −2 −1 −1

2 3 4

b Vertical translation 3 units upwards 7

y = 2x + 3

Performance task

22 a A nswers will vary. Complete answers should detail all aspects of the price and calculate the correct total.

3 2

x 1 y=2

−1

2 3 4

c Reﬂection across the x-axis and vertical dilation by a factor of 4

d T he population would increase at a slower rate, so there would be fewer rats each year compared to the current model (tripling). A.F.2e, A.F.2g

−4 −3 −2 −1

b r ≥ 250. c R(2) = 2250.

−2

21 a r = 250. This means that initially there were 250 rats.

(compression) y

b A nswer will vary. Students should use the percent decay formula f (x) = a(1 − r)x where a is the market value of their car, r is 0.03 and x is first 5 years and the 10 years. Waiting to buy the car may result in a higher purchase price due to inflation. It may not be worth waiting if the price increases outweigh the benefits of saving up. A.F.2e, A.F.2g, MP1, MP4

3 2

y = 2x

1 −2 −1 −1 −2 −3

x 1

3 4 5 6 y

1 2x 3

−4

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6 Polynomials & Factoring Big ideas • The properties of real numbers can be applied to many types of expressions. • A standard algorithm can be applied to rewrite many different kinds of expressions.

Chapter outline 6.01 6.02 6.03 6.04 6.05 6.06 6.07 6.08

Add and subtract polynomials (A.EO.2) Multiply polynomials (A.EO.2) Divide polynomials by a monomial (A.EO.2) Factor GCF (A.EO.2) Factor by grouping (A.EO.2) Factor trinomials (A.EO.2) Factor using appropriate methods (A.EO.2) Divide polynomials (A.EO.2) Topic 6 Assessment

662 681 707 718 734 748 761 779 790

Bees use hexagonal patterns in their honeycombs, which can be modeled using polynomials!

6. Polynomials & Factoring Topic overview Foundational knowledge Evaluating standards proficiency The skills book contains questions matched to individual standards. It can be used to measure proficiency for each. Students should be proficient in these standards. 8.PFA.1 — The student will represent, simplify, and generate equivalent algebraic expressions in one variable.

A.EO.1 — The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

Big ideas and essential understanding The properties of real numbers can be applied to many types of expressions. 6.01, 6.02 — Operations can be applied to polynomials in much the same way that they can be applied to real numbers.

A standard algorithm can be applied to rewrite many different kinds of expressions. 6.04 — Factoring out the greatest common factor provides the foundation for all other techniques for factoring quadratic expressions.

6.03, 6.08 — Polynomials can be divided using steps similar to those used when dividing real numbers.

6.05 — Factoring by grouping is an application of factoring out the greatest common factor. It is a standard algorithm that can be applied to factoring any factorable quadratic expression. 6.06 — The same standard algorithm can be applied to rewrite any factorable quadratic expression, though other methods may prove more efficient. 6.07 — The structure of an expression can provide information on the most efficient way to rewrite it.

Standards A.EO.2 — The student will perform operations on and factor polynomial expressions in one variable. A.EO.2a — Determine sums and differences of polynomial expressions in one variable, using a variety of strategies, including concrete objects and their related pictorial and symbolic models. 6.01 Add and subtract polynomials

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A.EO.2b — Determine the product of polynomial expressions in one variable, using a variety of strategies, including concrete objects and their related pictorial and symbolic models, the application of the distributive property, and the use of area models. The factors should be limited to five or fewer terms (e.g., (4x + 2) (3x + 5) represents four terms and (x + 1) (2x2 + x + 3) represents five terms). 6.02 Multiply polynomials

A.EO.2c — Factor completely first- and second-degree polynomials in one variable with integral coefficients. After factoring out the greatest common factor (GCF), leading coefficients should have no more than four factors. 6.04 Factor GCF 6.05 Factor by grouping 6.06 Factor trinomials 6.07 Factor using appropriate methods

A.EO.2d — Determine the quotient of polynomials, using a monomial or binomial divisor, or a completely factored divisor. 6.03 Divide polynomials by a monomial 6.08 Divide polynomials A.EO.2e — Represent and demonstrate equality of quadratic expressions in different forms (e.g., concrete, verbal, symbolic, and graphical). 6.06 Factor trinomials 6.07 Factor using appropriate methods

Future connections A2.EO.3 — The student will perform operations on polynomial expressions in two or more variables and factor polynomial expressions in one and two variables.

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6.01 Add and subtract polynomials Subtopic overview Lesson narrative In this lesson, students will build on their knowledge of the properties of operations with integers and apply it to a new context of polynomials. Students will use structure to identify the best approach for applying operations to polynomials in order to combine like terms when adding and subtracting polynomials. Students will use precise vocabulary when identifying types of polynomials. By the end of the lesson, students will be able to create and rewrite polynomial expressions to represent quantities for contextual situations involving combining like terms.

Learning objectives

6.01 Add and subtract polynomials

Students: Page 316

After this lesson, you will be able to… • add and subtract polynomials. • represent sums and differences of polynomials using pictorial models, including algebra tiles. • explain why addition and subtraction of polynomials produce another polynomial.

Add and subtract polynomials Polynomial expressions can be added and subtracted much like real numbers. Key vocabulary  degree (of a polynomial)  leading coefficient binomial Polynomial  leading term  monomial  polynomial The sum or difference of terms which have variables raised to non-negative integer powers and which have 



standard form (of a polynomial) coeﬃcients that are constant



trinomial

Exploration Essential understanding Operations can be applied to polynomials in much the same way that they can be applied to real numbers. Compare

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards. to

Mathematical process goals MPG1 — Mathematical Problem Solving

MPG5 — Mathematical Representations

Teachers can use concrete manipulatives or pictorial Teachers can incorporate this goal into their instruction 1. Create an addition problem like the example provided where the sum of the coeﬃcients is greater than representations (such as algebra tiles) to model the by giving students real-life situations that can be 9. What happens? addition and subtraction of polynomials. They can modeled using polynomial expressions. For instance, 2. Create and solve a subtraction problem using the vertical algorithm. Do polynomials behave the same as then encourage students to create their own symbolic they can present problems related to calculating the numbers when subtracting? representations of these problems. Additionally, teachers area of a complex shape or modeling the growth of can show how different representations of the same a population and guide students on how to apply the mathematical concept (like visual models and algebraic concepts adding and subtracting polynomials to solve of polynomials. We canof use algebra tiles to model sums and differences expressions) convey the same information. these problems. 2 2 The difference (6x + 4x − 5) − (4x − 2x + 3) can be modeled with algebra tiles. Lining up like terms, vertically, we can write: 662

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Content standards A.EO.2 — The student will perform operations on and factor polynomial expressions in one variable.

A.EO.2a — Determine sums and differences of polynomial expressions in one variable, using a variety of strategies, including concrete objects and their related pictorial and symbolic models.

Prior connections 8.PFA.1 — The student will represent, simplify, and generate equivalent algebraic expressions in one variable.

A.EO.1 — The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

Future connections A2.EO.3 — The student will perform operations on polynomial expressions in two or more variables and factor polynomial expressions in one and two variables.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 7 — 3.05 Equivalent algebraic expressions Grade 8 — 2.01 Represent algebraic expressions Grade 8 — 2.02 Simplify expressions and distributive property Algebra 1 — 1.01 Algebraic expressions

Tools You may ﬁnd these tools helpful: • Scientific calculator • Highlighter • Algebra tiles

Student lesson & teacher guide Add and subtract polynomials Students are introduced to definitions that will help them classify polynomials before moving on to an exploration that relates adding and subtracting multi-digit integers to adding and subracting polynomials.

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Students: Page 316

6.01 Add and subtract polynomials After this lesson, you will be able to… • add and subtract polynomials. • represent sums and differences of polynomials using pictorial models, including algebra tiles. • explain why addition and subtraction of polynomials produce another polynomial.

Add and subtract polynomials Polynomial expressions can be added and subtracted much like real numbers. Polynomial The sum or difference of terms which have variables raised to non-negative integer powers and which have coeﬃcients that are constant

Exploration Identifying like terms and combining horizontally or vertically Compare Targeted instructional strategies Students should be given a list of terms such as 3x2, 8x, 3x3, −x, 3, 8x2, −1

and be asked to group any terms together that they believe have features in common, and describe why any terms that did not get grouped are isolated. Once students feel comfortable identifying like terms, present students with a problem including the terms they grouped, such as: 1.

Create an addition problem like the example provided where the sum of the coeﬃcients is greater than (3x2 − x + 3) + (3x3 + 8x − 1 + 8x2) 9. What happens?

and solve a subtraction problemthe using the vertical algorithm. Doand polynomials behaveshown, the samemaking as Ask students2.to Create discuss the difference between original grouping task the problem sure numbers when subtracting? to point out the + in between the polynomials.

Show students that to combine like terms, the polynomials can be arranged horizontally or vertically with like We can use algebra tiles to model sums and differences of polynomials. terms grouped: The difference (6x2 + 4x − 5) − (4x2 − 2x 3+ 3) can2be modeled with algebra tiles. Lining up like terms, vertically, we can 3x + (3x + 8x2) + (−x + 8x) + (3 − 1) write:

can also be written as 3x2

−x + 3

+ (3x3 + 8x2 + 8x − 1) Ask students to try both strategies and discuss advantages and disadvantages to each. Then ask students what would be different in the solving of (3x2 − x + 3) − (3x3 + 8x − 1 + 8x2)

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Compare and connect English language learner support Present students with two polynomials to add and subtract, like (3x2 + 2x + 1) and (x2 − 4x + 3). Have students both add and subtract the polynomials using different methods, such as: • Vertical: Aligning like terms vertically. • Horizontal: Combining like terms in a single horizontal expression. Ask students to identify similarities in the process, such as combining like terms. They should also identify any differences in the approaches, such as ease of visualization or steps involved. They should then connect each method to different scenarios where each could be preferable, including whether the same method is easier for both addition and subtraction.

Color coding like terms Student with disabilities support Students have already learned how to add and subtract algebraic terms by combining any like terms. Adding and subtracting polynomials can be done in the same way. If a student has difficulty differentiating terms like x and x2, help students to recognize that they are not like terms. This can be done by highlighting or otherwise indicating the different types of terms in the polynomials to indicate that they should be combined independently of each other: (5x2 + 2x + 12) − (3x2 − 9x − 8)

When written in this way, it should be more obvious that we want to add the coefficients of x2 and the coefficients of x independently.

Misaligning terms when adding or subtracting vertically Address student misconceptions When using the vertical algorithm to add or subtract polynomials, an easy mistake to make is misaligning the terms, or having two different types of terms in the same column. This is more likely to happen when one or both of the polynomials have absent terms (equivalent to those terms having a coefficient of 0). For example, the sum (2x3 + x2 − 12) + (−3x2 + 11x − 6) may be misaligned in the vertical algorithm as

which can easily lead to an error.

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Add and subtract polynomials Polynomial expressions can be added and subtracted much like real numbers.

Exploration Polynomial The sum or difference of terms which have variables raised to non-negative integer powers and which have Students: Page 316 coeﬃcients that are constant

Exploration Compare

Create an addition problem like the example provided where the sum of the coeﬃcients is greater than 9. What happens?

Create and solve a subtraction problem using the vertical algorithm. Do polynomials behave the same as numbers when subtracting?

We can use algebra tiles to model sums and differences of polynomials. The difference (6x2 + 4x − 5) − (4x2 − 2x + 3) can be modeled with algebra tiles. Lining up like terms, vertically, we can write:

Suggested student grouping: In pairs Students compare a visual of vertical addition of polynomials to adding multi-digit numbers. Students should discover that terms in polynomials are like the place value in multi-digit numbers, but that we don’t carry tens to the next term like we do with place value. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Create an addition problem like the example provided where the sum of the coefficients is greater than 9. What happens? The 316 coefficients the terms that 1have a sum greater than 9 will just be the sum, and we do not carry the MathspaceofVirginia SOL Algebra mathspace.co tens place to the next term. 2. Create and solve a subtraction problem using the vertical algorithm. Do polynomials behave the same as numbers when subtracting? When adding polynomials, the solution is a polynomial. When subtracting polynomials, the solution is a polynomial. However, when subtracting polynomials where the term being subtracted has a larger coefficient, we will write the term’s difference as a negative number and we will not borrow from the next term like we do with multi-digit subtraction. Purposeful questions • What do you notice about the variable term in the solution when each term in the polynomials is added? • Why can we write 2x3 + 4x2 + 0x + 5 as 2x3 + 4x2 + 5? • What are the similarities between ‘carrying over’ in addition and ‘borrowing’ in subtraction for integers? Do these apply to polynomials?

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Polynomial The sum or difference of terms which have variables raised to non-negative integer powers and which have coeﬃcients that are constant

Possible misunderstandings Exploration • Students may ‘carry over’ or ‘borrow’ when adding or subtracting polynomials, in the same way that they would Compare with multi-digit numbers. Let students know these strategies are not necessary because polynomials are a collection of algebraic terms each with their own coefficient, so the place values in one term do not affect any other terms. • Students may add 4x2 with 3x2 by writing 7x4, so remind students that combining like terms such as 4x and 3xtowill not change the variable’s exponent, only the coefficient. This is a good time to review that 4x is equivalent to 4x1, but we do not write singular variables with an exponent. After the exploration, a visual explanation using tiles of adding polynomials in terms of 1. Create an addition problem like the algebra example provided where theand sumsubtracting of the coefficients is greater than combining like terms is stated. Then, students learn more definitions for classifying polynomials and their parts. 9. What happens? 2.

Create and solve a subtraction problem using the vertical algorithm. Do polynomials behave the same as

Students: Pagesnumbers 316–318 when subtracting?

The subtraction can be viewed as the expression: (6x2 + 4x − 5) + (−1) (4x2 − 2x + 3) 6x2 + 4x − 5

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−

−1

4x2 − 2x + 3 x2

−x

Using the opposites of the expression 4x2 − 2x + 3 with the algebra tiles, we get the expression: (6x2 + 4x − 5) + (−4x2 + 2x − 3) Equivalently, distributing the −1:

Creating zero pairs and combining like terms with the algebra tiles, we are left with the expression: 2x2 + 6x − 8 Therefore, the difference between the revenue from the gaming computers can be modeled by 2x2 + 6x − 8. 6x2 + 4x − 5 x2

−1 −1 −1 −1 −1

−4x2 − 2x + 3 −x2

−x2

−x

−1 −1 −1

2x + 6x − 8

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667

−1 −1 −1 −1 −1

−4x2 − 2x + 3 −x2

−x2

−1 −1 −1

2x + 6x − 8 x

−1 −1 −1 −1 −1 −1 −1 −1

Adding and subtracting polynomials creates more polynomials. The following vocabulary is helpful to know when working with polynomials: Standard form (of a polynomial)

Leading coeﬃcient

A way of writing a polynomial expression; The coeﬃcient of the leading term 6.01 Add and subtract polynomials 317 anxn + an−1xn−1 + … + a1x + a0, where n is mathspace.co Monomial a non-negative integer and each ai is a coeﬃcient. Adding and subtracting polynomials creates more polynomials. following vocabulary A The polynomial with one term is helpful to know when working withofpolynomials: Degree a polynomial Binomial The value of the highest exponent on a variable in the polynomial Standard form (of a polynomial)

Leading coeﬃcient A polynomial with two terms

A way of writing a polynomial expression; Leading term anxn + an−1 xn−1 + … + a1x + a0, where n is The term in a polynomial with theahighest exponent a non-negative integer and each i is a coeﬃcient. of the variable Degree of a polynomial

The coeﬃcient of the leading term Trinomial

The value of the highest exponent on a variable in the polynomial

Binomial

Leading term Examples Consider the in polynomial The term a polynomial with the highest exponent

Trinomial

Example 1

of the variable

Students: Page 318

Monomial A polynomial with three terms A polynomial with one term

A polynomial with two terms

A polynomial with three terms 3x − 6 + x2

a Rewrite the expression in standard form.

Create a strategy

Example 1

Recall that the standard form of a polynomial is written with the terms in order from the term with the highest variable exponent to the lowest. We can use the commutative property to change the order of the polynomials. Consider the polynomial

Apply the idea a Rewrite the expression in standard form.

3x − 6 + x2 x2 + 3x − 6

Create a strategy

b State the degree of the polynomial. Recall that the standard form of a polynomial is written with the terms in order from the term with the highest variable exponent to the lowest. We can use the commutative property to change the order of the polynomials.

Apply the idea

2 xApply + 3x the − 6 is a polynomial of degree 2. idea

Reﬂect and check

x2 + 3x − 6

Since the polynomial has 3 terms, it may be called a trinomial. b State the degree of the polynomial.

the the idea cApply Identify quadratic term, the linear term, and the constant term of the polynomial. 668

x2 + 3x − 6 is a polynomial of degree 2. Mathspace Virginia Apply the idea SOL Algebra 1 Teacher Edition Reﬂect and check mathspace.co 2 Reﬂect and check Polynomials of degree 2 are called quadratic polynomials. • Quadratic term: x Since the polynomial • Linear term: 3x has 3 terms, it may be called a trinomial. • Constant term: −6

Leading term

Trinomial

The term in a polynomial with the highest exponent A polynomial with three terms of the variable Adding and subtracting polynomials creates more polynomials. The following vocabulary is helpful to know when

working with polynomials:

Purpose Standard form (of a polynomial) Leading coeﬃcient Example 1 Show students to order the terms to write a polynomial incoeﬃcient standardofform. A wayhow of writing a polynomial expression; The the leading term anxn +the an−1polynomial xn−1 + … + a1x + a0, where n is Consider

Monomial Expected mistakes a non-negative integer and each ai is a coeﬃcient. 3x − 6 + x2 Students may not keep the sign of the term with the correctA term. Color coding terms or underlining terms to polynomial with one term Degree aexpression polynomialin standard form. Rewrite the include atheir signof will remind students which terms are positive and which terms are negative. Binomial

The value of the highest exponent on a variable in

the polynomial Create a students strategy A polynomial with two terms Reﬂecting with Recall that the standard form of a polynomial is written term with the terms in order from the with the highest variable Ask students how they could represent the constant as a term involving x. term What would the coeﬃcient be? Leading term exponent to the lowest. We can use the commutative propertyTrinomial to change the order of the polynomials. What would the exponent on the variable be? The term in a polynomial with the highest exponent A polynomial with three terms

of the variable Apply the idea

Students: Page 318

x2 + 3x − 6

b State the1 degree of the polynomial. Example

Apply the Consider theidea polynomial x2 + 3x − 6 is a polynomial of degree 2.

3x − 6 + x2

a Rewrite the expression in standard form.

Reﬂect and check

Since the polynomial has 3 terms, it may be called a trinomial.

Create a strategy

Recall that the standard form of a polynomial is written with the terms in order from the term with the highest variable exponent thequadratic lowest. We canthe uselinear the commutative change the order of the polynomials. c Identifytothe term, term, and theproperty constanttoterm of the polynomial.

PurposeApply Apply the the idea idea Reﬂect and check 2 x + 3x −of6 the leading 2 Show students that the highest exponent or the exponent term is the degree of polynomials. the polynomial. Polynomials of degree 2 are called quadratic • Quadratic term: x • Linear term: 3x

Expected b• mistakes State the degree Constant term: −6of the polynomial. Students may add the exponents of the terms and incorrectly state that the degree of the polynomial is 3. Apply the idea

Reﬂecting x2 + with 3x − 6students is a polynomial of degree 2. Ask students to consider whether the order of the terms will ever impact the type of polynomial represented. Reﬂect and check Virginia SOL Algebra 1 318 Mathspace mathspace.co Students:Since Page 318 the polynomial has 3 terms, it may be called a trinomial.

c Identify the quadratic term, the linear term, and the constant term of the polynomial.

Apply the idea

Reﬂect and check

• Quadratic term: x2 • Linear term: 3x • Constant term: −6

Polynomials of degree 2 are called quadratic polynomials.

Purpose 318 Mathspace Virginia SOL Algebra 1 Show students how each term in the polynomial is classiﬁed. A term with a variable exponent of 2 is quadratic, mathspace.co a term with a variable exponent of 1 is linear, and a term with a variable exponent of 0 is constant. Expected mistakes Students may incorrectly classify terms. For the constant term, it might help to note for students that the word constant means that something remains the same. Without a variable which can change in value, the number −6 remains the same. Reﬂecting with students Ask students to build their own trinomial using a quadratic, linear, and constant term. Mix up asking for various terms and ask students to build other types of polynomials. 6.01 Add and subtract polynomials mathspace.co

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Students: Page 319

Example 2 Consider the polynomials x3 − 6x + 2 and x2 + 9x + 7. a Find the sum of the two polynomials.

Create a strategy Since we want to ﬁnd the sum of the two polynomials, combine the like terms.

Apply the idea Sum = (x3 − 6x + 2) + (x2 + 9x + 7)

Add the polynomials together

= x3 − 6x + 2 + x2 + 9x + 7

Remove the parentheses (Associative Property)

= x3 + x2 + 3x + 9

Combine the like terms

Reﬂect and2check Example If we want to use the vertical algorithm method, we need to make sure we correctly align the like terms. Consider the polynomials x3 − 6x + 2 and x2 + 9x + 7. a Find the sum of the two polynomials.

Create a strategy Since we want to ﬁnd the sum of the two polynomials, combine the like terms. b Explain why the sum of two polynomials is also a polynomial.

Apply the idea

PurposeApply the idea 3 Sum = (x − 6x + 2) + (x2 + 9x + 7) Add the polynomials together Show students how to3add twois polynomials together. 2 By deﬁnition, a polynomial the sum or difference of terms which have variables raised to non-negative integer Remove the parentheses (Associative Property) = x − 6x + 2 + x + 9x + 7 powers and which that may beCombine real or complex. + x2 +coeﬃcients 3x + 9 the like terms = x3 have

Expected mistakes Adding one polynomial to the other is the same as adding more terms to one polynomial. This doesn’t change the Students may add the leading terms even though they are not like terms. fact that it is a polynomial, so the sum of two polynomials will always be a polynomial. Reﬂect and check

If wewith wantstudents to use the vertical algorithm method, we need to make sure we correctly align the like terms. Reﬂecting Reﬂect and check Ask students to explain why the sum of the polynomials has four terms when the given polynomials had three. We can use the same explanation for why the difference between two polynomials is also a polynomial, and we can extend this explanation to include the sum or difference of any number of polynomials.

Students: Page 319

b Explain why Example 3 the sum of two polynomials is also a polynomial. Simplify theidea expression: Apply the 5xterms + 1) − which (x2 + 7x − 10)variables raised to non-negative integer (3x2 − of By deﬁnition, a polynomial is the sum or difference have powers and which have coeﬃcients that may be real or complex.

Apply Addingthe oneidea polynomial to the other is the same as adding more terms to one polynomial. This doesn’t change the fact (3x that2 − it is the=sum two+ polynomials a polynomial. 5xa+polynomial, 1) − (x2 + 7xso − 10) 3x2 of − 5x 1 − x2 − 7x + will 10 always be Distribute the subtraction Reﬂect and check

= (3x2 − x2) + (−5x − 7x) + (1 + 10)

Group the like terms together

Simplify = 2x − 12x + 11 We can use the same explanation for why the difference between two polynomials is also a polynomial, and we can extend this explanation to include the sum or difference of any number of polynomials.

Example 3 Purpose Simplify the expression: Show students that polynomials are closed under addition. 2

(3x − 5x + 1) − (x + 7x − 10)

6.01 Add and subtract polynomials mathspace.co

Apply the idea (3x2 − 5x + 1) − (x2 + 7x − 10) = 3x2 − 5x + 1 − x2 − 7x + 10 2

670

Distribute the subtraction

= (3x − x ) + (−5x − 7x) + (1 + 10)

Group the like terms together

= 2x2 − 12x + 11

Simplify

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Expected mistakes Students may have diﬃculty articulating why this is true. Relate the addition of polynomials back to the addition of integers from the exploration. Review that integers are closed under addition because the sum of two integers is an integer. Polynomials are closed under addition because a polynomial is the sum of algebraic terms with non-negative Example 2 integer coeﬃcients, and the sum of two polynomials will simply be the sum of the algebraic terms from both Consider the polynomials x3 − 6x + 2 and x2 + 9x + 7. polynomials. a Find the sum of the two polynomials.

Reﬂecting with students Ask advanced Create alearners strategywhether they think it is impossible to find two polynomials whose sum is not a polynomial. students wetwo can sometimes disprove mathematical statements by providing a SinceExplain we want to to ﬁnd the sumthat of the polynomials, combine the like terms. counterexample to a claim. However, it is not possible for the sum of two polynomials to not be a polynomial so the idea no suchApply counterexample will exist in this case. Sum = (x3 − 6x + 2) + (x2 + 9x + 7)

Add the polynomials together

6x + 2zero + x2 + coefficients 9x + 7 Remove the parentheses (Associative Property) = x3 −with Include terms 3

+9 = x + x + 3xstrategies Targeted instructional

use with Example 2

Combine the like terms

When adding subtracting polynomials that do not have all matching terms, we can include these terms with Reﬂector and check zero as Ifthe to better match up the terms. This can be especially useful when using the vertical wecoeﬃcient want to use the vertical algorithm method, we need to make sure we correctly align the like terms. algorithm to add or subtract. Consider that 2x3 + x2 − 12 = 2x3 + x2 + 0x − 12 and

b Explain why the sum of two polynomials is also a polynomial.

−3x2 + 11x − 6 = 0x3 − 3x2 + 11x − 6

Apply the idea

so we can easily align our terms in a vertical algorithm: By deﬁnition, a polynomial is the sum or difference of terms which have variables raised to non-negative integer powers and which have coeﬃcients that may be real or complex. Adding one polynomial to the other is the same as adding more terms to one polynomial. This doesn’t change the fact that it is a polynomial, so the sum of two polynomials will always be a polynomial.

Reﬂect and check We can use the same explanation for why the difference between two polynomials is also a polynomial, and we can this 319–320 explanation to include the sum or difference of any number of polynomials. Students:extend Pages

Example 3 Simplify the expression: (3x2 − 5x + 1) − (x2 + 7x − 10)

Apply the idea (3x2 − 5x + 1) − (x2 + 7x − 10) = 3x2 − 5x + 1 − x2 − 7x + 10 = (3x2 − x2) + (−5x − 7x) + (1 + 10) 2

= 2x − 12x + 11

Distribute the subtraction Group the like terms together Simplify

6.01 Add and subtract polynomials mathspace.co

319

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671

Purpose Show students how to subtract polynomials. Expected mistakes Students may assume that if a term has no coefficient, nothing is actually subtracted. Remind students that the coefficient of a term without one shown is always 1. Reflecting with students Ask students to find the difference of the polynomials using the vertical algorithm and explain which method they prefer.

Distributing the subtraction

use with Example 3

Address student misconceptions When ﬁnding the difference between two polynomials, students may only apply the subtraction to the ﬁrst term of the subtracted polynomial instead of the whole expression. Remind students to use parentheses around the polynomials when they add or subtract them, as this will remind them to distribute the subtraction instead of only applying it to a single term. This is also an easy mistake to make when using the vertical algorithm to subtract two polynomials.

Students: Page 320

Purpose Show students various equivalent forms of the same expression. Expected mistakes Students may not understand how to write equivalent forms of the same expression when they are accustomed to writing an expression in its most simplified form. While the initial expression for the perimeter is not simplified, we can verify that it is equivalent to another expression that represents the perimeter of the yard. Reflecting with students Ask students to write at least two more equivalent expressions for the length of the fence. This might include an expression with required distribution. 672

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Students: Pages 320–321

The expression 12x2 + 6x + 6 is the same as having twelve x2 tiles, six x tiles, and 6 unit (or 1) tiles. 12x2 + 6x + 6 x2

x x x x x x

1 1 1

The expression (8x2 + 4) + (12x2 + 6x + 6) can be represented as: 8x2 + 4

12x2 + 6x + 6

x2 1 1 1

1 x x x x x x

1 1 1

Combine the like terms and count each type of tile. x2

x x x x x x

1 1 1

So, there are twenty x2, six x, and ten unit tiles. The result is 20x2 + 6x + 10.

Idea summary

6.01 Add and subtract polynomials mathspace.co

673

x2 x

1 1 x12 1

x+2

x2 x2

1 1 x x x xx2 x x x12 1 1 1

x x x x x x

x2 1 1 1

1 1 1

2 2 Combine the like(8x terms count each The expression + 4)and + (12x + 6x + 6)type canofbetile. represented as:

8x2 + 4

+ x2

x2 x

x2 2

x 1 1 1 1 x2

x x x x x x x2

x2 1 1 1

1 1 1 1 1 1 x2

x x x x x x

x x2

x2 2

Combine the like terms and count each type of tile. x2

x2 2

12x2 + 6x + 6 x2

x x2

1 1 1

So, there are twenty x2, six x, and ten unit tiles. The result is 20x2 + 6x + 10.

Idea summary

Purpose We add polynomials by combining like terms. We subtract polynomials by adding the negative terms. Show students how to add polynomials using algebra tiles. x2 x2 x2 x2 x2 1 1 1 1 Expected mistakes 6.01 Add and subtract polynomials 321 x x x x x x 1 1 1 Students may forget to combine like terms, resulting in a polynomial with unnecessary terms. Remind students mathspace.co 1 1 1 to look for and combine terms that have the same variable to the same power.

So, there are twenty x2, six x, and ten unit tiles.

Students: Page 321 2

The result is 20x + 6x + 10.

Idea summary We add polynomials by combining like terms. We subtract polynomials by adding the negative terms.

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321

Practice Students: Pages 322–325

What do you remember? 1

Choose the best word from this set: term, coefficient, variable, constant to describe:

674

The number 3 in the term 3x

The letter x in the term 6x

4x in the expression 4x + 7

The number 7 in the expression 5x + 7

The number −5 in 4 − 5x

The letter u in the term −12u

The number 8 in 6z + 8

−3y in 2 − 3y + 4z

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Describe each pair of terms as like or unlike. a

10p and 5p

3 and y

5n2 and 8n

5p and 5

10a and −9a

2ab and 6ba

15p and 15q

8 and 8z

8z and −8z

12b and −21b

13xy and 14yx

i 3

7m and 7m

Donna earns $15 per hour of work and is paid double for every hour worked on the weekend. At the end of a week, Donna calculates her pay for that week to be 15x + 30y dollars. a

What variable represents the number of hours Donna worked during the weekdays?

What variable represents the number of hours Donna worked during the weekends?

In the expression 3x2 + 5x + 7x2 − 2x, which terms are like terms? Explain your answer.

Write a simplified expression for each set of algebra tiles. a

b −x

−x

−x2

1 2

−1

1 x

−x

−1

Compare and contrast adding two-digit integers and adding binomials.

Let’s practice 7

Simplify: a

2a + 5a

10x + 6x

4b + 3b

12y − 3y

3c + 4c + 7c

15x − 6x − 2x

19b − 12b − 6b

−3n + 6n + 3n

8x − 3x + 7x

x+9+7

9x + 4x

12p − 9p

4m − 4m

5y + 6y

11 + y

17r

72r

m 2u2 + 9u2 8

State whether each expression is equal to 11y. a

9 + y + 10y − 9

6y − 5y

9y + 3y − 1

10y + 1

State whether each expression is equal to 8r + 9? a

9 + 8r

9r + 8

9r + 8 − r + 1

10r + 2 − 2r + 7

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Complete each step by writing the expression modeled by the algebra tiles. 1 x

x 2

x 3

1 −x

+ 1

−x

Consider the expression (2x2 − 4x + 7) − (5x2 + 3x − 2). Explain how to represent both polynomials with algebra tiles, perform the subtraction, and represent the resulting polynomial using the tiles.

Simplify: a c

(5x + 1) + (2x − 3) 3

(−6x − 2) − (4x − 3x )

(x + 1) − (2x − 1)

(−2x2 − x − 5) + (6x2 − 9x − 8)

(2x2 − 3x − 9) − (−6x2 + 6x − 2)

(9x3 − 3x + 2) − (−6x3 − 5x2 − 5x)

(2x2 + 1) + (x2 + 3x) + (4x − 5)

(−2x + 7) − (7x2 + 8x − 5) + 7x

(⬚x2 − 5x − 5) − (⬚x + 3) = 4x2 − 2x − 8

(4x + 1) − (⬚x⬚ + ⬚) = 0

Simplify: a

(−0.2 − 9.5x − 1.4x2) + (2.6 − 7.8x + 7.7x2)

(7.6x2 − 0.5x + 7.3) − (1.4x2 − 3.4x − 9.6)

(−8x6 + 9x4 − 11x3 − 13) + (10x6 + 5x5 − 3x3 − 2x2)

Fill in the blanks to make each equation true. a c

(x3 − x + 7) + (⬚x3 + ⬚x + 3) = 10

(⬚x⬚) + (2x3 − 3x + 1) = 2x3 + x + 1

If A (x) = −2x2 − 3, B (x) = −6x + 3 and C (x) = 6x2 + 2x, form a simpliﬁed expression for A (x) + B (x) + C (x).

If A (x) = 5x2 + 2, B (x) = −3x + 3 and C (x) = −2x2 + 7x, form a simpliﬁed expression for A (x) − B (x) − C (x).

If A (x) = −6x2 − 6, B (x) = 4x − 7 and C (x) = x2 − 5x, form a simpliﬁed expression for B (x) − A (x).

A rectangle with the given dimensions is to have a right triangle cut out from one corner. Write and simplify a polynomial sum or difference to model each of the following: a

An expression for the length represented by y.

An expression for the perimeter of the rectangle before the triangle has been removed.

If the area of the rectangle is 5x2 + 16x + 3 and the area of the triangle is x2 + 3x, determine the area leftover once the triangle has been removed.

2x 5x + 1 y x+3

A polynomial of degree m and a polynomial of degree n, with m ≥ n, are added together. Find the highest possible degree of the result.

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The revenue generated by Leticia’s seafood restaurant is modeled by R(m) = −3.2m2 + 10.9m + 990, and the profit of her restaurant is modeled by P(m) = 2.3m2 − 29.3m + 830, where m is the number of meals produced. Find the polynomial that models the costs of Leticia’s restaurant.

Joanita’s hat manufacturing business sells its hats exclusively through two retailers. The profit generated through selling at Just Stuff is modeled by A (q) = −0.5q2 + 22.5q + 510, and the profit generated through selling at Glorious Gifts is modeled by B (q) = 2.9q2 + 36.5q + 70, where q is the number of hats sold. Form an expression for the polynomial that models Joanita’s total profit.

Find a polynomial that represents the perimeter of a square with side length

A rectangle has a length of l = 3x + 5 and a perimeter of P = 12x. Find the width of the rectangle.

A piece of paper with dimensions 8.5 inches by 11 inches will have a square with length x cut out from each corner to form a box. Write and simplify a polynomial sum that can be used to represent the perimeter of the shape formed once the corners have been removed.

Maureen’s paddock has dimensions as shown in the diagram. All dimensions are in meters. a

Write a fully simpliﬁed expression for the perimeter of the paddock.

On Mondays, Maureen runs the entire perimeter of the paddock twice. Write a fully simpliﬁed expression for the distance she runs each Monday.

8 + 7x 18 − 3x 6x 6x − 2 12 − 7x 7x 6x

On Saturdays, Maureen goes for a much shorter run. The route is shown in the diagram by the thick, dark line. She runs from the Start to the End once.

Determine how much further she runs on Mondays than on Saturdays.

8 + 7x

End

18 − 3x 6x 6x − 2 12 − 7x 7x 6x

Start

Let’s extend our thinking 26

Simplify: a

8x + 6y − 2y − 4x

11m + 8n + 14m

9xy + 12yx

6p + 8q − 6p

2.5x + 9y − 5x + 10y

7a + 11a − 9b + b

8x − 7y − 6z + 10z

13m − 2.2n − 8m + 1.5n

−3s + 4t − 6t + 9s

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Simplify: a

(8a2b2 − 7a2b + 8ab − 11) + (−3a2b2 + 6a2b + 5ab)

(4x2y − 9xy2 + 4) + (−7x2y + 4xy2 + 8)

(3x2y2 − 2xy2 + 4y2) + (−9x2y2 + 3xy2 − 8y2)

Tom simplified the expression 7x + 6p − 4x + 2 and found it to be 3x + 6p + 2. Xanthe simplified the same expression and found it to be 11x + 6p + 2. Explain who is correct and why.

Show that the expression 12ab + 7c − 8ab − 9c is equivalent to the expression −6c + 2ab + 4c + 6ab − 4ba.

Determine whether each statement is always, sometimes, or never true. Explain your reasoning with examples. a

Two polynomials added together will result in a polynomial.

A linear function is a polynomial.

A polynomial added to a non-polynomial will result in a polynomial.

A non-polynomial added to a non-polynomial will result in a polynomial.

Is it ever possible that 8m + 5n = 13mn?

Explain how two trinomials can be added together to produce a binomial.

Given that (ax + 5) + (4x2 − 4x + 4) + (3x + 2) = 4x2 + 4x + 11 for all values of x, solve for a.

Consider [ax2 + (b − 5) x − 1] + [x2 − 5x + 2] = 2x2 + 2x + 1. a

Find the value of a.

Find the value of b.

Consider the following work: (−3x3 + 7x2 − x − 4) − (−2x3 + ax2 + bx + 6) = cx3 + 6x2 + 4x − 10 (−3 − 2) x3 + (7 − a) x2 + − (1 + b) x + (−4 − 6) = cx3 + 6x2 + 4x − 10 −5x3 + (7 − a) x2 + − (1 + b) x + (−10) = cx3 + 6x2 + 4x − 10 7−a=6

−(1 + b) = 4

a=1

−1 − b = 4 b = −3

Therefore, a = 1, b = −3, and c = −5. a

678

Identify and explain any mistakes.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Give the correct values for a, b, and c.

Answers

1 x

x −x −x

−x −x

What do you remember? 1 a Coeﬃcient

b Variable

c Term

d Constant

e Coeﬃcient

g Constant

h Term

2 a Like e Like i

Unlike

Variable

c Unlike

d Unlike

g Unlike

h Unlike

k Like

4 In the expression 3x2 + 5x + 7x2 − 2x, the terms 3x2 and 7x2 are like terms because they both have the variable x raised to the power of 2. The terms 5x and −2x are also like terms because they both have the variable x raised to the power of 1. 2

5 a 4x − x + 3

4x −2x + 3

2x + 3

(2x2 4x 7)

b y

11 1. Represent the polynomials with algebra tiles:

b Unlike

3 a x

(4x + 1) + (−2x + 2)

6.01 Add and subtract polynomials

b x +x

(5x2 3x 2)

x x x x

(2x2 4x 7) x2 1

5x2 3x 2

x x x x

x2 1

5x2 3x 2 x x x x

x2 1

When adding whole numbers we carry over excess from the sum if it goes into the next power of ten. When adding polynomial terms, there is never excess that goes into the next term. Also, when adding two-digit numbers, they always have the same place values (units and tens). But when adding binomials, they don’t necessarily have like terms (e.g. x3 + 2 and x2 + 4x).

1 1

4. Combine the like terms and count. 3x2 7x 9

x x x x x x x

12 a 7x − 2

d 4x2 − 10x − 13

i b 16x

c 7b

d 9y

e 14c

g b

h 6n

12x

x + 16

k 13x

m 11u

n 0

8 a Yes

b No f

b No

Yes

e Yes

3x + 7x − 4

2x3 − 3x2 + x − 1

h 15x3 + 5x2 + 2x + 2 j

−7x2 − 3x + 12

13 a 4x2 + 8x − 9 b 6.3x2 − 17.3x + 2.4 c 6.2x2 + 2.9x + 16.9 d 2x6 + 5x5 + 9x4 − 14x3 − 2x2 − 13

c Yes

d No

9 a Yes f

b −x + 2

c −10x3 + 3x2 − 2

g 8x2 − 9x − 7 Let’s practice

x x x

Differences:

e No

x x x

(2x2 4x 7)

When adding polynomials we combine like terms (terms with the same power of the variables). When adding whole numbers we combine digits with the same place-value (same power of ten). Also, a two-digit number has two place values to add and a binomial has two terms to add.

2. Distribute the negative sign to the second polynomial.

Similarities:

3. Remove two x2 tiles from each set since x2 − x2 = 0.

6 Answers will vary.

7 a 7a

14 a -1 and 1

b 4 and -3

c 4 and 1

d 4, 1 and 1

c No

d No

15 4x − 4x 16 7x2 − 4x − 1 17 6x2 + 4x − 1 18 a (5x + 1) − (2x) = 3x + 1 b (5x + 1) + (5x + 1) + (x + 3) + (x + 3) = 12x + 8 c (5x2 + 16x + 3) − (x2 + 3x) = 4x2 + 13x + 3

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19 m

d S ometimes true. If the terms that don’t fit the polynomial definition are equal, but with opposite signs, they will eliminate each other and the result will be a polynomial.

20 −5.5m + 40.2m + 160 21 2.4q2 + 59q + 580 22 x2 + 24x units

23 w = 3x − 5

31 Yes, if m = 1 and n = 1 the expression is true.

24 (8.5 − 2x) + x + x + (8.5 − 2x) + x + x + (11 − 2x) + x + x + (11 − 2x) + x + x = 39 25 a (22x + 36) m

b (44x + 72) m

c (35x + 36) m

26 a 4x + 4y

d 25m + 8n

e 21xy

g −2.5x + 19y

h 18a − 8b

8x − 7y + 4z

k 5m − 0.7n 2 2

j l

6s − 2t

27 a 5a b − a b + 13ab − 11 b −5xy2 − 3x2y + 12 c −6x2y2 + xy2 − 4y2 28 Tom is correct as the sign on the left of the coefficient tells us whether to add or subtract. −4x means we subtract 4x from 7x, not add. 29 12ab + 7c − 8ab − 9c = 4ab − 2c −6c + 2ab + 4c + 6ab − 4ba = 4ab − 2c Therefore 12ab + 7c − 8ab − 9c = −6c + 2ab + 4c + 6ab − 4ba 30 a A lways true. A polynomial is a collection of terms in the form mxn where m is a real number and n is a non-negative integer. When adding polynomials together, terms are either combined if they have the same value of n or left alone if not. That means all terms in the resulting polynomial will also be a collection of terms in the form mxn and will also be a polynomial. b A lways true. A linear function can only have a linear term in the form mx and/or a constant term in the form b, both of which are terms in a polynomial. c N ever true. At least one term in the non-polynomial will not combine with any of the terms in the polynomial and such terms will be left in the resulting sum unchanged, which makes the resulting sum a nonpolynomial.

For example:

680

32 If the three terms in each trinomial have the same exponents, so that there are three pairs of like terms, and one of the pairs of like terms have the same coefficient but with opposite signs, then those two terms will eliminate each other when added. For example, x2 + 2x + 1 and x2 − 2x + 4 would add together to give 2x2 + 5.

Let’s extend our thinking

For example:

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

33 a = 5 34 a a = 1

b b = 12

35 a W hen collecting coefficients for the x3 term, they forgot to subtract -2. When solving for b, they subtracted 1 instead of adding 1 to both sides. b a = 1, b = −5, and c = −1

6.02 Multiply polynomials Subtopic overview Lesson narrative In this lesson, students will build on their prior knowledge of the distributive property as well as area models to multiply two or more polynomials. Students will justify why polynomials are closed under multiplication using precise mathematical language and definitions. For binomials, students will use the structure of the expressions to recognize patterns and create generalizable formulas and help them evaluate products more efficiently. By the end of the lesson, students will be able to create and rewrite polynomial expressions to represent quantities for contextual situations involving multiplying polynomials. This skill will also prepare students for factoring quadratic trinomials in the next topic.

Learning objectives

6.02 Multiply polynomials

Students: Page 326

After this lesson, you will be able to… • multiply a monomial and a polynomial. • multiply polynomials. • represent polynomial multiplication using an area model. • explain why multiplication of polynomials produces another polynomial.

Multiply polynomials

Key vocabulary 

Exploration

difference of two squares



polynomial identity

Complete the area models for multiplication shown: 7

7x3

Essential understanding 3

3x Operations can be applied to polynomials in much the same way that they can be applied to real numbers.

3x2(7x3 + 9)

3(7 + 9) Complete the new area models for multiplication: Standards

This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards. 20

Mathematical process goals 10

MPG4 — Mathematical Connections

MPG3 — Mathematical Reasoning −3

−3 help students make mathematical Teachers can connections by relating the multiplication of polynomials to real-life situations, (x − 3)such (2x +as 1) calculating the volume of complex shapes or predicting population growth. They can also link current lessons to prior knowledge of algebraic expressions, showing how these concepts buildrelates upon each other. integers. Make a conjecture about how multiplying polynomials to multiplying

Teachers can foster mathematical reasoning by asking students to justify their steps when multiplying monomials and polynomials. They can challenge to (10 − 3) (20students + 1) analyze and evaluate different methods of multiplication, 1. theWhat doarea the area models have in common? including use of models, to determine the most What’s different about the area models? efficient2.approach. 3.

Consider the garden plot: 4x + 1

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MPG5 — Mathematical Representations Teachers can incorporate this goal by using concrete manipulatives (like algebra tiles) and pictorial representations (like area models) to demonstrate the multiplication of polynomials. They can also encourage students to use these different representations themselves, showing them how to visualize the process of polynomial multiplication and understand the relationships between the coefficients and exponents.

Content standards A.EO.2 — The student will perform operations on and factor polynomial expressions in one variable. A.EO.2b — Determine the product of polynomial expressions in one variable, using a variety of strategies, including concrete objects and their related pictorial and symbolic models, the application of the distributive property, and the use of area models. The factors should be limited to ﬁve or fewer terms (e.g., (4x + 2) (3x + 5) represents four terms and (x + 1) (2x2 + x + 3) represents ﬁve terms).

Prior connections 8.PFA.1 — The student will represent, simplify, and generate equivalent algebraic expressions in one variable.

A.EO.1 — The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

Future connections A2.EO.3 — The student will perform operations on polynomial expressions in two or more variables and factor polynomial expressions in one and two variables.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 8 — 2.02 Simplify expressions and distributive property Algebra 1 — 6.01 Add and subtract polynomials

Tools You may ﬁnd these tools helpful: • Scientific calculator • Graph paper • Highlighter

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Student lesson & teacher guide Multiply polynomials Students start with an exploration. In the exploration, students relate calculating the area of rectangles to calculating the area of rectangles using monomials and binomials in an effort to extend their prior knowledge of these properties to monomials and binomials.

Students: Page 326

6.02 Multiply polynomials After this lesson, you will be able to… • multiply a monomial and a polynomial. • multiply polynomials. • represent polynomial multiplication using an area model. • explain why multiplication of polynomials produces another polynomial.

Multiply polynomials Adjustable pre-drawn box method template Exploration Student with disabilities support

the area models for multiplication shown: To supportComplete students using the box method to multiply polynomials, provide students with an adjustable grid 7x3 be given 9 layout. Ideally, the template can7be done 9on grid paper and extra space can around the template so students can add lines or draw their own boxes on the grid paper. 3 3x2 A sample box method template is shown: 3x2(7x3 + 9)

3(7 + 9) Complete the new area models for multiplication:

2x 20

10 −3

−3

(10 − 3) (20 + 1)

(x − 3) (2x + 1)

1. also What do the from area models have in common? version of the template with outlines colored for the terms Students may benefit having a highlighted 2. and What’s different about area for models? of the problem a different colortheused the results of the multiplication. 3.

Make a conjecture about how multiplying polynomials relates to multiplying integers.

Consider the garden plot: 4x + 1

3x − 2

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Connecting the distributive property to polynomial multiplication Targeted instructional strategies To connect the distributive property to multiplying polynomials, display the following problems and have students distribute independently: 3x(x − 2) −5(x − 2) Write the two resulting expressions side-by-side and show adding and combining like terms. (3x2 − 6x) + (−5x + 10) 3x2 + (−6x − 5x) + 10 3x2 − 11x + 10 Students should then predict how to simplify (3x − 5) (x − 2) using the two original distribution problems.

Avoid missing terms with color coding or area models Address student misconceptions When multiplying polynomials where one factor has two or more terms and the other has three or more terms, students may miss terms. Consider encouraging students to write extra steps, use an area model, or annotate the question with different colors to help ensure no terms are missed. (3x2 + 4x − 5) (2x − 7) = 3x2(2x) + 4x(2x) − 5(2x) + 3x2(−7) + 4x(−7) − 5 − (7) Like terms = 6x3 + 8x2 − 10x − 21x2 − 28x + 35 Like terms = 6x3 − 13x2 − 38x + 35 3x2

−5

6x3

8x2

−10x

−7

−21x2

−28x

It is very common to see students simplify (a + b)2 to a2 + b2 and (a − b)2 to a2 − b2, which is incorrect. Challenge this misconception by having students show the work of applying the distributive property or giving an example like (10 − 4)2 = 62 = 36, but 102 − 42 = 100 − 16 = 84 to show it numerically.

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6.02 Multiply polynomials After this lesson, you will be able to… • multiply a monomial and a polynomial.

• multiply polynomials. Exploration

• represent polynomial multiplication using an area model.

explain why multiplication of polynomials produces another polynomial. Students: Page •326

Multiply polynomials Exploration Complete the area models for multiplication shown: 7

7x3

3x2

3x2(7x3 + 9)

3(7 + 9) Complete the new area models for multiplication:

2x 20

10 −3

−3

(10 − 3) (20 + 1)

(x − 3) (2x + 1)

What do the area models have in common?

What’s different about the area models?

Make a conjecture about how multiplying polynomials relates to multiplying integers.

Consider the garden plot: 4x + 1

Ideal student responses

3x − 2

Suggested student grouping: In pairs Students use area models to apply the distributive property and the product property for exponents. Students make connections between multiplying integers and polynomials. These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What do the area models have in common? The side lengths of the area models each have labels. The number of terms is the same as the number of side lengths. 2. What’s different about the area models? 326 Mathspace Virginia SOL Algebra 1 mathspace.co The first set of area models uses distribution that looks familiar and could be solved without the models. However, more multiplication is required with the second set of area models, so it’s harder to distribute in a familiar way. 3. Make a conjecture about how multiplying polynomials relates to multiplying integers. Multiplying polynomials is similar to multiplying integers, because the integer coefficient of each term is multiplied the same way an integer is multiplied. The difference between multiplying polynomials and integers is that the polynomials now have variables, which require properties of exponents. Purposeful questions • Can you find the product without using the area model at all? How does the model help us? • How can we find the area of one specific box in the model? • How would you multiply one term with another term? • Are the second set of area models more challenging or easier to you? Why is that?

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685

7x3

3x2

3x2(7x3 + 9)

3(7 + 9) Complete the new area models for multiplication: Possible misunderstandings

• Students might perform the distributive property without the model and then get confused when they are 20 presented with binomial × binomial. It1 is important to use the model with the distributive property so that x 10 is useful for any multiplication of polynomials. students see why a model −3

−3

After the exploration, students learn that area models can be used with polynomials that have a variety of number of terms. Area models are connected to algebra tiles and are used with contextual problems with visuals. Students − 3) (20 + 1) (x − 3) (2x + 1) note that polynomials are closed(10 under multiplication, similar to integers, and what to expect with the product of 1. What do the area models have in common? polynomials. What’s different about the area models?

Students: Pages 326–327 3. Make a conjecture about how multiplying polynomials relates to multiplying integers. Consider the garden plot: 4x + 1

3x − 2

The length of a rectangular garden plot is 4x + 1 feet. The width of the plot is 3x − 2 feet. The area can be represented as the product (4x + 1) (3x − 2). We can use models to simplify the product as a polynomial. 326

Mathspace Virginia SOL Algebra 1 4x + 1 mathspace.co

−x −x

3x − 2

An algebra tiles model

12x2

−2

−8x

−2

A box/area model

Notice the algebra tiles model shows each individual tile, but the box model combines some like terms together. Both models show the product of (4x + 1) (3x − 2) = 12x2 + 3x − 8x − 2. Which we can simplify by combining like terms to 12x2 − 5x − 2. a

The distributive property can be used to multiply two polynomials: Area models help us visualize the different terms from the distributive property. They can help us organize the multiplication of polynomials, so we don’t miss any terms. Then, we can combine like terms to get the simplest polynomial.

(a + b) (c + d) = ac + ad + bc + bd

The product of two polynomials will always result in a new polynomial where • The degree of the new polynomial will be the sum of the degrees of the multiplied polynomials. • The number of terms may vary from the original polynomials depending on how like terms are combined.

Example 1 686

Multiply 3x (2x2 − 5x + 4). Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Apply the idea

We can use the distributive property to get the product of the monomial 3x and the trinomial 2x2 − 5x + 4. 2

like terms to get the simplest polynomial.

Examples The product of two polynomials will always result in a new polynomial where The degree Students:•Page 327 of the new polynomial will be the sum of the degrees of the multiplied polynomials.

• The number of terms may vary from the original polynomials depending on how like terms are combined.

Example 1 Multiply 3x (2x2 − 5x + 4).

Apply the idea We can use the distributive property to get the product of the monomial 3x and the trinomial 2x2 − 5x + 4. 3x (2x2 − 5x + 4) = 3x(2x2) + 3x(−5x) + 3x(4) = 6x3 − 15x2 + 12x Since there are no more like terms and the expression is already in standard form, the ﬁnal answer is 6x3 − 15x2 + 12x.

Reﬂect and check 6x3 − 15x2 + 12x is considered a polynomial of degree 3, since 3 is the value of the highest exponent on a variable in the polynomial.

Purpose Show students how to multiply a monomial and a polynomial.

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327

Expected mistakes Students may multiply the coeﬃcients only. Remind students of the product property of exponents. Reﬂecting with students Ask students how they would explain aloud how to multiply the monomial and polynomial to a classmate.

Break down the distribution process

use with Example 1

Targeted instructional strategies Demonstrate and explain steps as they are applied. Some students may beneﬁt from seeing it in more steps with connections to prior learning. For example, −2x2 (4x3 − 2x + 3) = (−2x2) (4x3) − (−2x2) (2x) + (−2x2) (3) = −2 ⋅ 4x2+3 + 2 ⋅ 2x2+1 − 2 ⋅ 3x2 5

Distribute −2x2 Multiply coeﬃcients and add exponents

= −8x + 4x − 6x

Simplify operations

Students: Page 328

Example 2 Consider the polynomials 7y + 2 and 4y − 5. a Find the product of the two polynomials.

Create a strategy Multiply the polynomials using distribution.

Apply the idea We can create an area model to multiply the two polynomials: 7y

Combining each of the terms, we get:

28y2

−5

−35y

−10

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687

Consider the polynomials 7y + 2 and 4y − 5. a Find the product of the two polynomials.

Create a strategy Multiply the polynomials using distribution.

Apply the idea We can create an area model to multiply the two polynomials:

Example 2

Consider the polynomials 7y + 2 and 4y − 5. a Find the product of the two polynomials.

Create a strategy Multiply the polynomials using distribution.

28y2

−5

−35y

−10

Apply the idea

Combining each of the terms, we get: We can create an area model to multiply the2 two polynomials: 28y − 35y + 8y − 10 = 28y2 − 27y − 10 7y

Reﬂect and check

28y2

2 8y

We can also use the distributive property to get the product of 7y + 2 and 4y − 5. (7y + 2) (4y − 5) = 4y(7y + 2) − 5(7y + 2)

Distributive property

= 4y(7y) + 4y(2) − 5(7y) −35y −5 − 5(2)

Distributive property −10

= 28y2 + 8y − 35y – 10

Distributive property

Combine like terms = 28y2 − 27y – 10 Combining each of the terms, we get: Since the expression is already in standard form, the final answer is 28y2 − 27y – 10. 28y2 − 35y + 8y − 10 = 28y2 − 27y − 10 b Explain why the product of two polynomials is also a polynomial.

Reﬂect and check

PurposeApply We canthe alsoidea use the distributive property to get the product of 7y + 2 and 4y − 5. Show students how to multiply two binomials. A polynomial terms in −the form mxn where m isDistributive a real number and n is a non-negative integer. (7yis+a2)collection (4y − 5) =of 4y(7y + 2) 5(7y + 2) property = 4y(7y) + 4y(2) − 5(7y) − 5(2) Distributive property Reﬂecting with students non-negative integer exponents. together results in a sum of such products, by + 8y −multiplying 35y – 10 polynomials Distributive property = 28y2Since Ask students how what size area model to draw for multiplying two binomials. deﬁnition thethey resultknow is a polynomial expression.

We know that the product of two algebraic terms with non-negative integer exponents is an algebraic term with = 28y2 − 27y – 10

Combine like terms

the expression is already in standard form, the final answer is 28y2 − 27y – 10. Students:Since Page 328

Reﬂect and check

We can use this explanation to think about what happens when we perform multiple operations on polynomials. What happens weproduct add twoofpolynomials and multiply result by another polynomial? What if we multiply three b Explain whyifthe two polynomials is also athis polynomial. polynomials? Is our result still a polynomial?

Apply the idea A polynomial is a collection of terms in the form mxn where m is a real number and n is a non-negative integer. We know that the product of two algebraic terms with non-negative integer exponents is an algebraic term with non-negative integer exponents. Since multiplying polynomials together results in a sum of such products, by deﬁnition the result is a polynomial expression. 328

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Reﬂect and check

We can use this explanation to think about what happens when we perform multiple operations on polynomials. What happens if we add two polynomials and multiply this result by another polynomial? What if we multiply three polynomials? Is our result still a polynomial?

Purpose Show students that polynomials are closed under multiplication. 328

Mathspace

Virginia SOL Algebra 1

mathspace.co Expected mistakes Students might not be able to articulate why the statement is true. Refer to the exploration with the integer multiplication with area models. We know that the product of two integers is an integer, and that it is similar for the product of two polynomials.

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Reﬂecting with students Introduce advanced learners to the term “closure” of polynomials. We say that polynomials are closed under addition, subtraction, and multiplication because the sum, difference, or product of polynomials will also be a polynomial. Similarly, rational numbers are closed under addition, multiplication, and addition because the sum, difference, or product of rational numbers will also be a rational number.

Students: Page 329 Example 3 Inhar is designing a cubic storage container with odd-numbered side lengths. They decide to let 2x + 1 yards represent the length of each side. a Conﬁrm that the side length will always be odd.

Create a strategy Analyze the given side length.

Apply the idea Since 2x + 1 represents the side lengths of the container, we can note that twice any number is always an even product. If we add an odd number like 1 to that, the side length 2x + 1 will always be odd.

b Write an expression for the surface area of the storage container.

Purpose Create a strategy Show students why 2x + 1 must always be an odd number.

Draw and label a diagram of the storage container ﬁrst, then use it to calculate the surface area.

Expected mistakes Example 3 Apply the idea Students may have diﬃculty articulating why the statement is true. Reviewing why 2x + 1 will always be odd Inhar is designing a cubic storage container with odd-numbered side lengths. They decide to let 2x + 1 yards using examples of x is suﬃcient. represent the length of each side.

Reﬂecting with students 2x 2x 4x2 a Conﬁrm that2xthe side length will always be odd. Ask advanced learners whether 2x − 1, where x is a whole number, could represent odd side lengths. Createwhen a strategy (No, because x 1= 0, the side length would be negative.) What types could we use instead of 1 1 of2xnumbers Analyze the whole numbers sogiven that side thislength. expression would be a valid side length? (Natural numbers) 2x 1 2x 1 Extend the problem further by asking what expression could represent even-numbered side lengths, making Apply the sure to define theidea variable well example, where n isArea a natural number). We can calculate the area of (for one face on the 2n, storage of one face: 4x2 + 2x + 2x + 1 = 4x2 + 4x + 1 square

Since 2x + 1 represents the side lengths of the container, we can note that twice any number is always an even container, then multiply the polynomial expression by 6 yards If we add an odd number like 1 to that, the side length 2x + 1 will always be odd. faces on the cube-shaped container. Students:product. Page 329 Surface area of container: 6(4x2 + 4x + 1) = 24x2 + 24x + 6 square yards b Write an expression for the surface area of the storage container. c Write an expression for the volume of the storage container.

Create a strategy

Draw and a diagram of the storage container ﬁrst, then use it to calculate the surface area. Create a label strategy Use the formula for the volume of a cube to calculate the volume of the storage container.

Apply the idea

Apply the idea Since the formula for the volume of a cube is V = l ⋅ w ⋅ h, we can calculate the volume of the storage container as shown: V = (2x2x + 1) (2x + 1) (2x + 1)

2 Substitute expressions for h2x 4xand 2x l, w,

= (2x + 1) [(4x + 2x + 2x + 1)]

Distributive property

= (2x +11) (4x2 + 4x + 1)

Combine like terms

= (8x + 8x + 2x) + (4x + 4x + 1)

Distributive property

= 8x3 + 12x2 + 6x + 1 cubic yards

Combine like terms

We can calculate the area of one face on the storage container, then multiply the polynomial expression by 6 faces on the cube-shaped container.

Area of one face: 4x2 + 2x + 2x + 1 = 4x2 + 4x + 1 square yards Surface area of container: 6.02 Multiply polynomials 329 6(4x2 + 4x + 1) = 24x2 + 24x + 6 squaremathspace.co 6.02yards Multiply polynomials mathspace.co

c Write an expression for the volume of the storage container.

689

b Write an expression for the surface area of the storage container.

Example 3

Create a strategy

Inhar is designing a cubic storage container with odd-numbered side lengths. They decide to let 2x + 1 yards Draw and label a diagram of the storage container ﬁrst, then use it to calculate the surface area. represent the length of each side. a Conﬁrm that the side length will always be odd.

Apply the idea

Create a strategy Analyze the given side length. 2x

4x2

Apply the idea Since 2x + 1 represents the side lengths of the container, we can note that twice 1 is always an even 1 1 any 2x number product. If we add an odd number like 1 to that, the side length 2x + 1 will always be odd. 2x

b Write an expression forof the surface area the storage container. We can calculate the area one face on theofstorage Area of one face: 4x2 + 2x + 2x + 1 = 4x2 + 4x + 1 square container, then multiply the polynomial expression by 6 yards faces onathe cube-shaped container. Create strategy Surface area of container: 2

6(4xit to + calculate 4x + 1) = 24x + 24x +area. 6 square yards Draw and label a diagram of the storage container ﬁrst, then use the surface

Apply the idea

c Write an expression for the volume of the storage container.

Purpose Create a strategy Show students how 2x to write the expression for the surface area of the cubic container. 2 2x 4x 2x container. Use the formula for the volume of a cube to calculate the volume of the storage

Expected mistakes Apply the idea 1 Students may ﬁnd the1 area of one face of the container and assume that1 is 2x the surface area. Students may Since the formula for the volume of a cube is V = l ⋅ w ⋅ h, we can calculate the volume of the storage container as shown: attempt to calculate the volume 2x 1 of the storage container by multiplying the length, 2x 1 width, and height. V = (2x + 1) (2x + 1) (2x + 1)

Substitute expressions for l, w, and h

+ 2x 2x face + 1)] on the storageDistributive property =students (2x + 1)the [(4xarea Reﬂecting withcalculate We can of +one Area of one face: 4x2 + 2x + 2x + 1 = 4x2 + 4x + 1 square 2 + 4x + 1) Combine like terms given a speciﬁc value of x. = (2x + 1) (4x container, then multiply the polynomial expression by 6 Ask students to determine the surface area of the storage yards container 3 faces on the cube-shaped + 8x2 + 2x)container. + (4x2 + 4x + 1) = (8x

3 Students: Pages= 8x 329–330 + 12x2 + 6x + 1 cubic yards

Distributive property Surface area of container: 2 6(4x + 4x + 1) = 24x2 + 24x + 6 square yards Combine like terms

c Write an expression for the volume of the storage container. 6.02 Multiply polynomials mathspace.co

Create a strategy

329

Use the formula for the volume of a cube to calculate the volume of the storage container.

Apply the idea Since the formula for the volume of a cube is V = l ⋅ w ⋅ h, we can calculate the volume of the storage container as shown: V = (2x + 1) (2x + 1) (2x + 1) = (2x + 1) [(4x2 + 2x + 2x + 1)]

Substitute expressions for l, w, and h Distributive property

= (2x + 1) (4x2 + 4x + 1)

Combine like terms

= (8x3 + 8x2 + 2x) + (4x2 + 4x + 1)

Distributive property

= 8x3 + 12x2 + 6x + 1 cubic yards

Combine like terms

Reﬂect and check A labeled diagram of the storage container can help us conceptualize the problem.

6.02 Multiply polynomials mathspace.co

Height: (2x + 1) yards

Width: (2x + 1) yards Length: (2x + 1) yards

Example 4 690

Mathspace Virginia SOL Algebra 1 Teacher Edition Consider the diagram of the product of the expression (x − 1) (x − 4). mathspace.co (x − 4) x

−1

329

PurposeReflect and check A labeledhow diagram of the storage canpolynomial. help us conceptualize the problem. Show students to multiply morecontainer than one Expected mistakes Students may have difficulty starting the multiplication problem when seeing three polynomials. Show students that we start by multiplying two polynomials, then take the product and multiply it by the last polynomial. Height: (2x + 1) yards Reflecting with students Ask students to explain why the volume of the storage container is in cubic yards and the surface area of the Width: (2x + 1) yards storage container is in square yards. Length: (2x + 1) yards

Students: Page 330 Example 4

Consider the diagram of the product of the expression (x − 1) (x − 4). (x − 4) x

(x − 1)

−1

a Find the missing values on the diagram to complete the visual representation of multiplying (x − 1) (x − 4).

Create a strategy Use the algebra tiles on the left side and the algebra tiles on the top row to find the area of the tiles with missing values.

Apply the idea The diagram that shows the visual representation of multiplying (x − 1) (x − 4) is given by: (x − 4)

(x − 1)

−1

−x

−1

−x

Purpose Check students can use algebra tiles1to multiply two binomials. Virginia SOL Algebra 330 Mathspace mathspace.co

Expected mistakes Students may mistake the negative algebra tiles for positive ones, leading to an incorrect diagram. Discuss with students the importance of keeping track of negative values when multiplying algebraic expressions using algebra tiles.

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Students: Page 331 b Write the product of (x − 1) (x − 4).

Create a strategy Add all like terms of the algebra tiles from part (a).

Apply the idea (x − 1) (x − 4) = x2 − x − x − x − x − x + 1 + 1 + 1 + 1 2

= x − 5x + 4

Add all the algebra tiles Evaluate

Reﬂect and check You may encounter some products of polynomials that require combining exponents with a degree greater than 1. Recall the expanded form of the product property for exponents that says axm ⋅ bxn = abxm + n. So, we have:

5x3 ⋅ 4x2 = 5 ⋅ x ⋅ x ⋅ x ⋅ 4 ⋅ x ⋅ x =5⋅4⋅x⋅x⋅x⋅x⋅x = 20x5

Idea summary

Purpose Polynomials can be multiplied using the distributive property. Using an area model for multiplying polynomials Ensure students can write anofexpression from an algebraic diagram. helps keep track terms. Expected mistakes StudentsSpecial may writeproducts their x termof with a positive coefficient since they may count the number of tiles without binomials thinking of their value, resulting in a wrong simplified expression. Having students write out (−1 −1 −1 −1 −1) x after For some products of binomials, we can look for patterns to help us simplify more eﬃciently. writing out the algebra tiles will help clarify the operations needed.

Exploration Concrete-Representational-Abstract (CRA) Approach

use with Example 4

Targeted Considerinstructional the expansionsstrategies of the following binomials of the form (a + b) (a + b) = (a + b)2: 2

(x + 6) = xmodel 6x + 9 + 12x +the 36 multiplication of polynomials. • (x +by 3) (xengaging + 3) = x + students Concrete: Begin with algebra tiles to •physically 2 2 2 2 + 12rs + 9s2 + 10x + 25 = 4r • (x + 5) (x + 5) = x • (2r + 3s) Provide tiles that represent variables and constants: large squares for x , rectangles for x, and small squares for • (5s − 3)2 = 25s2 − 30s + 9 constants. Have students arrange one polynomial along the top of an area grid and the other along the side. do you notice about linearplace coeﬃcient thexproduct? For example,1. to What multiply (x + 2) by x + the 3, they tilesoffor and +2 along the top and tiles for x and +3 along 2. What do you notice about the constant of the product? the side. Students then fill in the grid by multiplying the corresponding tiles, creating a physical area model. 3. activity Is there helps a general rulesee for this type of product? This hands-on them how each term combines to form the product. Consider the expansions of the following binomials of the form (a + b) (a − b): Representational: Transition to drawing area models on paper to represent the multiplication visually. Instruct + 3s) (2r − 3s) = 4r2 − 9s2 • (x + 3) (x − 3) = x2 – 9 students to sketch a rectangle divided into sections based •on(2r the terms of the polynomials. Label the top of • (5s − 3) (5s + 3) = 25s2 − 9 • (x − 5) (x + 5) = x2 – 25 the rectangle with x and +22and the side with x and +3. Inside each section, have them write the product of • (x + 6) (x − 6) = x − 36 the corresponding terms (e.g., x × x = x2, x × 3 = 3x). Encourage them to shade or color-code different sections 1. What do you notice about the linear coeﬃcient of the product? to highlight like terms. This visual representation connects the physical tiles to abstract algebraic concepts, 2. What do you notice about the constant of the product? reinforcing their understanding of how the terms multiply. 3.

Is there a general rule for this type of product?

Abstract: Move on to multiplying polynomials using algebraic notation and symbols. Guide students through the distributive property without the aid of visuals: (x + 2) (x + 3) = x(x + 3) + 2x + 3). Simplify the expression step by step to get x2 + 3x + 2x + 6, and then combine like terms to obtain x2 + 5x + 6. Emphasize the patterns they observed in the concrete and representational stages. Practice additional problems to reinforce this process, helping students become comfortable with abstract manipulation of polynomials. 6.02 Multiply polynomials 331 mathspace.co

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Create a strategy Add all like terms of the algebra tiles from part (a).

Apply the idea

Connecting the stages: the concrete (x − 1) (x −Help 4) = x2students − x − x − xmake − x − x connections + 1 + 1 + 1 + 1 between Add all the algebra tiles tiles, the drawings, and the 2 abstract equations. Discuss= xhow the area model with tiles corresponds to their sketches and how both relate − 5x + 4 Evaluate to the algebraic steps. Encourage students to explain how each term in the final expression comes from the Reﬂectofand check terms in the polynomials. Use side-by-side comparisons to illustrate these connections. multiplication specific You may encounter some productsdeepen of polynomials require combining a degree on greater than 1. By linking all three stages, students their that understanding and exponents see how with operations polynomials mirror Recall the expanded form of the product property for exponents that says axm ⋅ bxn = abxm + n. those on real numbers, preparing them for more advanced concepts like factoring. So, we have:

5x3 ⋅ 4x2 = 5 ⋅ x ⋅ x ⋅ x ⋅ 4 ⋅ x ⋅ x =5⋅4⋅x⋅x⋅x⋅x⋅x

Students: Page 331

= 20x5

Idea summary Polynomials can be multiplied using the distributive property. Using an area model for multiplying polynomials helps keep track of terms.

Special products of binomials For some products of binomials, we can look for patterns to help us simplify more eﬃciently.

Special products of binomials Exploration

Students are informed that patterns may be relevant when multiplying polynomials, leading to special products. An exploration with special products of binomials follows. Consider the expansions of the following binomials of the form (a + b) (a + b) = (a + b)2: • (x + 3) (x + 3) = x2 + 6x + 9

• (x + 6)2 = x2 + 12x + 36

Critique, correct, + 10xclarify + 25 • (x + 5) (x + 5) = x and 2

• (2r + 3s)2 = 4r2 + 12rs + 9s2

English language • (5s − 3) = 25slearner − 30s +support 9 1. What you notice about the coeﬃcient of the product? Provide students withdospecial products oflinear binomials incorrectly expanded, both with and without using the 2. What do you notice about the constant of the product? identity rules. Examples could include: 3.

Is there a general rule for this type of product?

(x + 3)2 = x2 + 32

Distribute exponent

Consider the expansions of the following binomials of the form (a + b) (a − b): 2

=x +9

• (x + 3) (x − 3) = x2 – 9 22

25+ 4x + 16 (x (x − 5)−(x4)+ = 5) x= x− –4x (x +• 4) • (x + 6) (x − 6) =2x2 − 36

= x + 16

1. 2. 3.

Simplify

• (2r + 3s) (2r − 3s) = 4r2 − 9s2

• (5s term − 3) (5sin+ the 3) = 25s 9 of ( ) Distribute each first2 −set

Simplify

What do about the linear 2 you notice 2 2 coeﬃcient of the product?

(4x − 1) = (4x) + 2(4x)(1) + (1)

Square of a binomial (sum)

What do you notice about the constant of the product? 2

8x +for1 this type of product? Simplify = general 16x + rule Is there a

Students should work in pairs or small groups to identify the errors in mathematical reasoning or applying polynomial identities. Once the errors have been identified, students should expand the products correctly, adjusting the explanation steps as needed. Students should discuss with their partner or group and write a brief explanation of why the original steps were incorrect and how their corrected steps lead to the correct solution. 6.02 Multiply polynomials mathspace.co

331

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693

5x ⋅ 4x = 5 ⋅ x ⋅ x ⋅ x ⋅ 4 ⋅ x ⋅ x =5⋅4⋅x⋅x⋅x⋅x⋅x = 20x5

Exploration Idea summary Polynomials can be multiplied using the distributive property. Using an area model for multiplying polynomials

Students: Page 331 helps keep track of terms.

Special products of binomials For some products of binomials, we can look for patterns to help us simplify more eﬃciently.

Exploration Consider the expansions of the following binomials of the form (a + b) (a + b) = (a + b)2: • (x + 3) (x + 3) = x2 + 6x + 9

• (x + 6)2 = x2 + 12x + 36

• (2r + 3s)2 = 4r2 + 12rs + 9s2

• (x + 5) (x + 5) = x + 10x + 25 2

• (5s − 3) = 25s − 30s + 9 1.

What do you notice about the linear coeﬃcient of the product?

What do you notice about the constant of the product?

Is there a general rule for this type of product?

Consider the expansions of the following binomials of the form (a + b) (a − b): • (x + 3) (x − 3) = x2 – 9

• (2r + 3s) (2r − 3s) = 4r2 − 9s2

• (5s − 3) (5s + 3) = 25s2 − 9

• (x − 5) (x + 5) = x – 25 2

• (x + 6) (x − 6) = x − 36 1.

What do you notice about the linear coeﬃcient of the product?

What do you notice about the constant of the product?

Is there a general rule for this type of product?

Suggested student grouping: Small groups 331 Students are given two lists of expanded forms of special product examples: (a + 6.02 b) (aMultiply + b)mathspace.co =polynomials (a + b)2 and (a + b) (a − b). The goal is to look at patterns that appear with the given examples. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What do you notice about the linear coefficient of the product? For (a + b) (a + b) = (a + b)2, the linear coefficient of the product is twice the product of a and b. For (a + b) (a − b), the linear coefficient of the product is zero. 2. What do you notice about the constant of the product? For (a + b) (a + b) = (a + b)2, the constant of the product is b multiplied by itself. Note that the sign of the constant is positive, since the signs of b are the same in the original product. For (a + b) (a − b), the constant of the product is the b-term multiplied by itself. Note that the sign of the constant is negative, since the signs of b are positive and negative in the original product. 3. Is there a general rule for this type of product? In general, the multiplication pattern for each is (a + b)2

(a + b) (a + b) = a2 + 2ab + b2 (a + b) (a − b) = a2 − b2

Purposeful questions • Let’s look at the linear coeﬃcient of each product. What pattern do you see between the numbers in the binomial and the linear coeﬃcient?

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• Let’s look at the constant of each product. What pattern do you see between the numbers in the binomial and the linear coefficient? • Is your pattern true every time you multiply the binomials? Possible misunderstandings • For the first set of products, students might assume that the linear coefficient of the product (a + b)(a + b) is b multiplied by itself, but the examples (2r + 3s)2 = 4r2 + 12rs + 9s2 and (5s − 3)2 = 25s2 − 30s + 9 prove otherwise. Following the exploration, students are shown a visual representation of special products and are given the general rules for special binomial products and definitions for the square of a binomial and the product of a sum and difference. Some other common misconceptions are noted.

Students: Page 332 Some products of binomials follow special patterns. For example, consider the product of a binomial squared, (a + b)2: a

We can expand (a + b)2 to (a + b) (a + b) and represent them with an area model. Evaluating with this model and combining like terms, we get the product a2 + 2ab + b2. So, we have:

(a + b)2 = (a + b) (a + b) = a2 + 2ab + b2

Now consider the product of a sum and a difference, (a + b) (a − b):

−b

−b2

Notice that the term ab and −ab are opposites and combine to make zero. We call this a zero pair. So, (a + b) (a − b) = a2 − b2. A binomial of the form a2 − b2 is called a difference of two squares. If we remember the patterns for these special products, we can multiply two polynomials without using the distributive property. For binomials, we have the following special binomial products, which are called identities: Square of a Sum

Product of a sum and difference

(a + b)2 = a2 + 2ab + b2

(a + b) (a − b) = a2 − b2

Square of a Difference (a − b)2 = a2 − 2ab + b2

Note: (a + b)2 ≠ a2 + b2 and (a − b)2 ≠ a2 − b2.

Example 5 Multiply and simplify the following binomials. a (x − 4)2

Create a strategy

6.02 Multiply polynomials mathspace.co

695

Square of a Sum

Product of a sum and difference

(a + b)2 = a2 + 2ab + b2

(a + b) (a − b) = a2 − b2

Square of a Difference

Examples (a − b) = a − 2ab + b 2

Students: Pages 332–333

Note: (a + b)2 ≠ a2 + b2 and (a − b)2 ≠ a2 − b2.

Example 5 Multiply and simplify the following binomials. a (x − 4)2

Create a strategy We check ﬁrst whether (x − 4)2 is a special binomial product and identify its form.

Apply the idea 332

Mathspace

Virginia SOL Algebra 1

Since mathspace.co (x − 4)2 is a square of a binomial in the form (a − b)2, we use the formula and simplify the expression: (a − b)2 = a2 − 2ab + b2

Identity for the square of a binomial

(x − 4)2 = x2 − 2(x) (4) + 42

Substitute a = x and b = 4

= x2 − 8x + 16

Evaluate the multiplication and exponent

Reﬂect and check x

−4

We can also use an area model to find the product. Combining like terms, we get: (x − 4)2 = x2 + 16

−4x

−4

−4x

b (x + 4) (x − 4)

PurposeCreate a strategy Show students to use the special for the and square of its a binomial (difference) to substitute and We checkhow ﬁrst whether (x + 4) (x − 4) isproduct a special pattern binomial product identify form. solve a multiplication problem. Apply the idea

Expected mistakes Since (x + 4) (x − 4) is a product of a sum and difference, we use the formula and simplify the expression: Students may forget about the special product pattern and apply the exponent incorrectly to each term within Identity for the product of a sum and difference (a + b) (a − b) = a2 − b2 the parentheses, leading to x2 − 216. 2 (x + 4) (x − 4) = x − 4

Substitute a = x and b = 4

= x2 – 16

Evaluate the exponent Reflecting with students Ask students if they have any way that they recall the pattern without memorizing the entire a2 − 2ab + b2. Reflect and check Students might notice that when a = x, the linear term is twice the value of b in this specific example and the Using an area model, we see (x + 4) (x − 4) = x2 – 16. constant term is b multiplied by itself, noting the sign of b. 4

696

−4

−4x

−16

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co 6.02 Multiply polynomials mathspace.co

333

−4x

−4

−4x

Students: Page 333 b (x + 4) (x − 4)

Create a strategy We check ﬁrst whether (x + 4) (x − 4) is a special binomial product and identify its form.

Apply the idea Since (x + 4) (x − 4) is a product of a sum and difference, we use the formula and simplify the expression: (a + b) (a − b) = a2 − b2

Identity for the product of a sum and difference

(x + 4) (x − 4) = x2 − 42

Substitute a = x and b = 4

= x2 – 16

Evaluate the exponent

Reﬂect and check Using an area model, we see (x + 4) (x − 4) = x2 – 16. 4

−4

−4x

−16

Purpose Show students how to use the special product pattern for the product of a sum and a difference. 6.02 Multiply polynomials

333

Expected mistakes mathspace.co Students might forget that the linear term combines to zero if they choose to use distribution as opposed to the pattern to solve. Reﬂecting with students Ask students to provide their own examples of the product of a sum and a difference.

Students: Page 334 c (2x + 5) (2x − 5)

Create a strategy We check ﬁrst whether (2x + 5) (2x − 5) is a special binomial product and identify its form.

Apply the idea Since (2x + 5) (2x − 5) is a product of a sum and difference, we use the formula and simplify the expression: (a + b) (a − b) = a2 − b2

Identity for the product of a sum and difference

(2x + 5) (2x − 5) = (2x)2 − 52

Substitute a = 2x and b = 5

= 4x2 – 25

Evaluate the exponents

d Multiply and simplify: 3(2x + 5y)2

Create a strategy We check ﬁrst whether 3(2x + 5y)2 involves a special binomial product and identify its form.

Apply the idea

6.02 Multiply polynomials Since (2x + 5y)2 is a square of a binomial in the form (a + b)2, we use the formula and simplify the expression: mathspace.co (a + b)2 = a2 + 2ab + b2 2

Identity for the square of a binomial 2

3(2x + 5y) = 3 [(2x) + 2(2x) (5y) + (5y) ]

Substitute a = 2x and b = 5y and multiply by 3

697

Purpose Show students the special product pattern for the product of a sum and difference when the a-term has a coefficient other than one. c (2x + 5) (2x − 5) Expected mistakes Create a strategy Students may forget to square the coefficient of the a-term. Use distribution to show why we must also square We check first whether (2x + 5) (2x − 5) is a special binomial product and identify its form. the coefficients. Apply the idea

Reﬂecting with students Since (2x + 5) (2x − 5) is a product of a sum and difference, we use the formula and simplify the expression: Ask students how this problem may be more challenging than the problem in part (b). This is a good time for Identity for the product of a sum and difference (a + b) (a − b) = a2 − b2 students to identify expected mistakes on their own, such as forgetting to square the coeﬃcient of the a-term. 2 2 (2x + 5) (2x − 5) = (2x) − 5

Substitute a = 2x and b = 5

= 4x2 – 25

Students: Page 334

Evaluate the exponents

d Multiply and simplify: 3(2x + 5y)2

Create a strategy We check ﬁrst whether 3(2x + 5y)2 involves a special binomial product and identify its form.

Apply the idea Since (2x + 5y)2 is a square of a binomial in the form (a + b)2, we use the formula and simplify the expression: (a + b)2 = a2 + 2ab + b2

Identity for the square of a binomial

3(2x + 5y)2 = 3 [(2x)2 + 2(2x) (5y) + (5y)2]

Substitute a = 2x and b = 5y and multiply by 3

= 3 [4x2 + 20xy + 25y2]

Evaluate the exponents and multiplication

= 12x2 + 60xy + 75y2

Distributive property

Idea summary

Purpose Recognizing the patterns in special binomial factors may be helpful in multiplication problems and upcoming Show studentslessons. how to combinethethe square of a sum with distribution. Remember patterns: •

Square of a Sum: (a + b)2 = a2 + 2ab + b2

Expected mistakes • Square of a Difference: (a − b)2 = a2 − 2ab + b2 Students may •incorrectly the term that by the quantity. Show students that it is not Product of square a sum and difference: (a +isb)being (a − b) =multiplied a2 − b2 squared with the quantity. Reﬂecting with students Practice Give students example values of x and y to substitute into the expressions and conﬁrm that the initial problem is equivalent to its solution. What do you remember?

When and when not to square a number 1 Complete:

use with Example 5

Address student misconceptions

The distributive property states a(b + c) = a ⋅ ⬚ + a ⋅ ⬚

When squaring a term with a coeﬃcient, many students forget that the exponent also applies to the coeﬃcient. 2 a Evaluate 5(12 − 6) b Evaluate 5 ⋅ 12 − 5 ⋅ 6 This misconception would result in the following work: c Complete 5(12 − 6) = 5 ⋅ ⬚ − 5 ⋅ ⬚ (2x + 5) (2x − 5) = (2x)2 – 52 = 2x2 – 25

Substitute the given values Simplify

In this case, encourage students to go back and distribute the multiplication without using the pattern to check their answer. Note that (2x)2 = (2x) (2x) = 4x2, not 2x2. Students334 mayMathspace try to distribute the 31 before squaring the binomial in part (d), resulting in Virginia SOLinAlgebra 2 mathspace.co 2 (6x + 10y) = 36x + 120xy + 100y2 instead of 12x2 + 60xy + 75y2. Encourage students to notice that the square is not being applied to the 3, so if they distribute ﬁrst, they are actually squaring it which is incorrect. To further explain why this violates the order of operations, consider the expression 3 (2x + 5y) (2x + 5y) and that we would only distribute the 3 into one of the parentheses.

698

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Since (2x + 5y)2 is a square of a binomial in the form (a + b)2, we use the formula and simplify the expression: (a + b)2 = a2 + 2ab + b2

Identity for the square of a binomial

3(2x + 5y)2 = 3 [(2x)2 + 2(2x) (5y) + (5y)2] 2

= 3 [4x + 20xy + 25y ]

Students: Page 334 = 12x2 + 60xy + 75y2

Substitute a = 2x and b = 5y and multiply by 3 Evaluate the exponents and multiplication Distributive property

Idea summary Recognizing the patterns in special binomial factors may be helpful in multiplication problems and upcoming lessons. Remember the patterns: • • •

Square of a Sum: (a + b)2 = a2 + 2ab + b2 Square of a Difference: (a − b)2 = a2 − 2ab + b2 Product of a sum and difference: (a + b) (a − b) = a2 − b2

Practice What do you remember?

Practice 1

Complete:

The distributive property states a(b + c) = a ⋅ ⬚ + a ⋅ ⬚ Students: Pages 334–339 2

Evaluate 5(12 − 6)

Complete 5(12 − 6) = 5 ⋅ ⬚ − 5 ⋅ ⬚

Evaluate 5 ⋅ 12 − 5 ⋅ 6

What do you remember? 1

Complete:

The distributive property states a(b + c) = a ⋅ ⬚ + a ⋅ ⬚

a c

Evaluate 5(12 −Virginia 6) 334 Mathspace SOL Algebra 1

Evaluate 5 ⋅ 12 − 5 ⋅ 6

7wy ( y + w)

Complete 5(12 − 6) = 5 ⋅ ⬚ − 5 ⋅ ⬚

Distribute: a

−m (m + 1)

y ( y − 9)

y ( y − 4) + 10

e 4

mathspace.co

The area of the whole rectangle is 16 ⋅ 20 = 320 units2. a

Find the area of the shaded rectangle.

Find the area of the unshaded rectangle.

Find the sum of the two areas.

Does 16(18 + 2) = 16 ⋅ 18 + 16 ⋅ 2?

The area of this figure is represented by the expression ( y + 3) ( y + 6). a

Find the area of: Rectangle A

Rectangle B

iii Rectangle C

Rectangle D

Write an equivalent, simplified expression for the area of the figure.

6.02 Multiply polynomials mathspace.co

699

Match each product with an area model. A

9x2

−1

−3x

−1

x 3

9x2

−3

−3x

−9

(3x + 1) (3x − 1)

(−3x + 1) (3x + 1)

−3x

−9x2

−3x

iii

(x + 3) (x − 3)

9( y + 8) ( y + 2)

4x(5x(x − 3) + 2x)

(x + 3) (x + 3)

(−7y − 8) (−7y + 8)

Let’s practice 7

Simplify each product: a c

(7y − 6) (6y + 6) 2

(v + 5) (5v − 3v − 5)

e 8

Multiply and simplify: a

(x2 + 3)2

3x(7x − 5y)2

6(8x − 9y) (8x + 9y)

(11 − a) (11 + a) − 10

i 9

(x − 3)2 = x2 − ⬚ x + ⬚

(x + ⬚)2 = x2 + ⬚ x + 36

4(x + 8) − 2

7 + 5(3x + 4)

Simplify: a

700

(3x − 8) (3x + 8)

Complete the expansion of the perfect squares: a

5 + 3(x + 4)

8(3x + 4) − 6x

34 + 9(4x − 5)

5x − 8(2x − 3)

9x − 5(2x + 3)

−8(5x − 7) − 9

4y + 5 + 6( y − 9)

4(5(x − 3) + 2)

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Use the diagram and areas of rectangles to demonstrate that 4(x + 6) = 4x + 24

This rectangle has width x + 2 and length x + 6. Use the diagram and areas of rectangles to find a simplified expression for (x + 2) (x + 6).

Area I

Area II

Area III

Area IV

Multiply the following binomials using the algebra tiles. i

Find the missing values to complete the visual representation of multiplying the binomials.

Find the simplified expression for the binomials.

The diagram shows the visual representation to simplify (x + 2) (x − 3). (x − 3) x

−1

x (x + 2) 1

ii b

(x + 2) (x − 3) = ⬚

he diagram shows the visual representation to simplify T (x − 4) (x − 3).

(x − 3) x

−1

(x − 4)

−1

−1 −1

(x − 4) (x − 3) = ⬚

6.02 Multiply polynomials mathspace.co

701

The diagram shows the visual representation to simplify (x + 2) (x + 5).

(x + 5) x

(x + 2) 1

ii d

(x + 2) (x + 5) = ⬚

he diagram shows the visual representation to T simplify (x + 2) (2x − 1).

(2x − 1) x

−1

x (x + 2)

ii 14

(x + 2) (2x − 1) = ⬚

−1

Anette decides to use the area model to complete a polynomial multiplication question, (10x + 2) (10x + 3). To warm-up, she decides to try a simple integer example: 12(13) =

(10x + 2) (10x + 3) =

100

100 + 20 + 30 + 6 = 156

10x

100x

20x

30x

100x + 20x + 30x + 6 = 150x + 6

State the error in Anette’s polynomial product. Correct her work and solution.

Show how you would set up the area model to ﬁnd the product (10x + 3) (x2 + 10x + 2). Find the product using any method.

If Mackenzie creates a polynomial with degree 4 and Edgardo creates a polynomial with degree 3, state what we know about the product of their polynomials.

Sonya wants to extend the width and length of their vegetable patch by x ft. Their vegetable patch currently has a width of 5 ft and a length of 7 ft. For each of the following:

702

Write a simpliﬁed expression in terms of x.

State the unit of the answer to the expression from (i). Explain why you chose this unit.

The new length of the vegetable patch.

The area of the new vegetable patch.

The perimeter of the new vegetable patch.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

5 ft

x 7 ft

The diagram shows a square with side length x. a

What is the area of a square with side length x?

Write an expression for Area I.

Write an expression for Area II.

Show that x (x − 4) + 4x = x2

x 4

Area I

Area II

A square with side lengths measuring x − 1 centimeters has each side enlarged by a factor of 2. Write a simplified expression for the area of the new square.

Let’s extend our thinking 19

Find the missing values that make each equation true. a b c

(x + ⬚) (x − 7) = x2 − 3x + ⬚

(2x − 1) (⬚ x − 5) = 2x2 + ⬚ x + 5

(⬚ x + 3) (⬚ x + 6) = 12x2 + 30x + ⬚

Show the following results about special products are valid: a

(a + b)2 = a2 + 2ab + b2

(a + b) (a − b) = a2 − b2

(a − b)2 = a2 − 2ab + b2

Show how you can use one of the special products (a ± b)2 to ﬁnd each perfect square without a calculator. a

212

192

332

472

Write a simplified expression for the product of three consecutive integers, where the middle integer is m.

Consider the rectangle shown:

8st

Write a simplified expression for the area of the rectangle.

2(s + t)

A rectangular garden has a length that is one foot less than twice the width. a

Write and simplify an expression for the area of the rectangle.

A 2 foot border is to be placed all around the garden. Write and simplify an expression for the area of the border.

The landscaper designing the garden has 100 square feet of pavers to use for the border. Determine the largest garden the landscaper can border before running out of pavers. Assume the landscaper will only use positive-integer dimensions.

6.02 Multiply polynomials mathspace.co

703

A flat rate large box from USPS has dimensions of 1 ft × 1 ft × 5.5 in. If Jerome decides to put a layer of insulation in his box x inches thick, write a simplified polynomial expression that models a

The dimensions of the open volume left in the box in inches.

The volume he has left in his box to fill.

Consider the following problem: “Find two binomials whose product results in a polynomial with 5 terms”. a

Yao was given the problem, but they think it’s impossible. State whether you agree or disagree. Explain.

Create two different products of two polynomials that results in an answer with exactly 5 terms.

Determine whether each statement is always, sometimes, or never true. Explain your reasoning with examples. a

Two polynomials multiplied together will result in a polynomial

A term with a negative exponent is a polynomial

A polynomial multiplied by a non-polynomial will result in a polynomial

Explain how the area of a square given a binomial side length will always be a special product.

Polynomial multiplication can be used to ﬁnd the total area of a rectangle that has a border, where the length and width each represent a factor in the product.

704

Write an algebraic model that can be used to ﬁnd the total area of a square with an unknown side length and a border with an unknown border width.

Use your model to ﬁnd the total area of a square with a side length of 2m + 1 and a border width of 3.

Determine whether the conjectures are true or false. Create algebraic expressions to justify your response. a

The product of any two consecutive even numbers is one more than a perfect square.

The product of any two consecutive odd numbers is one less than a perfect square.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Answers

13 a i

(x 3) x

6.02 Multiply polynomials What do you remember? 1 a a(b + c) = a ⋅ b + a ⋅ c 2 a 30

(x 2)

b 30

c 5 (12 − 6) = 5 ⋅ 12 − 5 ⋅ 6 3 a −m2 − m

b y2 − 9y

c 7wy2 + 7w2y

d y2 − 4y + 10

e a3 + a2 + 2a

4 a 288 units2

ii (x + 2) (x − 3) = x2 − x − 6 b i

(x 4) 1

d Yes

5 a i A = y2

ii B = 3y

iii C = 6y

iv D = 18

b y2 + 9y + 18 A

b 32 units2

c 320 units2

6 i

(x 3)

ii D

iii C

iv B

Let’s practice 7 a 42y2 + 6y − 36

b 9y2 + 90y + 144

c 5v3 + 22v2 − 20v − 25

d 20x3 − 52x2

8 a x4 + 6x2 + 9

c i x

d 49y2 − 64 2

e 384x − 486y

111 − a2 (x 2)

x2 + 4xy + 4y2

9 a (x − 3)2 = x2 − 6x + 9

b (x + 6)2 = x2 + 12x + 36

10 a 4x + 30

b 15x + 27

c 3x + 17

d 18x + 32

e 36x − 11

g −x − 15

h −40x + 47

(x 5)

b 147x3 − 210x2y + 75xy2

c 9x2 − 64 2

ii (x − 4) (x − 3) = x2 − 7x + 12

10y − 49

−11x + 24

ii (x + 2) (x + 5) = x2 + 7x + 10 d i

(2x 1) x

20x − 52

11 Total area = 4(6 + x) Area small rectangle = 4x Area large rectangle = 24 Total area equals Area small rectangle plus Area large rectangle

(x 2)

Therefore 4(x + 6) = 4x + 24. 12 (x + 2) (x + 6) = x2 + 8x + 12

ii (x + 2) (2x − 1) = 2x2 + 3x − 2 14 a ( 10x) (10x) = 100x2, but Anette put 100x instead. Her final answer should have been 100x2 + 20x + 30x + 6 = 100x2 + 50x + 6.

Answers mathspace.co

705

10x

b (x + 4) (2x + 3) − x(2x − 1) = 12x + 12

10x

10x3

100x2

20x

c A garden with a width of 7′ and a length of 13′ will need 96 square feet of pavers to create a 2′ border.

3x2

30x

25 a T he new dimensions in inches are (12 − 2x), (12 − 2x), and (5.5 − 2x).

10x + 100x + 20x + 3x + 30x + 6 = 10x3 + 103x2 + 50x + 6 15 The result will be a polynomial with degree 7. 16 a i 7 + x ii The units will be ft, since we don’t need to change the units as we are only adding x ft to each side of the vegetable patch which is already measured in feet. b i x2 + 12x + 35 ii The units will be ft2, as we are finding the product of two lengths. c i 4x + 24 ii The units will be ft, as we are adding two lengths. 17 a A = x2

b x(x − 4)

c 4x

d A rea I plus Area II equals Area of Square. Therefore x(x − 4) + 4x = x2 18 4x2 − 8x + 4 cm2 Let’s extend our thinking 19 a 4 and -28

b 1 and -11

c A nswers may vary. 2, 6, and 18 is one possibility. 3, 4, and 18 is another. 20 a

b J erome will have a volume of (−8x3 + 118x2 − 552x + 792) in3 left to fill. 26 a A gree. The maximum number of terms that can be produced by multiplying two binomials is 2 · 2 = 4 if no terms combine. b A nswers will vary. A monomial multiplied by a five term polynomial will always results in a five term polynomial. Another possible solution is a binomial multiplied by a trinomial that results in a 4th degree polynomial with exactly one pair of terms combining. For example, (x2 + 3) (x2 − 2x + 1) = x4 − 2x3 + 2x2 − 6x + 3 27 a A lways true. A polynomial is a collection of terms in the form mxn where m is a real number and n is a nonnegative integer. When multiplying polynomials together, terms are multiplied, and if they are the same variable, their exponents are added together, so the exponents will still be non-negative integers. That means all terms in the resulting polynomial will also be a collection of terms in the form mxn and will also be a polynomial. b N ever true. A polynomial can never have terms with negative exponents, as a polynomial is a collection of terms in the form mxn where m is a real number and n is a non-negative integer. c S ometimes true. Consider the example (4x3 + x) · x−1 = 4x2 + 1. If a polynomial is multiplied by a non-polynonial of the form ax–b, where a is a real number, and b is a positive integer whose magnitude is less than or equal to the magnitude of the exponent of the term with the lowest degree in the polynomial, the result will be a polynomial. 28 The area of a square is A = s2 where s is the side length of the square. If s is a binomial such as (a + b) then the area of the square is the special product (a + b)2. 29 a Variable choices will vary.

Let x represent the side length of the square and let y represent the width of the border. Then the area of the square will always be (x + y)2 = x2 + 2xy + y2.

21 a (20 + 1)2 = (202 + 2(20) + 1) = 441 b (20 − 1)2 = (202 − 2(20) + 1) = 361 c (30 + 3)2 = (302 + 2(90) + 9) = 1089 d (50 − 3)2 = (502 − 2(150) + 9) = 2209 22 m3 − m 23 (16s2t + 16st2) units2 24 a If x feet represents the width of the garden, then the area will be x(2x − 1) = 2x2 − x square feet.

706

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

b Let x = (2m + 1) and let y = 3 so that (x + y)2 becomes (2x + 1 + 3)2 = (2x + 4)2. Then the total area is 4x2 + 16x + 16. 30 a False: (2x) (2x + 2) = 4x2 + 4x. Although 4x2 is a perfect square since both 4 and x2 are perfect squares, 4x will not be a perfect square if x is not and there is no constant value that would imply +1. Alternatively, note that 4x2 + 4x + 1 = (2x + 1)2. So the product is actually one less than a perfect square. b True. (2x − 1) (2x + 1) = 4x2 − 1 = (2x)2 − 1

6.03 Divide polynomials by a monomial Subtopic overview Lesson narrative When dividing a polynomial by a monomial, each term of the polynomial is divided by the monomial, after which each individual fraction left over is simplified using the rules of exponents. By the end of this lesson, students will be able to divide a polynomial by a monomial.

6.03 Divide polynomials by a Learning objective monomial Students: Page 340

After this lesson, you will be able to… • divide a polynomial by a monomial.

Divide by a monomial To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. To do this, use the quotient rule and divide coefficients and subtract exponents with the same base.

Key vocabulary 

 dividend common factor Dividing a polynomial by a monomial



divisor

Essential understanding

Polynomial division can be modeled with algebra tiles. Polynomials can be divided using steps similar to those used when dividing real numbers. −2x + 3

Standards

−x2

−x

−x2

This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards. 2x

Mathematical process goals 2 x −x

−x2

MPG4 — Mathematical Reasoning

MPG5 — Mathematical Representations 2

−4x + 6x Teachers can integrate this goal by using a variety of methods to represent mathematical ideas. showing students the connection between factoring and Create an area model where: For example, demonstrating the division of polynomials dividing in the context of polynomials. They can guide 1. The tiles on the inside add up to the dividend (numerator). by a monomial using concrete manipulatives and students to see that factoring a polynomial is essentially 2. The tiles on one side add up to the divisor (denominator). pictorial representations. They should also encourage the reverse process of multiplying, which is connected 3. The sum the tiles along can the other side mustby be the quotient (theto result of connections the division). between these different students make to division. Thisofunderstanding be reinforced such as between the symbolic notation highlighting the factored form of a polynomial Note: Finalhow answers are usually written without anycan negativerepresentations, exponents. and the concrete model. be used to simplify division problems. By relating these different mathematical procedures, students can start Example 1 to see the interconnectedness of various mathematical concepts and techniques. Simplify the following: 2 6x Teachers can integrate this goal into −4x their+instruction by

Create a strategy Apply the rule

Apply the idea .

6.03 Divide polynomials by a monomial mathspace.co

Divide each term by x

707

Content standards A.EO.2 — The student will perform operations on and factor polynomial expressions in one variable.

A.EO.2d — Determine the quotient of polynomials, using a monomial or binomial divisor, or a completely factored divisor.

Prior connections 8.PFA.1 — The student will represent, simplify, and generate equivalent algebraic expressions in one variable.

A.EO.1 — The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

Future connections A2.EO.3 — The student will perform operations on polynomial expressions in two or more variables and factor polynomial expressions in one and two variables.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lesson: Algebra 1 — 5.03 Quotient rule

Tools You may ﬁnd this tool helpful: • Scientific calculator

Student lesson & teacher guide Divide by a monomial Students are introduced to the method of dividing a polynomial by a monomial by dividing each term of the polynomial by the monomial separately. Area models with algebra tiles and written steps are then used to demonstrate a visual way of dividing a polynomial by a monomial.

708

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Students: Page 340

6.03 Divide polynomials by a monomial After this lesson, you will be able to… • divide a polynomial by a monomial.

Polynomial division can be modeled with algebra tiles. −2x + 3

−x2

−x

−x2

2x −x

−x

−4x2 + 6x

Create an area model where: 1. The tiles on the inside add up to the dividend (numerator). 2. The tiles on one side add up to the divisor (denominator). 3. The sum of the tiles along the other side must be the quotient (the result of the division). Note: Final answers are usually written without any negative exponents.

Example 1

Reviewing division of exponent rules and connections to polynomials Simplify the following:

Targeted instructional strategies In previous section, students practiced dividing monomialApply expressions Create a strategy the idea by dividing or reducing coeﬃcients, and subtracting exponents. Students should begin this section by practicing dividing monomial expressions, and a Apply the rule . Divide term bymonomial. x helpful exercise is to provide several monomial terms, and ask students to divide them by each the same For example: Simplify Divide each of the following by 2y:

Since there are no negative exponents and the expression is already in standard form, the ﬁnal answer is 3 4 −14y5 2y3 4y 3x−6y + 4x.

Ask students to then consider the following problem: 340

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Ask students to connect their answers to the previous exercise to the new problem, joining their answers together with addition or subtraction.

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Stronger and clearer each time English language learner support Students should respond to the following prompt: What steps do we take to divide a polynomial by a monomial?

Students begin by individually solving the problem and writing a description of their steps either after or next to the steps. In the first meeting, pair students to discuss their initial explanations. Encourage students to ask each other questions for clarity, such as: • What happens to each term of the polynomial? • How do you handle each part of the division? Students revise their explanations based on the discussion. Students should rotate pairs and follow the same editing structure for 2 − 3 rotations. Students should push for more precision, ensuring their step-by-step instructions are clear and concise. Have students write a final, detailed explanation of the process, incorporating feedback from their discussions.

Polynomial division by a monomial using color coding Student with disabilities support Students can visually follow the division process and recognize patterns using color coding. Ask students to highlight each term of a polynomial division problem in different colors. Guide them to divide each colored term by the monomial and use the same color to highlight the simplified terms. An example of this method is shown:

Having students mark their answer space with their colors ahead of time will also help students remember all terms as they simplify and write their final answer.

Disappearing Constants Address student misconceptions When simplifying, many students have trouble remembering when a value of 1 does not need to be written in the simpliﬁed polynomial expression. For example, many students will believe that the following the are same terms.

This will give a ﬁnal polynomial answer of 5x3 − 2x, when the ﬁnal answer should be 5x3 − 2x + 1. Encourage students to write out each term being divided separately, such as

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−4x2 + 6x

Create an area model where: 1. The tiles on the inside add up to the dividend (numerator).

2. The tiles on one side add up to the divisor (denominator). Examples

3. The sum of the tiles along the other side must be the quotient (the result of the division).

Students: Page 340

Note: Final answers are usually written without any negative exponents.

Example 1 Simplify the following:

Create a strategy Apply the rule

Apply the idea .

Divide each term by x Simplify Since there are no negative exponents and the expression is already in standard form, the ﬁnal answer is 3x4 + 4x.

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Purpose Divide a polynomial by a monomial with a coeﬃcient and power of 1. Expected mistakes Students may add an x to each term rather than take one away. Encourage students to write out the variables with the operations performed on the exponents. Reflecting with students Dividing by a monomial of x is similar to taking an x out of each term. Discuss with students how expanded form can be used to divide variables as a visual exercise.

Advanced learners: Closure of polynomials

use with Example 1

Targeted instructional strategies Introduce advanced learners to the term “closure” of polynomials if it was not introduced in the previous lessons. We say that polynomials are closed under addition, subtraction, and multiplication because the sum, difference, or product of polynomials will also be a polynomial. Then, ask students whether they think polynomials are closed under division. Remind them of the definition of a polynomial: The sum or difference of terms which have variables raised to non-negative integer powers and which have coefficients that are constant. If students only consider the examples in the lesson, they might incorrectly conclude that polynomials are closed under division. Provide them with an example like the one shown:

Then, ask questions such as: • Is the dividend a polynomial? How do you know? • Is the divisor a polynomial? • Is the quotient a polynomial? Why or why not? Students should use the negative exponent rule to show that the last term of the quotient has a negative exponent on the variable, so it is not a polynomial. Thus, a polynomial divided by another polynomial is not always a polynomial, so polynomials are not closed under division.

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Students: Page 341 Example 2 Simplify the following:

Create a strategy Apply the rule

Apply the idea Divide each term by 3y Simplify Since there are no negative exponents and the expression is already in standard form, the ﬁnal answer is 2y2 − 5y + 8.

Reﬂect and check We can check the answer by multiplying it with the monomial in the denominator. The product should be the numerator in the original expression. 3y (2y2 − 5y + 8) = 6y3 − 15y2 + 24y

Check

Example 3 Purpose triangle shown an area of 13n3 + 11n2 + 29n. Divide aThe polynomial by ahas monomial. Find a simpliﬁed polynomial expression for its height.

Expected mistakes Students may not take out the y from the 24y term when dividing by 3y. Encourage students to cross out variables where possible as a visual reminder when a term no longer needs a variable. Reﬂecting with students n In many cases, dividing by a monomial can be similar to taking out a greatest common factor. Give students a preview of factoring by expanding the polynomial’s terms into a product of primes and expanded variables, and Create a strategy demonstrate how dividing cancels out similar factors. Substitute the expressions into the area of triangle formula

Expand and cancel to divide a polynomial by a monomial Apply the idea Targeted instructional strategies

use with Example 2

Write the area of triangle formula

As an additional method of solution, show students expanding each term in the polynomial and cancel terms to 3 Substitute A = 13n + 11n2 + 29n and b = n find a solution. For example, walk students though each step of the problem shown: Multiply both sides by 2

Write as separate quotients

Evaluate the multiplication

Expand terms

Divide both sides by n

Evaluate the division Cancel matching factors in the Symmetric property of equality numerator and denominator

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Rewrite with remaining factors products 6.03 Simplify Divide polynomials by a monomial mathspace.co

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Since there are no negative exponents and the expression is already in standard form, the ﬁnal answer is 2y − 5y + 8.

Reﬂect and check We can check the answer by multiplying it with the monomial in the denominator. The product should be the numerator in the original expression. 2

3y (2y − 5y + 8) = 6y − 15y + 24y Students: Page 341

Check

Example 3 The triangle shown has an area of 13n3 + 11n2 + 29n. Find a simpliﬁed polynomial expression for its height.

Create a strategy Substitute the expressions into the area of triangle formula

Apply the idea Write the area of triangle formula Substitute A = 13n3 + 11n2 + 29n and b = n Multiply both sides by 2 Evaluate the multiplication Divide both sides by n Evaluate the division Symmetric property of equality

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Purpose Use the formula for area of a triangle and expressions to represent area and the base in order to solve for height. Reﬂecting with students Discuss with students how the process could be applied for the area of different polygons, such as a rectangle.

Students: Page 342

Idea summary When dividing a polynomial by a monomial, we divide each term of the polynomial by the monomial then simplify each individual fraction using the rules of exponents.

Practice What do you remember? 1

Determine whether each statement regarding the division of polynomials by monomials is true or false. Justify your conclusion. a

When dividing a polynomial by a monomial, it is possible to reduce the number of terms.

The degree of the polynomial will always decrease after dividing by a monomial

The result of dividing a polynomial by a monomial will have a constant term if the degree of the monomial matches the degree of any term in the polynomial.

The result of dividing a polynomial by a monomial will be another polynomial.

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Explain the difference between the two expressions: and

Simplify the following:

713

Practice Students: Pages 342–344

What do you remember? 1

Determine whether each statement regarding the division of polynomials by monomials is true or false. Justify your conclusion. a

When dividing a polynomial by a monomial, it is possible to reduce the number of terms.

The degree of the polynomial will always decrease after dividing by a monomial

The result of dividing a polynomial by a monomial will have a constant term if the degree of the monomial matches the degree of any term in the polynomial.

The result of dividing a polynomial by a monomial will be another polynomial.

Explain the difference between the two expressions: and

Simplify the following: a

Simplify the following:

Use algebra tiles to model the equation

Let’s Practice 6

Find the missing length which represents the quotient in each area model. a

−x

−x2

2x −x

−x

−4x

4x2 − 8x 714

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−6x2 + 4x

Simplify the following: a

(30j2) ÷ (35j )

(9w3v2 + 45w2v2 + 18wv) ÷ (9wv)

(b3 − 4b2 + 2b) ÷ (4b)

Fill in the blanks: a

Consider the following statement y5 ÷ y = y4 for all nonzero real numbers y. a

Determine whether the statement is true or false.

Is the statement still true when y is zero?

Consider the area of the following rectangles: Find a polynomial expression for its length. a

Area = (4x4 − 8x) square units

Area = (6x3 + 4x2 + 10x + 14) square units

? 2 4x

The triangle shown has an area of 12n3 + 18n2 + 5n. Find a polynomial expression for its height.

Let’s extend our thinking 12

Fill in the blanks to make a true algebraic statement. a

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Consider the following statement: x9 ÷ x3 = x3 for all nonzero real numbers x a

What is x9 ÷ x3 actually equal to?

Identify a nonzero real number for which the original statement is true.

Consider the problem

Identify and correct the error in each of the following student’s work. Lawrence:

Marika:

716

Create an example and describe the result when a polynomial is divided by a monomial that: a

Has a coeﬃcient larger than the coeﬃcients of the terms in the polynomial.

Has a variable with a larger degree than the terms in the polynomial.

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Answers

w2v + 5wv + 2

6.03 Divide polynomials by a monomial

n 3x2 + 4x

What do you remember?

p 6x3 − 4x2 + 8

r 4x3 − 7x2 + 3x + 6

1 a F alse. We can’t create a coeﬃcient of zero when dividing a single term of a polynomial by a monomial so no terms will eliminate. b False. If you divide by a constant monomial the degree will not change. c True. Suppose the monomial has degree n and the term has degree n. Then

will be a constant.

d F alse. If the degree of the monomial is larger than the degree of any of the terms in the polynomial the result will not be a polynomial. 2 In the ﬁrst expression we are dividing all terms in the numerator by 5. In the second expression only the constant term at the end is being divided by 5.

8 a

9 a True

b No

10 a (x3 − 2) units

b (3x3 + 2x2 + 5x + 7) units

11 24n2 + 36n − 10 units Let’s extend our thinking 12 a b

3 a

4 a

13 a x6

14 L awrence subtracted both the coefficients and the exponents of the variable. However, when dividing by a monomial the coefficients should be divided and the exponents should be subtracted. Marika, on the other hand, divided both the coefficients and the exponents when she should have divided the coefficients and subtracted the exponents.

3x − 4

5 x

−1

−x

6x2 − 8x

b 1

15 a E ach term of the resulting polynomial will have a fractional coeﬃcient.

b T he terms will have negative exponents and the expression will no longer be a polynomial.

Let’s practice 6 a −x + 2

b −3x + 2

7 a

c 3x

d 2p4

g 6x3 − 4x2 + 8

h 8z + 1

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6.04 Factor GCF Subtopic overview Lesson narrative In this lesson, students will learn about make generalizations about dividing polynomials by a monomial through the modeling of contextual situations involving dividing polynomial expressions by common factors and structures of polynomials. In the lesson, students will make mathematical connections to greatest common factors of integers and the distributive property to rewrite polynomial expressions in factored form. Students will be able to justify the equivalence of polynomial expressions and engage in error analysis. By the end of the lesson, students will be able to create polynomial expressions to represent quantities for contextual situations and work flexibly between standard and factored forms of polynomial expressions, as needed.

Learning objective Students: Page 345

Key vocabulary 

common factor



factor



greatest common factor (GCF)

Essential understanding Factoring out the greatest common factor provides the foundation for all other techniques for factoring quadratic expressions.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving

MPG3 — Mathematical Reasoning

Teachers can incorporate problem-solving into their lessons by presenting real-life scenarios that require factoring polynomials to solve. For example, propose a problem where students must use their knowledge of polynomials and factoring to determine the dimensions of a rectangular garden given its area and length. Encourage students to apply their skills to solve these problems and discuss the strategies they used.

Teachers can incorporate mathematical reasoning into their lessons by asking students to validate the steps they took to factor a polynomial. For example, after factoring out the GCF, students could be asked to use the distributive property to confirm the correctness of their factoring. This encourages them to think critically and make logical connections between mathematical procedures.

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MPG4 — Mathematical Connections Teachers can help students make connections by relating the process of factoring polynomials to prior knowledge of algebraic expressions and operations on polynomials. Additionally, teachers could highlight how factoring polynomials can be applied to other areas of math and real-world situations, such as calculating the area or volume of shapes.

Content standards A.EO.2 — The student will perform operations on and factor polynomial expressions in one variable. A.EO.2c — Factor completely first- and second-degree polynomials in one variable with integral coefficients. After factoring out the greatest common factor (GCF), leading coefficients should have no more than four factors.

Prior connections 8.PFA.1 — The student will represent, simplify, and generate equivalent algebraic expressions in one variable.

A.EO.1 — The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

Future connections A2.EO.3 — The student will perform operations on polynomial expressions in two or more variables and factor polynomial expressions in one and two variables.

Engage Activity Food hall trays

60 mins

Students will design various rectangular food trays with a specific area and find the length and width of each one.

Understanding and skills

Will use

Will develop

Dividing rational numbers.

Dividing a polynomial expression by a monomial with integer coefficients.

Adding, subtracting, and multiplying polynomials.

Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: Paper, pencil (recommended)

Support students with disabilities Support conceptual processing - understand mathematical relationships and make connections To support students in making connections between visualizing how to find the side lengths of a rectangle, remind students of the area model used for multiplying polynomials. This will help students check if their side lengths have the desired area as well as begin to generalize reasoning for how to divide polynomials by a monomial.

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Support for English language learners Critique, correct, and clarify Before students share their responses, display the following incorrect calculation and reasoning: A food tray with side lengths 10xy and 10x2y + 6x has an area of 20x3y + 60xy2. Ask students to identify the errors, critique the reasoning, and write a short explanation of how the students’ reasoning could be improved.

Classroom guide Hook

Notice and wonder

Students write observations about an equation where the left hand side is in factored form and the right hand side is in expanded form. There is a unknown factor on the left hand side of the equation.

•

5 mins

What do you notice? What do you wonder? (x + 3)(?) = x3 + 3x2 Slide 1 from Student Engage Activity

Implementation details Students may notice that the left side of the equation is in factored form, and the right side is in expanded form. They may wonder what the value of ? is, or wonder what values of x make the equation true. Encourage students to think about what the unkown value represented by the question mark would be. Highlight responses that explain why ? = x2.

Launch

5 mins

Amuse-me amusement parks plans to design rectangular food trays for the park’s food hall. They want to create multiple types of rectangular food trays with the same area. Each food tray will consist of two sections, which will vary in dimensions.

Slide 2 from Student Engage Activity

Give students time to read the information individually before forming groups. Important mathematical concepts: Length, width, area, dimensions Important contextual information: Food trays Suggested grouping: Form groups of 3 or 4 and assign numbers

Continue when Students have read the Launch and understand the context of the problem.

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Explore

Numbered heads together

•

35 mins

Anticipated strategies Students who have successfully completed the activity will have three different food trays with an area of 20x3y + 60xy2 and have determined the length and width of each food tray. Students will also have chosen one food tray and justified why they would use it at the amusement park. The greatest common factor of 20x3y + 60xy2 is 20xy. The common factor is composite so there are multiple food trays that could be created, such as with a width of 4xy and a length of 5x2 + 15y. There is no correct answer for which food tray to choose and encourage students to justify their reasoning. While x and y are unknown values, some students may plug in values for x and y and justify their choice by how narrow or wide the tray is. Students may justify their answer by choosing trays that are closer to a square in their proportions, or justify not selecting a tray because it is too long and narrow to reasonably use for putting food and drinks on it. Finding dimensions using an area model Students may use the rectangle shape with two sections from the launch to find the dimensions of the different food trays, which is similar to using an area model to find missing terms in a polynomial expression. Algebraically finding dimensions Students may notice common factors in the two terms and algebraically guess and check or multiply terms to determine dimensions that when multiplied produce the same area.

Misconceptions Using dimensions that do not produce equivalent areas What is the area of your food tray? How do you know? How can you check that the dimensions of the tray produce the correct area? What is the area formula for a rectangle?

Purposeful questions • What are the dimensions of your food trays? How do you know that they have equal area? • Which food tray would you pick to use? Why? • Can you justify that your food trays have the same area in another way? • What if you knew a side length and the area, how could you find the missing side length?

Continue when Students have determined the dimensions of each tray and showed that the area is correct for their length and width. Students have also picked one food tray for the amusement park and justified why they picked it.

Discuss

15 mins

Have a group discussion where groups can share their tray designs, as well as justify the area is correct for their chosen dimensions. Consider making connections from the discussion to dividing polynomials by monomials in general.

Discussion guide Choose several groups to share their food tray of choice, the dimensions of the food tray, and their process for how they found the dimensions of the tray. Write the dimensions of the food trays and summary of reasoning on the board as students share. Encourage students that worked more visually and more algebraically to share and compare methods. If students have chosen some of the food trays with the same dimensions, ask students to share any other food tray possible dimensions that have not been shared and any methods that have not yet been shared to determine the dimensions of the food tray. 6.04 Factor GCF mathspace.co

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Next, ask the class: Can you determine the side length of a rectangle if you know the area and one of the sides? And allow students time to respond and share their thoughts with the class.

Tools You may ﬁnd this tool helpful: • Scientific calculator

Student lesson & teacher guide Factor GCF Students begin the lesson with a review of what a greatest common factor (GCF) is, and learn about what part of variable expressions are part of a GCF. They then engage in an exploration determining the greatest common factor of two expanded variable expressions.

Students: Page 345

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Compare and connect English language learner support To support students in comparing and connecting different methods of finding the Greatest Common Factor (GCF) and applying it to various polynomial expressions, begin by introducing the two primary methods: prime factorization and listing factors. Students will compare and connect methods of factoring out a GCF using a common polynomial as an example, For example, 12x3 − 18x2 Split students into two groups, where one group finds the greatest common factor by prime factorization, and the other finds the greatest common factor by listing factors. Both groups should factor out the greatest common factor. Each group should create a display of their process that includes: • Steps for finding the GCF with the given method, both mathematically and in words. • Show the GCF being divided out of the polynomial expression. • Write the final factored polynomial expression. • Show how the GCF outside the parentheses can be distributed to recreate the original polynomial. Students should go on a gallery walk to compare the different methods, discussing the advantages and disadvantages of each. Next, guide students should test the comparisons of these methods by applying them to various polynomial expressions with different numbers of terms, variables, and exponents. Encourage students to discuss in groups the challenges and efficiencies of each method, noting which method they preferred for different types of polynomials. Have groups present their findings, fostering a classroom dialogue on why certain methods may be more effective in different scenarios. Students should reflect on these comparisons using sentence frames like: • I found the method of prime factorization useful because … • When dealing with polynomials with subtraction, I noticed … Finally, consolidate these discussions into a reference chart created collaboratively with student input, ensuring it evolves as a resource throughout the unit. This approach deepens students’ understanding of factoring out the GCF and helps them connect various methods to different polynomial formats.

Use area models to visualize factoring a GCF Student with disabilities support Using an area model to divide a polynomial by its GCF may be helpful in organizing a student’s thinking. For example, when factoring 2x out of 8x3 + 20x2 − 14x, the division

becomes the example shown, where the first term has been completed for you.

4x2

8x3

20x2

−14x

Show students how the size of the model can be adjusted for the number of terms, and how to ensure that terms are connected to the correct operations of + or −.

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Taking out a factor smaller than the greatest common factor Address student misconceptions Students might only factor out a common factor, rather than the greatest common factor. Encourage students to check all terms in the parentheses to ensure there are no remaining common factors. If there are any remaining common factors, they can be factored out in another step. For example: 1

8x2y3 + 12x3y5 = 2xy(4xy2 + 6x2y4) 2

Factor out the common factor of 2xy 2

= 2xy ⋅ 2xy (2 + 3xy ) Factor out 2xy2 from the parentheses

= 4x2y3(2 + 3xy2)

Simplify to notice the GCF was 4x2y3

Notice after step 1 that there is a still a common factor of 2xy2, so we can factor that out in another step and then simplify.

Concrete-Representational-Abstract (CRA) Approach Targeted instructional strategies Concrete: Begin by engaging students with physical manipulatives to explore factoring polynomials. Use algebra tiles or colored counters to represent the terms in a polynomial expression. For example, to represent the polynomial 6x + 12, provide six x tiles and twelve unit tiles. Have students physically group the tiles to find common factors. Encourage them to arrange the tiles into equal groups or rectangles, showing how the terms can be divided by the greatest common factor (GCF). This hands-on activity helps students see how common factors are shared among terms in a polynomial. Representational: Transition to the representational stage by having students draw pictures of the manipulative arrangements. Ask them to sketch the grouped tiles or rectangles they created. For instance, they can draw a rectangle divided into sections that represent the factors of the polynomial. Label the sides of the rectangle with the factors (e.g., one side labeled 6, the other side labeled (x + 2) for 6(x + 2). These drawings help students connect the physical grouping to a visual model, reinforcing the concept of factoring out the GCF. Abstract: Move on to the abstract stage by introducing the symbolic method of factoring polynomials. Teach students how to identify the GCF of the coefficients and variables in the terms. Using the example (6x + 12), guide them to see that the GCF is 6. Show them how to factor out the GCF: 6x + 12 = 6(x + 2). Provide practice with various polynomial expressions, having students factor out the GCF using algebraic notation. Emphasize how this symbolic process relates to the concrete and representational stages they’ve worked through. Connecting the stages: Help students make connections between all three stages by referring back to the manipulatives and drawings when working abstractly. Ask guiding questions like: • “How does factoring out the GCF here relate to the groups you made with the tiles?” • “Can you picture the rectangle you drew when you see this factored expression?” Encourage students to use the representation that makes the most sense to them. By linking the concrete materials, visual drawings, and abstract symbols, students can better understand and monitor their thinking about factoring polynomials.

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Exploration Students: Page 345

Suggested student grouping: In pairs In this exploration, students will be working on understanding the concept of expanding algebraic expressions and determining the greatest common factor (GCF). They will compare two expanded expressions to identify the common terms and calculate the GCF. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. How many y’s are common in both expressions? There are three y’s common in both expressions. 2. How many z’s are common in both expressions? There is one z common in both expressions. 3. What is the GCF of the two expressions? The GCF of the two expressions is y3z. Purposeful questions • How can exponents be used to represent the number of each type of variable common to both expressions? • How does the exponent of the variable greatest common factor connect to the exponents of the original two terms? Possible misunderstandings • Students may assume a greatest common factor must include a number other than 1 as a coefficient and include 5 and 2 in their GCF. They may also include the x3 since it will have the same exponent as the y3 in the GCF. Students are reminded of multiplying a monomial and a polynomial so that they can relate this to factoring a monomial out of a polynomial. This is described as the opposite of distribution. Students are then provided the steps for factoring out a GCF from a polynomial expression.

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Students: Page 345

Examples Students: Page 346 Example 1 Find the greatest common factor of the given terms. a 60 and 24.

Create a strategy List the prime factorization of 60 and 24, then determine the common factors that comprise the GCF.

Apply the idea The prime factorization of 60 is 2 ⋅ 2 ⋅ 3 ⋅ 5 The prime factorization of 24 is 2 ⋅ 2 ⋅ 2 ⋅ 3. The GCF is the product of the common factors: 2 ⋅ 2 ⋅ 3 = 12. Therefore, the GCF of 60 and 24 is 12.

Reﬂect and check We can also create factor trees for 60 and 24, then identify the common factors. Factor tree of 60

Factor tree of 24

24 30

2 15

2 3

12 6

2 5

60 = 2 ⋅ 2 ⋅ 3 ⋅ 5

24 = 2 ⋅ 2 ⋅ 2 ⋅ 3

The GCF is the product of the common factors: 2 ⋅ 2 ⋅ 3 = 12. Therefore, the GCF of 60 and 24 is 12.

b 60x3y2 and 24xy4.

Create a strategy 726

List the whole number factors of the coeﬃcients of 60x3y2 and 24xy4 and ﬁnd the expression with the lowest power of each of the variables. Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Apply the idea

Reﬂect and check

We know that the largest whole number that 60 and 24

We can also expand both expressions to find the common

The GCF is the product of the common factors: 2 ⋅ 2 ⋅ 3 = 12. Therefore, the GCF of 60 and 24 is 12.

Reﬂect and check We can also create factor trees for 60 and 24, then identify the common factors.

Purpose Factor tree of 60 Show students how to find the GCF of the integers. 60

Factor tree of 24 24

Expected mistakes 30 is the GCF because 2 it is a common 12 Students may state that a2common factor factor of the two integers, as opposed to the actual GCF. If you work through the prime factorization of the integers, show students that the 15 6 2 2 to get the GCF. common factors of the integers must be multiplied together in order 5 3 3 2 Reﬂecting with students Ask students to divide the original two numbers by the greatest common factor and then break down the results 60 = 2 ⋅ 2 ⋅ 3 ⋅ 5 24 = 2 ⋅ 2 ⋅ 2 ⋅ 3 into their prime factorizations. Ask students to notice whether there are any remaining common factors in the two factorizations. The GCF is the product of the common factors: 2 ⋅ 2 ⋅ 3 = 12.

Students:Therefore, Page 346 the GCF of 60 and 24 is 12. b 60x3y2 and 24xy4.

Create a strategy List the whole number factors of the coeﬃcients of 60x3y2 and 24xy4 and ﬁnd the expression with the lowest power of each of the variables.

Apply the idea

Reﬂect and check

We know that the largest whole number that 60 and 24 are divisible by is 12. The expression with the lowest power of each of the variables is xy2.

We can also expand both expressions to find the common factors:

Putting this together, the greatest common factor is 12xy2.

60x3y2 =

2 ⋅ 2 ⋅ 3 ⋅5⋅ x ⋅ x⋅ x⋅ y ⋅ y

24xy4 =

2 ⋅ 2 ⋅2⋅ 3 ⋅ x ⋅ y ⋅ y ⋅y⋅ y

So, the GCF is 2 ⋅ 2 ⋅ 3 ⋅ x ⋅ y ⋅ y = 12xy2.

346

Mathspace

Virginia SOL Algebra 1

mathspace.co Purpose Show students the GCF of the terms by focusing on the coefficients and then focusing on the variables.

Expected mistakes Students might assume that they need to multiply all of the variable terms in the expressions. Instead, expanding each of the variable terms and determining the common terms between each expression, similarly to finding a GCF of integers with prime factorization, will work. Reﬂecting with students Ask students why the lowest power of each of the variables would lead to the GCF of the variable terms. Use the expected mistake to show students why expanding and then multiplying the common variables between the expressions leads to the lowest power of each of the variables as the GCF.

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Finding greatest common factors with variables Targeted instructional strategies While students have experience finding the greatest common factor (GCF)of numbers, students would benefit from practice finding the GCF of terms with numbers and variables. To connect the process of finding the greatest common factor of numbers and variables, provide a pair of terms and show the prime factorization method. For example, 24y3 and 56y5: 24y3 = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ y ⋅ y ⋅ y

Expand into prime factorization

56y = 2 ⋅ 2 ⋅ 2 ⋅ 7 ⋅ y ⋅ y ⋅ y ⋅ y ⋅ y 24y3 = 2 ⋅ 2 ⋅ 2 ⋅

⋅y⋅y⋅y

56y5 = 2 ⋅ 2 ⋅ 2 ⋅

⋅y⋅y⋅y⋅

I dentify matching pairs of factors and cancel non-matching factors ⋅

=2⋅2⋅2⋅y⋅y⋅y 3

Write out product of matching factors

GCF of 24y and 56y = 8y

Simplify

Point out that the greatest common factor of terms with the same variables is the variable with the lower exponent.

Students: Page 347 Example 2 Factor the expression 8x2 + 4x.

Create a strategy Find the GCF and divide it out of each term.

Apply the idea

Reﬂect and check

The GCF of 8x2 and 4x is 4x.

Although the term 4x is in the original expression when it is factored out the second term does not become zero. Otherwise, when we check the answer by distributing the multiplication, 4x will be lost altogether.

Dividing out the GCF, we get: 8x2 ÷ 4x = 2x 4x ÷ 4x = 1 So we have:

We can check our factorization using the distributive property:

8x2 + 4x = 4x(2x) + 4x(1)

4x(2x + 1) = 4x(2x) + 4x(1) = 8x2 + 4x

= 4x(2x + 1)

Example 3 Purpose Factor thehow expression 3x(x −the 4) +GCF 7(x − 4). Show students to identify and then factor it. Expected mistakes Create a strategy StudentsThis may only part are of the GCF, such as think theyidentify havethe ﬁnished time, the factor expressions already factored. We 2x, can and use this to help GCF. factoring the expression. Remind students to always check if there are any remaining common factors required to factor. They can do Apply thefor idea this by checking any common variables in all the terms, and any common factors in the coeﬃcients of all the terms. In particular, notice that both terms 3x(x − 4) and 7(x − 4) have a factor of (x − 4). The remaining parts of each expression, 3x and 7, have no factors in common. So the GCF is (x − 4), which we can use to factor the expression:

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Idea summary

Reﬂecting with students Encourage advanced learners or all students to create their own polynomial expressions and show how to factor out the greatest common factor (GCF). Allow them to choose different coefficients or variables and challenge them to use a polynomial that is longer than a binomial. Ask them to explain why the GCF is what it is in each case and how factoring simplifies the expression.

Example 2 exponent rules and disappearing terms Misapplying

use with Example 2

Address student misconceptions Factor the expression 8x2 + 4x. Students may attempt to apply the laws of exponents to the coefficents as well as the exponents, which may a strategy lead to aCreate constant being eliminated from a final answer. This could result in a solution like: Find the GCF and divide it out of each term.

Apply the idea

Reﬂect and check

The GCF of 8x2 and 4x is 4x.

Although the term 4x is in the original expression when it is factored out the second term does not become zero. Dividing out the GCF, we get: Otherwise, when we check the answer by distributing the 2 8x ÷ 4x = 2x multiplication, will be lost altogether. Challenge this misconception by4x having students write the numerator4xand denominator in expanded form to ÷ 4x = 1 We can check our factorization using the distributive refresh their memories as to why we subtract the exponents when dividing. property: So we have: 8x2 + 4x = 4x(2x) + 4x(1)

Students: Page 347

4x(2x + 1) = 4x(2x) + 4x(1) = 8x2 + 4x

= 4x(2x + 1)

Example 3 Factor the expression 3x(x − 4) + 7(x − 4).

Create a strategy This time, the expressions are already factored. We can use this to help identify the GCF.

Apply the idea In particular, notice that both terms 3x(x − 4) and 7(x − 4) have a factor of (x − 4). The remaining parts of each expression, 3x and 7, have no factors in common. So the GCF is (x − 4), which we can use to factor the expression:

Idea summary

Purpose Follow these steps for factoring out a GCF: Show students1. an idea of what factoring by grouping, which will appear in a future lesson, will look like when Identify the GCF factoring a GCF. 2. Rewrite each term as a product of the GCF and the remaining factors 3. Rewrite the whole expression as a product of the GCF and the remaining factors in parentheses Expected mistakes Students may see that (x − 4) is common, but not necessarily see the expression as a factor. Remind students that a factor is a number or quantity that is multiplied, and we read (x − 4) as “the quantity of x minus 4.”

Reﬂecting with students Ask students to verify that the given expression is equivalent to its factored form by performing the 6.04 Factor GCF multiplication for both.

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Example 3 Factor the expression 3x(x − 4) + 7(x − 4).

Create a strategy

Abstract binomial GCF by drawing boxes This time, the expressions are already factored. We can use this to help identify the GCF.

use with Example 3

Student with disabilities support

Apply the ideato abstract problems that appear complex. For example, for binomial common factors, Encourage students In particular, notice termsas 3x(x − 4) and 7(x − 4) variable have a factor (x − 4).substituted back in. show them that they canthat beboth written a box or single andofthen The remaining parts of each expression, 3x and 7, have no factors in common. So the GCF is (x − 4), which we can

Make the explicit connection to the situation where the GCF is a monomial such as 3ax + 7a and have students use to factor the expression: factor to get a (3x + 7). We can also draw a box around the binomial so students consider it as a single term giving 3x⬚ + 7⬚ = ⬚ (3x + 7), eventually leading to 3x(x − 4) + 7 (x − 4) = (x − 4) (3x + 7).

Students: Page 347

Idea summary Follow these steps for factoring out a GCF: 1. Identify the GCF 2. Rewrite each term as a product of the GCF and the remaining factors 3. Rewrite the whole expression as a product of the GCF and the remaining factors in parentheses

Practice

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Students: Pages 348–350

What do you remember? 1

For each of these numbers: i

List the factors.

6 and 12

9 and 24

State the greatest common factor.

14 and 32

28 and 42

For these prime factorizations: 180 = 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5 600 = 2⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5 ⋅ Find the greatest common factor of 180 and 600.

730

For each of these algebraic expressions: i

List the factors.

3x2 and 2x

12xy4 and 24xy

State the greatest common factor.

14 and 7abc

x2y3 and x2y2

Identify the greatest common factor between the following sets of terms: a

8a and 9a

3x and 6x2

4b and b2

4y2 and 6y2

5p , 3p and p

45n, 55n2 and 20n2

4m2, −7m, 8m and −14m2

−42k, −21k2, −7k2 and −28k3

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Simplify the expressions: a

i 6

(b3 − 4b2 + 2b) ÷ (4b)

d (30j2) ÷ (35j )

Fill in the blanks to make a true algebraic statement.

c 7

Determine which of the following represent a factored form of the expression −12x + 20x2: a

−4 ⋅ 3x + 5

−2(6x − 10)

4x(5 − 3x)

x (−12 + 20x)

Let’s practice 8

Complete each factorization: a d

y2 + 5y = y(⬚ + ⬚)

−m2 + 19m = ⬚ (m − 19)

Factor the expressions:

2t2 + 2t = 2t(⬚ + ⬚)

−y2 − 2y = ⬚ ( y + 2)

3y2 + 6y = ⬚ ( y + 2) 8v − v2 = v(⬚ − ⬚)

6v + 30

−2s − 10

−12s + 10

y2 + 4y

2u2 − 8u

4t + 2t2

42x − x2

9z2 – 18z

r3 + r2 + 6r

w(w − 2) − (w − 2)

9(3x + 4) + 4(3x + 4)

8t(t − 3) + 9(t − 3)

Completely factor each of the polynomial. a

−6y4 + 14y3 − 10y2

14h4 + 28h6 + 56h3

18x2 − 24x + 36 − 72x4

−3a3 − 9a2 − 15a4

18m7 − 15m3 + 14m + 35m6

−d12 + d8 − d15

0.1h + 0.2h3 − 0.4h5

g 11

Xander was asked to factor the expression 35x2y + 10xy2 − 5xy. Identify his error. Xander: The greatest common factor is 5xy so the factored form of the expression is 5xy (7x + 2y).

Alex and Beth are both asked to factor −5x + 10y. Alex wrote down −5(x − 2y) and Beth wrote down 5(2y − x). Who is correct? Explain.

A farmer wants to create a set of adjacent fields which all have the same width. He plans to create the smallest field in the shape of a square, with the largest field 9 times the size of the smallest field, and the middle field to have a length that is 5 units more than the smallest field. a

Write expressions for the area of each ﬁeld.

Use factoring to determine the dimensions of the entire ﬁeld.

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Let’s extend our thinking 14

Identify the common factor of any two terms.

Identify the greatest common factor between the sets of terms:

45n, 55n2 and 20n2

4m2, −7m, 8m and −14m2

−42k, −21k2, −7k2 and −28k3

45n, 55n2y and 40mn3

3a2mn, 2ya4x and 5xma3

3a2bx, 4na4m and 12mba3

Fully factor each expression: a

44uv − 8u2v

−8w2 + 3w2y

5k2t + 40k2t2

49p2q − 28pq2

−16a2 − 18a2b

−30w2 − 25w2y

−10u v + 9uv

4x + 12 + 16yx

3x + 9 + 12yx

2x + 10x2y + 8yx

5ab2c + 25bc3 + 100abc

5a2b2 + 2ab − 3a2b2 − 4ab

4x y + 8xy + 12xz

m 30b2c + 10abc + 20c2 o

5x2y2 + 15yz2 + 25y2 + 60y

Create an expression with at least three factorizations. Then, write out each factorization including the fully factored form.

Explain how dividing monomials relates to factoring the greatest common factor.

Explain why the greatest common factor of the variables in any expression has the least possible exponent of any of the terms.

The rectangle shown has an area of 4x2 − 16x square units.

What expression describes the length of the rectangle? 4x

732

A property management company has a rectangular plot of land available for parking. The area of this plot is represented by the expression 20n + 5n2. a

Determine the dimensions of the block of land.

Given the the length is longer than the width for all n > 1, which expression is the length and which is the width?

The length is further divided into 5 equal sections, and one of these sections is fenced off for storage. What is the length of fencing needed?

You are to design a photo collage made of two large square photos with a side length of x and four smaller rectangular photos that have a height of x and a width of 4 inches. a

Find an algebraic expression for the area of the rectangle formed if the photos are all placed in a single row. Draw an example of what this arrangement would look like.

Fully factor your answer from part (a) and then draw a photo arrangement that would match these dimensions.

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Answers

b T he area of the entire field is the sum of each smaller field: x2 + 9x2 + x(x + 5). The factored form of this expression is x(11x + 5) so the width of the field is x and the length is 11x + 5.

6.04 Factor GCF What do you remember?

Let’s extend our thinking

1 a i 1, 2, 3, 6 ii 6

1, 2, 3, 4, 6, 12

b i 1, 3, 9 ii 3

1, 2, 3, 4, 6, 8, 12, 24

c i 1, 2, 7, 14 ii 2

1, 2, 4, 8, 16, 32

d i 1, 2, 4, 7, 14, 28 ii 14

1, 2, 3, 6, 7, 14, 21, 42

14 1

2 60 3 a x

b 12xy

c 7

d x2y2

4 a a

b 3x

c b

d 2y2

g m

h − 7k

e p

c 8p3

e 4x3 + 5x2 - 7

5 a

8z + 1

j 6 a 16a3

b 2x

c 14m3 and 10m

7 a No

b No

c No

d Yes

Let’s practice 8 a y(y + 5)

b 2t(t + 1)

c 3y(y + 2)

e −y(y + 2)

9 a 6(v + 5)

b −2(s + 5)

c −2(6s − 5)

d y(y + 4)

e 2u(u − 4)

2t(2 + t)

g x(42 − x)

h 9z(z − 2)

r(r2 + r + 6)

k 13(3x + 4)

(t − 3) (8t + 9)

d −m(m − 19)

(w − 2) (w − 1) 2

−5w2(6 + 5y)

g uv(−10u + 9v)

h 4(x + 3 + 4xy)

3(x + 3 + 4yx)

2x(1 + 5xy + 4y)

k 4x(xy + 2y2 + 3z2)

5bc(ab + 5c2 + 20a)

m 10c(3b2 + ab + 2c)

17 Answers will vary. For example: 2x2 + 4x has three factorizations: 2(x2 + 2x), x(2x + 4), and the fully factored form which is 2x(x + 2) 18 When the greatest common factor is a monomial, we can use monomial division to determine what expression is left after the factoring is complete. 19 The greatest common factor represents the largest value that can be divided evenly from each term. If the greatest common factor of a variable had an exponent larger than any of the original terms, then dividing it out would leave a negative exponent. For example, consider the expression x2 + x3. If we divide both terms by x2 (the smaller exponent) then we are left with 1 + x. If instead we try to divide both terms by x3 (the larger exponent) then we would be left with x-1 + 1. 20 x − 4 21 a 5n(4 + n), so the dimensions are 5n and 4 + n b Length = 5n, Width = 4 + n c E ach section is now n ⋅ (4 + n) and to fence this the perimeter is 8 + 4n units of fencing. 22 a x(2x + 16)

−d8(d4 − 1 + d7)

13 a L et the width of the smallest field be x so that its area is x2. The largest field has an area of 9x2 and the middle field has an area of x(x + 5).

h 0.1h(1 + 2h2 − 4h4)

12 They are both correct. If you expand the answer each gave you will see that they both produce −5x + 10y.

o 5y(x2y + 3z2 + 5y + 12)

n 2ab(ab − 1)

but Xander eliminated the term instead.

11 When factoring 5xy out of the term –5xy you get

c 5k2t(1 + 8t)

e −2a (8 + 9b)

b 14h (h + 2h + 4)

e m(18m − 15m + 14 + 35m )

b w2(3y − 8)

d 7pq(7p − 4q)

d −3a2(a + 3 + 5a2)

d 5n

16 a 4uv(11 − 2u)

c 6(3x2 − 4x + 6 − 12x4) 6

e a2

10 a −2y (3y − 7y + 5)

v(8 − v)

c −7k

b m

15 a 5n

b 2x(x + 8)

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6.05 Factor by grouping Subtopic overview Lesson narrative In this lesson, students will learn how to factor a polynomial expression using a method of grouping. They will be reminded of their previous skills of factoring out the greatest common factor in order to prepare them for the grouping method. In addition to learning the process of factoring by grouping, students will learn to identify the structures that allow for factoring by grouping and justify the equivalency of polynomial expressions.

Learning objective Students: Page 351

Key vocabulary 

factor by grouping

Essential understanding Factoring by grouping is an application of factoring out the greatest common factor. It is a standard algorithm that can be applied to factoring any factorable quadratic expression.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG3 — Mathematical Reasoning

MPG5 — Mathematical Representations

Teachers can incorporate this goal by encouraging students to use a variety of methods to represent their factoring by grouping process. This could include diagrams, flowcharts, or symbolic notation. Students could also be asked to translate between different representations, such as interpreting a written description of the factoring process in mathematical notation.

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Content standards A.EO.2 — The student will perform operations on and factor polynomial expressions in one variable. A.EO.2c — Factor completely first- and second-degree polynomials in one variable with integral coefficients. After factoring out the greatest common factor (GCF), leading coefficients should have no more than four factors.

Prior connections 8.PFA.1 — The student will represent, simplify, and generate equivalent algebraic expressions in one variable.

A.EO.1 — The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

Future connections A2.EO.3 — The student will perform operations on polynomial expressions in two or more variables and factor polynomial expressions in one and two variables.

Engage Activity Treasure ﬂoor

60 mins

Students will create a tiled ﬂoor design for an amusement park game and determine the area and dimensions of the rectangular shape created in multiple ways.

Understanding and skills

Will use

Will develop

Factoring greatest common factor (GCF).

Rewriting polynomials with four terms as the product of two linear factors.

Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: Pencil, paper • Download and print copies of the students graphic organizer from the student Launch slide.

Support for English language learners Collect and display As pairs are working, listen for and collect vocabulary, phrases, and methods students use for designing their tiled ﬂoor as well as ﬁnding the area and dimensions of their design and other rectangular space. Continue to update collected student language throughout the entire activity. Remind students to borrow language from the display as needed. Some terms and phrases may include: factoring, greatest common factor, grouping, multiply, divide

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Support students with disabilities Support visual-spatial - create and interpret visual representations Provide printed copies of the downloadable asset to students containing multiple tiles and the dimensions of each tile. This will help students visualize the design they are creating as well as support their algebraic justiﬁcation.

Classroom guide Hook

Co-craft questions

Students create questions about a design made up of rectangular pieces that ﬁt together.

•

5 mins

What mathematical questions could we ask about this image?

Implementation details Students may produce questions such as: • What are the dimensions of the overall rectangle? • What is the area of the overall rectangle? • What are the dimensions of the individual rectangles? • What is the area of each individual rectangle? Highlight student questions regarding the dimensions of the rectangle, area, and quantity of each piece.

Slide 1 from Student Engage Activity

Launch

5 mins

Amuse-me amusement park is creating a new attraction where patrons will search for treasure in an Egyptian tomb. One part of the attraction that has not been designed yet is the tiled floor. As part of the attraction, patrons must step on the correct tiles to avoid setting off a trap and losing the treasure. The four tiles that are to be used in the design are rectangular and have the following dimensions:

1 x2

x 8

Slide 2 from Student Engage Activity

Provide students with the downloadable asset which has each tile on it. You may wish to have students cut out rectangles to use while creating their design in the Explore. Suggested grouping: Form pairs

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Explore

Think-pair-share

•

25 mins

Anticipated strategies Use the manipulatives to design the tiled floor Students may use the manipulatives to create the floor design and find the area. Work algebraically from the start: Students may find the area of a rectangular design algebraically and then sketch the design.

Misconceptions Creating a non-rectangular design What would be the dimensions of your larger rectangular design? Can you show me visually and algebraically?

Purposeful questions • What is your design so far? Is it rectangular? How do you know? • What is the area of the rectangular shape created by using one of each type of tile? Can you show me visually and algebraically? • Can you write the area in any other way? • What is the area and dimensions of your own design? How do you know? • If you only knew the area of the design, could you ﬁnd the dimensions of the rectangles?

Continue when Students have created a design, labeled the dimensions, and calculated the area.

Discuss

25 mins

Discussion guide Have students share their tile ﬂoor designs as well as the area and dimensions of the larger rectangular shape. Ask students how they determined the area and dimensions. Encourage groups that found the dimensions visually to share and then ask groups that focused on the algebraic justiﬁcation. As an extension you may provide the following prompts: • Rewrite the area of your design in as many equivalent ways as possible • What is the area of the design made by using each tile exactly once? After using the extension prompts you may have a discussion about how you can verify several forms of an expression are equivalent.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lesson: Algebra 1 — 6.04 Factor GCF

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Tools You may ﬁnd these tools helpful: • Scientific calculator • Highlighter

Student lesson & teacher guide Factor by grouping Students begin with an exploration that involves factoring a GCF. Students are asked the possible factorization of an unfamiliar polynomial expression that involves factoring a GCF.

Students: Page 351

A greatest common factor of 1 Address student misconceptions Consider the example: Factor the expression 3a3 − 4a2 + 3a − 4. Solution: 3a3 − 4a2 + 3a − 4 = (3a3 − 4a2) + (3a − 4) 2

Split based on common factors

= a (3a − 4) + 1 (3a − 4)

Factor out the GCF (a2 and 1)

= (3a − 4) (a2 + 1)

Factor out the common binomial factor

Since (3a − 4) (a + 1) cannot be factored further, it is the ﬁnal answer. Some students may struggle with going from step 1 to step 2 as they are not sure what to factor out of 3a − 4 as there is not common factor other than 1. Encourage students to consider the reverse operation what we would need to multiply by to get back 3a and −4 if we were distributing the multiplication, to provide further scaffolding, next ask what the Multiplicative identity property says or have students ﬁll in the blank in 3a = ⬚ ⋅ 3a.

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Connecting greatest common factors and factoring by grouping Targeted instructional strategies To connect factoring out the greatest common factor and factoring by grouping, display the following binomials and have students factor out the greatest common factor of each: 6x2 + 9x and −4x − 12 Once students factor each binomial, display the factored forms and ask what the two binomials have in common. 6x2 + 9x = 3x(2x + 3) −8x − 12 = −4(2x + 3) Show students that polynomials with four terms can be separated into smaller polynomials to factor separately. Ask students to predict what steps factored 6x2 + 9x − 8x − 12 into (3x − 4) (2x + 3).

Critique, correct, clarify English language learner support Ask students to factor the expression −2a3 + 2 + 2a2 − 2a. Before students share their responses, display the following three worked solutions or share actual student responses anonymously. Solution A: −2a3 + 2a2 − 2a + 2 = −2 (a3 − a2 + a − 1) 2

Factor out −2 from all terms

= −2 (a (a − 1) + 1 (a − 1))

Factor out the GCF (a2 and 1)

= −2 (a − 1) (a2 + 1)

Factor out the common binomial factor

Solution B: −2a3 + 2a2 − 2a + 2 = −2 (a3 − a2 + a − 1)

= −2 (a2 (a − 1) + (a − 1)) 2

= −2 (a − 1) (a )

Factor out −2 from all terms Factor out the GCFs Factor out the common binomial factor

Solution C: −2a3 + 2a2 − 2a + 2 = (−2a3 + 2a2) + (−2a + 2) 2

= −2a (a − 1) + 2 (−a + 1)

Split based on common factors Factor out the GCF (−2a2 and 2)

Cannot be factored by grouping as the binomials −2a3 + 2a2 − 2a + 2 = −2a3 + 2a2 − 2a + 2 are not the same Invite students to identify the errors, critique the reasoning, and write a correct explanation. Invite one or two students to share their critiques and corrected explanations with the class. Listen for and amplify the language students use around the order of the steps and how to identify equivalent expressions. This will help students understand how to factor by grouping. Note that Solution A is a valid strategy that can help make the numbers smaller to work with at the beginning, but could introduce complications with the need for double parentheses, Solution B has an error on the last step as it does not recognize that 1 is the remaining factor, and Solution C should have factored out −2, not 2 from the second pair of terms to make the binomials the same. Emphasize that the binomials must be the same, but before saying it cannot be factored by grouping we should try to rearrange the order or check for a different common factor.

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Decomposition of factoring by grouping problem for students Student with disabilities support Provide a model of a detailed solution, highlighting and annotating the steps. Use Computational Thinking to decompose the problem into smaller, more manageable parts. Then Algorithmic thinking can be used to go through the factoring process. For example, the question x2 + 5x + 8x + 40 could be broken up into: 1. Factor the first pair: x2 + 5x. 2. Factor the second pair: 8x + 40. 3. Check that there is a common binomial factor. If not, go back and rearrange. 4. Make connections between the two parts and substitute your answers from parts (1) and (2) into x2 + 5x + 8x + 40. 4. Fully factor the expression from part (4).

Exploration Students: Page 351

Suggested student grouping: In pairs Students are presented with a polynomial containing four terms, and are asked to factor the GCF from the ﬁrst two terms, followed by factoring the GCF from the second two terms. Students may notice that the remaining binomials are also the same, and could factor the four-term polynomial using this information. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Based on your answers above, what do you think the factored form of x4 + 7x3 + 6x + 42 could be? Since both sets of terms has a common factor of (x + 7), the factored form of x4 + 7x3 + 6x + 42 could be (x + 7) (x3 + 6). Purposeful questions • What is the GCF of the ﬁrst set and the second set? How can we use this to write a factored form of the given polynomial? • Could you somehow combine your solutions for each set of factored terms?

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Possible misunderstandings • Students may not recognise the common binomial factor or how this can lead to the full factorization. Encourage students to rewrite the original expression in terms of the sets of factored terms and then look for what the two sets have in common. Temporarily replacing the binomial factor with a letter or symbol may help students identify the factorization. For example, replacing (x + 7) with A, we obtain Ax3 + 6A. Students learn a general form and procedure for factoring by grouping. Then, students are presented with an example to follow the method of factoring by grouping.

Students: Pages 351–352

Follow the steps shown with the example to factor by grouping: 1. Factor out a GCF, if possible

2. Rearrange the terms so that the first pair Follow the steps shown with the example to factor by grouping: and the second pair each have a common factor, ifout possible 1. Factor a GCF, if possible 3. Factor out a GCF from the binomial expressions 2. Rearrange the terms so that the first pair 4. Factor the common binomial and theout second pair each have a expression common factor, if possible 5. Verify that the final expression cannot be 3. Factor out a GCF from the binomial factored further, otherwise continue factoring expressions 4. Factor out the common binomial expression

Examples

Example 1

Students:Factor Page the352 expression 10x2 + 4x + 15x + 6.

5. Verify that the final expression cannot be factored further, otherwise continue factoring

Create a strategy

Example We arrange 1the terms ﬁrst, grouping those with common factors. We factor out the GCF on each pair and the common binomial factor afterward. Factor the expression 10x2 + 4x + 15x + 6. Apply the idea Create 2a strategy

10x + 4x + 15y + 6 = (10x2 + 4x) + (15x + 6) Group based on common factors We arrange the terms ﬁrst, grouping those with common factors. We factor out the GCF on each pair and the = 2x(5x + 2) + 3(5x + 2) Factor out each GCF (2x and 3) common binomial factor afterward. = (5x + 2) (2x + 3) Factor out the common binomial factor

Apply the+ idea Since (5x 2) (2x + 3) cannot be factored further, it is the ﬁnal answer. 10x2 + 4x + 15y + 6 = (10x2 + 4x) + (15x + 6)

Group based on common factors

Reﬂect and check

= 2x(5x + 2) + 3(5x + 2) Factor out each GCF (2x and 3) We can perform a midway check by grouping appropriately when we factor out the GCF from = (5x + 2)that (2x +we 3) are factoring Factor out the common binomial factor each set of binomials in the step 2x(5x + 2) + 3(5x + 2). Since (5x + 2) (2x + 3) cannot be factored further, it is the ﬁnal answer. If we factor out the GCF at this step and the binomial factors are not equivalent, then we will want to check that we factored out the GCF correctly. If the factoring is correct, we may need to try a different approach. There may be a Reﬂect and check better way to arrange the terms from the polynomial. We can perform a midway check that we are factoring by grouping appropriately when we factor out the GCF from Not every polynomial expression will be factorable, but we can try a few different approaches, checking our work 6.05 Factor by grouping each set of binomials in the step 2x(5x + 2) + 3(5x + 2). along the way. mathspace.co If we factor out the GCF at this step and the binomial factors are not equivalent, then we will want to check that we factored out the GCF correctly. If the factoring is correct, we may need to try a different approach. There may be a better way to arrange the terms from the polynomial.

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Apply the idea 10x2 + 4x + 15y + 6 = (10x2 + 4x) + (15x + 6)

Group based on common factors

= 2x(5x + 2) + 3(5x + 2)

Factor out each GCF (2x and 3)

= (5x + 2) (2x + 3)

Factor out the common binomial factor

Since (5x + 2) (2x + 3) cannot be factored further, it is the ﬁnal answer.

Reﬂect and check We can perform a midway check that we are factoring by grouping appropriately when we factor out the GCF from each set of binomials in the step 2x(5x + 2) + 3(5x + 2). If we factor out the GCF at this step and the binomial factors are not equivalent, then we will want to check that we factored out the GCF correctly. If the factoring is correct, we may need to try a different approach. There may be a better way to arrange the terms from the polynomial. Not every polynomial expression will be factorable, but we can try a few different approaches, checking our work along the way.

Purpose Show students how to factor a polynomial expression that contains more than one variable by grouping. Reﬂecting with students Challenge students to group the terms a different way and factor the expression by grouping. Students could rewrite the terms as 10x2 + 15x + 4x + 6. 10x2 + 4x + 15x + 6 = (10x2 + 15x) + (4x + 6) = 5x (2x + 3) + 2 (2x + 3) = (2x 1+ 3) (5x + 2) 352 Mathspace Virginia SOL Algebra

Group based on common factors Factor out each GCF (5x and 2) Factor out the common binomial factor

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The commutative property tells us that for a product, we can write the factors in either order, so this is also correct.

Encourage critical thinking about grouping

use with Example 1

Targeted instructional strategies Encourage students to consider why we sometimes reorder the terms to factor by grouping and sometimes we do not need to. Provide the following four orderings for the terms in equivalent expressions: • 6x3 + 10x2 − 27x − 45 • 6x3 − 27x + 10x2 − 45 • 6x3 − 45 − 27x + 10x2 • 6x3 − 27x − 45 + 10x2 Ask the students: 1. Which can be factored by grouping as they are, and which will need to be rearranged? 2. Which ones were possible as they were, but took more steps? 3. Is there anything else you notice or wonder? Encourage students to generalize this idea and apply it to other questions. Ideally, students would develop and be able to explain the conditions for applying different strategies.

Students: Page 353

Example 2 Show at least two different ways we can arrange and group the polynomial 4a2 − 10b + 5ab − 8a and factor it.

Create a strategy Determine if the polynomial has common factors between the ﬁrst and second pair of terms, then factor it and rearrange the polynomial so that the ﬁrst or second set of terms has a common factor, then factor it again.

Apply the idea Write the expression as 4a2 + 5ab − 8a − 10b and factor it.

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4a2 − 10b + 5ab − 8a = 4a2 + 5ab − 8a − 10b Mathspace 2Virginia SOL Algebra 1 2Teacher Edition 4a + 5ab − 8a − 10b = (4a + 5ab) + (−8a − 10b) mathspace.co

Rearrange the terms Group based on common factors

= a (4a + 5b) − 2(4a + 5b)

Factor out the GCF (a and − 2)

= (4a + 5b) (a − 2)

Factor out the common binomial factor

Show at least two different ways we can arrange and group the polynomial 4a2 − 10b + 5ab − 8a and factor it.

Apply the idea Write the expression as 4a2 + 5ab − 8a − 10b and factor it. 4a2 − 10b + 5ab − 8a = 4a2 + 5ab − 8a − 10b 2

4a + 5ab − 8a − 10b = (4a + 5ab) + (−8a − 10b)

Rearrange the terms Group based on common factors

= a (4a + 5b) − 2(4a + 5b)

Factor out the GCF (a and − 2)

= (4a + 5b) (a − 2)

Factor out the common binomial factor

Since (4a + 5b) (a − 2) cannot be factored further, it is the ﬁnal answer. Write the expression as 4a2 − 8a − 10b + 5ab and factor it. 4a2 − 10b + 5ab − 8a = 4a2 − 8a − 10b + 5ab

Rearrange the terms

4a2 − 8a − 10b + 5ab = (4a2 − 8a) + (−10b + 5ab)

Group based on common factors

= 4a(a − 2) + 5b(−2 + a)

Factor out the GCF (4a and 5b)

= (a − 2) (4a + 5b)

Factor out the common binomial factor

Since (a − 2) (4a + 5b) cannot be factored further, it is the ﬁnal answer.

Reﬂect and check Alternatively, we can group 4a2 and − 8a and 5ab and − 10b together and get the same answer. 4a2 + 5ab − 8a − 10b = (4a2 − 8a) + (5ab − 10b)

Group based on common factors

= 4a(a − 2) + 5b(a − 2)

Factor out the GCF (4a and 5b)

= (a − 2) (4a + 5b)

Factor out the common binomial factor

We can check the answer by multiplying the factored form (4a + 5b) (a − 2). (4a + 5b) (a − 2) = a (4a + 5b) − 2(4a + 5b) = 4a2 + 5ab − 8a − 10b

Distributive property Distributive property

Idea summary Purpose Follow these steps when factoring by grouping: Show students1. how toout factor the from same by grouping in more than one way. Factor the GCF thepolynomial expression, ifexpression possible 2. Arrange the terms so that the ﬁrst two have a common factor and the last two have a common factor,

Expected mistakes if possible Students may 3.factor a GCF of 2 from the second set of terms in the first example, instead of −2. This would lead Factor out the GCF for each pair of terms to different binomial factors, (4a + 5b) and (−4a − 5b). Emphasize that the binomials must be the same, and to be 4. Factor out the common binomial expression aware of the signs of terms when factoring. 5. Confirm that the binomial factors cannot be factored further, otherwise continue factoring Reflecting with students Ask students why the solutions for the factorization for both rearranged expressions are correct. The order of the binomials does not matter because of the commutative property of multiplication.

Concrete-Representational-Abstract (CRA) Approach Targeted instructional strategies

6.05 Factor by grouping 353 use with Example 2 mathspace.co

Concrete: Begin by engaging students with algebra tiles to represent each term in the polynomial 4a2 − 10b + 5ab − 8a. Assign different shapes or colors to represent each term: use large squares for a2, rectangles for ab, small squares for a, and circles for b. Include negative tiles or flip the tiles over to represent negative coefficients. Have students physically arrange the tiles to model the polynomial. Encourage them to experiment with grouping the tiles in different ways to find common factors, exploring at least two different arrangements.

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Show at least two different ways we can arrange and group the polynomial 4a2 − 10b + 5ab − 8a and factor it.

Representational: Transition to the representational stage by asking students to draw diagrams of the tile Apply the idea arrangements they created. They can sketch the shapes used for each term and label them appropriately. 2 Write to theclearly expression as 4a +groupings 5ab − 8a − 10b factoror it. boxing the grouped terms in their drawings. Encourage Guide them show the by and circling them to rearrange terms drawings to reflect Rearrange the different groupings they discovered with the tiles. + 5ab − 8ain= their 4a2 + 5ab − 8a − 10b the terms 4a2 − 10bthe 2 2 This helps students visualize the factoring process on paper. Group based on common factors 4a + 5ab − 8a − 10b = (4a + 5ab) + (−8a − 10b) a (4a + by 5b) demonstrating − 2(4a + 5b) Factortoout the GCF and − 2) Abstract: Move to the abstract= stage how write the (a polynomial and factor it using algebraic = (4a + 5b) (a − 2) Factor out the common binomial notation. Show how the physical groupings correspond to rearranging the terms infactor the expression. Write out Since (4a + 5b) − 2) cannot such be factored further, it is the answer. the steps to factor by(agrouping, as rearranging theﬁnal polynomial to 4a2 + 5ab − 8a − 10b, grouping terms, 2 10b + 5ab andthen factor it. 4a − 8a − factoringWrite outthe theexpression greatestascommon factors, and factoring out the common binomial factor. Encourage 2 2 students to connect back to their drawings manipulations with the tiles. This helps them 5ab −algebraic 8a = 4a − step 8a − 10b + 5ab Rearrangeand the terms 4a − 10b +each 2 the abstract symbols understand 4a how represent the concrete and visual actions they performed earlier. − 8a − 10b + 5ab = (4a2 − 8a) + (−10b + 5ab) Group based on common factors = 4a(a − 2)make + 5b(−2connections + a) Factor out theall GCF (4a and 5b) by discussing how each one Connecting the stages: Help students between three stages = (a − 2) (4a + 5b) Factor out the common binomial factor builds on the previous. Since (a − 2) (4a + 5b) cannot be factored further, it is the ﬁnal answer.

Ask guiding questions like: • “HowReﬂect did the way you grouped the tiles help you decide how to group the terms in your drawing?” and check • “CanAlternatively, you see how thegroup factors the equations match thesame groups you made with the tiles?” we can 4a2you and −pulled 8a andout 5abin and − 10b together and get the answer. Highlight how concrete manipulation of tiles led to the representational which in turn made the + 5ab − 8a − 10b = (4a2 − 8a) + (5ab − 10b) Group based on commondrawings, factors 4a2the abstract algebraic factoring clearer. reinforces theirFactor understanding shows = 4a(a −This 2) + 5b(a − 2) out the GCF and (4a and 5b) them how to apply these strategies to other problems = (a − 2) (4a + 5b) Factor out the common binomial factor We can check the answer by multiplying the factored form (4a + 5b) (a − 2). (4a + 5b) (a − 2) = a (4a + 5b) − 2(4a + 5b)

Students: Page 353

= 4a2 + 5ab − 8a − 10b

Distributive property Distributive property

Idea summary Follow these steps when factoring by grouping: 1. Factor out the GCF from the expression, if possible 2. Arrange the terms so that the ﬁrst two have a common factor and the last two have a common factor, if possible 3. Factor out the GCF for each pair of terms 4. Factor out the common binomial expression 5. Conﬁrm that the binomial factors cannot be factored further, otherwise continue factoring

6.05 Factor by grouping mathspace.co

Practice

353

Students: Pages 354–355

What do you remember? 1

For the expression 3(x + 7) + x(x + 7), what is the common factor?

Determine which expression need to be rearranged in order to factor by grouping. Do not factor. a

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a2 − 4a − 3a + 12

6x2 − 20 + 8x − 15x

1 + 100x2 − 10x − 10x

Write an algebraic expression that can be factored to (x + 3) (2x − 1).

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

7 + x − 28x − 4x2

Factor the expressions: a

a(a + 6) + 2 (a + 6)

2b(b − 3) − 5 (b − 3)

5x(x + 2) − (x + 2)

2x(x − 7) + 3 (7 − x)

5d(d + 3) + 20 (d + 3)

8y( y − 4) + 10(4 − y)

y( y + 5) + 7( y + 5)

a(a − 4) − 3(a − 4)

p( p − 3) + 6( p − 3)

5(q + 4) − q(q + 4)

5(r − 3) − r(r − 3)

6r(2r − s) − rs(2r − s)

m 7t(t + u) + 2u(t + u)

x( y − z) − w( y − z)

5y(4w + 3x) − z(4w + 3x)

8y( y − 4) + 3(4 − y)

Let’s practice 5

Factor the expressions: a

x2 + 5x + 8x + 40

x2 − 5x + 10x − 50

z2 − 7z + 2z − 14

2k2 + 12k + k + 6

2b2 + 6b + b + 3

3x2 − 10x + 3x − 20

−4y2 + 30y − 5y − 36

x2 − 3x + 8x − 24

x + 2x + 5x + 10

20a − 12a + 5a − 3

3y + 6y + 4y + 8

9t2 + 6t + 12t + 8

m 8x + xz − 16y − 2yz

24 + 3y + 8x + xy

7xy + wx + 7yz + wz

2mp + 6 + 3p + 4m

5mp + 6 + 2p + 15m

2x + 18yz + 12xy + 3z

2f (g + h) + (g + h)2

q 6

Factor the expressions: a

( y + 4) ( y + 7) + x ( y + 7)

Identify and explain the error: 6x − 21x3 + 14x − 4 = 3x(2 − 7x) + 2(7z − 2) = (3x + 2) (2 − 7x)

Complete the factoring process below and explain each step:: 7x + 7 + x + x2 = ⬚ (x + 1) + x (1 + ⬚) = ⬚ (x + 1) + x (⬚ + 1) = (x + 1) (⬚ + x)

The expression for the area of the rectangle shown is 3x2 + 18x + x + 6. Write the expression of the area in factored form.

3x2

18x

One expression for the area of the rectangle shown is m2 + 14m + 45. The rectangle is made up of four smaller rectangles. Use the diagram to express the area of the large rectangle in factored form.

6.05 Factor by grouping mathspace.co

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Find an expression for the total area of the rectangles in factored form: a

10m

Let’s extend our thinking 12

Factor the expressions: a

3y3 + 6y2 − 15y − 30

3x3 − x2 + 27x − 9

17x3 + 5x2 + 17x + 5

a3 + 5a2 + a + 5

Factor the expressions: a

8x(2y + 3w) − z(2y + 3w)

8z(5x2 + 4y) − (5x2 + 4y)

2x + xz − 40y − 20yz

50 + 5y + 10x + xy

12xy + wx + 12yz + wz

6y − yw + w2 − 6w

16ab + 6b2 − 32ac − 12bc

g 14

8xy + 4x − 6xy − 3x y

The polynomial expression x2 + 9x + 18 is factored by grouping and one of its factors is (x + 6). Rewrite the polynomial in the form x2 + ⬚ x + ⬚ x + 18 and factor the expression. For each polynomial: i

Find three pairs of values that make the polynomial factorable.

Determine what the pairs from part (i) have in common.

x3 − 3x2 + ⬚ x + ⬚

x3 + ⬚ x2 + x + ⬚

By rewriting 4x2 + 17x + 4 as an expression having four terms and factoring in pairs, factor the expression completely.

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Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Answers

Let’s extend our thinking 12 a 3(y + 2) (y2 − 5)

6.05 Factor by grouping

c (x + 1) (17x + 5) 13 a (2y + 3w) (8x − z)

What do you remember? 1 x+7 2 a Factorable as is

b Rearrange first

c Rearrange first

d Factorable as is

3 2x2 − x + 6x − 3 4 a (a + 6) (a + 2)

b (b − 3) (2b − 5)

c (5x − 1) (x + 2)

d (x - 7) (2x - 3)

e 5(d + 4) (d + 3)

g (y + 5) (y + 7)

2( y - 4) (4y - 5)

h (a − 4) (a − 3)

(p − 3) (p + 6)

(q + 4) (5 − q)

k (r − 3) (5 − r)

(2r − s) (6r − rs)

m (t + u) (7t + 2u)

n (y − z) (x − w)

o (4w + 3x) (5y − z)

p (y − 4) (8y − 3)

d (a + 5) (a2 + 1) b (5x2 + 4y) (8z − 1)

c (x − 20y) (2 + z)

d (5 + x) (10 + y)

e (12y + w) (x + z)

g x(2y + x) (4 − 3y)

h 2(8a + 3b) (b − 2c)

(6 − w) (y − w)

14 x + 6x + 3x + 18 = (x + 3) (x + 6) 15 a i

Answers will vary. 1 and -3 2 and -6 3 and -9

ii The second number is always the first number multiplied by -3. b i

Answers will vary. 1 and 1 2 and 2 3 and 3

ii They have to be the same number.

Let’s practice 5 a (x + 5) (x + 8)

b (x − 5) (x + 10)

c (z + 2) (z − 7)

d (2k + 1) (k + 6)

e (2b + 1) (b + 3)

g −(4y − 9) (y − 4)

h (x − 3) (x + 8)

16 (x + 4) (4x + 1)

(3x + 5) (x − 4)

(x + 2) (x + 5)

(4a + 1) (5a − 3)

k (3y + 4) (y + 2)

(3t + 2) (3t + 4)

m (x − 2y) (8 + z)

n (3 + x) (8 + y)

b (3x − 1) (x2 + 9)

o (x + z) (7y + w)

p (p + 2) (2m + 3)

q (p + 3) (5m + 2)

r (2x + 3z) (1 + 6y)

6 a (y + 7) (y + 4 + x)

b ( g + h) (2f + g + h)

7 (2 - 7x) ≠ (7x - 2) so either a factor of –3x needs to be taken out of the first group or a factor of -2 needs to be taken out of the second group. Factor out the GCF from 8 7x + 7 + x + x2 = 7(x + 1) + x(1 + x) each pair

= 7(x + 1) + x(x + 1)

= (x + 1) (7 + x) Factor out the binomial

Commutative property GCF from each group

9 (3x + 1) (x + 6) 10 (m + 5) (m + 9) 11 a (x + 6) (x + 3)

b (m + 5) (m + 10)

Answers mathspace.co

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6.06 Factor trinomials Subtopic overview Lesson narrative In this lesson, students will use the patterns and structures they have seen when factoring by grouping and multiplying two binomials in the context of factoring trinomials. Students will learn to recognize the role of the leading coefficient in choosing a factoring strategy. By the end of this lesson, students will be able to apply factoring by grouping to trinomials and be able to describe a factoring strategy for a polynomial based on the structure of the expression.

Learning objective

6.06 Factor trinomials

Students: Page 356

After this lesson, you will be able to… • factor trinomials completely.

Factor trinomials Trinomials can be rewritten as polynomials with four terms and factored by grouping.

Key vocabulary 

trinomial Exploration



zero pair

Consider the polynomial expressions factored by grouping below:

Essential understanding

The same standard algorithm can be applied to rewrite any factorable quadratic expression, though other methods may prove more efficient.

Standards This subtopic addresses 2023 Mathematics Standards of terms Learning 1. What patternsthe dofollowing you noticeVirginia between the original expression and the usedstandards. to rewrite the linear term?

Mathematical process goals 2. Choose one of the linear terms and rewrite the term in a different way than shown, then determine whether the Reasoning polynomial can still be factored by grouping. MPG3 — Mathematical

Teachers can incorporate this goal into their lessons by encouraging students to justify their steps when factoring by grouping. Students should be guided to use logical reasoning to explain why they arranged the terms in a certain When using the grouping method to factor a trinomial, the coefficients of the terms used to rewrite the linear term way, whya they grouped the terms as they did, and why factored out the common binomial. have sum equivalent to the linear coefficient from thethey original polynomial and a product equivalent to the product of the trinomial’s leading coefficient and constant. Steps in factoring a quadratic trinomial of the form ax2 + bx + c:

Content standards

1. Factor out any GCF. A.EO.2 —negative, The student will also perform operations onofand A.EO.2c — Factor completely first- and second-degree (If a is we can divide out a factor −1 before continuing.) factor polynomial expressions in one variable. polynomials in one variable with integral coefficients. 2. Find two numbers, r and s, that multiply to ac and add to b. 2 out the greatest common factor + sx +factoring c. 3. Rewrite the trinomial with four terms in the form ax + rx After A.EO.2e — Represent and demonstrate equality of (GCF), leading coefficients should have no more than 4. Factor by grouping. quadratic expressions in different forms (e.g., concrete, four factors. 5. Check whether the answer will not factor further and verify the factored form by multiplication. verbal, symbolic, and graphical). Remember to include any common factors divided out at the start, so each step results in an equivalent expression. Algebra tiles can also be useful in factoring. Consider the Mathspace 2 Virginia SOL Algebra 1 Teacher Edition expression 3x + 7x − 6 as the area of a rectangle. If we mathspace.co can find the lengths of this rectangle, then we will have two expressions that multiply to 3x2 + 7x − 6 because the area of a

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Prior connections 8.PFA.1 — The student will represent, simplify, and generate equivalent algebraic expressions in one variable.

A.EO.1 — The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

Future connections A2.EO.3 — The student will perform operations on polynomial expressions in two or more variables and factor polynomial expressions in one and two variables.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lesson: Algebra 1 — 6.05 Factor by grouping

Tools You may ﬁnd these tools helpful: • Scientific calculator • Highlighter

Student lesson & teacher guide Factor trinomials Students begin with an exploration into factoring trinomials by grouping.

Students: Page 356

6.06 Factor trinomials After this lesson, you will be able to… • factor trinomials completely.

Factor trinomials Trinomials can be rewritten as polynomials with four terms and factored by grouping.

Exploration Consider the polynomial expressions factored by grouping below:

6.06 Factor trinomials mathspace.co What patterns do you notice between the original expression and the terms used to rewrite the linear term?

Choose one of the linear terms and rewrite the term in a different way than shown, then determine

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Connecting the box model to factoring by grouping Targeted instructional strategies Many methods exist for factoring a trinomial into two binomials, but students may benefit from seeing the process of working backwards from a completed box model to the factored binomials. Display the polynomial x2 + 10x − 24, and tell students that the middle term can be rewritten as two x terms that multiply to be −24. Finding these two factors rewrites the trinomial as x2 + 12x − 2x − 24. These four terms can be written in a box model that students saw previously for multiplication. x2

+12x

−2x

−24

Taking the greatest common factor of each row and column gives us the terms inside the binomials that would multiply to be the polynomial written in the model. GCF x

GCF 2

GCF x

+12x

GCF −2

−2x

−24

−2x

−24

The greatest common factors of each row and column can be grouped to make the factored form of x2 + 10x − 24, (x − 2) (x + 12). x + 12 x

+ 12x

−2

−2x

−24

This connect factoring trinomials to the familiar process of multiplying polynomials.

Provide different factoring methods Student with disabilities support Students may have difficulties remembering the identities for the special products. Help them make connections to their previous work with factoring trinomials by encouraging them to connect to their prior knowledge of factoring by grouping or factoring by inspection. Have students write a difference of two squares, like a2 − 25, as a trinomial a2 + 0a − 25. Now they simply need to identify the two numbers whose product is −25 and whose sum is 0. Then they can rewrite the expression as a2 + 5a − 5a − 25 and proceed to factor using their preferred method. Many students will quickly become comfortable with the identities and will appreciate how much more efficient they are but others will prefer to use a method they are more familiar with. Any strategy that works should be acknowledged.

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Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Stronger and clearer each time English language learner support To support English Language Learners in understanding the lesson on factoring trinomials using the grouping method, start by having each student write out their steps for factoring a given trinomial, such as 3x2 + 2x − 8, and explain their reasoning behind each step. Emphasize moving away from trial and error by asking them to focus on why they chose specific pairs of factors. Next, have students pair up and exchange their written steps. Each student explains their process to their partner, who will then ask clarifying questions such as • Why did you choose these factors? • Can you explain the step where you grouped the terms? The goal is to encourage detailed explanations that avoid trial and error, focusing instead on the mathematical reasoning behind choosing factor pairs. Students should take note of any suggestions or new terms they learn during this exchange. After the first pairing, have students switch to new partners and repeat the process. This time, press for even more details, ensuring the explanation includes clear reasoning for each step. Questions might include • How do these factors multiply to the constant term? • How did you check your work? This helps solidify their understanding and ability to articulate the factoring process. Finally, students revise their initial written steps using the feedback from both partner discussions. The revised writing should clearly state how they identified pairs of factors, explain the grouping process in detail, use precise mathematical vocabulary, and ensure that the steps logically follow one another. For example, an initial attempt might simply say, “I guessed the factors of 3 and −8 and tried them out.” A refined explanation would be, “First, I multiplied 3 and −8 to get −24, then looked for factor pairs that multiplied to −24 and added to 2. I chose 6 and −4, then rewrote the trinomial as 3x2 + 6x − 4x − 8. Factoring the first two terms and the last two terms gives 3x(x + 2) − 4(x + 2). Grouping the terms makes the final factored form (3x − 4) (x + 2).” This iterative process helps students develop a clear, well-structured explanation of how to factor trinomials using the grouping method, emphasizing a systematic approach over trial and error.

Multiplying factored polynomials to check steps Address student misconceptions Students may lose track of or make errors with signs. For example, factoring x2 + 5x − 6 as (x + 2) (x + 3) or (x + 1) (x − 6). Challenge these misconceptions by having students ﬁnd the product of their answer to ensure the original question is returned. Encourage students to seek support from peers if they cannot identify the error. When c is negative in the standard form ax2 + bx + c students often mix up the signs of the two terms we decompose the middle term into, as they have opposite signs. Before beginning to factor by grouping encourage students to do a quick check that the polynomial they get after breaking up the middle term is indeed equal to the original polynomial. They could identify the error at the end by checking the product of the two binomials, but it is better to identify the error at the point of misconception.

6.06 Factor trinomials mathspace.co

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6.06 Factor trinomials After this lesson, you will be able to… • factor trinomials completely.

Exploration

Students:Factor Page 356 trinomials Trinomials can be rewritten as polynomials with four terms and factored by grouping.

Exploration Consider the polynomial expressions factored by grouping below:

What patterns do you notice between the original expression and the terms used to rewrite the linear term?

Choose one of the linear terms and rewrite the term in a different way than shown, then determine whether the polynomial can still be factored by grouping.

When using the grouping method to factor a trinomial, the coefficients of the terms used to rewrite the linear term have a sum equivalent to the linear coefficient from the original polynomial and a product equivalent to the product of Suggested student leading grouping: In pairs the trinomial’s coefficient and constant. 2 rewritten as polynomials with four terms, and then shown StudentsSteps are inpresented with three trinomials are factoring a quadratic trinomial of the that form ax + bx + c: how the1.expressions are factored by grouping. Students discover a pattern for rewriting trinomial expressions Factor out any GCF. in order to (Iffactor by grouping. A suggestion for implementation is to have students explore individually for a a is negative, we can also divide out a factor of −1 before continuing.) 2. Find two numbers, r and s, that multiply to ac and add to b. couple minutes before discussing with a partner.

3. Rewrite the trinomial with four terms in the form ax2 + rx + sx + c.

Ideal student responses 4. Factor by grouping. 5. Check whether may the answer notother factor further and verify the factored form by multiplication. These ideal responses differwill from correct student responses. Less formal responses can be connected with the more precise mathematical language presented here.

Remember to include any common factors divided out at the start, so each step results in an equivalent expression.

1. WhatAlgebra patterns do you notice original tiles can also be usefulbetween in factoring.the Consider the expression and the terms used to rewrite the linear term? expression 3x2 + 7x − 6 as the area of a rectangle. If we findterm the lengths of this rectangle, thenof wetwo will have two The can linear is rewritten as the sum linear terms in order to make the trinomial into a polynomial 2 expressions that multiply to 3x + 7x − 6 because the area a 2 with four terms. The product of the coeﬃcients of theofnew linear terms x2 is also xequivalent x2to the products of rectangle is A = l ⋅ w. the coeﬃcient of the trinomial’s quadratic term and the trinomial’s constant. We don’t yet know the side lengths of the rectangle, but we will

x x x −1 −1 2 2. Choose one of xthe linear terms and rewrite term take 3 of the tiles, 7 of the +x tiles, and 6 of thethe −1 tiles andin a different way than shown, then determine x x x −1 −1 whether the polynomial cana still be factored arrange them as closely into rectangle as we can.by grouping. x −1 −1 As long as start the by linear two whose coeﬃcients multiply to the same product as the We will liningterm up allisofrewritten the x2 tiles,as then putterms the x tiles underneath match the equal lengths. Finally,quadratic put the −1 tiles next to the tiles to matchconstant, the equal lengths. product of the to coeﬃcient of the trinomial’s term and thex trinomial’s it is possible to factor the polynomial bysome grouping. Notice we have empty spaces that need to be filled in.

Purposeful Mathspace Virginia SOL Algebra 1 356 questions mathspace.co • What relationship do you think there is between the color-coded terms? • What is the coeﬃcient of the term x2?

Possible misunderstandings • Students may not make a connection between the coeﬃcients of the new linear terms having a product equivalent to the product of the coeﬃcient of the trinomial’s quadratic term and the trinomial’s constant. This may need to be made explicit to students in order to determine how to resolve question 2. Students are introduced to a general procedure used for factoring by grouping given a trinomial. They are then shown how factoring polynomials can be represented by algebra tiles. They learn that a zero pair is formed when two terms cancel each other out.

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What patterns do you notice between the original expression and the terms used to rewrite the linear term?

Choose one of the linear terms and rewrite the term in a different way than shown, then determine whether the polynomial can still be factored by grouping.

Students: Pages 356–357

When using the grouping method to factor a trinomial, the coefficients of the terms used to rewrite the linear term have a sum equivalent to the linear coefficient from the original polynomial and a product equivalent to the product of the trinomial’s leading coefficient and constant. Steps in factoring a quadratic trinomial of the form ax2 + bx + c: 1. Factor out any GCF. (If a is negative, we can also divide out a factor of −1 before continuing.) 2. Find two numbers, r and s, that multiply to ac and add to b. 3. Rewrite the trinomial with four terms in the form ax2 + rx + sx + c. 4. Factor by grouping. 5. Check whether the answer will not factor further and verify the factored form by multiplication. Remember to include any common factors divided out at the start, so each step results in an equivalent expression. Algebra tiles can also be useful in factoring. Consider the expression 3x2 + 7x − 6 as the area of a rectangle. If we can find the lengths of this rectangle, then we will have two expressions that multiply to 3x2 + 7x − 6 because the area of a rectangle is A = l ⋅ w. We don’t yet know the side lengths of the rectangle, but we will take 3 of the x2 tiles, 7 of the +x tiles, and 6 of the −1 tiles and arrange them as closely into a rectangle as we can.

−1 −1

We will start by lining up all of the x2 tiles, then put the x tiles underneath to match the equal lengths. Finally, put the −1 tiles next to the x tiles to match the equal lengths.

−1 −1

Notice we have some empty spaces that need to be filled in. 356

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−x −x

−1 −1

Notice that x tiles will fit perfectly into the empty spaces. However, we don’t want to change the value of the expression so we need to make sure to add zero pairs. A zero pair is two values that add to 0. x and −x is a zero pair. Since there are 4 empty spaces for x tiles, we can fill 2 spaces with (positive) x tiles and 2 spaces with −x tiles. Technically this represents the expression 3x2 + 9x − 2x − 6 which is equivalent to 3x2 + 7x − 6 by combining like terms. x

−1 −1

−x −x

−1 −1

Now we can use the lengths of the sides of the rectangle to determine the expressions that can be multiplied together to create the original expression 3x2 + 7x − 6. The x2 tile has side lengths of x and x. The x tiles have a shorter side length of 1 and a longer side length of x. The −x tiles have a shorter side length of −1 and a longer side length of x. The shorter side length of the rectangle is x + 3 units and the longer side length is 3x − 2 units. This shows us: 3x2 + 7x − 6 = 3x2 + 9x − 2x + 6 = (x + 3) (3x − 2).

Example 1 Factor x2 + 10x − 24.

Create a strategy

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Examples The shorter side length of the rectangle is x + 3 units and the longer side length is 3x − 2 units. shows 357–358 us: Students:This Pages 3x2 + 7x − 6 = 3x2 + 9x − 2x + 6 = (x + 3) (3x − 2).

Example 1 Factor x2 + 10x − 24.

Create a strategy Since there are no common factors for all three terms, we proceed with ﬁnding the value of two integers that multiply to ac = (1) (−24) = −24 and add up to b = 10. After ﬁnding these integers, we use them to rewrite the middle term 10x as a sum of two terms and then factor the trinomial by grouping.

Apply the idea The factors of −24 are 1 and −24, −1 and 24, 2 and −12, −2 and 12, 3 and −8, −3 and 8, 4 and −6, −4 and 6. Among these factors, −2 and 12 are the pair that add up to 10. We can use this to rewrite the trinomial and factor by grouping as follows: x2 + 10x − 24 = x2 + 12x − 2x – 24

Rewrite polynomial with four terms

= x(x + 12) − 2(x + 12)

Factor each pair

= (x + 12) (x − 2)

Divide out common factor of (x + 12)

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There are no more common factors to be divided out, so the fully factored form of the polynomial is (x + 12) (x − 2).

Reﬂect and check We can perform a midway check that we are factoring by grouping appropriately when we factor out a GCF from each set of binomials in the step x(x + 12) − 2(x + 12). If we factor out a GCF at this step and the binomial factors are not equivalent, we may have split the linear term from x2 + 10x − 24 incorrectly or factored out a GCF incorrectly. This is an important place to stop and check that we are factoring appropriately. Also note that we could have also rewritten the polynomial as x2 − 2x + 12x − 24. This would have resulted in a different middle step in factoring by grouping but the same end result.

Example 2 Purpose 2 Show students Factor 3xhow − 27.to factor a trinomial by grouping. Expected mistakes Create a strategy StudentsWemay anyoffactor ignore that their sum mustasalso Remind can choose factor a GCF 3 out ofpair the and polynomial and write the polynomial 3(x2be − 9).equal Since to the10. linear term is students missing of the 2 pattern in the from theexploration. polynomial, we can write the polynomial as 3(x + 0x − 9). There are no common factors. We will ﬁnd the value of two integers that multiply to ac = (1) (−9) = −9 and add up to b = 0. After ﬁnding these integers, we use them

Reﬂecting with students to rewrite the middle term 0x as a sum of two terms. Then factor the trinomial by grouping. Challenge students to attempt to rewrite the trinomial expression using any of the factors of –24, ignoring idea that theirApply sum the must be equivalent to 10. For instance, students might atttempt to rewrite the expression as 2 The factors of −9 1 andit−9, and 9, 3 and −3. Among these factors, 3 and −3we are cannot the pair that upapproach to 0. x − 24x + 1x − 24 andare factor by−1grouping. Help students determine why useadd this for We can use this to rewrite the trinomial and factor by grouping as follows: factoring by grouping. The new expression will not be equivalent to the given expression. 3(x2 + 0x − 9) = 3(x2 + 3x − 3x − 9)

Rewrite polynomial with four terms

Show students how =to check their work 3[x(x + 3) − 3(x + 3)] own Factor each pair Targeted instructional strategies = 3(x + 3) (x − 3)

use with Example 1

Divide out common factor of (x + 3)

There are no more common factorsand to becheck dividedtheir out, so the fullyusing factored themultiplying polynomial ispolynomials. 3(x + 3) (x − 3). Encourage students to self-evaluate answer theform skillofof

(x + 12) (x − 2) = (x) (x − 2) + (12) (x − 2)

Reﬂect and check

Distribute x − 2

2 – 24 of squares (a Distribute = product x − 2xof+a12x Recall that the special difference + b) (a − b) x= and a2 − b12 . Notice that the factored form of the 2 2 − 3). binomial x − 9 = (x +=3)x(x + 10x – 24 Combine like terms

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If we factor out a GCF at this step and the binomial factors are not equivalent, we may have split the linear term from x2 + 10x − 24 incorrectly or factored out a GCF incorrectly. This is an important place to stop and check that we are factoring appropriately. Also note that we could have also rewritten the polynomial as x2 − 2x + 12x − 24. This would have resulted in a different middle step in factoring by grouping but the same end result.

Students: Pages 358–359

Example 2 Factor 3x2 − 27.

Create a strategy We can factor a GCF of 3 out of the polynomial and write the polynomial as 3(x2 − 9). Since the linear term is missing from the polynomial, we can write the polynomial as 3(x2 + 0x − 9). There are no common factors. We will ﬁnd the value of two integers that multiply to ac = (1) (−9) = −9 and add up to b = 0. After ﬁnding these integers, we use them to rewrite the middle term 0x as a sum of two terms. Then factor the trinomial by grouping.

Apply the idea The factors of −9 are 1 and −9, −1 and 9, 3 and −3. Among these factors, 3 and −3 are the pair that add up to 0. We can use this to rewrite the trinomial and factor by grouping as follows: 3(x2 + 0x − 9) = 3(x2 + 3x − 3x − 9)

Rewrite polynomial with four terms

= 3[x(x + 3) − 3(x + 3)]

Factor each pair

= 3(x + 3) (x − 3)

Divide out common factor of (x + 3)

There are no more common factors to be divided out, so the fully factored form of the polynomial is 3(x + 3) (x − 3).

Reﬂect and check Recall that the special product of a difference of squares (a + b) (a − b) = a2 − b2. Notice that the factored form of the binomial x2 − 9 = (x + 3) (x − 3). This can also be verified using algebra tiles: Algebra tiles key

358

−x

−1

+1 +1 +1 +1 +1 +1 +1 +1 +1

x 1 Mathspace Virginia SOL Algebra mathspace.co

−1

−x

−1 −1 −1 −1 −1 −1 −1 −1 −1

−1

−x

−1 −1 −1 −1 −1 −1 −1 −1 −1

−1

−x

−1 −1 −1 −1 −1 −1 −1 −1 −1

Notice the x terms form a total of 0. So, we know: 3x2 − 27 = (3x + 9) (x − 3) = 3(x + 3) (x − 3)

Example 3 Purpose 2 Show students a polynomial expression that is not a trinomial, but can be rewritten as a trinomial Factor 5xhow − 18xto+ factor 9. and factored by grouping. Create a strategy

Expected mistakes Since there are no common factors for all three terms, we proceed with ﬁnding the value of two integers that multiply Studentstomay thatadd they factor the expression by grouping and state the thatmiddle the polynomial ac = 5assume ⋅ 9 = 45 and up cannot to b = −18. After ﬁnding these integers, we use them to rewrite term −18x as cannot a sumRemind of two terms and then factor trinomial grouping. be factored. students that wethe have toolsbyfor factoring that we can use to help us factor, and that we should try to exhaust those options before giving up. Apply the idea

Thewith factorstudents pairs of 45 are 1 and 45, −1 and −45, 3 and 15, −3 and −15, 5 and 9, −5 and −9. Note that since the middle Reﬂecting term ofhow the trinomial is negative, need to consider negative positiveFor factors. Of thesestudents factors, −15 and −3 arethe Ask students they could refinewe this solution to be more and efficient. example, could use the pair that adds up to −18. special product of a sum and difference to help them factor the expression after factoring a GCF. We can use this to rewrite the trinomial and factor by grouping as follows: 5x2 − 18x + 9 = 5x2 − 15x − 3x + 9

Rewrite polynomial with four terms

= 5x(x − 3) − 3(x − 3)

Factor each pair to leave behind a common binomial

= (x − 3) (5x − 3)

Divide out the common factor of (x − 3)

6.06 Factor trinomials mathspace.co

There are no more factors to be taken out, so the fully factored form of the polynomial is (x − 3) (5x − 3).

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Compare connect This can alsoand be verified using algebra tiles:

use with Example 2

English language learner support −1

Ask students to identify similarities andAlgebra differences xbetween 2 −x asked to ﬁnd the product of (a − b) (a + b) and x being tiles key being asked to factor a2 − b2 (or any of the patterns). Consider using the following sentence stems as an added support: • “Factoring a difference of squares, a2 − b2, and multiplying the product of a sum and a difference, x x x +1 +1 +1 +1 +1 +1 +1 +1 +1 (a − b) (a + b), are similar because...” 2 2 • “Factoring a difference of squares, a − b , and multiplying the product of a sum and a difference, (a − b) (a + b), are different x because...” x x x x x x x x x x2 x2 x2 • “One thing that is the same between the two questions is...” • “One thing that is the different between the is...” −x −x −x two questions −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −x a2 − b2−x −x −1 squares, −1the −1 product −1 −1 −1 −1 −1 −1 and a difference, • “Factoring a difference of , and multiplying of −1 a sum −x −x −x −1 −1 −1 −1 −1 −1 −1 −1 −1 (a − b) (a + b), are related−1because...” Notice the x terms form a total of 0. So, we know:

Students: Pages 359–360

3x2 − 27 = (3x + 9) (x − 3) = 3(x + 3) (x − 3)

Example 3 Factor 5x2 − 18x + 9.

Create a strategy Since there are no common factors for all three terms, we proceed with ﬁnding the value of two integers that multiply to ac = 5 ⋅ 9 = 45 and add up to b = −18. After ﬁnding these integers, we use them to rewrite the middle term −18x as a sum of two terms and then factor the trinomial by grouping.

Apply the idea The factor pairs of 45 are 1 and 45, −1 and −45, 3 and 15, −3 and −15, 5 and 9, −5 and −9. Note that since the middle term of the trinomial is negative, we need to consider negative and positive factors. Of these factors, −15 and −3 are the pair that adds up to −18. We can use this to rewrite the trinomial and factor by grouping as follows: 5x2 − 18x + 9 = 5x2 − 15x − 3x + 9

Rewrite polynomial with four terms

= 5x(x − 3) − 3(x − 3)

Factor each pair to leave behind a common binomial

= (x − 3) (5x − 3)

Divide out the common factor of (x − 3)

There are no more factors to be taken out, so the fully factored form of the polynomial is (x − 3) (5x − 3).

Reﬂect and check We can check the answer by multiplying the factored form (5x − 3) (x − 3). 5x

−

5x2

−3x

−3

−15x

The polynomial 5x2 − 3x − 15x + 9 simpliﬁes to 5x2 − 18x + 9.

6.06 Factor trinomials mathspace.co

Idea summary Purpose Steps in factoring a quadratic trinomial: Show students1. how toout factor a trinomial when the leading coeﬃcient is not 1. Factor any GCF. (If a is negative, we can also divide out a factor of −1 before continuing.) 2. Find two numbers, r and s, that multiply to ac and add to b.

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Mathspace 3.Virginia SOL Rewrite theAlgebra trinomial1 Teacher with four Edition terms, in the form ax2 + rx + sx + c. mathspace.co 4. Factor by grouping. 5. Check whether the answer will not factor further and verify the factored form by multiplication. Remember to include any common factors divided out at the start, so each step results in an equivalent

359

Expected mistakes Students may incorrectly factor by missing positive or negative signs. Remind students to check that their common binomials are in fact common before moving on to completing their factorization. Reﬂect and check

We can the answer by multiplying the factored form (5x − 3) (x − 3). Reflecting withcheck students Prompt advanced learners to brainstorm and discuss5xsituations 3 where the factored form is more beneficial − than the expanded form of the polynomial. For example, it is difficult to tell whether the polynomial is prime or 5x2 −3xprime. composite, but the factored form shows that it isx obviously not Invite students to consider real-world problems where one form may provide more insights than the other. The simplest example is the area of a rectangle. The original expression is best for determining the area, but the 9 −15x −3 factored expression is best for finding the side lengths.

Students:The Page 360 polynomial 5x2 − 3x − 15x + 9 simpliﬁes to 5x2 − 18x + 9.

Idea summary Steps in factoring a quadratic trinomial: 1. Factor out any GCF. (If a is negative, we can also divide out a factor of −1 before continuing.) 2. Find two numbers, r and s, that multiply to ac and add to b. 3. Rewrite the trinomial with four terms, in the form ax2 + rx + sx + c. 4. Factor by grouping. 5. Check whether the answer will not factor further and verify the factored form by multiplication. Remember to include any common factors divided out at the start, so each step results in an equivalent expression.

Practice What do you remember? Practice

Complete this statement: To factor x2 + 9x + 18, we need to find two numbers whose product is ⬚ and whose sum is360–362 ⬚. Students: Pages 1

Given that a < b, find the values of a and b in each pair of equations:

What do youa remember? 1

Use the diagram and your knowledge of areas of rectangles to write the factored form of x2 + 6x + 4x + 24

Complete this statement: To factor x2 + 9x + 18, we need to find two numbers whose product is ⬚ and whose 6 x + sum is ⬚. Given that a < b, find the values of a and xb in each pair of 6x equations: x2 a

c 24

Use the diagram and your knowledge of areas of rectangles to write the factored form of x2 + 6x + 4x + 24 x 360

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+ 4

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Factor each quadratic expression completely: a

SOL

x2 + 2x + 3x + 6

3x2 − x + 12x − 4

For each of the quadratic equation: i

List all factors of the constant term.

Factor the expression completely.

x2 + 11x + 18

x2 + 17x + 72

m−5

x2 + 16x + 64

Which is a factor of 3m2 − 4m − 15? A

3m − 5

3m + 5

m+5

Let’s practice 7

Use the algebra tile model to factor the expression x2 − 2x − 8.

x x

−x −x −x −x

−1 −1 −1 −1 −1 −1 −1 −1

Factor the quadratic expressions completely: a

x2 − 8x + 15

x2 + 11x + 24

x2 + x − 90

x2 + 22x + 120

x − 34x − 72

x2 + x − 56

3x2 − 27x − 30

35m2 + 140m + 105

2x2 + 28x + 96

x2 + 10x + 25

x + 14x + 49

x2 − 16x + 64

m 81 + 18x + x2

36 − 12x + x2

−3x2 + 12x − 12

o 9

4x + 40x + 100

Brandon claims that the polynomial 3x2 + 15x − 42 will follow the factoring form of trinomials with a leading coeﬃcient a ≠ 1. Explain Brandon’s error, then fully factor the expression.

Draw a set of algebra tiles that shows that (2x + 1) (2x − 1) = 4x2 − 1.

Rewrite each of the quadratic expression in factored form (as a product of two linear factors): a

4x2 − 32x + 15 2

5x2 − 47x + 18 2

24x + 22x − 35

6x − 19x + 15

81x2 + 36x + 4

49x2 − 28x + 4

2x2 − 19x + 45 2

10x2 − 77x − 24

−6x + 13x − 5

−35x2 + 97x − 66

9x2 + 12x + 3

27 − 123x − 60x2

Fully factor each expression: a

60x2 − 70x − 100

2x4 − 25x3 + 42x2

A square has an area of x2 + 12x + 36. Determine the length of the sides of this square.

A cube has a surface area of 6x2 + 36x + 54. Find an expression for the length of one side of the cube.

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Fully factor each expression: a

27x2 − 48

x3 + 10x2 + 25x

x4 − 16

2x4 − 1250

Let’s extend our thinking 16

Consider the polynomials: 2x2 + x + 1, 3x2 − 2x − 4, 6x2 + 10x − 5, 12x2 − 12x + 1

19 20

What do the polynomials have in common?

How do you determine if a polynomial where a ≠ 1 is not factorable?

Using the digits 1 to 9 with no repeats, ﬁll in the blanks to create a factorable trinomial and provide the factored form: 4x2 + ⬚⬚ x + ⬚ Using the digits 1 to 9 with no repeats, ﬁll in the blanks to create a perfect square trinomial: ⬚ x2 + ⬚⬚ x + ⬚⬚

The expression 16x2 − 24x + ⬚ is a perfect square trinomial. Determine the missing value.

A photographer wants to put a border around her photo. a

Why might the photographer use x units to represent the width of the border?

Use factoring to ﬁnd expressions for the dimensions of the photograph with its border if the total area is 4x2 + 24x + 35 square units.

What is the area of the photograph? Explain how you reached your solution.

Factor the quadratic expression: 6ab2 − 36ab − 162a

Rewrite the quadratic expression in factored form: 3x2 − 24xy + 48y2

Consider the factorable polynomial 12x2 + bx − 6, where b is a positive integer. a

Find the largest possible value for b and list the factors.

Find the smallest possible value for b and list the factors.

Sheldon and Gabriella each factored 16x2 + 48x + 36. Whose work is incorrect? Identify and explain the error. Sheldon: 2

Gabriella:

16x + 48x + 36 = 16x + 24x + 24x + 36

16x2 + 48x + 36 = (4x + 6) (4x + 6)

= 8x(2x + 3) + 12(2x + 3)

= 2(2x + 3) (2x + 3)

= (8x + 12) (2x + 3) = 4(2x + 3) (2x + 3) 25

A shape is formed from a square with side lengths of 3x that has a smaller square, with side lengths of 4, cut out from the center. a

Write a simplified expression for the area of this shape.

A rectangle is created to have the same area found in part (a). Determine the dimensions for the rectangle.

4 3x

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Answers 6.06 Factor trinomials

c (x + 4) (x + 2) (x − 2)

b a = −9, b = 7

c a = −6, b = −4

b x(x + 5)2 d 2(x2 + 25) (x + 5) (x − 5)

Let’s extend our thinking

3 (x + 6) (x + 4) 4 a (x + 2) (x + 3)

16 a E ach polynomial has a leading coefficient a ≠ 1 and none of the polynomials can be factored.

b (3x − 1)(x + 4)

b F ind the product a · c, then look for factors of ac that can add or subtract to be equal to the value of b. If no such factors exist, the polynomial is not factorable.

5 a i 1, 2, 3, 6, 9, 18 ii (x + 2) (x + 9) b i 1, 2, 4, 8, 16, 32, 64 ii (x + 8) (x + 8)

17 4x2 + 12x + 5 = (2x + 1) (2x + 5) is one possible answer.

c i 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 ii (x + 8) (x + 9)

18 Answers may vary. 9x2 + 54x + 81 and 4x2 + 36x + 81 are two possibilities.

6 B

19 9 20 a S he would use x units to represent the width of her border if she did not know yet what size border she wanted.

Let’s practice 7 The algebra tiles total an area of x2 − 2x − 8 and the product of the width and length is (x − 4) (x + 2). 8 a (x − 5) (x − 3)

b T he expression for area factors as (2x + 5) (2x + 7) square units, so, the dimensions of the photograph are 2x + 5 units and 2x + 7 units.

b (x + 3) (x + 8)

c (x − 9) (x + 10)

d (x + 10) (x + 12)

e (x + 2) (x − 36)

g 3(x − 10) (x + 1)

h 35 (m + 1) (m + 3)

c I f the border is a width of x units, you would subtract 2x from each dimension since there is a border on both sides. That leaves the dimensions as 5 units and 7 units, so, the area of the photograph is 35 square units.

(x + 8) (x − 7)

2(x + 6) (x + 8)

(x + 5)2

(x − 8)2

m (9 + x)

n (6 − x)2

o 4(x + 5)2

p -3(x − 2)2

k (x + 7)

21 6a (b - 9) (b + 3)

9 Although this expression appears to have a leading coefficient of 3.3 is a common factor that can be factored out of the expression which leaves a trinomial where a = 1. The fully factored form is 3(x2 + 5x − 14) = 3(x + 7) (x − 2). 10 x

−1

−x

−1

23 a T he largest possible value for b is 71 and the factors are (12x − 1) (x + 6) b T he smallest possible value for b is 1 and the factors are (4x + 3) (3x − 2).

25 a 9x2 - 16 x2

b 3x + 4 and 3x - 4

x2 −1

11 a (2x − 1) (2x − 15)

b (5x − 2) (x − 9)

c (2x − 9) (x − 5)

d (10x + 3) (x − 8)

e (4x + 7) (6x − 5)

g (5 - 3x) (2x - 1)

h (7x − 11) (6 − 5x)

(9x + 2)2

22 3 (x - 4y)2

24 Gabriella did not pull out the common factor correctly. Since there is a factor of 2 in both factors of her expression, she needs to factor out a greatest common factor of 4.

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d −3(4x + 9) (5x − 1)

15 a 3(3x + 4) (3x − 4)

2 a a = 9, b = 10

c 3(3x + 1) (x + 1)

14 x + 3

1 18, 9

b x2(2x − 21) (x − 2)

13 x + 6

What do you remember?

12 a 10(6x + 5) (x − 2)

(3x − 5) (2x − 3) (7x - 2)2

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

6.07 Factor using appropriate methods Subtopic overview Lesson narrative In this lesson, students will explore structures of certain types of polynomials and their factored forms to make generalizations. Students will determine the best method for factoring polynomials and justify the use of that method. By the end of the lesson, students will be able to create polynomial expressions to represent quantities for contextual situations and work flexibly between various forms of polynomial expressions as needed.

Learning objectives Students: Page 363

Key vocabulary 

difference of two squares



perfect square trinomial

Essential understanding The structure of an expression can provide information on the most efficient way to rewrite it.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG5 — Mathematical Representations Teachers can incorporate this goal by encouraging students to use a variety of methods to represent their factoring by grouping process. This could include diagrams, flowcharts, or symbolic notation. Students could also be asked to translate between different representations, such as interpreting a written description of the factoring process in mathematical notation.

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Content standards A.EO.2 — The student will perform operations on and factor polynomial expressions in one variable.

A.EO.2e — Represent and demonstrate equality of quadratic expressions in different forms (e.g., concrete, verbal, symbolic, and graphical).

Prior connections 8.PFA.1 — The student will represent, simplify, and generate equivalent algebraic expressions in one variable.

A.EO.1 — The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

Future connections A2.EO.3 — The student will perform operations on polynomial expressions in two or more variables and factor polynomial expressions in one and two variables.

Engage Activity Special factoring

60 mins

Students will be writing their own factoring rules and applying them to perfect square trinomials and the difference of two squares.

Understanding and skills

Will use Factoring trinomials.

Will develop Recognizing special factoring patterns found in factoring perfect square.

Could extend Applying special factoring patterns found in factoring perfect square trinomials and the difference of two squares.

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Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: None.

Support students with disabilities Support memory - use previously taught skills and concepts Provide the formula(s) for expanding a quadratic in factored form: (x − a)(x − b) = x2 − (a + b) x + ab Where: • a and b are real numbers.

Support for English language learners Critique, correct, and clarify While students are discussing the calculations for factoring quadratic functions, display the following incorrect calculation and reasoning: The quadratic expression x2 + 8x + 12 has the factors (x − 3) and (x − 4). Ask students to identify the error, critique the reasoning, and write a short explanation of how the students’ reasoning could be improved.

Classroom guide Hook

Which one doesn’t belong

Students choose one of four binomial products.

•

5 mins

Which one doesn’t belong? Select one option.

Implementation details Highlight student responses that recognize differences in the structure of the expressions such as identical binomials, opposite signs, or similar values. Some students may choose to expand the binomial product or look for differences in the expanded forms.

(x + 1) (x + 1)

(x + 1) (x − 1)

(2x + 1) (2x − 1)

(x − 1) (x − 2)

Slide 1 from Student Engage Activity

Launch

5 mins

Remind students of what it means to factor a trinomial expression. Important mathematical concepts: Factoring, trinomials, quadratic expressions Suggested grouping: Form pairs.

Continue when Students have read the Launch and understand the context of the problem.

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Explore Anticipated strategies Find patterns Students may notice patterns in the terms that are listed in each of their selected expressions, and use those patterns to generalize rules about factoring. What are the relationships between unfactored form and factored form of the polynomials?

Think-pair-share

•

15 mins

Factor three of the following expressions: x2 − 64

x2 − 12x + 36

x2 − 16x + 64

x2 − 25

Slide 4 from Student Engage Activity

Create area models Students may use area models to try and determine how to factor each of their chosen polynomials. Then, they can look for patterns in the dimensions of each area model and summarize their ﬁndings.

Factor polynomials Students will factor three of the four polynomials. The polynomials and their corresponding factored forms are as follows: • x2 − 64 = (x + 8)(x − 8) • x2 − 12x + 36 = (x − 6)2 • x2 − 16x + 64 = (x − 8)2 • x2 − 25 = (x + 5)(x − 5) Students will also come up with their own factoring rules for the two types of polynomials, namely perfect square trinomials and the difference of two squares. Students will be apply their rules to determine what types of numbers make up the special polynomials that they’ve been working with.

Misconceptions Creating only one rule for both forms Do you notice anything different about your three polynomials? What about in their factored form? How would you describe each type of polynomial? Do you think we can deﬁne our factoring rule with just one rule? Or more?

Purposeful questions Use the following questions to check for understanding and encourage critical thinking: • Can you draw a picture or make a model to show that? • How could you prove that? • Is that true for all cases? • What would happen if..? What if not? • Do you see a pattern here? Can you make another trinomial that ﬁts the pattern?

Continue when All students have factored three polynomials and described rules for factoring.

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Discuss

35 mins

Begin with small group discssions, then share out to the whole class to formalize and iterate on their self-written factoring rules. Make connections to generalized factoring rules wherever possible.

Discussion guide The discussion should be used as a time for students to share their factoring rules, compare their results with their classmates’, and then iterate on their rules. Start the discussion by having each pair join with another pair to share their factoring rule(s) and the types of numbers that they need to have in order to create the required factors. Give groups time to adjust or update their rules after sharing and comparing. Next, invite each group to share their rules and patterns for which these rule apply. Group and display these examples with a visual divide between the two forms. For each rule type (perfect square trinomials and difference of two squares) ask students to produce example expression that the class can then try factoring according the directions in each rule described. If time permits, you may want to ask students to consider an expression like x2 + bc + (2c)2 and ask what we might be able to determine about the possible values for a if the factored form of the expression is known to be (x + a)2 (it can be shown that a and b must be even numbers). This will allow students to deepen their understanding of the generalized rules they produced.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Algebra 1 — 6.02 Multiply polynomials Algebra 1 — 6.04 Factor GCF Algebra 1 — 6.05 Factor by grouping Algebra 1 — 6.06 Factor trinomials

Tools You may ﬁnd this tool helpful: • Scientific calculator

Student lesson & teacher guide Factor using appropriate methods Students review special products that will help them with factoring. An exploration that requires analyzing patterns follows.

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Students: Page 363

Compare and connect English language learner support Ask students to identify similarities and differences between being asked to ﬁnd the product of (a − b) (a + b) and being asked to factor a2 − b2 (or any of the patterns). Consider using the following sentence stems as an added support: • “Factoring a difference of squares, a2 − b2, and multiplying the product of a sum and a difference, (a − b) (a + b), are similar because...” • “Factoring a difference of squares, a2 − b2, and multiplying the product of a sum and a difference, (a − b) (a + b), are different because...” • “One thing that is the same between the two questions is...” • “One thing that is the different between the two questions is...” • “Factoring a difference of squares, a2 − b2, and multiplying the product of a sum and a difference, (a − b) (a + b), are related because...”

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Connecting perfect squares to special products Targeted instructional strategies In previous grades, students learned that perfect squares are numbers that can be modeled as a square. Remind students of the pattern of perfect squares with a visual like the one shown. 12

Show students examples of models of special products using algebra tiles that form a square with their corresponding polynomials written underneath. Ask students to identify what shape each model has in common, and what they notice about the numerical values in the polynomials underneath.

−x

x2 + 6x + 9

x2 − 6x + 9

−x

−1

x2 − 9

−x

1 − 4x + 4x2

Students should predict how numerical values from the polynomial could look for the factored form given the patterns from the perfect square patterns and algebra tile models.

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Decompose the problem into smaller chunks and use an algorithm Student with disabilities support Encourage students to use Computational Thinking by decomposing the problem into smaller chunks and providing an annotated example. For example, when factoring the polynomial shown: 81x2 − 72x + 16 1. Fill in the box : 81x2 = ⬚2, call the value in the box a. 2. Fill in the box: 16 = ⬚2, call the value in the box b. 3. Check that 72x can be written as 2ab

4. Fill in the pattern: a2 − 2ab + b2 = (a − b)2 with the values of a and b from steps 1 and 2.

While this template is helpful for checking if a polynomial fits certain identities, students can also factor using the strategies previously used for other trinomials. While trinomials can be factored as is, when factoring difference of squares, a linear term with a coefficient of 0 can be added between the two terms. For example: a2 − 25 = a2 + 0a − 25

Rewrite with +0a

= a + 5a − 5a − 25

Find pair to rewrite for factoring by grouping

= a(a + 5) − 5(a + 5)

Factor out the GCF

= (a − 5) (a + 5)

Write as product

Avoid mistakes by checking that an expression is equivalent to a special product Address student misconceptions Students may notice that the ﬁrst and last terms are perfect squares and skip the step of checking whether or not the middle term does make it a perfect square trinomial. For example, when asked to factor 4x2 + 13x + 9 an incorrect solution could be: 1

4x2 + 13x + 9 = (2x)2 + 13x + 32

= (2x + 3) 2

Rewrite as a perfect square trinomial Use the pattern

While the correction solution is 4x + 13x + 9 = (4x + 9) (x + 1).

Exploration Students: Page 363

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Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Suggested student grouping: Small groups Students are presented with factored polynomials and their original polynomial form. Students should notice special products and patterns that arise between polynomials and their factored form. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What do you notice about the factored form of the given polynomials? The form of the factored polynomials can be generalized as (a ± b) (a ± b). There is a pattern in the quadratic term, the linear term, and the constant of each polynomial depending on its factored form. The quadratic term for each polynomial is a2. The linear term for each polynomial is the product ±2ab. The constant term for each polynomial is the product ±b2. These are a mixture of special products but they each follow a similar pattern between the two terms of the binomials that are multiplied together. Purposeful questions • What examples of factored binomials and their polynomials can you come up with based on the patterns you see here? • Do you prefer to memorize the algorithm for the special products or look for another pattern? Possible misunderstandings • Students may struggle to see patterns with few examples. Provide examples of other difference of two squares and perfect square trinomials to help students identify patterns.

Advanced learners: Analyzing structure to inform factoring methods Targeted instructional strategies Encourage advanced students or all students to analyze the structure of the polynomial expressions they are factoring to discover patterns within different types of polynomials. While working through the following examples, ask them to describe structures or identify patterns in the polynomial expressions. Ask questions like, “What similarities do you notice among these polynomials, and how do those similarities help in factoring them efficiently?” This will prompt students to derive generalizations and efficiently factor a polynomial based on the structures they observe. Additionally, this exercise helps students assess and articulate their problem-solving strategies. Students are provided with a procedure for factoring special products. They are also shown how models with algebra tiles connect to special products and their factored form.

Students: Pages 363–364

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For example, consider the product of a binomial squared, (a + b)2: a

We can expand (a + b)2 to (a + b) (a + b) and represent them with an area model. Evaluating with this model and combining like terms, we get the product a2 + 2ab + b2.

So, we have: a

(a + b)2 = (a + b) (a + b) = a2 + 2ab + b2

Follow these steps for determining if a trinomial is a perfect square trinomial and factoring: 1. Factor out the GCF 2. Determine a from the leading term and b from the constant term 3. Verify whether the linear term is equal to 2ab 4. If yes, use the structure of perfect square trinomials to write the factors Now consider the product of a sum and a difference, (a + b) (a − b): a

−b

−ab

−b2

Notice that the term ab and −ab are opposites and combine to make zero. We call this a zero pair. So, (a + b) (a − b) = a2 − b2. Follow these steps for determining if a binomial is a difference of two squares and factoring: 1. Factor out the GCF 2. Determine a from the leading term and b from the constant term 3. Verify whether a and b are perfect squares by identifying their square roots 4. If yes, use the structure of a difference of squares to write the factors

Examples Students: Page 365 Example 1 Factor 3p2 + 12p + 12 364

Mathspace Virginia SOL Algebra 1 mathspace.co

Create a strategy We can factor a GCF of 3 out of the polynomial and write the polynomial as 3( p2 + 4p + 4). Determine if the polynomial expression is a perfect square trinomial. Since a = p and b = 2 and the linear term is 2ab = 2( p) (2) = 4p, we can verify that this is a perfect square trinomial and we can use special products to factor.

Apply the idea Since p2 + 4p + 4 is a perfect square trinomial, we use the identity in factoring:

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Mathspace Virginia Edition Identity for a perfect square trinomial 2ab Algebra + b2 = (a1+Teacher b)2 a2 +SOL mathspace.co Substitute a = p and b = 2 = ( p + 2)2 There are no more common factors to be divided out, so the fully factored form of the polynomial is 3( p + 2)2.

Apply the idea Since p2 + 4p + 4 is a perfect square trinomial, we use the identity in factoring: a2 + 2ab + b2 = (a + b)2 = ( p + 2)2

Example 1

Identity for a perfect square trinomial Substitute a = p and b = 2

There are no more common factors to be divided out, so the fully factored form of the polynomial is 3( p + 2)2. Factor 3p2 + 12p + 12

Reﬂect and check Create a strategy If we instead were to factor by grouping, we will ﬁnd the value of two integers that multiply to ac = (1) (4) = 4 and add

We can a GCF of 3these out ofintegers, the polynomial write the polynomial as term 3( p2 4p + 4p Determine if theand then up to b =factor 4. After ﬁnding we useand them to rewrite the middle as+a4). sum of two terms polynomial expression is a perfect square trinomial. factor the trinomial by grouping.

Since a = p of and =2 and 4, the is 2ab = 2(factors, p) (2) = 2 4p, we2can that thisadds is a perfect The factors 4b are 1 and 2 linear and 2.term Among these and are verify the pair that up to 4. square We cantrinomial use this and to we can the usetrinomial special products to by factor. rewrite and factor grouping as follows: 3( p2 + 4p + 4) = 3( p2 + 2p + 2p + 4)

Apply the idea

Rewrite polynomial with four terms

= 3[p( p + 2) + 2( p + 2)] Factor each pair Since p2 + 4p + 4 is a perfect square trinomial, we use the identity in factoring: = 3( p + 2) ( p + 2) Divide out the common factor of ( p + 2) Identity for a perfect square trinomial a2 + 2ab + b2 = (a + b)2 There are no more common factors2 to be divided out, so the fully factored form of the polynomial is 3( p + 2)2. Substitute a = p and b = 2 = ( p + 2) There are no more common factors to be divided out, so the fully factored form of the polynomial is 3( p + 2)2.

Reflect and check PurposeExample 2 If we instead were to factor by grouping, we will find the value of two integers that multiply to ac = (1) (4) = 4 and add Show students how to factor a GCF and a perfect square trinomial with a sum. Fully factor 9x2 − 24x + 16. up to b = 4. After finding these integers, we use them to rewrite the middle term 4p as a sum of two terms and then

factor the trinomial by grouping. Expected mistakes Create a strategy factors of 4 the are 1general and 4, 2 and 2. of Among these factors, and 2 are the pair that the addsgeneral up to 4. We can use this special to StudentsThe may forget form a perfect square2 trinomial. Keeping forms of the 2 rewrite the trinomial and9xfactor by +grouping as follows: We check ﬁrst whether − 24x 16 is a special product and identify its type. products nearby may help students recall the patterns. 2 and b = 4 and 2the linear coeﬃcient is −2ab = 2(3x) (4) = −24x, we can verify that this is a perfect square Since a3(=p3x + 4p + 4) = 3( p + 2p + 2p + 4) Rewrite polynomial with four terms

Reﬂecting withand students trinomial we can use special to factor. = 3[p( p + 2)products + 2( p + 2)] Factor each pair Challenge students to factor the expression by grouping. Then, ask students which method they prefer for factoring. = 3( p + 2) ( p + 2) Divide out the common factor of ( p + 2) Apply the idea

are2 − no24x more common factors to betrinomial, divided out, so the of the polynomial is 3( p + 2) . Since 9x + 16 is a perfect square we can usefully the factored identity inform factoring: Students:There Pages 365–366 a2 − 2ab + b2 = (a − b)2 = (3x − 4)2

Identity for a perfect square trinomial Substitute a = 3x and b = 4

Example 2 Fully factor 9x2 − 24x + 16.

Create a strategy We check ﬁrst whether 9x2 − 24x + 16 is a special product and identify its type.6.07 Factor using appropriate methods 365 Since a = 3x and b = 4 and the linear coeﬃcient is −2ab = 2(3x) (4) = −24x, we can verify that this is amathspace.co perfect square trinomial and we can use special products to factor.

Apply the idea Since 9x2 − 24x + 16 is a perfect square trinomial, we can use the identity in factoring: a2 − 2ab + b2 = (a − b)2 = (3x − 4)2

Identity for a perfect square trinomial Substitute a = 3x and b = 4

Reﬂect and check We can check the answer by multiplying the factored form (3x − 4)2. (3x − 4)2 = 9x2 − 24x + 16

Identity for square of a binomial 6.07 Factor using appropriate methods mathspace.co

365

Example 3 Purpose Factor 1 −how x2. to factor a perfect square trinomial with a difference. Show students Create a strategy

6.07 Factor using appropriate methods mathspace.co Since it can be rewritten as (1)2 − (x)2 the polynomial is a difference of squares and we can use special products to factor. We check ﬁrst whether 1 − x2 is a special product and identify its type.

Apply the idea

771

Expected mistakes Students may ignore the negative symbol attached to 24x. Remind students to consider how the factoring changes with the signs of the terms. Reﬂecting with students Ask students how they can check that the factored form they wrote is correct. Students can multiply the binomials to conﬁrm that the product is the original polynomial.

Combining patterns for perfect square trinomials

use with Example 2

Student with disabilities support Some students will benefit from seeing a2 − 2ab + b2 = (a − b)2 and a2 + 2ab + b2 = (a + b)2 as a separate patterns, but students who find notation and manipulating symbols challenging may find it easier to consider that the second term in the parentheses can be positive or negative and that dictates the sign of the middle term. Some students will appreciated seeing the pattern in words instead of variables. First2 + 2 ⋅ First ⋅ Second + Second2 = (First + Second)2 In this case, First = 9x and Second = −4, so we can write it as: 81x2 − 72x + 16 = (9x)2 + 2 (9x) (−4) + (−4)2 2

Reﬂect and check = (9x − 4)

Notice that First = 9x and Second = −4 Substitute

We can check the answer by multiplying the factored form (3x − 4)2. (3x − 4)2 = 9x2 − 24x + 16

Identity for square of a binomial

Students: Page 366 Example 3 Factor 1 − x2.

Create a strategy We check ﬁrst whether 1 − x2 is a special product and identify its type. Since it can be rewritten as (1)2 − (x)2 the polynomial is a difference of squares and we can use special products to factor.

Apply the idea Since 1 − x2 is a difference of two squares, we use the identity in factoring: a2 − b2 = (a + b) (a − b)

Identity for a difference of two squares

= (1 + x) (1 − x)

Substitute a = 1 and b = x

Reﬂect and check We can check the answer by multiplying the factored form (1 + x) (1 − x). (1 + x) (1 − x) = 1 − x2

Identity for product of a sum and difference

We can also check our answer using algebra tiles: 1

−x

So, (1 − x) (1 + x) is equivalent to 1 − x + x − x2 = 1 − x2.

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Purpose Show students how to factor a difference of two squares. Expected mistakes Students may write the solution as (x + 1) (x − 1) because they are accustomed to writing the variable first in the binomials. Remind students that factored the pattern follows the root of the first square term followed by the root of the second square term. Reflecting with students Ask students what the factored form of 1 + x2 would be. Remind students that the sum of two squares is a prime polynomial that is not factorable.

1 is a perfect square

use with Example 3

Address student misconceptions Some students may not recognize 1 as a perfect square because it is its own square and square root. Supporting students with the step of going from 1 − x2 → 12 − x2 may be required. Also, it is important to have examples where the constant is ﬁrst as some students may not recognize this as a difference of squares as quickly.

Students: Page 367

Example 4 Factor 4m2 + 40m + 36.

Create a strategy We can factor a GCF of 4 out of the polynomial and write the polynomial as 4(m2 + 10m + 9). Determine if the polynomial expression is a perfect square trinomial. Since a = m and b = 3 and the linear term should be 2ab = 2(m) (3) = 6m and 6m ≠ 10m, we can verify that this trinomial is not a perfect square trinomial and instead factor by grouping.

Apply the idea The factors of 9 are 1 and 9, 3 and 3. Among these factors, 1 and 9 are the factor pair that adds up to 10. We can use this to rewrite the trinomial and factor by grouping as follows: 4(m2 + 10m + 9) = 4(m2 + m + 9m + 9)

Rewrite polynomial with four terms

= 4[m(m + 1) + 9(m + 1)]

Factor each pair

= 4(m + 1) (m + 9)

Divide out the common factor of (m + 1)

There are no more common factors to be divided out, so the fully factored form of the polynomial is 4(m + 1) (m + 9).

Reﬂect and check If the polynomial could not be rewritten as four terms and factored by grouping, we would determine that the polynomial is not factorable.

Idea summary Purpose By recognizing the patterns of factoring using special products, we can factor more eﬃciently: Show students• how to factor a GCF followed bya2grouping. Perfect square trinomials: a2 + 2ab by + b2factoring = (a + b)2 or − 2ab + b2 = (a − b)2 •

Difference of two squares: a2 − b2 = (a + b) (a − b)

Reflecting with students Ask students why factoring a GCF first may be important. Factoring a GCF first allows us to work with smaller numbers. Practice

What do you remember? 1

For each trinomial in the form ax2 + bx + c: i

Find two integer values that have a sum of b and a product of c.

Write the quadratic in factored form.

x2 + 2x − 24

x2 − 4x − 77

x2 − 17x + 66

6.07 Factor using appropriate methods mathspace.co

x2 + 10x + 24

773

= 4(m + 1) (m + 9)

Divide out the common factor of (m + 1)

There are no more common factors to be divided out, so the fully factored form of the polynomial is 4(m + 1) (m + 9).

Reﬂect and check the polynomial Students:If Page 367 could not be rewritten as four terms and factored by grouping, we would determine that the polynomial is not factorable.

Idea summary By recognizing the patterns of factoring using special products, we can factor more eﬃciently: • •

Perfect square trinomials: a2 + 2ab + b2 = (a + b)2 or a2 − 2ab + b2 = (a − b)2 Difference of two squares: a2 − b2 = (a + b) (a − b)

Practice What do you remember?

Practice 1

For each trinomial in the form ax2 + bx + c:

Find two integer values that have a sum of b and a product of c.

Students: Pages 367–370 ii Write the quadratic in factored form. a

x2 + 2x − 24

x2 − 4x − 77

x2 − 17x + 66

x2 + 10x + 24

each quadratic expression. The factored form of the expression is ( Ax + C) (Bx + D). Determine the What do 2youConsider remember? following:

i the product of A and B 2 For each trinomial in the form ax + bx + c:

the signs for A and B: opposite or the same

Write ivthethe quadratic inand factored form.or the same signs for C D: opposite

a −15x 24 + 14x – 16 x2 + 2x 2

x2 − 4x − 77

2 bc −45x 29x+−66 4 x2 −+17x

x2 + 10x + 24

Consider each quadratic expression. The factored form of the expression is ( Ax + C) (Bx + D). Determine the 6.07 Factor using appropriate methods 367 following: mathspace.co i

the product of A and B

the signs for A and B: opposite or the same

iii

the product of C and D

the signs for C and D: opposite or the same

15x2 + 14x – 16

−45x2 + 29x − 4

121m2 − 64

4 − 49y2

121x2 − 49y2

x2 − 16x + 64

49x2 − 28x + 4

Fully factor the quadratic expressions: a

n2 − 36

3t2 − 12

i 4

Find two integer values that have a sum of b and a product of c. iii the product of C and D

81 + 18x + x

4 − u2

5x2 − 320

36 − 12x + x

4x + 40x + 100

Describe the similarities and differences between factoring a polynomial in the form ax2 + bx + c where a = 1 or where a ≠ 1.

Let’s practice 5

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For each of the expression: i

State an appropriate strategy or combination of strategies for factoring. Explain your choice.

Factor the expression fully.

x2 + 17x + 72

3x2 + 3x − 60

x2 + 2x + 1

3n2 − 363

2x2 − 6x + 5x − 15

9x2 − 12x + 4

(x + 16)2 − x2

8p( p2− 9) −5 ( p2 − 9)

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Factor the expressions fully using appropriate techniques: a

y2 − 13y + 12

3x2 + 24x + 48

12x − 3x2 − 20 + 5x

9a2 − 100

−2x2 − 5x + 3

6x2 − 36x + 48

4x2 − 64

x2 − 6x + (x + 1) (x − 6)

A square carpet with side length x is modiﬁed by shortening one side by 6 inches and lengthening the other by 7 inches. a

Find the area of the carpet. Write your answer in factored form.

Find the area of the modiﬁed carpet in the form x2 + bx + c.

Describe the relationship between the values 7, 6, 1 and −42.

A square table of side length x has one of its dimensions decreased by 4. This can be expressed visually by the area model shown. x 4

Area I

Area II

Determine if the expressions are equivalent to the area x (x − 4):

Area I + Area II

x2 − Area II

Area I

x2 − Area I

A square rug originally had a side length of 2x inches. One of its dimensions is extended by 3 inches. We can model the area of the rug as a collection of rectangles, as shown in the diagram. The square on the left of the diagram has a side length of 2x inches and the short side of each rectangle is 1 inch.

Write a factored expression that represents the area of the rug.

Find the areas of each section from the rug.

Find the total area of the rug in terms of x. Give your answer in the standard form ax2 + bx + c.

The area of a quilt can be expressed as x2 + 6x + 4x + 24. a

Label the area model so that it represents the area of the quilt. Area I

Area II

Area III

Area IV

Express the area of the quilt in factored form.

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Suppose we want to write the trinomial x2 + bx + c in the form (x + p) (x + q). Read each condition and explain what must be true regarding the signs of p and q. a

c is negative.

c is positive and b is negative.

c and b are both positive.

Let’s extend our thinking 12

Determine whether each of the polynomial can be factored: a

x2 + 12x + 15

−x2 + 17x − 70

x + 9x + 5

c g

x2 + x − 42 2

6x − 13x − 5

x2 + 19x − 90

4x2 − 20x − 25

Find two digits (1 to 9) to ﬁll in the blanks and create a trinomial in the form ax2 + bx + c that a

x + 25

Maximizes the value of b. b Minimizes the value of c. (x + ⬚) (x + ⬚) (x + ⬚) (x + ⬚) Minimizes the value of b. (x + ⬚) (x − ⬚)

A quadratic polynomial is of the form ax2 + bx + c, where a is non-zero, and has a special factored form. a

If b = 0, identify the form that the quadratic could be. State what must be true about the sign of c.

If b < 0, identify the form that the quadratic could be. State what must be true about the sign of c.

Factor the expressions fully using appropriate techniques: a

4b2 − 81c2

x2y2 − 36x2

9a2 + 24ab + 16b2

−2x4 + 2x3 + 24x2

x2 − x2y + xy − x

80x4 + 92x3 + 24x2

15x2y + 50xy − 40y

xy2 + 4x

81 − n4

4x3 + 16x2 − x − 4

Rewrite the expression (a2 − b2) (c2 − d2) as a difference of two squares.

Maribel is crafting a large blanket to give to her parents for their anniversary. Her plan is to combine a patchwork of family photographs with the ﬂags from her parents’ home countries. She hasn’t determined the exact size of the blanket yet, and instead has expressions for the possible dimensions of each piece of fabric. (2x + 3)

(x + 3)

(x + 6)

(x)

(x + 3) (x + 11)

776

(2x − 5)

Write an expression for the total area of the blanket. Show at least three possible factorizations for the area.

Find a value for x that creates a blanket with reasonable dimensions. Be sure to include units.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Answers

7 a (x − 6) (x + 7)

6.07 Factor using appropriate methods What do you remember? 1 a i -4, 6

ii (x - 4) (x + 6)

b i 7, -11

ii (x + 7) (x - 11)

c i -6, -11

ii (x - 6) (x - 11)

d i 6, 4

ii (x + 6) (x + 4)

2 a i 15

ii The same

iii –16

iv Opposite

b i −45

ii Opposite

iii −4

iv Opposite

3 a (n − 6) (n + 6)

b (2 - u) (2 + u)

c (11m + 8) (11m − 8)

d (2 + 7y) (2 - 7y)

e 3(t + 2) (t - 2)

c T he sum of 7 and -6 is 1. The product of 7 and -6 is -42. 8 a No

b Yes

c Yes

9 a 2x(2x + 3) b Area of square rug is 4x2 in2

Area of each rectangle is 2x in2

c 4x2 +6x in2 10 a

+ 4

5(x − 8) (x + 8)

h (x - 8)

(9 + x)

j 2

k 4(x + 5)

b (x + 6) (x + 4)

(6 − x)2 2

(7x − 2)

11 a Exactly one of p or q must be negative. b Both p and q must be positive.

4 In both cases, we are looking for a pair of binomials where the leading coefficients of the factors will multiply to be a and the constant terms will multiply to be c. The difference is that in a polynomial where a ≠ 1 the factors of the constant c won’t add up to the coefficient of the linear term, b.

Let’s extend our thinking

Let’s practice

13 a (x + 9) (x + 8) = x2 + 17x + 72

5 a i F inding two numbers that have a sum of 17 and a product of 72 ii (x + 9) (x + 8) b i C ommon factor then find two numbers that have a sum of 1 and a product of -20 c i Perfect square trinomial ii (x + 1)2 d i Common factor then difference of squares

e No

b Yes

c Yes

d No

g Yes

h No

b (x + 2) (x + 1) = x2 + 3x + 2 c (x + 1) (x − 9) = x2 − 8x − 9 14 a If b = 0, the quadratic might be able to be factored as a difference of two squares. For this to be the case, we must also have c < 0.

15 a (2b + 9c) (2b - 9c) c (3a + 4b)2

e i Factoring by grouping

d -2x2 (x − 4) (x + 3)

ii (2x + 5) (x − 3)

e x(x - 1) (1 − y) or -x(x − 1) (y − 1)

i Perfect square trinomial

ii (3x − 2)2

4x2 (5x + 2) (4x + 3)

g 5y (3x − 2) (x + 4)

g i Difference of squares

h x(y2 + 4)

ii 32(x + 8) h i Binomial common factor, then difference of squares ii (8p − 5) ( p − 3) (p + 3)

c (3x − 5) (4 − x)

12 a No

b x2 (y + 6) (y − 6)

ii 3(n + 11) (n − 11)

6 a (y − 1) (y − 12)

c Both p and q must be negative.

b If b < 0, the quadratic might be able to be factored as a perfect square. For this to be the case, we must also have c > 0.

ii 3(x − 4) (x + 5)

d No

g (11x + 7y) (11x – 7y) i

b x2 + x − 42

(9 + n2) (3 + n) (3 − n)

(x + 4) (2x + 1) (2x - 1)

b 3(x + 4)

d (3a − 10) (3a + 10)

e −(2x − 1) (x + 3)

g 4(x + 4) (x − 4)

h (x − 6) (2x + 1)

6(x − 2) (x − 4)

Answers mathspace.co

777

17 a T he total area of the blanket is 9x2 + 45x + 54. Many factorizations exist. Three possible answers are: (3x + 6) (3x + 9), 9(x + 2) (x + 3), or 9(x2 + 5x + 6) b A nswers will vary. Since dimensions can’t be zero or negative, all answers must have x > 2.5. For example, if x = 10 inches, the overall dimensions for the blanket would be 36 inches by 39 inches and each piece would have dimensions of 21 inches, 15 inches, 23 inches, 13 inches, 16 inches and 10 inches which will all reasonably display a photograph.

778

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

6.08 Divide polynomials Subtopic overview Lesson narrative In this lesson, students will learn how to divide two polynomials using the process of factoring and dividing out the common factors. Students will review and use a variety of types of factoring in order to simplify the polynomial expression, including factoring out negatives and the greatest common factor.

Learning objective

6.08 Divide polynomials

Students: Page 371

After this lesson, you will be able to… • divide a polynomial by a binomial or factored divisor.

Divide polynomials Dividing polynomials involves a process known as algebraic manipulation. We can view our dividend as the Key vocabulary 

numerator of a fraction and our divisor as the denominator.  divisor dividend . The dividend is 3x2 + 7x − 6 and the divisor is 3x − 2. We can use algebra tiles Consider the expression: to model the division. We will create a rectangle to represent the dividend, and the side lengths of the rectangle will represent its factors.

Essential understanding

We already know one of the factors is the divisor so we can make Polynomials can be using steps x divided x x −1 −1 similar to those used when dividing real numbers. one of the side lengths 3x − 2.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG2 — Mathematical Communication

MPG4 — Mathematical Connections

Teachers can integrate this goal by relating the concept Teachers can integrate this goal by asking students to of dividing polynomials to concepts previously learned, write out their thought process when solving problems, Next we fillsuch in the with 3 of the multiplying, x2 tiles, 7 of and the positive asrectangle adding, subtracting, factoring using the correct xmathematical notation and vocabulary. x x −1 −1 2 6 of the −1 tiles to represent the dividend 3xconcept + 7x − 6, They can also show how this They should encourage students to explain their x tiles, andpolynomials. to line up with equal lengths. is related to tiles other disciplines, such as physics or reasoning when dividing out common factors and making sure x2

Notice thateconomics, there are some empty that we need to fill into with which oftenspaces use polynomial functions simplifying the resulting expression. This can involve zero pairs. model real-world scenarios. group discussionsx or presentations to the class. x x −1 −1 −1 −1

MPG5 — Mathematical Representations x

−1 −1

Teachers can integrate this goal by encouraging students to use different representations to understand and solve problems involving polynomial division. This could include using physical manipulatives, drawing diagrams, or using symbolic notation. For example, teachers could ask students to use algebra tiles to physically represent the process of dividing polynomials. x

−1 −1

−x −x

The empty spaces are the right size for x tiles. Since we need to add zero pairs so that we don’t change the6.08 value of the expression 779 Divide polynomials mathspace.co we will fill 2 of the spaces with positive x tiles and the other 2 spaces with −x tiles.

−1 −1

Now we can see that the length of the left side of the rectangle is

Content standards A.EO.2 — The student will perform operations on and factor polynomial expressions in one variable.

A.EO.2d — Determine the quotient of polynomials, using a monomial or binomial divisor, or a completely factored divisor.

Prior connections 8.PFA.1 — The student will represent, simplify, and generate equivalent algebraic expressions in one variable.

A.EO.1 — The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

Future connections A2.EO.3 — The student will perform operations on polynomial expressions in two or more variables and factor polynomial expressions in one and two variables.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Algebra 1 — 6.04 Factor GCF Algebra 1 — 6.05 Factor by grouping Algebra 1 — 6.06 Factor trinomials Algebra 1 — 6.07 Factor using appropriate methods

Tools You may ﬁnd this tool helpful: • Scientiﬁc calculator

Student lesson & teacher guide Divide polynomials Students are presented vocabulary for dividing and a step-by-step process for dividing a polynomial by another polynomial using algebraic tiles. Steps for dividing algebraically are also provided.

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Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Students: Pages 371–372

6.08 Divide polynomials After this lesson, you will be able to… • divide a polynomial by a binomial or factored divisor.

Divide polynomials Dividing polynomials involves a process known as algebraic manipulation. We can view our dividend as the numerator of a fraction and our divisor as the denominator. . The dividend is 3x2 + 7x − 6 and the divisor is 3x − 2. We can use algebra tiles

Consider the expression:

to model the division. We will create a rectangle to represent the dividend, and the side lengths of the rectangle will represent its factors. x

−1 −1

We already know one of the factors is the divisor so we can make one of the side lengths 3x − 2.

Next we fill in the rectangle with 3 of the x2 tiles, 7 of the positive x tiles, and 6 of the −1 tiles to represent the dividend 3x2 + 7x − 6, making sure to line up tiles with equal lengths. Notice that there are some empty spaces that we need to fill in with zero pairs.

−1 −1

−x −x

The empty spaces are the right size for x tiles. Since we need to add zero pairs so that we don’t change the value of the expression we will fill 2 of the spaces with positive x tiles and the other 2 spaces with −x tiles.

−1 −1

Now we can see that the length of the left side of the rectangle is

−1 −1

x + 3. This means that

−1 −1

6.08 Divide polynomials mathspace.co

371

6.08 Divide polynomials mathspace.co

781

We can approach this algebraically by following these steps: 1. Completely factor both the numerator and denominator. 2. Identify all common factors that are present in both the numerator and the denominator. These could be monomial or binomial factors. 3. Divide out all common factors from the numerator and denominator. 4. Simplify the resulting expression.

Example 1

Expanding terms to demonstrate division Factor and simplify:

Targeted instructional strategies The process ofadivision Create strategycan be demonstrated as canceling out matching factors in the numerator and denominator after expanding intoand their prime factors. We need to factor both theterms numerator denominator by ﬁrst factoring the greatest common factor of the terms in Begin byeach demonstrating an example with a monomial, such as expression. Expand and divide out common factors

Apply the idea

Factor the numerator and denominator

Write out remaining factors

Divide out the common factors Simplify

Connect this process to different representations of factors without expanding, such as a term outside of Simplify common factors to 1 parentheses. Reﬂect and check

common When we have multiple commonDivide factors, out it is as simple asfactors dividing out each factor separately.

Simplify

Finally, demonstrate Example 2 this with factored forms of trinomials from previous sections. Factor and simplify:

Create a strategy

Divide out common factors Write out remaining factors

This canWe becan used introduction toadivision problems showing a2factorable in the numerator and a use as thean formula for factoring difference of two squares: A2 − B = ( A + B) ( A −trinomial B) binomial in the denominator. Apply the idea Factor the numerator and denominator

Information gap Divide out the common factors and simplify to 1 English language learner support common factors totrinomial 1 Pair students and give each a card with Simplify a different factorable with the same binomial factor in the

denominator. For example, one student receives

, while the other gets

. Students

should not show each other their cards at any point. Students should be told that the goal is to find the sum of their binomials after dividing. Each student factors their numerator independently, such as (2x + 1) (x + 3) and (3x + 1) (x + 3) from the original example, respectively. They must then communicate to identify the common binomial factor and simplify their expressions. Once theySOL divide, can work together to combine like terms and find the sum of their 372 Mathspace Virginia Algebrathey 1 binomials. mathspace.co This collaboration ensures mutual understanding and enhances their mathematical communication skills. Afterward, pairs present their solutions, explaining their factoring process and simplification, fostering engagement and deeper comprehension.

782

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Provide a table of important vocabulary and connect with examples Student with disabilities support Create a table with key vocabulary terms about quotients of polynomials. Provide an example that exhibits all key terms and help students write definitions of each with corresponding parts from the displayed example. Encourage students to use these terms in their mathematical discussions and written work.

Term Numerator Denominator Dividend Divisor Factor Polynomial

Definition

Example 3x2 + 4x − 15 and (3x − 5) (x + 3) 3x − 5 3x2 + 4x − 15 and (3x − 5) (x + 3) 3x − 5 3x − 5 and x + 3 3x2 + 4x − 15

Misunderstanding ‘dividing out common terms’ Address student misconceptions A frequent misconception is that students often think of dividing out common terms as “canceling” them, which implies those terms become zero.

It’s crucial to emphasize that when we divide out identical terms from the numerator and denominator, we are not making them disappear or turn into zero. Instead, they simplify to 1, as anything divided by itself is 1. We can approach this algebraically by following these steps: 1. Completely factor both the numerator and denominator. 2. Identify all common factors that are present in both the numerator and the denominator. These could be monomial

or binomial factors. Examples

3. Divide out all common factors from the numerator and denominator.

Students:4.Page 372 Simplify the resulting expression. Example 1 Factor and simplify:

Create a strategy We need to factor both the numerator and denominator by ﬁrst factoring the greatest common factor of the terms in each expression.

Apply the idea Factor the numerator and denominator Divide out the common factors Simplify common factors to 1

Reﬂect and check When we have multiple common factors, it is as simple as dividing out each factor separately.

Example 2 Factor and simplify:

6.08 Divide polynomials mathspace.co

783

Purpose We can approach this algebraically by following these steps: Check students can simplify rational expressions by factoring and dividing out common factors in the numerator 1. Completely factor both the numerator and denominator. and denominator. 2. Identify all common factors that are present in both the numerator and the denominator. These could be monomial or binomial factors. Expected mistakes

3. might Divide out common from the numerator andbefore denominator. A student notallfactor thefactors expressions completely dividing out common factors. 4. Simplify the resulting expression. For example, after identifying the greatest common factor (GCF) of the numerator as 2, they might factor out the 2 but neglect to further factor the quadratic expression. This could lead them to represent the numerator as 2 (x2 + 5x − 50) instead of 2 (x + 10) (x − 5). Example 1

Advanced learners: Thinking about restrictions to variables Factor and simplify:

use with Example 1

Targeted instructional strategies a strategy In upperCreate level mathematics, students will be expected to state restrictions on variables before performing We need to factor both the numerator and denominator by ﬁrsttofactoring the greatest common of thethrough terms in the any type of division of polynomials. Introduce this concept advanced learners whilefactor working each expression. examples, and encourage them to list the restrictions to the variables.

For this Apply example, ask students to consider what would happen if we substituted x = −10 into the original the idea expression. Because we cannot divide by 0, the expression would be undefined. In addition, if x = −10, , so we wouldn’t be able to divide By first stating that x ≠ −10, the expression

Factor the numerator and denominator

Divide out the common factors

, so we can simplify the expression. This is why restricting

Simplify common factors to 1

the variables is an important part of the simplification process. Reﬂect and check

When we have multiple common factors, it is as simple as dividing out each factor separately.

Students: Page 372 Example 2

Factor and simplify:

Create a strategy We can use the formula for factoring a difference of two squares: A2 − B2 = ( A + B) ( A − B)

Apply the idea Factor the numerator and denominator Divide out the common factors and simplify to 1 Simplify common factors to 1

Purpose Show students how to divide polynomials when factoring using an appropriate method, such as factoring a difference of squares. 372 with Mathspace Virginia SOL Algebra 1 Reﬂecting students mathspace.co Ask students to consider if there’s a difference between “simplifying” and “canceling out” terms. This can serve as a segue to address common misconceptions.

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Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Displaying list of common special factors and factoring methods

use with Example 2

Targeted instructional strategies Having a list of these special factors and methods displayed prominently in the classroom or students’ study area can serve as a handy reference and reinforcement of these key concepts. The list can include patterns like the difference of squares: a2 − b2 = (a − b)(a + b), perfect square trinomials: a2 + 2ab + b2 = (a + b)2 and a2 − 2ab + b2 = (a − b)2, and methods such as factoring a GCF and factoring by grouping. Encourage students to refer to this list when working on problems, and to practice recognizing these patterns in different problems. This will help them develop a strong understanding of factoring methods and improve their problem-solving eﬃciency.

Students: Page 373 Example 3 Factor and simplify:

Create a strategy Start by factoring the numerator. Then, we can divide out common factors.

Apply the idea Factor numerator Divide out common factors Simplify common factors to 1

Idea summary Purpose To divide polynomials: Check students simplify rational expressions by factoring and dividing out common factors in the numerator 1. can Completely factor the numerator and denominator and denominator. 2. Divide out all common factors between the numerator and denominator 3. Simplify the resulting expression (if necessary)

Reﬂecting with students Students may not factor out a greatest common factor from the polynomial, which may keep students from identifying the matching factor. Students may get

and think there is no factor that divides out of

Practice

the problem. Students should check their binomials in the numerator to see if any additional common factors can be taken out. What do you remember?

Develop an algorithm for dividing polynomials 1 Choose all of the following that are correct steps in dividing a polynomials.

use with Example 3

Targeted instructional strategies a Divide out all common factors from the numerator and denominator.

b Identify allThinking common factors that arestudents present into both the numerator and the denominator. Incorporate Algorithmic by guiding develop a procedure for dividing polynomials. c through Completely factorexamples both the numerator and denominator. Begin by going a few in with whole group instruction and then have students examine the Subtractathe factors from the numerator denominator. solutions andddevelop setcommon of steps. Encourage them to and apply this algorithm to various examples, refining their e Simplify the resulting expression. understanding with each iteration. For example: 2

State whether each of the following factors could be divided out of:

1. Factor the numerator polynomial completely.

2. Factor the denominator polynomial completely.

a 3

3x + 1

x−1

State whether each of the following factors could be divided out of:

x2 − 2 6.08 Divide polynomials mathspace.co

785

Factor and simplify:

Create a strategy Start by factoring the numerator. Then, we can divide out common factors.

Apply the idea 3. Identify any common factors shared by the numerator and denominator.

4. Divide out (cancel) the common factors fromFactor bothnumerator the numerator and denominator 5. Simplify the resulting expression to obtain the final simplified quotient. Divide out common factors

Students: Page 373

Simplify common factors to 1

Idea summary To divide polynomials: 1. Completely factor the numerator and denominator 2. Divide out all common factors between the numerator and denominator 3. Simplify the resulting expression (if necessary)

Practice What do you remember? Practice 1

Choose all of the following that are correct steps in dividing a polynomials.

Students: Pages 373–375 a Divide out all common factors from the numerator and denominator. b

Identify all common factors that are present in both the numerator and the denominator.

Completely factor both the numerator and denominator.

Simplify the resulting expression.

What do youd remember? Subtract the common factors from the numerator and denominator. 1

Choose of the following are correct steps a polynomials. 2 all State whether each ofthat the following factors couldinbedividing divided out of: a

Divide out all common factors from the numerator and denominator.

Identify all common factors that are present in both the numerator and the denominator.

Completely factor both the numerator and denominator.

a 2the common factorsbfrom 3x +the 1 numerator and c denominator. x−1 Subtract

Simplify the resulting 3 State whether each expression. of the following factors could be divided out of:

n+2

3x + 1

n−1

n2 − 1

x−1

n−1

6.08 Divide d polynomials x2 − 2 mathspace.co

373

n+1

n+2

n2 − 1

Assume all variables are non-zero: i

What is the greatest common factor of the numerator and denominator?

Write the numerator and denominator as a product of the greatest common factor and the remainder.

iii

Simplify fully.

786

n+1

State whether each of the factor could be divided out of:

a 4

x2 − 2

State whether each of the factor could be divided out of: a

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Fill in the blanks: a

Let’s Practice 6

Simplify: a

Which polynomial expression is equivalent to this expression if s ≠ −1?

A 8

3s − 5

5 −4s

5 − 4s2

6 − 4s2

Simplify: a

Factor and simplify:

Simplify:

Consider the problem

Identify and correct the error in each of the following student’s work. Andre:

Liz:

6.08 Divide polynomials mathspace.co

787

Find the quotient of the following expressions: a

Ralph rode his bike for a distance of (4d2 − 6d) km over a time of (3d2 + 2d) hours. What is the simpliﬁed expression for the speed he traveled?

Let’s extend our thinking 14

Explain why

Fill in the blanks to make a true algebraic statement. a

does not equal . Illustrate your answer with an example.

What would be the result of dividing 2g2 − 10g + 12 by g2 − 5g + 6?

Mark divided a polynomial using the following steps. Do you agree with his solution? Why or why not? Factor and Simplify: Completely factor the numerator and denominator

Divide out the common factors

Rewrite in simplest form

Find the quotient of the expressions: a

At a landfill site, a hole in the shape of a rectangular prism is to be dug out. The length of the rectangular cross-section measures 2x meters and the width is (x + 1) meters. If they need the volume of the landfill to be 2x(x2 − 4x − 5) cubic meters, find the expression for the depth of the hole.

Create and solve an example of polynomial division where the degree of the denominator is greater than that of the numerator.

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Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Answers

9 a

10 a 2

d z−5

6.08 Divide polynomials What do you remember? 1 a This is a step in dividing polynomials b This is a step in dividing polynomials c This is a step in dividing polynomials d T his is not a step in dividing polynomials. Factors have to do with how many times a term will go into another term using multiplication, not subtraction. e This is a step in dividing polynomials 2 a N o. This is not a common factor because it is not a factor of the denominator.

2(z − 6)

11 Andre forgot to factor n2 − 1. This mean his answer still has a common factor of n + 1 left in both the numerator and denominator. Liz incorrectly divided out 4n2 from the numerator and only 4 from the denominator. 12 a 2x − 1

b Y es. This is a common factor because it is a factor of the numerator and the denominator.

e 13

d N o. This is not a common factor because it is not a factor of the denominator.

Let’s extend our thinking

b N o. This is not a common factor because it is not a factor of the denominator. c Y es. This is a common factor because it is a factor of the numerator and the denominator. You ﬁrst have to completely factor the numerator. d N o. This is not a common factor because it is not a factor of the denominator. 4 a i 4 b i b+4

iii

does not equal

because they

c Y es. This is a common factor because it is a factor of the numerator and the denominator.

3 a Y es. This is a common factor because it is a factor of the numerator and the denominator.

km/hr

14 The expression

represent different mathematical relationships and values for different inputs of x. Let’s consider an example where x = 2. For

, when we substitute x = 2, we get

. However, for , regardless of the value of x, the fraction remains constant at while

= 5. So, in this example,

= 5, which clearly shows that they are

not equal. 15 a 5a2 − 5a − 10

b (3y − 1) (2y2 + 1)

16 2 17 Mark is not correct. While he has the right steps, he did not complete factor the numerator and x + 3 is still left as a common factor

5 a

18 a 3x − 1

19 x − 5 20 Example:

Let’s practice 6 a x−2

In general, the degree of the denominator must always remain greater than the degree of the numerator, by the same amount as the original problem.

b 2x − 5

b x+3

c m+4

d m−8

7 B 8 a x−4 e −(a + 11)

(h + 2)

h u−4

x+3

a+3

k m+4

m−8

m −1

n −5

o −(a + 11)

p (h + 2)

r 2(q + 11)

Answers mathspace.co

789

Topic 6 Assessment: Polynomials & Factoring 1

Simplify each expression. a

(3x3 + 2x − 8) + (9x5 − 5x3 + 2x + 3)

(x2 − 4x + 6) (6x − 5)

Revenue is the proﬁt minus the cost. The revenue generated by Leticia’s seafood restaurant is modelled by R (m) = −3.2m2 + 10.9m + 990, and the proﬁt of her restaurant is modelled by P (m) = 2.3m2 − 29.3m + 830, where m is the number of meals produced. Find the polynomial that models the cost of Leticia’s restaurant.

Simplify each product: a

12a (3a3 − 7)

(9 − 8r)2

15a

6b2

105

39b

Factor each polynomial completely: a e

Find an expression for the total area of the following rectangles in factored form: a

x2 + 18x + 81

q3 − 7q2 + q − 7

2x3 + 16x2 + 30x

y2 − y − 132

6t − 27

A cube has a surface area of 6x2 + 48x + 96. Find an expression for the length of one side of the cube.

The volume of the box that a telescope comes in can be represented by the expression 21m2 + 14m − 56. Find possible dimensions of the box.

Simplify each expression. a

SOL

What is the quotient of (16x2 + 10x − 44) and (2x + 4)? Assume the denominator does not equal zero. A

32x3 + 84x2 − 48x − 176

16x2 + 12x − 40

8x − 11

8x + 11

The rectangle shown below has an area of 36x4 − 24x2 square units. Find a polynomial expression for its length. ? 6x

790

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

The volume V of the rectangular prism is given by V = x3 + 5x2 − 9x − 45. Find an expression for the missing dimension.

x+5 x+3 ?

SOL

When factored completely, identify the factors of this polynomial: 6x2 + 5x − 6 2 3

SOL

3x − 2 3x + 2

x−2 x+2

2x + 3 x+3

Which of the following binomials is factor of x2 + 3x − 18? A

x−3

x−6

x+3

x+6

Performance task 14

The form of an expression can help you to identify and interpret its parts in order to solve real-world problems. For each of the following, choose the form of the expression that is best for solving the problem, explain why it is useful, and then use it to ﬁnd the solution. i

Wallace needs to find the volume of a shipping box so he can know how many cookies he can fit inside to send to his grandmother. He also needs to make sure each of the dimensions of the box is less than 12 in so it can fit in her post office box. The height, h, of the box is 6 in. A h3 + 4h2 − 5h

h (h + 5) (h − 1)

−5h + h2 (h + 4)

Orli wants to buy two nice outdoor rugs to cover her patio and needs to know the area they should each be. Let x = 3 ft. 4x − 5

4x 3x 3x

A (4x − 5) (4x) + (3x) (7x − 5) B (4x − 5) (7x) + (3x) (3x) C x (37x + 35)

Topic 6 Assessment: Polynomials & Factoring mathspace.co

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Answers

Performance task

Topic 6 Assessment: Polynomials & Factoring 1 a 9x5 − 2x3 + 4x − 5 3

c 6x − 29x + 56x − 30

b d 3x2 − 2x − 5

A.EO.2a, A.EO.2b, A.EO.2d

A.EO.2a

c 81 − 144r + 64r2

b d

A.EO.2b

A.EO.2a, A.EO.2b, A.EO.2c A.EO.2d, A.EO.2e, MP1, MP5

4 a (a + 15) (a + 7)

b (3b + 2) (2b + 13)

A.EO.2b, A.EO.2e b (q − 7) (q2 + 1)

5 a c ( y − 12) ( y + 11) 2

e (x + 9)

d 3(2t − 9) f

2x (x + 5) (x + 3)

A.EO.2c 6 x+4 A.EO.2c 7 7, 3m − 4 and m + 2 A.EO.2c 8 a x+4

b 2x + 3

A.EO.2d 9 C A.EO.2d 10 6x3 − 4x A.EO.2d, A.EO.2e 11 x − 3 A.EO.2d, A.EO.2e 12 3x + 2 and 2x + 3 A.EO.2c 13 A, D A.EO.2c

792

, the factored form clearly shows all 3 dimensions of B the box which will allow Wallace to easily check that none of them exceeds 12 in. He will then be able to quickly multiply the dimensions to ﬁnd the volume. The dimensions are 6 in, 11 in, and 5 in so the box can be mailed to his grandmother. The volume is 330 in3.

ii A or B because there are two ways she could lay the rugs. These two expressions show the dimensions of each individual rug. The third expression shows the dimensions of a rectangular patio that has the same area as Orli’s but a rug with those dimensions would not be appropriate. For expression A the area of the rugs is 84 ft2 and 144 ft2. For expression B the area of the rugs is 147 ft2 and 81 ft2.

2 −5.5m2 + 40.2m + 160

3 a 36a4 − 84a

14 i

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

7 Quadratic Functions Big ideas • A family of functions is defined by a unique set of characteristics shared by all functions that belong to that family. These characteristics give insight into the types of real-world situations that a function models. • There are many ways to represent a function (equation, table, graph, written description, etc.). The way a function is represented can affect what conclusions can be made. • Functions provide a representation for how related quantities vary. This makes functions a good way to represent many real world situations.

Chapter outline 7.01 7.02 7.03 7.04 7.05

Characteristics of quadratic functions (A.F.2) Quadratic functions in factored form (A.F.2) Quadratic functions in vertex form (A.F.2) Quadratic functions in standard form (A.F.2) Compare linear, quadratic, and exponential functions (A.F.1, A.F.2) Topic 7 Assessment

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The word “quadratic” comes from “quad”, meaning square, due to the squared term in the equation!

7. Quadratic Functions Topic Overview Foundational knowledge Evaluating standards proficiency The skills book contains questions matched to individual standards. It can be used to measure proficiency for each. Students should be proficient in these standards. 8.PFA.3 — The student will represent and solve problems, including those in context, by using linear functions and analyzing their key characteristics (the value of the y-intercept (b) and the coordinates of the ordered pairs in graphs will be limited to integers).

Big ideas and essential understanding A family of functions is defined by a unique set of characteristics shared by all functions that belong to that family. These characteristics give insight into the types of real-world situations that a function models. 7.01 — Quadratic functions have a linear rate of change. The features of a quadratic function can give insight into the real-world scenario the function represents; particularly the intercepts and vertex.

There are many ways to represent a function (equation, table, graph, written description, etc.). The way a function is represented can affect what conclusions can be made. 7.02 — Different representations of a function may highlight or hide different characteristics but they do not change the function itself. The factored form of a function highlights the x-intercepts.

7.03 — Different representations of a function may highlight or hide different characteristics but they do not change the function itself. The vertex form of a quadratic function highlights the coordinates of the vertex and as a result the axis of symmetry.

Functions provide a representation for how related quantities vary. This makes functions a good way to represent many real-world situations. 7.05 — Linear and exponential functions can be distinguished by their rate of change.

7.04 — Different representations of a function may highlight or hide different characteristics but they do not change the function itself. The standard form of a quadratic function highlights the y-intercept. 7.05 — All of the functions in a given family share certain characteristics that can be identified from their equations, graphs, or input/output pairs.

Standards A.F.1 — The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.

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A.F.1a — Determine and identify the domain, range, zeros, slope, and intercepts of a linear function, presented algebraically or graphically, including the interpretation of these characteristics in contextual situations. 7.05 Compare linear, quadratic, and exponential functions

A.F.1f — Graph a linear function in two variables, with and without the use of technology, including those that can represent contextual situations. 7.05 Compare linear, quadratic, and exponential functions A.F.1g — For any value, x, in the domain of f, determine f (x), and determine x given any value f (x) in the range of f, given an algebraic or graphical representation of a linear function. 7.05 Compare linear, quadratic, and exponential functions A.F.2 — The student will investigate, analyze, and compare characteristics of functions, including quadratic and exponential functions, and model quadratic and exponential relationships. A.F.2b — Given an equation or graph, determine key characteristics of a quadratic function including x-intercepts (zeros), y-intercept, vertex (maximum or minimum), and domain and range (including when restricted by context); interpret key characteristics as related to contextual situations, where applicable. 7.01 Characteristics of quadratic functions 7.02 Quadratic functions in factored form 7.03 Quadratic functions in vertex form 7.04 Quadratic functions in standard form 7.05 Compare linear, quadratic, and exponential functions A.F.2c — Graph a quadratic function, f (x), in two variables using a variety of strategies, including transformations f (x) + k and kf (x), where k is limited to rational values. 7.01 Characteristics of quadratic functions 7.02 Quadratic functions in factored form 7.03 Quadratic functions in vertex form 7.04 Quadratic functions in standard form 7.05 Compare linear, quadratic, and exponential functions

A.F.2d — Make connections between the algebraic (standard and factored forms) and graphical representation of a quadratic function. 7.02 Quadratic functions in factored form 7.04 Quadratic functions in standard form A.F.2e — Given an equation or graph of an exponential function in the form y = abx (where b is limited to a natural number), interpret key characteristics, including y-intercepts and domain and range; interpret key characteristics as related to contextual situations, where applicable. 7.05 Compare linear, quadratic, and exponential functions A.F.2f — Graph an exponential function, f (x), in two variables using a variety of strategies, including transformations f (x) + k and kf (x), where k is limited to rational values. 7.05 Compare linear, quadratic, and exponential functions A.F.2g — For any value, x, in the domain of f, determine f (x) of a quadratic or exponential function. Determine x given any value f (x) in the range of f of a quadratic function. Explain the meaning of x and f (x) in context. 7.02 Quadratic functions in factored form 7.03 Quadratic functions in vertex form 7.04 Quadratic functions in standard form 7.05 Compare linear, quadratic, and exponential functions A.F.2h — Compare and contrast the key characteristics of linear functions ( f (x) = x), quadratic functions ( f(x) = x2), and exponential functions ( f (x) = bx) using tables and graphs. 7.05 Compare linear, quadratic, and exponential functions

A2.F.1 — The student will investigate, analyze, and compare square root, cube root, rational, exponential, and logarithmic function families, algebraically and graphically, using transformations.

A2.EI.6 — The student will represent, solve, and interpret the solution to a polynomial equation.

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7.01 Characteristics of quadratic functions Subtopic overview Lesson narrative In the lesson, students will examine graphs, equations, and contextual situations of quadratic functions to identify key characteristics, including the domain and average rate of change intervals. By the end of the lesson, students will be able to make sense of problems and analyze context to interpret key features of quadratic graphs and tables and reason abstractly and quantitatively to find solutions by connecting key features to the quantities they represent.

Learning objectives Students: Page 378

Key vocabulary 

axis of symmetry



parabola



quadratic function

 vertex

Essential understanding Quadratic functions have a linear rate of change. The features of a quadratic function can give insight into the realworld scenario the function represents; particularly the intercepts and vertex.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG3 — Mathematical Reasoning Teachers can enhance students’ mathematical reasoning skills by asking them to justify their solutions to problems or explain why certain characteristics are true for quadratic functions. For example, they could ask why the graph of a quadratic function is a parabola or why the domain of a quadratic function is always all real numbers.

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MPG5 — Mathematical Representations Teachers can encourage students to use a variety of representations by having them graph quadratic functions or represent them symbolically. They can also use technology, such as graphing calculators or online graphing tools, to help students explore and visualize quadratic functions and their key characteristics. For instance, they can encourage students to use these tools to observe how changes in the values of a, b and c affect the shape and position of the parabola. Additionally, they can have students represent real-world situations, like the trajectory of a thrown ball, using quadratic equations and graphs.

A.F.2c — Graph a quadratic function, f (x), in two variables using a variety of strategies, including transformations f (x) + k and kf (x), where k is limited to rational values.

Prior connections A.F.1 — The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.

Engage Activity Social networks Students will determine the number of social connections in a social network based on the number of users.

Understanding and skills

Will develop Identifying the pattern for a quadratic function represented visually or in a table. Finding missing outputs to complete a table representing a quadratic function.

Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: Paper, pencil, graphing calculator (recommended)

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Support students with disabilities Support language - write explanations of mathematical thinking To help students be able to explain their graphs, equations, or tables, use sentence starters such as: • “The number of friends one person can have is ⬚.” • “If each friend is connected with everyone else, then adding one more person will ⬚.”

Support for English language learners Three reads Have students read the task aloud. On the ﬁrst read, ask students to describe the situation. Prompt: Students read the problem. Students think/write: Answer the question “What is the problem about?” Answers may look like: • How many connections can you have between x people? • How to display the same information in different ways. • What is the total number of possible connections on a social network? Share: Students are called upon to discuss their answers with the class. On the second read, ask students to interpret the question. Prompt: Students read the problem. Students think/write: Answer the question “What does an answer look like?” Answers may look like: • An equation. • A graph. • A table. On the third read, have students identify important information. Prompt: Students read the problem. Students think/write: Answer the question “What are the important pieces of information given in the question?” Answers may look like: • There are 10, 30, 100, 1000 users. • Connections are two way. If they are friends with you, you are friends with them.

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Classroom guide Hook

Notice and wonder

Students write observations about two different sets of social network connections.

•

5 mins

What do you notice? What do you wonder? Facebook

Twitter

Implementation details Encourage students to describe the different possible connections in a social network, like the ones shown in the image. Highlight responses that relate to the total possible connections in the network based on the number of users shown as the activity focuses on students exploring the number of connections (“I follow you, you follow me” type connections) based on the number of users in the social network.

Slide 1 from Student Engage Activity

Launch

5 mins

Have you ever wondered ‘What is the total number of possible connections in a social network?’ You will work with your group to investigate this question. You can assume that the social network consists of friends such that, “I follow you, you follow me”, are the only possible connection types. Facebook

Twitter

Slide 2 from Student Engage Activity

Provide time for students to read the instructions and prompt them to consider think the number of connections in a social network with 1, 2, or 3 people. Before forming groups, ask a few students to share the number of connections possible. Invite students to display for the class how they modeled their solution and encourage multiple representations in recording the number of connections, such as in a table or visual representation. Important mathematical concepts: Quadratic functions. Important contextual information: Social networks, users, and “I follow you, you follow me” connections. Suggested grouping: Form groups of 3 or 4 and assign roles

Continue when Students have read the Launch and understand the context of the problem.

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Explore

Group roles

•

35 mins

In your groups, create a presentation showing how to determine the total possible connections in social networks with different numbers of users. Make sure to include the following in your group’s work: • Number of total possible connections for 10, 30, 100, and 1000 users. • At least two different representations that show how the number of connections changes based on the number of users. Slide 4 from Student Engage Activity

Students will be creating diagrams, tables, graphs, or evaluating a function or rule in order to ﬁnd out how many connections various amounts of social network users can have.

Anticipated strategies Create a visual representation Create a diagram or model Students may create a diagram or visual model, similar to the images shown in the Hook, to show different connections based on the number of users. Encourage students to generalize their ﬁndings as it will become increasingly diﬃcult with more users. Create a table Students may create a table to record the number of possible connections. Users in social network 1 2 3 4 5 10 30 100 1000

Total connections 0 2 6 12 20 90 870 9900 999 000

Create a graph Students may create a graph that shows the number of possible connections in terms of the number of users in the social network.

Total connections

80 70 60 50 40 30 20 10

Users in social network 1 2 3 4 5 6 7 8 9 10

Create a rule Students may determine a rule to ﬁnd the number of possible connections in a social network. Rule: n(n − 1) = n2 − n

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Misconceptions Misunderstanding connection type and working with a nonquadratic relationship. What are possible connections in this problem? How many connections would there be on the social network if there were 1 person, 2 people, or 3 people as users? Why? How do you know?

Purposeful questions Use the following questions to check for understanding and encourage critical thinking: • How many connections are there when there are 10, 30, or 100, 1000 users? How do you know? • Is there another way you can represent the number of connections in the social network based on the number of users? • Can you explain in words how you see the number of connections changing based on the number of users?

Continue when Students have determined the number of total possible connections for 10, 30, 100, and 1000 users and used at least two different representations that show how the number of connections changes based on the number of users.

Discuss

15 mins

Have a class discussion in order to determine how many connections different amount of social network users can have. Consider sequencing the strategies presented by the methods used for determining connections, starting with diagrams, then tables, graphs, and ﬁnally, rules.

Discussion guide Invite students to share their methods for determining the number of connections based on the different number of users in a social network. Encourage groups to share different representations, such as a visual representation, table, graph, or rule. If the class did not use one of the representations as justiﬁcation, ask the class if there are any other representations that could have been used and provide groups time to generate or discuss the missing representation. Note that students are not expected to write equations or a rule on their own, but it is a good extension question that students may have explored on their own or to have them think about for the ﬁrst time in the discussion. Ask the class about how many connections would be in a social network with n users to encourage this generalization. Next, ask groups how the relationship they have observed compares to relationships they have learned before (linear and exponential). Ask students to describe in words how they see the number of connections changing based on the number of users. Quadratic relationships will be introduced formally in the next lesson, but students may describe that the change in increase for the number of connections increases by a constant value for each consistent change in number of users. Extension: • Research a social network that you or someone you know uses. Find the number of users and calculate the possible number of total connections. Is the number of total possible connections close to the number of actual connections for an average user? Why or why not? • Draw at least three models that show different possible connection types (such as “I follow you, but you don’t follow me back” and “We both follow each other”). For each model, record the total connections and the number of users. How are these models similar or different to one another?

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Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lesson: Algebra 1 — 2.04 Characteristics of functions

Tools You may ﬁnd these tools helpful: • Graphing calculator • Graph paper

Lesson supports The following supports may be useful for this lesson. More speciﬁc supports may appear throughout the lesson:

Elaborate on the axis of symmetry Targeted instructional strategies Students may benefit from further discussion about the axis of symmetry. Relate the axis of symmetry to lines of symmetry from previous grades. Remind students about the definition of symmetry, and encourage them to check that if a parabola is symmetric, they can fold one side over and it overlaps the other side. The overlapping points will have the same y-value and are the same x-distance from the line of symmetry. Teachers can make more connections by relating the equation of the axis of symmetry to equations of vertical lines from Topic 3.

Always, sometimes, or never English language learner support Have students consider whether the following statements are always, sometimes, or never true to check their understanding of the vocabulary present in this topic. • A zero occurs when a function intercepts the x-axis (answer: always) • A quadratic function has a single y-intercept (answer: always) • The vertex is a maximum point on the graph (answer: sometimes - it may instead be the minimum point on the graph) • The domain of a quadratic function is limited by the position of the vertex (answer: never) • The vertex is on the axis of symmetry (answer: always) • A quadratic function has two zeros (answer: sometimes - it depends on the location of the vertex and the direction the parabola opens)

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Label key features on a quadratic graph Student with disabilities support Provide students with example graphs of quadratic functions to draw, highlight, and label key features including domain, range, maximum, minimum, intercept, and vertex. These features can be color-coded and annotated for reference for future problems. domain

domain

x-intercepts

y-intercept

range

vertex (minimum)

y-intercept

x-intercepts

range

Positive leading coefficient

Negative leading coefficient

Quadratic functions are symmetric about the vertex Address student misconceptions 9

When writing the range of a function such as f (x) = (x − 3)2 + 2 from a graph or an equation, a common error a student may make is to say the range is less than or equal to 3, the x-coordinate of the vertex.

8 7 6

This shows the student has a misconception in associating domain and range with the independent and dependent variables, respectively.

5 4 3 2 1 −1

x 1

For students with this misconception, have them practice with discrete points in identifying the domain and range and then continue practicing with continuous graphs.

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Student lesson & teacher guide Characteristics of quadratic functions Students learn new characteristics that are attributed to quadratic functions. Key features such as the axis of symmetry, vertex, domain and range, and number of x-intercepts are described and accompanied by a graph illustrating the feature.

Students: Pages 378–379

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Concrete-Representational-Abstract (CRA) Approach Targeted instructional strategies Concrete: Engage students with a hands-on activity to model quadratic functions in a real-world context. Use a rolled piece of paper or a ball to demonstrate the path of an object thrown into the air. Have students measure and record the height of the object at different time intervals using a tape measure and stopwatch. They can place markers or sticky notes at each measured point to visualize the path. This physical movement helps students experience the shape of a quadratic function through the object’s motion. Representational: Transition from the hands-on activity to visual representations of the data collected. Guide students to plot the recorded points on graph paper, creating a coordinate plane. Have them draw the curve that connects the points, forming a parabola. Encourage students to identify and label key features on their graphs, such as the vertex, x-intercepts, and y-intercepts. Abstract: Move to the symbolic representation by introducing the quadratic equation that models the object’s motion. Teach students how to write the equation in the form y = ax2 + bx + c using the data from their graphs. Work with them to calculate the values of a, b, and c based on the points they’ve plotted. Discuss how to find the domain and range from the equation and interpret these in the context of the real-world scenario. Solve problems using algebraic methods, reinforcing how the equation relates back to the physical activity and the graph. Connecting the stages: Help students make connections between the concrete activity, their graphs, and the equation. Encourage them to reflect on how the motion of the object (concrete) corresponds to the shape of the graph (representational) and how both are described by the quadratic equation (abstract). Ask guiding questions like: • “How does the highest point of the ball relate to the vertex of the parabola?” • “What do the x-intercepts tell us about the object’s motion?” This integration reinforces their understanding and allows them to choose the most helpful representation when solving problems.

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Examples Students: Pages 379–380

Apply the idea

Reﬂect and check 9

Having the vertex in your table is useful, since it tells you where the parabola has a minimum or maximum. Sometimes the table values you select will not include the vertex of the function, depending on the quadratic function being graphed. If you plot your initial table values and find you are unsure where the parabola changes direction, you can add additional values to your table until you can identify the vertex.

8 7 6 f (x)

5 4 3 2 1 −2

−1

x 1

Note that the quadratic function has one x-intercept, at x = 1.

b State the axis of symmetry.

PurposeCreate a strategy Apply the idea Show students how to graph a quadratic function by constructing a table of values. The axis of symmetry is a vertical line that passes y 9

through the vertex.

8 Expected mistakes 7 Students may assume that they should know how to graph a quadratic function by reading its equation like the 6 way they graphed linear functions. However, when learning to graph linear functions, a basic approach is by 5 f (x) creating a table of values. 4 3 2 1 −2

−1

x 1

The axis of symmetry is x = 1.

Example 2 Consider the graph of the quadratic function g(x):

y 9 7.01 Characteristics of8 quadratic functions 7 mathspace.co 6 g(x) 5 4

809

and find you are unsure where the parabola changes direction, you can add additional values to your table until you can identify the vertex.

4 3 2 1

Students: Page 380−2 −1

x 1

Note that the quadratic function has one x-intercept, at x = 1.

b State the axis of symmetry.

Create a strategy

Apply the idea

The axis of symmetry is a vertical line that passes through the vertex.

8 7 6 5

f (x)

4 3 2 1 −2

−1

x 1

The axis of symmetry is x = 1.

Example 2 Purpose Consider the graph of the quadratic function g(x): Show students how to determine the axis of symmetry from a graph.

y 9 8 7 Expected mistakes 6 g(x) 5 Students may state the axis of symmetry as the x-value of the vertex, and not an equation. Remind students that 4 the axis of symmetry is a line, so it is always an equation. 3 2 1 Reflecting with students x −5 −4equations, −3 −2 −1 ask1 them 2 3to reflect After students have drawn a number of graphs of quadratic functions from given −1 −2 on any patterns or shortcuts they found to drawing the correct graph. −3

Encourage students to look for corresponding similarities in questions and answers. For example, students may a Find the x-intercepts and y-intercept. connect the constant term of the equation to the y-intercept, or connect the sign of the leading coefficient to whetherCreate the vertex is a maximum or minimum. a strategy To ﬁnd the x-intercepts, locate the places where the parabola crosses the x-axis.

Advanced learners: Exploratory quadratic graphs To ﬁnd the y-intercept, locate the place whereanalysis the parabolaof crosses the y-axis.

use with Example 1

Targeted instructional strategies

After students have graphed the quadratic function using a table of values, encourage advanced learners to Virginia SOL Algebra 1 380 Mathspace mathspace.co analyze the patterns and symmetry of the graph more deeply. Pose open-ended questions like, “What patterns do you notice in the y-values of the points?” and “How do the points relate to the axis of symmetry?” or “How can these patterns help you predict other points?” Students should recognize that the axis of symmetry can be used to find mirror points. Additionally, they may notice that the y-values on either side of the axis of symmetry increase by odd numbers. Make students aware that this is only true for some quadratics (ones where the leading coefficient is 1), which they will explore more when they learn about quadratic equations.

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4 3 2 1 −2

Students: Pages 380–381

−1

x 1

The axis of symmetry is x = 1.

Example 2 Consider the graph of the quadratic function g(x):

9 8 7 6 5 4 3 2 1 −5 −4 −3 −2 −1

−1 −2 −3

g(x)

x 1

a Find the x-intercepts and y-intercept.

Create a strategy To ﬁnd the x-intercepts, locate the places where the parabola crosses the x-axis. To ﬁnd the y-intercept, locate the place where the parabola crosses the y-axis.

Apply the idea 380

Mathspace

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We can identify the intercepts on the graph: mathspace.co 9 8 7 6 5 4 3 2 1 −5 −4 −3 −2 −1

Apply the idea We can identify the intercepts on the graph:

g(x)

x 1

−1 −2 −3 9

From the graph we can see the there are two x-intercepts at (−4, 8 0) and (2, 0), and there is one y-intercept at (0, 8). 7 6 g(x) 5 b Determine the domain and range. 4 3 PurposeCreate a strategy 2 Show students how to identify the x- to and of a 1quadratic function from its graph. To find the domain of g(x), we want findy-intercepts all possible x-values for which g(x) x could be graphed. −5 values 1 2 of3a parabola affects the range of the function, −4 −3 of−2 −1 The vertex To find the range, we want to find all possible g(x). −1 Expected mistakes as it will be the maximum or minimum value of g(x). −2 Students may write the coordinates of the intercepts backwards, such as the y-intercept as (8, 0). Review the −3

coordinates theidea points along each axis. Applyofthe

We can see that for a parabola, there are no restrictions on which x-values can be graphed as each side of the

the graph see the there are two x-intercepts at (−4, 0) and (2, 0), and there is one y-intercept at (0, 8). Students:From Page 381 we can parabola continues infinitely in either x direction. Domain: {x − ∞ < x < ∞} b the domain range. ThisDetermine parabola opens down,and so the y-value of the vertex is the maximum value of the function. The parabola continues infinitely in the negative y direction. Range: {aystrategy y ≤ 9} Create To find the domain of g(x), we want to find all possible x-values for which g(x) could be graphed.

Reflect and check

To ﬁnd the range, we want to ﬁnd all possible values of g(x). The vertex of a parabola affects the range of the function, For the domain, we may also see it written as “all real values of x” or in interval notation as “(−∞, ∞)” instead of using as it will be the maximum or minimum value of g(x). inequality or set notation.

Apply the idea We can see that for a parabola, there are no restrictions on which x-values can be graphed as each side of the parabola continues inﬁnitely in either x direction. 7.01 Characteristics of quadratic functions Domain: {x − ∞ < x < ∞} mathspace.co This parabola opens down, so the y-value of the vertex is the maximum value of the function. The parabola continues inﬁnitely in the negative y direction. Range: { y y ≤ 9}

811

Create a strategy To find the domain of g(x), we want to find all possible x-values for which g(x) could be graphed. To find the range, we want to find all possible values of g(x). The vertex of a parabola affects the range of the function, as it will be the maximum or minimum value of g(x).

Apply the idea We can see that for a parabola, there are no restrictions on which x-values can be graphed as each side of the parabola continues inﬁnitely in either x direction. Domain: {x − ∞ < x < ∞} This parabola opens down, so the y-value of the vertex is the maximum value of the function. The parabola continues inﬁnitely in the negative y direction. Range: { y y ≤ 9}

Reflect and check For the domain, we may also see it written as “all real values of x” or in interval notation as “(−∞, ∞)” instead of using inequality or set notation.

Purpose Show students how to determine the domain and range of a quadratic function from its graph. Expected mistakes Some students may incorrectly exclude the vertex from the range. Clarify for students that the vertex should be included in the range. Students may state the maximum value of the range as 8, which is the location of the y-intercept. Point out to students that the graph continues to the vertex, which has a maximum y-value of 9.

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c Describe what happens to the graph as x gets very large and positive.

Create a strategy We can look at the graph and see what is happening for larger and larger values of x. We may need imagine the graph extending beyond what is shown.

Apply the idea As x gets very large, the graph continues down and the function values are negative with a very large size. It is decreasing faster and faster. y 5 −4

−2

−5

g(x) 2

x 4

−10 −15 −20 −25 −30 −35

Reflect and check The function values for g(x) also become large and negative as x becomes large and negative.

Example 3 Purpose To buildThe conﬁdence with the shape of aabove parabola. graph shows thedescribing height, y (in feet), of a softball ground

Softball throw

x seconds after it was thrown in the air.

Height in feet, y 14 12

812

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

10 8 6 4

381

−15 −20 −25 −30 −35

Reflecting with students Consider reflecting with students by using technology that allows students to zoom out so they can “see” more of the parabola, especially for “very large x-values” as suggested in the problem. Reflect and check

function values for g(x) also become large and negative as x becomes large and negative. Students:The Pages 382–383

Example 3 The graph shows the height, y (in feet), of a softball above ground x seconds after it was thrown in the air.

Softball throw Height in feet, y 14 12 10 8 6 4 2 Time in seconds, x 0.5

1.5

2.5

3.5

a Find the y-intercept and describe what it means in context.

Create a strategy We want to ﬁnd the place where the parabola crosses the y-axis. Once we ﬁnd the y-intercept, we want to connect this to the context of the softball. Since the y-axis represents the height of the softball in feet above the ground, we can use it to identify the height of the softball at 0 seconds.

Apply the idea 382

Mathspace Virginia SOL Algebra 1 mathspace.co

We can identify the y-intercept on the graph: Softball throw Height in feet, y 14 12 10 8 6 4 2 0.5

1.5

Time in seconds, x 2.5 3 3.5

The y-intercept is (0, 6). The y-intercept tells us that the softball was thrown from a height of 6 feet above the ground.

b Find the value of the x-intercept and describe what it means in context.

Purpose Create a strategy Show students how to interpret the y-intercept of a quadratic function in context. We want to ﬁnd the place where the parabola crosses the x-axis.

Once we ﬁnd the x-intercept, we want to connect this to the context of the softball. Since the x-axis represents the time in seconds after being thrown, we can use it to identify how many seconds the softball hits the ground.

Apply the idea We can identify the x-intercept on the graph: Softball throw Height in feet, y 14 12 10

7.01 Characteristics of quadratic functions mathspace.co

813

10 8 6 4

Expected mistakes 2 Students may state that the ball was 6 feet above the ground zero seconds after the ball was thrown. While this Time in seconds, x is technically a correct description, we should0.5 consider is actually 1 1.5 what 2 2.5 3 3.5 happening in the context. The ball was thrown from that height initially. The y-intercept is (0, 6).

Students:The Page 383tells us that the softball was thrown from a height of 6 feet above the ground. y-intercept b Find the value of the x-intercept and describe what it means in context.

Create a strategy We want to ﬁnd the place where the parabola crosses the x-axis. Once we ﬁnd the x-intercept, we want to connect this to the context of the softball. Since the x-axis represents the time in seconds after being thrown, we can use it to identify how many seconds the softball hits the ground.

Apply the idea We can identify the x-intercept on the graph: Softball throw Height in feet, y 14 12 10 8 6 4 2 Time in seconds, x 0.5

1.5

2.5

3.5

The x-intercept is (3, 0). The x-intercept tells us that the softball hits the ground 3 seconds after it was thrown in the air.

Purpose 7.01 Characteristics of quadratic functions mathspace.co Show students how to interpret the x-intercept of a quadratic function in context.

383

Reﬂecting with students Ask students to identify the other x-intercept which they can do using the axis of symmetry. Then, explain why it is not discussed in the problem. The other x-intercept is not shown because it occurs during a negative time period. Time can be negative in different contexts, but, here, it doesn’t make sense as this is before the ball was in motion.

814

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Students: Page 384 c Find the value of the vertex and describe what it means in context.

Create a strategy

Apply the idea

In order to ﬁnd the vertex, we want to ﬁnd the maximum point of the parabola.

We can identify the vertex on the graph: Softball throw

Once we ﬁnd the vertex, we want to connect this to the context of the softball. We know that the x-value of the vertex represents time in seconds after the softball is thrown and the y-value of the vertex represents the height of the softball above ground in feet.

Height in feet, y 14 12 10 8

c Find the value of the vertex and describe what it means in context. 6 4

Create a strategy

Apply the idea

In order to ﬁnd the vertex, we want to ﬁnd the maximum point of the parabola.

2 We can identify the vertex on the graph:

0.5

Time in seconds, x 1Softball 1.5 2throw 2.5 3 3.5

Height in feet, y

The vertex14is (1.25, 12).

After 1.25 seconds, the softball reaches a maximum 12 height of 12 feet above the ground. 10

d State the domain and describe what it means in context.

8 6

Purpose Create a strategy Apply the4idea Show students how to interpret the vertex of a quadratic function in context.

The domain of the context should be reasonable. We can Domain: 0 ≤2 x ≤ 3 Time seconds, x use the graph of the function to determine the domain The domain of the function starts at in x= 0 seconds when Expected mistakes 0.5 1 1.5 2 2.5 3 3.5 and explain its meaning in context. the softball was recorded from where it was initially Students may have trouble finding the value halfway between 1 and 1.5. Encourage students to discuss the thrown. The domain of the function ends at x = 3 seconds scaling of the graph and what each of the lines represents. Remind half of 0.5 is 0.25. The is students (1.25,lands 12). on whenvertex the softball the ground. After 1.25 seconds, the softball reaches a maximum Students: Page 384 height of 12 feet above the ground.

Idea summary d State the domain and what it means context. From the graph of describe a quadratic function, weincan identify key features including: •

Domain and range

Create• a strategy x- and y-intercepts

Apply the idea

The domain of the context shouldfunction be reasonable. • Maximum or minimum value We can Domain: 0 ≤ x ≤ 3 use the•graph of the function to determine the domain Vertex The domain of the function starts at x = 0 seconds when and explain its meaning in context. • Axis of symmetry the softball was recorded from where it was initially thrown. The domain of the function ends at x = 3 seconds when the softball lands on the ground.

Idea summary

Purpose From the graph of a quadratic function, we can identify key features including: Show students• how to interpret the domain of a quadratic function in context. Domain and range •

x- and y-intercepts

Expected mistakes • Maximum or minimum function value Students may •assume Vertex that the domain of the function is −∞ < x < ∞. Point out to students that for this context, 384 Mathspace Virginia SOL Algebra 1 • Axis of symmetry we should bemathspace.co intentional about what the domain is. Negative x-values are not valid here as the ball was not set in motion until x = 0. The ball cannot continue traveling downward after hitting the ground, so x-values after x = 3 do not make sense either.

7.01 Characteristics of quadratic functions mathspace.co

384

Mathspace

Virginia SOL Algebra 1

815

Create a strategy

Apply the idea

The domain of the context should be reasonable. We can Domain: 0 ≤ x ≤ 3 use the graph of the function to determine the domain The domain of the function starts at x = 0 seconds when and explain its meaning in context. the softball was recorded from where it was initially thrown. The domain of the function ends at x = 3 seconds Students: Page 384 when the softball lands on the ground.

Idea summary From the graph of a quadratic function, we can identify key features including: • • • • •

Domain and range x- and y-intercepts Maximum or minimum function value Vertex Axis of symmetry

Practice Students: Pages 385–392 384

Mathspace Virginia SOL Algebra 1 mathspace.co

What do you remember? SOL

Which table of values best represents the rule shown? The square of the sum of x and 3 is equal to y. A

x 4 5

y 13 14

x 4 5

y 25 34

x 4 5

y 19 28

x 4 5

y 49 64

Choose the graph that has each set of characteristics: a • Axis of symmetry at x = −1 • x-intercepts: (−7, 0), (5, 0) Select the graph that represents the function. A y 25 20 15 10 5

−6−5−4−3−2 −1 −5 −10 −15 −20 −25 −30 −35

816

x 1 2 3 4 5 6 7 8

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

25 20 15 10 5 −7 −6−5−4−3−2 −1 −5 −10 −15 −20 −25 −30 −35

x 1 2 3 4 5 6 7

C y 25 20 15 10 5

−6−5−4−3−2 −1 −5 −10 −15 −20 −25 −30 −35

x 1 2 3 4 5 6 7 8

25 20 15 10 5 −7 −6−5−4−3−2 −1 −5 −10 −15 −20 −25 −30 −35

x 1 2 3 4 5 6 7

• Vertex is a maximum • x-intercept: (6, 0) • y-intercept: (0, −36) Select the graph that represents the function. x 1 2 3 4 5 6 7 8 9 10 11 12 13 14

y A

B −12−11−10−9−8−7 −6−5−4 −3−2−1 −5

−5 −10

−10

−15

−20

−25

−30

−35

y x 1 2 3 4 5 6 7 8 9 10 11 12 −5

x −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 y −5

−10

−15

−20 −25 −30 −35

y x

−20 −25 −30 −35

• No x-intercept • Axis of symmetry: x = 10 • Range: y ≤ −6

7.01 Characteristics of quadratic functions mathspace.co

817

Select the graph that represents the function. y A

x 1 2 3 4 5 6 7 8 9 1011 121314151617

−5

−10

−15

−20

−25

−30

−35

C y

x 1 2 3 4 5 6 7 8 9 1011 121314151617

−5

−10

−15

−20

−25

−30

−35

Consider the graph of the quadratic function. State the number of x-intercept(s) the quadratic function has based on the graph.

818

1 2 3 4 5 6 7 8 9 10 11 121314151617

5 y 4 3 2 1 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8 −9

For each table, complete the following: i

Graph the quadratic function shown in the following tables.

Find the coordinates of the vertex.

iii

Determine whether the vertex is a maximum or minimum point.

Determine the axis of symmetry.

State the number of x-intercept(s).

x y

0 −7

1 −2

2 1

3 2

4 1

5 −2

6 −7

x y

−7 11

−6 6

−5 3

−4 2

−3 3

−2 6

−1 11

Determine how many x-intercept(s) the equation x2 + 64 = 0 has.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

x 1

2 3 4 5 6 7

Let’s practice 6

Consider the graph of the function y = f (x).

State the number of x-intercept(s) of the function.

Determine the domain of the function.

Determine the range of the function.

Describe the behavior of the function for large values of x.

8 6 4 2

−8 −6 −4 −2 −2

−4 −6 −8 −10

Consider the function g(x) = −(x + 5) (x + 1). a

Copy and complete the table. x g(x)

−5

−4

−3

−2

−1

Determine the equation of the axis of symmetry.

Determine if the graph will have a maximum or a minimum.

Graph the function.

Determine the number of x-intercept(s) based on the graph.

Consider the function h(x) = x2 − 4x + 4. a

Copy and complete the table: x h(x)

−6

−1

Find the coordinates of the x- and y-intercepts.

Find the coordinates of the vertex.

Determine the domain and range.

Determine the equation of the axis of symmetry.

Determine if the graph will have a maximum or minimum.

Graph the function.

Zahra jumps off a diving platform and the path of her dive is modeled by the function f (x) = −x2 + 2x + 8, where f (x) is her height in meters above the pool, and x is the horizontal distance in meters from the edge of the diving platform. a

Select the graph of the function: A y

7 6

2 −6

−4

−2

x 2

5 4

−4

−6

−8

x 1

7.01 Characteristics of quadratic functions mathspace.co

819

C y

−6

−4 −6 3

Find the height of the diving platform.

Find the maximum height of Zahra’s dive.

Determine and interpret the domain and range of the function.

A frisbee is thrown upward and away from the top of a cliff. The height, y meters, of the frisbee at time, x seconds, is given by the equation y = −20(x − 6) (x + 2). a

Select the graph of the function: A y

360

300

240

180

120

x 1

−4

D 320

240

160

−6

−4

−2 −80

−160

−240

−320

Determine the height at which the frisbee is thrown.

Find the maximum height the frisbee reached.

Determine the domain and range of the function.

−2 −80

320

80 −6

C y

820

−8

x 2

−2

2 1

−4

x 2

For each of the quadratic function, find the: i

x-intercept(s)

y-intercept

iii

vertex

y = x2 − 4x + 4

y = (x + 4) (x − 2)

y = x2 − 2x − 3

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

y = −x2 − 2

For each of the following quadratic functions, find the: i a

y = x + 2x + 1

y = −(x − 5) (x + 1)

Range

y = (x − 3)2

y = −x2 − 2x − 1

y = 3(x − 2)2 + 1

Find the range when the domain is {−6, −1, 0, 5, 7} for each of the quadratic function: a

Domain

y = x2 − 2x + 3

y = −x2 + 3x − 1

y = −2(x + 1) (x − 4)

Use the graph of f (x) to evaluate for the following values. a

x = −5

x = −2

f (x) = −9

f (x) = 0

−9 −8 −7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8 −9

The graph shows the height of a soccer ball above ground, in feet, after it is kicked in terms of x seconds. a

Find the y-intercept.

Describe what the y-intercept means in context.

Find the x-intercept.

Describe what the x-intercept means in context.

Find the coordinates of the vertex.

Describe what the vertex means in context.

f (x) x 1

y 10 8 6 4 2 x 1

A clothing company is designing a new jacket and wants to determine how to maximize their proﬁt once the jacket is ready to be sold. The graph represents the total proﬁt, P, the shop will make at each price point, x, the jacket could sell for. y 700 600 500 400 y = P(x)

300 200 100

Find the value of the vertex and describe what the vertex means in context.

Determine the domain that results in a proﬁt for the clothing company.

Determine the corresponding range of proﬁt.

The manager believes that selling a jacket at a higher price will always result in a larger proﬁt. Explain how increasing the price of the jacket affects the proﬁt. 7.01 Characteristics of quadratic functions mathspace.co

821

Let’s extend our thinking 17

Sketch the graph of quadratic equations with the following key features: a b • Has a maximum function value • Axis of symmetry at x = 1 • x-intercept: (5, 0) • Vertex is a minimum • y-intercept: (0, −25) • x-intercepts: (−2, 0), (4, 0) c

• No x-intercept • Axis of symmetry: x = 8 • Range: y ≤ −9

Elise is the owner of a restaurant. Elise wants to install new wooden ﬂoors in several rooms. The rooms in the restaurant are square. The wood costs $6.25 per square foot. The cost of the ﬂooring in terms of its side length is shown by the quadratic function C(x) = 6.25x2. a

Determine how much Elise should expect to spend on ﬂooring if the room has side lengths of 12 ft.

Determine how much the price would change if the side lengths decreased by 3 ft.

Graph the given quadratic model, C(x) = 6.25x2. Make sure to choose appropriate labels and scale.

Describe what changes and what stays the same about the graph of the quadratic model if the cost per square foot decreases.

Graham, Habib, and Joel throw or kick footballs around the same time. The vertical height of Graham’s football is shown in the graph. The function G(t) represents the vertical distance of the football above the ground, in feet, and t represents time, in seconds. 16 14 12 10 8 6 4 2

y = G(t)

t 1

The vertical height of Habib’s football is shown in the table. H(t) represents the vertical distance of the football above the ground, in feet, and t represents time, in seconds. t H(t)

0.172 0

2 14

3 16

4 14

5 8

5.828 0

The height of Joel’s football can also be modeled with a quadratic function that has the following key features. Let J(t) represent the vertical distance of the football above the ground, in feet, and t represent time, in seconds. • y-intercept: (0, 5) • Vertex at (1, 5.5) • t-intercept: (4.317, 0) Use the above information to complete the following:

822

Graph the three quadratic functions on the same coordinate plane.

Determine whose football reached the ground the quickest after being kicked or thrown. Explain your answer.

Determine whose football reached the greatest height. Explain your answer.

Describe what G(0) and H(0) mean in context.

Habib claims that his football reaches the maximum height the quickest. Determine whether or not Habib is correct. Explain your answer.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Rafael is trying to explain the design for a painting to his friend, over the phone. Describe how Rafael could use key features of quadratic functions to share his idea.

Write the equation of a quadratic function that has the following key features and sketch a graph of the function:

Two x-intercepts that have the same absolute value but opposite signs

One x-intercept that is a fraction and the other is a prime number

One unique x-intercept that is less than 1 but greater than 0

x-intercepts at (7, 0) and (−2, 0), and has a y-intercept at (0, 14)

x-intercepts at (−1, 0) and (−10, 0), and has a y-intercept at (0, 30)

Determine how many unique quadratic equations exist for each of the key feature: a

x-intercepts at (−7, 0) and (10, 0)

One unique x-intercept at (−4, 0) and y-intercept at (0, −20)

Vertex at the origin and passes through (2, 14)

x-intercepts at (15, 0) and (−7, 0), and is symmetric about the y-axis

Ori models his golf shot using the quadratic equation: y = −x2 + 10x − 16 where y is the height of the ball (in yards) and x is the time after placing the ball on the ground (in seconds). Use graphing technology to explore this function. Describe and justify a real-world problem involving this equation which has: a

One viable x-intercept

No viable x-intercept

Two viable x-intercepts

7.01 Characteristics of quadratic functions mathspace.co

823

Answers

7.01 Characteristics of quadratic functions

2 −10 −8 −6 −4 −2 −2

What do you remember?

x 2

−4

1 a D

−6

2 a B

b A

c D

−8 −10

3 Two 4 a i

e Two

8 a

4 2

−8 −6 −4 −2 −2

-1

h(x)

6 8

b (2, 0), (0, 4)

−4

c (2, 0)

−6

Range: y ≥ 0

d Domain:

−8

e x=2

−10

f ii (3, 2)

iii Maximum iv x = 3

b i

v Two

Minimum

g 12

10 8

−6 −4 −2 −2

4 2

−10 −8 −6 −4 −2 −2

2 4

x 2

8 10

−4

9 a B ii (−4, 2)

iii Minimum

iv x = −4

v Zero

b 8m

5 None

c 9m

Let’s practice

d Domain: {x ∣ 0 ≤ x ≤ 4}, this means that during the dive she traveled 4 meters horizontally. Range: {y ∣ 0 ≤ y ≤ 9}, this means that during the dive her maximum height was 9 meters above the pool and that her dive ended when she entered the pool.

6 a Two b c d As x gets large the function increases more and more quickly. The function values will get extremely large and positive. 7 a

-6

-5

-4

-3

-2

-1

g(x)

-5

b x = -3

c Maximum

10 a A c 320 m

b 240 m d Domain: [0, 6], Range: [0, 320]

11 a i (2, 0)

ii (0, 4)

iii (2, 0) b i (−4, 0) and (2, 0) ii (0, −8) c i (−1, 0) and (3, 0)

iii (−1, −9) ii (0, −3)

iii (1, −4) d i None

ii (0, −2)

iii (0, −2)

824

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

12 a i All real numbers

ii y ≥ 0

b i All real numbers

ii y ≤ 9

c i All real numbers

ii y ≥ 0

d i All real numbers

ii y ≤ 0

13 a {3, 6, 18, 38, 51}

b {−55, −29, −11, −5, −1}

c {−100, −48, −12, 0, 8}

d {13, 28, 76, 193}

14 a f (−5) = −5

b f (−2) = −8

c x = −3

d x = −6 and 0

c Any graph that has the key features

y x 2

10 12

−5

15 a (0, 6)

−10

b The y-intercept represents the height from which the soccer ball is kicked.

−15

c (3, 0) d The x-intercept represents the time at which the soccer ball hits the ground.

b The price would decrease by $393.75

e (1, 8) f

18 a Elise should expect to pay $900

fter 1 second, the soccer ball reached the maximum A height of 8 feet above the ground.

c 1000 cost ( y) 900 800 700 600 500 400 300 200 100

16 a ( 25, 625). If the price per jacket is $25, then the maximum profit of $625 is reached. b 0 < x < 50 c 0 < y ≤ 625 d T he profit is increasing over the interval of the domain: 0 < x < 25. The profit is decreasing over the interval of the domain: 25 < x < 50. A higher priced jacket does not necessarily mean a greater profit for the company. The profit is at its maximum when the jacket costs $25. The profit model is a quadratic function, so the profit will only decrease after it reaches the maximum. Let’s extend our thinking 17 a Any graph that has the key features

y 5 x −5

x 5

−5 −10 −15 −20 −25

d T he decrease in in price per square foot makes the parabola wider. The x and y-intercept, domain and range, vertex, axis of symmetry, end behavior, and intervals of the domain for where the function is positive/negative and increasing/decreasing stay the same. 19 a

16 14 12 10 8 6 4 2

y G (t)

H (t) J (t)

t 1

b J oel’s football reached the ground the quickest as the football hit the ground in 4.317 seconds. It took Habib’s football 5.828 seconds to reach the ground and it took Graham’s football just under 7 seconds to hit the ground.

−5

feet (x) 2

c H abib’s football reaches the greatest height of 16 ft. Graham’s football reaches a maximum height of 15 ft and Joe’s football reaches a maximum height of 5.5 ft. d G (0) and H(0) represent the initial heights at which Graham and Habib throw the football, in this case 6 ft and 5 ft, respectively. e H abib is incorrect. Joel’s football reaches the maximum height after 1 second, while Habib’s football takes 2.828 seconds to reach the maximum height and Graham’s football takes 3 seconds to reach the maximum height.

Answers mathspace.co

825

20 Rafael could have his friend draw a coordinate plane with the x-axis being the water and the y-axis being the left side of his drawing. Rafael could tell his friend how to scale the x and y-axis. Once this is complete, Rafael can describe the vertices for the 8 major parabolas, each being its own quadratic function. Describing key features, such as the domain, range, if it has a maximum or minimum, and vertex location would allow Rafael to share his design idea.

d −(x + 2) (x − 7) = y 15 10 5 x −8 −6 −4 −2

−4

−12 −10 −8 −6 −4 −2 −10

−6

−20

−8

−30 −40

22 a An infinite number of quadratic equations b One quadratic equation x 1

d No quadratic equation exists a Determine the time that the ball will hit the ground. This problem has only one viable x-intercept because we know that the golf ball hits the ground after being in the air, so a non-viable x-intercept is at (2, 0) and a viable x-intercept at (8, 0).

b Determine the times when the ball is on the ground.

16 14 12

This problem has two viable x-intercepts at (2, 0) and (8, 0) and there is no restriction on when this happens.

10 8 6 4 2

c D etermine the times when the ball is on the ground a minute after the ball is hit. x 2

826

c One quadratic equation 23 Answers may vary.

c (5x − 4)2 = y

x 2

−50

b (3x + 5) (x − 1) = y

−1 −2 −3 −4 −5

−8 −6 −4 −2 −2

5 4 3 2 1

e 3(x + 10) (x + 1) = y

−3 −2 −1

4 6

−5

21 a (x − 3) (x + 3) = y

y 20

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

This problem has no x-intercept because at (2, 0) and (8, 0), they are both outside of the time period that we are interested in.

7.02 Quadratic functions in factored form Subtopic overview Lesson narrative In this lesson, students will learn how the factored form of a quadratic function relates to some of its key characteristics. Students will examine the structures of factored equations, compared to the graphs and use repeated reasoning to make generalizations between the general factored form of a graph and the x-intercepts. Students will make sense of problems by analyzing context to interpret key features and then reason abstractly and quantitatively to create quadratic models using a choice of tools and methods, such as graphs, tables, and diagrams. Students will use precision when labeling and creating scales for graphs. By the end of the lesson, students will be able to represent quadratic contextual situations with graphs, tables, and equations in factored form, as well as justify and interpret their models.

Learning objectives Students: Page 393

Key vocabulary 

factored form



factor



root



solution (to an equation)



x-intercept



zero (of a function)

Essential understanding Different representations of a function may highlight or hide different characteristics but they do not change the function itself. The factored form of a function highlights the x-intercepts.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG1 — Mathematical Problem Solving Teachers can integrate this goal into their instruction by providing students with practice problems that challenge them to apply the concept of factoring polynomials and the factored form of a quadratic function to solve complex 7.02 Quadratic functions in factored form mathspace.co

827

problems. For example, they could provide a real-world problem that involves the area of a rectangular plot with a fixed perimeter, which can be represented by a quadratic function, and ask students to factor the function to find the dimensions of the rectangle that maximize the area. MPG4 — Mathematical Connections

MPG5 — Mathematical Representations

Teachers can incorporate this goal by highlighting the connections between the concepts they are teaching and previous lessons. For instance, they could draw attention to the similarities and differences between factoring polynomials and factoring quadratic functions, and between the graphical representation of linear and quadratic functions. They can also show how mathematical concepts can be applied in different subjects and real-world contexts, like the relationship between quadratic functions and areas of rectangular plots in physical scenarios.

Teachers can integrate this goal by instructing students to represent quadratic functions in factored form both algebraically and graphically. They can also encourage the use of various methods and tools to visualize these representations, such as graphing calculators or online graphing tools. Teachers can further emphasize the importance of understanding the relationships between different representations, such as the link between the factored form of a quadratic function and its graph.

A.F.2c — Graph a quadratic function, f (x), in two variables using a variety of strategies, including transformations f (x) + k and kf (x), where k is limited to rational values.

A.F.2d — Make connections between the algebraic (standard and factored forms) and graphical representation of a quadratic function. A.F.2g — For any value, x, in the domain of f, determine f (x) of a quadratic or exponential function. Determine x given any value f (x) in the range of f of a quadratic function. Explain the meaning of x and f (x) in context.

Prior connections A.F.1 — The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.

Future connections A2.EI.6 — The student will represent, solve, and interpret the solution to a polynomial equation. A2.F.2 — The student will investigate and analyze characteristics of square root, cube root, rational, polynomial, exponential, logarithmic, and piecewise-defined functions algebraically and graphically.

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Engage Activity Creating a business plan

60 mins

Students will create a business plan based on a factored, quadratic function.

Understanding and skills

Will use

Will develop

Graphing linear functions.

Graphing a quadratic function given in factored form.

Finding and interpreting key features of quadratics.

Determining and interpreting key features of a quadratic function.

Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: Graph paper and pencil

Support students with disabilities Support conceptual processing - self monitor understanding and ask clarifying questions Have students reﬂect on their own learning using questions from KWL strategy: “What do I Know? What do I Want to learn? What have I Learned?” Answers may look like: Know • I know that a negative x2 value will give a downwards facing curve. • I know that P is the proﬁt and x is the amount spent. • I know the function is quadratic. Want to learn • I want to learn why the function was given in factored form. • I want to learn how to graph the function. • I want to learn what materials were used to make the piñatas. Learned • I learned that when a factor is equal to zero the function equals zero. • I learned when the function equals zero it goes through the x-axis.

Support for English language learners Collect and display As pairs are working, listen for and collect vocabulary, phrases, and methods students use for creating a business plan for Rosaria. Consider grouping language for each part of the process (calculating material cost and proﬁt, graphing the equation, and interpreting the information to give advice). Continue to update collected student language throughout the entire activity. Remind students to borrow language from the display as needed. Some terms and phrases may include: cost, proﬁt, quadratic, function, key points, minimum, maximum, intercepts, points, and coordinate plane.

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Classroom guide Hook

Notice and wonder

Implementation details

Profit

Students might notice that as costs increase, there is a maximum profit which is achieved, followed by a decline back to none or negative profit. Students may also notice more general observations, such as the fact that there is a relationship between profits and costs, and they may wonder what specific situation is being represented by the function. Key observations that students might make: • The profit begins and ends after a certain amount of money is spent. • The maximum amount of profit is between two x-intercepts. • Why does the profit go back down when more money is spent? • Is there a way to increase the profit?

5 mins

What mathematical questions could we ask about these graphs? 60 y

40 Profit

Students write observations about two quadratic graphs showing cost and proﬁts.

•

10 x 0

10 20 Cost

x 0

10 20 Cost

Slide 1 from Student Engage Activity

Launch

5 mins

Rosaria is starting a piñatas business with her two sisters, Valentina and Paola, to sell at the local swap meet. She wants to make sure that the business will be profitable before sinking too much time and energy into the project. With some help from their math teacher, Rosaria has represented the cost versus profits with the following quadratic equation: P = −0.06 (x − 105) (x − 230) where P is profit and x is money spent on materials. Slide 2 from Student Engage Activity

Ask students to share whether they have ever started a business, even something as informal as a lemonade stand, tutoring, or babysitting. Call upon students’ experiences to ask what kind of plans they made before starting the business. Discuss that businesses usually put together plans to predict the costs and profits they can expect under various situations. If time allows, have students brainstorm what concrete components they would want to be part of a business plan. This will make it easier for groups to determine what their finished product will look like. Important mathematical concepts: Quadratic equation, x-intercepts, points, key points, maximum, minimum Important contextual information: Profit, costs, piñata, materials, business plan Suggested grouping: Form pairs

Continue when Students have read the Launch and understand the context of the problem.

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Explore

Think-pair-share

•

35 mins

Students will graph the quadratic equation, identify the maximum proﬁts, and provide business advice.

Anticipated strategies Graph a function Graph a factored quadratic Students will graph a quadratic function given an equation in factored form. The graph looks like the following: 275

Profits

250

Identify key points on a graph Students will identify key points on their graphs in order to answer contextual questions. The x-intercepts are at 105 and 230 and the maximum is at (167.5, 234.375).

225 200 175 150 125 100

Interpret the graph and equation

Come up with business advice

50 25

As for the business advice, students can draw certain information from Costs 25 50 75 100 125 150 175 200 225 250 their graph, including but not limited to: • What is the maximum profit that Rosaria can earn based on this model? $234.38 in profit according to this model. • How much money does Rosaria need to spend on her business before she begins to make a profit? $110. • How much money does Rosaria need to spend on her business to make the maximum profit? $167.50. • How much money does Rosaria need to spend before she stops making a profit? $230.

Misconceptions Graphing the quadratic. What technology do we have available to graph functions? How much is reasonable for Rosaria to spend when starting her business and how can we choose input values to help us graph the function? What should this graph look like when we are done? Choosing a scale. What are the key features of a quadratic? Are these key features visible on your graph? Should they be? Which key features have meaning in this context?

Purposeful questions Use the following questions to check for understanding and encourage critical thinking: • Does it make sense for proﬁts to increase as the costs increase? Why does that work? • Why do the proﬁts decrease? • What do all the key points on the graph represent? • Why is the quadratic equation presented in factored form? • Can you create connections between the equation and the key points? • Did we need to graph the function to create a business plan?

Continue when All students have graphed the function and interpreted 2-3 key points on the graph as they relate to Rosaria’s business.

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Discuss

15 mins

Use a gallery walk for students to share their work with the class leading to a whole class discussion. Consider making connections from the discussion to the factored form of quadratic equations in general.

Discussion guide The discussion should be two-fold: First, about the interpretation of the proﬁt versus cost graph, and second, about the formalization of the relationship between the factored quadratic formula and the x-intercepts. Begin the discussion with a gallery walk so that students can see other partners’ business plans. Have partners explain how they plotted their graphs and what their graphs mean. You can ask students to identify key points, like x-intercepts and maximums, and use that information to talk about the proﬁts versus costs of Rosaria’s business. Some students may have noticed a relationship between the x-intercepts and the factored form of the quadratic, and some students may have been able to concretely identify and name a relationship. If any partners have done this, be sure to emphasize the connection once they have presented to the class. As you wrap up the discussion around Rosaria’s business plan, you can ask students to take a step back from the scenario and notice connections between the key eatures of the graph and the factored quadratic equations. Students may be able to recognize that the x-intercepts of the graph are easily found by looking at the equation and seeing that the values are the two zeros on the graph of the function.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Algebra 1 — 6.06 Factor trinomials Algebra 1 — 7.01 Characteristics of quadratic functions

Tools You may ﬁnd these tools helpful: • Graphing calculator • Graph paper

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Student lesson & teacher guide Quadratic functions in factored form Students begin with an exploration of a graph that represents the vertical height of an object over time.

Students: Page 393

Finding the vertex from factored form Targeted instructional strategies Students may be unsure of how to determine the coordinates of the vertex from just the intercepts. Ask students where they think the axis of symmetry would be with respect to the x-intercepts. Point out to students that the x-intercepts have the same y-value, so they must be symmetric about the vertex. This means that the average of their x-values will give us the axis of symmetry. In other words, the axis of symmetry is halfway between the x-intercepts.

Incorrectly identifying the sign of the x-intercepts Address student misconceptions Students may incorrectly identify the sign of the x-intercepts from an equation in factored form. This is particularly common for equations with negative roots, like y = (x + 2) (x + 3). Ask students what values of x would result in the equation giving y = 0 and guide them to the conclusion that x = −2 and x = −3 are the correct values for the x-values of the x-intercepts. Help students make connections between the intercepts and the zeros, reminding students that if either factor is equal to zero, then y will also be equal to zero. In this example, x = −2 and x = −3 are the values that will make the equation equal to zero.

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Compare and connect English language learner support Encourage students to connect previously learned concepts, such as x-intercepts, direction of parabola opening, axis of symmetry and vertex, to the values in the factored form for quadratic functions. This can be done by asking them to connect features of the equation to its corresponding graph. For example, the equation

has the graph: y 3 2 1 x −4 −3 −2

−1

−1 −2 −3

Students may make connections such as: • The values of 1 and 3 in the equation match the magnitudes of the x-intercepts on the graph. • The signs of the 1 and 3 in the equation are opposite from the signs of their matching intercepts on the graph. Providing students with multiple examples to compare and connect can help students familiarize themselves with the key features of the factored form for quadratic function equations.

Exploration Students: Page 393

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Suggested student grouping: Small groups Students are given the context of a graph of a water balloon’s vertical height over time and the equation of the function in standard form, y = ax2 + bx + c. Students relate the given function to the factored form of the function and should discover that the factored form highlights the x-intercepts of the function. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Shorena says that the function y = −3(x + 1) (x − 2) is equivalent to the given function. How can we determine if she is correct? We can multiply the factors in the equation and determine if the equation is equivalent to the given function. 2. How do the key features of the graph relate to the context? Graphically, we see that the water balloon has been thrown from 6 feet above the water. This is the y-intercept. We also see that the x-intercepts are at (−1, 0) and (2, 0). In the context of the problem, the water balloon hits the pool after 2 seconds. The vertex on the graph is at approximately (0.5, 6.7), meaning the water balloon’s maximum height is about 6.7 feet after half of a second. 3. How might the function y = −3(x + 1) (x − 2) relate to the graph? The x-intercepts on the graph are at (−1, 0) and (2, 0). We can see the opposite of these numbers in the equation. The −3 in the equation indicates that the graph will open downward. Purposeful questions • What are the key features of the graph? Can you ﬁnd each of these from the given equations? • Why do you think that part of the graph has a dashed curve while the other part of the graph has as solid curve? • Which x-intercept is irrelevant to the context of the problem? How do you know? Possible misunderstandings • Students may assume that since the functions are in different forms that the functions themselves are different and so there is no way to show they both represent the same graph. Remind students that we learned about multiplying polynomials in the previous chapter and can apply those skills here to Shorena’s polynomial function. Students are introduced to the factored form of a quadratic function and how it relates to its graph. An approach to graphing a quadratic function given in factored form is presented.

Students: Pages 393–394

1 −4 −3 −2 −1

−1

x 1

−4 −3 −2 −1

−1

x 1

−2

−2 −3

−3

−4

If a > 0, then the quadratic function opens upwards and has a minimum value.

If a < 0 then the quadratic function opens downwards and has a maximum value.

The x-intercepts are the points where f (x) = 0, so we refer to x1 and x2 as the zeros of the function. • (x1, 0) and (x2, 0) are the x-intercepts of the function y = f (x) 7.02 Quadratic functions in factored form • x1 and x2 are zeros of the function mathspace.co • (x − x1) and (x − x2) are factors of the function y = f (x) • x1 and x2 are solutions or roots of the equation f (x) = 0

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−2

−2 −3

−3

−4

If a > 0, then the quadratic function opens upwards and has a minimum value.

If a < 0 then the quadratic function opens downwards and has a maximum value.

The x-intercepts are the points where f (x) = 0, so we refer to x1 and x2 as the zeros of the function. • (x1, 0) and (x2, 0) are the x-intercepts of the function y = f (x) • x1 and x2 are zeros of the function • (x − x1) and (x − x2) are factors of the function y = f (x) • x1 and x2 are solutions or roots of the equation f (x) = 0

x x1

To draw the graph of a quadratic function, we generally want to ﬁnd three different points on the graph, such as the x- and y-intercepts. 8

Since the graph of a quadratic function has a line of symmetry passing through the vertex, we know the vertex lies halfway between the two x-intercepts.

6 4 2 −8 −6 −4 −2 −2

x 2

We can also determine the direction in which the graph opens by identifying if the scale factor, a, is positive or negative.

−4 −6 −8

Examples Students: Page 395

Example 1 Consider the graph of a quadratic function: 4 394

Mathspace Virginia SOL Algebra 1 mathspace.co

2 1 −4 −3 −2 −1

−1

−2 −3 −4

a Identify the coordinates of the x- and y-intercepts of the function.

Create a strategy The x-intercepts occur when y = 0 and the y-intercept occurs when x = 0.

Apply the idea 4

3 2 1 −4 −3 −2 −1

−1

−2 −3

836

The x-intercepts are (−2, 0) and1 Teacher (3, 0). Edition Mathspace Virginia SOL Algebra mathspace.co The y-intercept is (0, 2).

b Find the equation of the quadratic function in factored form.

−4

x 1

a Identify the Example 1 coordinates of the x- and y-intercepts of the function.

Create strategy Considerathe graph of a quadratic function: 4

The x-intercepts occur when y = 0 and the y-intercept occurs when x = 0.

3 2 y 4 1

Apply the idea

3 −4 −3 −2 −1 2 −1

1 −2 −4 −3 −2 −1

−3 −1 −4

4 x

−2 −3

a Identify the coordinates of the x- and y-intercepts of the function. The x-intercepts are (−2, 0) and (3, 0).

−4

The y-intercept is (0, 2). Create a strategy The x-intercepts occur when y = 0 and the y-intercept occurs when x = 0. b Find the equation of the quadratic function in factored form.

Apply the idea

PurposeCreate a strategy 3 Show students to identifyfor the x- and a graphed function. Substitutehow the x-intercepts x and x iny-intercepts the equation yof = a(x − x ) (x − x quadratic ), then use any other point 2 on the graph to 1

substitute for x and y and solve for a.

Reﬂecting with students x −4 −3 −2 −1 1 2 3 4 Ask students features they may already know about the equation of the graphed function. Students may Apply what the idea −1 be able Since to point out thatofthe factor will beform negative the graph opens the x-values the scale x-intercepts areof−2the andquadratic 3, we knowfunction that the factored will be:because −2 downward. Some students may attempt to use the of the equation. y =intercepts a(x + 2) (x − to 3) build the factored form −3 Thesome x-intercepts (−2, 0) ﬁnd anda(3, −4 for value of are a. We can by0). substituting in the coordinates of the y-intercept into the function.

Students:The Pages 395–396 y-intercept is (0, 2).

b Find the equation of the quadratic function in factored form.

Create a strategy Substitute the x-intercepts for x1 and x2 in the equation y = a(x − x1) (x − x2), then use any other point on the graph to substitute for x and y and solve for a.

Apply the idea

7.02 Quadratic functions in factored form mathspace.co

Since the x-values of the x-intercepts are −2 and 3, we know that the factored form will be:

395

y = a(x + 2) (x − 3) for some value of a. We can ﬁnd a by substituting in the coordinates of the y-intercept into the function. To ﬁnd a: Factored form Substitute (0, 2) Evaluate the addition and subtraction Evaluate the multiplication Divide both sides by −6 The equation of the quadratic function in factored form: 7.02 Quadratic functions in factored form mathspace.co

395

Example 2 Purpose Show students to writefunction: the equation of a quadratic function in factored form given its graph. Consider how the quadratic 2

y = 2x + 4x − 48 Expected mistakes Stateinitially the coordinates of the x-intercepts. Studentsa may write the factored form of the quadratic as y = a(x − 2) (x + 3), since the x-intercepts are (−2, 0) and (3, 0), and ignore the signs of the general form, which subtracts the x-values of the x-intercepts.

Create a strategy

In the factored form y = a(x − x1) (x − x2), the values of x1 and x2 are the x-values of the x-intercepts. The y-value of the x-intercepts is y = 0. Factor the quadratic, then determine its x-intercepts. We can factor out a GCF of 2, so that the equation becomes y = 2(x2 + 2x − 24).

7.02 Quadratic functions in factored form Since there are no common factors for the remaining three terms and the trinomial is not a perfect square trinomial, mathspace.co we proceed to factor by grouping by ﬁnding the value of two integers that multiply to ac = (1) (−24) = −24 and add up to b = 2. After ﬁnding these integers, we use them to rewrite the middle term 2x as a sum of two terms.

837

To ﬁnd a: Factored form Substitute (0, 2) Evaluate the addition and subtraction

Reflecting with students Evaluate the multiplication Ask students if substituting the y-intercept (0, 2) intobythe Divide both sides −6 factored form to find the scale factor was the only approach to calculating the scale factor. Point out to students that any point on the graph could be substituted equation the quadratic factored form:a. into the The equation tooffind the valuefunction of theinscale factor,

Students: Page 396

Example 2 Consider the quadratic function: y = 2x2 + 4x − 48 a State the coordinates of the x-intercepts.

Create a strategy In the factored form y = a(x − x1) (x − x2), the values of x1 and x2 are the x-values of the x-intercepts. The y-value of the x-intercepts is y = 0. Factor the quadratic, then determine its x-intercepts. We can factor out a GCF of 2, so that the equation becomes y = 2(x2 + 2x − 24). Since there are no common factors for the remaining three terms and the trinomial is not a perfect square trinomial, we proceed to factor by grouping by ﬁnding the value of two integers that multiply to ac = (1) (−24) = −24 and add up to b = 2. After ﬁnding these integers, we use them to rewrite the middle term 2x as a sum of two terms.

Apply the idea The factor pair whose sum is 2 is −4 and 6. We can use this to rewrite the trinomial and factor by grouping as follows: 2(x2 + 2x − 24) = 2(x2 − 4x + 6x − 24)

Rewrite polynomial with four terms

= 2[x(x − 4) + 6(x − 4)]

Factor each pair

= 2(x − 4) (x + 6)

Divide out common factor of (x − 4)

There are no more common factors to be divided out, so the fully factored form of the quadratic function is y = 2(x − 4) (x + 6). The x-intercepts are (4, 0) and (−6, 0).

Reﬂect and check Notice that x + 6 is the same as x − (−6).

Purpose Show students how to factor a quadratic function in order to identify its x-intercepts. Expected mistakes Students may incorrectly factor x2 + 2x − 24. Remind students that we can check that our factoring is correct by Mathspace Virginia Algebra 1 396 the multiplying factored formSOL and conﬁrming that it is equivalent to the original polynomial. mathspace.co

Reﬂecting with students Ask students what other resources they can use to check that their solution is correct. Students may graph the given function and graph the function they wrote in factored form using technology to conﬁrm that the graphs are the same.

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Students: Page 397 b Determine the coordinates of the y-intercept.

Create a strategy The y-value of the y-intercept is the result when x = 0. We can substitute x = 0 into the factored form to ﬁnd this value.

Apply the idea To ﬁnd the y-value of the y-intercept: y = 2(x − 4) (x + 6) Given quadratic function b Determine the coordinates of the y-intercept. y = 2(0 − 4) (0 + 6) Substitute x = 0 y = 2(−4) (6) Create a strategy

Evaluate the subtraction and addition

y = y-intercept −48 The y-value of the is the resultEvaluate when x =the 0. multiplication We can substitute x = 0 into the factored form to ﬁnd this value. The y-intercept is (0, −48).

Apply the idea Reﬂect and check To ﬁnd the y-value of the y-intercept: The original function which we canfunction identify without making any calculations. y = 2(xshows − 4) (x the + 6)y-intercept, Given quadratic y = 2(0 − 4) (0 + 6)

Substitute x = 0

y = coordinates 2(−4) (6) Evaluate the subtraction and addition c Determine the of the vertex. y = −48 Evaluate the multiplication

a strategy PurposeCreate The y-intercept is (0, −48). The vertex lies to on algebraically the axis of symmetry, so the x-coordinate of the will befunction exactly inin the middle between Show students how ﬁnd the y-intercept from a vertex quadratic factored form.

the two x-intercepts. Reflect and check We can find the middle value by taking the average of 4 and −6. We can then substitute this x-coordinate value into the function to find the y-coordinate. Students:The Page 397 original function shows the y-intercept, which we can identify without making any calculations.

Apply the idea To the x-coordinate: c find Determine the coordinates of the vertex. so the x-coordinate of the The average of 4 and −6 is half way between them. We can calculate that Create a strategy vertex and the axis of symmetry is x = −1. To find the y-coordinate: The vertex lies on the axis of symmetry, so the x-coordinate of the vertex will be exactly in the middle between the two x-intercepts. middle value by taking the average of 4 and −6. We can then substitute this y = 2(x − 4)We (x +can 6) find the Given quadratic function x-coordinate the functionSubstitute to find thex y-coordinate. y =value 2(−1 −into 4) (−1 + 6) = −1 y = 2(−5) (5)

Apply they idea = −50

Evaluate the subtraction and addition

Evaluate the multiplication To ﬁnd the x-coordinate: The vertex is (−1, −50). The average of 4 and −6 is half way between them. We can calculate that vertex and the axis of symmetry is x = −1. d Draw the graph of the function. To ﬁnd the y-coordinate: y = 2(x − 4) (x + 6)

Create a strategy

so the x-coordinate of the

Given quadratic function

y = 2(−1 − 4) (−1 + 6) Substitute x = −1 The scale factor is 2 which is positive, so the graph will open up. We can draw the graph through any three points y = 2(−5) (5) Evaluate the subtraction and addition that we know are on it. y = −50 Evaluate the multiplication The vertex is (−1, −50). d Draw the graph of the function.

PurposeCreate a strategy The scale factor is 2 which is positive, so the graph will open up. We can draw the graph through any three points Show students how to ﬁnd the vertex of a quadratic function from factored form. that we know are on it.

Expected mistakes 7.02 Quadratic functions in factored form 397 Students may expect that the vertex should somehow be highlighted in the given form or mathspace.co the factored form of the quadratic function. Remind students that while we can identify some key features of quadratic functions from factored forms, the vertex is not one of them.

7.02 7.02 Quadratic functions in factoredinform 397 form Quadratic functions factored mathspace.co

mathspace.co

839

Apply the idea To find the x-coordinate: The average of 4 and −6 is half way between them. We can calculate that vertex and the axis of symmetry is x = −1. Reflecting with students To find the y-coordinate:

so the x-coordinate of the

Challenge students to visualize what the graph of the function looks like now that they’ve calculated the y = 2(x − 4) (x + 6) Given quadratic function intercepts and the Point making connections between the form of the quadratic function and its y =vertex. 2(−1 − 4) (−1 + 6)out that Substitute x = −1 graph is a skill that is useful when verifying that our mathand is correct. y = 2(−5) (5) Evaluate the subtraction addition y = −50

Evaluate the multiplication

Students: Pages 397–398 The vertex is (−1, −50).

d Draw the graph of the function.

Create a strategy The scale factor is 2 which is positive, so the graph will open up. We can draw the graph through any three points that we know are on it.

Apply the idea

Reflect and check 20

Any three points is enough to draw the graph, but knowing where the vertex is can make it easier since the vertex is on the axis of symmetry.

−8 −6 −4 −2 −10

−20 −30 7.02 Quadratic functions in factored form mathspace.co

−40 −50

397

−60

Apply the idea

Reflect and check 20

Any three points is enough to draw the graph, but knowing where the vertex is can make it easier since the vertex is on the axis of symmetry.

10 PurposeExample 3 x −4 key −2 features 2 4 6of 8a quadratic function to graph it. Show students how−8to−6 use

Identify the characteristics−10 of h(x) =

(3x + 2) (x − 7).

−20

Reﬂecting with students a Identify the factors of the function. −30 Challenge students to consider how they would write the factored form equation from the graph if they were −40 Createthe a strategy initially given graph as opposed to the equation. −50

The function is given in factored form y = a(bx − x1) (x − x2) where a, (bx − x1) and (x − x2) are the factors. −60

Students: Page 398

Apply the idea

Reﬂect and check

The factors of h (x) are , (3x + 2) and (x − 7).

is the greatest common factor (GCF) of a (bx − x1) (x − x2)

Example 3

but is still a factor of the function.

Identify the characteristics of h(x) = (3x + 2) (x − 7). b Identify the roots of the function. a Identify the factors of the function.

Create a strategy Create a strategy

The roots of the function are the x values, x1 and x2, where h(x) = 0. The function is given in factored form y = a(bx − x1) (x − x2) where a, (bx − x1) and (x − x2) are the factors.

Apply the idea Apply the idea

Reﬂect and check

To solve for the roots algebraically, set each of the variable factors equal to zero. The factors of h (x) are , (3x + 2) and (x − 7). is the greatest common factor (GCF) of a (bx − x1) (x − x2) 3x + 2 = 0 and x − 7 = 0 but is still a factor of the function. Rearrange each equation to isolate the term with the variable. 3x = −2 and x = 7 b Identify the roots of the function. Isolate the variable by dividing by the coeﬃcient of x.

PurposeCreate a strategy and x = 7 x= Demonstrate to students howaretothe identify the factors of a function The roots of the function x values, x and x , where h(x) = 0. given in the form y = a(bx − x1) (x − x2). The roots of h(x) are x =

Apply the idea Reflect and check To solve for the roots algebraically, set each of the variable factors equal to zero. Notice in part (a) we also identified the factor of3x +but it to find roots. That is because a factor without 2 =we 0 did andnot x − use 7=0

840

a variable will notSOL result in a root becauseEdition. Mathspace Virginia Algebra 1 Teacher Rearrange each equation to isolate the term with the variable. mathspace.co 3x = −2 and x = 7 398

Mathspace

Virginia SOL Algebra 1

Isolatemathspace.co the variable by dividing by the coeﬃcient of x.

The function is given in factored form y = a(bx − x1) (x − x2) where a, (bx − x1) and (x − x2) are the factors.

Apply the idea

Reﬂect and check

The factors of h (x) are , (3x + 2) and (x − 7).

is the greatest common factor (GCF) of a (bx − x1) (x − x2)

Students: Pages 398–399

but is still a factor of the function.

b Identify the roots of the function.

Create a strategy The roots of the function are the x values, x1 and x2, where h(x) = 0.

Apply the idea To solve for the roots algebraically, set each of the variable factors equal to zero. 3x + 2 = 0 and x − 7 = 0 Rearrange each equation to isolate the term with the variable. 3x = −2 and x = 7 Isolate the variable by dividing by the coeﬃcient of x. x= The roots of h(x) are x =

and x = 7

Reﬂect and check Notice in part (a) we also identiﬁed the factor of a variable will not result in a root because

but we did not use it to ﬁnd roots. That is because a factor without .

We can substitute these values back into the equation to check our answers. If we substitute get h Mathspace (x) = 0, thenVirginia we know roots 398 SOLour Algebra 1 are correct. mathspace.co

Let’s start with the root

and x = 7 and

. Substitute Evaluate the multiplication Evaluate inside the parentheses Zero product property

Next, let’s try the root x = 7. Substitute x = 7 Evaluate the multiplication Evaluate inside the parentheses Zero product property Evaluating for each root gave an output of 0 conﬁrming that both are in fact roots of the function. c Identify the zeros of the function.

Apply the idea PurposeCreate a strategy The zeroshow of a function are the same roots. given in In factored part (b) weform. solved for the roots, x1 and x2, and got x = Show students to find the roots ofasa its function , 7. These are also the zeros of the function. d State the x-intercepts of the function.

Create a strategy The points (x1, 0) and (x2, 0) are the x-intercepts for h(x).

Apply the idea The zeros or roots of h(x) are

Reﬂect and check We can check our x-intercepts by graphing h(x).

and 7.

h(x)

These are the x-values of the x-intercepts. The y-value of any y-intercept is 0 because the x-axis is at x = 0. The x-intercepts of the function are at

and (7, 0).

x −2 7.02 Quadratic 2 4 functions 6 in factored form −2 mathspace.co −4

841

Evaluate the multiplication Evaluate the multiplication Evaluate inside theinside parentheses Evaluate the parentheses Zero product Zeroproperty product property

Students:Evaluating Page forthe each root Next, let’s 399 try root x =gave 7. an output of 0 conﬁrming that both are in fact roots of the function. Substitute x = 7 c Identify the zeros of the function.

Evaluate the multiplication

Create a strategy

Apply the idea

Evaluate inside the parentheses The zeros of a function are the same as its roots. In part (b) we solved for the roots, x1 and x2, and got x = Zero product property , 7. These are also the zeros of the function. Evaluating for each root gave an output of 0 conﬁrming that both are in fact roots of the function. d State the x-intercepts of the function. c Identify the zeros of the function.

PurposeCreate a strategy Show students zeros a the function are the same as its roots. Create athat strategy Apply the idea The points (x1, the 0) and (x2, 0)ofare x-intercepts for h(x). The zeros of a function are the same as its roots.

Students:Apply Page the399 idea

The zeros or roots of h(x) are

In part (b) we solved for the roots, x1 and x2, and got x =

Reﬂect and are check , 7. These also the zeros of the function. We can check our x-intercepts by graphing h(x).

and 7.

d State the function. The y-value of These arethe thex-intercepts x-values ofof the x-intercepts. any y-intercept is 0 because the x-axis is at x = 0.

h(x) x

Create a strategy

−2

The x-intercepts of the function are at and (7, 0). The points (x1, 0) and (x2, 0) are the x-intercepts for h(x).

−2

Reﬂect and check−4

Apply the idea The zeros or roots of h(x) are

We can check our x-intercepts by graphing h(x). −6

and 7.

These are the x-values of the x-intercepts. The y-value of any y-intercept is 0 because the x-axis is at x = 0. The x-intercepts of the function are at

and (7, 0).

−8

h(x) x

−2

There are two points where the parabola crosses the x-axis, at x =

−2

and x = 7. −4 7.02−6 Quadratic functions in factored form mathspace.co

399

−8

There are two points where the parabola crosses the x-axis, at x =

and x = 7.

7.02 Quadratic functions in factored form mathspace.co

399

Purpose Help students understand how to find the x-intercepts of a function.

Students: Page 400 Example 4 The graph of a quadratic function has x-intercepts at (−2, 0) and (1, 0) and passes through the point (−3, −2). Write an equation in factored form that models this quadratic.

Create a strategy To write the equation for this quadratic in factored form we need to ﬁrst identify the roots or zeros of the equation. We can then substitute these values for x1 and x2. The x-intercepts of the function are at (−2, 0) and (1, 0), so we know the equation has roots/zeros of x = −2 and x = 1.

842

Mathspace Virginia Algebra 1 Teacher We can put theseSOL in the factored form asEdition the factors: mathspace.co y = a(x + 2) (x − 1) We can ﬁnd a by substituting the coordinates of the additional point, (−3, −2), into the function. To ﬁnd a:

Apply the idea Since the zeros are x = −2 and x = 1, we can identify the factors by rearranging those equations so they are equal to 0: By adding 2 to both sides of x = −2 and subtracting 1 from both sides of x = 1 we get: x + 2 = 0 and x − 1 = 0 We can put these in the factored form as the factors: y = a(x + 2) (x − 1) We can ﬁnd a by substituting the coordinates of the additional point, (−3, −2), into the function. To ﬁnd a: y = a(x + 2)(x − 1)

Factored form

−2 = a(−3 + 2)(−3 − 1)

Substitute x = −3 and y = −2

−2 = a(−1)(−4)

Evaluate the addition

−2 = 4a

Evaluate the multiplication

Divide both sides by 4

Substituting the value we found for a, the equation of the quadratic function in factored form is:

Reﬂect and check Checking the graph of the equation, we can see that it satisﬁes the given information. 10

8 6 4 2 −4 −3 −2 −1 −2

x 1

−4 −6 −8

Purpose Show students how to write a quadratic equation in factored form given the x-intercepts and a point on the graph. Mathspace Virginia SOL Algebra 1 Students:400 Page 401 mathspace.co

Example 5 Find the equation that models the graph shown.

y −6

−4

−2

x 2

−2 −4 −6

Create a strategy This quadratic function only has 1 x-intercept, which is also the vertex. When this happens, the function is in the form f (x) = a(x − x1)2. Remember, we need an additional point, like the y-intercept, to ﬁnd the exact equationintofactored this 7.02 Quadratic functions form function. mathspace.co

Apply the idea 2

843

−4 −6

Apply the idea Since the x-intercept is at (−2, 0), the function takes the form f (x) = a(x + 2)2. Next, we can use the y-intercept of (0, −1) to ﬁnd the value of the leading coeﬃcient. Given equation Substitute x = 0 and y = −1 Evaluate the addition Evaluate the exponent Divide both sides by 4 The equation of the graph is

Reﬂect and check We could have used any point on the parabola to solve for the scale factor, a. There is another point at (−6, −4). We would substitute x = −6 and y = −4, then the equation would take the form −4 = a(−6 + 2)2.

Notice that this is the same thing we got earlier because no matter which points we substitute in we will get the same function because they are all points on the same parabola.

Purpose Show students how to find the equation of a quadratic function given its graph.

Advanced learners: Engaging in natural extensions to7.02 deepen understanding Quadratic functions in factored form Targeted instructional strategies

401 use with Example 5 mathspace.co

Extend the problem for advanced learners by adding additional constraints or posing new questions. For example, prompt them to find another quadratic function that shares the same vertex and has the same shape but opens upward instead of downward. Then, encourage students to explore how altering the parameters in a quadratic equation impacts its graph. Ideally, students should notice that the new graph is simply a reflection of the original across the x-axis, so the only thing that change in the equation is the sign of the leading coefficient, a. Engaging in these natural extensions deepens their understanding of the relationship between algebraic expressions and their graphical representations.

844

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Students: Page 402 Example 6 A cannonball is ﬁred from the edge of a cliff which is 15 meters above sea level. The peak of the cannonball’s arc is 20 meters above sea level and 10 meters horizontally from the cliff edge. The cannonball lands in the sea 30 meters away from the base of the cliff. The path of the cannonball is shown on the following graph,

Example but the axes6have not been labeled.

A cannonball is ﬁred from the edge of a cliff which is a meters Label the axessea of the graph match thecannonball’s information provided. 15 above level. The to peak of the arc is 20 meters above sea level and 10 meters horizontally from the cliff edge. The cannonball lands in the sea 30 meters Create a strategy away from the basematch of thethe cliff.information, we want to make sure that the axes labels and scales accurately represent To make the graph the path make on sense for the context. The pathof ofthe thecannonball cannonballand is shown the following graph,Both axes will have meters as their units. but the axes have not been labeled.

Apply the idea

Reﬂect and check

We can see that the path on the graph starts at a point Another way to show the scale of the axes is to label a Label the axes of the graph to match the information provided. on the vertical axis and ends at a point on the horizontal some key points. For example: axis. So, we can make the vertical axis represent the y (m) Create a strategy (10, 20) height, with y = 0 being sea level, and the horizontal axis To make the graph match the information, we want to make sure that the axes labels and scales accurately represent represent distance, with x = 0 being the edge of the cliff. (0,have 15) meters as their units. the path of the cannonball and make sense for the context. Both axes will We can then add values onto the axes to show that the cannonball starts at the edge of the cliff at (0, 15), reaches Apply idea Reﬂect and check x (m) its peakthe at (10, 20), and then falls into the sea at (30, 0). We can see that the path on the graph starts at a point Another way to show the scale of the axes is(30, to 0) label y (m) on the vertical axis and ends at a point on the horizontal some key points. For example: 20 axis. So, 15 we can make the vertical axis represent the y (m) (10, 20) height, with y = 0 being sea level, and the horizontal axis 10 represent distance, with x = 0 being the edge of the cliff. 5

We can then add values onto the axes to show that x (m)the 25(0, 15), 30 reaches cannonball starts5at the10edge15of the20cliff at its peak at (10, 20), and then falls into the sea at (30, 0).

(0, 15)

x (m) (30, 0)

y (m)

20 b Determine the factored equation which models the path of the cannonball. 15

Purpose Create a10strategy Show students 5how to use context to appropriately label a graph and its scale.

To match the graph in part (a), we can let x represent the horizontal distance from the cliff, and let y represent the x (m) height above sea level.

10 15 20 25 30 5 Students:ToPages 402–403 ﬁnd the factored equation that models the cannonball, we need to know both x-intercepts and the scale factor.

We know that one of the x-intercepts is at x = 30, and that the vertex is at x = 10. Remember that the vertex lies on the axisDetermine of symmetry a quadratic function, somodels we canthe usepath this to the other x-intercept. b theoffactored equation which of find the cannonball. We can find the scale factor by substituting any point into the equation (that isn’t an x-intercept) and solving for the scale factor that makes the equation true. Create a strategy To match the graph in part (a), we can let x represent the horizontal distance from the cliff, and let y represent the height above sea level. To find the factored equation that models the cannonball, we need to know both x-intercepts and the scale factor. We know that one of the x-intercepts is at x = 30, and that the vertex is at x = 10. Remember that the vertex lies on the axis of symmetry of a quadratic function, so we can use this to find the other x-intercept. We find the scale factor by substituting any point into the equation (that isn’t an x-intercept) and solving for the Mathspace Virginia SOL Algebra 1 402 can mathspace.co scale factor that makes the equation true.

402

Mathspace Virginia SOL Algebra 1 mathspace.co

7.02 Quadratic functions in factored form mathspace.co

845

Apply the idea Since both x-intercepts have the same y-value, they will be mirrored across the axis of symmetry. Since x = 30 is 20 more units than x = 10, the other x-intercept will be at 20 less units than x = 10. So, the other x-intercept is at x = −10. If we let the scale factor of the equation be a, then our equation will be: y = a(x + 10) (x − 30) We can ﬁnd the scale factor by substituting in a point on the graph (let’s use the y-intercept) and solving for a. Model equation Substitute (0, 15) Evaluate the addition and subtraction Evaluate the multiplication Divide both sides by −300 So, the equation which models the path of the cannonball is:

c A second cannonball is ﬁred, and this one can be modeled by the equation:

Purpose Apply idea Show students how to write the where factored form of alanded. quadratic function from its graph, even if both x-intercepts Usethe this model to predict the cannonball Since both x-intercepts have the same y-value, they will be mirrored across the axis of symmetry. Since x = 30 is are not shown. 20 moreaunits than x = 10, the other x-intercept will be at 20 less units than x = 10. So, the other x-intercept is at Create strategy

x =mistakes −10. Expected We can use what we know about the general factored form of the equation to provide information about the If we letstruggle the scale of the equation be x-intercept a, then our equation willseeing be: Studentscannonball’s may to identify the other without it on the graph. Remind students that path.factor quadratic functions are symmetrical and we can yuse that information to determine other points on a graph that = a(x + 10) (x − 30) Apply the idea aren’t shown. We can ﬁnd the scale factor by substituting in a point on the graph (let’s use the y-intercept) and solving for a. a value is negative, we know this function sense for a cannonball, it willequation StudentsSince maythe also forget to include the scale factorwill in open theirdownward. factored This formmakes and incorrectly concludeasthat Model equation upwards to a maximum vertical height before falling back down, due to gravity. The x-intercepts will be at −12 and is y = (x arc + 10) (x − 30). Remind students to check their equation by substituting any known points, that are not 27. Since the cannonball is being ﬁred Substitute away from (0, the15) cliff in the positive x-direction, we know that x = −12 is a nonx-intercepts, into the equation. viable solution. So, the second cannonball lands in the sea 27 meters from the base of the cliff. Evaluate the addition and subtraction

Evaluate the multiplication Reflecting with students Ask students to notice the sign of a. Is this what expect, based on the graph? Why? Students should Divide boththey sideswould by −300 Idea be able to point outsummary that a negative value means the graph opens downward. So, the To equation which models cannonball is: write the equation of the path graphofofthe a quadratic function in factored form, substitute the x-intercepts for x1

and403 x2 in the equation y = a(x − x1) (x − x2), then use any other point on the graph to substitute for x and y and Students: Page solve for a, the scale factor.

c A second cannonball is ﬁred, and this one can be modeled by the equation:

Use this model to predict where the cannonball landed.

Create a strategy We can use what we know about the general factored form of the equation to provide information about the cannonball’s path.

Apply the idea

7.02 Quadratic functions in factored form mathspace.co

403

Since the a value is negative, we know this function will open downward. This makes sense for a cannonball, as it will arc upwards to a maximum vertical height before falling back down, due to gravity. The x-intercepts will be at −12 and 27. Since the cannonball is being ﬁred away from the cliff in the positive x-direction, we know that x = −12 is a nonviable solution. So, the second cannonball lands in the sea 27 meters from the base of the cliff.

Idea summary 846

To write the equation of the graph of a quadratic function in factored form, substitute the x-intercepts for x1 and x2 in the equation y = a(x − x1) (x − x2), then use any other point on the graph to substitute for x and y and Mathspace Virginia SOL Algebra 1 Teacher Edition solve for a, the scale factor. mathspace.co

Apply the idea Since both x-intercepts have the same y-value, they will be mirrored across the axis of symmetry. Since x = 30 is

Purpose20 more units than x = 10, the other x-intercept will be at 20 less units than x = 10. So, the other x-intercept is at x = −10. how to use a quadratic equation in factored form to solve a problem in context about an Show students If we let the scale factor of the equation be a, then our equation will be: x-intercept. y = a(x + 10) (x − 30)

Reflecting with students We can ﬁnd the scale factor by substituting in a point on the graph (let’s use the y-intercept) and solving for a. Ask students to examine the factored equation. Why does substituting −12 in for x cause the y-value to be 0? equationto be zero? Students have not yet been introduced to the Why does substituting 27 in for x cause Model the y-value Substitute (0, 15) explain what happens when any one term in a product zero-product property, but should be able to informally Evaluate the addition and subtraction is zero. Evaluate the multiplication

Use key features to indicateDivide a scale both sides by −300

use with Example 6

Student with disabilities support So, the equation which models the path of the cannonball is:

If students ﬁnd it diﬃcult or laborious to add the scale of the axes to their graph, instead recommend that they label some key points which can be used to indicate scale. y (m)

c A second cannonball is ﬁred, and this one can(10, be 20) modeled by the equation:

20 15

15) the cannonball landed. Use this model to predict(0, where 10

Create a strategy 5 about the general factored form of the equation to provide information about the We can use what we know cannonball’s path. x (m) (30, 0) 5

Apply the idea

Since the a value is negative, we know this function will open downward. This makes sense for a cannonball, as it will arc upwards to a maximum vertical height before falling back down, due to gravity. The x-intercepts will be at −12 and 27. Since the cannonball is being ﬁred away from the cliff in the positive x-direction, we know that x = −12 is a nonStudents:viable Page 403So, the second cannonball lands in the sea 27 meters from the base of the cliff. solution.

Idea summary To write the equation of the graph of a quadratic function in factored form, substitute the x-intercepts for x1 and x2 in the equation y = a(x − x1) (x − x2), then use any other point on the graph to substitute for x and y and solve for a, the scale factor.

Practice Students: Pages 404–408 7.02 Quadratic functions in factored form mathspace.co

What do you remember? 1

State the factored form for a quadratic equation and describe its graphical features.

Describe the axis of symmetry of a quadratic function.

For each of the following quadratic equations, state the roots: a

y = (x − 1) (x − 4)

y = (x − 3) (x + 7)

y = x(x − 4)

403

y = 2(x − 1) (x + 5)

7.02 Quadratic functions in factored form mathspace.co

847

For each of the quadratic function, determine the y-value of the y-intercept: y = 3(x − 1) (x − 8)

a 5

y = x(x + 9)

Consider the graph of a function.

Select the equation that represents the function: A

y = x(x + 5)

y = x(5 − x)

y = −x(x − 5)

y = x(x − 5)

−2 −1

x 1

−1

−2 −3 −4 −5 −6 −7

Consider the graph of a function.

9 8 7 6 5 4 3 2 1

Select the equation that represents the function: A

h = (2 − t) (t − 4)

h = −(2 − t) (t − 4)

h = (t + 2) (t − 4)

h = −(t − 2) (t + 4)

−6 −5 −4 −3 −2 −1 −1 −2

Use the table of values to graph the function. x y

0 12

2 0

4 −4

6 0

8 12

Consider the function Select the graph that represents the function: A

6 5 3 2

848

3 2

−3 −2 −1 −1

x 1

2 3 4 5 6 7

−7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4

−2 −3

−5

−4

−6

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

x 1

2 3

2 3 4

6 5

3 2 1

4 3 2 1

−3 −2 −1 x

−7 −6 −5 −4 −3 −2 −1 −1 −2

−1

x 1

2 3 4 5 6 7

−2 −3

2 3

−4 −5

−3 −4

−6

Graph a quadratic function with x-intercepts of (5, 0) and (−3, 0), a y-intercept of (0, 60), and its vertex is a maximum.

Let’s practice 10

For each quadratic function: i

State the coordinates of the x-intercepts.

Determine the axis of symmetry.

iii

Determine the coordinates of the vertex.

Graph the function.

y = (x − 2) (x + 4)

y = −2(x − 1) (x − 5)

For each graph: i

State the zeros.

State the y-value of the y-intercept.

iii

Write the equation of the quadratic function in factored form.

2 −2 −1

−2

−8 −6 −4 −2 −2

x 1

x 2

−4

−6

−8

For each of the quadratic function, find the: i

Solutions

a c

y-intercept

iii

vertex

y = (x − 3) (x + 1)

y = (x + 2) (x − 5)

y = (x − 4) (x + 6)

y = −(x − 1) (x + 2)

For each of the quadratic function, find the: i

Domain

Range

y = (x − 1) (x − 2)

y = −(x + 3) (x − 4)

y = (x − 2) (x − 2)

y = −(x + 1) (x + 1)

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849

Find the range when the domain is {−6, −1, 0, 5, 7} for each of the quadratic function: a

y = (x + 4) (x − 1)

y = −(x − 3) (x + 2)

y = −2(x − 2) (x + 3)

y = 3(x + 1) (x − 2)

Use the function f (x) = −2(x + 4) (x − 1) to evaluate for the following values: a

x = −5

x = −2

x =

f (x) = 0

A quadratic function has x-intercepts of (5, 0) and (−3, 0), and a y-intercept of (0, 60). Write the equation of the function in factored form.

A quadratic function passes through the points (−2, 0), (9, 0) and (0, −6). Write the equation of the function in factored form.

The zeros of a quadratic function are x = −4 and x = −7. The graph of the function passes through the point (−3, 12). Write the equation of the function in factored form.

Satellite dishes follow the shape of a parabola to optimally receive signals. Winston models the cross section of a satellite dish with the points (−2, 0) and (2, 0) being the edges of the satellite and the x-axis representing the top opening of the satellite dish. The satellite dish is a

- foot deep.

Select the graph that represents the problem: A y

1 x

x −2

−1

−2

−1

−2

C y

1 x

−2

850

−1

x −2

−1

−2

Find the vertex of the satellite dish.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Xia notices that the Sunshine State Arch is in the shape of a quadratic function. They know that the arch has a height of 110 ft and the feet of the arch are 100 ft apart. Xia chooses to let the x-intercepts of the arch be the origin and (100, 0). a

Determine the coordinates of the vertex of the arch.

Write the equation of the quadratic function for Xia’s representation of the Sunshine State Arch in factored form.

Find the domain and range of the Sunshine State Arch.

Let’s extend our thinking 21

Create a quadratic function in the form f (x) = (x − ⬚) (x − ⬚) that meets the given condition. a

Two positive x-intercepts.

Two negative x-intercepts.

One positive and one negative x-intercept.

One positive x-intercept.

One x-intercept at (0, 0).

Rewrite the quadratic functions from the previous question in the form f (x) = ax2 + bx + c and determine the similarities and differences between the values of a, b, and c depending on the type of solution the quadratic has.

Ami throws a javelin forward in a parabolic arc from the ground. Using photos that her friend Maryellen is taking from the stands, she determines that 10 yards horizontally from where she threw the javelin, it reaches a maximum height 5 yards above the ground. Ami models her throw with the origin of a coordinate plane being the point 20 yards behind her.

For Ami’s model, state the roots.

Write the equation for the quadratic function modeling the path of Ami’s throw in factored form.

Burnell jumps up and off a 4 meter high springboard into the diving pool below. Burnell’s jump can be represented by the equation y = −2(x − 2) (x + 1) where y is Burnell’s height above the water and x is the horizontal distance from the springboard towards the pool (both in meters).

Graph the equation modeling Burnell’s jump. Label any x- and y-intercepts.

Use the model to predict where Burnell will enter the water. Explain your answer.

Wilson tosses an eraser into the air and counts how long it takes for the eraser to return to his hand. He estimates that it takes 2 seconds and that he is tossing the eraser 6 ft into the air. Wilson models the height of the eraser above his hand in feet as a function h of time t in seconds, letting t = 0 be when he tosses the eraser and h = 0 be the height of his hand.

Explain how to ﬁnd the intercepts of the function and state them.

Assuming that the path of the eraser is symmetric going up and coming down, state the coordinates of the vertex of the function.

Graph the function. Choose appropriate labels and scales.

Write the equation of the function in factored form.

A quadratic function has the factored form equation y = k(x − x0) (x − 12). If the function has a vertex at the point (8, 15), determine the values of x0 and k. Justify your answers.

7.02 Quadratic functions in factored form mathspace.co

851

Answers 7.02 Quadratic functions in factored form

Let’s practice

What do you remember?

6 2

The values of x1 and x2 are the x-values of the x-intercepts of the quadratic function. The scale factor of the function is represented by a.

−8 −6 −4 −2 −2

3 a x = 1, 4

b x = 3, −7

c x = 0, 4

d x = 1, −5

4 a y = 24

4 6

−8 −10

ii x = 3

b i (1, 0) and (5, 0) iii (3, 8) iv

b y=0

6 D y

8 7 6 5 4 3 2 1 −2 −1 −1 −2

x 1

2 3 4 5 6 7 8

11 a i x = 1, 4 x

−4 −2 −2 −4

ii y = 8

iii y = 2(x − 1) (x − 4) b i x = −2, 6

8 10

ii y = 4

iii

8 A

12 a i x = 3, −1

ii (0, −3)

iii (1, −4)

b i x = −2, 5

ii (0, −10)

iii (1.5, −12.25)

c i x = 4, −6

ii (0, −24)

iii (−1, −25)

d i x = 1, −2

ii (0, 2)

iii (−0.5, 2.25)

9 Answers may vary 70 60 50 40 30 20 10 −4 −3 −2 −1 −10 −20 −30

13 a i All real numbers

x 1 2 3 4 5 6

b i All real numbers

ii y ≤ 12.25

c i All real numbers

ii y ≥ 0

d i All real numbers

ii y ≤ 0

14 a {−6, −4, 14, 36, 66}

b {−36, −14, 4, 6}

c {−100, −48, 12}

d {−6, 0, 54, 120}

15 a f (−5) = −12 c

b f (−2) = 12 d x = −4 and x = 1

16 y = −4(x − 5) (x + 3) 17 18 y = 3(x + 4) (x + 7) 19 a C

852

−4

5 D

x 2

−6

2 The axis of symmetry of a quadratic is the vertical line that can be drawn so each side of the quadratic function mirrors the other. This line passes through the vertex of the quadratic function.

16 14 12 10 8 6 4 2

1 y = a(x − x1) (x − x2)

ii x = −1

10 a i (2, 0) and (−4, 0) iii (−1, −9)

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

20 a (50, 110)

b Since y represents Burnell’s height above the water, he will enter the water when y = 0. In other words, the x-intercepts of the model represent where Burnell might enter the water. This will be at either (–1, 0) or (2, 0).

c Domain: 0 ≤ x ≤ 100 Range: 0 ≤ y ≤ 110 Let’s extend our thinking 21 a A nswers should be in the form of f (x) = (x − a) (x − b) where a and b are positive real numbers. For example, f (x) = (x − 5) (x − 3) has positive x-intercepts b A nswers should be in the form of f (x) = (x + a) (x + b) where a and b are positive real numbers. For example, f (x) = (x + 1) (x + 2) has negative x-intercepts c A nswers should be in the form of f (x) = (x + a) (x − b) where a and b are positive real numbers. For example, f (x) = (x − 4) (x + 1) has one negative and one positive x-intercept d A nswers should be in the form of f (x) = (x − a) (x − a) or f (x) = (x − a)2 where a is a positive real number. For example, f (x) = (x − 6)2 has one positive x-intercept e A nswers should be in the form of f (x) = x(x − b) where b is any real number. For example, f (x) = x(x + 1) has one x-intercept at (x, 0) 22 The functions will be dependent on the values chosen in the previous question. The signs of b and c will be different depending on the signs of the x-intercepts. Specifically: when both x-intercepts have the same sign c is positive. When the x-intercepts have different signs c is negative. b is positive when both x-intercepts are positive and b is negative when both x-intercepts are negative. When the x-intercepts are positive and negative, b will be either positive or negative depending on which x-intercept is larger. The coefficient a is not influenced by the values of the x-intercepts in this form. 23 a x = 20, 40 24 a

We also know that x represents Burnell’s horizontal distance from the springboard towards the pool. Since Burnell is jumping into the pool, the x-value when he enters the water must be positive. This means that Burnell will enter the water at the coordinates (2, 0), which is 2 meters horizontally from the springboard. The other x-intercept, (–1, 0) is a non-viable solution because it has a negative x-value. 25 a T he horizontal intercepts represent the times when the eraser is 0 ft above Wilson’s hand. This will be exactly when Wilson tosses the eraser and when it returns to his hand. Since Wilson tosses the eraser at t = 0 and it takes 2 seconds for it to return to his hand, the horizontal intercepts are (0, 0) and (2, 0). Since one of the horizontal intercepts is at the origin, we have the vertical intercept is also (0, 0). b (1, 6) c

feet (h) 7 6 5 4 3 2 1 1

seconds (t) 3

d y = −6x(x − 2) 26 Since the vertex lies on the axis of symmetry, we know that 8 is the average of 12 and x0. So then, we must have x0 = 4. If we substitute the coordinates of the vertex into the function, we get: 15 = k(8 − 4) (8 − 12) If we solve this equation we get

y 4 (0, 4) 3 2 1 (−1, 0) −1

(2, 0) 2 3

Answers mathspace.co

853

7.03 Quadratic functions in vertex form Subtopic overview Lesson narrative In this lesson, students will use technology and coordinate grids and repeated reasoning to make generalizations about the relationship of quadratic equations in vertex forms and the graphs key characteristics. Students will also explore an interactive visual model to represent the process of completing the square. By the end of the lesson, students will be able to examine equations in vertex form to identify and explain the transformations of the parent quadratic function and work flexibly between quadratic equations in vertex form and their graphs to answer questions about contextual situations.

Learning objectives Students: Page 409

Key vocabulary 

completing the square



dilation



horizontal translation



perfect square trinomial



reflection



vertex form



vertical translation

Essential understanding Different representations of a function may highlight or hide different characteristics but they do not change the function itself. The vertex form of a quadratic function highlights the coordinates of the vertex and as a result the axis of symmetry.

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Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG2 — Mathematical Communication

MPG4 — Mathematical Connections

The goal can be integrated by encouraging students to express their reasoning and solutions clearly using mathematical language. This could involve explaining their thought process in determining the vertex, x-intercepts, and y-intercept of a quadratic function in vertex form, or discussing how they graphed the function.

Teachers can help students make connections between different areas of mathematics and between mathematics and other disciplines by relating the concept of a vertex and the form of a quadratic function to real-world situations. They should also connect the different forms of quadratic functions (standard, vertex, and factored forms) and how they can be used interchangeably to analyze and interpret key characteristics of the function.

MPG3 — Mathematical Reasoning This goal can be integrated into instruction by prompting students to justify their steps in converting a quadratic function from standard form to vertex form, or in determining the key characteristics of a quadratic function given its vertex form. They should use logical reasoning to analyze their own and others’ arguments and to determine whether conclusions are valid.

A.F.2c — Graph a quadratic function, f (x), in two variables using a variety of strategies, including transformations f (x) + k and kf (x), where k is limited to rational values.

Prior connections A.F.1 — The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.

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855

Engage Activity Kicks for Charity

60 mins

Students will use an applet to play a soccer game and try to determine the quadratic function that guarantees the soccer ball passes through a given vertex.

Understanding and skills

Will use Identifying key features of graphs.

Will develop Writing the equation of a quadratic functions in vertex form when given a graph, table, or written description. Identifying the effect of a given transformation on a quadratic function in vertex form.

Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: None

Support students with disabilities Support organization - collect and record data Provide students with the following table: Point 1

Point 2

Equation that goes through points

This will help students make observations regarding trigonometric ratios and angles.

Support for English language learners Compare and connect with discussion supports Ask students to reﬂect on the differences between the easier and harder versions of the applet. Consider sharing the following sentence prompts: • The difference between the levels is ... • I can get a star on the easy version because ... • I could use equations to solve for the star because ...

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Classroom guide Hook

Notice and wonder

Students write observations about an applet with three, adjustable quadratic functions.

•

5 mins

What do you notice? What do you wonder? Explore the applet. 6

Implementation details Encourage students to explore the applet and highlight student responses about how the parameters a, h, and k affect each parabola. Use the hook as an opportunity to review key features of graphs of quadratic functions, such as vertical and horizontal shifts, stretching and compressing of the parent function, and the graph’s concavity.

−6

−4

−2

−4

Launch Explore the applet. 13 12 11 10 9 8 7 6 5 4 3 2 1

13 12 11 10 9 y = 2.6 x82 y = (x + 2)2 y = x2 − 2.5 7 13 6 Slide 1 from Student Engage Activity 12 5 11 4 5 mins 10 3 9 2 8 1 7 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 610 11 12 13 14 15 16 2 5 4 3 8 2 −0.125 1

y = a(x − h) + k

Kick

Random

Make harder

Kick Random Make harder

y = a(x − h)2 + k

Kick

Make harder

y = a(x − h)2 + k 8

−0.125 3

7.03 Quadratic functions in vertex form mathspace.co

−0.125

Random

−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Slide 2 from Student Engage Activity

−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 3

857

Ask the class if anyone plays soccer or follows the Miami FC team or other soccer teams, calling upon student knowledge and interest in soccer will increase engagement in the task. Allow students time to explore the soccer challenge applet and to read the instructions of the game. Note: It may increase student buy-in if you encourage excitement around the charity and $10, 000 aspect of the task. Students may want to choose a charity they would contribute to, and you can create a chart on the board for students to record each time they are able to ‘earn’ $10, 000 by correctly ﬁnding the quadratic equation to model the kick of the soccer ball. Important mathematical concepts: Quadratic equation, parameters, vertical and horizontal shifts, stretching and compressing of functions, concavity, and transformations. Important contextual information: Miami FC soccer team and charity. Suggested grouping: Form pairs

Continue when Students have read the Launch and understand the context of the problem.

Explore

Think-pair-share

•

35 mins

Students will come up with at least three equations to represent a soccer ball going through different points.

Anticipated strategies Connect transformations to quadratics Students may remember how to transform other functions, and connect how a, h, and k affect the parent quadratic function y = x2. Trial, error, and improvement Students may explore the applet and notice that (h, k) is the vertex of the parabola and that a affects the vertical stretch and compression through testing various kicking attempts. With this information, students can create equations. Algebraic testing Students may test whether the vertex (h, k) satisﬁes the equation algebraically. Some students may notice that (h, k) is the vertex and that a vertically stretches or compresses the function. Other students may ﬁnd (h, k) and then plug in another point on the function to solve for a so that the function passes through the vertex and the other desired point. Encourage students to use multiple strategies and record their observations.

Misconceptions Not generalizing how a, h, and k affect the quadratic function. Have you tried changing one parameter at a time? What would the equation and graph be for each case? Can you record this somewhere? What do you notice?

Purposeful questions Use the following questions to check for understanding and encourage critical thinking: • What is the goal of this kicking challenge? • What is your plan to adjust the equation so that the kick hits the goal or is closer to hitting the goal? What makes you say that? • What observations do you notice about the graph and equation? • How do the parameters affect the graph?

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Continue when Students have completed three kicking challenges, recorded the equation for each, and generalized the process of ﬁnding the function that should model the path of the soccer ball.

Discuss

15 mins

Have a group discussion where some partners’ can share their equations and can be tested in front of the class. Consider sequencing the strategies presented from transformations, to trial and error, to algebraic testing.

Discussion guide Invite students to share their generalizations of the process for ﬁnding the function that should model the path of the soccer ball. Write student responses on the board and ask the class to generalize the class observations as needed, such as how a, h and k affect the function. Test out some of the equations that students share to ensure that they do hit the target. If you are keeping track of the scores, mark down the winnings of partners’ at this time. Choose one generalization for the class to test out. Display the explanation and ask groups to try and follow the directions to ﬁnd the equation for another simulation of the kicking challenge. As an extension you may wish to give students the following prompt: Compare the following functions and how the parameters affect the functions: y = ax2 y = (bx)2

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Algebra 1 — 6.07 Factor using appropriate methods Algebra 1 — 7.01 Characteristics of quadratic functions Algebra 1 — 7.02 Quadratic functions in factored form

Tools You may ﬁnd these tools helpful: • Graphing calculator • Clear plastic sheets • Tracing paper • Blank coordinate grid

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Student lesson & teacher guide Vertex form The following supports may be useful for this section. More specific supports may appear throughout the lesson:

Students: Page 409

Relate the transformations in vertex form back to the parent quadratic function Targeted instructional strategies Relate the different constants and coeﬃcients in the vertex form of a quadratic function to the transformations of y = x2, explaining why these transformations should follow a particular order (or otherwise be taken into account). Consider the general vertex form y = a(x − h)2 + k. To get this function from y = x2, we apply the following transformations: 1. If a is negative, then we reﬂect across the x-axis. 2. Vertical stretch or compression by a factor of the magnitude of a. 3. Horizontal translation by h units, left or right depending on the sign of h. 4. Vertical translation by k units, up or down depending on the sign of k. Let students experiment with performing these steps in different orders. Explain that steps 1, 2 and 3 can be done interchangeably, but steps 1 and 2 should come before step 4. This is because vertical translation moves the vertex off the x-axis, which changes how the other transformations affect the graph.

Break down vertex form Student with disabilities support To incorporate decomposition, guide students to break down vertex form into its individual components: a, h, and k. Examine how each parameter affects the graph by focusing on one at a time. For instance, keep h and k constant and vary a to explore changes in the graph’s width and direction. The applet from the exploration can be used to emphasize the impact of each parameter.

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Compare and connect English language learner support Encourage students to connect previously learned concepts, such as x-intercepts, direction of parabola opening, axis of symmetry and vertex, to the values in the vertex form for quadratic functions. This can be done by asking them to connect features of the equation to its corresponding graph. For example, the equation

has the graph: 6

5 4 3 2 1 −1

−1

x 1

−2

Students may make connections such as: • The values of 1 and 3 in the equation match the magnitudes of the coordinates of the vertex on the graph. • The

in the equation matches the scale factor in the graph compared to y = x2.

• The y-value of the y-intercept, 5.5, is not one of the coeﬃcients or constants in the equation. Providing students with multiple examples to compare and connect can help students familiarize themselves with the key features of the factored form for quadratic function equations.

Clarify left and right translations Address student misconceptions Students may suggest that f (x) = (x + 5)2 is a horizontal translation of 5 units to the right. Challenge this misconception using a graphing tool to show that f (x) = (x + 5)2 has a vertex at (−5, 0), and so has been translated to the left from the parent function f (x) = x2. Alternatively, use a table of values to see what happens to the vertex and other key features.

Exploration Students: Page 409

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861

Suggested student grouping: Small groups Students explore on a GeoGebra applet which shows how changing a, h, and k in the vertex form of a quadratic equation will change its graph. Students are not formally introduced to vertex form y = a(x − h)2 + k, but will draw conclusions about the vertical stretches and compressions that a changes on the graph of the function, the horizontal translations of h, and the vertical translations of k. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here 1. What happens to the orange graph as the slider moves from a positive to a negative number? As the orange slider which changes the coefficient of x2 moves, we can see that positive coefficients will keep the graph of the function opening upward, while negative coefficients change the function to a downward-opening graph. In addition, ∣a∣ > 1 stretches and 0 < ∣a∣ < 1 compresses the graph vertically. 2. What happens to the blue graph as the slider moves from a positive to a negative number? As the blue slider changes, we can see that positive numbers in the slider are subtracted, and the graph has a horizontal shift to the right. Negative numbers are represented with an addition sign in the equation, and the graph has a horizontal translation to the left when the numbers on the slider are negative. 3. What happens to the red graph as the slider moves from a positive to a negative number? As the red slider moves, we can see that the graph vertically translates above the x-axis when the slider moves to positive numbers, and vertically translates the y-intercept below the x-axis when the slider moves to negative numbers. Purposeful questions • Does the slider change the location of the vertex? If yes, how are they connected? • Does the slider change the shape of the parabola? • Can you describe the effect of each slider in terms of translations, dilations or reﬂections? Possible misunderstandings • Since the vertex form of a quadratic function shows the value in parentheses being subtracted, students may be confused as they try to describe how the graph changes as the blue graph’s slider moves from a positive number to a negative number. Point out to students that when they move the slider physically up, the number for the equation is positive, even though it’s being subtracted. When we move the slider physically down, the number for the equation is negative, even though we are seeing addition in the equation itself. The sliders in the applet will all move to positive numbers at the top and negative numbers as we slide them down.

Mislabeling the signs of h and k in the vertex Address student misconceptions Students may incorrectly identify the sign of the coordinates of the vertex from an equation in vertex form y = a (x − h)2 + k as either (−h, k) or (h, −k). This is likely to happen if students are expecting the two variables of h and k to have the same sign, as with x1 and x2 in the factored form. Ask students to check if the vertex they have found is a point on the curve by substituting the coordinates back into the equation.

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Help visualize transformations with physical tools Student with disabilities support For students who have diﬃculty visualizing the transformations of the parent function x2, prepare physical manipulatives that can replicate these transformations. For example, graph different dilations of x2, such as 2x2 and

, onto sheets of clear plastic grid paper or

tracing paper. Provide a coordinate plane with the same size grid squares over which students can lay these sheets and move them around to visualize transformations. y

a=1 x

a = 0.5

y = x2 y 4

a=2

2 x

1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

y = 2x2

Coordinate plane

Explain how the manipulatives can represent different transformations: • Horizontal and vertical translation can be represented by moving the sheet up, down, left or right on the coordinate plane. • Reﬂection across an axis can be represented by physically ﬂipping the sheet across that axis. • Vertical stretch and compression can be represented by swapping out the current sheet for one with the correct scale factor without changing the position of the vertex. Additionally or alternatively, students can use a GeoGebra applet with sliders for a, h and k to visualize what aspect of the graph each variable affects. https://www.geogebra.org/m/Cav9fa27 Students are introduced to the vertex form of a quadratic function and review how the graph of a quadratic function in vertex form relates to its graph.

7.03 Quadratic functions in vertex form mathspace.co

863

Students: Pages 409–411

y 4

y = x2 + 2

3 2

y = x2

−4 −3 −2 −1 −1

This graph shows y = x2 translated vertically up by 2 to get y = x2 + 2, and down by 2 to get y = x2 − 2.

y = x2 − 2

−2

Similarly, a parabola can be horizontally translated by increasing or decreasing the x-values by a constant number. However, the x-value together with the translation must be squared together. That is, to translate y = x2 to the left by h units we get y = (x + h)2. 6 5

y y = x2

y= (x + 2)

3 2

This graph shows y = x2 translated horizontally left by 2 to get y = (x + 2)2 and right by 2 to get y = (x − 2)2.

y= (x − 2)

−4 −3 −2 −1 −1

−2

A parabola can be dilated by multiplying every y-value by a constant number greater than 1. So to expand the parabola y = x2 by a scale factor of a we get y = ax2. We can compress a parabola by using a scale factor between 0 and 1. 6

y = x2

4 y = 2x2 3

This graph shows y = x2 vertically expanded by a scale factor of 2

2 1 −4 −3 −2 −1

x 1

−1

to get y = 2x2 and compressed by a scale factor of 2 to get y =

−2

3 2

y = x2

1 −4 −3 −2 −1 −1

x 1

−2 −3

864

−4

y = −x2

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co 410

Mathspace Virginia SOL Algebra 1 mathspace.co

Finally, we can reflect a parabola across the x-axis by multiplying by −1. So to reflect y = x2 across the x-axis we get y = −x2. Notice that reflecting will change the parabola from opening up to opening down.

This graph shows y = x2 vertically expanded by a scale factor of 2

2 1 −4 −3 −2 −1

x 1

−1

to get y = 2x2 and compressed by a scale factor of 2 to get y =

−2

3 2

y = x2

−4 −3 −2 −1 −1

−2 −3

y = −x2

−4

The x-value of the vertex, h, represents the horizontal translation; the y-value, k, represents the vertical translation; 410 Mathspace Virginia SOL Algebra 1 and the leading coeﬃcient, a, represents the shape of the parabola and the direction it opens. The parent function of mathspace.co a quadratic is f (x) = x2, so writing these translations in function notation becomes af (x − h) + k = a(x − h)2 + k. As an example, consider the graph of y = (x − 2)2 − 3 • Translation of the parent function 3 units down and 2 units right 3 with a vertex at (2, −3) 2 • Axis of symmetry x = 2 (0, 1) The x-value of the vertex, translation; represents vertical translation; • y-intercept at the (0, 1)y-value, can be k, calculated by the substituting x = 0 into 1 h, represents the horizontal x the shape of the parabola and the direction it opens. The parent function of and the leading coeﬃcient, a, represents the function −4 is−3f (x) −2= −1 1 2 these 3 4 translations in function notation becomes af (x − h) + k = a(x − h)2 + k. a quadratic x2, so writing 4

−1

−2

As an example, consider the graph of y = (x − 2)2 − 3 • Translation of the parent function 3 units down and 2 units right 3 −4 with a vertex at (2, −3) 2 • Axis of symmetry x = 2 (0, 1) • y-intercept at (0, 1) can bethe calculated by substituting = 0 into 1 function from its graph, we When writing a quadratic can begin by identifying vertex and substituting xthese values x the function into the function for h and k. Then, we can use another point on the parabola, like the y-intercept, to help us find the 4 −3

−4 −3 −2 −1

value of a.

−1

−2

Example 1

−3 −4

Consider the following function: Examples When writing a quadratic function from its graph, we can begin by identifying the vertex and substituting these values m(x) = (x − 2)2 + 8 into the function for h and k. Then, we can use another point on the parabola, like the y-intercept, to help us find the Students: Page 411 value of the a. vertex. a Find

Create a strategy Example 1

Apply the idea

The function is given in vertex form y = a (x − h)2 + k Consider following where thethe vertex is the function: point (h, k). m(x) = a Find the vertex.

In the equation m(x) = (x − 2)2 + 8 the x-coordinate of vertex is h = 2 and the y-coordinate of vertex is k = 8. (xWith − 2)2h += 82 and k = 8, the ordered pair of the vertex is at (2, 8).

Create strategy b Stateathe domain.

Apply the idea

The function is given in vertex form y = a (x − h)2 + k where vertex is the point (h, k). Createthe a strategy

In the equation m(x) = (x − 2)2 + 8 the x-coordinate of Apply isthe vertex h =idea 2 and the y-coordinate of vertex is k = 8.

To ﬁnd the domain of m(x), we want to ﬁnd all possible x-values for which m(x) could be graphed.

We know parabola, therepair areof nothe restrictions With h = 2that andfor k =a 8, the ordered vertex is at on which x-values can be graphed as each side of the (2, 8). parabola continues inﬁnitely in either x direction. Domain: {x∣ −∞ < x < ∞}

b State the domain.

Purposec State the range. Create a strategy Apply the idea See if students can identify the x- and y-coordinates of the vertex from an equation.

To find the domain of m(x), we want to find all possible We know that for a parabola, there are no restrictions Create strategy Apply the idea can be graphed as each side of the x-valuesafor which m(x) could be graphed. on which x-values Expected mistakes To find the range, we want to find all possible values of This parabola opensinfinitely down, so y-value of the vertex parabola continues in the either x direction. Studentsm(x). may say the of vertex is (−2, 8). Help students connections back to the function’s translations to The vertex a parabola affects the range of the make is the maximum value of the function. The parabola Domain: {x∣ −∞ < x < ∞} 2 understand the that (x − 2) moves the function to the right, meaning the x-value of the vertex function, as it will be the maximum or minimum value of continues infinitely in the negative y direction.will be positive. m(x). Range: {y∣ y ≤ 8} c State the range.

7.03 Quadratic functions in vertex form mathspace.co

Create a strategy

Apply the idea 7.03 Quadratic functions in vertex form

To ﬁnd the range, we want to ﬁnd all possible values of

This parabola opens down, so the y-value of the vertex

mathspace.co

411

865

Create a strategy

Apply the idea

The function is given in vertex form y = a (x − h)2 + k where thethe vertex is the function: point (h, k). Consider following

In the equation m(x) = (x − 2)2 + 8 the x-coordinate of vertex is h = 2 and the y-coordinate of vertex is k = 8.

Example 1

m(x) =

Students: Page 411

a Find the vertex. b Stateathe domain. Create strategy

(xWith − 2)2h += 82 and k = 8, the ordered pair of the vertex is at (2, 8).

Apply the idea 2

The function is given in vertex form y = a (x − h) + k Createthe a strategy where vertex is the point (h, k). To ﬁnd the domain of m(x), we want to ﬁnd all possible x-values for which m(x) could be graphed.

In the equation m(x) = (x − 2)2 + 8 the x-coordinate of Apply the idea vertex is h = 2 and the y-coordinate of vertex is k = 8. We know that for a parabola, there are no restrictions With h = 2 and k = 8, the ordered pair of the vertex is at on which x-values can be graphed as each side of the (2, 8). parabola continues inﬁnitely in either x direction. Domain: {x∣ −∞ < x < ∞}

b State the domain. c State the range.

Create a strategy

Apply the idea

To find the domain of m(x), we want to find all possible We know that for a parabola, there are no restrictions PurposeCreate a strategy Apply the idea x-valuesunderstanding for which m(x) could be graphed. onfunction. which x-values can be graphed as each side of the See students’ of the domain of a quadratic To find the range, we want to find all possible values of This parabola opens down, so the y-value of the vertex m(x). The vertex of a parabola affects the range of the Students:function, Pageas411 it will be the maximum or minimum value of m(x).

parabola continues inﬁnitely in either x direction. is the maximum value of the function. The parabola Domain: {x∣ −∞ < x < ∞} continues inﬁnitely in the negative y direction. Range: {y∣ y ≤ 8}

c State the range. in vertex form Apply the idea 7.03 Quadratic functions mathspace.co

Create a strategy To ﬁnd the range, we want to ﬁnd all possible values of m(x). The vertex of a parabola affects the range of the function, as it will be the maximum or minimum value of m(x).

411

This parabola opens down, so the y-value of the vertex is the maximum value of the function. The parabola continues inﬁnitely in the negative y direction. Range: {y∣ y ≤ 8}

Purpose Check if students are able to identify the range given an equation.

7.03 Quadratic functions in vertex form mathspace.co

411

Reflecting with students Students may not consider the sign of the leading coefficient and assume the vertex is a minimum, leading them to reverse the direction of the inequality. Remind them that when the leading coefficient, or the a-value is negative, the parabola is reflected across the x-axis.

Students: Pages 412–413 d Draw the graph of the function.

Create a strategy The scale factor is

which is negative, so the graph will open down. We can draw the graph through any three

points that we know are on it.

Apply the idea 8

Start by plotting the vertex, found in part (a), on the graph.

vertex

6 4 2

−8 −6 −4 −2 −2

−4 −6 −8

We can solve for the x-intercepts by substituting m(x) = 0 and solving for x.

vertex

866

4 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co 2 x

−8 −6 −4 −2 −2

x −8 −6 −4 −2 −2

−4 −6 −8

We can solve for the x-intercepts by substituting m(x) = 0 and solving for x.

vertex

6 4 2

−8 −6 −4 −2 −2

−4 −6

x = 2 + 4 and x = 2 − 4

−8

x = 6 and x = −2 We can solve for the y-intercept by substituting x = 0 and solving for m(x).

6 4 2

−8 −6 −4 −2 −2

−4 −6 −8

Then, we can get an additional point by reﬂecting the y-intercept across the axis of symmetry.

6 4 2

−8 −6 −4 −2 −2

−4 −6 −8 412

Mathspace Virginia SOL Algebra 1 mathspace.co

To ﬁnish the drawing of the graph of the function, connect the points plotted to draw the parabola.

6 4 2 −8 −6 −4 −2 −2

x 2

−4 −6 −8

Reﬂect and check Technically we only need 3 points to graph a parabola, but ﬁnding more points can make our graph more precise.

Purpose Example 2 Check if students can draw the graph of a function from its equation. The table of values represents a quadratic function.

−4

−3

−2

−1

p(x) −5 0 3 4 3 0 −5 Expected mistakes a Write the function p(x) in vertex form. Students may make the mistake of thinking the y-intercept is at (0, 8). Have students substitute x = 0 into the equation to verifya the y-intercept. Point out that y = 8 is the y-value of the vertex, not the y-intercept. Create strategy We can use the fact that a quadratic function has symmetry about its vertex to identify the location of the vertex from the table.

Apply the idea

7.03 Quadratic functions in vertex form Looking at the values of p(x), we can see that it has a maximum value of 4 and falls off symmetrically on either side. mathspace.co So, we know that the vertex is the point (−1, 4), so we can use that to set up the equation p(x) = a(x + 1)2 + 4

867

2 −8 −6 −4 −2 −2

x 2

−4 −6

Reflecting with students−8 To graph a function, it may be easier for students to find a few points that lie on the graph. They could do this with theReﬂect intercepts or by substituting any value of x into the equation. and check Technically we only need 3 points to graph a parabola, but ﬁnding more points can make our graph more precise.

Students: Page 413 Example 2

The table of values represents a quadratic function.

x p(x)

a Write the function p(x) in vertex form.

−4 −5

−3 0

−2 3

−1 4

0 3

1 0

2 −5

Create a strategy We can use the fact that a quadratic function has symmetry about its vertex to identify the location of the vertex from the table.

Apply the idea Looking at the values of p(x), we can see that it has a maximum value of 4 and falls off symmetrically on either side. So, we know that the vertex is the point (−1, 4), so we can use that to set up the equation p(x) = a(x + 1)2 + 4 We can now ﬁnd the value of a by substituting any other pair of values from the table, such as (0, 3). Doing so, we get 3 = a(0 + 1)2 + 4 which we can solve to get a = −1. So, the quadratic function shown in the table of values is p(x) = −(x + 1)2 + 4

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Purpose Show students how to use a table of values to write the equation of a quadratic function in vertex form. Expected mistakes Students may write the equation f (x) = (x + 1)2 + 4 without considering the leading coefficient. Ask them to describe how the y-values change as x increasing. This should help them realize that the parabola opens downward. Reflecting with students Ask students if using a table is enough information for them to write the equation of the function in vertex form or if they prefer to also have a graph of the function. Allowing students to be resourceful in their problem solving will give them confidence to find solutions using their preferred methods.

Students: Page 414 b Determine the value of p(x) when x = 6.

Create a strategy We can substitute x = 6 into the equation that was created in part (a) in order to predict what p(x) will be equal to.

Apply the idea The equation found in part (a) is p(x) = −(x + 1)2 + 4

Original equation

p(6) = −(6 + 1)2 + 4

Substitute x = 6

= −(7)2 + 4

Evaluate the addition

= −49 + 4

Evaluate the exponent

= −45

Evaluate the addition

This means that when x = 6, p(x) is equal to −45.

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Mathspace Virginia Algebra 1 Teacher Edition c Determine theSOL x- and y-intercepts of the function. mathspace.co

Create a strategy

Apply the idea

Use the table of values to identify the x-intercepts when

The x-intercepts occur at (−3, 0) and (1, 0).

b Determine the value of p(x) when x = 6.

Create a strategy We can substitute x = 6 into the equation that was created in part (a) in order to predict what p(x) will be equal to.

Apply the idea The equation found in part (a) is p(x) = −(x + 1)2 + 4

Original equation

p(6) = −(6 + 1)2 + 4

Substitute x = 6

= −(7)2 + 4

Evaluate the addition

= −49 + 4

Evaluate the exponent

= −45

Evaluate the addition

This means that when x = 6, p(x) is equal to −45.

c Determine the x- and y-intercepts of the function.

Purpose b Determine the valuethe of p(x) whenofx a = 6. Create ahow strategy Apply the Show students to find value function in vertex form atidea a given value of x. Use the table of values to identify the x-intercepts when

The x-intercepts occur at (−3, 0) and (1, 0).

Create strategy Reflecting students y = 0with anda the y-intercept when x = 0. The y-intercept occurs at (0, 3). We canhow substitute x =the 6 into the equation that wasx created in part (a) in only order using to predict p(x)ofwill be equal to. Ask students to find value of p(x) when = 6 if they were thewhat table values. Students should be able to follow a pattern in order to find the unknown value of the function without the equation. Apply the idea Example 3 The equation found xin part −4 (a) is −3

−2

−1

The quadratic = 2x has been3transformed new −12 + 4−5 Original equation p(x) function = −(xp(x) + 1)2f (x) 0 4 3to produce 0 a −5 quadratic function g(x), as 2 shown in the graph: Substitute x = 6 p(6) = −(6 + 1) + 4

−21

−32

−45

= −(7) + 4

Evaluate the addition −1 −3

= −49 + 4

Evaluate the exponent

= −45

Evaluate the addition

−5

−7

−9

−11

y 8

f (x)

−13

Students should be able to see that each value subtracts the next odd integer to find the value. g(x)

means that when x = 6, p(x) is equal to −45. Students:This Page 414

x −8

−6

−4

−2 −2

c Determine the x- and y-intercepts of the function.

Create a strategy a Describe the transformation from f (x) to g(x). Use the table of values to identify the x-intercepts when y = 0 and y-intercept when x = 0. Apply thethe idea

Apply the idea The x-intercepts occur at (−3, 0) and (1, 0).

Reﬂect and check The y-intercept occurs at (0, 3).

The function g(x) has the same shape and size as f (x), but We could conﬁrm that g(x) is a horizontal shift left by has been shifted to the left. Comparing the vertices of evaluating f (x + 6) = 2(x + 6)2: the two parabolas, we can see that this is a translation of g(−8) = 8 and f (−8 + 6) = 2(−8 + 6)2 = 2(−2)2 = 2(4) = 8 Example 3 Purpose6 units to the left. g(−6) = 0 and f (−6 + 6) = 2(−6 + 6)2 = 2(0)2 = 2(0) y= 0 The quadratic f (x) = 2x2the hasintercepts been transformed to produce a new table form. Show students howfunction to determine of a function 8 =8 g(−4) given = 8 andin f (−4 + 6) = 2(−4 + 6)2 = 2(2)2 = 2(4) f (x) quadratic function g(x), as shown in the graph: 6 Reflecting with students Ask students to explain how to find the intercepts from the graph or equation. We know that from the graph 4 or equation, the y-intercept is the point where x = 0, and the x-intercept is the point where y = 0. The equation g(x) 2 would require some algebraic working, while a graph would require some analyzation. x

414

Mathspace Virginia SOL Algebra 1 mathspace.co

−8

Plot the points from the table to help visualize key features Student with disabilities support

−6

−4

−2

use with Example 2

−2

Students may have difficulty determining the shape of the relationship from a table of values, as this requires a Describe the transformation from f (x) to g(x). the awareness of a lot of different numeric values at once. Support students by advising them to plot the points on a coordinate and help identify key features of the relationship. Apply theplane idea to see the shape of the relationship Reflect and check The function g(x) has the same shape and size as f (x), but We could confirm that g(x) is a horizontal shift left by has been shifted to the left. Comparing the vertices of evaluating f (x + 6) = 2(x + 6)2: the two parabolas, we can see that this is a translation of g(−8) = 8 and f (−8 + 6) = 2(−8 + 6)2 = 2(−2)2 = 2(4) = 8 6 units to the left. g(−6) = 0 and f (−6 + 6) = 2(−6 + 6)2 = 2(0)2 = 2(0) = 0 g(−4) = 8 and f (−4 + 6) = 2(−4 + 6)2 = 2(2)2 = 2(4) = 8

7.03 Quadratic functions in vertex form mathspace.co

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Mathspace

Virginia SOL Algebra 1

869

c Determine the x- and y-intercepts of the function.

Create a strategy

Apply the idea

Use the table of values to identify the x-intercepts when y = 0 and the y-intercept when x = 0.

The x-intercepts occur at (−3, 0) and (1, 0).

Students: Page 414

The y-intercept occurs at (0, 3).

Example 3 y

The quadratic function f (x) = 2x2 has been transformed to produce a new quadratic function g(x), as shown in the graph:

f (x)

6 4 g(x)

2 x

−8

−6

−4

−2

2 −2

a Describe the transformation from f (x) to g(x).

Apply the idea

Reﬂect and check

Purpose Show students how to describe a transformation of a quadratic function given its graph. 414

Mathspace

Virginia SOL Algebra 1

Reflecting with students mathspace.co Ask students how they can explain that the graph has not had a stretch or compression. Remind students that just eyeballing and explaining that they do not see a stretch or compression is not enough evidence. Students may argue that other points on the original graph have direct translations to the translated graph.

Students: Page 415 b Write the equation of the function g(x) in vertex form.

Create a strategy

Apply the idea

Remember that vertex form for a quadratic is g(x) = a(x − h)2 + k, where the vertex is at the point (h, k).

f (x) = 2x2 has been translated 6 units to the left to produce g(x), and we can see that its vertex is at (−6, 0). So, g(x) has can be written as g(x) = 2(x + 6)2.

c Describe the transformations of the graph of f (x) resulting in the function h(x) = −2(x + 6)2 + 3.

PurposeCreate a strategy Apply the idea Show students toupward write the equation vertex form given an downward equationand andbeits translation. Since f (x)how opens and has a vertexof at a(0,function 0), the in The function h(x) will open a reﬂection of vertex form of h(x) gives information about its vertex and

f (x) across the x-axis, since the value of a became negative.

Expected mistakes direction of its opening. The vertex of h(x) is (−6, 3). The vertex became a Students may incorrectly write the function as g(x) = (2x +maximum 6)2, placing that thetovertical valuethe and scale shiftedfactor the graph upshows 3 units and thePoint left 6 out units.to students that the scale factor should be stretch from the parent function inside of the parentheses. placed outside of the set of parentheses when the quadratic function is written in vertex form. d Sketch the graph of h(x).

Create a strategy Use the description of the transformations of g(x) from part (c) and graph of g(x) to sketch h(x).

870

Apply the idea

Mathspace Virginia SOL Algebra 1 Teacher Edition Start by reﬂecting g(x) across the x-axis using points from its graph. mathspace.co g(x)

y 8

Perform a vertical shift 3 units up. g(x)

y 8

b Write the equation of the function g(x) in vertex form.

Create a strategy

Apply the idea

Remember that vertex form for a quadratic is g(x) = a(x − h)2 + k, where the vertex is at the point (h, k).

f (x) = 2x2 has been translated 6 units to the left to produce g(x), and we can see that its vertex is at (−6, 0). So, g(x) has can be written as g(x) = 2(x + 6)2.

Students: Page 415

c Describe the transformations of the graph of f (x) resulting in the function h(x) = −2(x + 6)2 + 3.

Create a strategy

Apply the idea

Since f (x) opens upward and has a vertex at (0, 0), the vertex form of h(x) gives information about its vertex and direction of its opening.

The function h(x) will open downward and be a reﬂection of f (x) across the x-axis, since the value of a became negative. The vertex of h(x) is (−6, 3). The vertex became a maximum value and shifted the graph up 3 units and to the left 6 units.

b Write the equation of the function g(x) in vertex form.

Create a strategy

Apply the idea

d Sketch the h(x).for a quadratic is Remember thatgraph vertexofform g(x) = a(x − h)2 + k, where the vertex is at the point (h, k).

f (x) = 2x2 has been translated 6 units to the left to produce g(x), and we can see that its vertex is at (−6, 0). So, g(x) has can be written as g(x) = 2(x + 6)2.

PurposeCreate a strategy Show students how to describe the transformation ofpart a function given its to transformed Use the description of the transformations of g(x) from (c) and graph of g(x) sketch h(x). function written in vertex form. c Describe the transformations of the graph of f (x) resulting in the function h(x) = −2(x + 6)2 + 3. Apply the idea

Expected mistakes Create a strategy Start by reﬂecting g(x) across the x-axis using points from itsApply graph.the idea Perform a vertical shift 3 units up. StudentsSince mayf (x) forget that changing the sign of the scale factor of a function written in vertex form will reflect the opens upward and has a vertex at (0, 0), the The function h(x) will open downward and be ya reﬂection of y g(x) for quadratic functions functionvertex vertically. Remind students that the scale factor works in the samesince waythe it value works g(x)about8its vertex and f (x) across form of h(x) gives information the x-axis, of a became negative. 8 direction of itsform. opening. written in factored The vertex of h(x) is (−6, 3). The vertex became a 6 6 maximum value and shifted the graph up 43 units and to the left 6 units.

Students: Page 415

2 −8

−6

d Sketch the graph of h(x).

−4

−2

x −8

−2

−6

−4

Create a strategy

−4

−2

x 2

−2 −4

h(x)

−6

Use the description of the transformations−8of g(x) from part (c) and graph of g(x) to sketch h(x).

−8

Apply the idea

The graph of h(x) follows: Start by reﬂecting g(x) across the x-axis using points from its graph. y

Perform a vertical shift 3 units up.

g(x)

6 2 −8 −6

−4

−2

−6 x

−4

−2

2 −8

−6

−4

−6

−4 h(x)

−8

−6

x 2

−4

h(x)

y 8 6

−8

g(x)

−2

x 2

−4 −6

−8

The graph of h(x) follows: y

7.03 Quadratic functions in vertex form mathspace.co

415

7.03 Quadratic functions in vertex form mathspace.co

415

8 6 4 2 −8

−6

−4

h(x)

−2

x 2

−4 −6 −8

7.03 Quadratic functions in vertex form mathspace.co

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Purpose Show students how to graph a transformation of a function written in vertex form. Develop decomposition skills by performing one transformation at a time. Expected mistakes Students may sketch graphs that are inaccurate by attempting to eyeball the reflection and translation. Encourage students to mark at least three points on the graph of h(x) that they can transform and keep the shape of the graph. The more points they can follow, the more accurate their graph will be when it gets to its final location on the coordinate plane. Reflecting with students Make students aware that the order of transformations matters. If we translate and then reflect, we would get a different graph. g(x)

6 2 −8

−6

−4

−2

g(x)

4 2

x −8

−6

−4

−2

−4

−6

−8

x 2

Then, reflect across the x-axis.

First, perform shift 3 units up.

Notice this is not the same graph from the original order of transformations.

Students: Page 416 Example 4 Meri throws a rock into Crescent Lake. The height of the rock above ground is a quadratic function of time. The rock is thrown from 4.4 ft above ground. After 1.5 seconds, the rock reaches its maximum height of 24 ft. Write the quadratic equation in vertex form.

Create a strategy The maximum height of the rock at 24 feet indicates that this is where the vertex of the function is located. This occurs at 1.5 seconds. Substitute the vertex of the graph and the y-intercept into the vertex form of a quadratic function to determine the equation.

Apply the idea The vertex is located at (1.5, 24), so we can substitute h = 1.5 and k = 24. The rock is thrown from 4.4 feet, so the y-intercept is located at (0, 4.4) and we can substitute x = 0 and y = 4.4. y = a(x − h)2 + k 2

4.4 = a(0 − 1.5) + 24

Vertex form of a quadratic function Substitute x = 0, y = 4.4, h = 1.5, and k = 24

−19.6 = a(−1.5)2

Subtract 24 from both sides

−19.6 = a(2.25)

Evaluate the exponent

−8.7 = a

Divide by 2.25 on both sides

The quadratic equation in vertex form that models the rock’s height above the ground as a function of time is y = −8.7(x − 1.5)2 + 24.

Idea summary The vertex form of a quadratic function is:

f (x) = a(x − h)2 + k 872

Mathspace Virginia SOL Algebra 1 Teacher Edition (h, k) coordinates of the vertex mathspace.co a scale factor Changing the values of a, h, and k will transform the graph in different ways: •

a: vertical stretch or compression

Example 4 Meri throws a rock into Crescent Lake. The height of the rock above ground is a quadratic function of time. The rock is thrown from 4.4 ft above ground. After 1.5 seconds, the rock reaches its maximum height of 24 ft. Write the Purposequadratic equation in vertex form.

Show students how to write a quadratic function in vertex form based on a contextual situation. Create a strategy

Expected Themistakes maximum height of the rock at 24 feet indicates that this is where the vertex of the function is located. This y of a quadratic occurs 1.5know seconds. Substitute the vertex of values the graph y-intercept intointo the vertex form Students may atnot how to substitute the ofand thethe given context function to determine the equation. an equation without visualizing the graph. Allow students to sketch the path 25 1.5s, 24 ft Example 4 of the rock on a coordinate grid and label its key features before attempting Apply the idea 20 to write its equation. The vertex at Crescent (1.5, 24), so we The can substitute h =rock 1.5 and k = ground 24. Meri throwsisalocated rock into Lake. height of the above is a quadratic function of time. The rock is thrown from 4.4 ftfrom above 1.5 seconds, the rockatreaches its maximum height of x24 Write the The rock is thrown 4.4ground. feet, so After the y-intercept is located (0, 4.4) and we can substitute = ft. 0 and y= 4.4. 15 quadratic equation in vertex form. 2 Vertex form of a quadratic function y = a(x − h) + k 4.4 = a(0 − 1.5)2 + 24

Create a strategy

Substitute x = 0, y = 4.4, h = 1.5, and k = 24

Subtract 24 from both sides −19.6 = a(−1.5) 5 4.4isftlocated. This The maximum height of the rock at 24 feet indicates that this is where the vertex of the function −19.6 = a(2.25) Evaluate the exponent occurs at 1.5 seconds. Substitute the vertex of the graph and the y-intercept into the vertex form of a quadratic = a the equation. Divide by 2.25 on both sides function to −8.7 determine The quadratic equation in vertex form that models the rock’s height above the ground as a function 1of time2is Students: Page 4162 y = −8.7(x 1.5) + 24. Apply the− idea

x 3

The vertex is located at (1.5, 24), so we can substitute h = 1.5 and k = 24. The rock is thrown from 4.4 feet, so the y-intercept is located at (0, 4.4) and we can substitute x = 0 and y = 4.4. = a(x − h)2 + k Idea ysummary

Vertex form of a quadratic function

+ 24 Substitute 4.4 = a(0 − 1.5) The vertex form of a quadratic function is: x = 0, y = 4.4, h = 1.5, and k = 24 Subtract 24 from both sides 2

−19.6 = a(−1.5)2

f (x) = a(x − h) + k

−19.6 = a(2.25)

Evaluate the exponent (h, k) coordinates of the vertex −8.7 = a Divide by 2.25 on both sides a scale factor The quadratic equation in vertex form that models the rock’s height above the ground as a function of time is Changing the values of a, h, and k will transform the graph in different ways: y = −8.7(x − 1.5)2 + 24. • a: vertical stretch or compression • h: horizontal translation • k: vertical translation

Idea summary The vertex form of a quadratic function is:

f (x) = a(x − h)2 + k

Completing the square (h, k)

coordinates of the vertex

CompletingInteractive the square exploration a scale factor

Explore online to answer the question Changing the values of a, h, and k will transform the graph in different ways:

• a: vertical stretch or compression Exploration mathspace.co • h: horizontal translation •

k: vertical translation

Students: Page Use the416 interactive exploration in 7.03 to answer the question. 1.

What patterns do you notice when working through the process of completing the square?

Completing the square 416

Interactive exploration

Mathspace Virginia SOL Algebra 1 Explore online to answer the question mathspace.co

mathspace.co Use the interactive exploration in 7.03 to answer the question. 1.

416

What patterns do you notice when working through the process of completing the square?

Mathspace Virginia SOL Algebra 1 mathspace.co

7.03 Quadratic functions in vertex form mathspace.co

873

Suggested student grouping: Individual Students explore an applet where they create an incomplete square, and then use a perfect square trinomial to ‘complete’ the square. After creating several incomplete squares and analyzing the process by which the GeoGebra completes the square, students should notice that by creating a perfect square trinomial and adding or subtracting the missing number of pieces from the square, they can write a polynomial in vertex form that represents the incomplete square. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What patterns do you notice when working through the process of completing the square? The number added or subtracted to the factored form of the perfect square trinomial is equivalent to the number of pieces of the square in surplus or missing. Each time we complete the square, the constant of the perfect square trinomial is both added and subtracted from the polynomial. Purposeful questions • How many squares would complete the red square? How many more or less squares are needed to complete the red square? • Where does the quadratic term on the left side of the equation come from? Where does the linear term on the left side of the equation come from? Where does the constant term on the left side of the equation come from? • What is happening when we check the box to complete the square? Possible misunderstandings • Students may not see any pattern in completing the square with the applet. Walking through an example with the entire class, and annotating the applet and the polynomials will help students see where the visual model ﬁts with the algebraic representation. Students may be able to predict how to come up with the perfect square trinomial and added or subtracted constant themselves after engaging with the applet and following a pattern a few times with the class or one-on-one. Students learn the process of completing the square in order to rewrite a quadratic function in vertex form.

Students: Page 417 Completing the square is a method we use to rewrite a standard quadratic expression in vertex form. Completing the square allows us to rewrite our equation so that it contains a perfect square trinomial. A perfect square trinomial takes on the form A2 + 2AB + B2 = ( A + B)2, which is the same format we need to have an equation in vertex form. For quadratic equations where a = 1, we can write them in perfect square form by following these steps: 1 2

Rewrite the x term

Add and subtract

Factor the perfect square trinomial

Match the completed square to vertex form

to keep the equation balanced

If a ≠ 1, we can ﬁrst divide through by a to factor it out. The quadratic equation, when rewritten by completing the square, becomes the vertex form of a quadratic equation.

Example 5 Consider the following equation:

874

y = x2 − 4x + 6 Mathspace Virginia SOL Algebra 1 Teacher Edition a Rewrite the equation in vertex form by completing the square. mathspace.co

Create a strategy We’ll follow the standard complete the square method and stop working once our equation is in vertex form,

Factor the perfect square trinomial

Match the completed square to vertex form

a ≠ 1, we417 can ﬁrst divide through by a to factor it out. Students:If Page The quadratic equation, when rewritten by completing the square, becomes the vertex form of a quadratic equation.

Example 5 Consider the following equation:

y = x2 − 4x + 6

a Rewrite the equation in vertex form by completing the square.

Create a strategy We’ll follow the standard complete the square method and stop working once our equation is in vertex form, y = a(x − h)2 + k.

Apply the idea Original equation Add and subtract Evaluate the exponents Factor x2 − 4x + 4 Evaluate the subtraction We’ve completed the square and the equation is now in vertex form.

Reﬂect and check We must add and subtract

to the right side of the equation so that the value of the equation does not change.

Purpose Show students how to complete the square in order to rewrite a quadratic function given in standard form as vertex form. Expected mistakes Students may add the value of

7.03to Quadratic functions vertex form to one side of the equation, but forget subtract it asinwell.

417

mathspace.co

Reflecting with students Challenge advanced learners to rewrite the equation y = 2x2 − 4x + 6 in vertex form by completing the square. They will need to factor the leading coefficient out first, which gives y = 2(x2 − 2x + 3). When they complete the square, they will need to multiply the constant term by 2 when moving it outside of the parentheses. They should end up with the equation y = 2(x − 1)2 + 4.

Students: Page 418 b Determine the vertex of the quadratic function and if it is a minimum or maximum.

Create a strategy

Apply the idea

Use the vertex form of the quadratic function from part (a) to determine its vertex and whether it is a minimum or maximum value.

The quadratic function in vertex form is y = (x − 2)2 + 2, meaning the vertex is located at the point (2, 2). Since the value of a is 1, we know that the graph opens upward and the vertex is a minimum value.

c Sketch the graph of the parabola.

Create a strategy We can ﬁnd key points of our parabola to sketch it. In part (a) we found the x-value of the vertex, and in part (b) we found the vertex form of our equation which shows us the y-value of our vertex. We can also use the vertex form to consider the x- and y-intercepts. 7.03 Quadratic functions in vertex form mathspace.co

Apply the idea

The y-value of the vertex is 2 since the vertex form of the equation is y = (x − 2)2 + 2. This means that the vertex is located at (2, 2). We know the scale factor a is 1, so the parabola opens upward. Since the vertex is above the x-axis

875

Purpose Show students how the to identify thequadratic vertex in vertexand form and determine the direction in which the graph opens. b Determine vertex of the function if it is a minimum or maximum. Expected mistakes Create a strategy Apply the idea 2 StudentsUse may howoftothedetermine if the vertex is a minimum or maximum value.form Emphasize how theforget vertex form quadratic function from part The quadratic function in vertex is y = (x − 2) + 2,the leading coefficient this key and feature of the (a) tohighlights determine its vertex whether it is agraph. minimum or meaning the vertex is located at the point (2, 2). maximum value.

Since the value of a is 1, we know that the graph opens upward and the vertex is a minimum value.

Students: Page 418 c Sketch the graph of the parabola.

Apply the idea The y-value of the vertex is 2 since the vertex form of the equation is y = (x − 2)2 + 2. This means that the vertex is located at (2, 2). We know the scale factor a is 1, so the parabola opens upward. Since the vertex is above the x-axis and opens up, the graph will not cross the x-axis and there are no x-intercepts. Find the y-intercept: y = (x − 2)2 + 2

Vertex form of the equation

y = (0 − 2)2 + 2

Substitute x = 0

y=6

Evaluate

Therefore, the y-intercept is (0, 6).

We can use the vertex and y-intercept to sketch the parabola, remembering there is an axis of symmetry at the vertex.

7 5 4 3 2 1

x 1

Idea summary Purpose A quadratic function in standard form can be converted to vertex form by completing the square: Show students1 how to graph a quadratic function from vertex form. 2 Rewrite the x term Expected mistakes Students may attempt to use only the vertex to sketch a parabolic graph. Remind students that we can ﬁnd 3 Add and to keep the equation balanced other key features of the function and plot them. Forsubtract this example, having two points was enough but since we know that parabolas are symmetric, we can identify that the point (4, 6) is a third point that we can use to help Factor the perfect square trinomial us sketch the 4function. 5

Match the completed square to vertex form

Reﬂecting with students ≠ 1,can we can divide by athat to factor it out. Ask students ifIf awe tell ﬁrst from the through equation there are no x-intercepts and therefore no need to substitute y = 0 and attempt to solve. Since the equation indicates that the graph opens upwards and has been translated vertically418upwards there willSOL be Algebra no x-intercepts. Mathspace Virginia 1 mathspace.co

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Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

y = (x − 2) + 2 Vertex form of the equation The value of k 2is not always the y-coordinate of the y-intercept Substitute x = 0 y = (0 − 2) + 2 Address student misconceptions y=6

Evaluate

use with Example 5

StudentsTherefore, may identify +2 as a constant term in the equation and incorrectly equate this with being the y-value 8 the y-intercept is (0, 6). 7found by substituting x = 0 into the of the y-intercept. Remind students that the y-value of the y-intercept can be We can use the vertex and y-intercept to sketch the parabola, remembering 6 equation. there is an axis of symmetry at the vertex. 5

Point out to students that in the vertex form of a quadratic function, there is another constant term inside the 4 parentheses which we need to take into account. 3 2 1

Students: Page 418

x 1

Idea summary A quadratic function in standard form can be converted to vertex form by completing the square: 1 2

Rewrite the x term

Add and subtract

Factor the perfect square trinomial

Match the completed square to vertex form

to keep the equation balanced

If a ≠ 1, we can ﬁrst divide through by a to factor it out.

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Practice Students: Pages 419–424

What do you remember? 1

Determine the axis of symmetry of this quadratic function. 8

6 4 2 −6 −4 −2 −2

x 2

−4 −6 −8

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877

Write an equation for each quadratic function in vertex form. a

−2 −1 −2

12 10

2 1

−12

x h(x) 0 −6 1 −1 2 2 3 3 4 2 5 −1 6 −6

−14

j(x) 11 6 3 2 3 6 11

−6 −5 −4 −3 −2 −1 0

For the quadratic equations: i

Rewrite the equation in vertex form by completing the square.

Identify the coordinates of the vertex.

y = x2 − 4x + 3

y = −x2 + 6x − 2

y = −x2 − 5x – 4

y = −2x2 + 8x − 7

y = x2 is horizontally translated 10 units to the right and vertically translated 2 units up.

y = x2 is horizontally translated 9 units to the left and is vertically stretched by a factor of 9 units.

y = x2 is reﬂected across the x-axis and vertically translated 8 units down.

For each of the following, state whether the transformation from the parent function f (x) = x2 to the function g(x) is a translation up, down, left, or right: g(x) = (x − 8)2

g(x) = x2 − 5

g(x) = (x + 9)2

Consider the graph of a function. y 5 x −12 −10 −8 −6 −4 −2

2 −5

−10 −15 −20 −25

878

Write the equation of the transformed graph in vertex form.

a 6

−10

−8 −7 −6 −5 −4 −3 −2 −1 −2

2 3 4 5 6 7 8

−8

−6

−4

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

g(x) = x2 + 0.75

Select the equation that represents the function:

y = −(x + 5)2 + 25

y = (x − 5)2 + 25

y = (x + 5)2 − 25

y = −(x + 5)2 − 25

Consider the table of values of a function. x y

1 −6

2 0

3 2

4 0

5 −6

Sketch a graph of the function labeling any points of interest. 8

Consider the function Sketch a graph of the function labeling any points of interest.

Let’s practice 9

For each of the graphs: i

Describe the transformation from y = x2.

Write the equation. y 8

6 6

2 −2

x 2

−2 −4

−4

−6 −4 −2

−2

x −6

−4

−2

7.03 Quadratic functions in vertex form mathspace.co

879

−6 −4 −2

−2 −4

y 6 4 2 x −2

x 2

−2

−4

For each equation: i

Describe the transformations from y = x2.

Sketch the graph of the parabola.

y = (x − 3)2 − 4

y = 3(x + 1)2 + 4

y = −(x + 3)2 − 6

y = −0.5(x − 6)2 + 1

Each of the following describes the transformations of a function from the parent function f (x) = x2. Write the equation of each transformed function in vertex form. a

Vertical stretch by a scale factor 2 and translated right 1 unit and down 3 units.

Reflection over the x-axis, vertically compressed by a scale factor of 3, and translated right 2 units and up 2 units.

Reflection over the x-axis, vertically stretched by a scale factor of 3, and translated left 5 units and down 4 units.

Vertically stretched by a scale factor of 3, translated left 1 unit, and translated up 2 units.

Sketch the graph of each parabola. a

14 12 10 8 6 4 2

y = (x + 3)2 − 9

y = −(x − 2)2 + 1

y = −(x + 4)2 − 2

Write the vertex form equation that represents each of the graph: a

4 2 −2

−2

x 2

−2

−4

−6

−8

2 −2

y 2

x −6 −4 −2

880

−2

−4

−6

−8

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

For each of the following: i a

Rewrite the equation in vertex form. 2

y = x − 4x + 2

y = x + 10x

Sketch the graph of the parabola.

y = x2 + 4x – 4

y = x2 + 5x + 3

Explain how the maximum or minimum value of a quadratic function can be found by completing the square.

For each of the quadratic function: i

State whether the transformation from f (x) to g(x) is a horizontal translation, vertical translation, vertical stretch, or vertical compression by k units.

State the value of k.

10 g ( x ) 8

10 8

4 2

g(x) −10 −8 −6 −4 −2 −2 −4 −6

x f ( x) 2 4 6 8 10

−10 −8 −6 −4 −2 −2 −4 −6 −8

10 8 6 4 2 −10 −8 −6 −4 −2 −2 −4 −6

f ( x) x 2 4 6 8 10

−8 −10

−10

f ( x) g( x) x 2 4 6 8 10

−8 −10

For each quadratic function: i

Determine the coordinates of the vertex.

Determine the x-intercept.

iii

Determine the y-intercept.

Draw a graph of the quadratic function.

m(x) = (x − 3)2

p(x) = −(x − 1)2 – 7

n(x) = (x + 4)2 − 1

r(x) = 3(x + 5)2

y = −2(x − 2)2 + 3

y = −0.5(x + 2)2 + 1

For each of the quadratic function, find the: i

x-intercept(s)

y-intercept

iii

vertex

y = (x − 1)2 − 4

y = (x + 2)2 − 1

y = (x − 3)2 + 5

Range

y = (x − 4)2 − 2

For each of the quadratic function, find the: i a

Domain 2

y = 2(x − 3) + 4

y = −3(x + 1) + 5

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Find the range when the domain is {−6, −1, 0, 5, 7} for each of the quadratic function: a

y = 3(x + 1)2 − 2

y = −2(x − 3)2 + 4

y = 4(x + 2)2 + 1

y = −3(x − 4)2 − 5

Rodney observed that the water stream of a fountain is in the shape of a parabola. This water stream lands on an underwater spotlight. He models the path of the water stream with the maximum height of 8 feet, represented by the vertex (4, 8) and the underwater spotlight, represented by the point (8, 0). a

Select the graph that represents the problem: A y

−8 −6 −4 −2 −2

x 2

4 3

−4

−6

−8

x 1

C y

−16 −12 −8 −4 −2

4 3

−4

−6

1 4

x 4

8 12 16

−8

x 2

8 10 12 14 16 18

Determine appropriate labels and units for the axes.

Determine the domain and range of the function.

For each quadratic equation: i

Determine the coordinates of the vertex.

Determine the equation of the axis of symmetry.

iii

Determine if it has a maximum or minimum.

Determine the domain and range.

Draw a graph of the quadratic function.

Ashtyn throws a rock into Crescent Lake. The height of the rock above ground is a quadratic function of time. The rock is thrown from 4.4 ft above ground. After 1.5 seconds, the rock reaches its maximum height of 24 ft. Write the quadratic equation in vertex form.

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A roller coaster has a part that is modeled by a quadratic function. Hau loves roller coasters and wants to be able to build a small model with his 3D Printer. Assume that the roller coaster passes through points (0, 0) and (57, 0) and reaches a maximum height of 90 ft. Help Hau build a model of the roller coaster by writing an equation for the parabola in vertex form.

Karima and Riley are playing soccer. Karima just kicked a soccer ball and Riley is playing goalie 2 ft in front of the goal post. The following diagram shows the situation on a coordinate plane. The soccer ball is kicked at (1, 0.6) and the goal post is 13 ft away from Karima’s back foot. The maximum height of the soccer ball is 5.5 ft and this occurs when it is 8 ft away from Karima.

Height (ft)

(8, 5.5) 5

(1, 0.6) 0

Distance (ft)

Write a quadratic function that models the height of the soccer ball in feet in terms of the horizontal distance from Karima’s starting position.

Riley jumps to try to block the soccer ball and will block soccer balls that are between 4 ft and 8 ft high. State whether Riley will be able to block Karima’s kick. Explain your answer.

Let’s extend our thinking 26

Determine how many quadratic equations could share a common vertex.

Consider the diagram: a

Describe a situation that could be modeled by the following three quadratic functions. For each part, make sure to identify the vertex and another possible point on the function.

Write equations for the quadratic functions from your above description.

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Consider the family of quadratic equations of the form y = ⬚(x − ⬚)2 − ⬚ where the boxes contain the integers 2, 3, and 5. a

Write a quadratic equation with the lowest possible minimum value.

Describe the similarities and differences between members of the family.

Compare and contrast the following functions, given that a is a positive real number: f (x) = −3a(x − h)2 + k and g(x) = a(x − h)2 − k

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Answers 7.03 Quadratic functions in vertex form What do you remember?

e i V ertical compression by a scale factor 2, a horizontal translation left 2 units, and a vertical translation down 3 units. ii y = 0.5(x + 2)2 − 3 f

1 x=2 2 a f (x) = (x + 3)2 + 5

b g(x) = −(x − 4)2 − 3

d j(x) = (x + 3)2 + 2

c h(x) = −(x − 3) + 3

i R eflection over the x-axis, vertical stretch by a scale factor 2, a horizontal translation right 3 units, and a vertical translation up 4 units.

ii y = −2(x − 3)2 + 4

3 a i y = (x − 2)2 – 1

ii (2, –1)

b i y = −(x − 3)2 + 7

10 a i H orizontal shift 3 units to the right and vertical shift 4 units down.

ii (3, 7)

c i y = −(x + 2.5)2 + 2.25

ii (−2.5, 2.25)

d i y = −2(x − 2)2 + 1

ii (2, 1)

4 a y = (x − 10)2 + 2

b y = 9(x + 9)2

c y = −x − 8 5 a Right

b Down

c Left

−1 −2 −3 −4 −5

d Up

6 C 7

3 2 1

−1

−1 −2 −3

x 1

7 6 5 4 3 2 1

x 1

b i R eflection about the x-axis, horizontal shift 3 units to the left and vertical shift 6 units down. ii

x y 1

−5 −4 −3 −2 −1 −5

−4 −5

−6

−7

10 9 8 7 6 5 4 3 2 1 −9 −8 −7 −6 −5 −4 −3 −2 −1

−8 −9

c i H orizontal shift 1 unit to the left, vertical shift 4 units up and dilated (compressed) by a factor of 3. x 1

Let’s practice 9 a i Vertical translation up 4 units. ii y = x2 + 4 b i Horizontal translation right 1 unit. ii y = (x − 1)2 c i R eflection over the x-axis and a vertical translation up 6 units.

11 10 9 8 7 6 5 4 3 2 −5 −4 −3 −2 −1

x 1 2 3 4

ii y = −x2 + 6 d i V ertical stretch by a scale factor 3 and a horizontal translation left 2 units. ii y = 3(x + 2)2

Answers mathspace.co

885

d i H orizontal shift 6 units to the right, vertical shift 1 unit up and dilated (compressed) by a factor of 0.5. ii

4 3 2 1 −1 −2 −3 −4 −5

y 5

4 3 2

x 1 2 3 4 5 6 7 8 9

1 x −3 −2 −1

13 a y = −1(x − 1)2 + 4

b y = −2(x + 2)2 + 3

d y = −3(x + 1)2 + 2

c y = −0.5(x − 4) − 2 11 a g(x) = 2(x − 1)2 −3 c g(x) = −3 (x + 5)2 − 4 12 a

14 a i y = (x − 2)2 − 2

d g(x) = 3(x + 1)2 + 2

−6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8 −9

1 −4 −3 −2 −1 −1

x 1

−2 −3 −4

b i y = (x + 5)2 − 25 ii

y 5

y 1

2 −5

1 2 3 4 5 6 7

−10

−1

−15

−2

−20

−3

−25

−4

c i y = (x + 2)2 − 8 ii

y x −9 −8 −7 −6 −5 −4 −3 −2 −1 −2 −4 −6 −8 −10 −12 −14 −16 −18

886

−10 −8 −6 −4 −2

x −6−5−4−3−2−1

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

6 4 2 −7 −6 −5 −4 −3 −2 −1 −2 −4 −6 −8

x 1

2 3

d i (–5, 0)

d i

ii (–5, 0)

90 80 70 60 50 40 30 20 10

4 3 2 1

−4 −3 −2 −1 −1

−2

−9 −8 −7 −6 −5 −4 −3 −2 −1 −10

−3

iii (0, 75) y

x 1

−4

15 When completing the square, a quadratic is converted to vertex form. In vertex form, a(x – h)2 + k, the value (h, k) is the vertex of the graph.

18 a i (1, 0) and (−1, 0)

ii (0, −3)

b i (−2, 0)

ii (0, 3)

iii (−2, −1)

c i (3, 0)

ii (0, −4)

iii (3, 5)

d i (2, 0)

ii (0, 1)

iii (2, 3)

ii k = 3

19 a i All real numbers

ii y ≥ 4

b i Horizontal translation left

ii k = 5

b i All real numbers

ii y ≤ 5

c i Vertical compression

c i All real numbers

ii y ≥ −2

d i All real numbers

ii y ≤ 1

16 a i Vertical translation up

17 a i (3, 0) iv

ii (3, 0) 9 8 7 6 5 4 3 2 1

−2 −1 −1

iii (0, 9)

20 a {82, 1, −2, 73, 142}

c {65, 13, 1, 105, 169}

b {−62, −14, 4, −42, −92} d {−107, −29, −5, −80, −158}

21 a B b x-axis: Height (in feet), y-axis: Distance (in feet). c Domain: 0 ≤ x ≤ 8 Range: 0 ≤ y ≤ 8 iii Maximum

ii (−5, 0), (−3, 0)

18 16 14 12 10 8 6 4 2 −9 −8 −7 −6 −5 −4 −3 −2 −1 −2

c i (1, –7)

−4 −3 −2 −1 −2 −4 −6 −8 −10 −12 −14 −16 −18

iv Domain: − ∞ < x < ∞ Range: y ≤ 1 v

y x 1 2 3 4 5 6 7 8 9

iii (0, –8) x

2 3 4

1 −1 −1 −2 −3 −4 −5 −6 −7 −8 −9

y 1

iii (0, 15)

ii None 2

ii x = 4

22 a i (4, 1)

x 1 2 3 4 5 6 7 8

b i (–4, –1)

iii (1, −4)

b i

iii Minimum

iv Domain: Range: y ≥ −20

Answers mathspace.co

887

28 a y = 2(x − 3)2 − 5 or y = 3(x − 2)2 − 5

4 −4 −3 −2 −1 −4

x 1

2 3

−8 −12 −16 −20 −24

b S imilarities between all equations: The parabolas open up, vertical stretch by a factor greater than 1, vertex is in quadrant 4. Differences: The coordinates of the vertex and vertical stretch factor change based on the equation. Examples: y = 2(x − 3)2 − 5, vertex at (3, –5), vertical stretch by a factor of 2

y = 3(x − 2)2 − 5, vertex at (2, –5), vertical stretch by a factor of 3

y = 5(x − 2)2 − 3, vertex at (2, –3), vertical stretch by a factor of 5

25 a h(x) = −0.1(x − 8)2 + 5.5 b Y es, Riley is standing at 11 ft, so h(11) = 4.6 ft. Riley will be able to block the kick because the soccer ball is above 4 ft and less than 8 ft. Let’s extend our thinking 26 An infinite amount of quadratic equations share a common vertex because there are an infinite amount of values for the leading coefficient, a. 27 a A ny situation that has three vertices and three points described to fit the provided diagram. The height of a fish above water in feet in terms of the horizontal distance from a kayak is shown. The height of fish starts to be tracked at (-1, 0), when the fish is 1 ft to the left of the kayak. The fish jumps over the kayak and the maximum height of 7 ft occurs 2 ft away from the kayak. The fish hits the water 5 ft to the right of the kayak. The fish then goes 9 ft below water when the fish is 9.5 ft to the right of the kayak. The fish jumps back out of the water 14 ft to the right of the kayak. The maximum height of the shorter jump is 1.5 ft when 15.25 ft to the right of the kayak. b

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29 Both f (x) and g(x) are quadratic functions. Both functions have an axis of symmetry at x = h. f (x) has a maximum value of y = k and g(x) has a minimum value of y = –k. f (x) is reflected over the x-axis and opens down while g(x) opens up. f (x) has a vertical stretch by a factor of 3a and g(x) has a vertical stretch by a factor of a.

7.04 Quadratic functions in standard form Subtopic overview Lesson narrative In this lesson, students will use technology and coordinate grids to make generalizations about the relationship of quadratic equations in standard forms and the graph’s key characteristics. Students will use values in the standard form of a quadratic to determine the axis of symmetry, vertex, and draw the graph. By the end of the lesson, students will use the symmetry of quadratic functions to work flexibly between quadratic equations in standard form and their graphs to answer questions about contextual situations.

Learning objectives Students: Page 425

Key vocabulary 

axis of symmetry



standard form (of a quadratic equation)



vertex

Essential understanding Different representations of a function may highlight or hide different characteristics but they do not change the function itself. The standard form of a quadratic function highlights the y-intercept.

Standards This subtopic addresses the following Virginia 2023 Mathematics Standards of Learning standards.

Mathematical process goals MPG4 — Mathematical Connections To integrate mathematical connections into their instruction, teachers can have students relate what they are learning about quadratic functions to other topics in mathematics or other disciplines. For example, they can discuss how quadratic functions are used in physics to model the motion of objects under the force of gravity. Teachers can also encourage students to make connections between the different forms of quadratic functions and their graph representations.

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MPG5 — Mathematical Representations Teachers can incorporate mathematical representations into their instruction by having students graph quadratic functions in standard form and analyze the effects of changing the values of a, b, and c. They can also have students use technology, such as graphing calculators or online graphing tools, to explore and visualize the relationship between the equation and the graph of the quadratic function. Additionally, they can ask students to explain the relationship between the standard form of a quadratic function and its key characteristics, such as vertex, x-intercepts, (zeros), y-intercepts, maximum or minimum value, and domain and range.

A.F.2c — Graph a quadratic function, f (x), in two variables using a variety of strategies, including transformations f (x) + k and kf (x), where k is limited to rational values.

Prior connections A.F.1 — The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.

Engage Activity Planetary trebuchets

60 mins

Students will investigate an applet modeling projectiles launched with different parameters for gravity, initial velocity, and initial height.

Understanding and skills

Will use Knowing key features of a quadratic.

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Will develop Understanding that the initial height, or y-intercept, of a quadratic function is equal to the constant value in its standard form equation. Understanding that the initial velocity and gravity both influence the coefficient of the linear term in the standard form equation of a quadratic. Understanding that the force of gravity is directly related to the coefficient of the quadratic term in a standard form equation.

Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: None.

Support students with disabilities Support language - write explanations of mathematical thinking Use sentence starters such as: • The projectile behaves differently ... • The velocity of the projectile affects ... • The x-intercept of the function denotes ... to help students to test their ideas.

Support for English language learners Critique, correct, and clarify Have students consider which of the following statements are always, sometimes, or never true to check their understanding of the vocabulary: • The coeﬃcient of the x2 term represents the gravity of the planet. (answer: sometimes) • The coeﬃcient of the x term represents the velocity. (answer: sometimes) • The vertex is the maximum or minimum of the function. (answer: always)

Classroom guide Hook Students choose one of four quadratic equations in standard form with varying values of the coeﬃcients and number of terms in each equation.

Which one doesn’t belong

•

5 mins

Which one doesn’t belong? Select one option.

Implementation details

y = x2 + 6x + 8

y = x2 − 8

Encourage students to connect the values of each term with what it might mean for the graph of the quadratic.

y = x2 + 8x − 16

y = 2x2 + 6x − 8

Ask students what they notice about the coeﬃcients and constants in each equation. Ask how these values affect the graph of the equation as well as its solutions.

Slide 1 from Student Engage Activity

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Launch

5 mins

Allow students time to explore how the applet works. If needed, discuss the vocabulary terms used to describe quadratic functions.

Consider the following applet.

Velocity

Important mathematical concepts: Quadratic functions, x- and y-intercepts, constant, standard form, coeﬃcient, maximum, minimum, concavity, vertex, axis of symmetry, parameters

Height

y = −0.8 x2 + 2 x + 5

Important contextual information: Trebuchets, catapult, projectile, initial height, velocity, gravity

Moon

Mars

Earth

Jupiter

Throw

Suggested grouping: Form pairs

Slide 2 from Student Engage Activity

Continue when Students have explored the applet and understand how the applet functions.

Explore

Think-pair-share

•

35 mins

Students will explore the applet and determine how the key features of the quadratic are connected to the features of the projectile, as well as come up with an equation modeling its ﬂight path.

Anticipated strategies Isolate applet parameters Velocity, height, and planets Students may isolate applet parameters. They can start by moving the velocity slider and making note of how both the equation, each feature of the parabola, and the ﬂight path are related. Then they can repeat the observations with the height slider and the planet selection.

Isolate each feature x- and y-intercepts, vertex, axis of symmetry, and so forth Students can focus on one feature at a time, such as the y-intercept, and move each projectile parameter while observing if and how that feature changes before moving onto the next feature. Some key findings that students may derive with either strategy are as follows: • The constant value of the standard form equation is the y-intercept of its graph influences the height of the vertex and the location of the x-intercept. • The initial velocity of the function is the coefficient of the linear term and directly changes the location of the axis of symmetry, vertex, and x-intercepts. • The coefficient of the quadratic term is related to the planet the projectile is on and changes the location of the axis of symmetry, vertex, and x-intercepts. 892

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Misconceptions Misattributing features What did you observe that led you to this conclusion? What other factors may have inﬂuenced this behavior?

Purposeful questions Use the following questions to check for understanding and encourage critical thinking: • What features were all inﬂuenced by the planet choice? What do they have in common? • What features were not inﬂuenced by the planet choice? What do they have in common? • Are there any patterns to how the features of the quadratic changed as the slider changed?

Continue when Students have listed connections between the key features of the quadratic and its standard form equation.

Discuss

15 mins

Have a class discussion where pairs present their ﬁndings and compare with other groups. Consider making connections from the discussion to standard form equation components.

Discussion guide Invite pairs to share their ideas and observations and survey how many pairs reached similar conclusions. Begin by inviting pairs to answer the following: • How did each of the sliders and buttons on the applet change the equation? • What do you think each of these represented? Then, begin a discussion on what information is provided by the standard form of a quadratic equation: • If given an equation like this, what about the quadratic would you know? • What features of the quadratic may be harder to determine from this equation? • If you wanted to graph this equation without technology, what strategies would you try? As an extension you may wish to give students the following prompt: • How could we ﬁnd the key features of the quadratic if we were only given its standard form equation?

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Algebra 1 — 4.02 Substitution method Algebra 1 — 7.02 Quadratic functions in factored form Algebra 1 — 7.03 Quadratic functions in vertex form

Tools You may ﬁnd these tools helpful: • Graphing calculator • Graph paper

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Student lesson & teacher guide Quadratic functions in standard form Students learn about the standard form of a quadratic function and how to identify some key features when given a quadratic function in standard form, including how to calculate the axis of symmetry and vertex.

Students: Page 425

Relate the equation for the axis of symmetry back to vertex form Targeted instructional strategies Students may ﬁnd the formula for the axis of symmetry to be unintuitive and thus have diﬃculty remembering it. It may help to show students how the formula is derived. This would also be a good extension activity for advanced learners. The formula for the axis of symmetry can be derived by rearranging the standard form into the vertex form by completing the square.

We can see from the vertex form that the axis of symmetry is at

Let students know that they do not have to learn this proof, just that the ﬁrst step can be suﬃcient for noticing that the x-coordinate of the vertex should have a negative sign when in terms of a and b.

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Compare and connect English language learner support Create displays of the three following methods to graph a quadratic function. Alternatively, students could each create their own display and a teacher can choose a few that highlight different characteristics. Consider the equation f (x) = −x2 + 2x + 3: 7 6 5 4 3 2 1

1. Find the axis of symmetry using the formula 2. Find the vertex 3. Find the y-intercept 4. Find a third point using the axis of symmetry x

−5 −4 −3 −2 −1 −1

5. Sketch the graph, labeling the key features

1 2 3 4 5 6 7 8

−2 −3 −4 −5 −6

7 6 5 4 3 2 1

1. Identify the y-intercept

2. Write the equation in factored form: f (x) = −(x + 1) (x − 3) 3. Identify the x-intercepts 4. Sketch the graph, labeling the key features x

−5 −4 −3 −2 −1 −1

1 2 3 4 5 6 7 8

−2 −3 −4 −5 −6

7 6 5 4 3 2 1 −5 −4 −3 −2 −1 −1

1. Create a table of values

x y

−1 0

0 3

1 4

2 3

3 0

2. Sketch the graph x 1 2 3 4 5 6 7 8

−2 −3 −4 −5 −6

Pair up students to discuss which method they preferred and why, as well as any similarities or differences in the two methods. For further investigation, present students with a quadratic equation that does not have any real roots and ask students to discuss why one method works while the other cannot.

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Use GeoGebra to visualize how coefficients affect the graph Student with disabilities support Students may struggle to understand how each value in the standard form equation affects the graph of the equation. Provide students with some sort of visual support which demonstrates how the change in each value affects the graph. This can be achieved through a variety of examples or with an interactive manipulative such as a GeoGebra applet. • a is the scale factor, so it affects the direction of the opening and vertical dilation • b affects the position of the axis of symmetry (together with a) • c is the constant term, so it affects the vertical position of the graph

Remind students of the equation for the vertex Address student misconceptions When using the formula for the axis of symmetry, students may forget

finds the x-value of the vertex,

not the actual vertex. Students often forget they can find the y-value by substituting the x-value back into the original equation.

Exploration Students: Page 425

Suggested student grouping: In pairs Students explore an applet that allows them to change a, b, and c of a quadratic equation in standard form using sliders. Students should notice how the different values transform the parent function and be able to identify some of the key features that the standard form of a quadratic function highlights. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What happens to the graph as the value of a changes? As a becomes larger, the function experiences a vertical stretch. There is a vertical compression when 0 < ∣a∣ < 1. Then, the graph reﬂects across the x-axis when a < 0.

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2. What happens to the graph as the value of b changes? Changing b affects the location of the vertex with respect to the y-axis. When b = 0, the vertex of the parabola lies on the y-axis. When a > 0, b shifts the graph down and to the left for b > 0 and down and to the right for b < 0. When a < 0, b shifts the graph up and to the right for b > 0 and up and to the left for b < 0. Changing b does not affect the shape of the parabola. 3. What happens to the graph as the value of c changes? As the value of c changes, the graph experiences a vertical translation. For positive values of c, the graph is translated up c units. For negative values of c, the graph is translated down c units. The value of c is the y-coordinate of the y-intercept. Purposeful questions • Do any of the sliders have a similar effect as any of the parameters of factored form or vertex form? • Does the slider change the location of the vertex? • Does the slider change the shape of the parabola? • Can you describe the effect of each slider in terms of translations, dilations or reﬂections?” Possible misunderstandings • Students may not try a variety of values of the parameters in combination with the one they are changing and over generalize the observed effect. For example, if the value of b is set to zero while altering the parameter c, a student may come to the incorrect conclusion that c gives the y-coordinate of the vertex in all cases.

Students: Pages 425–426

For example, if we have the function g(x) = 3x2 + 12x − 15 where a = 3, b = 12, and c = −15 we can start by ﬁnding the x-coordinate: Equation for the x-coordinate of the vertex Substitute a = 3 and b = 12 Evaluate the multiplication Evaluate the division We can substitute the x-coordinate of the vertex into the original equation in order to ﬁnd the y-coordinate of the vertex. g(x) = 3x2 + 12x − 15

Original function

= 3 (−2)2 + 12 (−2) − 15

Substitute x = −2

= 3 (4) + 12 (−2) − 15

Evaluate the exponent

= 12 − 24 − 15

Evaluate the multiplication

= −27

Evaluate the subtraction

The coordinates of the vertex of g(x) are (−2, −27). We can conﬁrm this by looking at the graph: −6 −5 −4 −3 −2 −1 −5 −10 −15

y 1

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Equation for thefunction x-coordinate of the vertex Original

g(x) = 3x2 + 12x − 15 2

15 Substitute x = b−2= 12 = 3 (−2) + 12 (−2) −Substitute a = 3 and = 3 (4) + 12 (−2) − 15 Evaluate the exponent = 12 − 24 − 15

Evaluate the multiplication Evaluate the multiplication

= −27 Evaluate the subtraction Evaluate the division The coordinates of the vertex of g(x) are (−2, −27). conﬁrm this byinlooking at ﬁnd the the graph: We can substitute the x-coordinate of the vertex intoWe thecan original equation order to y-coordinate of the vertex. g(x) = 3x2 + 12x − 15

Original function

= 3 (−2)2 + 12 (−2) − 15

−6 −5 −4 Substitute x =−3 −2−2 −1

= 3 (4) + 12 (−2) − 15

Evaluate the exponent−5

= 12 − 24 − 15

Evaluate the multiplication −10

= −27

Evaluate the subtraction

−15

The coordinates of the vertex of g(x) are (−2, −27). We can conﬁrm this by looking at the graph: −20 −6 −5 −4 −3 −2 −−125 (−2, −27)

We can also see here that the axis of symmetry is the line:

y 1

−5 −10 −15

The axis of symmetry always passes through the vertex.

−20 − 25

Example 1

(−2, −27)

For the quadratic function = 3x 6xsymmetry + 8: We can also see here that ythe axis−of is the line: Examples 2

a Identify the axis of symmetry.

Students: Page 426

The axisaofstrategy symmetry always passes through the vertex. Create We will use the formula

Apply the idea

so we need to identify

Equation for axis of symmetry

Example the values of1 a and b from the equation.

a = y3,=b3x = 2−6 For the quadratic function − 6x + 8:

Substitute b = −6 and a = 3

a Identify the axis of symmetry.

Evaluate the multiplication

Create a strategy

Apply the idea

We will use the formula

so we need to identify

the values of a and b from the equation. 426

Evaluate the division

Evaluate the multiplication Equation for axis of symmetry The axis of symmetry is x = 1.

a = 3, b = −6

Substitute b = −6 and a = 3

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Evaluate the multiplication Evaluate the division Evaluate the multiplication The axis of symmetry is x = 1.

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Purpose Show students how to calculate the axis of symmetry of a quadratic function when given the function in standard form.

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Students: Page 427 b State the coordinates of the vertex.

Create a strategy Once we have the x-coordinate of the vertex from the axis of symmetry, we can substitute it into y = 3x2 − 6x + 8 and evaluate to get y. From part (a), we know that the axis of symmetry is x = 1 so the x-coordinate of the vertex is x = 1.

Apply the idea y = 3x2 − 6x + 8

Given equation

y = 3(1)2 − 6(1) + 8

Substitute x = 1

y = 3(1) − 6(1) + 8

Evaluate the exponent

y=3−6+8

Evaluate the multiplication

y=5

Evaluate the subtraction and addition

The vertex is (1, 5). b State the coordinates of the vertex. c State the coordinates of the y-intercept.

a strategy PurposeCreate Create a strategy Apply the idea Once we have the x-coordinate ofof the vertex fromto the axis of symmetry, we can it into y = 3x2 −in 6x + 8 and Show students how to use the axis symmetry calculate the vertex of substitute aofquadratic function Since the y-intercept occurs when x = 0, substitute In this case, the value c in the equation is 8. standard form. evaluate to get y. x = 0 into the equation and evaluate y.

So, we have that the coordinates of the y-intercept are

From part (a), we know that the axis of symmetry is x = 1 so the x-coordinate of the vertex is x = 1. Expected mistakes When we are given an equation in standard form, (0, 8). Studentsthemay struggle to begin ﬁnding y-value of the y-intercept will be y =the c. vertex. Remind students that the x-coordinate of the vertex is the the idea value ofApply x for the axis of symmetry, which we have already calculated. y = 3x2 − 6x + 8 Given equation d Draw parabola. x = 1 6(1) corresponding +8 Substitute ygraph = 3(1)2of− the Reﬂecting withastudents

y = 3(1) 6(1) + 8 Evaluate exponent Challenge students to −complete the square andthe convert the equation in standard form to vertex form: =3−6+8 Evaluate the multiplication y = 3(x −Create 1)2 + 5.a ystrategy

We have all the key features we need to create a graph. For more accuracy, we can use the axis of symmetry and y=5 Evaluate the subtraction and addition y-intercept to ﬁnd another point. This point will be a reﬂection of the y-intercept across the axis of symmetry. Students:The Page vertex427 is (1, 5). We know that the parabola will open upwards because a > 0. cApply State theidea coordinates of the y-intercept. the Axis of symmetry: x = 1

Create a strategy

Vertex: (1, 5)Apply the idea Since the y-intercept occurs when x = 0, substitute y-intercept: In (0,this 8) case, the value of c in the equation is 8. 8 x = 0 into the equation and evaluate y. So,(2, we8)have that the coordinates of the y-intercept are Another point: When we are given 6 an equation in standard form, the y-value of the y-intercept will be y = c.

(0, 8).

d Draw a graph2 of the corresponding parabola. x PurposeCreate a strategy −1 1 2 3 Show students how to identify the y-intercept of a quadratic function in standard form. We have all the key features we need to create a graph. For more accuracy, we can use the axis of symmetry and

y-intercept to check find another point. This point will be a reflection of the y-intercept across the axis of symmetry. Reflect and

Expected mistakes We know that the will open upwards a >equation 0. From the graph weparabola can identify that the vertexbecause form of the would be: Students may mistake the y-coordinate of the vertex for the y-intercept. Point out to students that while this may 2 y = 3(x − 1) + 5 sometimes bethe true, it is not always true for the vertex of a function and its y-intercept. Apply idea Axis of symmetry: x = 1

Vertex: (1, 5) y-intercept: (0, 8)

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Another point: (2, 8) 6 4 2 x −1

Reﬂect and check

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Create a strategy

Apply the idea

Since the y-intercept occurs when x = 0, substitute x = 0 into the equation and evaluate y.

In this case, the value of c in the equation is 8.

When we are given an equation in standard form, y-value427 of the y-intercept will be y = c. Students:thePage

So, we have that the coordinates of the y-intercept are (0, 8).

d Draw a graph of the corresponding parabola.

Create a strategy We have all the key features we need to create a graph. For more accuracy, we can use the axis of symmetry and y-intercept to ﬁnd another point. This point will be a reﬂection of the y-intercept across the axis of symmetry. We know that the parabola will open upwards because a > 0.

Apply the idea Axis of symmetry: x = 1

Vertex: (1, 5) y-intercept: (0, 8)

Another point: (2, 8) 6 4 2 x −1

Reﬂect and check From the graph we can identify that the vertex form of the equation would be: y = 3(x − 1)2 + 5

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Purpose Show students how to use at least three points on the graph of the function given in standard form to graph the quadratic function. Reﬂecting with students Ask students to explain why (2, 8) is a good third point to include on the graph. Since a quadratic is symmetrical about the axis of symmetry, we can see that the point (2, 8) is mirrored with the y-intercept.

Develop processes for graphing from standard form

use with Example 1

Targeted instructional strategies Encourage students to develop an algorithm for graphing quadratic functions in standard form. Begin by going through several examples as a group and then encourage them to write a set of steps that could be used for other examples. By following this sequential process, students will apply algorithmic thinking to connect the equation with its graph. For example: 1. Identify the coefficients a, b, and c in the quadratic equation y = ax2 + bx + c. 2. Use a to determine the direction of opening. 3. Calculate the axis of symmetry using x =

4. Find the vertex by substituting the axis of symmetry as the x-value into the original equation to find the corresponding y-value. The vertex is at (x, y). 5. Determine the y-intercept by evaluating the equation when x = 0 or noting it is (0, c). 6. Select additional x-values on either side of the axis of symmetry and compute their y-values. 7. Plot the axis of symmetry, the vertex, the y-intercept, and any other points. 8. Draw the parabola that passes through all the plotted points.

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Offer a technology alternative for graphing

use with Example 1

Student with disabilities support Let students know that they do not need to draw perfect graphs. Set a minimum requirement for graphing, such as only labeling the vertex, y-intercept, and drawing a roughly parabolic shape. For graphs of standard form quadratic equations, students only need to label the vertex and one other point on the curve to define the parabola. This other point will usually be the y-intercept as it is easy to find. For students with difficulty drawing graphs, allow the use of graphing technology such as Desmos or GeoGebra, or pair up students with one describing the graph and the other drawing.

Students: Page 428

Example 2 Naomi is playing a game of Kapucha Toli, where to start a play, a ball is thrown into the air. Naomi throws a ball into the air from a height of 6 feet, and the maximum height the ball reaches is 12.25 feet after 1.25 seconds. a Sketch a graph to model the height of the ball over time.

Create a strategy To sketch a graph, we’ll use key points found by using the given information. We’ll also use the units which are given, being feet and seconds. We will let x represent the time since the ball was tossed in seconds. We will let y represent the height of the ball in feet.

Apply the idea It’s given that the ball is thrown from a height of 6 feet. This means that at 0 seconds, the height of the ball is 6 feet. So our y-intercept is (0, 6). We’re told the maximum height of the ball is at 12.25 feet after 1.25 seconds. The maximum height will occur at the vertex of the graph, so the vertex is (1.25, 12.25). This also means that our axis of symmetry is x = 1.25. We can use the axis of symmetry to determine a second point on the graph, the point across the axis of symmetry from the y-intercept. The point is (2.5, 6). We can sketch our graph by plotting the y-intercept, the vertex, and the point found with our axis of symmetry. Now we need to identify an appropriate scale. Kapucha Toli 13 Height in feet ( y) 12 11 10 9 8 7 6 5 4 3 2 1 Time in seconds (x) −1

We know that our graph will not go above y = 12.25 feet and that any part of the graph that goes below the x-axis will not be viable, so graphing −1 ≤ y ≤ 13 going up by 1 will show the full picture. We know that time starts at x = 0 and the ball is on the way back down at x = 2.5, so graphing 0 ≤ x ≤ 4 going up by 1 or 0.5 should be suﬃcient. This is an appropriate way to label the axes.

Now, we can graph the height of the ball over time. Kapucha Toli 13 Height in feet ( y) 12 11 10 9 8 7 6 5 4 3 2 1 Time in seconds (x) −1

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6 5 4 3 2 1

This is an appropriate way to label the axes.

Time in seconds (x)

−1

Now, we can graph the height of the ball over time. Kapucha Toli 13 Height in feet ( y) 12 11 10 9 8 7 6 5 4 3 2 1 Time in seconds (x) −1

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Purpose Show students how to sketch the graph of a quadratic function given in context. Expected mistakes Students may leave out units on the axes. Remind students that labeling a graph for a problem in context will give the audience an idea of what is happening in the problem. Reﬂecting with students Ask students to identify the domain of the function for the context. The domain is 0 ≤ x ≤ 3, since negative time does not make sense in the context of the problem.

Students: Page 429 b Predict when the ball will be 3 feet above the ground.

Create a strategy We can use the sketch of our graph to predict when the ball will be at 3 feet.

Apply the idea We can draw a horizontal line from y = 3 across until we reach the graph. After that, we can draw vertical line until we reach the x-axis to determine after how many seconds the ball is at 3 feet.

We hit the x-axis around x = 2.7. Therefore, the ball is 3 ft above the ground after about 2.7 seconds.

Kapucha Toli 13 Height in feet ( y) 12 11 10 9 8 7 6 5 4 3 2 Time in seconds (x) 1 −1

Reﬂect and check When reading from a graph, we often have to estimate. Any prediction between 2.6 and 2.9 would be reasonable in this case. c Write a quadratic equation in standard form to model the situation.

PurposeCreate a strategy Show students how to make prediction graph. To write the equation, we a can use the keyusing pointsaand the graph we’ve sketched in previous parts. 902

Apply the idea

Mathspace Virginia SOL Algebra 1 Teacher Edition Since we know 3 points, we can use the standard form and substitution in order to solve for a, b, and c for our mathspace.co standard form quadratic equation which is of the form y = ax2 + bx + c. We know that c represents the y-value of the y-intercept which is (0, 6): y = ax2 + bx + 6

7 6 5 4 3 2 1

We hit the x-axis around x = 2.7. Therefore, the ball is 3 ft above the ground

Time in seconds (x)

Reflecting with students 2 1 3 after about 2.7 seconds. −1 Challenge students to give another way to determine when the ball is 3 feet above the ground. Students could find the Reflect equation the function and substitute y = 3 in order to solve for x. andofcheck When reading from a graph, we often have to estimate. Any prediction between 2.6 and 2.9 would be reasonable in

Students:this Pages case. 429–430

c Write a quadratic equation in standard form to model the situation.

Create a strategy To write the equation, we can use the key points and the graph we’ve sketched in previous parts.

Apply the idea Since we know 3 points, we can use the standard form and substitution in order to solve for a, b, and c for our standard form quadratic equation which is of the form y = ax2 + bx + c. We know that c represents the y-value of the y-intercept which is (0, 6): y = ax2 + bx + 6 Now, we can substitute our other two points to solve for a and b. Next, we can substitute in (1.25, 12.25). y = ax2 + bx + 6

Standard form of a quadratic with c = 6

12.25 = a(1.25)2 + b(1.25) + 6

Substitute (1.25, 12.25)

12.25 = 1.5625a + 1.25b + 6

Evaluate the exponent

6.25 = 1.5625a + 1.25b

Subtract 6 from both sides

Since we have two unknowns, we’ll have to use our ﬁnal point to create a second equation. We’ll now substitute in (2.5, 6). y = ax2 + bx + 6 2

Standard form of a quadratic with c = 6

6 = a(2.5) + b(2.5) + 6

Substitute (2.5, 6)

6 = 6.25a + 2.5b + 6

Evaluate the exponent

0 = 6.25a + 2.5b

Subtract 6 from both sides

Now, we have two equations with two unknowns. We can solve this system using the substitution method. Let’s ﬁrst isolate b in our second equation. Second equation

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Subtract 6.25a from both sides Divide by 2.5 on both sides Evaluate the division Now, we can use the this in our ﬁrst equation, letting b = −2.5a. 6.25 = 1.5625a + 1.25b

First equation

6.25 = 1.5625a + 1.25(−2.5a)

Substitute b = −2.5a

6.25 = 1.5625a − 3.125a

Evaluate the multiplication

6.25 = −1.5625a

Combine like terms

−4 = a

Divide both sides by −1.5625

Therefore a = −4. Finally, we can use b = −2.5a to solve for b. b = −2.5a b = −2.5(−4)

Substitute a = −4

b = 10

Evaluate the multiplication

Therefore b = 10. Now it’s time to piece it all together. Since a = −4, b = 10, and c = 6, we know that our equation in standard form is: y = −4x2 + 10x + 6

7.04 Quadratic functions in standard form Vertex form of a quadratic with vertex (1.25, 12.25) mathspace.co Substitute in (0, 6) Evaluate the parentheses Subtract 12.25 from both sides

903

b = −2.5(−4)

Substitute a = −4

b = 10

Evaluate the multiplication

Therefore b = 10. Now it’s time to piece it all together. Since a = −4, b = 10, and c = 6, we know that our equation in standard form is: y = −4x2 + 10x + 6

Reﬂect and check An alternative and simpler solution is to use the vertex to write it in vertex form and then using the intercept to solve for a. Vertex form is y = a(x − h)2 + k. The vertex is (1.25, 12.25), this gives us: y = a(x − 1.25)2 + 12.25 We can then substitute in the point (0, 6) and solve for a. Vertex form of a quadratic with vertex (1.25, 12.25) Substitute in (0, 6) Evaluate the parentheses Subtract 12.25 from both sides Divide by 1.5625 on both sides Evaluate the division So, now we have the equation: y = −4(x − 1.25)2 + 12.25 Now, we need to convert to standard form: y = −4(x − 1.25)2 + 12.25

Equation in vertex form

y = −4(x2 − 2.5x + 1.5625) + 12.25

Expand the binomial

y = −4x2 + 10x − 6.25 + 12.25

Distributive property

y = −4x2 + 10x + 6

Combine like terms

We get the same answer of: y = −4x2 + 10x + 6.

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Purpose mathspace.co Show students how to write an equation of a quadratic function in standard form from its graph’s key features. Expected mistakes Students may struggle to ﬁnish solving for a and b when they have an equation with two unknown variables. Remind students that solving systems of equations is how we approach solving for two unknowns, because it is possible to solve when we have two equations with two unknowns.

Three reads

use with Example 2

English language learner support Advise students to read through the instructions a few times, focusing on gathering different information each time in order to build up understanding of what the question is asking. On the first read, students should aim to identify the scenario presented in the question. Ask students, “What do you think is happening in this question?” or “Can you explain what this question is about?” On the second read, students should aim to interpret the problem by anwering questions like, “What is the question asking you to find?” and “What information should be included in the answer?” On the third read, students should look for important information in the instructions. In this question, the important information includes: • The ball was thrown initially from a height of 6 feet. • The maximum height the ball reaches is 12.25 feet. • The ball reaches its maximum height after 1.25 seconds. Students can be prompted by framing these as questions like “What might be the variables in this situation?” or “Which pieces of information can you use to draw a graph?”

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Students: Pages 431–432 Example 3 y

Write the standard form equation of the function shown on the graph.

8 6 4 2 −2 −2

x 2

10 12

−4

Create a strategy Use the vertex to ﬁrst write the equation in vertex form, then use another point on the parabola to solve for a, and ﬁnally convert to standard form.

Apply the idea Vertex form is y = a (x − h)2 + k. The vertex is (8, −2), this gives us: y = a(x − 8)2 + (−2) or y = a(x − 8)2 − 2 We can then substitute in the point (4, 6) and solve for a. Vertex form equation Substitute in x = 4 and y = 6 Evaluate the subtraction Evaluate the exponent Add 2 to both sides Divide by 16 on both sides Simplify the fraction Substituting

we get the equation:

Now, we need to convert to standard form: Equation in vertex form Expand the binomial Distributive property Combine like terms is the standard form equation of the function on the graph.

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Reﬂect and check Alternatively, we could have used the zeros and factored form. Using the zeros of 6 and 10, we can write the equation in factored form y = (x − x1) (x − x2).This gives us: y = a(x − 6)(x − 10) We can then substitute in the point (4, 6) and solve for a. Factored form equation

Reﬂect and check

Substitute x = 4 and y = 6

Evaluate the subtraction Alternatively, we could have used the zeros and factored form. Evaluate the multiplication Using the zeros of 6 and 10, we can write the equation in factored form y = (x − x1) (x − x2).This gives us: Divide by 12 onyboth = a(xsides − 6)(x − 10) thesolve fraction We can then substitute in the pointSimplify (4, 6) and for a. So, now we have the equation:

Factored form equation Substitute x = 4 and y = 6 Evaluate the subtraction

Evaluate Now, we need to convert to standard form: the multiplication Equation in both factored form Divide by 12 on sides Multiply the binomials Simplify the fraction So, now we have the equation:

Combine like terms Distributive property

Now, we need to convert to standard form: Equation in factored form

Example 4 Purpose Multiply the binomials Show students how toout write standard form equation of a function using the vertex and another point on the The whale jumps the the water at 3 seconds and reenters the water after 6.5 seconds. Combine 49 feetlike afterterms 4.75 seconds. Determine the function.The whale reaches a maximum height of equation in standard form that models the whale’s jump. Distributive property

Students: Pages 432–433 Example 4

The whale jumps out the water at 3 seconds and reenters the water after 6.5 seconds. The whale reaches a maximum height of 49 feet after 4.75 seconds. Determine the equation standard form that models the whale’s jump. Create ainstrategy If we think of the water level as the x-axis, then the moments where the whale exits and reenters the water would represent the x-intercepts. The maximum height is the vertex of the parabola formed by the whale’s jump path.

Apply the idea Using the x-intercepts of 3 and 6.5, we can create the following equation, where x represents the time in seconds and y represents the height of the jump in feet:

Create a strategy

y = a(x − 3)(x − 6.5)

If we think of the water level as the x-axis, then the moments where the whale exits and reenters the water would 432 Mathspace Virginia SOL Algebra 1 represent the x-intercepts. The maximum height is the vertex of the parabola formed by the whale’s jump path. mathspace.co

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Create a strategy If we think of the water level as the x-axis, then the moments where the whale exits and reenters the water would represent the x-intercepts. The maximum height is the vertex of the parabola formed by the whale’s jump path.

Apply the idea Using the x-intercepts of 3 and 6.5, we can create the following equation, where x represents the time in seconds and y represents the height of the jump in feet: y = a(x − 3)(x − 6.5) Next, we can substitute the values of the maximum point, which occurs at (4.75, 49), to ﬁnd the value of a. 432

Mathspace Virginia Algebra y=a (x − 3)SOL (x − 6.5) 1 mathspace.co

Equation for whale’s path

49 = a(4.75 − 3)(4.75 − 6.5)

Substitute y = 49 and x = 4.75

49 = a(1.75)(−1.75)

Evaluate the subtraction

49 = −3.0625a

Evaluate the multiplication

−16 = a

Division property of equality

Now, we know the factored form of the equation that models the whale’s jump: y = −16(x − 3)(x − 6.5) The last step is to get it into standard form. We can do this by multiplying all the factors together. y = −16 (x − 3) (x − 6.5) 2

Equation for whale’s path

y = −16(x − 6.5x − 3x + 19.5)

Distributive property

= −16x2 + 104x + 48x − 312

Distributive property

= −16x2 + 152x − 312

Combine like terms

The equation in standard form that models the whale’s jump is y = −16x2 + 152x − 312.

Reflect and check Notice that the parabola formed by the whale opens downward. If we did not have another point to help us find the value of a, our parabola would have been facing upward. Using the vertex helped us find a negative value for a which is what made the parabola face downward.

Idea summary Purpose The standard form of a quadratic equation highlights the y-intercept of a quadratic function. Show students how to model a real-life situation using a quadratic function in standard form. 2

y = ax + bx + c

Expected mistakes a scale factor b linear coeﬃcient Students might write the factors incorrectly, using addition instead of subtraction. Remind students that if x = 3 is c y-value of the y-intercept a zero, then we must rewrite this equation in terms of zero to find its factor. The axis of symmetry is the line:

Zero x=3 Equation in terms of zero x−3=0 The axis of symmetry is also Corresponding factor (x the − 3)x-coordinate of the vertex. To ﬁnd the y-coordinate, you substitute the x-coordinate back into the original function. Therefore, the coordinates of the vertex are:

Reflecting with students Before examining the solution strategy, ask students to brainstorm how the context might relate to a quadratic equation. What would the variables be? How does the initial information given in the problem relate to the graph of a quadratic function? Invite students to sketch a quick graph of the whale’s jump to help them understand the problem. Then ask them to outline a strategy to find the solution.

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The equation in standard form that models the whale’s jump is y = −16x2 + 152x − 312.

Reflect and check Notice that the parabola formed by the whale opens downward. If we did not have another point to help us find the of a,433 our parabola would have been facing upward. Using the vertex helped us find a negative value for a Students:value Page which is what made the parabola face downward.

Idea summary The standard form of a quadratic equation highlights the y-intercept of a quadratic function.

y = ax2 + bx + c a scale factor b linear coeﬃcient c y-value of the y-intercept The axis of symmetry is the line:

The axis of symmetry is also the x-coordinate of the vertex. To ﬁnd the y-coordinate, you substitute the x-coordinate back into the original function. Therefore, the coordinates of the vertex are:

Practice Students: Pages 434–438

What do you remember? 1

Describe the basic shape of a parabola.

Given that the standard form of a quadratic is y = ax2 + bx + c: i

908

What does the sign of a tell us?

433

What does changing the value of c do?

State the coordinates of the vertex.

y = −x2 − 4x − 9

For each quadratic function: i

Determine the axis of symmetry.

iii

State the coordinates of the y-intercept.

y = x2 − 4x + 8

Rewrite each equation in standard form. a

7.04 Quadratic functions in standard form mathspace.co

y = (3x − 1) (2x + 1)

y = 2(x − 4)2 + 1

y = (x − 6) (x + 6)

y = 5(x − 1)2 − 5

f (x) = 4x2 − x + 1

For each equation: i

Find f (3).

Find f (−5).

iii

Find

f (x) = 2x2 + 3x − 4

f (x) = −x2 + 2x + 10

f (x) = −3x2 − 5

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Consider the table of values of a function. x y

−6 −8

−5 −9

−4 −8

−3 −5

−2 0

−1 7

Select the graph that could represent the function: A

3 −12 −10 −8 −6 −4 −2 −3

10 12

−6

−9

−12

−15

−18

3 −12 −10 −8 −6 −4 −2 −3 −6 −9

−2 −3

−6

−2 −3 −6 −9

x 2

10 12

Let’s practice 6

Consider the graph of a function. 2

Select the equation that represents the function: A

y = x2 − 6x + 4

y = x + 6x − 4

y = x2 − 6x − 4

y = x2 + 6x + 4

−7 −6 −5 −4 −3 −2 −1 −1

x 1

−2 −3 −4 −5 −6

For each quadratic function: i

Determine the axis of symmetry.

State the coordinates of the vertex.

iii

State the coordinates of the y-intercept.

Draw a graph of the corresponding parabola.

y = x2 − 2x + 5

y = 2x2 + 24x + 75 y = 4x2 − 64

y = −x + 6x − 8

y = 2x2 + 2x + 9

f 7.04 Quadratic functions in standard form mathspace.co

909

For each quadratic function: i

State the coordinates of the y-intercept.

iii

Draw a graph of the corresponding parabola.

Determine the coordinates of the x-intercept(s).

y = x2 − 3x − 10

y = x2 − 9

y = 4 − 3x − x2

y = 2x2 + 12x + 18

y = 4x2 + 8x − 5

y = −6x2 + 10x + 4

Consider the function y = −2x2 + 10x − 3 where x represents time in minutes and y represents the height of a ball from the ground in feet.

Draw a graph of the corresponding parabola. Make sure to label the axes.

Using your graph, predict when the ball will be 3 ft above the ground.

Write an equation in standard form to represent the parabola. a

−1−2 −4 −6 −8 −10

14 12 10 8 6 4 2

x 1

−4 −6 −8 −10

14 12 10 8 6 4 2

4 2 −4 −3 −2 −1 −2

x 1

−4 −6 −8 −10

−4 −3 −2 −1−2

2 3 4 5 6 7 8

−4 −3 −2 −1 −2 −4 −6 −8 −10

Darnell throws a bag of cookies to his friend, Ike, from a height of 3.25 ft. The cookies reach a maximum height of 19.25 ft, 1 second after being thrown. a

Determine the equation in standard form that represents the situation.

Graph the corresponding parabola. Make sure to label the axes.

If Ike caught the cookies after 2 seconds, determine the height of the catch.

Carliss is starting a summer car wash business. Her profit function relates the total profit to the rate she charges for each car wash. The rate and profit are in dollars. P(x) = −x2 + 50x − 95

910

Carliss wants to make at least $525 this summer. Determine if she could make enough money based on her quadratic proﬁt model. Explain your reasoning.

Draw a graph of P(x).

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

The table shows points on a quadratic function. 2

Another quadratic function has the equation B(x) = x − 2x − 8.

x A(x)

−3 −5

−2 −8

−1 −9

0 −8

2 0

Determine what the quadratic functions have in common and what is different. 14

Orland and Delﬁno are comparing the vertex of the following graph and equation of quadratic functions: 5

g(x) = −0.5x2 + x + 7.5

f ( x)

3 2 1

−6−5−4−3−2 −1 −1 −2

1 2 3 4 5 6 7 8

−3 −4 −5

Identify and explain their error(s): Orland’s work: Finding axis of symmetry for g(x)

1 2 3

Finding y-value of vertex

4 5 Vertex of g(x) is (−1, 6) and vertex of f (x) is (1, −4) Delﬁno’s work: 1

Finding axis of symmetry for g(x)

2 3

Finding y-value of vertex

4 5 Vertex of g(x) is (2, 7.5) and vertex of f (x) is (1, −5)

Let’s extend our thinking 15

Kinsey wants to understand how vertex form and standard form of a quadratic function are related. Consider the vertex form of the function. y = a(x − h)2 + k a

Expand a(x − h)2 + k and write the equivalent equation in standard form.

State the value of b in the equation from part (a).

State the value of c in the equation from part (a).

Explain how the equations are related.

7.04 Quadratic functions in standard form mathspace.co

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Beau is competing at his high school swim meet and dives off a springboard that is 3 ft above the pool surface. He reaches a maximum height of 8 feet after 1 second. a

Determine the equation in standard form that represents Beau’s diving path.

Determine after how many seconds Beau will be at the same height as the springboard again.

If the pool is 12 feet deep, determine how many seconds after jumping it will take Beau to reach the bottom of the pool. Explain your reasoning.

Fill in the boxes to create a quadratic equation with the lowest possible minimum value. Use the digits 1 to 5 at most once. y = ⬚ x2 + ⬚ x + ⬚

A Happy Birthday banner is modeled by the quadratic function h(x) = 0.2x2 − x + 1.25 where x is the distance from the left side of the banner and h, is the height above the ground. Both x and h are in feet. Currently, the lowest part of the banner touches the ground. Immanuel is trying to create a virtual card and needs to know how to affect the look of the banner by changing parts of the quadratic function.

912

Determine a domain and range for the quadratic function that models the initial Birthday banner. Explain your reasoning.

Describe to Immanuel how changes to the quadratic function affect the shape of the Birthday banner.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Answers

b i x = –6

7.04 Quadratic functions in standard form

ii (–6, 3)

iii (0, 75)

y 80 70

What do you remember?

60 50

1 a A parabola is a ∪-shaped curve that opens either upward or downward.

40 30

b i I f the coefficient a is positive, the curve is ∪-shaped opening upward. If a < 0, then the curve is ∩-shaped, opening downwards. ii Changing the constant term c in a quadratic equation shifts the entire graph vertically. If c is positive, the graph moves upward, and if c is negative, the graph moves downward. The magnitude of the shift depends on the value of c. 2 a i x=2

20 10 −12 −10 −8 −6 −4 −2

c i x=3 iv

ii (3, 1)

iii (0, –8)

4 2

−6 −4 −2 −2

ii (2, 4)

iii (0, 8)

b i x = –2

ii (–2, –5)

iii (0, –9)

−6

c x2 − 36

d 5x2 − 10x

−10

4 a i f (3) = 23

ii f (−5) = 31

iii

b i f (3) = 7

ii f (−5) = −25

iii

c i f (3) = −32

ii f (−5) = −80

iii

d i f (3) = 34

ii f (−5) = 106

iii

3 a 6x + x − 1

x 2 4

2 4

8 10

−4

b 2x − 16x + 33

−8 −12

d i x=0 iv

ii (0, –64)

iii (0, –64)

−8 −6 −4 −2 −10

4 6

−20 −30 −40

5 C

−50 −60

Let’s practice

−70

6 D 7 a i x=1 iv

ii (1, 4) 14

iii (0, 5)

e i x = –0.5 iv

ii (–0.5, 8.5) 18

10 8

8 6

6 4 2 −2

16 14 12 10

−6 −4 −2

iii (0, 9)

4 2

x 2

−4 −3 −2 −1

i x=3

x 1

2 3

ii (3, 2.25)

iii (0, 0)

4 y 3 2 1 −8 −6 −4 −2−1

x 2

4 6

−2 −3 −4 −5 −6 −7 −8 −9

Answers mathspace.co

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8 a i (0, –10)

ii (–2, 0) and (5, 0)

iii

2 −6 −4 −2

y 2

−2

i (0, 4)

iii

y 8

−4

−6

−8

−10

−12 −14

−2

y x

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8 −9

9 a

1 2 3 4 5

10 9 8 7 6 5 4 3 2 1

x (min) 6

y (ft)

c i (0, 4)

ii (–4, 0) and (1, 0) 7 6 5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3

x 1

2 3 4 5

b Because the y-axis represents height in feet, we can look at when our graph has a y-value of 3. This occurs twice, when x is around 0.7 and 4.3. Because x is in terms of minutes, we can say that the ball will be 3 ft above the ground around 0.7 and 4.3 minutes. b y = x2 + x − 5

c y = −3x2

d y = 2x2 + 2x − 4

11 a y = −16x2 + 32x + 3.25 20 18 16 14 12 10 8 6 4 2

ii (–3, 0) y

x 1 2 3

e i (0, –5)

ii (–2.5, 0) and (0.5, 0) 1 −1 −2 −3 −4 −5 −6 −7 −8 −9

sec 1

−7 −6 −5 −4 −3 −2 −1 −2

−4 −3 −2 −1

b feet

18 16 14 12 10 8 6 4 2

iii

10 a y = −x2 + 10x − 10

d i (0, 18) iii

914

ii (–3, 0) and (3, 0) 1

iii

−1 −2

b i (0, –9) iii

and (2, 0)

x 1

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

c 3.25 ft 12 a C arliss could make $525 because the maximum profit is $530 if each car wash costs $25. b

y 500 400 300 200 100

x 5

15 20 25 30 35 40 45 50

13 Both quadratic functions have a minimum value −9, opens upwards, and have the same y-intercept at (0, −8). A(x) has an axis of symmetry of x = −1 and B(x) has an axis of symmetry of x = 1. 14 Orland incorrectly used 0.5, when the a value of the quadratic function is −0.5. This error creates an incorrect axis of symmetry and y-value for the vertex. Delfino used an incorrect equation to find the axis of symmetry, as the equation is

Delfino forgot the 2

in the denominator. This error creates an incorrect axis of symmetry and y-value for the vertex. Delfino labeled the vertex of f (x) incorrectly as the vertex is (1, –4). The vertex of g(x) is (1, 8) and the vertex of f (x) is (1, –4)

16 a y = −5x2 + 10x + 3

c A fter 3 seconds. Because Beau starts at 3 ft above the pool surface, we know that the pool surface is at a height of 0 feet. Therefore, 12 feet deep would be the same thing as y = −12. We can plug this into our equation and see that x = 3, meaning that this will happen 3 feet after jumping. 17 y = x2 + 5x + 2 18 a Domain: 0 ≤ x ≤ 5. This is because the y-intercept is (0, 1.25) and the vertex is (2.5, 0). Parabolas are symmetric about the axis of symmetry and since the shape of the sign is symmetric, the sign must be 5 ft wide.

Let’s extend our thinking 15 a y = ax2 − 2ahx + ah2 + k b y = ax2 + (−2ah)x + (ah2 + k) so b = −2ah c y = ax2 + (−2ah)x + (ah2 + k) so c = ah2 + k d V ertex form and standard form are equivalent quadratic equations in different forms. The equations give us different information more efficiently. We can find the vertex, (h, k), efficiently using vertex form. We can find the y-intercept, c, efficiently by substituting in 0 for x and evaluating the corresponding output using standard form. We can also find the axis of symmetry,

b 2 seconds

Range: 0 ≤ y ≤ 1.25

This is because the leading coefficient is positive, so the minimum value occurs at the vertex. The maximum height of the banner would be at the end points of the banner and the left end point occurs at the y-intercept. b C hanging the constant, c, will translate the quadratic function vertically up or down. Changing the linear coefficient, b, will move the vertex. Changing the leading coefficient, a will widen or narrow the parabola and change the vertex. Students are encouraged to create graphs of several cases to make generalizations.

in standard form and use it to find the y-value of the vertex. In both forms, the leading coefficient, a, is the vertical stretch/compression factor and tells us if the parabola will open up or down.

Answers mathspace.co

915

7.05 Compare linear, quadratic, and exponential functions Subtopic overview Lesson narrative In this lesson, students will compare key characteristics that linear, quadratic, and exponential functions have in common, such as intercepts, domain, and range. Students will examine the similarities and differences in rates of change for linear and exponential functions. Students will examine tables of values and explore graphs to compare the key characteristics of these functions. By the end of the lesson, students will know the characteristics of a linear, quadratic, and exponential functions and identify their differences.

7.05 Compare linear, quadratic, and exponential functions

Learning objective Students: Page 439

After this lesson, you will be able to… compare key characteristics of linear, quadratic, and exponential functions using graphs and tables.

Comparing functions In this lesson, we will use our prior knowledge of linear, quadratic, and exponential functions to identify key features and compare various functions represented in different ways.

Key vocabulary 

characteristic (of a function)

Exploration

Consider the table:

Essential understanding x

y = 3x y = 3x2 y = 3x 1 certain3characteristics 3 that can be 3 identified from their equations, graphs, All of the functions in a given family share 2 6 12 9 or input/output pairs. Linear and exponential functions can be distinguished by their rate of change. 3 9 27 27 5 15 125 243 1. Compare the three functions and how they change as x increases. Standards

This subtopic addresses the following 2023 Mathematics Learning standards. The way a function is represented canVirginia affect the characteristics weStandards are able toofidentify for the function. Different representations can highlight or hide certain characteristics. Remember that key features of functions include: Mathematical process goals • how the function increases or decreases • domain and range • vertex • xy-interceptsProblem Solving MPG3 — Mathematical Reasoning MPG1 —and Mathematical • maximum or minimum value(s) This goal can be integrated by asking students to reason Teachers can integrate this goal by posing complex, One way to compare functions is to look at growth rates as the x-values increase over regular intervals.and In order to about the properties of linear, quadratic, exponential open-ended problems related to linear, quadratic, and compare the growth rates of quadratics with those of exponential or linear functions, we will look only at the half of functions, and how these properties influence the shape exponential functions. For example, they might ask the quadratic that is increasing. of the graph and the behavior of the function. Teachers students to model a real-world situation using each type y which could also challenge students to use logical reasoning to of function, then compare the models to determine predict the effect of certain transformations on each type is most appropriate. They could also prompt students to of function. use their knowledge of these functions to create 30 their own problems and then solve them. g (x) 916

20 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co 10

f (x) h (x)

MPG4 — Mathematical Connections To achieve this goal, teachers could highlight the connections between different types of functions, such as how the rate of change in a linear function relates to the slope of the line, or how the vertex of a quadratic function relates to its maximum or minimum value. Teachers could also emphasize the connections between mathematical concepts and real-world applications, such as how exponential functions can model population growth or compound interest.

A.F.2c — Graph a quadratic function, f (x), in two variables using a variety of strategies, including transformations f (x) + k and kf (x), where k is limited to rational values. A.F.2e — Given an equation or graph of an exponential function in the form y = abx (where b is limited to a natural number), interpret key characteristics, including y-intercepts and domain and range; interpret key characteristics as related to contextual situations, where applicable. A.F.2f — Graph an exponential function, f (x), in two variables using a variety of strategies, including transformations f (x) + k and kf (x), where k is limited to rational values.

A.F.1f — Graph a linear function in two variables, with and without the use of technology, including those that A.F.2g — For any value, x, in the domain of f, can represent contextual situations. determine f (x) of a quadratic or exponential function. A.F.1g — For any value, x, in the domain of f, Determine x given any value f (x) in the range of f of a determine f (x), and determine x given any value f (x) quadratic function. Explain the meaning of x and f (x) in the range of f, given an algebraic or graphical in context. representation of a linear function. A.F.2h — Compare and contrast the key A.F.2b — Given an equation or graph, determine characteristics of linear functions (f (x) = x), quadratic key characteristics of a quadratic function including functions (f (x) = x2, and exponential functions (f (x) = bx) x-intercepts (zeros), y-intercept, vertex (maximum or using tables and graphs. minimum), and domain and range (including when restricted by context); interpret key characteristics as related to contextual situations, where applicable.

A2.F.1 — The student will investigate, analyze, and compare square root, cube root, rational, exponential, and logarithmic function families, algebraically and graphically, using transformations.

7.05 Compare linear, quadratic, and exponential functions mathspace.co

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Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Algebra 1 — 2.04 Characteristics of functions Algebra 1 — 3.03 Slope-intercept form Algebra 1 — 5.08 Characteristics of exponential functions Algebra 1 — 7.01 Characteristics of quadratic functions

Tools You may ﬁnd these tools helpful: • Graphing calculator • Highlighters • Spreadsheet application

Lesson supports The following supports may be useful for this lesson. More speciﬁc supports may appear throughout the lesson:

Concrete-Representational-Abstract (CRA) Approach Targeted instructional strategies Concrete: Engage students with hands-on activities to explore linear, quadratic, and exponential functions. Use physical objects like stacking blocks, rubber bands, and paper folding. For linear functions, have students create a staircase with blocks, adding one block at each step to show constant growth. For quadratic functions, use rubber bands stretched between pegs on a pegboard to form parabolic shapes. For exponential functions, demonstrate layering paper folds to show how quickly the number of layers grows. These activities help students physically experience how each function behaves differently. Representational: Transition to visual representations by having students draw what they observed. For the linear function, they can draw a straight line showing constant growth. For the quadratic function, guide them to sketch a parabola based on the shape created with rubber bands. For the exponential function, have them plot the number of paper layers against the number of folds to create an exponential curve. Encourage students to label key features like intercepts, slope, and curvature. Abstract: Move to working with equations and tables of values. Teach students the general forms: y = mx + b for linear, y = ax2 + bx + c for quadratic, and y = a ⋅ bx for exponential functions. Have them create tables of values by choosing x-values and calculating y-values using the equations. Discuss how the rate of change differs: constant for linear, changing for quadratic, and increasing rapidly for exponential functions. Solve problems that require identifying the domain, range, and intercepts from the equations.

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Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Connecting the stages: Help students make connections between the physical activities, their drawings, and the equations. Ask guiding questions like: • “How does the staircase model relate to the straight line graph and the linear equation?” • “What does the rapid increase in paper folds tell us about exponential growth?” Encourage them to see how each stage represents the same concept in different ways. This will help them understand the characteristics of each function type and how to distinguish between them when using graphs and tables.

Student lesson & teacher guide Comparing functions Students begin with an exploration that involves comparing the patterns of different types of relationships.

Students: Page 439

7.05 Compare linear, quadratic, and exponential functions After this lesson, you will be able to… compare key characteristics of linear, quadratic, and exponential functions using graphs and tables.

Exploration the table: of functions and review which are easiest to identify ListConsider key features 2 x

y=3 y = 3x y = 3x 1 3 3 3 2 they 6 12 to compare 9 for different functions. In pairs, Provide students with a list of key features can be asked 3 9 27 a table of values, equation, or graph. have them identify strategies for determining a chosen key27feature from 5 15 125 243 Encourage students to consider if their approach would be different for linear, exponential or quadratic functions. x

Targeted instructional strategies

Compare the three functions and how they change as x increases.

It can also be helpful to examine which key features are easiest to identify from each of the different representations. Use a group discussion to have students consider each key feature. The way a function is represented can affect the characteristics we are able to identify for the function. Different • how the function increases or decreases • domain and range representations can highlight or hide certain characteristics. Remember that key features of functions include: • how the function increases or decreases • domain and range • vertex • x- and y-intercepts • vertex • x- and • maximum or y-intercepts minimum value(s) • maximum or minimum value(s)

Some possible be:is to look at growth rates as the x-values increase over regular intervals. In order to One way answers to comparecould functions compare the growth rates of quadratics with thosefunctions of exponential or linear functions, look only at the halfspecified of The domain of linear, quadratic, and exponential is “all real values ofwe x,”will unless otherwise the quadratic that is increasing. or determined by a constraint. It is easiest to see this with the graph. In a table of values, we can assume the y pattern continues unless otherwise stated. 30 g (x) 20

7.05 Compare linear, quadratic, and exponential functions f (x) mathspace.co h (x)

919

To identify the y-intercept, it is easiest from equations in the form y = mx + b, y = ax2 + bx + c, or y = abx, as we can read the y-intercept directly from these forms. We can also see it fairly easily from the table if it is one of the given values, but otherwise we first have to find the equation or extrapolate from the table. From the graph, we need to look where the function crosses the y-axis. All of these functions will have exactly one y-intercept.

Collect and display English language learner support Use the “Collect and Display” routine to help students connect their own words to precise mathematical vocabulary when comparing linear, quadratic, and exponential functions. During discussions, listen for phrases students might use, such as “going up in a straight line,” “making a U-shaped curve,” “getting steeper quickly,” or “leveling off eventually.” Record these phrases on a chart or board under headings like “Students’ Descriptions.” Next to each student’s phrase, write the corresponding mathematical term—like “linear function,” “parabola,” “exponential growth,” or “asymptote”—under a heading like “Mathematical Vocabulary.” Include simple sketches or diagrams to illustrate each concept. Encourage students to refer to this visual display when analyzing graphs and tables, and prompt them to use the precise vocabulary in their explanations. This approach supports English language learners by bridging their informal language to academic terms, enhancing their understanding of both mathematical concepts and the language used to describe them.

Key features not appearing on the graph Address student misconceptions Students may think that they cannot compare a key feature if it is not present in the graph. For example, that they cannot compare the number of x-intercepts for the two graphs: 8

6 4 2 −8 −6 −4 −2 −2

x 2

−4 −6 −8

Challenge this misconception by asking students how they would answer the question “identify the … for f (x)” to remind them that “none” or “zero” or “never” are valid answers.

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Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Use graphic organizers to highlight key features of functions Student with disabilities support Remind students of the key characteristics of functions that will be addressed in this subtopic. Provide a resource sheet or graphic organizer to help them draw connections and complete examples. 4

3 2 1

−4 −3 −2 −1 −1

• Domain: (−∞, ∞) • Range: (−∞, ∞) • x-intercept: (0, 0) • y-intercept: (0, 0) • Increasing: m > 0 • Decreasing: m < 0 • Constant rate of change

−2 y = x2 −3 −4

4 3

y y = x2

2 1

−4 −3 −2 −1 −1

• Domain: (−∞, ∞) • Range: [0, ∞) • x-intercept: (0, 0) • y-intercept: (0, 0) • Decreases at a decreasing rate when x < 0 • Increases at increasing rate when x > 0

−2 −3 −4

3 2 y = 2x

−4 −3 −2 −1 −1

x 1

• Domain: (−∞, ∞) • Range: (0, ∞) • x-intercept: None • y-intercept: (0, 1) • Increases at increasing rate • Asymptote at y = 0

−2 −3 −4

7.05 Compare linear, quadratic, and exponential functions mathspace.co

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exponential functions After this lesson, you will be able to… compare key characteristics of linear, quadratic, and exponential functions using graphs and tables.

Exploration Comparing functions

Students:InPage 439 this lesson, we will use our prior knowledge of linear, quadratic, and exponential functions to identify key features and compare various functions represented in different ways.

Exploration Consider the table: x 1 2 3 5 1.

y = 3x 3 6 9 15

y = 3x 3 9 27 243

y = 3x2 3 12 27 125

Compare the three functions and how they change as x increases.

The way a function is represented can affect the characteristics we are able to identify for the function. Different representations can highlight or hide certain characteristics. Remember that key features of functions include: • how the function increases or decreases • domain and range Suggested grouping: Small groups • vertex • x-student and y-intercepts maximum minimum Students• are given or a table of value(s) values and the equations for three different types of patterns: a linear, a quadratic, One way to compare functions to look at how growth rates as thechange x-values based increaseon over regular intervals. In order to which and an exponential. Students will is compare the values the functions in the table, compare rates growth of quadratics with those of exponential or linear functions, we will look only at the half of aligns with how the wegrowth saw their patterns in previous chapters. the quadratic that is increasing.

Ideal student responses

7.05 Compare linear, quadratic, and exponential functions

These ideal responses may differ from other correct student responses. Less formal responses can be 30 language presented here. connected with the more precise mathematical g (x)

1. Compare the three functions and how they change as x increases. f (x) For y = 3x, the values are increasing at a20constant rate. For y = 3x2 and y = 3x, the values will increase at an increasing rate, exponential h (x) the quadratic function. Afterwith this the lesson, you will be function able to… increasing faster than compare key characteristics of linear, 10 quadratic, and exponential functions using graphs and tables.

Purposeful questions

• Do you notice a pattern in the y-values of the function as x increases? x Comparing functions • Would a graph of each function help you to compare the functions and how they change as x increases?

−1 1 quadratic, 2 3 and 4 5 6functions to identify key features In this lesson, we will use our prior−2knowledge of linear, exponential and compare functions represented in different ways. at a greater rate. But, as x continues to increase, PossibleNotice misunderstandings startingvarious at x = 0, g(x) is greater than h(x) and is increasing

the quadratic function g(x) ishow increasing at a slowerchange rate thanand the exponential function, • Students may not compare the patterns instead state that and theeventually functionsthe allexponential increase. Point function will overtake quadratic have function. Exploration out to students that thethe functions different growth patterns and encourage them to identify which Notice that no quadratic, matter what the are, an exponential growth function will always exceed a linear or quadratic pattern is linear, andintercepts exponential. Consider the table: growth function as values of x become larger. x y = 3x

y = 3x y = 3x2 7.05 Compare linear, quadratic,functions. and exponential functions 1 3 3 3 Students are reminded of the growth patterns of linear, quadratic, and exponential A list of the439 key features mathspace.co 2 6 12 9 of functions is presented as review. 3 9 27 27 5 15 125 243 Students: Page 439 1.

Compare the three functions and how they change as x increases.

30 g (x)

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Mathspace Virginia SOL Algebra 1 Teacher Edition 20 mathspace.co 10

f (x) h (x)

30 g (x) f (x)

h (x) 10 x −2

−1

Notice starting at x = 0, g(x) is greater than h(x) and is increasing at a greater rate. But, as x continues to increase, the quadratic function g(x) is increasing at a slower rate than the exponential function, and eventually the exponential function will overtake the quadratic function. Notice that no matter what the intercepts are, an exponential growth function will always exceed a linear or quadratic growth function as values of x become larger. 7.05 Compare linear, quadratic, and exponential functions mathspace.co

439

Examples Students: Page 440

Example 1 Which of the following functions increases the fastest for very large values of x? • y=9⋅x • y = 3x • y = 2x2 • y = 4x

Create a strategy Remember that exponential functions (b > 1) will always have a greater rate of change when compared to linear and quadratic functions as x increases toward infinity. Then, we must compare the constant factor b in the equation y = a ⋅ b x.

Apply the idea

Reﬂect and check

As x increases toward infinity, we know that our greatest rate of change is from one of the exponential functions, y = 3x or y = 4x. Remembering our lesson on characteristics of exponential functions, a greater constant factor, b, will result in a greater rate of change as x continues to increase. Therefore, y = 4x increases the fastest for very large values of x.

How would you approach this problem if exponential equations had 2 different leading coefficients? For example, would y = 2 ⋅ 4x or y = 4 ⋅ 2x have a greater rate of change as x increases toward infinity?

Example 2 Purpose Consider the functions shown. Assume that the domain of f is all real numbers. Check student understanding of different types of functions algebraically and their rates of change. • Function 1:

• Function 2:

Reflecting with students x −1 0 1 2 3 4 5 y Challenge advanced learners to further articulate why exponential functions with larger bases grow faster than 12 f (x) −3.75 −2 −0.25 1.5 3.25 5 6.75 those with smaller bases, and why exponential functions eventually surpass 10 linear and quadratic functions, regardless of the coefficients. An example student response is shown: 8 6 4 2

g (x) x

7.05 Compare linear, quadratic, and exponential functions −8 −6 −4 −2 2 4 6 8 mathspace.co −2 −4

923

Exponential functions with larger bases grow faster because their outputs multiply by the base each time Apply the idea Reﬂect and check x increases. For example, in y = 3x, the output triples with each increase in x, while in y = 4x, it quadruples. As x increases toward infinity, we know that our How would you approach this problem if exponential Multiplying by 4rate grows faster multiplying by 3. Exponential functions eventually linear greatest of change is than from one of the exponential equations had 2 different leading outgrow coefficients? For functions x x xgrows faster x becausefunctions, linear functions grow by adding a constant amount, and multiplying than adding. y = 3 or y = 4 . Remembering our lesson example, would y = 2 ⋅ 4 or y = 4 ⋅ 2 have a greater rate on characteristics exponential functions, ato greater of change asfunctions x increases toward infinity? Quadratics grow by an ofamount proportional x2, but exponential double, triple, etc., with each unit constant factor, b, will result in a greater rate of change increase in x. x as x continues to increase. Therefore, y = 4 increases the fastest for very large values of x.

Students: Page 440 Example 2

Consider the functions shown. Assume that the domain of f is all real numbers. • Function 1:

• Function 2:

−1

f (x)

−3.75

−2

−0.25

1.5

3.25

6.75

10 8 6

g (x)

4 2 −8 −6 −4 −2 −2

x 2

−4

a Determine which function has a higher y-intercept.

Create a strategy Remember that the y-intercept of a function occurs when x = 0. We can use this to evaluate the y-intercept of f and identify the y-intercept of g.

Apply the idea For f, we can see from the table that f (0) = −2. For g, we can see from the graph that g(0) = −3. So, the y-intercept of f is the point (0, −2) and the y-intercept of g is the point (0, −3), and therefore f has a higher y-intercept.

440

Mathspace Virginia SOL Algebra 1 mathspace.co

Purpose Show students how to find the y-intercept for types of functions. Reflecting with students Ask students to explain what the y-intercept represents in a function, and how it can be identified from a graph and a table of values.

Students: Page 441 b Determine which function will be greater as x gets very large.

924

Create a strategy

Apply the idea

We can consider how quickly each function changes to get an idea of how it will behave for very large values of x.

We can see that f (x) has a constant rate of change (slope) of 1.75. This is a linear function. From the graph, we can see that g(x) increases at an increasing rate as x increases. So, we can see that Function 2 will eventually surpass Function 1 as x gets very large.

Example 3

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co Consider functions representing three options to earn money one of the following ways: Option 1 You are given $2

Option 2

Option 3

Purpose Check that students can identify function types and understand an increasing quadratic function will always surpass a linear function. b Determine which function will be greater as x gets very large.

Reﬂecting with students a strategy idea the growth of a function. Highlight Discuss Create with students the concept of rate of change and Apply how itthe affects We can consider how quickly each function changes We can see thatand f (x) an has increasing a constant rate of change the difference between a constant rate of change in a linear function rate of change in a to get an idea of how it will behave for very large values (slope) of 1.75. This is a linear function. From the graph, quadratic function. of x.

we can see that g(x) increases at an increasing rate as x increases. So, we can see that Function 2 will eventually surpass Function 1 as x gets very large.

Students: Page 441

Example 3 Consider functions representing three options to earn money one of the following ways: Option 2 Days

Total Amount

$16

$25

$36

Option 3 Total Amount (in dollars)

Option 1 You are given $2 each day

60 50 40 30 20 10 1

2 3 4 5 6 7 8 9 10 Days

Note: Option 3 starts with $2 on day one and doubles each day after this. a Find the equation that represents each option, where x is the number of days that have passed.

Create a strategy For each option, we can consider how the total amount of money changes as the days progress and derive an equation to represent the relationship.

Apply the idea We can see that Option 1 has a constant rate of change regardless of the interval we considered. So, Option 1 can be represented by the linear function, f (x) = 2x. Now, observing the table of values for Option 2, we can see that the total amount is just the square of the number of days passed. So, Option 2 can be represented by the function f (x) = x2. Finally, the relationship for Option 3 is represented in the graph, but also described to us. Since we are told that the function starts at $2 and is doubled each day, we can see that Option 3 is just represented by the function f (x) = 2x.

Reﬂect and check If the relationship between the days passed and the total amount weren’t directly obvious in Option 2, we could have tested the data provided in the table to rule out a linear or exponential relationship. For a linear relationship, the rate of change between any two points must be equal. We can check that this wasn’t true for Option 2. So, we could have then tested if it represented an exponential relationship. For an exponential relationship, the ratio of between two points, a unit apart, must be equal. We can see that for Option Therefore, we could see that Option 2 represented neither a linear or exponential relationship.

7.05 Compare linear, quadratic, and exponential functions mathspace.co

441

Purpose Show students how to derive equations from different types of growth patterns, including linear, quadratic, and exponential.

7.05 Compare linear, quadratic, and exponential functions mathspace.co

925

Reflecting with students Have students consider how they could write a formula for each function in a spreadsheet to quickly make a table of values. As an extension for advanced students, encourage them to define each relationship both explicitly in terms of n and recursively. For example: • Explicit form: 1 2 3 4 5

A Day 1 = A2 + 1 = A3 + 1 ⋮

B Option 1 2 = 2 ∗ A3 = 2 ∗ A4 ⋮

C Option 2 1 = A32 = A42 ⋮

D Option 3 2 = 2A3 = 2A4 ⋮

B Option 1 2 = B2 + 2 = B3 + 2 ⋮

C Option 2 1 = C2 + 2 ∗ A3 − 1 = C3 + 2 ∗ A4 − 1 ⋮

• Recursively: 1 2 3 4 5

A Day 1 = A2 + 1 = A3 + 1 ⋮

D Option 3 2 = D2 ∗ 2 = D3 ∗ 2 ⋮

Students: Page 442 b Find the value of each option at 8 days, 12 days, and 14 days.

Create a strategy Construct a table of values with the amounts of money gained with each option.

Apply the idea Days

Option 1 Total

Option 2 Total

Option 3 Total

$16

$10

$25

$32

$12

$36

$64

$14

$49

$128

$16

$64

$256

$18

$81

$512

$20

$100

$1024

$22

$121

$2048

$24

$144

$4096

$26

$169

$8192

$28

$196

$16 384

At 8 days, Option 1 will make $16, Option 2 will make $64, and Option 3 will make $256. At 12 days, Option 1 will make $24, Option 2 will make $144, and Option 3 will make $4096. At 14 days, Option 1 will make $28, Option 2 will make $196, and Option 3 will make $16 384.

Reﬂect and check We could calculate the total amount of money on days 8, 12 and 14 using the functions found in part (a), instead of constructing a table.

c Determine which option will be greater for larger and larger values of x.

926

Mathspace Virginia SOL Algebra 1 Teacher Edition Create a strategy mathspace.co Use the table comparison from part (b) to determine which option will be greater for larger and larger values of x.

Apply the idea

Reﬂect and check

$20

$100

$1024

$22

$121

$2048

$24

$144

$4096

$26

$169

$8192

Purpose 14 $28 $196 $16 384 Show students how to determine the value of each option on specific days using different representations: At 8tabular, days, Option will make $16, Option 2 will make $64, and Option 3 will make $256. numerical, and1graphical. At 12 days, Option 1 will make $24, Option 2 will make $144, and Option 3 will make $4096.

Reﬂecting with students At 14 days, Option 1 will make $28, Option 2 will make $196, and Option 3 will make $16 384. Ask students: “If you could choose one of these options to earn money for 30 days, which one would you choose Reﬂect and why?” and check We could calculate the total amount of money on days 8, 12 and 14 using the functions found in part (a), instead of

Students:constructing Page 442 a table.

c Determine which option will be greater for larger and larger values of x.

Create a strategy Use the table comparison from part (b) to determine which option will be greater for larger and larger values of x.

Apply the idea

Reﬂect and check

As x gets larger and larger, we can see that Option 3, the exponential option, will be far greater than Options 1 or 2.

An exponential function will always exceed a linear or quadratic function as values of x become larger.

Purpose Students demonstrate their understanding of exponential growth compared to linear and quadratic growth. Reﬂecting students 442 with Mathspace Virginia SOL Algebra 1 Ask studentsmathspace.co to describe the differences in the growth patterns among the three options. Why does the exponential function grow faster?

Visual supports for understanding functions

use with Example 3

Student with disabilities support Use graphs to represent the functions for each option to earn money. The students can observe how the total amount of money changes with the number of days for each option and compare the graphs to understand which option will be greater for larger and larger values of x. Students can also use color-coding to differentiate between the three options. For example, they can use one color to represent Option 1, another color for Option 2, and a third color for Option 3. This can help them visually distinguish between the different options and their rates of change.

Students: Page 443

Idea summary It is important to be able to compare the key features of functions whether they are represented in similar or different ways: • • •

domain and range x- and y-intercepts maximum or minimum value(s)

• •

how the function increases or decreases vertex

Practice What do you remember? 1

For each pair of functions, determine which function y is changing more rapidly: • Function 1: • Function 2: quadratic, and exponential functions a 7.05 Compare linear, mathspace.co x x −1 0 1 2 0 1 2 3 y y 3 10 17 24 −1 3 7 11 b

• Function 3:

• Function 4:

927

Practice Students: Pages 443–447

What do you remember? 1

For each pair of functions, determine which function y is changing more rapidly: • Function 1: • Function 2: a x 0 1 2 3 y 3 10 17 24 b

−1 −1

x y

0 3

1 7

• Function 3: • Function 4: y 5

1 1

−2

−3 −4

−5

x −3 −2 −1 0 1 y 3 6 7 6 3

−2

−1

2 3 4 5

2 2

3 2 1 −5 −4 −3 −2 −1 −1

x 1

−2

928

−5 −4 −3 −2 −1 −1

2 3 4 5

For each pair of table and its graph, determine whether a linear or quadratic function could represent it: a

−5 −4 −3 −2 −1 −1

2 11

−4 −3 −2 −1 −1

x 1

−2 −3 −4

For large values of x, does the function f (x) = 3x − 1 or g(x) = 3x2 − 1 increase at a faster rate?

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

The graphs of f (x), g(x) and h(x) are shown. Use the graphs to complete the following:

Identify the graphs as exponential, linear, or quadratic.

Evaluate each function for x = 2.

Which function do you think will have the largest value at x = 100? Explain

Approximate the value of x for each function when the function value is 12.

h (x) f (x)

12 g (x)

8 4 2

x 1

The population of two different bacteria, labeled J and K, are given by the shown table of values: Bacteria J: t (Time in days) P (Population)

0 1

1 120

2 480

3 1080

0 1

1 50

2 2500

3 1.25 × 105

4 1920

Bacteria K: t (Time in days) Q (Population)

4 6.25 × 106

The population P of bacteria J at time t can be modeled using the general equation P(t) = at2, where a ≥ 0. By using the table of values, graph P(t) for t > 0.

The population Q of bacteria K at time t can be modeled using the general equation Q(t) = bt, where b > 1. By using the table of values, graph Q(t) for t > 0.

Determine which population of bacteria is growing faster.

Let’s practice 6

For each pair of functions, determine which has the greater y-intercept: • Function 1: • Function 2: a 2 y = 4x + 6x + 3 y = 4x + 6 b

• Function 3: • Function 4: y 5

1 −5 −4 −3 −2 −1 −1

x 1

2 3 4 5

−5 −4 −3 −2 −1 −1

−2

−3 −4

−5

x 1

2 3 4 5

• Function 1: The line with a slope of 4 that crosses the y-axis at (0, 6). • Function 2: The parabola given by the equation y = x2 + 4.

7.05 Compare linear, quadratic, and exponential functions mathspace.co

929

• Function 3:

• Function 4:

x 2 4 6 y 2 −2 −6

y 5 4 3 2 1 x −2

• Function 5: x y

2 19

4 35

−1

• Function 6: y = 4x + 6

6 51

Consider each pair of functions: • Function A: y = −4x + 3

• Function B: y 12 10 8 6 4 2 −4 −2 −2

x 2

10 12

−4

930

Determine how many x-intercept(s) function A has.

Determine how many x-intercept(s) function B has.

Which function has a smaller value for f (4)?

The parabola C is given by

and the exponential function D is given by y = −8x + 1.

Graph the functions on the same coordinate plane.

Determine which function has the lower y-intercept.

The parabola E is given by y = 12(x − 1)2 − 4 and the exponential function F is given by y = 3x + 4. a

Graph the functions on the same coordinate plane.

State which function increases at a faster rate for very large values of x.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

The parabola J is given by y = −9x2 + 20. The table shows the function values for exponential function K: x y

−2

−1

135

Determine if the exponential function K is increasing or decreasing.

Graph the functions J and K on the same coordinate plane.

Determine what transformation we can apply to the function y = 3x to produce function K.

Which function has a larger value of y when x = 6? What is the that y-value?

The line P is given by

and the parabola Q is given by y = −(x − 1) (x − 4).

Graph the line P and the parabola Q on the same coordinate plane.

Identify how many times P and Q intersect.

Identify which function has the higher function value at x = 0.

The table of values for the function P and for the function Q are provided. Function P:

Function Q:

x x −2 −1 0 1 2 y y 9 6 3 0 −3

0 6

1 3

2 2

Determine what type of functions Function P and Function Q are.

Graph the functions on the same coordinate plane.

As x gets very large, determine which function will have the greater value.

3 3

4 6

Let’s extend our thinking 13

Some friends decide to go camping for the weekend. They cannot all ﬁt in one car so some of them catch a bus to the campground, which is 450 km from home. Those in the car started driving at 8:00 AM and arrived at the campground at 3:30 PM, driving at a constant speed. The bus also drives at a constant speed and takes the same route as the car. Its distance in kilometers ( y) from home x hours after leaving is given by the equation y = 71x. a

Determine the speed of the car, in kilometers per hour.

Determine the speed of the bus, in kilometers per hour.

Determine which vehicle was traveling faster.

Consider the functions f (x) = 2x, g(x) = 2x2 and h(x) = 2x for x ≥ 0. a

Describe the pattern of how each of the functions increases. Explain how you identiﬁed the patterns.

Compare how the three functions increase as x gets very large.

7.05 Compare linear, quadratic, and exponential functions mathspace.co

931

Two companies Crest Corporation and Mint Corporation are operating mines. Crest Corporation’s operations are such that the total amount mined by the nth week is given by the equation C = 10n2. The total amount mined by Mint Corporation over time is shown in the graph. Amount mined (thousands of metric tons) 10 8 6 4 2 5

Week 60

If mining operations for both companies were to only last at most a year, determine which company will have mined the most minerals in that time.

Determine if the two corporations will ever mine the same amount at the same time after the ﬁrst week.

At the point of intersection, the total quantity of minerals remaining in both mines is equal. If both mining companies continue to operate in the same way indeﬁnitely, determine which company will exhaust their mine ﬁrst. Explain your reasoning.

During a sudden viral outbreak, scientists must decide between two antivirals to try and control the situation. In a laboratory, they apply Adravil and Felicium to two samples of the virus, each containing 200 microbes. They keep track of the number of microbes in each sample, and notice that the number of microbes using Adravil is increasing by a constant amount of 12 each hour. The table shows the results for Felicium. Number of hours (t) Number of microbes using Felicium

0 200

3 600

6 1800

9 5400

Determine which antiviral will better control the number of microbes. Explain your choice.

The new antiviral, Tretonin, shows the preliminary results: Number of hours (t) Number of microbes using Tretonin

0 200

3 202

6 208

9 218

If the trends from the ﬁrst 9 hours continue in the future, determine which treatment will be better of the short-term, and which will be better over the long-term. Justify your answer.

932

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Answers

8 a

14 12 10 8 6 4 2

7.05 Compare linear, quadratic, and exponential functions What do you remember? 1 a Function 1

b Function 3

2 a Quadratic

b Linear

3 a g(x)

8 6 4 2 2 4 6

2 4 6 8 10 12

b Exponential function D

4 a Linear function: f (x)

Exponential function: g(x)

Quadratic function: h(x)

b Function F

9 a y 25 20

b f (2) = 8, g(2) = 4, h(2) = 8

c g (x) because it is exponential it increases more quickly and will eventually surpass the linear and quadratic functions.

10 F

d When f (x) = 12, x = 3.

−3 −2 −1

−5

When g(x) = 12, x ≈ 3.5.

When h(x) = 12, x ≈ 2.5. 5 a

10 a Increasing

P(t)

300

y 15

250

200

150

−3 −2 −1 −5

100 50

−10

t 1

1 2 3 4 5 J

−15

c Dilate y = 3x vertically by a factor of 5.

Q(t) 300

d T he exponential function K has the larger y-value of y = 3645.

250 200

11 a y

b Twice

150

100

t 1

−4 −3 −2 −1 −2

c T he population of bacteria K is growing faster than the population of J.

−4

x 1 2 3 4 5 6 Q P

−6 −8

Let’s practice 6 a Function 2

b Function 4

d Function 3

e Function 6

7 a 1

b 0

c Function 1

c They are equal at x = 0.

c Function A

Answers mathspace.co

933

12 a Function P is a linear function and function Q is a quadratic function. b

10 8 6 4 2 −5 −4 −3 −2 −1 −2 −4 −6 −8 −10

Q x 1 2 3 4 5 P

Let’s extend our thinking 13 a 60 km/h

b 71 km/h

c Bus 14 a W e can ﬁll out a table of values for each function and calculate how much they increase as x increases by 1. Increase in f (x)

Increase in g(x)

f (x)

g(x) 0

h(x)

Increase in h(x)

128

The linear function, f (x), increases at a constant rate which is the slope of the line. The quadratic function, g (x), increases at a constantly increasing rate. The increase in g (x) is increasing by 4 each time. The exponential function, h (x), is increasing at an exponentially increasing rate. b T he graphs of f (x) (green), g (x) (blue) and h (x) (purple) can help us to see how they increase comparatively.

y 6 5 4 3 2 1

x 1

934

x 1

c Function Q

100 90 80 70 60 50 40 30 20 10

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

For very small values, 0 < x < 1, linear function is generally increasing the most quickly and the exponential is increasing the slowest. For about 1 < x < 2.7, the quadratic function increases more rapidly than the linear, and the exponential increases the slowest. For about 2.7 < x < 6.4, the quadratic is increasing the most quickly, followed by exponential, and linear is the slowest. For large positive values of x, the exponential increases the most quickly, followed by quadratic, and linear is the slowest. 15 a Crest Corporation b Yes, they will. c M int Corporation.The amount mined by Mint Corporation can be modeled by an exponential function, and that of Crest Corporation can be modeled by a quadratic function. Before the point of intersection the quadratic function was above the exponential function, but after the point of intersection the exponential function will increase much more rapidly than the quadratic function and so Mint Corporation will exhaust their mine first. 16 a A dravil, because the bacteria are only growing linearly, increasing at 12 per hour, while with Felicium they are growing exponentially, tripling every hour, which tends to infinity much quicker. b I n the short-term, Tretonin is better than Adravil in the first 24 hours because it is growing by less than 12 microbes per hour, but the rate of growth is increase by 6 every three hours. In the long-term, Adravil is better than Tretonin because after 24 hours it will be increasing at a faster rate than Adravil as it is quadratic growth versus linear.

Topic 7 Assessment: Quadratic Functions SOL

A function is represented by this rule. 12

Three less than one-fourth the square of a number x is y.

Plot three points on the grid that are represented by this rule. Each point must have coordinates and integers.

6 3

−12 −9 −6 −3 −3

9 12

−6 −9 −12

SOL

Which equation could represent a graph with x-intercepts of (8, 0) and (−3, 0)? A

SOL

y = x2 − 5x + 24

y = x2 + 5x + 24

y = x2 − 5x − 24

f (x) = x2 − 6x + 9

f (x) = x2 + 6x + 9

f (x) = x2 − 6x + 8

f (x) = x2 + 8x + 16

Select every function that has exactly one zero. A

SOL

y = x2 + 5x − 24 f (x) = 3(x − 4) 2

f (x) = 2(x − 4)

f (x) = 4x2 − 36

f (x) = x − 2

Select every function that has an x-intercept at 2. A

f (x) = 3(x − 6)

f (x) = x2 − 36

f (x) = 2(x + 2)2

f (x) = x2 − 4

A function f is described: • f (x) = (x − 4)2 + 7 • The domain of f is all real numbers greater than 2. The range of f is all real numbers greater than or equal to A

The graph of h(x) shows the height of a bird diving to catch a ﬁsh, in meters, x seconds after beginning the dive.

Find and interpret the y-intercept.

Find and interpret h(2).

Find the coordinates of the vertex and describe what it means in context.

State the domain of the function and describe what it represents in context.

h(x)

20 15 10 5

x 1

For each quadratic function: i

State the coordinates of the y-intercept.

Determine the axis of symmetry.

iii

Determine the coordinates of the vertex.

State the domain and range.

y = −2 (x − 1) (x − 5)

y = (x − 2)2 + 4

y = −x2 − 4x − 9 Topic 7 Assessment: Quadratic Functions mathspace.co

935

For each function: i

Graph the function.

State the domain and range.

f (x) = − (x + 3) (x − 5)

g (x) = 0.5(x + 3)2 − 2

h(x) = 3x2 − 12x + 12

p (x) = 12x2 − 27

For each function: i

Describe the transformations from y = x2.

Sketch the graph.

y = 3x2 − 7

An object is thrown into the air. The height (in feet), h, reached by the object after t seconds is modeled by a quadratic function. The table of values shows some points on the function: t h (t)

1 7

2 12

3 15

4 16

5 15

6 12

7 7

8 0

Write the equation of the quadratic function in vertex form.

Determine and interpret h(5.5).

Find the maximum height reached by the object.

Determine how long it will take the object to hit the ground.

Kyle throws a ball to his friend, Rick, from a height of 2.25 ft. The ball reach a maximum height of 18.25 ft, 1 second after being thrown. a

Determine the equation in standard form that represents the situation.

Graph the corresponding parabola.

If Rick caught the ball after 2 seconds, determine the height of the catch. and g(x) = 3 (x − 5) (x − 4):

For a

Determine which function has the lower minimum value.

Identify which parabola has the higher function value at x = −2.

Consider these functions: Function B:

Function A: x a(x)

−3 −5

−2 1

−1 3

0 1

1 −5

2 −15

3 −29

8 6

y b(x)

4 2 −1 −2 −4 −6 −8

936

Determine which function has the higher maximum value.

Determine which function has a lower y-intercept.

Find a(1).

Find x given that b(x) = 4.

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

x 1 2 3 4 5 6 7 8 9 10

Mary got a job offer from a company. The company offered her an income of $156 000 for the ﬁrst year, and a projected income increase of $12 400 each year after that. Alternatively, she can start her own business, where she estimates her initial annual income to be $52 000 for the ﬁrst year, increasing by 38% each year after that.

Determine whether or not Mary’s income for each situation could be modeled by a linear, quadratic, or exponential model.

Find her income after ﬁve years for both options.

The table shows the net proﬁt (in thousands of dollars) over time of two restaurants where x is the time in weeks: x Restaurant A Restaurant B

1 2 9

2 4 16

3 8 25

4 16 36

5 32 49

6 64 64

7 128 81

Determine the net proﬁt for each restaurant in week 1.

Determine when the net proﬁt of both restaurants will be equal.

Assuming both restaurants stay in business for a very long time, which restaurant would you expect to reach a proﬁt of $1 million ﬁrst? Explain your reasoning.

Prossy is starting a summer car wash business. Her revenue function represents the amount of money she makes in dollars for x car washes. R (x) = −x2 + 48x + 60 Prossy spent $68 on car wash supplies, and it costs her $7 per car for other related fees. a

Write a function that represents Prossy’s proﬁt.

Prossy wants to make at least $400 this summer. Determine if she could make enough money based on her proﬁt model. Explain your reasoning.

Draw a graph of P (x).

Performance task 17

A diver dives off of a boat, following a parabolic path to her deepest point and returns to the surface. Let x represent her distance, in feet, from the boat and let y represent her depth, in feet. Each of the quadratic equations represent the diver’s path. Standard form: y = 0.15x2 − 4.05x + 10.8 Factored form: y = 0.15 (x − 3) (x − 24) Vertex form:

y = 0.15(x − 13.5)2 − 16.537

For each of the following, ﬁnd the requested value and explain which form you used and why: a

The deepest depth the diver reaches under the water.

The diver’s depth when she is 10 ft away from the boat.

The height the diver jumped from.

The distance from the boat the diver was when she hit the water and when she came up for air.

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A soccer ball is tossed into play by a player who is 6 ft tall. It has an initial upward velocity of 20 ft/s and the force of gravity causes the ball to change position by an additional −32 ft/s2. Description Starting height Travels upwards at 20 ft/s Pulled downward at 32 ft/s2

Term in model

−16x2 , where x is the time in seconds,

Note: The standard model for projectile motion is given by 2

v0 is the initial upward velocity, h is the starting height, and g is 32 ft/s .

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Complete the table above and write a quadratic equation to model this relationship.

What form of the quadratic equation will help you identify how long it takes the ball to reach its maximum height? Write the equation in that form and ﬁnd the time.

What form of the quadratic equation will help you identify how long it takes the ball to hit the ground (assuming it doesn’t come in contact with any players ﬁrst)? Write the equation in that form and ﬁnd the time.

What form of the quadratic equation will help you identify the height of the ball 0.5 seconds after it is thrown? Write the equation in that form and ﬁnd the height.

Draw a graph of the function. What information from each form of the quadratic equation helped you to draw the graph?

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Answers

8 a i

18 16 14 12 10 8 6 4 2

Topic 7 Assessment: Quadratic Functions 1

(8, 13)

−6−5−4−3−2−1 −2 −4 −6

6 3

x (4, 1) 3 6 9 12 −12 −9 −6 −3 −3 (0, −3)

1 2 3 4 5 6

ii Domain: All real values of x

−6

Range: y ≤ 16

−9 −12

b i 5

4 3

A.F.2c

2 D

A.F.2b

−7 −6 −5 −4 −3 −2 −1 −1 −2

3 A, C, D, E

−3

A.F.2b

ii Domain: All real values of x

4 A, C, F

Range: y ≥ −2

A.F.2b

c i 5 B

6 4

6 a The y-intercept is y = 45. This means that initially the bird is 45 m above the water.

3 2 1

b h (2) = 5. This means that after 2 seconds the bird is 5 m above the water.

−2 −1 −1

c (3, 0)

d Domain: 0 ≤ x ≤ 6

A.F.2b, A.F.2g ii x = 3

iii (3, 8)

iv Domain: All real values of x Range: y ≤ 8 ii x = 2

iii (2, 4)

iv Domain: All real values of x Range: y ≥ 4 ii x = −2

iv Domain: All real values of x Range: y ≤ −5

2 3 4 5 6

ii Domain: All real values of x Range: y ≥ 0

The domain represents the seconds of the bird’s dive. The dive lasted a total of 6 seconds.

c i (0, −9)

x 1

−2

After 3 seconds, the bird has reached the water and caught the ﬁsh.

b i (0, 8)

A.F.2b

7 a i (0, 10)

x 1

iii (−2, −5)

d i

−4 −3 −2 −1 −3 −6 −9 −12 −15 −18 −21 −24 −27

x 1

2 3 4

ii Domain: All real values of x Range: y ≥ −27 A.F.2b, A.F.2c, A.F.2d

A.F.2b

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9 a i V ertical dilation by a factor of 3 (stretch), vertical translation 7 units downwards ii 8

d b(3) − 4 and b(7) = 4 A.F.2b, A.F.2g, A.F.2h

4 2

−4 −3 −2 −1 −2

14 a I ncome in year n with the company is modeled by a linear function.

2 3 4 y = 3x2 − 7

−4

Income in year n with her own business is modeled by an exponential function.

−6 −8

b C ompany: Her income after ﬁve years would be $218 000.

b i Vertical dilation by a factor of

(compression),

reﬂection across the x-axis, vertical translation 4 units upwards, horizontal translation 1 unit to the left 8 4 2

−4 −3 −2 −1 −2 1 y = − ( x + 1)2 + 4 2 −4

2 3 4

10 a h (t) = − (t − 4)2 + 16

= (−x2 + 48x − 60) − (7x + 68)

= −x2 + 41x − 8

b h (5.5) = 13.75 ft. The height of the object after 5.5 seconds have elapsed is 13.75 ft. c 16 ft d 8 seconds A.F.2b, A.F.2g 11 a y = −16x2 + 32x + 2.25

450 P (x) 400 350 300 250 200 150 100 50 5

x 10

A.F.2c, A.F.2g Performance task 17 a − 16.537 ft or 16.537 ft below the water. This is from the y-coordinate of the vertex and can be easily found from the vertex form. Time in seconds 3

c 2.25 ft A.F.2c, A.F.2d, A.F.2g 12 a f (x) b g(x) A.F.2b, A.F.2g, A.F.2h

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c R estaurant A because the net proﬁt is doubling each week while Restaurant B’s net proﬁt is increasing at a slower rate.

b P rossy could make $400 because the maximum proﬁt is $412.25 if she washes 20.5 cars.

A.F.2c

15 a Restaurant A: $2000, Restaurant B: $9000

16 a P (x) = R (x) − C (x)

−8

A.F.1g, A.F.2g, A.F.2h

A.F.2g, A.F.2h

−6

20 Height in feet 18 16 14 12 10 8 6 4 2

Business: Her income after ﬁve years would be $260 254.82.

b T he net proﬁt of Restaurant A and Restaurant B will be equal after 6 weeks which is $64 000.

b Function B c a(1) = −5

13 a Function B

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

b − 14.7 ft or 14.7 ft below the water. This can be found pretty easily from any of the forms by substituting 10 for x. Since we are substituting 10 the calculations from standard form are probably the simplest since the powers of 10 simplify the multiplication. Vertex form is the most complex to calculate mentally because of the squaring of a decimal. c 1 0.8 ft, this can be found from looking at the c value of the standard form equation since it is the y-intercept.

d 3 ft and 24 ft, these can be found from the factored form equation because each factor shows an x-intercept. In this case the x-intercepts are at x = 3 and x = 24. x = 3 is closer to the boat so that must be where she entered the water, making x = 24 the location where she came up for air. A.F.2d, A.F.2g, A.EO.2e, MP2, MP5 18 a

Description

Term in model

Starting height

Travels upwards at 20 ft/s

20t

Pulled downward at 16 ft/s2

−16t2

Answers will vary. Forms can include: y = −16x2 + 20x + 6, y = −2 (4x + 1) (2x − 3), y = −16 (x − 0.625)2 + 12.25, or another equivalent form of this same equation. b V ertex form: y = −16 (x − 0.625)2 + 12.25, 0.625 seconds. c A nswers will vary. Any form is appropriate but standard form creates the most complex calculations while factored form is the least complex. 1.5 s. d A nswers will vary. Any form is appropriate but vertex form creates the most complex calculations. 12 ft. e

14 y 13 12 11 10 9 8 7 6 5 4 3 2 1

x 0.5

1.5

Standard form: y-intercept

Vertex form: coordinates of the vertex

Factored form: Positive x-intercept

A.F.2b, A.F.2d, A.F.2g, MP1, MP4, MP5

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8 Quadratic Equations Big ideas A standard algorithm can be followed to solve a wide range of equations. This algorithm is reliable and useful in a variety of situations, but there is often a more efficient method that can be used based on the structure of the equation.

Chapter outline 8.01 8.02 8.03 8.04 8.05

Solve quadratics using graphs and tables (A.EI.3, A.F.2) Solve quadratics by factoring (A.EI.3, A.F.2) Solve quadratics using square roots (A.EI.3) Solve quadratics using the quadratic formula (A.EI.3) Solve quadratics using appropriate methods (A.EI.3) Topic 8 Assessment

946 968 985 1008 1034 1047

The arc that a whale makes when it leaps out of the water can be described by a quadratic equation. The highest point of the jump is the vertex of the parabola.

8. Quadratic Equations Topic overview Foundational knowledge Evaluating standards proficiency The skills book contains questions matched to individual standards. It can be used to measure proficiency for each. Students should be proficient in these standards. A.EI.1 — The student will represent, solve, explain, and interpret the solution to multistep linear equations and inequalities in one variable and literal equations for a specified variable. A.EO.1 — The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

Big ideas and essential understanding A standard algorithm can be followed to solve a wide range of equations. This algorithm is reliable and useful in a variety of situations, but there is often a more efficient method that can be used based on the structure of the equation. 8.01 — Quadratic equations can be solved using a variety of methods. Graphing or creating a table of values can be an efficient method when the equation has integer solutions. 8.02 — Quadratic equations can be solved using a variety of methods. Factoring can be an efficient method when the equation is factorable and is made up of constants and coefficients that are not large in value.

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8.03 — Quadratic equations can be solved using a variety of methods. Using square roots can be an efficient method when the equation only has two terms and the constant term is a perfect square. Completing the square can be an efficient method for an equation that cannot be factored. 8.04 — Quadratic equations can be solved using a variety of methods. The quadratic formula can be used to solve any quadratic equation, but it is not always the most efficient method and is best used for equations that cannot be easily graphed or factored. 8.05 — There are many methods that can be used to solve a quadratic equation. The structure of the equation can give insight into which method might be the most efficient.

Standards A.EI.3 — The student will represent, solve, and interpret the solution to a quadratic equation in one variable. A.EI.3a — Solve a quadratic equation in one variable over the set of real numbers with rational or irrational solutions, including those that can be used to solve contextual problems. 8.01 Solve quadratics using graphs and tables 8.02 Solve quadratics by factoring 8.03 Solve quadratics using square roots 8.04 Solve quadratics using the quadratic formula 8.05 Solve quadratics using appropriate methods A.EI.3b — Determine and justify if a quadratic equation in one variable has no real solutions, one real solution, or two real solutions. 8.01 Solve quadratics using graphs and tables 8.04 Solve quadratics using the quadratic formula A.EI.3c — Verify possible solution(s) to a quadratic equation in one variable algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context. 8.01 Solve quadratics using graphs and tables 8.02 Solve quadratics by factoring 8.03 Solve quadratics using square roots 8.04 Solve quadratics using the quadratic formula 8.05 Solve quadratics using appropriate methods

A.F.2 — The student will investigate, analyze, and compare characteristics of functions, including quadratic and exponential functions, and model quadratic and exponential relationships. A.F.2c — Graph a quadratic function, f (x), in two variables using a variety of strategies, including transformations f (x) + k and kf (x), where k is limited to rational values. 8.01 Solve quadratics using graphs and tables A.F.2d — Make connections between the algebraic (standard and factored forms) and graphical representation of a quadratic function. 8.02 Solve quadratics by factoring A.F.2g — For any value, x, in the domain of f, determine f (x) of a quadratic or exponential function. Determine x given any value f (x) in the range of f of a quadratic function. Explain the meaning of x and f (x) in context. 8.01 Solve quadratics using graphs and tables 8.02 Solve quadratics by factoring

Future connections A2.EI.2 — The student will represent, solve, and interpret the solution to quadratic equations in one variable over the set of complex numbers and solve quadratic inequalities in one variable.

A2.F.1 — The student will investigate, analyze, and compare square root, cube root, rational, exponential, and logarithmic function families, algebraically and graphically, using transformations.

A2.EI.5 — The student will represent, solve, and interpret the solution to an equation containing a radical expression.

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8.01 Solve quadratics using graphs and tables Subtopic overview Lesson narrative In this lesson, students will explore the connections between the graphs and tables of equivalent equations. Students will analyze the structures of equivalent equations and the relationships between their graphs and tables in order to explain how equivalent equations can be used to solve quadratic equations of the form f (x) = c for a real constant, c. By the end of the lesson, students will be able to interpret and represent contextual situations using quadratic equations, tables, or graphs and then, using the assistance of technology when necessary, solve them by finding intersections between linear constants and quadratic functions or by finding the zeros of equivalent expressions.

Learning objectives Students: Page 450

Key vocabulary 

equivalent equations



quadratic equation



root (of an equation)



solution (to an equation)



vertex



x-intercept



zero (of a function)

Essential understanding Quadratic equations can be solved using a variety of methods. Graphing or creating a table of values can be an efficient method when the equation has integer solutions.

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Standards This subtopic addresses the following Virginia Standards of Learning for Mathematics standards.

Mathematical process goals MPG1 — Mathematical Problem Solving

MPG4 — Mathematical Connections

To integrate this goal, teachers can encourage students to approach the concept of quadratic functions as a problem-solving task. They can ask students to identify what they know about quadratic functions and what they need to know in order to graph them. When introducing the concept of solving a quadratic equation, teachers can present it as a problem to be solved and discuss various strategies for finding the solutions.

To integrate this goal, teachers can help students make connections between their prior knowledge of quadratic functions and the new concept of solving quadratic equations. They can also help students see the connections between different methods of solving quadratic equations, such as using tables and using graphs. When presenting real-world problems, teachers can highlight the connections between the mathematical concepts and their practical applications.

MPG5 — Mathematical Representations Teachers can integrate this goal by demonstrating how to represent quadratic functions and solutions to quadratic equations using tables and graphs. They can also encourage students to use these representations in their own problem-solving processes. When discussing the types of solutions to quadratic equations, teachers can guide students in representing these different possibilities visually on a graph. In solving contextual problems, teachers can help students represent the problem using a quadratic equation and then use graphs or tables to find the solutions.

Content standards A.EI.3 — The student will represent, solve, and interpret the solution to a quadratic equation in one variable. A.EI.3a — Solve a quadratic equation in one variable over the set of real numbers with rational or irrational solutions, including those that can be used to solve contextual problems. A.EI.3b — Determine and justify if a quadratic equation in one variable has no real solutions, one real solution, or two real solutions. A.EI.3c — Verify possible solution(s) to a quadratic equation in one variable algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context.

Prior connections A.EI.1 — The student will represent, solve, explain, and interpret the solution to multistep linear equations and inequalities in one variable and literal equations for a specified variable. A.EO.2 — The student will perform operations on and factor polynomial expressions in one variable.

A.EO.4 — The student will simplify and determine equivalent radical expressions involving square roots of whole numbers and cube roots of integers. A.F.1 — The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships. 8.01 Solve quadratics using graphs and tables mathspace.co

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A2.F.1 — The student will investigate, analyze, and compare square root, cube root, rational, exponential, and logarithmic function families, algebraically and graphically, using transformations.

A2.F.2 — The student will investigate and analyze characteristics of square root, cube root, rational, polynomial, exponential, logarithmic, and piecewise-defined functions algebraically and graphically.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Algebra 1 — 7.01 Characteristics of quadratic functions Algebra 1 — 7.03 Characteristics of quadratic functions

Tools You may ﬁnd these tools helpful: • Graphing calculator • Graph paper • Spreadsheet application

Lesson supports The following support may be useful for this lesson. More speciﬁc supports may appear throughout the lesson:

Concrete-Representational-Abstract (CRA) Approach Targeted instructional strategies Concrete: Engage students with physical manipulatives to model quadratic equations. Use a large floor grid or coordinate plane and have students physically place markers or sticky notes at points that satisfy a given quadratic equation. For example, provide students with numbered cards for x and have them calculate y using the equation y = x2 or another simple quadratic. They can then place their markers at the corresponding points on the grid. This hands-on activity helps students see the shape of the parabola forming as they plot more points. Representational: Transition to visual representations by guiding students to create tables of values for quadratic equations. Have them select values for x, calculate the corresponding y-values, and record them in a table. Then, using graph paper, students can plot these points on a coordinate plane. Encourage them to connect the points smoothly to reveal the parabola’s shape. Abstract: Move to solving quadratic equations using algebraic symbols and methods. Teach students how to find the roots of the equation by setting f (x) = c and solving for x. Introduce the quadratic formula and discuss how it can be used to find exact solutions. Encourage students to analyze the equation to determine the number of real solutions by calculating the discriminant. This abstract work connects to the graphs and tables they created, as the solutions correspond to the points where the graph intersects the x-axis.

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Connecting the stages: Help students make connections between the physical activity, their graphs, and the algebraic equations. Discuss how the points they plotted with manipulatives match the points on their graphs and represent solutions to the equation. Ask guiding questions like: • “How does the shape we formed on the floor relate to the parabola on your graph?” • “What do the points where the graph crosses the x-axis tell us about the solutions to the equation?” By linking the concrete, representational, and abstract stages, students can better understand how each method represents the same mathematical concepts.

Student lesson & teacher guide Solving quadratic equations using graphs and tables Students explore solving quadratic equations using tables and graphs. The terms roots and zeros of the function are used to describe the x-intercepts. Students learn to solve quadratic equations by writing them in an equivalent form, where the y-value is set to zero, and then finding the roots.

Students: Pages 450–451

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In the graph and table shown, we see x = 0 is the only solution. This is because x = 0 is the only value that makes x2 = 0 true. If we tried to ﬁnd the solution to x2 = −2 there would be no real solutions, because squaring any non-zero real number will give a positive result. The roots, or zeros, in a quadratic function occur when f (x) = 0. The method we will use to solve a problem such as x2 = 4 is by creating an equivalent equation by rearranging it so it is equal to 0 and then identifying the x-intercepts. x2 = 4

Given equation

x −4=0

Subtract 4 from both sides

Next, we can replace the 0 in the equation with y to get y = x2 − 4. The graph of this equation is the graph of y = x2 shifted down 4 units so the graph of y = x2 − 4 is: 4

f (x)

3 2 1

−4 −3 −2 −1 −1

−2 −3 −4

We can see that the graph crosses the x-axis at −2 and 2, so the solutions to x2 = 4 are −2 and 2. We can check this using substitution: (−2)2 = 4 and (2)2 = 4 We can follow this process to solve any quadratic equation graphically. In other words, for any function f (x) = c, for some real number constant, c, we can write the equivalent equation f (x) − c = 0, and find the x-intercepts of g(x) = f (x) − c to solve for x. A quadratic equation can have one, two or no real solutions.

One real solution

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Two real solutions

No real solutions

We can check this using substitution: (−2)2 = 4 and (2)2 = 4 We can follow this process to solve any quadratic equation graphically. In other words, for any function f (x) = c, for some real number constant, c, we can write the equivalent equation f (x) − c = 0, and find the x-intercepts of g(x) = f (x) − c to solve for x. A quadratic equation can have one, two or no real solutions. y

One real solution

Two real solutions

No real solutions

Think about the nature of the zeros Targeted instructional strategies Before attempting to ﬁnd the zeros or roots of a quadratic equation, ask students whether they can determine 8.01 Solve quadratics using graphs and tables 451 if the equation has any x-intercepts or not. mathspace.co Use a few examples to demonstrate to students that a quadratic equation will only have real zeros if: • The parabola opens upwards and has a vertex below the x-axis, or • The parabola opens downwards and has a vertex above the x-axis, or • The vertex is on the x-axis This can be checked by ﬁnding the coordinates of the vertex and comparing them to the sign of the leading coefficient.

Intercepts are not solutions to all quadratic equations Address student misconceptions It is important for students to recognize that “solutions” is not synonymous with “x-intercepts”. Students should only look for the intercepts if the equation is set equal to zero. Rather than “x-intercepts”, it would be better to only refer to them as “zeros” as this implies the equation must be set equal to zero first. To reinforce this idea, teachers can provide a visual example of a graph that is not equal to zero and one that is. Use the two graphs to highlight the differences and make connections to the solutions. 7 6 5 4 3 2 1 −4 −3 −2 −1 −1 −2 −3 −4

x 1

2 3 4 Equation x2 − 1 = 3

Solutions are x = 2 and x = −2

7 6 5 4 3 2 1 −4 −3 −2 −1 −1 −2 −3 −4

x 1

2 3 4 Equation x2 − 1 = 0

Solutions are x = 1 and x = −1

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Support perseverance - move through a frustrating point Student with disabilities support Since students do not know any algebraic methods for solving quadratic equations at this stage, it can be easy to get frustrated with methods that are imprecise or unreliable. For example, the table of values method may take a very long time or not work if there are no real solutions. Give students guidance on where to start when solving problems in order to minimize frustration from a lack of direction, while also being sure to set problems which can be solved relatively efficiently.

Compare and connect English language learner support Students may struggle with the subtle differences between the vocabulary of “solutions”, “x-intercepts”, and “zeros”. Use the two graphs and key features below to highlight the differences. 9

Equation: x2 = 9

7 6

•

5 4 2 1

−2 4

−1 1

0 0

1 1

1 −8

3 0

2 4

3 9

−4 −3 −2 −1

−4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8 −9

−3 9

• Solutions of x = −3 and x = 3 • x-intercept at x = 0 • Zero at x = 0

x y

y x

Equation: x2 − 9 = 0 •

x y

−3 0

−1 −8

0 −9

• Solutions of x = ±3 • x-intercepts at x = ±3 • Zeros at x = ±3

Give students time in pairs to discuss comparisons between the two. Encourage students to connect the similiarities and differences between the vocabulary terms through conversation with their peers. Some guiding questions include: • What differences do you notice between the graphs? • What differences do you notice between the equations and the key features? • Are the solutions always the zeros or x-intercepts? Then as a class, discuss the connections between solutions, x-intercepts, and zeros. Summarize the findings of your discussion by highlighting thee differences between these terms. • Solutions are any values that make an equation true. • Zeros are solutions to the equation when it is set equal to zero. • x-intercepts are the solution when y = 0.

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Examples Students: Page 452 Example 1 Solve the equation 2x2 = 18.

Create a strategy We can write an equivalent equation set equal to zero, and then use a table to ﬁnd the zeros of the new function.

Apply the idea If we set this equation equal to zero, we would get 2x2 = 18

Given equation

2x − 18 = 0

Subtraction property of equality

When building a table, we want to choose values within a suitable range so we don’t have to do too many calculations. Start by ﬁnding the values in the domain −4 ≤ x ≤ 4. If the y-value (also called the function value) is zero for any of these x-values, then we have found a solution to the corresponding equation. In the table, we are looking for the entries where y = 0. x y

−4 14

−3 0

−2 −10

−1 −16

0 −18

1 −16

2 −10

3 0

4 14

We can see the equation has solutions of x = −3, x = 3, which we can also write as x = ±3.

Reﬂect and check We can see that the table of y-values has both positive and negative values. Whenever this is the case for a function of the form f (x) = ax2 + bx + c, we know that the equation 0 = ax2 + bx + c must have two real solutions, and the corresponding parabola will have two x-intercepts.

Example 2 Purpose Consider how the function = (xequivalent − 2)2 − 9. Show students to usey an equation to solve a problem and demonstrate how a table can be used a Draw graph of the function. to find the zerosa of a quadratic equation. Expected mistakes Create a strategy StudentsThe may substitute for xform instead finding the values thatWe make y equal xto= 0. Or, students may only function is given0inin vertex so weof know the vertex is at (2, −9). can substitute 0 to find the y-intercept (0, −5). We cannot findlook other for points on the curve by substituting other values, and by filling a table of values. find oneatsolution and a second solution. Remindinstudents to visualize the shape and orientation of the graph. Ask them to think about what it means to find y = 0 on the graph. Apply the idea

y Reﬂecting with students x y 6 Ask students to describe other ways they could solve this problem. They could make a table for y = 2x2 and 4 0 −5 then ﬁnd where y = 18. They could divide both sides of the equivalent2equation by 2 to get x2 − 9 = 0 and x recognize this as a difference of1 squares −8 and factor the equation. −4−3−2 −1 −2

452

−9

−8

−5

1 2 3 4 5 6 7 8 9

−4 −6 −8 −10

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Visualizing quadratic equations

use with Example 1

Student with disabilities support Example 1 For students who struggle with abstract concepts, visualizing quadratic equations can be helpful. Start by Solve the equation 2x2 equal = 18. to zero and creating a table of values. This table can then be used to plot the setting the given equation equation on a graph. Create a strategy

Discuss withuse students solutions anfunction. equation that y We can write an equivalent equation set equal to zero, and then a table tohow ﬁnd the the zeros of the of new 16

is equal to 0 correspond to the points where the parabola intersects the x-axis.

Apply the idea12

If we set this equation equal to zero, we would get 8 2x2 = 18

Given equation

2x2 − 18 = 0

These points are also known as the zeros of the equation.

Subtraction property of equality x

1 want 2 to 3 choose 4 When values within a suitable range so we don’t have to do too many −4 −3building −2 −1a table, we −4by ﬁnding the values in the domain −4 ≤ x ≤ 4. If the y-value (also called the function value) is zero calculations. Start for any of these −8 x-values, then we have found a solution to the corresponding equation. In the table, we are looking for the entries where y = 0.

−12

x y

−4 14

−3 −16 0

−2 −10

−1 −16

0 −18

1 −16

2 −10

3 0

4 14

We can see the equation has solutions of x = −3, x = 3, which we can also write as x = ±3.

Encourage students to use this visual approach whenever they are solving quadratic equations. It can help them better understand concept of solutions and how to find them. Reﬂect and the check We can see that the table of y-values has both positive and negative values. Whenever this is the case for a function of the form f (x) = ax2 + bx + c, we know that the equation 0 = ax2 + bx + c must have two real solutions, and the Students:corresponding Pages 452–453 parabola will have two x-intercepts.

Example 2 Consider the function y = (x − 2)2 − 9. a Draw a graph of the function.

Create a strategy The function is given in vertex form so we know the vertex is at (2, −9). We can substitute x = 0 to find the y-intercept at (0, −5). We can find other points on the curve by substituting in other values, and by filling a table of values.

Apply the idea x

−5

−8

−9

−8

−5

4 2 −4−3−2 −1 −2

x 1 2 3 4 5 6 7 8 9

−4 −6 −8 −10

Reﬂect and check It is important when drawing graphs1 to clearly show the key features such as the vertex and the intercepts by 452 Mathspace Virginia SOL Algebra mathspace.co choosing appropriate scales for the axes.

b Determine the solution(s) to the equation 0 = (x − 2)2 − 9.

Create a strategy We can find the solution(s) by looking at the graph from part (a) and identifying where it crosses the x-axis.

954

Apply the idea

Mathspace Virginia SOL Algebra 1 Teacher Edition The solutions to the equation can be found at the x-intercepts of the graph we mathspace.co drew in part (a).

6 4 2

Purpose Students demonstrate that they can use vertex form to create a graph and include all the key features of a quadratic function. Reflecting with students Ask students how they could draw the graph if they did not recognize the equation was in vertex form. Remind Reflect andfunction, check we can always create a table of values to find points and plot the curve. them that, for any It is important when drawing graphs to clearly show the key features such as the vertex and the intercepts by scales for the axes. Students:choosing Page appropriate 453 b Determine the solution(s) to the equation 0 = (x − 2)2 − 9.

Create a strategy We can find the solution(s) by looking at the graph from part (a) and identifying where it crosses the x-axis.

Apply the idea The solutions to the equation can be found at the x-intercepts of the graph we drew in part (a).

4 2 −4−3−2 −1 −2

x 1 2 3 4 5 6 7 8 9

−4 −6 −8 −10

The solutions are x = −1 and x = 5.

Reﬂect and check We can verify the solutions by substituting each one into the equation for x and substituting 0 for y. If the right side of the equation evaluates to 0 then it is a solution. First, let’s check the solution x = −1 0 = (x − 2)2 − 9 2

0 = (−1 − 2) − 9

Original equation Substitute in x = −1

0 = (−3)2 − 9

Simplify inside parenthesis

0=9−9

Evaluate the square

0=0

Subtract

0 = 0 is a true statement, so x = −1 is a solution to 0 = (x − 2)2 − 9. Now, let’s check our other solution, x = 5 0 = (x − 2)2 − 9

Original equation

0 = (5 − 2)2 − 9

Substitute in x = 5

0 = (3) − 9

Simplify inside parenthesis

0=9−9

Evaluate the square

0=0

Subtract

0 = 0 is a true statement, so x = 5 is also a solution to y = (x − 2)2 − 9.

Purpose Show students that this is an equivalent expression to the previous equation, set equal to 0, and that they can ﬁnd the roots to solve. Expected mistakes 8.01 Solve quadratics using graphs and tables 453 mathspace.co Students may give their answer in coordinates, (−1, 0) and (5, 0), which does not answer the question. Ask students what form the solutions to the equation should be in.

8.01 Solve quadratics using graphs and tables mathspace.co

955

Finding non-integer solutions using technology

use with Example 2

Targeted instructional strategies As an extension for advanced learners, ask them to find the solution(s) to the equation 5 = (x − 2)2 − 9. Encourage them to use computational thinking and consider tools like spreadsheets. Here is a possible solution: 1. Open a spreadsheet program, such as GeoGebra, GoogleSheets, Excel, Desmos, etc. 2. Fill in headings A 1 2

B (x-2)^2-9

3. Use the graph from part (b), notice that y = 5 between −2 < x < −1 and 5 < x < 6. Start by estimating the negative solution where −2 < x < −1. Fill in the cells using a formula and then dragging down. Emphasize that an equals sign must preceed the formula. A x -2 =A2+0.1

1 2 3

B (x-2)^2-9 =(A2-2)^2-9

B (x-2)^2-9 7 6.21 5.44 4.69

4. Fill down A 1 2 3 4 5

x -2 -1.9 -1.8 -1.7

5. Refine to the interval of −1.8 < x < −1.7 1 2 3 4 5 6 7 8

A x -1.8 =A2+0.01 -1.78 -1.77 -1.76 -1.75 -1.74

B (x-2)^2-9 5.44 5.3641 5.2884 5.2129 5.1376 5.0625 4.9876

6. Refine to the interval of −1.75 < x < −1.74 A 1 2 3 4 5 6 7 8 9 10 11

x -1.75 =A2+0.001 -1.748 -1.747 -1.746 -1.745 -1.744 -1.743 -1.742 -1.741

B (x-2)^2-9 5.0625 5.055001 5.047504 5.040009 5.032516 5.025025 5.017536 5.010049 5.002564 4.995081

7. Estimate the solution to be in the interval −1.742 < x < −1.741. 8. Repeat for the positive solution. Or to be more efficient, use symmetry.

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Students: Page 454

Purpose Show students how to interpret a point on a graph in a real-world context. Reﬂecting with students Remind students that f (x) = y. So, when we are given f (8), we are really being given an x-value. When given notation like this, they must solve for y, by finding the value of y when x = 8.

8.01 Solve quadratics using graphs and tables mathspace.co

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Students: Page 455

Purpose Given the graph of a quadratic function in a context, show students how to use the mathematical model to ﬁnd solutions to a contextual problem. Reﬂecting with students Ask students to consider why we look at y = 38 rather than looking at the x-intercepts. Students should be encouraged to consider the context of the problem and what the variables x and y represent. Relate the graph to the context and highlight that y represents height, and the known value in this part, which helps them to answer the given question.

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Students: Page 455

Purpose To show students how to estimate a viable solution from a graph and determine the number of viable solutions based on the context of the problem.

Students: Page 456

Example 4 Identify the number of real solutions each quadratic function has. a

y 6 5 4 3 2 1 −4 −3 −2 −1

x 1

−1

Create a strategy

Apply the idea

Real solutions correspond with x-intercepts. How many x-intercepts does this function have?

This quadratic never crosses the x-axis so there are no x-intercepts. The function has 0 real solutions.

y 4 3

Purpose 2 To show students that a quadratic function has no real solutions if its graph does not intersect the x-axis. 1 x

−1 −1

−2 −3

Create a strategy Real solutions correspond with x-intercepts. How many x-intercepts does this function have?

Apply the idea

Reﬂect and check

The quadratic crosses the x-axis in two different spots. y 4 3 2

We can determine from the graph the exact value of the two real solutions. solutions are x = 2 and x = 5. and tables 8.01The Solve quadratics using graphs

mathspace.co

959

−1

Create a strategy

Apply the idea

Real solutions correspond with x-intercepts. How many

This quadratic never crosses the x-axis so there are no x-intercepts.

does this function have? Students:x-intercepts Page 456

The function has 0 real solutions. b

y 4 3 2 1

−1 −1

−2 −3

Create a strategy Real solutions correspond with x-intercepts. How many x-intercepts does this function have?

Apply the idea

Reﬂect and check

The quadratic crosses the x-axis in two different spots. y

We can determine from the graph the exact value of the two real solutions. The solutions are x = 2 and x = 5.

4 3 2 1

−1 −1

−2 −3

This quadratic has two real solutions.

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Purpose To show students that the number of real solutions corresponds to the number of x-intercepts shown on the graph of the function.

Students: Page 457

Idea summary We solve a quadratic equation by creating an equivalent equation by making your equation set equal to 0. The solutions to a quadratic equation are any values that make the equation true. If the equation is equal to 0, the solutions are called roots of the equation or zeros of the function. These correspond to the x-intercepts of the graph. For any function f (x) = c, for some real number constant, c, we can write the equivalent equation f (x) − c = 0, and ﬁnd the x-intercepts of g(x) = f (x) − c to solve for x.

Practice What do you remember? 1

Using the given tables, ﬁnd the solutions to the following equations: a

960

x2 + 7x + 12 = 0

x −6 −5 −4 −3 −2 Mathspace Virginia y SOL 6 Algebra 2 1 Teacher 0 0Edition2 mathspace.co b 3x2 − 27 = 0

−1 6

−4

−3

−2

−1

Practice Students: Pages 457–460

What do you remember? 1

Using the given tables, ﬁnd the solutions to the following equations: a

x2 + 7x + 12 = 0 x y

−4 0

−3 0

−2 2

−1 6

−4 21

−3 0

−2 −15

−1 −24

0 −27

1 −24

2 −15

4 21

x2 + 8x + 12 = 5

4x2 = 4

(x − 3)2 = 7

6x2 = 2x − 9

Using the given graphs, ﬁnd the solutions to the following equations: a

x2 + 2x − 8 = 0

15 10 5 x −7 −6 −5 −4 −3 −2 −1 −5

2x2 − 12x + 16 = 0 18 16 14 12 10 8 6 4 2

1 2 3 4 5

−2 −1 −2 −4

−10

3 0

Rewrite each equation so that it is equal to zero: a

−5 2

3x2 − 27 = 0 x y

−6 6

x 1

2 3 4 5 6 7

For the function y = x2 + 10x + 21: a

Copy and complete the table of values. x y

−8

−7

−6

−5

−4

−3

−2

Draw the graph of the equation.

Use the models from parts (a) and (b) to ﬁnd: i

The value of x when y = −4.

The value of x when y = 5.

8.01 Solve quadratics using graphs and tables mathspace.co

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Determine the number of real solutions each quadratic function has: a

−2 −1

−1

−2

−1

−3

−1

−4

−2

−4 −3 −2 −1

−2

3 2 1

−1

3 2 1

−4 −3 −2 −1

x 1

−1

−2

−3

−4

−5

−6

Let’s practice 6

For each function: i

Complete the table of values.

Set each function equal to zero and use the table to determine the solution(s) to the equation.

a x

−2

−1

c x y e

962

−1

f 4

−1

x y

−1

(x − 5)2 = 4

y = (x − 5) (x − 1) x y

−2

x y

y = x2 − x − 2 x y

−2

Solve the following equations by drawing a graph of the corresponding function: a

x2 − 15x + 54 = 0

−(x + 5)2 + 9 = 0

(x − 3) (x + 2) = 0

(x − 1)2 = 0

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

x2 − 15x + 50 = 0

For each function: i

Draw the graph of the corresponding function.

Use the graph to determine the solution(s) to the equation.

0 = x2

0 = −x2 + 9

3x2 = 3

0 = − (x − 3)2

x2 + 12x = −32

(x − 3) (x − 2) = 0

2x2 − 2x = 12

(x − 5)2 = 0

A compass is accidentally thrown upward and out of an air balloon at a height of 100 feet. The height, y, of the compass at time x, in seconds, is given by the equation: y = −20x2 + 80x + 100

Graph the relationship y = −20x2 + 80x + 100.

Find the time it takes for the compass to hit the ground.

When will the compass be 160 feet high?

A frisbee is thrown upward and away from the top of a hill that is 120 yards tall. The height, y, of the frisbee at time x in seconds is given by the equation y = −10x2 + 40x + 120. This equation is graphed.

y 160 140

Determine when the frisbee reaches a height of 120 yards.

120

Determine how many seconds it takes for the frisbee to hit the ground.

100

Determine the height reached by the frisbee after 2 seconds.

Verify your answer to part (b) by substituting values into the equation.

80 60 40 20

x 1

Beth throws a pebble vertically upwards. After t seconds, its height h feet above the ground is given by the formula h = 18t − 2t2. This function has been graphed as shown.

h 40 35

Explain what the point at (4.5, 40.5) represents in context of the problem.

Find the values of t where the graph of h intercepts the horizontal axis.

Explain what the intercepts you found in part (b) represent in context of the problem.

Identify when the pebble is 36 feet above the ground.

30 25 20 15 10 5 t 1

2 3 4 5 6 7 8 9

Use the given table to answer the following: x y

−3 9

−2 4

−1 1

0 0

1 1

2 4

3 9

Find f (−2)

Find x when f (x) = 1

Find the domain when the range is {9, 0}.

Find the range when the domain is {−2, 2, 3}.

8.01 Solve quadratics using graphs and tables mathspace.co

963

Use the graph to answer the following. a

Find f (−1)

Find x when f (x) = 0

Find the domain when the range is {3, 4}.

Find the range when the domain is {1, −1, −2}.

2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

Let’s extend our thinking 14

An object is released 900 meters above ground and falls freely. The distance the object is from the ground is modeled by the formula d = 900 − 4.9t2, where d is the distance in meters that the object falls and t is the time elapsed in seconds. Javier graphed the given equation, the line d = 450, and the point of intersection as shown below.

d 900 800 700 600

Explain the point (9.583, 450) in context of the problem.

500

Use the graph to estimate how many seconds it takes for the object to reach a height of 0.

400

Explain how you could find a more precise answer to part (b). Find a more precise answer.

(9.583, 450)

300 200 100 t 2

The kinetic energy E of a moving object is given by speed in meters per second.

where m is its mass in kilograms and v is its

Graph this equation for a vehicle with a mass of 1400 kg using a calculator or other technology.

Use the graph to estimate the velocity when the kinetic energy is 137 200 J.

The formula for the surface area of a sphere is S = 4π r2, where r is the radius in centimeters. a

Graph the relationship S = 4π r2.

Use your graph to estimate the surface area of a sphere with radius 5.5 cm.

Use your graph to estimate the radius of a sphere with a surface area of 150 cm2.

A rectangle has width a, height b, and area A. The graph shows the A 100 possible values of the area A (vertical axis) plotted against the width a 90 (horizontal axis). The equation corresponding to the graph is A = a (20 − a). a

List the values of a where the graph intercepts the horizontal axis.

Explain why it is not possible to have a rectangle with a width of a = 20.

Find the width and height of the rectangle corresponding with the largest possible value of A.

80 70 60 50 40 30 20 10

a 2 4 6 8 10 12 14 16 18 20

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Answers

8.01 Solve quadratics using graphs and tables

–2

–1

–2

ii x = –1, x = 2 What do you remember?

7 a

1 a x = −4, x = −3

b x = −3, x = 3

2 a x2 + 8x + 7 = 0

b 4x2 − 4 = 0

c (x − 3) − 7 = 0

d 6x − 2x + 9 = 0

3 a x = −4, x = 2 4 a

b x = 2, x = 4

-8

-7

-6

-5

-4

-3

-2

-3

-4

-3

x 2

y 8 7 6 5 4 3 2 1 x

x = 6, x = 9 b

y 8 6

−9 −8 −7 −6 −5 −4 −3 −2 −1−1

−2 −3 −4

2 x

c i x = –5

−8

ii x = –8 and x = –2

5 a Two solutions

b No solution

c One solution

d Two solutions

-2

-1

-2

20 10 x 2

-1

-2

-8

-18

-2

-1

-6

–3

–4

–3

5 4 3 2 1 −1 −2 −3 −4 −5

ii x = 0, x = 4 e i

x = 5, x = 10

ii x = –1, x = 1 d i

ii x = 0 c i

−2

ii x = 0 b i

−4

x = −8, x = −2

Let’s practice 6 a i

−6

–3

–4

–3

x 1

2 3 4 5 6 7 8 9

x = 3, x = 7

ii x = 1, x = 5

Answers mathspace.co

965

4 3 2 1

c i

−4 −3 −2 −1−1

2 3 4

−2 −3 −4 −5 −6 −7

x 1

ii x = 1, x = -1 7 6 5 4 3 2 1

d i

y 2 x −2

−2 −4

−4 −3 −2 −1 −1 −2

−6

2 3 4

−8

ii x = 3

x = 1

e i 8 a i

9 8 7 6 5 4 3 2 1

6 4 2 8

−4 −8

x 2

ii x = -4, x = -8 f

−3 −2 −1

−6

9 8 7 6 5 4 3 2 1

−8 −6 −4 −2 −2

ii x = 0 b i

−3 −2 −1

9 8 7 6 5 4 3 2 1 −1−1 −2

x 1

ii x = 3, x = -3

x 1

2 3 4 5 6 7

ii x = 2, x = 3 g i

8 6 4 2 −3 −2 −1 −2 −4 −6 −8 −10 −12

966

−4 −3 −2 −1 −3 −6

x = –2, x = 3 f

24 21 18 15 12 9 6 3

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

x 1

3 4 5

Let’s extend our thinking

ii x = 3, x = -2 h i

9 8 7 6 5 4 3 2 1

14 a A fter 9.583 seconds, the object will have a height of 450 meters. b About 13.5 seconds c Substitue d = 0 into the original equation and solve for t.

x 2 3 4 5 6 7 8 9

ii x = 5 180 160 140 120 100 80 60 40 20

Substitute d = 0

4.9t2 = 900

9 a

0 = 900 − 4.9t2

Add 4.9t2 to both sides

t ≈ 183.67

Divide both sides by 4.9

t ≈ 13.55

Take the square root of both sides

A more precise answer is 13.55 seconds. b v = 14 m/s

15 a E 200 000

180 000 160 000 140 000 120 000 100 000 80 000 60 000 40 000 20 000

x 1

b 5 seconds c At x = 1 second and x = 3 seconds 10 a Initially, at t = 0 and after 4 seconds. b 6 seconds

v 2 4 6 8 10 12 14 16

16 a 800

700 600 500

c 160 yards

400

d y = −10x2 + 40x + 120

Original equation

0 = −10(6) + 40(6) + 120 Substitute y = 0 and x = 6 0 = −360 + 240 + 120 0 = 0

Evaluate the multiplication Evaluate the addition

300 200 100

S 1

11 a A fter 4.5 seconds, the pebble is 40.5 feet above the ground.

b T he actual answer is about S = 380, so their answer should be close to this.

The highest distance the pebble reaches above the ground is 40.5 feet.

c T he actual answer is about r = 3.45 cm, so their answer should be close to this.

b t = 0, t = 9

17 a a = 0, a = 20

c The times when the pebble is on the ground. d After 3 seconds and 6 seconds. 12 a f (−2) = 4 c Domain: {3, 0} 13 a f (−1) = 4 c Domain: {−1, 0}

b x = −1 d Range : {4, 9}

b T here is no corresponding rectangle because its area would be 0 which is not possible. c T he largest area is 100. The width and height of the corresponding rectangle would both be 10.

b x=1 d Range : {0, 3, 4}

Answers mathspace.co

967

8.02 Solve quadratics by factoring Subtopic overview Lesson narrative In the lesson, students will discover how the zero product property can be used to solve quadratic equations in factored form. They will also connect the factors and x-intercepts to write the equation of a quadratic. By the end of the lesson, students will be able to examine the structure of quadratic equations to recognize when they can be solved using factoring and the zero product property.

Learning objectives Students: Page 461

Key vocabulary 

factored form



x-intercept



zero product property

Essential understanding Quadratic equations can be solved using a variety of methods. Factoring can be an efficient method when the equation is factorable and is made up of constants and coefficients that are not large in value.

Standards This subtopic addresses the following Virginia Standards of Learning for Mathematics standards.

Mathematical process goals MPG2 — Mathematical Communication

MPG3 — Mathematical Reasoning

This goal can be integrated by encouraging students to communicate their reasoning when solving problems. Teachers can ask students to explain the steps they used to solve a quadratic equation by factoring, and to justify their choice of factoring techniques. Student learning can be deepened by discussing, justifying, conjecturing, reading, writing, presenting, and listening to mathematical ideas during the lesson.

Teachers can integrate this goal by encouraging students to use logical reasoning when solving problems. For instance, when teaching the Zero Product Property, teachers can ask students to justify why, if the product of two factors is zero, then at least one of the factors must be zero. Teachers can also encourage students to use reasoning to evaluate the validity of solutions to quadratic equations

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MPG5 — Mathematical Representations Teachers can integrate this goal by encouraging students to use different representations for solving quadratic equations by factoring. For example, teachers can demonstrate how to represent a quadratic equation in standard form and then factor the quadratic expression. Then, teachers can encourage students to use symbolic notation to apply the Zero Product Property and solve the resulting linear equations for the variable. Additionally, teachers can present real-world problems that can be modeled with quadratic equations, encouraging students to create and interpret mathematical representations in context.

Content standards A.EI.3 — The student will represent, solve, and interpret the solution to a quadratic equation in one variable. A.EI.3a — Solve a quadratic equation in one variable over the set of real numbers with rational or irrational solutions, including those that can be used to solve contextual problems. A.EI.3c — Verify possible solution(s) to a quadratic equation in one variable algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context.

A.F.2 — The student will investigate, analyze, and compare characteristics of functions, including quadratic and exponential functions, and model quadratic and exponential relationships. A.F.2g — For any value, x, in the domain of f, determine f (x) of a quadratic or exponential function. Determine x given any value f (x) in the range of f of a quadratic function. Explain the meaning of x and f (x) in context. A.F.2d — Make connections between the algebraic (standard and factored forms) and graphical representation of a quadratic function.

Prior connections A.EO.2 — The student will perform operations on and factor polynomial expressions in one variable.

Engage Activity A whale’s jump

60 mins

Students will determine the key moments in a whale’s jump and describe at least two methods that could be used to ﬁnd when the whale reaches a height of 40 ft above the water.

Understanding and skills

Will use Factoring quadratic expressions. Identifying solutions of equations from graphs and tables.

8.02 Solve quadratics by factoring mathspace.co

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Will develop Identifying appropriate solutions to quadratic equations in a real-world context. Connecting the factors of a quadratic equation with its solutions. Solving quadratic equations by factoring.

Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: Graphing technology, pencil, paper

Support students with disabilities Support memory - use previously taught skills and concepts Provide the formulas for the various forms of quadractic functions: Vertex form: f (x) = a(x − h)2 + k Where: • (h, k) are the coordinates of the vertex • The sign of a indicates the direction of opening of the graph • The value of a is the scale factor of the quadratic function Factored form: g(x) = a(x − x1)(x − x2) Where: • x1 and x2 are the x-values of the x-intercepts • The sign of a indicates the direction of opening of the graph • The value of a is the scale factor of the quadratic function Standard form: h(x) = ax2 + bx + c Where: • The sign of a indicates the direction of opening of the graph • The value of a is the scale factor of the quadratic function • b helps us to ﬁnd the axis of symmetry and vertex using the formula • c is the value of y-intercept

Support for English language learners Collect and display As pairs are working, listen for and collect vocabulary, phrases, and methods students use to ﬁnd the coordinates for when the whale is at certain points along its jump. Consider grouping language for the various strategies. Continue to update collected student language throughout the entire activity. Remind students to borrow language from the display as needed. Some terms and phrases may include: solutions, factors, quadratic equation, factoring, graphing, completing the square, maximum, vertex, x-intercepts, setting equation equal to 0, and solve.

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Classroom guide Hook Students compare the similarities and differences between substituting a zero into the standard form and factored form of a quadratic equation.

Open questions

•

5 mins

What are the similarities and differences between these equations? Equation 1:

Implementation details

x2 + 3x − 10 = 0 (2)2 + 3(2) − 10 = 0

Equation 2: Highlight student responses that (x + 5) (x − 2) = 0 compare the forms of the quadratic (2 + 5) (2 − 2) = 0 expressions on the left hand side of the equation; the ﬁrst equation is in Slide 1 from Student Engage Activity standard form and the second equation is in factored form. Encourage students to explain what x = 2 represents in both equations, since it is a solution for each equation.

Launch After students read the information, ask students what they think a ‘key moment’ means in terms of the whale’s jump, and what the key moments might be. Ensure that students can articulate that the whale breaches the water’s surface, reaches a maximum height before going back below the surface. It may be useful to show a quick video clip of breaching whales.

5 mins

Jessie decides to join a whale-watching tour and manages to record a whale jumping out of the water.

Important mathematical concepts: Quadratic, factors, maximum, minimum Important contextual information: Breach, water surface Suggested grouping: Form pairs

Slide 2 from Student Engage Activity

Explore

Think-pair-share

•

25 mins

Students who have successfully completed the activity will have times for the key moments in the whale’s jump with justiﬁcation and at least two different methods to determine when the whale reaches a height of 40 ft above the water. Encourage students to use multiple methods to solve for the times and note these different strategies for sequencing in the class discussion.

8.02 Solve quadratics by factoring mathspace.co

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Anticipated strategies Solve graphically Students may solve graphically for the times when the whale is at key moments of its jump.

50 45 40 35 30 25 20 15 10 5

Height above water in feet

Solve using a table of values

Time in seconds 4 5 6 7

Students may solve for the times when the whale is at key moments of its jump by using a table of values.

Solve by factoring Students may solve for the times when the whale is at key moments of its jump by factoring and connecting the factors to its solutions. One option for the factored form of the model of the whale jump: f (x) = −(x − 3)(16x − 104)

Solve by completing the square Students may solve for the times when the whale is at key moments of its jump by completing the square and converting to vertex form. Vertex form of the model of the whale jump: f (x) = −16(x − 4.75)2 + 49 Key moments in the whale’s jump modeled by f (x) = −16x2 + 152x − 312: • Leaves water at 3 seconds. • Re-enters water at 6.5 seconds. • When the times when the whale leaves and re-enters the water are inputed into the function and evaluated, the output of the function is 0 as these are the x-intercepts of the function. • Highest peak of the jump is 49 ft above water at 4.75 seconds (a humpback whale jumps on average 45 − 50 ft above water!). • Height of 40 ft at 4 and 5.5 seconds.

Misconceptions Misinterpreting the x and y coordinates What do the inputs of the function represent? What do the outputs represent? In your solution, which value is the height of the whale? Which is the amount of time elapsed? How do you know?

Purposeful questions Use the following questions to check for understanding and encourage critical thinking: • When does the whale leave and re-enter the water? How do you know? • When does the whale reach its maximum height? What is the whale’s maximum height during the jump? How do you know? • How will you solve for when the whale reaches a height of 40 ft above the water? Can you solve the equation? Can a table be used to solve? A graph? Explain. • Can you solve the quadratic equation using another method? 972

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Continue when Students have determined when the whale leaves the water, re-enters the water, and reaches the highest point in the jump, described what happens when they input the time when the whale leaves and re-enters the water in the function, and described at least two methods that could be used to ﬁnd when the whale reaches a height of 40 ft above the water.

Discuss

25 mins

Begin with partner presentations. Consider sequencing the presentations by grouping different representations together and displaying at least one example of each strategy (graphing, table of values, algebraic methods) to compare.

Discussion guide Start by inviting pairs to share the key features of the whale’s jump that they found. Ask a few groups to explain their method for solving for the times of these key features and what happens when the times that the whale leave and re-enters the water are inputed into the function. Next, ask how groups determined the time when the whale is 40 ft above the water. Start with groups that solved graphically or by using a table of values, then ask groups who solved algebraically by completing the square or factoring to share their work. If no groups solved by factoring, ask the class what other methods could be used to solve for the desired time. Ask students to connect the algebraic solutions to the solutions in the graph. Speciﬁcally, what is known about the graph at its x-intercepts? How can this be related back to the equation? Must it be written in a speciﬁc form to reveal the intercepts? Similarly, ask how to connect the vertex of the graph to the equation. Is there a way to rewrite the equation to highlight the vertex? It is possible that students may not know the connections yet between rewriting the equation and the solutions to the whale problem. You may wish to do a small example with students of an equation that can be easily rewritten, such as y = x2 + 4x + 4 or even have students work backwards from an equation that is already factored such as y = (x + 2)(x + 2) and try rewriting it in as many ways as possible. By using a small example, students may be able to understand that it is possible to do the same with the function of the whale’s height, even if they don’t yet have the procedural skills built for rewriting the function.

Tools You may ﬁnd these tools helpful: • Graphing calculator • Graph paper

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Student lesson & teacher guide Solve quadratics by factoring Students start by exploring the concept of the zero product property.

Students: Page 461

Concept development: the zero product property Targeted instructional strategies To help students better understand the zero product property, have them consider the equation a(x − b)(x − c) = 0 Ask the question “What values of x make this equation true?” and ask students to justify their answer. Point out that the left-hand side of the equation is a product of three factors, and that any product including zero will be equal to zero. As such, if any of the factors are equal to zero then the equation is true.

Stronger and clearer each time English language learner support Ask students to write or otherwise communicate an explanation for the question: “Why are the factors of the factored form related to the solutions of the corresponding equation?” Put students into pairs and instruct them to present their explanation to their partner. Give enough time for students to give feedback and discuss an explanation together. A sample student response may be: “The factors of the factored form of the equation are related to the solutions of the corresponding equation because of the Zero Product Property. When we factor the equation, we can use the Zero Product Property to set each factor equal to zero. Solving each of these equations for x will give us the solutions to the original equation.” After their discussion, give students time to refine their explanation using the results of their discussion and any feedback they received.

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Recall steps for factoring Student with disabilities support Review the different methods for factoring trinomials. Remind students of the steps for factoring: 1. Set the equation equal to zero 2. Factor out the GCF, if one exists 3. Factor by grouping, by using the box method, or by using any other appropriate strategy 4. Check that the answer will not factor further and verify the factored form by multiplication or the identities of special products: • (a + b)2 = a2 + 2ab + b2 • (a − b)2 = a2 − 2ab + b2 • (a + b) (a − b) = a2 − b2

Focusing on process instead of concepts Address student misconceptions Students may incorrectly determine the solutions to an equation as the values they find during the solution process. For example, they may identify the factors as the solutions, rather than solving to find the roots: (x + 3) (x + 2) Or, they may find the factors that sum to the coefficient of the linear term and state those values as their answer: x=3 x=2 Refer students back to the question they are trying to answer. Ask them to explain what part of the process they are at, and explain the purpose of that step in the process.

Exploration Students: Page 461

Suggested student grouping: In pairs Students explore the idea of inverse operations when solving equations set equal to zero. Students will rediscover the concept behind the zero product property. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. What values of the variables make each of the following equations true? z = 7; a = 0; d = −4; either x = 0 or y = 0.

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Purposeful questions • When the operation between a constant and a variable is addition or subtraction, what property do you use to ﬁnd the variable that makes the value zero? (additive inverse) • When the operation between a constant and a variable is multiplication, what property do you use to ﬁnd the variable that makes the value zero? (zero property of multiplication) • When there is both multiplication and addition/subtraction in an equation, what properties allow you to isolate the variable? (multiplicative identity, additive identity) • When there is a product of two unknowns set equal to zero, what must be true? Why? (at least one of the variables must be zero; zero property of multiplication.) Possible misunderstandings • Students may know how to solve the problems, but not be able to identify the arithmetic properties involved. • For the last equation, students may think both x and y must be equal to zero at the same time. Help students articulate that either value being equal to zero (with any combination of real values for the other variable) will cause the entire product to equal zero. Students will discover how the zero product property can be used to solve quadratic equations in factored form. They will also connect the factors and x-intercepts to write the equation of a quadratic.

Students: Page 461

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Examples Students: Page 462

Purpose Make students aware that with equations set equal to zero, after factoring by grouping, they can use the zero product property to find the solutions. Expected mistakes A student may not apply the zero product property, and instead determine that the solutions are x = 11 and x = −5, which are the constants of the factors. Remind students that after factoring the polynomial, we still need to set each factor equal to zero to solve for x.

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Students: Pages 462–463

Apply the idea 3x2 + 3x − 10 = 8

Given equation

3x2 + 3x − 18 = 0

Subtraction property of equality

Now, we can solve by factoring. 3(x2 + x − 6) = 0 2

Factor out the GCF

3(x + 3x − 2x − 6) = 0

Rewrite the trinomial as a polynomial with four terms

3[x(x + 3) − 2(x + 3)] = 0

Factor each pair of terms

3(x + 3) (x − 2) = 0 x + 3 = 0 and x − 2 = 0 x = −3 and x = 2

Divide out common factor of (x + 3) Zero product property Addition property of equality

Reﬂect and check We can check our answers by substituting them back into the original equation to see if they make the equation true. We will check x = −3 ﬁrst. 3x2 + 3x − 10 = 8

Original equation

Substitute x = −3

3(−3) + 3(−3) − 10 = 8 27 − 9 − 10 = 8 8=8

Evaluate the multiplication Evaluate the subtraction

This is a solution to the equation. Now, we will check x = 2. 3x2 + 3x − 10 = 8

Original equation

3(2)2 + 3(2) − 10 = 8

Substitute x = 2

12 + 6 − 10 = 8 8=8

Evaluate the multiplication Evaluate the subtraction

This also satisﬁes the equation, so it is a solution.

Example 2 Purpose Make students aware an into equation needs that itbyisthe setequation equal to Luis throws a ball that straight the air. The pathto of be the rewritten ball can be so modeled y = zero −5x2 +before 14x + 3 factoring. where x Show students how factor and equation when leading coeﬃcient notlong 1. will it represents the to time the ball is insolve the airainquadratic seconds and y represents the the height of the ball in meters. is How take the ball to hit the ground?

Expected mistakes StudentsCreate may try to factor the left side of the equation as written, without first setting the equation equal to zero. a strategy Remind The students that factoring helps us solve a itproblem if we apply the zero productThe property, question is asking us to only find the time (the x-value) takes for the ballcan to hit the ground (the y-value). ground which a height 0. In other words, the question is asking us to solve the equation −5x2 + 14x + 3 = 0. requiresrepresents an equation setofequal to zero. When we factor, we usually have a positive leading coefficient. To begin, we can factor out −1 which will give us a positive leading coefficient.

Memory support: use a multiplication chart

use with Example 1

Student with disabilities support For students who are less confident in their mental factoring, offer an alternative method to factoring by grouping. The box method offers a visual approach to factoring that may be beneficial.

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3x2 + 3x − 10 = 8

Original equation

3(2)2 + 3(2) − 10 = 8

Substitute x = 2

12 + 6 − 10 = 8

Evaluate the multiplication

8=8

Evaluate the subtraction

also satisﬁes the equation, so it is a solution. Students:This Pages 463–464

Example 2 Luis throws a ball straight into the air. The path of the ball can be modeled by the equation y = −5x2 + 14x + 3 where x represents the time the ball is in the air in seconds and y represents the height of the ball in meters. How long will it take the ball to hit the ground?

Create a strategy The question is asking us to find the time (the x-value) it takes for the ball to hit the ground (the y-value). The ground represents a height of 0. In other words, the question is asking us to solve the equation −5x2 + 14x + 3 = 0. When we factor, we usually have a positive leading coefficient. To begin, we can factor out −1 which will give us a positive leading coefficient.

Apply the idea −(5x2 − 14x − 3) = 0 5x2 − 14x − 3 = 0

Factor out −1 Divide both sides by −1

Next, we need to ﬁnd two numbers that multiply to ac = 5 ⋅ −3 = −15 and add to b = −14. The factor pair that satisﬁes these conditions is −15 and 1. Rewrite the polynomial with 4 terms Factor by grouping Factor out the GCF of (x − 3)

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Zero product property Addition property of equality Division property of equality The x-values represent time, so a negative value does not make sense since we cannot go backward in time. This means

is a nonviable solution, and x = 3 is the only viable solution.

The ball hit the ground after 3 seconds.

Reﬂect and check As we can see from the graph,

is an x-intercept.

But because it does not make sense in context, it is not a solution to the problem. We can picture Luis standing at the y-axis when he throws the ball since x = 0 would represent the present moment.

12 10 8 6 4 2 x 1

Idea summary Purpose We can use the zero product property to solve quadratic equations by ﬁrst writing the equation in Make studentsfactored awareform: that various contextual situations may require finding the input values that cause the a (x − x1) (x − x2) = 0 output values to be zero. Students will demonsrate using factoring by grouping to solve a contextual problem. then setting each factor equal to zero and solving for x.

Expected mistakes A student may not realize that “the ground” is represented by y = 0 as it represents a height of 0 m. Have students refer back to the independent and dependent variables in this context, then consider what information was given in the problem. Finally, have them relate that information to the given variables.

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Reflecting with students Apply the idea What if we wanted to find out what time the ball was at a particular height, rather than when it hit the ground? −(5x2 − 14x − 3) = 0 Factor out −1 Ask students to describe how they could solve the problem. (They would need to rewrite the equation so it is Divide both sides by −1 5x2 − 14x − 3 = 0 equal to zero, then solve.) Next, we need to find two that multiply to ac = 5 ⋅whether −3 = −15 and to b = −14. factor pair that satisfies Then, extend the problem by numbers having students consider theadd factoring byThe grouping method work for these conditions is −15 and 1. any height chosen. Point out that not every equation will end up with values that can be factored. Challenge polynomial 4 terms advanced learners to find a height otherRewrite than 0thethat would with allow a factoring solution. Factor by grouping Factor out the GCF of (x − 3)

Three reads

use with Example 2

Zero product property English language learner support Addition property of equality

Advise students to read through the instructions multiple times, focusing on gathering different information each Division of equality time in order to build up their understanding ofproperty what the question is asking. The x-values representshould time, so aim a negative value does not make sense since we cannot backwardAsk in time. On the first read, students to identify the scenario represented in thego equation. students is a nonviable solution, and x = 3 is the only viable solution. This means “What do you think is happening in this question?” The ball hit the ground after 3 seconds.

On the second read, students should aim to interpret the problem by answering questions like “What is the questionReﬂect askingand you to find?” or “What information should be included in the answer?” check y

12 On the third students for In this question, important As weread, can see from the should graph, look is animportant x-intercept. information in the instructions. information includes: 10 But because it does not make sense in context, it is not a solution to the • x represents thecan time theLuis ballstanding is in the problem. We picture at air the y-axis when he throws the ball 8 since x = 0 would represent the present moment. • y represents the height of the ball in meters 6 • The ground represents a height of zero 4

Students can be prompted with questions like “What does each variable represent in this problem?” 2

x 1

Students: Page 464

Idea summary We can use the zero product property to solve quadratic equations by ﬁrst writing the equation in factored form: a (x − x1) (x − x2) = 0 then setting each factor equal to zero and solving for x.

Practice Students: Pages 465–466

What do you remember? 464

Mathspace

Virginia SOL Algebra 1

Solve themathspace.co following equations by using the zero product property: a

x (x − 9) = 0

2m(m − 8) = 0

c (5c − 12) = 0

(k − 3) (k − 5) = 0

( y − 6) ( y + 11) = 0

(3x − 9) (2x − 5) = 0

5(x + 5) (x − 5) = 0

(7a − 2)2 = 0

x = −5, x = −7

Consider the quadratic equation:

x2 + 2x − 35 = 0

Select the solution of the quadratic equation. A 980

x = −5, x = 7

x = 5, x = −7

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

x = 5, x = 7

Consider the quadratic equation:

x2 − 4x − 12 = 0

Select the solution of the quadratic equation. A 4

x = −6, x = −2

x = −6, x = 2

x = 6, x = −2

x = 6, x = 2

Determine the x-intercept(s) for each of the following quadratic function: a

y = (x + 8) (x + 4)

y = − (x − 10)2

y = (x − 3) (x + 2)

y = (1 − x) (x + 5)

y = (x − 6) (x − 24)

y = − (x − 8) (x + 2)

y = − (x + 13)2

y = x (x + 3)

Let’s practice 5

Solve the following equations by factoring: a

6x2 + 54x = 0

4y − 8y2 = 0

x2 − 5x − 14 = 0

f 2 + 6f − 55 = 0

h2 + 19h + 88 = 0

x2 − 20x + 100 = 0

x2 + 8x − 20 = 0

x2 − 13x − 114 = 0

Solve the following equations by ﬁrst rearranging, and then factoring: a

2y2 = 9y + 5

3x2 − 14x = −8

x2 − 3x − 10 = 0

x2 + 7x + 12 = 0

x2 + 3x = 28

x2 − 11x + 19 = −5

x(x + 2) − 48 = 0

m2 = 3m + 10

x2 = 4 − 3x

10y = y2 + 24

x2 + 6x + 8 = 0

Solve: a

x(x + 18) + 80 = 0

x − 12x = −20

m + 5m = 14

−m2 − 7m = −18

−n2 − 5n = −84

−6y = y + 8

Solve the quadratic equations and verify your solution(s) by graphing. a

Solve the quadratic equations by factoring. Justify your work. a

x2 − 14 = 5x

x2 + x − 42 = 0

x2 − 9x = −20

x2 − 35 = 2x

Software engineers are designing a self-serve checkout system for a supermarket. They notice that the traﬃc through the store during the day is described by the function C = −t (t − 12) where C is the number of customers and t is the number of hours after the store opens. To meet the peak demand, the engineers allow for an extra checkout machine to automatically turn on when the number of customers ﬁrst reaches 32 people, and to automatically turn off when it next falls below 32 people.

Find the times t when the number of customers is equal to 32 people.

Determine how many hours it takes the extra checkout machine to turn on after opening.

Determine how many hours the extra machine will be on for.

Delores needs a sheet of paper x in by 12 in for an origami alligator. The local art supply store only sells square sheets of paper. The lower portion of the image shows the excess area A of paper that will be left after Delores cuts out the x in by 12 in piece. The excess area, in square inches, is given by the equation A = x (x − 12)

Determine the lengths for which there will be no excess area.

Determine the value of x that will allow Delores to make an origami alligator with the least amount of excess paper.

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The area of a rectangle is 60 cm2. If the area can be expressed as A = 17x − x2, what are the dimensions of the rectangle?

For each function: i

Determine the solution(s).

y = −x2 + 5x + 6

y = x2 − 8x + 12

Write the equation in factored form.

y = −x2 + 8x − 16

y = x2 + 5x

Let’s extend our thinking 14

The school football field is in the shape of a rectangle and has stadium seating all the way around the field that is of a uniform width. If the total area of the field and the stadium is 8400 yd2, determine the width of the stadium seating. x

50 yd

100 yd

A = 8400 yd2 15

The sum of the series 1 + 2 + 3 + 4 + … + n is given by

Determine a method to solve for n given any

sum, and then ﬁnd the number of integers required for a sum of 66.

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Answers 8.02 Solve quadratics by factoring

x2 − 11x + 19 = −5

x − 11x + 24 = 0

Given equation Given equation

(x2 − 8x − 3x + 24) = 0 Rewrite the trinomial as a polynomial with four terms

What do you remember? 1 a x = 0, x = 9

[x(x − 8) − 3(x − 8)] = 0 b m = 0, m = 8

Factor each pair of terms

(x − 8)(x − 3) = 0 Divide out common factor of (x − 8)

x − 8

0 and x − 3 = 0

Zero product property

d k = 3, k = 5

e y = 6, y = −11

8 a x = −8, −10

b x = 6, −8

c m = 5, −2

d x = −4, 1

g x = 5, x = −5

8 and x = 3

Addition property of equality

e x = 2, 10

2 B

g y = −2, −4

h y = 4, 6

3 C

m = 2, −9

b (10, 0)

9 a (x + 7)(x − 6) = 0

c (3, 0), (−2, 0)

d (1, 0), (−5, 0)

e (6, 0), (24, 0)

(8, 0), (−2, 0)

g (−13, 0)

h (0, 0), (−3, 0)

4 a (−8, 0), (−4, 0)

5 a x = 0, x = −9

c x = 7, x = −2

d f = −11, f = 5

e h = −8, h = −11

g x = 2, −10

h x = 19, −6

6 a x = −2, x = 7 c 2

x − 3x − 10 = 0

7 a

x = 10

Given equation

0 and x + 2 = 0

x2 + 7x + 12 = 0

5 and x = −2

5, x = 4

x 7

Factor each pair of terms

x − 5

x 2 4 6 8

polynomial with four terms

(x − 5)(x + 2) = 0 Divide out common factor of (x − 5)

(x − 5)(x − 4) = 0

d g = 2, g = 6

b x2 − 9x + 20 = 0

(x2 − 5x + 2x − 10) = 0 Rewrite the trinomial as a [x(x − 5) + 2(x − 5)] = 0

n = −12, 7

−7, x = 6

−8 −6 −4 −2 −5 −10 −15 −20 −25 −30 −35 −40 −45

Let’s practice

m = −7, 2

4 3

Zero product property

Addition property of equality

−1 −1

Given equation

2 3 4 5 6 7

(x + 3x + 4x + 12) = 0 Rewrite the trinomial as a

[x(x + 3) + 4(x + 3)] = 0

polynomial with four terms

c x2 − 2x + 35 = 0

Factor each pair of terms

(x − 7)(x + 5) = 0

(x + 3)(x + 4) = 0 Divide out common factor of (x + 3)

x + 3

0 and x + 4 = 0

−3 and x = −4 x2 + 3x = 28

c 2

x + 3x − 28 = 0

Zero product property Addition property of equality Given equation Subtraction property of equality

(x + 7x − 4x − 28) = 0 Rewrite the trinomial as a polynomial with four terms

[x(x + 7) − 4(x + 7)] = 0

Factor each pair of terms

(x + 7)(x − 4) = 0

Divide out common factor of (x + 7)

x + 7

0 and x − 4 = 0

Zero product property

−7 and x = 4

7, x = −5 5

−8 −6 −4 −2 −5

y x 2 4 6 8

−10 −15 −20 −25 −30 −35

Addition property of equality

Answers mathspace.co

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Let’s extend our thinking

d (x + 4)(x + 2) = 0

−4, x = −2

14 10 yards 4

quadratic equation

3 2 1

−9 −8 −7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4

10 a t = 4, t = 8

b 4 hours

c 4 hours 11 a x = 0, x = 12

b 12 in

12 5 cm × 12 cm 13 a i (−1, 0), (6, 0)

ii y = −(x − 6) (x + 1)

b i (6, 0), (2, 0)

ii y = (x − 2) (x − 6)

c i (4, 0)

ii y = −(x − 4)2

d i (0, 0), (−5, 0)

ii y = x(x + 5)

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15 To find the number of integers we want to solve the

Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

We can rearrange this

equation into the form n2 + n − 2S = 0. We can factor this and then use the zero product property to determine the two solutions. To find the number of integers required for a sum of 66 we substitute S = 66 giving the equation n2 + n − 132 = 0. Factoring this gives (n − 11) (n + 12) = 0 which has solutions of n = 11, n = −12. We can exclude the solution of n = −12 as we are only considering the positive solution.

8.03 Solve quadratics using square roots Subtopic overview Lesson narrative In this lesson, students will examine the structure of the equations and make generalizations leading to an understanding of using inverse operations to solve certain forms of quadratic equations using square roots. Students will also examine the completing the square method of solving quadratic equations. By the end of the lesson, students will be able to examine the structure of quadratic equations to complete the square and solve using the square root property.

Learning objectives Students: Page 467

Key vocabulary 

completing the square



perfect square



perfect square trinomial



radicand



simplified radical form



square root

Essential understanding Quadratic equations can be solved using a variety of methods. Using square roots can be an efficient method when the equation only has two terms and the constant term is a perfect square. Completing the square can be an efficient method for an equation that cannot be factored.

Standards This subtopic addresses the following Virginia Standards of Learning for Mathematics standards.

Mathematical process goals MPG1 — Mathematical Problem Solving Teachers can integrate this goal in their instruction by providing students with a variety of practice problems that require solving quadratic equations using square roots and completing the square. They can encourage students to apply the steps outlined in the lesson and to connect their solution methods to their prior knowledge of simplifying radicals. 8.03 Solve quadratics using square roots mathspace.co

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MPG3 — Mathematical Reasoning Teachers can integrate this goal by asking students to justify their steps in solving the problems. They can also present situations where students have to decide whether a given solution to a quadratic equation is valid or not, thereby applying their reasoning skills.

A.EI.3c — Verify possible solution(s) to a quadratic equation in one variable algebraically, graphically, and with technology to justify the reasonableness of answer(s). Explain the solution method and interpret solutions for problems given in context.

Prior connections A.EO.2 — The student will perform operations on and factor polynomial expressions in one variable.

A.EO.4 — The student will simplify and determine equivalent radical expressions involving square roots of whole numbers and cube roots of integers.

A2.EI.5 — The student will represent, solve, and interpret the solution to an equation containing a radical expression.

Engage Activity Designing tiles

60 mins

Students will design tiles and solve for exact measurments of tile pieces based on a range of area to introduce solving quadratic equations by completing the square.

Understanding and skills

Will use Solving quadratic equations using square roots.

Will develop Writing algebraic expressions for area of tiles. Solving quadratic equations to find exact measurements of tile pieces based on a range of areas.

Could extend Solving quadratic equations by completing the square visually and algebraically.

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Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: None.

Support students with disabilities Support organization - solve multistep or complex problems Chunk the task into smaller parts: for example, freely designing square tiles and designing square tiles using the applet. Ask a group member to volunteer to be the time keeper for the activity or keep track of time for the whole class and remind them when they should be transition from one part of the work to the next. Use a visual display to show the chunked sections of time for students.

Support for English language learners Three reads Have students read the task aloud. On the first read, ask students to describe the situation. Prompt: Students read the problem. Students think/write: Answer the question “What is the problem about?” Answers may look like: • Leroy is designing floor tiles. • Leroy is creating square tiles for a construction project. Share: Students are called upon to discuss their answers with the class. On the second read, ask students to interpret the question. Prompt: Students read the problem. Students think/write: Answer the question “What does an answer look like?” Answers may look like: • A design of square tiles. • Combination of tiles that are in the shape of a square. On the third read, have students identify important information. Prompt: Students read the problem. Students think/write: Answer the question “What are the important pieces of information given in the question?” Answers may look like: • The dimensions of each piece. • The design must be in the shape of a square. • Can only use one of piece 1 in the design.

Classroom guide Hook Students compare the similarities and differences between a quadratic binominal set equal to a perfect square and a quadratic trinomial set equal to a perfect square.

Open questions

•

5 mins

What are the similarities and differences between these equations? x2 + 4x = 49 x2 + 4x + 4 = 49 Slide 1 from Student Engage Activity

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Implementation details Students may start by describing similarities and differences about the structure of equation, such as the fact that both equations have a constant on the right hand side of the equation, but one equation has three terms while the other has two. Encourage students to think about how these quadratics might be solved using methods they currently know.

Launch

5 mins

Allow students to read the instructions individually before forming pairs. It may be helpful to ask students to think about the area of each piece before forming pairs.

Leroy has a summer job in construction and is helping to tile a floor. Leroy is designing different squares to be reviewed by his manager.

Important mathematical concepts: Area model for quadratic expressions Suggested grouping: Form pairs

x x

x 1

Slide 2 from Student Engage Activity

Continue when Students have read the Launch and understand the context of the problem.

Explore In the first stage of the Explore, students are designing square tiles using any combination of square and rectangular pieces so long as it contains exactly one x × x piece.

Think-pair-share

Repeat a pattern Students may design a single square, such as the design provided, then copy that design into larger and larger squares to meet the criteria of having five different designs.

Students may expand the given design by adding additional x × 1 pieces and 1 × 1 pieces to fill in the space. Answers will vary depending on the tile designed by each student.

Large tile size

Slide 5 from Student Engage Activity

Utilize square pieces Students may only use the square pieces to design their square tiles. Encourage these students to consider different designs that still incorporate the rectangular piece.

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25 mins

Use the applet to design square tiles.

Anticipated strategies

Expand the design

•

Misconceptions Using the visual proportions to “match” dimensions How many 1 × 1 pieces would you need to align to match a side length of x? How can we be sure two side lengths match up using these specific tiles? In the second stage of the Explore, students are designing square tiles using the specific combination of x × x with an equal number of x × 1 to the right and below it.

Anticipated strategies Guess and check Students may input different values of x into their expression until they reach a square area between 100 and 150 square units.

Solve algebraically Students may write an expression for the area of the tile as (x + b)2 or x2 + 2bx + b2 and set it equal to a value between 100 and 150 to solve.

Solve graphically Students may write an equation for the area of the tile and use graphing technology to determine the inputs for when their quadratic function is between 100 and 150 square units.

Misconceptions Miscalculating the area of the main tile pieces What are the dimensions of each tile piece? What are the areas of each tile piece? Check in with groups to make sure they understand the area of each piece: • x2 square units • x square units • 1 square unit

Combining non-like terms What is the expression for the area of your tiles? What are like terms in the area expression and visually in your tile?

Expanding polynomial expressions incorrectly What is the connection between your tile and the algebraic expression for area? How can the area expression be rewritten to represent the length and width of your tile?

Purposeful questions Use the following questions to check for understanding and encourage critical thinking: • What is the relationship between the number of rectangular pieces and 1 × 1 square pieces? • What are the different ways you could write the expression for the area of the square design? • Are there other ways to solve this problem? • What restrictions does needing a square tile place on the problem?

Continue when Students have five different tile designs and have solved for exact dimensions that meet the area criteria.

Discuss

25 mins

Have a whole class gallery walk. Consider making connections between knowing the number of 1 × 1 square pieces and the process of completing the square.

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Discussion guide Have groups share their tile designs and justifications, either through presentations or a gallery walk. This allows students to generalize and see multiple examples of the process of visually completing the square. Once designs are being displayed, ask groups to share their process and reasoning for finding the exact measurements of the large square and rectangular pieces. Ask students if they can determine which method of solving quadratic equations were explored through the task.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Algebra 1 — 5.06 Simplify radicals Algebra 1 — 7.03 Quadratic functions in vertex form

Tools You may ﬁnd this tool helpful: • Scientiﬁc calculator

Student lesson & teacher guide The square root property Students are guided through the steps for solving equations using square roots. They are introduced to the square root property, then they are shown how to simplify square roots by relating back to properties of exponents.

Students: Pages 467–468

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We can simplify using properties of exponents, properties of radicals, or a perfect square factor: Properties of exponents When we simplified radicals using rational exponents we did the following. First, we converted the radical to a rational exponent. Then, we found the prime factors of 24 and applied properties of exponents to simplify.

Properties of radicals We can follow a similar process for this method, except we can leave the expression in radical form. First, we ﬁnd the prime factors of 24, then we can use properties of radicals to simplify.

Perfect square method This is the quickest method for simplifying a radical. Instead of ﬁnding all the prime factors of 24, we want to ﬁnd the largest perfect square factor of 24. Then, we can use the multiplication property of radicals to simplify.

Example 1 Solve the following equations by using square roots: a x2 = 9

Create a strategy In this equation we have 9 being equal to the square of x. This is equivalent to x being equal to the square root of 9.

8.03 Solve quadratics using square roots mathspace.co

Apply the idea x = ±3

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Scaffold by providing steps for solving equations using square roots Student with disabilities support We can simplify using properties of exponents, properties of radicals, or a perfect square factor:

For students who have difficulty remembering the procedure for solving using the square root method, provide Properties of exponents the reasoning for each step of work as a scaffold. For example: 2(x − 4)2 + 11 = 29 ⬚=⬚

⬚=⬚ ⬚=⬚

⬚=⬚

When we simplified radicals using rational exponents we did the following. First, we converted the radical to a rational exponent. Then, we sides found the prime factors of 24 and applied properties of Subtract ⬚ from both exponents to simplify.

Divide both sides by ⬚

Evaluate the square root of both sides Add ⬚ to both sides

There are two solutions for square root equations Address student misconceptions Properties of radicals

Remind students not to forget the ± symbol whenWe taking the asquare root of for both ofexcept an equation. can follow similar process thissides method, we can It can leave the expression in radical form. First, wegives ﬁnd theboth primea factors help to take some time at the start of the lesson to review why taking the square root positive and of 24, then we can use properties of radicals to simplify. negative result. Consider that (−10)2 = 100 and (10)2 = 100, so if x2 = 100 then we have x = ±10.

Perfect square method

Examples

This is the quickest method for simplifying a radical. Instead of ﬁnding all the prime factors of 24, we want to ﬁnd the largest perfect square factor of 24. Then, we can use the multiplication property of radicals to simplify.

Students: Page 468 Example 1

Solve the following equations by using square roots: a x2 = 9

Create a strategy In this equation we have 9 being equal to the square of x. This is equivalent to x being equal to the square root of 9.

Apply the idea x = ±3

Reﬂect and check Checking our answers:

(−3)2 = 9 (3)2 = 9

Both answers satisfy the equation.

Purpose Show students how to solve an equation by evaluating the square root of both sides. 468

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Expected mistakes mathspace.co Students may state that the solution is only positive 3. Students may be guided using questions such as “Is three the only number that when squared results in nine?” or “What is the sign of the product of two negative numbers?”

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Students: Page 469 b 4x2 − 27 = 0

Create a strategy We can begin by isolating the variable, but we will need to simplify the radical since 27 is not a perfect square.

Apply the idea Given equation Addition property of equality Division property of equality Square root property Division property of radicals From here, we can factor 27. Our goal is to separate it into factors where one is a perfect square. 27 = 9 ⋅ 3 where 9 is a perfect square. b 4x2 − 27 = 0

Multiplication property of radicals

Create a strategy

Evaluate the radicals We can begin by isolating the variable, but we will need to simplify the radical since 27 is not a perfect square.

Reﬂect and check Apply the idea

For nearly all of our work with solutions to functions and equations, it is standard practice to leave our ﬁnal Given equation expression in exact form. In questions involving applications Addition of quadratics, we may be asked to evaluate the square root at the very end using property of equality a calculator, then approximate to a speciﬁc number of decimal places. Division property of equality c (x − 2)2 − 100 = 0

Square root property

Purpose Division property of radicals Create a strategy Show students how to solve an equation that involves basic inverse operations for solving equations alongside In order to use square roots to solve, the squared expression must be isolated. In this example we want to isolate the From here, we can factor 27. Ourroots. goal is to separate it into factors where one is a perfect square. 27 = 9 ⋅ 3 where 9 is evaluating and simplifying square term (x − 2)2. a perfect square.

Expected mistakes

Apply the idea

Multiplication property of radicals

(x − that 2)2 − the 100 =solution 0 Given Students may state is equation . Remind students to determine if the numbers inside of a radicand (x − 2)2 = 100

Add 100 to both sides

the radicals can continue to be simpliﬁed, either Evaluate by checking their prime factorization or considering perfect square factors. x − 2 = ±10

Take the square root of both sides

Reflect and Reflecting with students This leaves uscheck with two equations x − 2 = 10 and x − 2 = −10. Add 2 to solve both equations and we find that the solutions are xof=our −8, x = check 12. For nearly allthey work with solutions to functions it is standard practice to leavethe ourfinal final solutions into Ask students how can that their solutionand is equations, correct. Students may substitute expression in exact form. the original equation to check that the equation is valid. In questions involving applications of quadratics, we may be asked to evaluate the square root at the very end using

calculator,469–470 then approximate to a speciﬁc number of decimal places. Students:a Pages c (x − 2)2 − 100 = 0

Create a strategy In order to use square roots to solve, the squared expression must be isolated. In this example we want to isolate the 8.03 Solve quadratics using square roots 469 term (x − 2)2. mathspace.co

Apply the idea (x − 2)2 − 100 = 0 (x − 2)2 = 100 x − 2 = ±10

Given equation Add 100 to both sides Take the square root of both sides

This leaves us with two equations x − 2 = 10 and x − 2 = −10. Add 2 to solve both equations and we ﬁnd that the solutions are x = −8, x = 12. 8.03 Solve quadratics using square roots mathspace.co

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c (x − 2)2 − 100 = 0

Create a strategy In order to use square roots to solve, the squared expression must be isolated. In this example we want to isolate the term (x − 2)2.

Apply the idea (x − 2)2 − 100 = 0 (x − 2)2 = 100 x − 2 = ±10

Given equation Add 100 to both sides Take the square root of both sides

This leaves us with two equations x − 2 = 10 and x − 2 = −10. Add 2 to solve both equations and we ﬁnd that the solutions are x = −8, x = 12.

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Purpose Show students how to solve an equation in vertex form using square roots. Expected mistakes Students may state that the solutions of the equation are x = ±12, assuming that they do not need to separate the equation into two equations to see how the equations should be solved.

Students: Pages 470–471

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Now checking the next solution: Original equation Substitute Evaluate the multiplication Evaluate the subtraction Evaluate the exponent Both answers satisfy the original equation.

Example 2 Purpose State a quadratic that has the given solutions. Show students how toequation solve an equation using square roots where the coeﬃcient of x is not 1. a

Reflecting with students The ± in the equation may confuse students and make them less confident in moving forward. Point out to Create a strategy students that this equation simplifies to two linear equations. We can work backwards from solving to find the equation that had the given solutions. For students that arethe still unsure, Now checking next solution:separate into two linear equations sooner, like so: Apply the idea

Original equationGiven equation Given solutions Square root property Substitute Add 1 to both sides Separate into two equations Square Evaluateboth the sides multiplication Add 8 to both sides Evaluate the subtraction

Divide both sides by 3 Reﬂect and check

Evaluate the exponent This is one equation, but we could also subtract 7 from both sides to get an equivalent equation with the same solutions. Both answers satisfy the original equation. Students:(xPage =0 + 1)2 − 7471

Example 2

State a quadratic equation that has the given solutions.

Create a strategy

a We can use a similar process as the previous problem, but this time we need to multiply both sides by 3 ﬁrst.

Create a strategy Apply the idea We can work backwards from solving to ﬁnd the equation that had the given solutions. Given solutions

Apply the idea

Multiply 3 to both sides Given solutions Subtract 5 from both sides Add 1 to both sides Square both sides Square both sides

Reflect and check Reflect and check When completing the square, fractional solutions come from equations where a ≠ 1. The 3 in the denominator came fromisthe coefficient x we inside thealso parentheses. This one equation,of but could subtract 7 from both sides to get an equivalent equation with the same solutions. (x + 1)2 − 7 = 0 8.03 Solve quadratics using square roots mathspace.co

Create a strategy

471

8.03 Solve quadratics using square roots mathspace.co

We can use a similar process as the previous problem, but this time we need to multiply both sides by 3 ﬁrst.

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Create a strategy We can work backwards from solving to ﬁnd the equation that had the given solutions.

Apply the idea

Purpose Given solutions Show advanced learners how to work backwards to construct a quadratic equation from its solutions. Add 1 to both sides Square both sides Reflecting with students Ask students to walk through solving their final equation. Point out to students that the operations they use to Reflect and check solve the equation are simply the inverse operations they used to build it. This is one equation, but we could also subtract 7 from both sides to get an equivalent equation with the same solutions. 2

=0 + 1) − 7471 Students:(xPage

Create a strategy We can use a similar process as the previous problem, but this time we need to multiply both sides by 3 ﬁrst.

Apply the idea Given solutions Multiply 3 to both sides Subtract 5 from both sides Square both sides

Reﬂect and check When completing the square, fractional solutions come from equations where a ≠ 1. The 3 in the denominator came from the coeﬃcient of x inside the parentheses.

8.03 Solve quadratics using square roots

471

mathspace.co Purpose Provide an opportunity for advanced learners to deepen their understanding of solutions to quadratic equations by working backwards to construct a quadratic equation from its solutions.

Expected mistakes Students may think they perform the multiplication by 3 on both sides last, since it appears in the denominator. Remind students that when solving an equation that has a number in the denominator as shown, we can only access the terms in the numerator by performing the multiplicaiton of the denominator term on both sides of the equation ﬁrst.

Students: Page 472

Example 3 A square ﬁeld has perpendicular lines drawn across it dividing it into 36 equal sized smaller squares. If the total area of the ﬁeld is 225 square feet, determine the side length of one of the smaller squares.

Create a strategy We know that there are 36 smaller squares in total on a larger square grid, so there must be 6 by 6 smaller squares on the grid. If we let the side of a smaller square be x, then the side of the larger square can be 6x. This gives us the quadratic equation (6x)2 = 225. We can then solve this equation by taking square roots.

Apply the idea Write the equation Take the square root of both sides Divide both sides by 6 Simplifying the expression gives us the solutions x = ±2.5. We can exclude the negative solution as the length of the square must be positive. So, the smaller square has a side length of 2.5 feet.

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Mathspace SOL Algebra 1 Teacher Edition Reﬂect Virginia and check mathspace.co In most real-life applications, we will exclude the negative solution as it will be non-viable for the context.

of the ﬁeld is 225 square feet, determine the side length of one of the smaller squares.

Reﬂect and check In most real-life applications, we will exclude the negative solution as it will be non-viable for the context.

Idea summary Purpose When we use the square root property, we always include the ± symbol to denote the positive and negative root. Show studentsWehow and solve equation using square roots can to usewrite the following factsan to simplify radical expressions, for a, bgiven ≥ 0: a context, and determine extraneous solutions. Multiplication property of radicals

Expected mistakes Division property of radicals Students may state that the side length of each square is ±2.5 feet. Remind students to consider if their 3 in the context of a problem. For this problem, students need to determine whether solutionsExample make sense −2.5 feet makes sense as a distance.

A square ﬁeld has perpendicular lines drawn across it dividing it into 36 equal sized smaller squares. If the total area Completing the square of the ﬁeld is 225 square feet, determine the side length of one of the smaller squares. Completing the square is a method we use a quadratic expression so that it contains a perfectuse square Draw a visual representation oftoarewrite problem in context with Example 3 trinomial which can be factored as A2 + 2AB + B2 = ( A + B)2. We used this method in a previous lesson to convert Create awith strategy Student disabilities support a quadratic equation from standard form to vertex form. We will now learn to use the completing the square method We know that there are 36 smaller squares in total on a larger square grid, so there must be 6 by 6 smaller squares combined the square root property to solvein quadratic For students thatwith cannot visualize the scenario this equations. on the grid. If we let the side of a smaller square be x, then the side of the larger square can be 6x. This gives us the problem,quadratic help them create representation of equation what is by taking square roots. equation (6x)2a=visual 225. We can then solve this

Exploration happening. This shows the student that the square field is 225 in2 Apply the idea being subdivided into 36 equal parts, each part being a Consider the equation x2 + 6x = 11. perfect square. Write the equation 1. Try to develop a method for turning the left-hand side of the equation into a perfect square trinomial. Show them how there are 6 of the smaller along Take thesquares square root of both sides 2. Remember that we need to keep both sides of the equation balanced. After making the perfect square the sides, where each square has a side length of x.byThis bothissides 6 trinomial, check that yourDivide equation still balanced. means the sides of the larger square have a length of 6x, 3. How could we solve the equation 2in this form? us the solutions meaningSimplifying the areathe of expression the large gives square is (6x) . 2x = ±2.5. We can exclude the negative solution as the length of the

x x

− 5length = 0? of 2.5 feet. 4. must How youSo, apply method to xhas− a10x square becould positive. theyour smaller square side

Reﬂect and check

Students:InPage 472 most real-life applications, we will exclude the negative solution as it will be non-viable for the context. 472

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Idea summary When we use the square root property, we always include the ± symbol to denote the positive and negative root. We can use the following facts to simplify radical expressions, for a, b ≥ 0: Multiplication property of radicals Division property of radicals

Completing the square

Completing the square Completing the square is a method we use to rewrite a quadratic expression so that it contains a perfect square

trinomial which can be factored as A2 + 2AB + B2 = ( A + B)2. We used this method in a previous lesson to convert

Students recall that the process of completing the square leads to perfect square trinomials, and these equations can a quadratic equation from standard form to vertex form. We will now learn to use the completing the square method be solved using square roots. Anroot exploration an quadratic equationequations. that may be solved by completing the square follows. combined with the square property toof solve

Exploration Consider the equation x2 + 6x = 11. 1.

8.03 Solve quadratics using square roots mathspace.co

Try to develop a method for turning the left-hand side of the equation into a perfect square trinomial.

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We can use the following facts to simplify radical expressions, for a, b ≥ 0: Multiplication property of radicals Division property of radicals

Students: Page 472

Completing the square Completing the square is a method we use to rewrite a quadratic expression so that it contains a perfect square trinomial which can be factored as A2 + 2AB + B2 = ( A + B)2. We used this method in a previous lesson to convert a quadratic equation from standard form to vertex form. We will now learn to use the completing the square method combined with the square root property to solve quadratic equations.

Exploration Consider the equation x2 + 6x = 11. Concrete-Representational-Abstract (CRA) Approach

Targeted instructional strategies 1. Try to develop a method for turning the left-hand side of the equation into a perfect square trinomial. 2. Remember that we need towith keepphysical both sidesmanipulatives of the equation balanced. After making the perfect square Concrete: Begin by engaging students to explore quadratic equations. Use algebra trinomial, check that your equation is still balanced. 2 tiles to represent x , x, and constant terms on either side of the equation. Have students build squares using 3. How could we solve the equation in this form? the tiles to visualize the process of completing the square. For example, provide tiles to model the equation 4. How could you apply your method to x2 − 10x − 5 = 0? 2 x + 2x = 8 and guide students to figure out how many unit tiles are needed to form a perfect square on the left side. Encourage them to rearrange the tiles to see how adding a certain number of unit tiles completes the square, but remind them that whatever is added to one side of an equation should also be added on the other side. 472

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mathspace.co Finally, rearrange the tiles on the right side into a square and ask students how many unit tiles the x could be replaced with in order to make both squares the same size.

Representational: Transition to drawing representations of the algebra tiles. Have students sketch the shapes they used, with squares representing x2, rectangles for x, and unit squares for constants. Use grid paper to help them keep proportions accurate. Guide them in illustrating the steps of completing the square, showing how they add squares to complete the larger square. Encourage them to label their drawings with the corresponding algebraic terms.

Abstract: Move on to solving quadratic equations using algebraic symbols and notation. Teach students how to use inverse operations and the square root property to solve equations. Show them how completing the square transforms an equation like x2 + 2x = 8 into (x + 1)2 = 9. Then, show how the sides of the squares on either side must then be equal, so x + 1 = 3. Provide practice problems where they apply these steps to solve different quadratics. Encourage them to connect the algebraic steps back to the manipulatives and drawings they used earlier. This helps them see how the abstract equations relate to the concrete and representational stages. Include examples of solved equations, highlighting each step of the process.

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Division property of radicals

Completing the square Exploration Completing the square is a method we use to rewrite a quadratic expression so that it contains a perfect square

trinomial which can be factored as A2 + 2AB + B2 = ( A + B)2. We used this method in a previous lesson to convert

Students:a Page 472 quadratic equation from standard form to vertex form. We will now learn to use the completing the square method combined with the square root property to solve quadratic equations.

Exploration Consider the equation x2 + 6x = 11.

472

Try to develop a method for turning the left-hand side of the equation into a perfect square trinomial.

Remember that we need to keep both sides of the equation balanced. After making the perfect square trinomial, check that your equation is still balanced.

How could we solve the equation in this form?

How could you apply your method to x2 − 10x − 5 = 0?

Mathspace Virginia SOL Algebra 1 mathspace.co

Suggested student grouping: In pairs Students are presented with an equation. Students may use a previously-learned approach to attempt to answer the questions, leading them to eventually complete the square and solve the equation. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Try to develop a method for turning the left-hand side of the equation into a perfect square trinomial. In order to make x2 + 6x into a perfect square trinomial, we would need to add a constant term to it. The constant term that makes a perfect square trinomial is 9, so we could rewrite the left side as x2 + 6x + 9 and then (x + 3)2. 2. Remember that we need to keep both sides of the equation balanced. After making the perfect square trinomial, check that your equation still balanced. To keep the equation balanced, we should also add 9 to the right side of the equation, making the equation (x + 3)2 = 20. This is a reasonable action to keep the equation balanced. Thinking back to completing the square with the vertex form of a quadratic function, adding values that keep the equation balanced should be a familiar part of completing the square. 3. How could we solve the equation in this form? This equation is now in a form that we can use square roots to solve. We start by evaluating the square root of both sides, then we add 3 to both sides. 4. How could you apply your method to x2 − 10x − 5 = 0? Using the same method, we could move 5 to the right side of the equation, and then create a perfect square trinomial on the left side using x2 − 10x. After keeping the equation balanced and factoring the trinomial, use square roots to solve for x. Purposeful questions • What does the expression x2 + 6x need in order to be classiﬁed as a perfect square trinomial? • How do we keep equations balanced? • How is this similar to completing the square for quadratic functions that we want to convert to vertex form? Possible misunderstandings • Students may forget that introducing terms to an equation is mathematically valid, as long as we perform the same operation on both sides of the equation.

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After the exploration, students learn a step-by-step process for completing the square given a polynomial equation with a leading coefficient of 1. After completing the square, we can solve equations by using square roots, which students learned how to do earlier in the lesson.

Students: Page 473 For quadratic equations where a = 1, we can write them in perfect square form by following these steps: 1 2

Subtract c from both sides

Rewrite the x term

For quadratic equations where a = 1, we can write them in perfect square form by following these steps: to both sides

4 1

Add

2 5

Subtract c from both sidestrinomial Factor the perfect square

Rewrite the x term 3 If a ≠ 1, we can ﬁrst divide through by a to factor it out. Note that when we were using completing the square to write to anboth equation sidesin vertex form, we keep the constant term Add 4 on the same side of the equation as the variable terms. Then, to maintain equivalency and complete square, we term. This results in Factor all the the terms beingsquare on thetrinomial same side of the equation so we can perfect

add and subtract the same 5

identify the vertex of the parabola. If a ≠if1,we wewant can ﬁrst divide by we a tokeep factorthe it out. But, to solve thethrough equation, x terms together and move the constant term to the other side of Note that when we were using completing the square to write an equation in vertex form, we keep the constant term the equation. Then the term is added to both sides of the equation to maintain equivalency and create a perfect on the same side of the equation as the variable terms. Then, to maintain equivalency and complete square, we trinomial. This gives us a squared factor on one side and a constant term on the other side of the equation, allowing term. This results allcan therewrite terms being on the by same side of the so wewe can addtoand the root sameproperty us usesubtract the square to solve for x. Ifin we an equation completing theequation square, then can identify the vertex of the parabola. solve it using square roots. But, if we want to solve the equation, we keep the x terms together and move the constant term to the other side of

Example 4 Then the the equation.

term is added to both sides of the equation to maintain equivalency and create a perfect

Examples trinomial. This gives us a squared factor on one side and a constant term on the other side of the equation, allowing Solve the following quadratic equations by completing the square.

us to use the square root property to solve for x. If we can rewrite an equation by completing the square, then we can

Students:solve 473 18x + square 32 = 0 roots. a Page x2 it+ using Create a strategy

Example 4

To solve an equation by completing the square, start by moving the constant term to the other side of the equation. We will complete the square by ﬁnding half the coeﬃcient of the x term, squaring it, and adding it to both sides of the Solve the following quadratic equations by completing the square. equation. Once we’ve completed the square, we can solve. a x2 + 18x + 32 = 0

Apply the idea Create a strategy 2

x + 18x + 32 = 0 Given equation To solve an equation by completing the square, start by moving the constant term to the other side of the equation. Subtract 32 from both sides x2 + 18x = −32 We will complete the square by finding half the coefficient of the x term, squaring it, and adding it to both sides of the equation. Once we’ve completed the square, we can solve. Since the coefficient of the x-term is 18, we will need to add to both sides of our equation.

Apply the idea x2 + 18x + 32 = 0 x2 + 18x = −32

Complete the square Given Factor equation the perfect square trinomial Subtract fromroot bothofsides Take the 32 square both sides

This us with two equations 9 we = 7 will andneed x + 9 to = −7. byequation. subtracting 9, so we get Sinceleaves the coeﬃcient of the x-term xis+18, addWe can solve to both both equations sides of our the solutions x = −2 and x = −16. Complete the square Factor the perfect square trinomial Take the square root of both sides This leaves us with two equations x + 9 = 7 and x + 9 = −7. We can solve both equations by subtracting 9, so we get the solutions x = −2 and x = −16. 8.03 Solve quadratics using square roots 473 mathspace.co

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473

Purpose Show students how to solve a polynomial equation by completing the square and using square roots. Reﬂecting with students Ask students if they have a method to recall the the following ﬁll-in-the-blanks:

step of completing the square. For instance, offer students

x2 + 18x + ⬚ = −32 + ⬚ (x + ⬚)2 = −32 + ⬚

Some students may point out that this is simply the algorithm for turning x2 + 18x into a perfect square trinomial and keeping the equation balanced. The term, in this case 81, is the value that makes x2 + 18x + 81 a perfect square trinomial and therefore keeps the equation balanced.

Students: Page 474 b 2x2 − 10x + 7 = 0

Create a strategy In this example, the coeﬃcient of x2 is 2, so we will need to divide this coeﬃcient out before completing the square. We can then perform steps similar to the previous example.

Apply the idea Given equation Divide both sides by 2 Subtract The coeﬃcient of x is

from both sides

Taking half of −5 and squaring it gives us

so this is the value that

completes the square. Complete the square Factor the left side, evaluate the addition on the right side Square root property Division property of radicals Multiplication property of radicals Evaluate the radicals This leaves us with two equations: Next, we add

to solve both equations, and we ﬁnd that the solutions are

and

Reﬂect and check These can also be combined into one fraction:

Idea summary

Purpose Completing the square can be used to solve any quadratic in the form ax2 + bx + c = 0, but it is easiest to use Show studentswhen howa to polynomial equation with a leading coeﬃcient ≠ 1 by completing the square and = 1 solve and b isaeven. using square roots.

8.03 Solve quadratics using square roots 1001 mathspace.co

Expected mistakes Students may try to complete the square without factoring the leading coefficient from the equation. Remind students that we need to have a leading coefficient of one before trying to create a perfect square trinomial. Reflecting with students Ask students to explain why this example is more complex than the example in part (a). This example requires us to factor the leading coefficient, and the term that creates a perfect square trinomial is a fraction. We also need to simplify radicals in this example.

Provide a step-by-step procedure for when the leading coefficient is not 1 use with Example 4

Targeted instructional strategies Students can complete the square for any quadratic equation in standard form with the steps: 1

b 2x2 − 10x + 7 = 0

Divide both sides by a

Create a strategy In this example, the coeﬃcient of x2 is 2, so we will need to divide this coeﬃcient outboth beforesides completing the square. 3 Subtract from We can then perform steps similar to the previous example.

Rewrite the x-term

Apply the idea

Given equation

5 Add Divide both sides by 2

Subtract

to both sides

from both sides

Factor the trinomial

Taking half of −5 and squaring it gives us is the value that The coeﬃcient of x is Notice that the constant term in the standard form does not affect the process so of this completing the square. completes the square.

Since there are many algebraic manipulations in these steps, it can be very helpful for students to repeat the Complete the squareof algorithms in practice can model to students how to procedure using numerical examples. Seeing examples use it without being prompted. Factor the left side, evaluate the addition on the right side

Compare and connect

Square root property

use with Example 4

English language learner support Division property of radicals Help students understand the similarities and differences between solving quadratic equations by factoring and Multiplication property of radicals students take to arrive at the answer. solving by completing the square. Highlight the different approaches For example, ask students to compare the solutions of x2 + 18x + 32 = 0 obtained by factoring and by Evaluate the radicals completing the square. Then ask students to connect the steps in each method to the structure of the quadratic equation. This leaves us with two equations: This routine will help students to see how both methods are based on the principle of balancing equations, but Next, we add to solve both equations, and we ﬁnd that the solutions are and use different strategies to isolate the variable x. Reﬂect and check can474 also be combined into one fraction: Students:These Page

Idea summary Completing the square can be used to solve any quadratic in the form ax2 + bx + c = 0, but it is easiest to use when a = 1 and b is even.

1002 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co 474

Mathspace

Virginia SOL Algebra 1

Practice Students: Pages 475–477

What do you remember? 1

Fully simplify each square root. a

x2 = 25

x2 = 81

x2 = 49

x2 − 25 = 0

x2 = 100

x2 = 1

x2 + 5x + ⬚

State the quadratic equation that has the given solutions: x = ±5

Complete the following expressions so they form a perfect square trinomial. a

Solve the following equations by using square roots:

a 4

x2 − ⬚ x + 16

x2 − ⬚ x + 1

x2 − x + ⬚

x2 − 5x + ⬚ = (x − ⬚)2

Complete the square by ﬁnding the missing values. a

x2−

+ ⬚ = (x − ⬚)2

x2 + 4x + ⬚ = (x + ⬚)2

Solve the following quadratic equations by completing the square: a

x2 + 18x + 32 = 0

x2 − 6x + 8 = 0

x2 − 9x + 8 = 0

x2 − 2x − 32 = 0

25y2 = 36

5x2 − 45 = 0

(4x + 3)2 = 64

5( p2 − 3) = 705

(x + 2)2 = 20

2(x − 3)2 = 8

x2 − 8x − 9 = 0

x2 + 14x − 51 = 0

Let’s practice 7

Solve the following equations by ﬁnding square roots: a

x2 − 5 = 31

(m − 7)2 = 81

(x − 10)2 = 26

Solve the following equations by completing the square: a

x2 + 2x − 8 = 0

x2 − 6x + 5 = 0

2x2 − 12x − 32 = 0

4x2 + 11x + 7 = 0

Solve the following quadratic equations by completing the square. Express your answers in simplest form. a

x2 + 11x + 5 = 0

6x2 + 48x + 24 = 0

x2 − 7x + 8 = 0

x2 + 22x + 9 = 0

x2 + 24x + 5 = 0

5x2 + 55x + 3 = 0

2x2 + 5x + 1 = 0

State the quadratic equation that has the given solutions: a

(4 − d)2 = 9

Solve each quadratic equation below using either the square root property or by completing the square. Justify each step work. a

x2 + 5 = 30

x2 + 12x + 32 = 0

x2 + 8x + 5 = 0

(8x + 9)2 = 256

8.03 Solve quadratics using square roots 1003 mathspace.co

Solve each quadratic equation below using either the square root property or by completing the square. Verify your work by graphing or with substitution. a

(x − 5)2 = 36

x2 + 13x + 36 = 0

2x2 − 12x − 54 = 0

(5x − 3)2 = 49

Harry is using a diving board to dive into a swimming pool. The distance from his head to the surface of the water can be represented as (x − 7) (x + 7) = 147. Find the viable solution to the quadratic equation.

Consider the equation x2 + 24x = 10. Janessa tried to solve the equation by completing the square. 1 2 3 4 a

Identify the mistakes she made.

Solve the equation correctly.

On the graph of y = x2 − 4, there are two points where y = 12. Without drawing the graph, find the x-coordinates of these two points.

On Earth, the equation d = 4.9t2 is used to ﬁnd the distance (in meters) an object has fallen through the air after t seconds. Willow is sky diving and wants to release her parachute once she has fallen 400 m. Determine the time it will take her to fall 400 m, rounding your answer to the nearest second.

The kinetic energy E of a moving object is given by speed in meters/second.

where m is its mass in kilograms and v is its

If a vehicle weighing 1600 kilograms has kinetic energy E = 204 800, determine what speed it is moving. 18

Eduardo is trying to solve the equation (x + 9)2 = 25. He thinks that the equation is equivalent to x + 9 = 5. Nicolette, however, thinks he has performed the operations incorrectly, and that he should have subtracted 9 from both sides of the equation ﬁrst giving an equivalent equation of x2 = 16. Describe any errors Eduardo and/or Nicolette have made, and ﬁnd the correct solution to the equation.

Let’s extend our thinking 19

The revenue y (in millions of dollars) of a company x years after it ﬁrst started is modeled by y = 12.5x2 − 64x + 135 a

Use this equation to predict the number of years it will take for the revenue to reach $1 167 million.

Describe another method you could use to calculate this.

1004 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Sauya’s teacher gave her a square and a rectangle and asked her to use them to create a larger square. x

Sauya cut the rectangle in half, and placed the two pieces on either side of the square as shown below.

Determine the area of the smaller square that, when added, completes the larger square.

Find the total area of the larger square.

Complete the square of the quadratic equation, ax2 + bx + c = 0, to ﬁll in the blanks: (x + ⬚ )2 = ⬚

8.03 Solve quadratics using square roots 1005 mathspace.co

Answers

b x2 + 12x + 32 = 0

8.03 Solve quadratics using square roots What do you remember? 1 a

x + 12x = −32

(x + 6)2 = 4 Factor the perfect square trinomial

x + 6 = ±2 Take the square root of both sides x = −6 ± 2

2 a x = 5, x = −5

b x = 9, x = −9

c x = 10, x = -10

d x = 1, x = -1

e x = 7, x = −7

c x2 + 8x + 5 = 0

Given equation

x2 + 8x = −5

Isolate x

Subtract 5 from both sides

3 a x = 25 4 a 8

Subtract 32 from both sides

x2 + 12x + 36 = −32 + 36 Complete the square

x = −4 or x = −8

x = 5, x = –5

Given equation

b x =5 b 2

5 a

b 2

c x + 4x + 4 = (x + 2) 6 a x = −2, x = −16 c x = 8, x = 1

Complete the square

Factor the perfect square trinomial

Take the square root of both sides

Subtract 4 from both sides

Given equation

b x = 4, x = 2

Take the square root of both sides

Evaluate the square root

Consider the positive case

Subtract 9 from both sides

Divide both sides by 8

Let’s practice 7 a x = 6, x = −6

b a = 15, a = −15

d x = 3, x = −3

Consider the negative case

e m = 16, m = −2

d = 1, d = 7

Subtract 9 from both sides

h p = −12, p = 12

Divide both sides by 8

12 a x = 11 or x = −1

x = 5, 1

8 a x = 2, x = -4

b x = 5, x = 1

c x = 9, x = −1

d x = 3, x = -17

e x = 8, x = -2

9 a

5 −4 −2 −5 −10 −15 −20 −25 −30 −35 −40 −45

x 2 4 6 8 10 12 14

b x = −4 or x = −9 x2 + 13x + 36 = 0 2

h 2

10 a (x − 2) = 6 or x − 4x + 4 = 6 b (2x + 1)2 = 3 or 4x2 + 4x + 1 = 3 11 a

Given equation

Subtract 5 from both sides

Simplify the right side

Take the square root of both sides

Evaluate the square root

1006 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

(−4) + 13(−4) + 36 = 0 16 − 52 + 36 = 0 0=0 (−9)2 + 13(−9) + 36 = 0 81 − 117 + 36 = 0 0=0

c x = 9 or x = −3 2x2 − 12x − 54 = 0

20 a b Adding the areas of each rectangle, we get

2(9)2 − 12(9) − 54 = 0 2(81) − 108 − 54 = 0 162 − 108 − 54 = 0

After simplifying, the equation becomes

0=0 2x2 − 12x − 54 = 0 2(−3)2 − 12(−3) − 54 = 0

2(9) + 36 − 54 = 0 18 + 36 − 54 = 0 0=0 d x = 2 or x =

y −1

−5 −10 −15 −20 −25 −30 −35 −40 −45

13 x = 14 14 a I n step 1, she added 144 to the left-hand side, but did not add it to the right-hand side. In step 3, she did not include a plus or minus sign in front of the radical. b 15 x = 4, x = –4 16 t = 9 seconds 17 v = 16 meters/second 18 Eduardo has incorrectly taken the square root of both sides. When taking the square root of something, we need to consider both the positive and the negative case, so an equivalent equation would be x + 9 = ±5. Nicolette has incorrectly subtracted 9 from both sides before taking the square root. The correct equation to solve is x + 9 = ±5 giving an answer of x = –9 ± 5. Let’s extend our thinking 19 a 12 years b W e could draw the graph of the quadratic function y = 12.5x2 − 64x + 135 and also the line y = 1167 and find their points of intersection. The x-coordinates of the intercepts would be the required year. (Discounting the negative value.)

Answers 1007 mathspace.co

8.04 Solve quadratics using the quadratic formula Subtopic overview Lesson narrative In this lesson, students will use completing the square from the previous lesson to derive the quadratic formula. Students will solve equations using the quadratic formula and they will explore how the discriminant of the quadratic equation determines the types of solutions. By the end of the lesson, students should be able to interpret quadratic models to solve problems in contextual situations to an appropriate degree of precision. Students will also be able to explain why the quadratic formula works as well as describe the relationships between the quadratic equation, the graph of the function, and the number and types of solutions.

Learning objectives Students: Page 478

Key vocabulary 

discriminant



quadratic formula

Essential understanding Quadratic equations can be solved using a variety of methods. The quadratic formula can be used to solve any quadratic equation, but it is not always the most efficient method and is best used for equations that cannot be easily graphed or factored.

Standards This subtopic addresses the following Virginia Standards of Learning for Mathematics standards.

Mathematical process goals MPG1 — Mathematical Problem Solving During the lesson, teachers can integrate problem-solving by providing students with a variety of practice problems that require solving quadratic equations using the quadratic formula. Teachers can encourage students to apply the steps outlined in the lesson and connect their solution methods to their prior knowledge of radicals. Teachers can also present real-world problems that can be modeled and solved using quadratic equations, allowing students to apply their mathematical skills in a practical context. 1008 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

MPG2 — Mathematical Communication

MPG3 — Mathematical Reasoning

Teachers can promote mathematical communication by encouraging students to explain their thought processes while solving problems. They can ask students to justify their choice of solution method and explain the reasoning behind their steps. Teachers can also ask students to explain the role of the discriminant within the quadratic formula and how it determines the type and number of real solutions.

To enhance mathematical reasoning, teachers can ask students to use logical reasoning to determine the value of the discriminant and the number and type of real solutions. They can also encourage students to justify the steps in the quadratic formula and evaluate the validity of their solutions.

Content standards A.EI.3 — The student will represent, solve, and interpret the solution to a quadratic equation in one variable.

A.EI.3b — Determine and justify if a quadratic equation in one variable has no real solutions, one real solution, or two real solutions.

A.EI.3a — Solve a quadratic equation in one variable over the set of real numbers with rational or irrational solutions, including those that can be used to solve contextual problems.

Prior connections A.EO.1 — The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

A.EO.4 — The student will simplify and determine equivalent radical expressions involving square roots of whole numbers and cube roots of integers.

Engage Activity Conditions and roots

60 mins

Students will use various methods to solve quadratic equations to determine whether statements are always, sometimes, or never true. Students will determine the number and nature of solutions to a quadratic equation through investigating different conditions related to the discriminant.

Understanding and skills

Will use Solving quadratic equations by completing the square.

8.04 Solve quadratics using the quadratic formula 1009 mathspace.co

Will develop Determining the different types of roots visually (repeated real root, distinct real roots, no real roots). Determining the different types of roots algebraically. Connecting the value of b2 − 4ac to determining the different types of roots algebraically.

Could extend Deriving the quadratic formula and explain how different values for the discriminant affects the solutions.

Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: Paper and pencil

Support students with disabilities Support conceptual processing - self monitor understanding and ask clarifying questions Have students reflect on their own learning using questions from KWL strategy: “What do I Know? What do I Want to learn? What have I Learned?” Answers may look like: Know - ask during the Launch • I know what the variables a, b, and c stand for in a quadratic equation. • I know how to solve quadratic equations. • I know about the different types of solutions a quadratic equation can have. Want to learn - ask during the second stage of the Explore • I want to learn what roots each condition will result in. • I want to learn how a graph can show me roots of a quadratic equation. • I want to learn about the relationship between coefficients and the roots of a quadratic. Learned - ask after the class discussion • I have learned that when b2 − 4ac = 0, the roots are always one repeated real root. • I have learned that when a > 0, the roots sometimes are real roots. • I have learned that when b2 − 4ac < 0, the roots are never real roots. • I have learned that when

, the roots are sometimes distinct, real roots.

• I have learned that when c = 0, the roots are always real roots. • I have learned that when b = 0, the roots are sometimes one, repeated real roots.

Support for English language learners Collect and display As pairs are working, listen for and collect vocabulary, phrases, and methods students use for testing conditionresult pairs. Consider grouping language for each part of the process (meeting the conditions, defining the number of roots). Continue to update collected student language throughout the entire activity. Remind students to borrow language from the display as needed. Some terms and phrases may include: quadratic equation, solutions, roots, coefficients, repeated real roots, real roots, distinct roots.

1010 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Classroom guide Hook

Open questions

Students write observations about the applet displaying a quadratic function and its type of roots.

The applet does not have numbers, motivating students to find the solutions for their equation. Ask students to be precise when finding the solutions. If students are struggling to get started, encourage them to use the applet to visualize different types of roots before starting to solve algebraically. Highlight different student responses that address the different types of roots seen visually in the applet as well as solutions students have found algebraically so far; no real roots, a repeated real root, two distinct real roots. If no one has one type of root, provide an example of a quadratic equation so students see all possibilities and have a chance to connect the graph to the quadratic equation and its solutions.

−1

y = −x2 −x

−1

No real roots

10 5

−3

−2

−1

−5 −10

Slide 1 from Student Engage Activity

Launch Students should read the instructions of the task individually before forming groups. This allows students an opportunity to read each condition and ask questions related to terminology or the goal of the task before moving into groups. Important mathematical concepts: Coefficients of a quadratic function, standard form, roots, complete the square, discriminant Suggested grouping: Form groups of 4 and assign roles

5 mins

What do you notice? What do you wonder? Explore the applet.

Implementation details Encourage students to connect what they have learned in the previous subtopic about solving quadratic equations by completing the square and the number and nature of solutions to a quadratic equation.

•

5 mins

This activity relates to the quadratic equation ax2 + bx + c = 0 where a, b, and c are real numbers with a ≠ 0. Choose a condition from each column to investigate: b2 − 4ac = 0

c=0

b2 − 4ac > 0

b=0

b2 < 4ac

a<0

Slide 2 from Student Engage Activity

Continue when Students have read the Launch and understand the context of the problem.

8.04 Solve quadratics using the quadratic formula mathspace.co

1011

Explore

Team roles

Anticipated strategies

•

25 mins

Use the applet to create quadratics that meet the conditions.

Using graphs Students sketch graphs or use the applet and determine validity of result based on the condition visually. Encourage students to come up with as many different examples as possible before deciding if something is “always” or “never” true. Once students have created multiple examples as justification, prompt them to generalize their observations about the graph, the equations that match the condition, and the number and nature of the roots.

−1

y = −x

−1

−x

No real roots

−1

D = b2 − 4 ac = ( −1)2 − 4( −1)( −1)

−1

= −3 10 5

−3

−2

−1

Using algebra

−5

Students complete the square for any quadratic equation that matches the condition (deriving the quadratic formula in some cases) and determine validity of result.

−10

Slide 4 from Student Engage Activity

Students will have a range of responses for their written and visual communication to explain their group’s conclusions. Encourage multiple responses and forms for justification. Note the strategies each group is using to highlight in the class discussion. Some example results: • b2 − 4ac = 0: Always has one repeated real root • b2 < 4ac: Always has no real roots • b2 − 4ac > 0: Always has two distinct real roots • a < 0: Sometimes has distinct real roots, one repeated real root, no real roots • c = 0: Sometimes has distinct real roots or one real root and never has no real roots • b = 0: Sometimes has two distinct real roots, sometimes has repeated real roots, sometimes has no real roots

Misconceptions Generalizing without testing all possible options Are these all of the possible types of solutions based on this condition? How do you know?

Not including positive and negative values when multiplying inequality by unknown variable What are the possible values for a? How could these possible values impact the inequality?

Purposeful questions Use the following questions to check for understanding and encourage critical thinking: • What does always, sometimes, or never true mean in this context? • How can the graph of the quadratic equation be used to understand the statements? • Can you show this algebraically as well as graphically (or vice versa)? • Which statements so far are always, sometimes, or never true? • Are there any patterns you are noticing about parts of the quadratic equation and the number and type of roots?

Continue when Students have investigated the roots for two conditions. 1012 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Discuss

25 mins

Start with a group discussion. Consider sequencing the strategies presented from graphs, to using algebra.

Discussion guide Ask groups to share their findings for their chosen statements. Start by asking a group to start the class discussion. Encourage groups to build on other classmate’s ideas, such as if one group has a visual explanation for why a statement is always, sometimes, or never true ask if a group had a different way of justifying their group’s decision. If groups do not have an algebraic explanation, ask the class and return to the algebraic justification at the end of the discussion or during the start of the next class. Continue the class discussion to cover all six statements. If as a class some statements were not investigated, discuss them as a class now. As an extension you may wish to give students the following prompt: Return to the hook and derive the quadratic formula by completing the square. Ask the class what they notice about parts of the formula. You can introduce the discriminant (b2 − 4ac, what is under the radical in the quadratic formula) and ask students to use their observations from class to try and generalize when the quadratic equation will have no real solutions, one real solution, and two real solutions. This is optional, as this will be discussed in more detail in the next lesson. • b2 − 4ac < 0: The equation has no real solutions, the graph does not cross the x-axis • b2 − 4ac = 0: The equation has one repeated real solution, the graph touches the x-axis at one point • b2 − 4ac > 0: The equation has two distinct real solutions, the graph crosses the x-axis at two points

Tools You may ﬁnd these tools helpful: • Graphing calculator • Highlighter

Student lesson & teacher guide The quadratic formula Students begin with an exploration of solving quadratic equations using any method.

8.04 Solve quadratics using the quadratic formula 1013 mathspace.co

Students: Page 478

Compare and connect English language learner support The “Compare and Connect” strategy encourages students to identify, compare, and contrast different mathematical approaches, representations, concepts, and language, fostering their awareness and enhancing their understanding through reflective discussions. Ask students to solve a quadratic equation using the square root method and compare their results to the quadratic formula. Consider the equation (3x + 5)2 − 34 = 0, which has the standard form 9x2 + 30x − 9 = 0. Solving using the square root method gives the solutions:

Solving using the quadratic formula gives the solutions:

Ask students which method they ﬁnd more comfortable and why. Encourage students to use key vocabulary like “square roots,” “substituting,” “rearranging,” etc. when explaining their reasons.

1014 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Check for reasonableness using a graph Targeted instructional strategies Encourage students to check the reasonableness of their answers by graphing the function . For example, graphing f (x) = 16x2 − 24x − 3 can show that it has x-intercepts at the expected coordinates. This can lead into a discussion about why graphing is not always appropriate in the case of irrational solutions. y 15 10 5 −2

x 2

−5 −10 −15

Double-check the signs in the quadratic formula Address student misconceptions When students ﬁrst use the quadratic formula, forgetting the correct signs can be a common mistake, either with the leading negative sign or the ± sign before the square root. Remind students to double-check their solutions by substituting them back into the starting equation. It can aso help to have the quadratic formula written out for students in a common place or resource from which they can copy or check.

Exploration Students: Page 478

8.04 Solve quadratics using the quadratic formula 1015 mathspace.co

Suggested student grouping: In pairs Students are given two quadratic equations to solve using any preferred method. A suggestion for implementation is to allow students to attempt the problems independetly, then pairing up with a classmate who may have solved each equation using a different method. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Which method did you choose and why? Students may state that they solved equations by graphing, completing the square, or factoring. 2. Did you ﬁnd it challenging to solve the problems? If so, why? When an equation is not factorable or is not written in a form that is set up to solve using square roots, it may be more challenging to solve or plan a strategy for solving. Graphing technology allows us to solve these equations easily, but we cannot always ﬁnd the exact solution. Purposeful questions • Is it possible to factor the polynomial to solve the equation? • If you could not graph the polynomial, how would you solve the equation? • What methods for solving quadratic equations have we learned about in this course? Possible misunderstandings • Students may state methods that we can use to solve quadratic equations without actually solving the equations in the exploration. Encourage students to use those methods to solve the equations. Students are presented with a new method for solving quadratic equations: the quadratic formula.

Students: Page 478

1016 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Examples Students: Page 479 Example 1 The standard form of a quadratic equation is ax2 + bx + c = 0. a Derive the quadratic formula by solving this equation for x.

Create a strategy To solve for x, we can complete the square.

Apply the idea

Division property of equality Subtraction property of equality Complete the square Factor the left side Evaluate the exponent Evaluate the subtraction Square root property Evaluate the square root Subtraction property of equality Evaluate the addition

Reﬂect and check The standard form of a quadratic equation represents any quadratic equation. Since we solved this equation for x, this formula can be used to ﬁnd the solution to any quadratic equation.

b Use the quadratic formula to solve the equation −x2 + 6x − 8 = 0

PurposeCreate a strategy Advanced learners how theis quadratic formula from tothe standard ofis,a so quadratic equation by First, we need show to make sureto thederive equation in standard form and equal zero. This oneform already we can see that athe = −1, b = 6, and c = −8. We can substitute these values into the quadratic formula to solve for x. completing square. Expected mistakes Students may attempt to begin completing the square without factoring the coefficient of x2 first. Point out to students that the coefficient of x2 must be 1 when completing the square. Reflecting with students Remind students that if we know the formula we are attempting to derive, it may be helpful to look at the 8.04steps Solve quadratics using formula we are attempting to work toward as we work through the to derive it. the quadratic formula 479 mathspace.co

8.04 Solve quadratics using the quadratic formula 1017 mathspace.co

Evaluate the addition

Reﬂect and check The standard form of a quadratic equation represents any quadratic equation. Since we solved this equation for x, formula479–480 can be used to ﬁnd the solution to any quadratic equation. Students:this Pages b Use the quadratic formula to solve the equation −x2 + 6x − 8 = 0

Create a strategy First, we need to make sure the equation is in standard form and equal to zero. This one already is, so we can see that a = −1, b = 6, and c = −8. We can substitute these values into the quadratic formula to solve for x.

Apply the idea Quadratic formula with a = −1, b = 6, and c = −8 Evaluate the exponent and multiplication Evaluate the subtraction

8.04 Solve quadratics using the quadratic formula mathspace.co

479

Evaluate the square root Now, we can separate this into the two answers:

When we simplify, we ﬁnd the answers to be x = 2 and x = 4.

Reﬂect and check Since the answers are rational, we could have solved the quadratic equation by factoring. −x2 + 6x − 8 = 0

Given equation

−(x − 6x + 8) = 0

Factor out −1

Apply the idea2

−(x − 4) (x − 2) = 0

Factor the trinomial Quadratic formula with a = −1, b = 6, and c = −8

Using the zero product property, we get the answers x = 4 and x = 2. Evaluate the exponent and multiplication c Use the quadratic formula to solve the equation 5x2 = 8x + 1. Evaluate the subtraction

PurposeCreate a strategy Evaluate the square root Show students how to solve an equation using the quadratic formula. Before using the quadratic formula, we need to get the equation in the form ax2 + bx + c = 0. Then, we can correctly Now, we can separate this into the two answers: identify the values of a, b, and c.

Expected mistakes StudentsApply may the forget ideaor skip using negative signs when initially working through the quadratic formula to solve When we simplify, we ﬁnd the answers to be x = 2 and x = 4. 2 parentheses around values that we substituted for variables will help us keep a problem. Remind students 5x that = 8x + 1 Given equation track of the signage. 5x2 − 8x − 1 = 0 Subtraction property of equality Reﬂect and check

Now we see a are = 5,rational, b = −8, cwe = −1. Since thecan answers could have solved the quadratic equation by factoring. Reﬂecting with students 2 −x + 6x − 8 = 0 equation Ask students to check their answer using Given another method for solving a quadratic equation. Students may solve Quadratic with a = 5, b = −8, and c = −1 2 − 6x + 8) = 0 Factor out formula −1 −(x the equation by graphing, factoring, or completing the square. −(x − 4) (x − 2) = 0

Factor the trinomial Evaluate the exponent and multiplication

Students:Using Pages 480–481 the zero product property, we get the answers x = 4 and x = 2. Evaluate the addition c Use the quadratic formula to solve the equation 5x2 = 8x + 1. Product of radicals

Create a strategy

Evaluate the square root Before using the quadratic formula, we need to get the equation in the form ax2 + bx + c = 0. Then, we can correctly identify the values of a, b, and c. Simplify by a factor of 2

Apply the idea The answers are

5x2 = 8x + 1 2

5x − 8x − 1 = 0

Given equation Subtraction property of equality

Now we can see a = 5, b = −8, c = −1.

480

Mathspace Virginia SOL Algebra 1 mathspace.co

Quadratic formula with a = 5, b = −8, and c = −1

1018 Mathspace Virginia SOL Algebra 1 Teacher Edition Evaluate the exponent and multiplication mathspace.co Evaluate the addition

c Use the quadratic formula to solve the equation 5x2 = 8x + 1.

Create a strategy Before using the quadratic formula, we need to get the equation in the form ax2 + bx + c = 0. Then, we can correctly identify the values of a, b, and c.

Apply the idea 5x2 = 8x + 1

Given equation

5x − 8x − 1 = 0

Subtraction property of equality

Now we can see a = 5, b = −8, c = −1. Quadratic formula with a = 5, b = −8, and c = −1 Evaluate the exponent and multiplication Evaluate the addition Product of radicals Evaluate the square root Simplify by a factor of 2 The answers are

Reﬂect and check When the answer is irrational, then the quadratic formula or completing the square are the only methods we could 480 Mathspace Virginia SOL Algebra 1 use tomathspace.co solve the quadratic equation.

PurposeExample 2 Show students how to solve a quadratic equation by ﬁrst converting it to standard form and then using the Solve: quadratic formula. 2 a 5x − 15x + 2 = 0

Expected mistakes Create a strategy Students may assume that a = 5, b = 8, and c = 1, without conﬁrming that the equation is in the form Rearrange the equation into the form ax2 + bx + c = 0, then use the quadratic formula. ax2 + bx + c = 0 ﬁrst. Remind students that the equation must be set equal to zero to use the quadratic formula. Apply the idea

Reﬂecting with students The equation 5x2 − 15x + 2 = 0 is already in standard form so we can identify a, b, and c. We can see that a = 5, Ask students the number of x-intercepts the function y = 5x2 − 8x − 1 would have if we graphed it on a b = −15, and c = 2 and substitute them into the quadratic formula. coordinate plane. We know that when the equation is equal to zero, we have two solutions for x. This would Reﬂect and check formula mean that graphically when y = 0, there are twoQuadratic x-intercepts.

When the answer is irrational, then the quadratic formula or completing the square are the only methods we could use to solve the quadratic equation. Students: Page 481 Substitute a = 5, b = −15, c = 2 Simplify the adjacent signs

Example 2 Evaluate the multiplication

Solve: a 5x2 − 15x + 2 = 0

Evaluate the exponent

Create a strategy

Evaluate the subtraction

Rearrange the equation into the form ax2 + bx + c = 0, then use the quadratic formula. and . So, the solutions are

Apply the idea The equation 5x2 − 15x + 2 = 0 is already in standard form so we can identify a, b, and c. We can see that a = 5, 2 b = 10 −15, and+c 2m = 22and = msubstitute + 8m + 9them into the quadratic formula. − 6m Quadratic formula

Create a strategy 2

Rearrange the equation into the form ax + bx + c = 0, and use the quadratic formula. Substitute a = 5, b = −15, c = 2

Apply the idea 10 − 6m + 2m2 = m2 + 8m + 9 2

10 − 6m + m = 8m + 9 10 − 14m + m2 = 9 1 − 14m + m2 = 0 m2 − 14m + 1 = 0

Simplify adjacent signs Original the equation Subtract m2 from both 8.04 sidesSolve quadratics using the quadratic formula 1019 mathspace.co Evaluate the from multiplication Subtract 8m both sides Subtract 9 from both sides Evaluate the exponent Write in descending order

Solve: the answer is irrational, then the quadratic formula or completing the square are the only methods we could When use to2solve equation. − 15xthe + 2 quadratic =0 a 5x

Create a strategy Rearrange the equation into the form ax2 + bx + c = 0, then use the quadratic formula.

Example 2

Apply the idea

Solve: The equation 5x2 − 15x + 2 = 0 is already in standard form so we can identify a, b, and c. We can see that a = 5, a 5x2 − 15x + 2 = 0 b = −15, and c = 2 and substitute them into the quadratic formula.

Create a strategy

Quadratic formula Rearrange the equation into the form ax2 + bx + c = 0, then use the quadratic formula. Substitute a = 5, b = −15, c = 2

Apply the idea The equation 5x2 − 15x + 2 = 0 is already in standard form we can identify Simplify thesoadjacent signs a, b, and c. We can see that a = 5, b = −15, and c = 2 and substitute them into the quadratic formula. Evaluate the multiplication Quadratic formula Evaluate the exponent Substitute a = 5, b = −15, c = 2 Evaluate the subtraction Simplify the adjacent signs and

So, the solutions are

. Evaluate the multiplication Evaluate the exponent

b 10 − 6m + 2m2 = m2 + 8m + 9

Purpose Evaluate the subtraction Create ahow strategy Show students to use the quadratic formula to find the roots of a quadratic equation. Rearrange the equation into the form ax2 + bx + c = 0, and use the quadratic formula. and . So, the solutions are

Students: Pages 481–482 Apply the idea

10 − 6m + 2m2 = m2 + 8m + 9 b 10 − 6m + 2m2 = m2 2+ 8m + 9 10 − 6m + m = 8m + 9 2

10 − 14m + m = 9 Create a strategy

Original equation Subtract m2 from both sides Subtract 8m from both sides

= 0the form ax2 + bx +Subtract 9 use fromthe both sides formula. 1 −equation 14m + m2into Rearrange the c = 0, and quadratic 2 Write in descending order m − 14m + 1 = 0

Apply the idea 10 − 6m + 2m2 = m2 + 8m + 9

Original equation

Subtract m2 from both sides

Subtract 8m from both sides

10 − 6m + m = 8m + 9 10 − 14m + m = 9 1 − 14m + m2 = 0

Subtract 9 from both8.04 sides Solve quadratics using the quadratic formula

m2 − 14m + 1 = 0

Write in descending order

481

mathspace.co

Now we can see that a = 1, b = −14, and c = 1 so we can substitute these values into the quadratic formula. Quadratic formula 8.04 cSolve Substitute a = 1, b = −14, = 1 quadratics using the quadratic formula

481

mathspace.co

Simplify the adjacent signs Evaluate the multiplication Evaluate the exponent Simplify the expression inside the square root Simplify the square root Divide out the common factor of 2 So, the solutions are

and

Example 3 1020 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co A ball is launched from a height of 80 ft with an initial velocity of 107 ft per second. Its height, h feet, after x seconds is given by h = −16x2 + 107x + 80

Evaluate the exponent Simplify the expression inside the square root

Purpose Simplify the square root Challenge students to rearrange a quadratic equation into standard form and solve it using the quadratic formula. Divide out the common factor of 2

Students:So,Pages 482–483 the solutions are

and

Example 3 A ball is launched from a height of 80 ft with an initial velocity of 107 ft per second. Its height, h feet, after x seconds is given by h = −16x2 + 107x + 80 Determine the number of seconds it will take the ball to reach the ground. Explain your reasoning.

Create a strategy On the ground the ball will have a height of 0 ft, so h = 0. This means we want to solve the equation 0 = −16x2 + 107x + 80. Since the numbers are large, the quadratic formula is an appropriate method for solving.

Apply the idea For this equation, a = −16, b = 107, and c = 80. Substituting into the quadratic equation, we get Quadratic formula with a = −16, b = 107, and c = 80 Evaluate the exponent and multiplication Evaluate the addition Product of radicals Evaluate the square root Therefore, The two solutions, rounded to two decimal places, are x = −0.68 and x = 7.37. Remember that x represents seconds. We can exclude the negative solution as it is outside the domain, which is x ≥ 0, since time cannot be negative. The ball will reach the ground after 7.37 seconds. 482

Mathspace

Virginia SOL Algebra 1

mathspace.co Reﬂect and check

In many real-world situations, negative numbers do not make sense. Always check that your answers satisfy the constraints of the variables.

Example 4 Purpose StudentsThe demonstrate how solve equation quadratic andofdetermine appropriate amount of litter in a to park at thean end of the dayusing can bethe modeled againstformula the number people who an visited the solutionpark in context. that day by the equation: Expected mistakes is the number of piecesinofexact litter and P israther the number people. Studentswhere mayLleave their solution form, thanofsimplifying and using a decimal. Remind students that for problems that do not specifywho rounding orpark theifform solution, particularly contextual problems, Determine the number of people visited the thereof arethe 20 pieces of litter at the end for of the day. students should use their judgment to make a decision about how to present a solution that is mathematically Create strategy reasonable for athe problem. In this case, writing the seconds in decimal form makes more sense than writing We want to ﬁnd the number of people, P, when there are 20 pieces of litter at the end of the day, L = 20. seconds in exact form. We can do this by substituting L = 20 into the equation, rearranging the equation into quadratic standard form, and

thenwith usingstudents the quadratic formula to solve for P. Reflecting Ask students why they can solve the equation for two solutions, but may only use one that is counted as valid. the idea Point outApply the connection between the graph of the function having two x-intercepts, which occur when h = 0. Model equation Then refer to the context of the problem again. Substitute in L = 20 Multiply both sides8.04 by −50 Solve quadratics using the quadratic formula 1021 mathspace.co Add 1000 to both sides Quadratic formula with a = 1, b = −73, and c = 850

Three reads

use with Example 3

English language learner support Advise students to read through the instructions a few times, focusing on gathering different information each time in order to build up their understanding of what the question is asking. On the first read, students should aim to identify the scenario presented in the question. Ask students “What do you think is happening in this question?” or “Can you explain what this question is about?” On the second read, students should aim to interpret the problem by answering questions like “What is the question asking you to find?” and “What information should be included in the answer?” The two solutions, rounded to two decimal places, are xinformation = −0.68 and xin= 7.37. Remember thatIn x represents seconds. On the third read, students should look for important the instructions. this question, the We can exclude the negative solution as it is outside the domain, which is x ≥ 0, since time cannot be negative. important information includes: the equation for the height of the ball, h represents the height of the ball, and x The ball will reach the ground after 7.37 seconds. is the number of seconds after the ball is thrown. Students can be prompted by framing these as questions like “What does each variable in the equation represent?”

Reﬂect and check

In many real-world situations, negative numbers do not make sense. Always check that your answers satisfy the constraints of the variables.

Students: Page 483 Example 4

The amount of litter in a park at the end of the day can be modeled against the number of people who visited the park that day by the equation:

where L is the number of pieces of litter and P is the number of people. Determine the number of people who visited the park if there are 20 pieces of litter at the end of the day.

Create a strategy We want to ﬁnd the number of people, P, when there are 20 pieces of litter at the end of the day, L = 20. We can do this by substituting L = 20 into the equation, rearranging the equation into quadratic standard form, and then using the quadratic formula to solve for P.

Apply the idea Model equation Substitute in L = 20 Multiply both sides by −50 Add 1000 to both sides Quadratic formula with a = 1, b = −73, and c = 850 Evaluate the operations The two solutions (rounded to two decimal places) are P = 58.46 and P = 14.54. Since we are counting the number of people who visited the park, we want to round to the nearest whole number. If there are 20 pieces of litter in the park at the end of the day, then either 58 or 15 people visited the park that day.

Reﬂect and check In real-life applications where we are counting whole objects, we want to round to the nearest integer, so our solution makes sense in context.

1022 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

8.04 Solve quadratics using the quadratic formula mathspace.co

483

Purpose Students demonstrate how to substitute a value into an equation given with a context, and convert the equation to standard form in order to solve for the variable using the quadratic formula. Students interpret the solution to the equation in context. Expected mistakes Students may attempt to substitute the value of 20 as P and solve the equation for L. Point out to students that writing out or highlighting what variables in contextual problems represent is a skill for keeping track of whether the math we work through makes sense.

Students: Page 484

Idea summary For any quadratic equation of the form 0 = ax2 + bx + c where a ≠ 0 and a, b, and c are real numbers, the quadratic formula can be used to solve for x.

The discriminant Idea summary 2

The radicand the quadratic formula is called discriminant. For anyinquadratic equation of the form 0the = ax + bx + c where a ≠ 0 and a, b, and c are real numbers, the The discriminant quadratic formula can be used to solve for x.

Students learn that a specific expression within the quadratic formula is called the discriminant.

Students: Page 484

b2 – 4ac

discriminant

The discriminant can be used to determine the number and type of solutions to any quadratic equation.

The discriminant

The radicand in the quadratic formula is called the discriminant. Interactive exploration Explore online to answer the questions

mathspace.co 2

b – 4ac discriminant Use the interactive exploration in 8.04 to answer these questions. 1. What do each theto sliders represent? The discriminant can be of used determine the number and type of solutions to any quadratic equation. 2.

How many types of roots can there be?

How do the typesexploration of roots relate to the value of the discriminant, D? Interactive

Explore to answerrelate the questions How do online the x-intercepts to the value of the discriminant, D?

Why the discriminant determines the number of real roots mathspace.co Student with disabilities support

The values of a, b, and c affect the value of the discriminant. The type of number the discriminant is affects the

number and type of immediately x-intercepts the graph. Some students may not thethese discriminant Use the interactive explorationon ingrasp 8.04 towhy answer questions. can be used to determine the number and nature of the1. solutions to a quadratic equation. Quadratic equations can have 3 types of solutions: 2 real solutions, 1 real solution, or no real solutions. The value of What do each of the sliders represent? the discriminant quickly reveals which type of solution a quadratic equation has.

How many of roots can therequadratic be? Highlight the2.square root types component of the formula and ask students what would happen to the ytypes Discriminant (> 0): 3. radicand How do the of roots relate to the value of the discriminant, D? solutions if the was positive, negative, or zero. 13 4.

12 x-intercepts relate to the value (2)2 − 4(−1) (12) b2 −of4ac How do the the=discriminant, D?= 52

11 Solutions: 10 9 The values of a, b, and 8 c affect the value of the discriminant. The type of number the discriminant is affects the If necessary, remind students that the square number and type of7x-intercepts on the graph.root of a negative number cannot be a real number, and the x-intercepts: 6 square root of zero is zero. Quadratic equations can have 3 types of solutions: 2 real solutions, 1 real solution, or no real solutions. The value of 5 4 the discriminant quickly reveals which type of solution a quadratic equation has. Point out that the radicand 3 is the discriminant, so its value can be used to determine the number and nature of 2 y Discriminant (> 0): the roots. 13 1 x 12 b2 − 4ac = (2)2 − 4(−1) (12) = 52 −3 −2 −1 11 1 2 3 4 5 Solutions: 10 9 8.04 Solve quadratics using the quadratic formula 1023 8 mathspace.co 7 x-intercepts: 6 SOL Algebra 1 484 Mathspace Virginia mathspace.co 5 4

The discriminant The radicand in the quadratic formula is called the discriminant.

Exploration Students: Page 484

b2 – 4ac

discriminant

The discriminant can be used to determine the number and type of solutions to any quadratic equation.

Interactive exploration Explore online to answer the questions

mathspace.co Use the interactive exploration in 8.04 to answer these questions. 1.

What do each of the sliders represent?

How many types of roots can there be?

How do the types of roots relate to the value of the discriminant, D?

How do the x-intercepts relate to the value of the discriminant, D?

The values of a, b, and c affect the value of the discriminant. The type of number the discriminant is affects the number and type of x-intercepts on the graph. Quadratic equations can have types of solutions: 2 real solutions, 1 real solution, or no real solutions. The value of Suggested student grouping: In 3pairs the discriminant quickly reveals which type of solution a quadratic equationThe has.applet substitutes the values from Students use a GeoGebra applet to manipulate a quadratic function. the function into the discriminant. Students discover that when y Discriminant (> 0):the discriminant is positive there are two 13 x-intcerepts, when the discriminant is zero there bis2 one x-intercept, and when the discriminant is negative there 12 − 4ac = (2)2 − 4(−1) (12) = 52 11 are no x-intercepts. Solutions:

10 9 Ideal student responses8 7 These ideal responses may differ from other correct student responses. Less formal responses can be x-intercepts: 6 5 connected with the more precise mathematical language presented here. 4 1. What do each of the3sliders represent? 2 2 Each slider represents 1 a coeﬃcient or constant in the standard form of a quadratic function: y = ax + bx + c. x The gray slider represents −3 −2 −1 1 the 2 3value 4 5of a, the red slider represents b, and the blue slider represents c.

2. How many types of roots can there be? There are three different types of solutions: one real root with one x-intercept, two real roots with two x-intercepts, and no real roots with no x-intercepts. 484 Mathspace Virginia SOL Algebra 1 mathspace.co 3. How do the types of roots relate to the value of the discriminant, D? When the discriminant is zero, there is one real root. When the discriminant is positive, there are two real roots. When the discriminant is negative, there are no real roots.

4. How do the x-intercepts relate to the value of the discriminant, D? When the discriminant is zero, there is one x-intercept. When the discriminant is positive, there are two x-intercepts. When the discriminant is negative, there are no x-intercepts. Purposeful questions • Where do the values in the discriminant, D, come from? • What happens to the discriminant when the graph intersects the x-axis exactly once? Possible misunderstandings • Students may state that there are only two types of roots when moving the graph of the quadratic function. Use purposeful questions to guide students to graph a parabola that will intersect the graph exactly once as well. Students compare the graphs of quadratic equations to the equations’ real solutions when y = 0, the discriminant, and the x-intercepts.

1024 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Use the interactive exploration in 8.04 to answer these questions. 1.

What do each of the sliders represent?

How many types of roots can there be?

How do the types of roots relate to the value of the discriminant, D?

4. How do the x-intercepts relate to the value of the discriminant, D? Students: Pages 484–485

Discriminant (> 0):

b2 − 4ac = (2)2 − 4(−1) (12) = 52 Solutions:

x-intercepts:

x 1

The square root of a positive number is a real number, so the plus or minus sign ensures there will always be 2 real solutions when the discriminant is positive. 484

Discriminant (= 0):

Mathspace Virginia SOL Algebra 1 y mathspace.co

b2 − 4ac = (−2)2 − 4(1) (1) = 0

Solution:

x2 − 2 x + 1 = 0

3 2

x-intercept: (1, 0)

1 x −1

The square root of zero is zero. This eliminates the radical part of the quadratic equation, leaving only

which

will result in a single value. So, when the discriminant is zero, there will be one real solution. Discriminant (< 0):

b2 − 4ac = (−2)2 − 4(1) (2) = −4

Solutions: 3 2

x-intercepts: None

1 x −1

No real number gives us a negative number when squared. Therefore, there are no real solutions when the discriminant is negative.

Example 5 Use the discriminant to determine the number and nature of the solutions of the following quadratic equations: a 2x2 − 8x + 3 = 0

Create a strategy For this equation, we have a = 2, b = −8, c = 3.

Apply the idea The discriminant is (−8)2 − 4(2) (3) = 40. Since it is positive, the equation has two real solutions. 8.04 Solve quadratics using the quadratic formula 1025 mathspace.co

Reﬂect and check

Two real solutions means the function has two x-intercepts.

x-intercepts: None

1 x −1

Examples real number gives us a negative number when squared. Therefore, there are no real solutions when the Students:NoPage 485 discriminant is negative.

Example 5 Use the discriminant to determine the number and nature of the solutions of the following quadratic equations: a 2x2 − 8x + 3 = 0

Create a strategy For this equation, we have a = 2, b = −8, c = 3.

Apply the idea The discriminant is (−8)2 − 4(2) (3) = 40. Since it is positive, the equation has two real solutions.

Reﬂect and check Two real solutions means the function has two x-intercepts.

Purpose Make students aware that a positive discriminant will lead to an equation with two real 8.04 Solve quadratics using the solutions. quadratic formula

485

mathspace.co

Reﬂecting with students Students may be unsure how the discriminant for the problem indicates the number of real solutions when beginning to evaluate it. Help students visualize why this makes sense when the discriminant is positive, negative, or zero by graphing y = 2x2 − 8x + 3 using technology and pointing out where y = 0 on the graph of the function. This is where the polynomial expression is equal to zero and we can visualize the number of x-intercepts.

Students: Page 486 b −5x2 + 6x − 2 = 0

Create a strategy For this equation, we have a = −5, b = 6, c = −2.

Apply the idea The discriminant has a value of (6)2 − 4(−5) (−2) = −4. Since it is negative, the equation has no real solutions.

Reﬂect and check If the discriminant is negative, then the formula will involve taking the square root of a negative number which will result in no real solutions. This quadratic will not intercept the x-axis. We can also see that if 4ac > b2 the discriminant will be negative, and the corresponding equation will have no real solutions.

c x2 − 3x + 9 = 3x

Purpose Create a strategy Make students aware that a negative discriminant will lead to an2 equation with no real solutions. Before identifying our variables, the equation must be in the form ax + bx + c = 0, so we need to begin by subtracting 3x from both sides. Reﬂecting with students x2 − 6x + 9 = 0

Ask students to write their own quadratic equations that will lead to a negative discriminant. Students may see a For this equation, we have a = 1, b = −6, c = 9. pattern in the equation itself as they proceed to attempt to write their own equations that lead to equations with no real solutions. Apply the idea Reﬂect and check The discriminant has a value of (−6)2 − 4(1) (9) = 0, so the equation has one real solution.

1026 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

When the discriminant is zero, the quadratic equation simpliﬁes to be

which is equal to the x-value of

the vertex. This means the vertex lies on the x-axis and the x-coordinate of the vertex is the only solution to the quadratic equation.

Create a strategy Before identifying our variables, the equation must be in the form ax2 + bx + c = 0, so we need to begin by subtracting 3x from both sides. x2 − 6x + 9 = 0 For this equation, we have a = 1, b = −6, c = 9.

Apply the idea

Reﬂect and check

The discriminant has a value of (−6)2 − 4(1) (9) = 0, so the equation has one real solution.

When the discriminant is zero, the quadratic equation simpliﬁes to be

which is equal to the x-value of

the vertex. This means the vertex lies on the x-axis and b −5x2 + 6x − 2 = 0

the x-coordinate of the vertex is the only solution to the quadratic equation.

Create a strategy For this equation, we have a = −5, b = 6, c = −2.

Idea summary

Purpose Apply The the idea discriminant, b2 − 4ac, can help us determine the type and number of solutions to a quadratic equation Make studentswithout aware that a discriminant of zero will lead to an equation with one real solution. 2 equation needing to solve the fully.

The discriminant has a value of (6) − 4(−5) (−2) = −4. Since it is negative, the equation has no real solutions. • b2 − 4ac > 0 two real solutions Expected mistakes b2 − 4ac = 0 one real solution Reﬂect• and check Students may •forget we can only solutions use the quadratic formula when equations are in the form ax2 + bx + c = 0. 2 b − 4acis<negative, 0 no realthen If the discriminant the formula will involve taking the square root of a negative number which will result in no real solutions. This quadratic will not intercept the x-axis.

Help in determining theand sign of the discriminant usenowith We canstudents also see thatsee if 4acpatterns > b2 the discriminant will be negative, the corresponding equation will have real Example 5 Targeted solutions. instructional strategies

Ask students if they can identify when the discriminant will always be positive. Draw their attention to the 2 possiblec signs of +the x2 − 3x 9 = two 3x terms, b and 4ac. Remind students that the square of a real number must be positive, so if 4ac is negative then −4ac will be positive.Create If this aisstrategy the case, then the discriminant must also be positive. Ask students to consider when 4ac is Before identifying our variables, the equation must be in the form ax2 + bx + c = 0, so we need to begin by negative. Direct students to the fact that the sign of a product is determined by the signs of the numbers being subtracting 3x from both sides. multiplied. In this case, 4ac will be negative when a x2and − 6x c+ have 9 = 0 opposite signs. For this equation, we have a = of 1, ba=quadratic −6, c = 9. equation in standard form with a and c having opposite signs will This means that the discriminant Virginia SOL Algebra 1 486 Mathspace always have real solutions.

Applymathspace.co the idea

Reﬂect and check

For further understanding, ask students if they can explain why this is the case (the explanation involves the The discriminant has a value of (−6)2 − 4(1) (9) = 0, so the When the discriminant is zero, the quadratic equation directionequation in which the parabola opens and the y-intercept). has one real solution. simpliﬁes to be which is equal to the x-value of the vertex. This means the vertex lies on the x-axis and

Students: Page 486

the x-coordinate of the vertex is the only solution to the quadratic equation.

Idea summary The discriminant, b2 − 4ac, can help us determine the type and number of solutions to a quadratic equation without needing to solve the equation fully. • • •

b2 − 4ac > 0 two real solutions b2 − 4ac = 0 one real solution b2 − 4ac < 0 no real solutions

8.04 Solve quadratics using the quadratic formula 1027 mathspace.co

Practice Students: Pages 487–490

What do you remember? 1

Is each statement true or false? a

Any quadratic equation that can be solved by completing the square can also be solved by the quadratic formula.

Any quadratic equation that can be solved by factoring can also be solved by the quadratic formula.

The equation 3x2 + 3x − 7 = 0 can be solved using the quadratic formula.

The quadratic formula will always give 2 unique solutions.

For the equation 5x2 − x − 2 = 0, we would set a = 5, b = 1,and c = −2.

The standard form of a quadratic equation is ax2 + bx + c = 0. Find the values of a, b and c in the following quadratic equations: a

x2 − 6x + 5 = 0

−4x2 + 15x − 8 = 0

2x2 + 9x = 0

−5(x − 3)2 + 4 = 0

x2 + 7x = 10

Jeremy is using the quadratic formula for the equation 9y2 = 8y. He has correctly identified that b = 8, what are the values of a and c?

Given the quadratic equation x2 − 4x + k = 0, where k is a constant, if one of the roots is other root?

The solutions of a quadratic equation are 9 and −9. What can be said about the value of b2 − 4ac?

Consider the quadratic equation:

, what is the

x2 + 3x − 5 = 0

Select the solution to the quadratic equation.

Consider the quadratic equation:

x2 − 16x = 5x − 9

Select the solution to the quadratic equation. A

Let’s practice 8

For the following equations: i

Find the value of the discriminant.

State the number of real solutions for the equation.

3x2 − 5x + 7 = 0

2x − 2x = x − 1

b f

x2 − 4 = 0 2

c 2

4x − x = x − 5

1028 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

−x2 − 8x − 16 = 0

+ 3x + 9 = 0

For the following equations: i

Determine the number of real solutions. Explain your reasoning.

Find the real solution(s) of the equation.

x2 − 8x − 48 = 0

4x2 − 4x + 1 = 0

x2 + 5x +

x2 + 12x + 36 = 0

Find the value of the discriminant.

State the number and nature of the solutions to the equation.

x2 + 6x = −90

4x2 = 6x − 7

2x2 − 2x = x − 1

13 + x2 = −7x 2

−x2 − 5x = 8

5x + 13x + 10 = x − 7x − 15

−7 + 3x = 2x2 − 5

1.8x2 + 5.2x − 2.3 = 0

3x(x + 4) = −3x + 4

x2 + 11x + 28 = 0

6 − 9x = −2x2 − 4x

Solve each equation below using the quadratic formula. Justify your work. Leave your answer(s) in exact, simplified form. a

x2 − 7x + 9 = 0

x2 − 5x − 2 = 0

−2x2 − 15x − 4 = 0

3x2 + 9x − 4 = 0

5x = (x − 5) (3x + 3)

12 − 8m + 2m2 = m2 + 12m + 15

Solve the following equations using the quadratic formula. Round your answers to two decimal places.

x2 − 4x + 7 = 0

For the following equations:

−5x − 15x + 3 = 0 2

3n = 2n − 2n + 7

Solve each equation below using the quadratic formula. Verify your solution(s) by graphing. a

x2 + 5x + 6 = 0

x2 − 5x + 6 = 0

2x2 + 7x + 3 = 0

4x2 − 17x − 15 = 0

2x2 + 6x − 8 = 0

4x2 − 10x + 4 = 0

Yuri is playing baseball and hits a homerun with an initial velocity of 101 ft/s, from a height of 3 ft. After x seconds, its height (in feet) is given by h = −16x2 + 101x + 3 LaDeana is in the crowd and catches the homerun ball in the stands from a height of 15 ft above the ground. Determine the number of seconds after Yuri hits the ball that LaDeana catches it. Round your answer to two decimal places.

The stopping distance of a car when the brakes are applied can be modeled by the equation where s is the stopping distance in feet and u is the initial speed in miles per hour.

Determine the speed the car was travelling if its stopping distance was 100 ft, rounding your answer to the nearest hundredth of a mile.

It takes 399 ft to stop when travelling at the speed limit of 70 mph. If it takes Ray 496 ft to stop, determine how fast over the speed limit Ray was driving.

The number of customers at a restaurant can be estimated by the equation y = −x2 + 28x − 159 where y is the number of people and x is the hour of the day (in 24-hour time). Determine the opening hours of the restaurant. Explain your reasoning.

8.04 Solve quadratics using the quadratic formula 1029 mathspace.co

Let’s extend our thinking 17

For each of the given solutions to a quadratic equation: i

Find the values of a, b and c.

Write down the quadratic equation that has these solutions. b

a 18

Use the discriminant to match each quadratic equation below with the correct graph. Explain your reasoning a

5x2 − 10x − 35 = 0

9 8 7 6 5 4 3 2 1 −3 −2

−1

2x2 + 8x + 8 = 0

x2 − x + 1 = 0

y −3 −2 −1

−10

6 5

−20

4 3

−30 x 1

2 1

−40 −5 −4 −3 −2

−1

Consider the equation in terms of x: mx2 − 3x − 5 = 0 a

Given that it has two unique solutions, determine the possible values of m.

There is one value of m that must be eliminated from the range of solutions found in the previous part. Determine the solution and explain why it must be eliminated.

Find the values of n for which x2 − 8nx + 1296 = 0 has one solution.

With reference to the discriminant, explain what determines the nature of the solutions to a quadratic equation.

Determine the range of values of the constant k such that the equation 3x2 + kx + 12 = 0 has no real solutions. Justify your answer.

Use an algebraic method to show that the graphs of the functions f (x) = 3x2 − x + 8 and g (x) = −x2 + 2x − 4 do not intersect.

A quadratic equation has two real solutions whose difference is k, for some positive value k. Determine algebraically a simpliﬁed expression for the discriminant of this equation.

Consider a right-angled triangle with side lengths x units, x + p units and x + q units, ordered from shortest to longest. No two sides of this triangle have the same length. a

Complete the statement:

p and q have lengths such that 0 < ⬚ < ⬚.

Find the discriminant of this quadratic equation.

Determine the number of real solutions.

Find the value of x when q = 2p. Give your answer in terms of p.

Write a quadratic equation in standard form that describes the relationship between the sides of the triangle in terms of x.

x+p

x+q

1030 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Answers

11 a No real solutions

8.04 Solve quadratics using the quadratic formula

d No real solutions

e x = 0.39, x = −3.28

12 a What do you remember? 1 a True

b True

b No real solutions

x = 0.25, x = −5.25 Given equation

Quadratic formula with c True

a = 1, b = −7, and c = 9

d False

e False

Evaluate the exponent

2 a a = 1, b = −6, c = 5

b a = −4, b = 15, c = −8

c a = 1, b = 7, c = −10

e a = 2, b = 9, c = 0

and multiplication

Evaluate the subtraction

a = −5, b = 30, c = −41

3 a = −9, c = 0

Given equation

5 b2 − 4ac > 0

Quadratic formula with a = 1, b = −5, and c = −2

6 A

Evaluate the exponent and multiplication

7 C

Evaluate the addition

Let’s practice 8 a i Δ = −59

ii 0

b i Δ = 16

ii 2

c i Δ=0

ii 1

d i Δ = −9

ii 0

e i Δ=1

ii 2

i Δ = −59

ii 0

9 a i 2 real solutions because the discriminant is 256, greater than 0. ii x = 12, x = −4 b i 1 real solution because the discriminant is 0. ii

ii No real solutions d i 2 real solutions because the discriminant is 9, greater than 0. ii x = −7, x = −4 e i 2 real solutions because the discriminant is 16, greater than 0. ii

Given equation

Quadratic formula with a = −2, b = −15, and c = −4 Evaluate the exponent and multiplication Evaluate the subtraction Distribute the negative sign

Given equation

Quadratic formula with a = 3, b = 9, and c = −4 Evaluate the exponent and multiplication Evaluate the addition

i 1 real solution because the discriminant is 0.

Given equation

Quadratic formula with a = −5, b = −15, and c = 3

ii x = −6 10 a i 0

c i N o real solutions because the discriminant is -12, less than 0.

ii The equation has 1 real solution.

b i –76

ii The equation has 0 real solutions.

c i 1

ii The equation has 2 real solutions.

d i -23

ii The equation has 0 real solutions.

Evaluate the exponent and multiplication Evaluate the addition Distribute the negative sign

Answers 1031 mathspace.co

13 a x = −2, −3 Given equation

Expand the right-hand side

Subtract 5x from both sides to set the equation to zero

2 1 −4 −3 −2 −1

Combine like terms

−3

b x = 2, 3

Given equation

Subtraction property of equality Quadratic formula with a = 1, b = 2, and c = −7 Evaluate the exponent and multiplication Evaluate the addition

13 y 12 11 10 9 8 7 6 5 4 3 2 1 −1

x 1

2 3 4 5 6 7

c x = −4, 1

8 y 6 4 2

Evaluate the square root

−4 −3 −2 −1−2 −4 −6 −8 −10 −12 −14 −16 −18

Simplify by a factor of 2

h Given equation Subtraction property of equality Combine like terms

Evaluate the square root Simplify by a factor of 2

1032 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

2 1 −1

−1 −2 −3

Product of radicals

2 3 4

Quadratic formula with a = 1, b = −20, and c = −3

Evaluate the addition

x 1

Evaluate the exponent and multiplication

−1

−4

Evaluate the exponent and multiplication

x 1

−2

Quadratic formula with a = 3, b = −17, and c = −15

Evaluate the addition

−4

x 1

21 W hen the discriminant is positive there are two distinct real solutions. If the discriminant is of the form n2 for some integer n, then it will have integer solutions.

If the discriminant is equal to zero there will be one real solution. This is because the ± sign applies only to the square root of the discriminant. Consider that plus zero and minus zero give the same result.

3 2 1

−4 −3 −2 −1 −1

A negative discriminant means that there will be no real solutions, because the square root of a negative number is not a real number.

−2 −3 −4

22 T he quadratic will have no real solutions when the discriminant is less than zero. This will occur when b2 < 4ac. This gives us k2 < 144. Taking the square root of both sides gives us −12 < k < 12.

15 10 5 −3 −2 −1 −5 −10 −15 −20 −25 −30 −35

23 T he two graphs will intersect when f (x) = g(x), so we can equate the two equations and solve.

x 1

2 3 4 5

1 3x2 − x + 8 = −x2 + 2x – 4 Equating the two functions 2 4x2 − 3x + 12 = 0 Rearranging to set equation equal to zero

We can determine the number of real solutions by finding the discriminant. Substituting in values for a, b and c 1 b2 − 4ac = (−3)2 − 4⋅4⋅12

2 b2 − 4ac = −183 Evaluating the numerical expression

14 6.19 seconds

3 0 > −183 Comparing to zero

15 a 27.84 mph

4 0 > b2 − 4ac

b Ray was driving 10 mph over the speed limit. 16 I f we let y = 0, we can solve the equation −x2 + 28x − 159 = 0 using the quadratic formula to get:

Substitution property of equality

As the determinant is less than zero there are no real number solutions and therefore the two functions do not intersect at all. 24 A ssume the roots are α and α + k, then expand the factored form into the standard form.

Evaluating this gives us the approximate solutions x = 7.92 and x = 20.08. Since it makes more sense for a restaurant to open and close on the hour, we can round to the nearest hour to determine that the restaurant has opening hours from 8 am to 8 pm (converting to 12-hour time).

17 a i a = −7, b = 5, c = 10

ii −7x2 + 5x + 10 = 0

b i a = 9, b = 22, c = 3

ii 9x2 + 22x + 3 = 0

b C

c A

19 a b T he solution of m = 0 must be excluded. The denominator in the quadratic formula is equal to 2a and if m = 0 then 2a = 0 but we cannot divide by zero. 20 n = ±9

in factored form using the known roots

2 x2 − (α + k) x − αx + α(α + k) = 0 Expand using the distributive property

3 x2 − (2α + k) x + (α2 + αk) = 0 Combine like terms This gives us b = −(2α + k) and c = α2 + αk Substituting into b2 − 4ac

Let’s extend our thinking

18 a B

Writing quadratic equation 1 (x − α) (x − (α + k)) = 0

1 b 2 − 4ac = (−(2α + k))2 – 4 ⋅ 1 ⋅ (α2 + αk) Substituting in values for a, b and c 2 b 2 − 4ac = 4α2 + 4αk + k2 – 4α2 – 4αk E xpand using the distributive property 3 b 2 − 4ac = k2

Combining like terms

25 a 0 < p < q b x2 + 2(p − q)x + p2 − q2 = 0 c 8q(q − p) d Two real solutions e x = 3p

Answers 1033 mathspace.co

8.05 Solve quadratics using appropriate methods Subtopic overview Lesson narrative In this lesson, students will summarize the different methods they have learned for solving quadratics. They will analyze quadratic equations in different forms to discuss which method is most ideal for solving each form and why. By the end of the lesson, students should be able to create and interpret quadratic models to solve problems in contextual situations, choosing their tools and using appropriate degree of precision.

8.05 Solve quadratics using Learning objectives appropriate methods Students: Page 491

After this lesson, you will be able to… • use the structure of a quadratic expression to identify ways to rewrite it. • choose appropriate methods for solving quadratic equations based on the structure of the quadratic expression. • create and solve quadratic equations for real-world contexts.

Solving quadratic equations using appropriate methods We have several methods we can use to solve quadratic equations. To determine which method is the most suitable Key vocabulary we need to look at the form of the quadratic equation.



quadratic equation

Graphing

Advantages: Helps us visualize the quadratic and its key features

Disadvantages: Only best when intercepts are integers, in which case it could have been factored instead

Essential understanding

Equation form: Any form is ﬁne if using technology, otherwise it is x There are many methods that can be used to solvebest a quadratic The of the equation can give in a form equation. that is equal to structure 0 Solution Solution insight into which method might be the most efficient.

Standards This subtopic addresses the following Virginia Standards of Learning for Mathematics Factoring Advantages: This is usually the fasteststandards. method

Mathematical process goals

Disadvantages: Not all polynomials are factorable, some factorable polynomials are diﬃcult to factor

MPG1 — Mathematical Problem Solving

Equation form: 2 apply different methods to solve quadratic equations. They can Teachers can integrate this goal by asking students axto + bx + c = 0 where a, b, c are small provide a variety of problem situations that require the use of different solution methods such as graphing, factoring, Square root property Simpleststudents method for vertex form square roots, and the quadratic formula. Teachers Advantages: can also encourage to solving developequations their owninproblem-solving or equations missing an x-term strategies and to articulate their reasoning. Disadvantages: Few equations are given in this form Equation form: x2 = k or a(x − h)2 = k Completing the square

Advantages: Can be used to solve any quadratic equation

1034 Mathspace Virginia SOL Algebra 1 Teacher Edition Disadvantages: Requires more steps than other methods, fractions mathspace.co

make it diﬃcult Equation form: x2 + bx + c = 0 where b is even

MPG2 — Mathematical Communication

MPG3 — Mathematical Reasoning

Teachers can encourage students to communicate their mathematical thinking by explaining their process in solving quadratic equations, discussing their choice of methods, and justifying their solutions. They can also ask students to articulate the reasoning behind their choice of methods, and to share their strategies and reasoning with their peers.

Teachers can integrate this goal by asking students to analyze the given quadratic equation, determine the most efficient method for solving it, and justify their choice. They can also ask students to evaluate their solutions and verify the correctness of their reasoning.

Content standards A.EI.3 — The student will represent, solve, and interpret A.EI.3c — Verify possible solution(s) to a quadratic the solution to a quadratic equation in one variable. equation in one variable algebraically, graphically, and with technology to justify the reasonableness of A.EI.3a — Solve a quadratic equation in one variable answer(s). Explain the solution method and interpret over the set of real numbers with rational or irrational solutions for problems given in context. solutions, including those that can be used to solve contextual problems.

Prior connections A.EO.2 — The student will perform operations on and factor polynomial expressions in one variable. A.EO.4 — The student will simplify and determine equivalent radical expressions involving square roots of whole numbers and cube roots of integers.

A.F.2 — The student will investigate, analyze, and compare characteristics of functions, including quadratic and exponential functions, and model quadratic and exponential relationships.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Algebra 1 — 8.01 Solve quadratics using graphs and tables Algebra 1 — 8.02 Solve quadratics by factoring Algebra 1 — 8.03 Solve quadratics using square roots Algebra 1 — 8.04 Solve quadratics using the quadratic formula

Tools You may ﬁnd these tools helpful: • Scientiﬁc calculator • Graphing calculator • Strategy comparison graphic organizer 8.05 Solve quadratics using appropriate methods 1035 mathspace.co

Student lesson & teacher guide Solving quadratic equations using appropriate methods Students review the various methods for solving quadratic equations that they learned about through the chapter. Some of these methods may have advantages and disadvantages, but students are reminded that no single method is the only way to solve quadratic equations.

Students: Pages 491–492

8.05 Solve quadratics using appropriate methods After this this lesson, lesson, you you will will be be able able to… to… After •• use the structure of a quadratic expression use the structure of a quadratic expression to to identify identify ways ways to to rewrite rewrite it. it. choose appropriate appropriate methods methods for for solving solving quadratic quadratic equations equations based based on on the the structure structure of of the the quadratic quadratic •• choose expression. expression. •• create create and and solve solve quadratic quadratic equations equations for for real-world real-world contexts. contexts.

Solving Solving quadratic quadratic equations equations using using appropriate appropriate methods methods We have have several several methods methods we we can can use use to to solve solve quadratic quadratic equations. equations. To To determine determine which which method method is is the the most most suitable suitable We we we need need to to look look at at the the form form of of the the quadratic quadratic equation. equation. Advantages: Advantages: Helps Helps us us visualize visualize the the quadratic quadratic and and its its key key features features Disadvantages: Only best when intercepts are integers, in Disadvantages: Only best when intercepts are integers, in which which case it it could could have have been been factored factored instead instead case

Graphing Graphing y y

Solution Solution

x x Solution Solution

Factoring Factoring

Equation Equation form: form: Any Any form form is is ﬁne ﬁne if if using using technology, technology, otherwise otherwise it it is is best in in a a form form that that is is equal equal to to 0 0 best

Advantages: Advantages: This This is is usually usually the the fastest fastest method method Disadvantages: Not all polynomials are Disadvantages: Not all polynomials are factorable, factorable, some some factorable factorable polynomials are are diﬃcult diﬃcult to to factor factor polynomials Equation Equation form: form: ax22 + + bx bx + + cc = =0 0 where where a, a, b, b, cc are are small small ax

Square root root property property Square

Advantages: Simplest Simplest method method for for solving solving equations equations in in vertex vertex form form Advantages: or or equations equations missing missing an an x-term x-term Disadvantages: Disadvantages: Few Few equations equations are are given given in in this this form form 2 2 Equation Equation form: form: xx2 = =k k or or a(x a(x − − h) h)2 = =k k

Completing Completing the the square square

Advantages: Can Can be be used used to to solve solve any any quadratic quadratic equation equation Advantages: Disadvantages: Requires more steps than other methods, Disadvantages: Requires more steps than other methods, fractions fractions make it it diﬃcult diﬃcult make 2 Equation Equation form: form: xx2 + + bx bx + + cc = =0 0 where where b b is is even even

1036 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

8.05 8.05 Solve Solve quadratics quadratics using using appropriate appropriate methods methods mathspace.co mathspace.co

491 491

ax2 + bx + c = 0 where a, b, c are small Advantages: Simplest method for solving equations in vertex form or equations missing an x-term

Square root property

Disadvantages: Few equations are given in this form Equation form: x2 = k or a(x − h)2 = k Advantages: Can be used to solve any quadratic equation

Completing the square

Disadvantages: Requires more steps than other methods, fractions make it diﬃcult Equation form: x2 + bx + c = 0 where b is even

Advantages: Can be used to solve any quadratic equation

Quadratic formula

Disadvantages: Can be time-consuming, many opportunities to 8.05although Solve quadratics usingsimplify appropriate methods 491 make miscalculations, calculators its use mathspace.co

Equation form: ax2 + bx + c = 0 where a, b, c are large

There is not one correct method for solving a quadratic equation. You would not be wrong by using one method over another; it is just easier, sometimes more practical, to use some methods over others.

Example 1 For the following quadratic equations, ﬁnd solution using an eﬃcient methods method. Justify which method you used. Provide a graphic organizer forthecomparing solving 2

+ 12 = 0 a x − 7x instructional Targeted strategies

We can Create solve aaquadratic strategy equation in a variety of ways. Have students consider which thinking strategy or conceptThe they can In particular, students can ifcompare contrast graphic like leading use. coeﬃcient of x2 is 1, so we can check this can beand easily factored.strategies The factors using of 12 area ±1, ±2, ±3, organizer ±4, ±6, ±12, and we want to ﬁnd two factors that have a product of 12 and sum to −7. Astemplates. the product is positive but the sum is the Strategy comparison graphic organizer found in our lesson support negative, we know both factors must be negative.

Support the comparison by asking questions like: • WhatApply possible strategies could I use to solve this problem? the idea • OfSince the strategies know, which seem to best particular problem? the equationI can be factored by grouping, we ﬁt willthis factor the equation and solveWhy? it. • Will the chose work? Why? How can change problem thisform strategy The strategy two factorsI that havealways a product of 12 and a sum of −7 areyou −3 and −4. Wethe cangiven write the equationso in that factored (x −work? 4) (x − 3) = 0, which gives us two solutions x = 3 and x = 4. doesasnot • Could the equation have been solved with fewer steps or in a simpler way? Reﬂect and check

In general, if the coeﬃcients are small, and especially if a = 1, it is worth checking to see if we can easily factor the equation to solve.

Allow students to solve quadratics using their preferred method Address student misconceptions b x2 − 11 = 21

Students may have the misconception that there is a “correct” method for solving a quadratic equation, and that they may be marked incorrectly for using a “less eﬃcient” or “less appropriate” method. Create a strategy 2

Heretowe have bphrasing = 0, and can easily positions isolate the xone , which means we solve this by than using another square roots. Make sure avoid which method as can being better and communicate to students that the most appropriate method is the one which they ﬁnd most comfortable.

Apply the idea

If a student to always thewillsame for using everysquare problem, ask them why they ﬁnd it most Sinceprefers we can easily isolateuse x2, we solve method the equation roots as follows: comfortable. Take this as an opportunity to identify potential gaps or weaknesses in a student’s knowledge and Given equation review concepts with them if appropriate. After they have gained some conﬁdence in the prerequisite skills, Add 11 to both sides encourage the student to try using the other methods. Evaluate the square root of both sides Factor 32 Multiplication property of radicals Evaluate the radical giving us two solutions:

492

Mathspace Virginia SOL Algebra 1 mathspace.co

and

8.05 Solve quadratics using appropriate methods 1037 mathspace.co

Support organization - provide a flowchart for decision-making Student with disabilities support Encourage all students to use Algorithmic Thinking to develop a flowchart that shows how to identify an appropriate strategy. Some students may require additional support to make flowcharts. This could include indicating what features they should look for or providing them with all of the possible strategies and questions written on paper for them to move around and draw the arrows between. While some students may benefit from being given a flowchart along with explicit instruction of how to use it. For example: Can we rearrange into vertex form?

Is the equation in standard form?

Yes Square root method

Can it be easily factored?

Yes

Quadratic formula

Factoring method

No Rearrange to standard form

Note that different students may require different flowcharts to reflect their personal strengths and weaknesses with each method. For example, a student who is comfortable with square roots, but has difficulty finding factor pairs may prioritize the factoring method last.

Stronger and clearer each time English language learner support Present an equation for all the students to try and solve. Then, ask students to explain their approach to solving the equation to a partner. Encourage students to communicate their reasoning with reference to “key features” of the equation and “operations” of particular methods. After giving the students some time to discuss with their Quadratic formula Advantages: Can be used to solve any quadratic equation partner, present a similar equation to the students and ask them to try and solve it. Disadvantages: Can be time-consuming, many opportunities to make miscalculations, although calculators simplify its use

Examples

Equation form: ax2 + bx + c = 0 where a, b, c are large

is not one correct method for solving a quadratic equation. You would not be wrong by using one method over Students:There Page 492 another; it is just easier, sometimes more practical, to use some methods over others.

Example 1 For the following quadratic equations, ﬁnd the solution using an eﬃcient method. Justify which method you used. a x2 − 7x + 12 = 0

Create a strategy The leading coeﬃcient of x2 is 1, so we can check if this can be easily factored. The factors of 12 are ±1, ±2, ±3, ±4, ±6, ±12, and we want to ﬁnd two factors that have a product of 12 and sum to −7. As the product is positive but the sum is negative, we know both factors must be negative.

Apply the idea Since the equation can be factored by grouping, we will factor the equation and solve it. The two factors that have a product of 12 and a sum of −7 are −3 and −4. We can write the equation in factored form as (x − 4) (x − 3) = 0, which gives us two solutions x = 3 and x = 4.

Reﬂect and check In general, if the coeﬃcients are small, and especially if a = 1, it is worth checking to see if we can easily factor the equation to solve. 1038 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co b x2 − 11 = 21

Disadvantages: Can be time-consuming, many opportunities to make miscalculations, although calculators simplify its use The leading coeﬃcient of x2 is 1, so we can check Equation if this canform: be easily factored. The factors of 12 are ±1, ±2, ±3, ±4, ±6, ±12, and we want to ﬁnd two factors that have a product of 12 and sum to −7. As the product is positive but the sum is 2 ax + bx + c = 0 where a, b, c are large negative, we know both factors must be negative.

Create a strategy

There is not one correct method for solving a quadratic equation. You would not be wrong by using one method over

Apply another;the it isidea just easier, sometimes more practical, to use some methods over others.

Since the equation can be factored by grouping, we will factor the equation and solve it. The two factors Example 1 that have a product of 12 and a sum of −7 are −3 and −4. We can write the equation in factored form as (x − 4) (x − 3) = 0, which gives us two solutions x = 3 and x = 4.

For the following quadratic equations, ﬁnd the solution using an eﬃcient method. Justify which method you used.

Reﬂect and check + 12 =0 a x2 − 7x

In general, if the coefficients are small, and especially if a = 1, it is worth checking to see if we can easily factor the equation solve. Create atostrategy The leading coefficient of x2 is 1, so we can check if this can be easily factored. The factors of 12 are ±1, ±2, ±3, ±4, ±6, ±12,xand want to find two factors that have a product of 12 and sum to −7. As the product is positive but the sum is 2 b − 11we = 21 negative, we know both factors must be negative.

Purpose Create a strategy Show students how to use factoring to solve a2 quadratic equation when an expression can be factored. Apply the idea Here we have b = 0, and can easily isolate the x , which means we can solve this by using square roots. Since the equation can be factored by grouping, we will factor the equation and solve it.

Expected mistakes The twothe factors have a product of 12 and a sum of −7 are −3 and −4. We can write the equation in factored form ideathatequation StudentsApply using different assume they are wrong because the method asmay (x − 4)solve (x − 3)the = 0, which gives us twoasolutions x =method, 3 and x = and 4. we can easily isolate x2, we will solve the equation using square roots as follows: used in Since the problem is factoring. Remind students to use whichever method of solving equations is preferred equation given for x. and check and thatReﬂect their solutions should matchGiven the solutions In general, if the coeﬃcients are small, Add and 11 toespecially both sidesif a = 1, it is worth checking to see if we can easily factor the

solve. Students:equation Pagesto 492–493

Evaluate the square root of both sides Factor 32

b x2 − 11 = 21

Multiplication property of radicals

Create a strategy

Evaluate the radical

giving and Here we have b =us 0, two and solutions: can easily isolate the x2, which means we can solve this by using square roots.

Apply the idea Since we can easily isolate x2, we will solve the equation using square roots as follows: Given equation 492

Mathspace Virginia SOL AlgebraAdd 1 11 to both sides mathspace.co

Evaluate the square root of both sides Factor 32 Multiplication property of radicals Evaluate the radical giving us two solutions:

and

Reﬂect and check In general, if we can easily rearrange the equation into the form (x − h)2 = k for some positive value of k then solving using square roots is a suitable method. 492

Mathspace Virginia SOL Algebra 1 mathspace.co 2

c 3x − 24x + 20 = 5

Purpose Create a strategy Show students how to use square roots to solve a quadratic equation.

For most of the methods we know, the quadratic needs to be equal to zero ﬁrst. We can subtract 5 from both sides, then check see if factoring can be used.

Reflecting with students Ask students Apply whether the idea a solution in decimal form would be acceptable. Discuss with students the appropriate and expected times to offer exact versus decimal solutions if neither istospecified in the problem. Since the trinomial is equal to a constant, we will first set the equation equal zero and attempt to factor theThis is at the discretion of theThen, teacher. trinomial. we can determine an approach that is appropriate for solving this equation. 3x2 − 24x + 20 = 5

Given equation

3x2 − 24x + 15 = 0

Subtract 5 from both sides

Factor the GCF of 3

x2 − 8x + 5 = 0

Divide by 3 on both sides

3(x − 8x + 5) = 0

From here, we can see that the equation cannot be factored further. Since a = 1 and b is even, we can use completing the square to solve. 8.05 Solve quadratics using appropriate methods 1039 Subtraction property of equality mathspace.co Complete the square Factor the left side, evaluate the right side

Reﬂect and check In general, if we can easily rearrange the equation into the form (x − h)2 = k for some positive value of k then solving square roots is a suitable method. Students:using Page 493 c 3x2 − 24x + 20 = 5

Create a strategy For most of the methods we know, the quadratic needs to be equal to zero ﬁrst. We can subtract 5 from both sides, then check see if factoring can be used.

Apply the idea Since the trinomial is equal to a constant, we will ﬁrst set the equation equal to zero and attempt to factor the trinomial. Then, we can determine an approach that is appropriate for solving this equation. 3x2 − 24x + 20 = 5

Given equation

3x2 − 24x + 15 = 0

Subtract 5 from both sides

3(x2 − 8x + 5) = 0 x2 − 8x + 5 = 0

Factor the GCF of 3 Divide by 3 on both sides

From here, we can see that the equation cannot be factored further. Since a = 1 and b is even, we can use completing the square to solve. Subtraction property of equality Complete the square Factor the left side, evaluate the right side Evaluate the square root of both sides Add 4 to both sides

Reﬂect and check The quadratic formula could have been used, but it may have been more time-consuming, especially if we didn’t factor out the GCF ﬁrst. Original equation set equal to 0 Substitute a, b, c into quadratic formula Evaluate the division Multiplication property of radicals Evaluate the radical Evaluate the division

Purpose Show students how to solve a quadratic equation by completing the square. Reﬂecting with students Ask students what their preferred method for solving this equation is and why. Point out to students that they 8.05 Solve quadratics using appropriate 493 may even choose to graph the function y = x2 − 8x + 5 and determine the x-intercepts to solvemethods the equation for mathspace.co when y = 0.

Support explanations - use sentence stems to help students justify their reasoning Student with disabilities support

use with Example 1

Help students justify why they chose a particular method by providing them with the sentence frames: • I can see that a key feature of the equation is... • I know that the ... method is appropriate when the equation is... • I ﬁnd it most comfortable to use the ... method when...

1040 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Students: Page 494 Example 2 A rectangular enclosure is to be constructed from 100 meters of wooden fencing. The area of the enclosure is given by A = 50x − x2, where x is the length of one side of the rectangle. If the area is 525 m2, determine the side lengths.

Create a strategy We can set up and solve a quadratic equation, 50x − x2 = 525. Since the values are large we will try solving this problem with the quadratic formula. The two solutions will be the side lengths of the enclosure.

Apply the idea Rearranging the equation into standard form we get x2 − 50x + 525 = 0. We can solve this using the quadratic equation: Quadratic formula Substitute a = 1, b = −50, c = 525 Evaluate the operations Evaluate the square root and

This leaves us with two values,

Evaluating each expression for x we

get x = 35 and x = 15 as the side lengths of the rectangular enclosure.

Reﬂect and check We can conﬁrm our answer is correct by checking the conditions of the problem. We had 100 meters of fencing and 2(35 + 15) = 100. We needed the area to be 525 m2 and 35(15) = 525 as required. Since there are two rational solutions, the quadratic equation was also factorable: x2 − 50x + 525 = (x − 35) (x − 15), but these factors are not immediately obvious.

Idea summary

Purpose Below is a list of the easiest method to use and the form of the quadratic equation for which we should use it: Show students how to use a context to write and solve an equation with the quadratic formula. Easiest equation form Expected mistakes Graphing Any form is fine when using technology Students may struggle to write an equation that represents the context. Writing expressions as words may help Factoring ax2 + bx + c = 0 where a, b, c are small students translate from the words to mathematics. 2 2 Square root property x = k or a(x − h) = k 2

x + bx + c = 0 where b is even Reflecting withCompleting studentsthe square 2 Quadratic formula ax + bx + ccorrect = 0 where b, c are large Ask students to check if the solutions they find are bya,checking the conditions of the problem. We had 2 100 meters of wooden fencing and an area of 525 m , so the sum and product of the two solutions should match these values. Since 2(35 + 15) = 100 and 35 ⋅ 15 = 525, we can confirm that this solution is correct.

Advanced learners: Exploring optimization

use with Example 2

Targeted instructional strategies Extend the problem for advanced students by asking them to find the maximum possible area for the enclosure Mathspace Virginia SOL Algebra 1 with the494 given perimeter. Students should recognize that the area function is a downward facing quadratic mathspace.co function, so the y-value of its vertex represents the maximum area. Prompt students to calculate the value of x at the vertex and interpret its meaning in the context of the problem. This exploration allows students to connect quadratic equations to real-world optimization and deepen their understanding of solving quadratics for values other than the x-intercepts.

8.05 Solve quadratics using appropriate methods 1041 mathspace.co

Students:x2Page 494 − 50x + 525 = (x − 35) (x − 15), but these factors are not immediately obvious.

Idea summary Below is a list of the easiest method to use and the form of the quadratic equation for which we should use it: Easiest equation form Any form is fine when using technology ax2 + bx + c = 0 where a, b, c are small x2 = k or a(x − h)2 = k x2 + bx + c = 0 where b is even ax2 + bx + c = 0 where a, b, c are large

Graphing Factoring Square root property Completing the square Quadratic formula

Practice Mathspace Virginia SOL Algebra 1 Students:494 Pages 495–497 mathspace.co

What do you remember? 1

Solve the following equations: a

(x − 3)2 = 64

(2 − x)2 = 81

x (x + 7) = 0

(10x − 9)2 = 0

(x − 6) (x + 7) = 0

x2 − 8x + 15 = 0

x2 − 4x − 22 = 0

The formula for the surface area of a sphere is S = 4π r2, where r is the radius. Determine the radius of a sphere that has a surface area of 804 in2, rounding your answer to two decimal places.

Consider the quadratic equation: x2 − 3x − 108 = 0 Select the solution to the quadratic equation. A

x = 9, x = −12

x = −9, x = 12

x = −9, x = −12

x = 9, x = 12

Consider the quadratic equation: x2 − 200 = −79 Select the solution to the quadratic equation.

A square lot has an area of 289 m2. Find the length of one side of the square lot if A = s2. A

s = 144.5 m

s = 72.25 m

1042 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

s = 17 m

Let’s practice 6

For the following quadratic equations, ﬁnd the solution using an eﬃcient method. Justify which method you used. a

x2 − 10 = 15

x − 7x = 0

25y2 = 36 2

5k − 17k + 13 = 0

(x − 6)2 − 2 = 0

24x2 = 71x − 35

x2 + 27x + 23 = 3x − 40

3x2 − 36x + 33 = 0

3x − 12x − 36 = 0

4x2 − 13x + 2 = 0

x2 + 18x + 32 = 0

(x + 5)2 − 2 = 15

−4 + 2x2 − 5x = 0

4x2 + 5x + 1 = 0

x2 + 9x + 20 = 0

x2 − 2x − 15 = 0

11x + x + 5 = 0

x2 − 4x = −1

3x2 − 2x + 5 = 10x + 1

Solve the following equations. Justify your work. 4x2 = 2 + 8x

x2 − 18x = −6

Solve the following equations. Verify your solution(s) by graphing or substitution. a

a 9

x2 − 7x + 6 = 0

Solve the following equations:

i 8

x + 24x + 63 = 0

x2 − 4x = 32

2x2 + 3x − 5 = 0

x2 = −2x + 24

The given rectangle has a length of L = 56y + 11 and a width of W = 5y2: a

Write the perimeter of the ﬁgure in terms of y.

If the perimeter is equal to 630, ﬁnd y.

At time t seconds, the distance, s, traveled by an object moving in a straight line is given by

where u is its starting speed and a is its acceleration. When u = 16 and a = 8, ﬁnd how long it would take for the object to travel 128 m. 12

The base of a triangle is 3 m more than twice its height. The area of the triangle is 115 m2. Let x be the height of the triangle. a

Find the height by solving for x.

Find the length of the base.

A rectangular swimming pool is 16 m long and 6 m wide. It is surrounded by a pebble path of uniform width x m. The area of the path is 104 m2. a

Find an expression for the area of the path in terms of x.

Write an equation and solve for x, the width of the path.

16 m x

14 Harry is using a diving board to dive into a swimming pool. The distance from his head to the surface of the water can be represented as (x − 7) (x + 7) = 147. Select the viable x-value to the quadratic equation. A

x = 14

x = −14

x = 49

x=7

8.05 Solve quadratics using appropriate methods 1043 mathspace.co

Let’s extend our thinking 15

For each of the following equations determine, without solving them, the most eﬃcient method for solving them. Explain your thinking. a

x2 − 3x + 2 = 0

8x = x2

16x2 − 81 = 0

Executives at the Widget Emporium are discussing whether to merge their company with the Trinket Bazaar, a large competitor. Market analysis shows that the extra revenue the company will receive can be modelled by the equation R = 0.25t2, and the extra costs by the equation C = 3.5t. R and C are measured in thousands of dollars and t is measured in months after the merger.

Find the times at which the extra revenue R will match the extra cost C.

The executives decide that they can only afford to operate at a loss for one year. Based on this requirement, state whether you would advise that the Widget Emporium merge with the Trinket Bazaar.

An interplanetary freight transport company has won a contract to supply the space station orbiting Mars. They will be shipping stackable containers, each carrying a fuel module and a water module, that must meet certain dimension restrictions. The design engineers have produced a sketch for the modules and container, shown below. The sum of the heights of both modules equal to the height of the container.

663 cm x

6426 cm x2 17 cm

Water Module

Fuel Module

Container

Write an equation that equates the height of the container and the sum of the heights of the modules.

Find the tallest possible height of the container, rounding your answer to two decimal places. Explain your method.

The shaded area in the rectangle has a uniform width and an area of 11 ft2:

3 x+1 4

Determine a quadratic equation to represent this situation.

Find the value of x, rounding your answer to two decimal places.

1044 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Answers 8.05 Solve quadratics using appropriate methods What do you remember? 1 a x = 11, x = -5

e x = 6, x = −7

g x = 5, x = 3

g x = −2, x = −16

8 a Rearrange the equation

b x = 11, x = −7

c x = 0, x = −7

e x = 6, x = −2

4x2 − 8x − 2 = 0

m = 0, m = −5

Use the quadratic formula: Substitute b = −8, a = 4, c = −2

2 r = 8.00 in

Move 8x and 2 to the left side

Evaluate the multiplication inside the square root

3 B 4 D

Simplify

5 D b Rearrange the equation x2 − 18x + 6 = 0

Let’s practice 6 a x = 5, x = −5 Using the method of taking square roots. We can add 10 to both sides to get x2 = 25. b x = 6, x = 1

Use the quadratic formula: Substitute b = −18, a = 1, c = 6

Using the method of factoring. Using the fact −6 − 1 = −7 and −6 · −1 = 6, we can rewrite as (x − 6) (x − 1).

Move 6 to the left side

Evaluate the multiplication inside the square root Simplify

c Using the method of taking square roots. We can

c Use the quadratic formula:

divide both sides by 25 to get Substitute b = −2, a = 1, c = −15

d Using the quadratic formula as a ≠ 1 and it is not easily factorable. e x = −3, x = −21 Using completing the square, we can add subtract 63 from both sides and then add

to both sides

of the equation giving x2 + 24x + 144 = 81 which can be rewritten in the form (x + 12)2 = 81 which can then be solved by taking square roots. f

x = 0, x = 7 We can easily factor this quadratic x(x − 7).

g Using the quadratic formula as a ≠ 1 and it is not easily factorable. h x = −5, x = −4

c x = −3, x = −21

Simplify

d Rearrange the equation x2 − 4x + 1 = 0

Move 1 to the left side

Use the quadratic formula: Substitute b = −4, a = 1, c = 1

Evaluate the multiplication inside the square root Simplify

9 a x = 8, −4 82 − 4 ⋅ 8 = 32

Using the method of factoring. Using the fact 4 + 5 = 9 and 4 · 5 = 20, we can rewrite as (x + 4) (x + 5). 7 a

Evaluate the multiplication inside the square root

32 = 32

−4 − 4 ⋅ −4 = 32

32 = 32

Substitute x = 8 Evaluate Substitute x = −4 Evaluate

d x = 11, x = 1

Answers 1045 mathspace.co

Let’s extend our thinking

b 2 ⋅ (1)2 + (3 ⋅ 1) − 5 = 0 0=0

15 a U sing the method of factoring. The factors of this quadratic are whole numbers and can be found by considering the factors of the constant.

Evaluate x = 1 Evaluate

Substitute

Evaluate

b U sing the method of factoring. We can see that there is a factor of x on both sides of the equation. c U sing the method of taking square roots. We can isolate x2 on one side of the equation and everything else on the other.

c x = 4, −6

d U sing the quadratic formula. The coefficients are not easily factored.

25 20 15

16 a 0 and 14 months

10 5

−9−8−7−6−5−4−3−2−1 5

1 2 3 4 5 6 7 8 9

b N o, after one year the company will still be spending more money than it is making. 17 a b Solving the equation found in part (a) for x and multiplying entire equation by x2 gives 17x2 − 663x + 6426 = 0. We can factor this as 17(x − 18) (x − 21). Solving the quadratic gives two solutions x = 18 and is x = 21. Now, the height of the containter

10 15 20 25

inversely proportional to x so we want to choose the least solution of x, that is x = 18. Substituting x = 18 into gives us a height of 36.83 cm

9 y 8 7 6 5 4 3 2 1 −4 −3 −2 −1−1 −2 −3 −4 −5 −6 −7 −8 −9

10 a 10y2 + 112y + 22

18 a (4 + x) (3 − x) = 11

x 1

b y=4

11 t = 4 12 a x = 10 2

b 23 m 2

13 a 4x + 44x m

b x=2

14 A

1046 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

b x = 0.61

Topic 8 Assessment: Quadratic Equations 1

Using the given table, ﬁnd the solutions to the equation x2 − 2x − 8 = 0: x y

−4 16

−3 7

−2 0

−1 −5

0 −8

1 −9

2 −8

3 −5

4 0

Find the solutions to f (x) = 3x2 + 21x + 30, using the graph provided. 12 10 8 6 4 2 −8 −7 −6 −5 −4 −3 −2 −1 −2 −4 −6 −8

x 1 2

Solve the following equations: a

x2 + 6x − 27 = 0

4x2 − 29x = −30

5 + 6x = 2x2

x2 − 81 = 0

(x + 4)2 = 121

The area of a rectangle is 32 cm2. If the dimensions can be expressed as 12 − x and x, ﬁnd the dimensions of the rectangle.

On Earth, the equation d = 4.9t2 is used to ﬁnd the distance, in meters, an object has fallen through the air after t seconds. Kevin is sky diving and wants to release his parachute once he has fallen 510 m. Determine the time it will take him to fall 510 m, rounding your answer to the nearest second.

The revenue of a toy manufacturing company is modeled by the function r(t) = −4t2 + 450t where t is time in days and r(t) is the revenue in dollars. A company’s break-even is obtained when the revenue ﬁrst reaches $7500. Find the time it takes the company to reach break-even. Round your answer to the nearest hundredth.

SOL

Find the discriminant of the equation 5x2 − 4x + 2 = 0 and state what it reveals about the solutions.

Fill in the blanks with your answers.

The solutions to 2x2 − 8x + 6 = 0 are: ⬚ and ⬚

Determine whether or not the following equations have real solutions: a

SOL

x2 + 5x − 14 = 0

4x2 − 8x + 12 = 0

Identify if each function has exactly one zero. Justify your reasoning. a

f (x) = 7x2 − 3

f (x) = −2(x + 5)(x + 2)

f (x) = 9(x − 6)

f (x) = x2 + 6x + 8

f (x) = x2 − 4x + 16

−1 and −12

−2 and −6

What are the real roots of x2 − 8x + 12 = 0? A

2 and 6

1 and 12

Topic 8 Assessment: Quadratic Equations 1047 mathspace.co

SOL

What values of x are solutions of 3x2 + 7x = 10? A

1 and

10 and

and −1

The graph of y = x2 − 4x − 12 is shown.

10 8 6 4 2

What are the solutions to x2 − 4x − 12 = 0? A

x = 2 and x = −16

x = 0 and x = −12

x = 6 and x = 2

x = −2 and x = 0

and −10

−4

−2 −2

x 2

−4 −6 −8 −10 −12 −14 −16

Look at function g: g(x) = 8x2 − 18 Which set contains only the zeros of function g? A

SOL

{−18, 8}

{−18, 0, 8}

The graph of y = −x2 − 2x + 24 is shown.

26 y 24 22 20 18 16 14 12 10 8 6 4 2

On the grid, identify each solution to −x2 − 2x + 24 = 0.

−8 −6 −4 −2−2 −4

x 2

Consider the function h: h(x) = x2 + 9x − 36

State a number that is a zero of the function h?

Explain at least two different ways to verify that the solution from part (a) is a zero of the function.

Let h(x) = −4x2 + kx + 16. a

If h(1) = −3, what is the value of h(−3)?

How many zeros does the function have?

A scientist dropped an object from a height of 350 feet. She recorded the height of the object in 0.5-second intervals. Her data is shown. a

Find the value of f (x) when x = 3.

Explain the meaning of the answer from part (a) and what x = 3 represents in context.

1048 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

x 0.0 0.5 1.0 1.5 2.0 2.5

f (x) 350 345 335 315 285 245

SOL

Which of these functions has exactly two different zeros? A

h(x) = x2 − 6x + 9

k(x) = x2 + 9x + 20

Performance task 20

Ursula is launching a pumpkin off the edge of the physics building at her school with a small catapult. a

Determine an equation, deﬁning any variables, that models the path of the pumpkin given the following information: • The physics building is 24 m high. • When the pumpkin is 2 m from the building, it is 44 m high. • When the pumpkin is 3 m from the building, it is 30 m high.

Use an eﬃcient method to ﬁnd how far from the building the pumpkin hits the ground. Explain your method.

Ursula wants to do a demonstration where she launches the pumpkin into a target. If the target is 2 m high, how far from the building does she need to place it so it gets hit by the pumpkin? Explain.

Topic 8 Assessment: Quadratic Equations 1049 mathspace.co

Answers

11 A A.EI.3a

Topic 8 Assessment: Quadratic Equations

12 A A.EI.3a

1 x = −2, x = 4

13 C

A.EI.3a

2 x = −2, x = −5

14 C

A.EI.3a 3 a x = 3, x = −9

A.EI.3a

c x = 9, x = −9

d x = 7, x = −15

15 x = −6, 4 A.EI.3a 16 a x = −12, or 3 b Answers may vary. One possible solution:

A.EI.3a 4 8 cm ⋅ 4 cm A.EI.3a 5 t = 10 seconds

1. Substitute the values of x into the equation h(x) = x2 + 9x − 36 and check if the result is 0.

2. Factor the equation h(x) = x2 + 9x − 36 into (x + 12)(x − 3) and check if the values of x make either factor equal to 0.

A.EI.3a

A.EI.3a, A.EI.3c

6 20.35 days

17 a h(−3) = 25

A.EI.3a 7 The discriminant is −24 which means that the solutions to the equation are non-real solutions.

18 a f (3) = 195 b T he object is 195 feet above the ground at 3 seconds after it was dropped. The x = 3 represents the time in seconds.

A.EI.3b 8 1 and 3 A.EI.3a

A.F.2g

9 a Real solutions

19 D

b No real solutions

A.EI.3b

A.EI.3b 10 a No, it has two zeros. The function is a quadratic with zeros at

b 2 zeros

A.F.2g, A.EI.3b

and

b Y es, it has exactly one zero. The function is a linear function, and all linear functions have exactly one zero. In this case, the zero is at x = 6. c N o, it has two zeros. The function is a quadratic with a positive leading coeﬃcient (1), so it opens upwards. The discriminant (b2 − 4ac) is positive, indicating that there are two distinct real zeros.

Performance task 20 a y = −8x2 + 26x + 24 where x represents the horizontal distance from the edge of the building and y represents the vertical height above the ground. b S ubstitute 0 for y to represent a height of 0 m when the pumpkin is on the ground. The right hand side of the equation is factorable which gives the equation 0 = −2 (4x + 3) (x − 4). Solving gives x =

and x = 4.

The negative solution does not make sense in terms of this context so the pumpkin must hit the ground 4 m from the building.

d N o, it has no zeros. The function is a quadratic with a positive leading coeﬃcient (1), so it opens upwards. The vertex is above the x-axis, so it never crosses the x-axis.

c S ubstituting 2 for y in the original equation gives the equation 2 = −8x2 + 26x + 24.

e N o, it has two zeros. The function is a quadratic with a negative leading coeﬃcient (−2), so it opens downwards. There are two distinct zeros at x = −5 and x = −2.

Putting the equation in standard form gives 0 = −8x2 + 26x + 22. It can then be solved using the quadratic formula. She needs to place the target about 3.95 m from the building.

A.EI.3b

1050 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

A.EI.3, A.F.2g, MP1, MP3, MP4, MP5

9 Data Analysis Big ideas • Collecting and analyzing data can inform predictions and decisions, as long as the data is based on a valid sample. • Different representations of data highlight different characteristics of the data. • Many sets of bivariate data can be modeled using familiar functions.

Chapter outline 9.01 9.02 9.03 9.04 9.05

Data and sampling (A.ST.1) Scatterplots (A.ST.1) Linear regression (A.ST.1) Quadratic regression (A.ST.1) Analyze bivariate data (A.ST.1) Topic 9 Assessment

1056 1085 1114 1144 1168 1193

Data analysis helps experts track and study wildlife patterns to protect endangered species.

9. Data Analysis Topic overview Foundational knowledge Evaluating standards proficiency The skills book contains questions matched to individual standards. It can be used to measure proficiency for each. Students should be proficient in these standards. 8.PS.3 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on scatterplots.

A.F.1 — The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.

A.F.2 — The student will investigate, analyze, and compare characteristics of functions, including quadratic and exponential functions, and model quadratic and exponential relationships.

Big ideas and essential understanding Collecting and analyzing data can inform predictions and decisions, as long as the data is based on a valid sample. 9.01 — Representative samples are crucial if a data set will be used to make predictions and decisions.

Many sets of bivariate data can be modeled using familiar functions. 9.03, 9.04, 9.05 — The relationship between the variables in a set of bivariate data reveals the type of function that best models the data.

Different representations of data highlight different characteristics of the data. 9.02 — Correlation does not imply causation.

Standards A.ST.1 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on representing bivariate data in scatterplots and determining the curve of best fit using linear and quadratic functions. A.ST.1a — Formulate investigative questions that require the collection or acquisition of bivariate data. 9.01 Data and sampling 9.02 Scatterplots 9.03 Linear regression 9.04 Quadratic regression 9.05 Analyze bivariate data

1054 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

A.ST.1b — Determine what variables could be used to explain a given contextual problem or situation or answer investigative questions. 9.01 Data and sampling 9.02 Scatterplots 9.03 Linear regression 9.04 Quadratic regression 9.05 Analyze bivariate data

A.ST.1c — Determine an appropriate method to collect a representative sample, which could include a simple random sample, to answer an investigative question. 9.01 Data and sampling 9.02 Scatterplots 9.03 Linear regression 9.04 Quadratic regression 9.05 Analyze bivariate data A.ST.1d — Given a table of ordered pairs or a scatterplot representing no more than 30 data points, use available technology to determine whether a linear or quadratic function would represent the relationship, and if so, determine the equation of the curve of best fit. 9.02 Scatterplots 9.03 Linear regression 9.04 Quadratic regression 9.05 Analyze bivariate data A.ST.1e — Use linear and quadratic regression methods available through technology to write a linear or quadratic function that represents the data where appropriate and describe the strengths and weaknesses of the model. 9.03 Linear regression 9.04 Quadratic regression 9.05 Analyze bivariate data

A.ST.1f — Use a linear model to predict outcomes and evaluate the strength and validity of these predictions, including through the use of technology. 9.03 Linear regression 9.05 Analyze bivariate data A.ST.1g — Investigate and explain the meaning of the rate of change (slope) and y-intercept (constant term) of a linear model in context. 9.03 Linear regression 9.05 Analyze bivariate data A.ST.1h — Analyze relationships between two quantitative variables revealed in a scatterplot. 9.02 Scatterplots 9.03 Linear regression 9.04 Quadratic regression 9.05 Analyze bivariate data A.ST.1i — Make conclusions based on the analysis of a set of bivariate data and communicate the results. 9.02 Scatterplots 9.03 Linear regression 9.04 Quadratic regression 9.05 Analyze bivariate data

Future connections A2.ST.1 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on univariate quantitative data represented by a smooth curve, including a normal curve.

A2.ST.2 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on representing bivariate data in scatterplots and determining the curve of best fit using linear, quadratic, exponential, or a combination of these functions.

9. Data Analysis 1055 mathspace.co

9.01 Data and sampling Subtopic overview Lesson narrative In this lesson, students will learn about the data cycle, forming an investigative question, and analyze various methods of data collection and their impact on data quality. It expands upon previous knowledge of univariate data to create a definition of bivariate data and how it can be collected. The lesson distinguishes between populations and samples, and reviews different sampling methods for gathering responses to investigative questions. By the end of the lesson, students will understand the data cycle and various sampling methods, exploring the advantages and disadvantages of each method, considering factors like cost, time, and accuracy.

Learning objectives Students: Page 500

Key vocabulary 

bias



bivariate data



cluster sampling



convenience sampling



data cycle



dependent variable



independent variable



investigative question



measurement



observation



population



sample



sample survey



simple random sampling



statistical question



statistical variable



stratified sampling



survey



systematic sampling



univariate data

Essential understanding Representative samples are crucial if a data set will be used to make predictions and decisions.

Standards This subtopic addresses the following Virginia Standards of Learning for Mathematics standards.

Mathematical process goals MPG1 — Mathematical Problem Solving Teachers can integrate this goal by posing real-world problems that require the application of bivariate data analysis. For example, presenting a situation where students need to collect and analyze data on two different variables, such as height and weight, to identify any existing relationship. 1056 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

MPG2 — Mathematical Communication

MPG3 — Mathematical Reasoning

This goal can be integrated by having students explain their thought process when formulating investigative questions or choosing appropriate sampling methods. Teachers can encourage students to use mathematical language and notation in their explanations. Students can also be asked to present their findings to the class, effectively communicating their mathematical reasoning.

Teachers can facilitate this goal by asking students to justify their choice of variables and sampling methods in the given context. They should reason why their chosen methods are appropriate for the investigative question at hand. This encourages the use of logical reasoning and critical thinking.

MPG5 — Mathematical Representations Teachers can integrate this goal by asking students to visually represent the bivariate data they collect, such as using scatterplots. They should make connections between the visual representation and the mathematical concepts it represents. For instance, students can be asked to represent the relationship between two variables in a graph, and interpret its meaning in the context of the problem.

Content standards A.ST.1 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on representing bivariate data in scatterplots and determining the curve of best fit using linear and quadratic functions.

A.ST.1b — Determine what variables could be used to explain a given contextual problem or situation or answer investigative questions. A.ST.1c — Determine an appropriate method to collect a representative sample, which could include a simple random sample, to answer an investigative question.

A.ST.1a — Formulate investigative questions that require the collection or acquisition of bivariate data.

Engage Activity Populations and samples

60 mins

Students will look at a conclusion made from a sample and identify points of bias. They will simulate a survey of a population and make conclusions based on the results.

9.01 Data and sampling 1057 mathspace.co

Understanding and skills

Will use

Will develop

Using data from a random sample to make predictions about a population.

Estimating a population total or percentage using data from a sample survey. Analyzing a sample for bias.

Could extend Determining a way to sample without bias.

Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: None.

Support students with disabilities Support memory - use math vocabulary This task will include vocabulary such as: survey, data, simulate, poll, bias, population, percentage, conclusion, random sample, and analyze. Preview vocabulary terms prior to the lesson and have students deﬁne the vocabulary words on a resource sheet, or provide resource sheets for the vocabulary terms.

Support for English language learners Discussion supports Provide the following sentence frames for students to use precise language when discussing the validity of Javier’s survey: • “I noticed ⬚, so I ⬚.” • “If ⬚ then ⬚ because ⬚.” • “I know ⬚ because ⬚.” • ⬚ reminds me of ⬚ because ⬚.”

Classroom guide Hook Students create questions about a poster created for a student body election.

Implementation details

Co-craft questions

9/10 students want pizza for lunch everyday!

I CAN MAKE IT HAPPEN. VOTE FOR

JAVIER!

Slide 1 from Student Engage Activity

1058 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

5 mins

What mathematical questions could we ask about this situation?

We want students to consider where the statistics on the poster came from and question their validity. Some possible questions include: • How many students were surveyed? • How was the survey conducted? • Which students were surveyed? • How did Javier ﬁnd or select these students? • Is this sample representative of all students at Javier’s school?

•

Launch

5 mins

When Javier ran for student council Vice President, he wanted to research topics that his fellow students care about. To find this information, he conducted a survey of his friends. Javier stated that: “I asked 20 students and 18 of them said they want pizza for lunch everyday. I decided to say ‘9 out of 10’ on the poster because it was catchy, but it means the same thing.” Slide 4 from Student Engage Activity

Before starting the task, ask students to deﬁne the terms sample and population. Ask students to share what they think makes a “good” sample for conducting a survey. Important mathematical concepts: Sample, population, survey, simulate Suggested grouping: Form pairs

Continue when Students have read the Launch and understand the context of the problem.

Explore

Think-pair-share

In this task, students will be asked a number of openended questions. Students will be asked to articulate whether or not they think Javier’s survey was large enough to represent the school’s population and whether there was any bias introduced by only polling his friends.

•

25 mins

Use the applet to investigate the problem.

Survey size

Take

Anticipated strategies Estimate a population total Students will use a sample percentage to work out a population total. Based on the given information about students stating they want pizza for lunch every day, students may calculate

to estimate the proportion of the whole student Slide 5 from Student Engage Activity population who wants pizza for lunch each day. Students may choose to round this value to 2107 or 2108 as you cannot have a fraction of a student, or they may keep the exact value of 2107.8 since it is an estimate and not an actual quantity of students.

Analyze a sample Students will consider a sample and the corresponding statistics to draw conclusions about the validity of the conclusions made. Some observations students might make are: • Javier only polled his friends so the statistic

might not be representative of the whole population.

• If Javier polled a different group of 20 students he might see different results. • Javier polled 20 people total and this could be too few (or too many) people to poll.

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Simulate a sample Students will simulate sampling of a population using the applet. During the simulation, students may observe: • The proportion of the sample that prefers pizza for lunch everyday can vary from one simulation to the next. • In many simulations, the proportion of students who want pizza for lunch every day is less than , which was Javier’s claim. • For small simulations (about n < 20), there is much more variability in the results of the simulation.

Misconceptions Conﬂating survey bias with sample size issues What does it mean for survey to have bias? Can you think of any examples? How does the sample size affect the validity of a survey? Is it possible to survey too many or too few people? Is this the same idea as bias? What issues might there be Javier only surveying his friends? Do you think this contributes to any bias? What percent of the school did Javier survey? Do you think this enough to accurately represent the views of the entire school? Why or why not? If not, how many would you poll to feel comfortable representing the views of the school?

Purposeful questions Use the following questions to check for understanding and encourage critical thinking: • What sample size do you think is ‘big enough’? Is there is sample size that is too big or too small? • Is there bias in Javier’s original survey? What evidence supports/refutes this? • Did the simulation(s) change your opinion on how big of a sample should be surveyed? Explain.

Continue when Students have determined how big of a sample is ‘big enough’ for Javier’s survey, and whether they would do anything differently if they were to replicate the poll.

Discuss

25 mins

Pairs will begin by ﬁlling in a class data chart, then having a whole class discussion of each pair’s results. Consider sequencing the strategies presented from estimating a population total, to analyzing a sample, to simulating a sample, to a discussion on bias.

Discussion guide Create a class chart where pairs will ﬁll in how big of a sample they would poll if they were to recreate Javier’s survey. Begin by calling on several pairs to explain how they determined how many is ‘enough’ to survey. Allow students to critique one another’s answers, and if any groups have signiﬁcantly higher or lower sample sizes than the others, spend some time discussing. Some questions you can ask to promote discussion: • For small samples, say less than 10% of the school population: Consider the simulations you did in the activity. How many of them with a sample size of (insert student answer of 10% or less) were representative of Javier’s survey? Was there more variability in the outcomes of the small samples? How do you know? • For large samples, say 1000 or more: What logistical issues arise from having a very large sample? Does it take longer to survey? Do the pros outweigh the cons? Next, ask students to share if there were any other issues with the survey besides the sample size. Ask for examples of what kind of bias occurs when you only ask your friends for their opinion and not other groups of people. Some questions you can ask to promote discussion: • Does it result in many of the same opinions being overrepresented? • What happens when other voices aren’t included? Students may wish to share other examples of bias they have heard about in real world situations, such as advertisements, politics, or articles making health and medical claims. As an extension, you may provide students with the prompt: • Design a method of population sampling which results in a representative data set and reduces the amount of bias in the survey. 1060 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 8 — 3.04 Independent and dependent variables Grade 8 — 4.01 Data collection and sampling

Lesson supports The following supports may be useful for this lesson. More speciﬁc supports may appear throughout the lesson:

Understanding the data cycle Targeted instructional strategies Start with a brief discussion about the data cycle, which includes planning, collecting, processing, and interpreting data. Ask students to list each part of the data cycle and describe in their own words what each part entails. This image could be posted publicly as a reminder. Possible responses could be: • Formulate questions • Finding a context we want to investigate • Write a statistical question for the investigation • Collect or acquire data • Deciding whether we need univariate or bivariate data • Write survey questions and give it to a random sample of people • Measure objects or quantities • Plan an observation strategy • Design a scientific experiment • Research online

Formulate questions

Analyze and communicate results

The data cycle

Collect or acquire data

Organize and represent data

• Organize and represent data • Draw a graph or diagram • Analyze and communicate results • Summarize findings • Answer the statistical question

The data cycle is cyclical Address student misconceptions Students may mistakenly believe that the data cycle is a linear process. It’s crucial to correct this misconception by explaining that the data cycle is a circular process, with each step feeding into the next. This is also called an iterative process. If data is interpreted and new questions arise, the cycle begins again with new planning. Emphasize that iterative processes that refine the product or conclusion are widely applicable. For example, in coding, code can often be improved to be more efficient, or a bakery can make micro-adjustments to their recipes.

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Student lesson & teacher guide Formulate questions for bivariate data Students will learn about bivariate data, its importance, and how it allows for the exploration of relationships between two variables. They will learn how to formulate an statistical question that focuses on the relationship between these variables, and understand the importance of identifying the correct variables for accurate data analysis.

Students: Pages 500–501

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Variables

Bivariate data

Variables are quantities or qualities that can be measured or classified.

Bivariate data is data that is collected from two different variables and compared against each other. This data is typically numerical.

Independent variable

Example: Age versus height, or 1-mile time versus 5-mile time

The variable that is varied or controlled to explore the effect it has on the dependent variable. Dependent variable The variable that depends on the independent variable. We typically want to explore the effect that the independent variable has on the dependent variable.

Person

Age (years)

Art Kumi Isla Daria Xia

30 40 50 60 70

Systolic blood pressure (mmHg) 121 140 134 154 146

For a study about heart health, a person’s systolic blood pressure is measured against their age. Their age is the independent variable and can be any value. Their systolic blood pressure is the dependent variable that is recorded against their age. Notice that each person’s age and systolic blood pressure make a pair of values in the bivariate data set.

To start working with bivariate data, we need to formulate a statistical question. Statistical question A statistical question that can be answered by collecting data and whose answer may vary depending on the sample the data is collected from. Also called an investigative question. Statistical question Is there a relationship between age and systolic blood pressure? Do test scores increase as the amount of time studying increases? Does how long a pen lasts impact the cost of the pen?

Not statistical question What is your blood pressure? How long did you study and what was your grade? Are there pens under five dollars that will last all year?

A statistical question is different from a survey question that is asked to those in the people in a study. We need to make sure that questions are not leading people to answer a particular way. This means not using emotive language or suggesting a particular answer. Good survey question Do you watch soccer? How would you rate your meal? What was your average speed driving here?

Leading question Do you watch the most popular sport in the world, soccer? What did you think of the meal from the outstanding chef? Did you do the wrong thing and go over the speed limit to get here?

Notice how the good questions are very neutral and the leading questions may encourage people to respond in a particular way.

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501

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Examples and non-examples of statistical questions Targeted instructional strategies To formulate effective statistical questions, students may benefit from a discussion of what makes a statistical question “good” or “bad.” Using the examples and non-examples from the lesson: Statistical question Not statistical question Is there a relationship between age and systolic blood What is your blood pressure? pressure? Do test scores increase as the amount of time How long did you study and what was your grade? studying increases? Does how long a pen lasts impact the cost of the pen? Are there pens under five dollars that will last all year? Discuss with students why these are good or bad examples. Some discussion points might be: • The second statistical question makes the intent of the investigation clear; we want to determine if there is an increasing trend between test scores and study time. To answer the question, we will need to collect bivariate data from various people, represent the data, and analyze the trend in the data. • “What is your blood pressure?” is not a statistical question because there is a single response to this question. This is actually a survey question because it would help us collect data, but it does not lead us to investigate the data. After the discussion, encourage students to create their own examples and non-examples of statistical questions.

Critique, correct, and clarify English language learner support Display the leading survey question, “Did you love the movie The Boys in the Boat?” In pairs, students can discuss why the question is leading how they could rewrite the question to be a better survey question. Prompt student discussion with questions such as, “Which part of the question makes it leading?” or “When you read the question, does it make you feel a certain way about the movie?” After discussing with their partners, allow students to write an argument for why they think the question is leading and write a better survey question. After a few minutes, invite students to share their arguments and survey questions with the class. Better survey questions might be: • What did you think to the movie The Boys in the Boat? • What did you like or dislike about the movie The Boys in the Boat? • Would you recommend watching the movie The Boys in the Boat?

Examples Students: Page 502

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Purpose Determine if students can identify questions that require bivariate data to answer.

Students: Pages 502–503

Each of these questions is clear and concise, focusing specifically on the relationship between temperature and ice cream sales. They each propose a different aspect of the relationship to investigate, making them effective statistical questions.

Reﬂect and check After one round of the data cycle we may formulate a new question to further explore the topic.

Example 3 In a study conducted at a high school, students’ study times (in hours) and their corresponding test scores (out of 100) were recorded for one particular examination. The data is to be analyzed to understand the relationship between study time and test scores. a Identify the variables involved in this scenario.

Create a strategy

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A variable is any characteristic, number, or quantity that can be measured or counted. There are two types of variables: dependent and independent. The dependent variable is what is being measured or

Purpose This example checks whether students can formulate clear and relevant statistical questions to explore a relationship between two variables, in this case, temperature and ice cream sales. Expected mistakes Students may write questions that are vague or not directly related to the relationship between temperature and ice cream sales. For instance, a question like “Does weather affect ice cream sales?” is too broad as it doesn’t specify the aspect of weather (in this case temperature) being considered. Eachwith of these questions is clear and concise, focusing specifically on the relationship between temperature and ice Reﬂecting students cream sales.the They each propose a different aspectbe of the relationshiptotodiscuss investigate, making them the effective After formulating statistical question, it would interesting with students kindstatistical of data that questions. could be collected to answer these questions and how they might go about analyzing it. This could lead to a deeper understanding of the investigative process. Reﬂect and check After one round of the data cycle we may formulate a new question to further explore the topic.

Students: Page 503 Example 3

In a study conducted at a high school, students’ study times (in hours) and their corresponding test scores (out of 100) were recorded for one particular examination. The data is to be analyzed to understand the relationship between study time and test scores. a Identify the variables involved in this scenario.

Create a strategy A variable is any characteristic, number, or quantity that can be measured or counted. There are two types of variables: dependent and independent. The dependent variable is what is being measured or observed (the outcome), while the independent variable is what is being manipulated or changed (the likely cause).

Apply the idea In this case, the two variables are study time and test scores. The independent variable is the study time. This is because it is the variable that we think will cause changes in the test scores. The dependent variable is the test score. This is because it may change in response to changes in the study time. Test scores are what we are interested in predicting or explaining.

Reflect and check It’s important to consider potential confounding variables in any study. A confounding variable is an outside influence that may impact one or both of the variables. These may lead to a false conclusion. For example, the difficulty of the test, the student’s previous knowledge, and other external factors (like health, sleep, etc.) could all potentially impact a student’s test score.

b Rewrite this question so it is not leading and it could be used to accurately collect data.

Purpose “We believe students who study more do better on tests. How much time did you spend studying last night? Did you do well on the test?” Check that students can identify independent and dependent variables in a real-world scenario. Create a strategy We need to make sure that the question is not leading people to answer a particular way. This means not using emotive language or suggesting a particular answer.

Apply the idea “How much time did you spend studying last night? Did you do well on the test?”

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503

Students: Page 503

b Rewrite this question so it is not leading and it could be used to accurately collect data. “We believe students who study more do better on tests. How much time did you spend studying last night? Did you do well on the test?”

Create a strategy We need to make sure that the question is not leading people to answer a particular way. This means not using emotive language or suggesting a particular answer.

Apply the idea “How much time did you spend studying last night? Did you do well on the test?”

Purpose Check if students can rewrite a question to make it non-leading.

Students: Page 504

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503

Collect data using samples Students will learn about the concept of sampling in data collection and its various methods including simple random, systematic, stratified, and cluster sampling. They will understand the strengths and weaknesses of each method and how to choose the appropriate method for their data collection. The lesson also explains how to maintain the representativeness of the sample.

Students: Pages 504–506

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Best sampling method Address student misconceptions Students may believe that there is a “best” sampling method. Correct this misconception by explaining that the method will depend on the specific situation and research question. Discuss the advantages and disadvantages of each method in different scenarios to reinforce this point.

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Concrete-Representational-Abstract (CRA) Approach Targeted instructional strategies Concrete: Engage students with hands-on activities to explore different sampling methods. Begin by creating a physical representation of a population. For example, fill a large container with a variety of colored beads or marbles, where each color represents a different characteristic within the population. Introduce various sampling methods: • Simple Random Sampling: Have students close their eyes and randomly select a specific number of beads from the container, ensuring each bead has an equal chance of being chosen. • Systematic Sampling: Line up the beads and instruct students to select every nth bead (e.g., every 5th bead) from the sequence. • Stratified Sampling: Divide the beads into groups based on color (strata). Then, have students randomly select a proportional number of beads from each group to represent the entire population. • Cluster Sampling: Group the beads into clusters (e.g., small bags or sections) and randomly select entire clusters to include in the sample. Encourage students to physically perform each sampling method, allowing them to experience how samples are collected and how different methods can yield different results. Representational: Transition to visual representations of the sampling methods. Guide students to create diagrams, charts, or graphs that illustrate each method: • Simple Random Sampling: Draw the entire population and use random dots or highlights to indicate the randomly selected samples. • Systematic Sampling: Depict the population in a line or grid format and mark every nth item that is selected. • Stratified Sampling: Illustrate the population divided into distinct strata based on characteristics, showing the proportional samples taken from each group with colored sections. • Cluster Sampling: Represent the population divided into clusters, highlighting the entire clusters that are selected for the sample. These visual tools help students understand the process and rationale behind each sampling method. Encourage them to label their diagrams clearly and use different colors or symbols to distinguish between groups and samples. Abstract: Move on to discussing the sampling methods using formal definitions and statistical concepts: • Simple Random Sampling: Explain that every member of the population has an equal chance of being selected. Discuss its benefits in reducing bias and how randomness is achieved. • Systematic Sampling: Describe how selecting every nth member can simplify the sampling process but may introduce bias if there is a hidden pattern in the population. • Stratified Sampling: Emphasize how dividing the population into strata ensures representation from all subgroups, improving the accuracy of the sample in reflecting the population’s diversity. • Cluster Sampling: Explain that selecting entire clusters can be more practical and cost-effective, especially with large populations spread over wide areas, but may increase sampling error if clusters are not representative. Introduce terms like “population,” “sample,” “bias,” and “representativeness.” Encourage students to compare the advantages and disadvantages of each method, considering factors such as ease of use, cost, time efficiency, and accuracy Connecting the stages: Help students make connections between the concrete activities, their visual representations, and the abstract concepts. Ask guiding questions such as: • “How did physically selecting beads help you understand the concept of randomness in sampling?” • “What did you notice when comparing the different sampling methods during the activity?” Encourage students to relate their diagrams to the hands-on activities by explaining how each visual representation maps onto the steps they performed. Facilitate discussions on how the sampling method chosen can impact the data collected and the conclusions drawn from it. By connecting all three stages, students can better monitor their thinking and make informed decisions about which sampling method to use in different investigative scenarios. This holistic understanding reinforces the importance of choosing appropriate sampling methods to collect representative data for predictions and decisions.

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Collect and display English language learner support As students are working, note how students describe the concepts of “sample,” “population,” “sampling methods” and how they relate this to “representative sample” and “potential biases”. Collect the different ways that students find to understand these concepts and display them in a common place for the students to access. If students do not come up with alternative ways to word these concepts and are confused by them, suggest some of your own. For example: Population: • Everyone we wish we could survey • Who or what the survey is about • Can be people, animals, objects, or cases Sample: • Some people or objects from the population • Subset of the population • Represents the population • Who we can actually access to survey Sampling methods • Strategies for selecting a subset of individuals from a population • Ways to collect data from a group Representative sample • A subset of a statistical population that accurately reflects the members of the entire population • A small quantity that accurately reflects the larger entity Potential biases • Factors that may skew the sampling process, leading to unrepresentative results • Unfair influence that may affect the outcome of the study

Provide visuals and specific examples for each sampling method Student with disabilities support For each type of sampling method, include an explanation of the definition, a real-world example and an illustration to support student understanding. Here are some examples that could be used for each sampling method: • Simple random sampling • Definition: Explain that in random sampling, each member of the population has an equal chance of being selected for the sample. • Real-world example: Describe how names might be drawn from a hat to decide who will serve on a committee. • Illustration: Population Sample

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• Convenience sampling • Definition: Explain that a convenience sample involves only collecting data from people who are easy to reach or within your group of acquaintances. • Real-world example: A student might only survey students in their class. • Illustration:

• Systematic sampling • Definition: Describe how in systematic sampling, a starting point is randomly chosen, and then every nth member of the population is selected for the sample. • Real-world example: A factory might inspect every 10th item coming off the assembly line to check for defects. • Illustration:

Systematic Sampling

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• Stratified sampling • Definition: Explain how stratified sampling involves dividing the population into homogeneous subgroups and then taking a random sample from each subgroup. • Real-world example: A television network wanting to get a balanced view of people’s opinions on a new TV show might take a sample that includes a set number of viewers from different age groups. • Illustration: Population

Strata

Sample

• Cluster sampling • Definition: Explain how cluster sampling involves dividing the population into groups. Then, entire groups are randomly selected. • Real-world example: At a conference, attendees are seated at tables with 10 people each. The person leading the conference randomly invites 3 tables of people to help with a demonstration. • Illustration: Population Clusters

Sample group (2 Clusters)

Clusters

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Examples Students: Page 506

Purpose Evaluate students’ ability to choose an optimal sampling method given the specific scenario. Reﬂecting with students Help students better understand experiments by describing that there are typically two groups in an experiment: the control group and the experimental group. • The control group is the “normal” group. In this example, “those using a standard treatment” would be considered the control group. • The experimental group are the ones with the “new” routine or the ones “testing” a theory. In this example, “patients using a new dosage” are in the experimental group. By having a control group and an experimental group, we can determine that any changes observed in the experimental group are caused by the “new routine” or the thing being tested.

Students: Pages 506–507

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Purpose Evaluate students’ ability to choose an optimal sampling method given the specific scenario.

Students: Page 507

Purpose Evaluate students’ ability to choose an optimal sampling method given the specific scenario. Expected mistakes Students might say an observational study would be best for collecting data. While an observation is valid, it would be difficult to collect this data by observing people. For example, how would they know which people had been visiting for a month or less? And how could they get a random sample of those people?

Students: Page 507

Purpose Determine if students’ are able to identify the target population in a given scenario.

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Students: Page 507

Purpose Evaluate students’ ability to choose the best collection method for a given scenario.

Students: Page 508 c Explain why doing a sample of the children at a park one Monday morning would not be a good sample.

Create a strategy For the sample to be representative of the population, different children of varying ages with various playground habits should be included.

Apply the idea This sample would be convenient, so is a convenience sample which is not representative. This sample likely wouldn’t include school age children who would be at school, not at the playground on a Monday morning. It also wouldn’t include children who don’t regularly get to go to the playground. Finally, there may not be many children at the park, but the population of children might be large in comparison.

Example 6 Purpose Check students’ a good sample.the question “Is there a relationship between the frequency of Dr. Jane is understanding a health researcherofand she formulated exercise and overall health among working adults in Washington, DC?” She wants to collect data for her research. Choose an appropriate sampling method.

Create a strategy When selecting a sampling method, Dr. Jane needs to consider several factors such as the size of her target population, the resources she has available, and potential biases that could influence the results.

Apply the idea An appropriate sampling method for this study could be stratified sampling. Considering Washington, DC’s large and diverse population, stratified sampling would ensure that all segments of the population are represented in the sample. Dr. Jane could divide the population into different strata based on factors like age, occupation, or zipcode, and then randomly select participants from each stratum.

Reﬂect and check While stratified sampling can provide a representative sample, it can be more complex and time-consuming to 1076 Mathspace Virginia SOL Algebra 1 Teacher Edition implement, and it might not be feasible if information about the different strata is not readily available. mathspace.co An alternative method might be simple random sampling, where every individual in the population has an equal chance of being selected. However, this method might not guarantee that all segments of the population are adequately represented, especially for a diverse population.

Students: Page 508 Example 6

Dr. Jane is a health researcher and she formulated the question “Is there a relationship between the frequency of exercise and overall health among working adults in Washington, DC?” She wants to collect data for her research. Choose an appropriate sampling method.

Reﬂect and check While stratified sampling can provide a representative sample, it can be more complex and time-consuming to implement, and it might not be feasible if information about the different strata is not readily available. An alternative method might be simple random sampling, where every individual in the population has an equal chance of being selected. However, this method might not guarantee that all segments of the population are adequately represented, especially for a diverse population.

Purpose To assess students’ understanding of how to select an appropriate sampling method based on the characteristics of the population and the statistical question. Reflecting with students Remind students that the goal of stratified sampling is to ensure that all segments of the population are represented. While a simple random sample is a valid sampling method, it could lead to certain subgroups being underrepresented in the sample. Ask students what examples of subgroups of the population might look like and why it is important to consider the diversity of the population. For example, subgroups might be adults above 50, immigrants or refugees, or Mathspace Algebra students508 between the Virginia ages SOL of 16 and1 24. It is important to consider different subgroups as frequency of exercise mathspace.co affect people’s bodies in different ways.

Advanced learners: Discussion of various sampling methods

use with Example 6

Targeted instructional strategies Invite students to consider multiple sampling methods that Dr. Jane could use for her research beyond stratified sampling. Encourage them to think about the methods not mentioned in the lesson (cluster sampling, systematic sampling, and convenience sampling) and discuss how each could be implemented in the context of Washington, DC’s working adults. Ask open-ended questions such as, “How might using cluster sampling affect the representativeness of Dr. Jane’s data?” or “What are the potential benefits and drawbacks of using systematic sampling in this scenario?” By exploring various methods and their implications, students can develop a more comprehensive understanding of how sampling choices impact the validity and reliability of research findings.

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Collect and display

use with Example 6

English language learner support As students are working, note how students describe the various sampling methods. Collect the different ways that students find to understand these concepts and display them in a common place for the students to access. If students do not come up with alternative ways to word these concepts and are confused by them, suggest some of your own. For example: • Random sampling • Picking members of a population by chance • Every member has an equal chance to be selected • No system for how the sample is chosen • Systematic sampling • Using a consistent pattern for choosing the sample • Count the number of people/objects, then choose the multiples of a number for the sample • Stratified sampling • Dividing a population into categories, then randomly picking members from each category • Ensuring all segments of the population are represented • Cluster sampling • Dividing a population into groups, then randomly picking entire groups • All members of the selected groups are included in the sample Encourage students to use these alternative ways of understanding and expressing these concepts as they discuss the problem and work towards a solution.

Students: Page 509

Idea summary After we formulate a clear statistical question, we use the data cycle to collect, show, and explain information. To get data, we can use methods like: • • • • •

Watching (Observation) Measuring Asking questions (Survey) Doing experiments Acquiring existing secondary data

Sampling methods are techniques to collect data from a representative subset of the population, known as a sample. • •

Population: every member of a group. Sample: a subset of the population.

Types of sampling methods include: • • • •

Simple Random Sampling: every member of the population has an equal chance of being selected. Systematic Sampling: involves selecting every nth member of the population. Stratified Sampling: dividing the population into subgroups, and then selecting a separate random sample from each subgroup. Cluster Sampling: the population is divided into groups, or clusters. Then, a random sample of clusters is selected, and all members within selected clusters are included in the sample.

The type of sampling method chosen can greatly influence the quality of data collected and the conclusions drawn from it.

Practice What do you remember? 1 What is the difference between bivariate data and univariate data? 1078 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co 2 Give an example of a real-world situation where the relationship between two variables can be investigated using bivariate data. 3

State whether each statistical question could be answered by collecting univariate data or bivariate data:

Practice Students: Pages 509–513

What do you remember? 1

What is the difference between bivariate data and univariate data?

Give an example of a real-world situation where the relationship between two variables can be investigated using bivariate data.

State whether each statistical question could be answered by collecting univariate data or bivariate data:

How are the weights of students on the wrestling team distributed?

Is there a relationship between iron levels in soil and weed growth?

What shoe sizes are the most common at each of my local schools?

Is taxable income related to latitude of home address?

Does the number of days children spend in daycare affect the number of days spent home sick?

Is there a relationship between the amount of natural sunlight in a classroom and students’ exam results?

Typically how old are people when they learn to skate on ice? Does it vary by country?

State whether each statement about statistical questions is true or false. a

The question must have a yes or no answer.

The question allows for surveys to be conducted.

The answers to question may vary from one person to another.

The answers to question requires only numerical values.

They are only used for bivariate data

They should include your hypothesis for what you think the answer will be.

State whether each of the following questions are statistical questions: a

Which city is the capital of France?

How far away is the moon from the Earth, right now?

Which is a typical maximum temperature during the summer in Chesterfield, VA?

How much do kittens weigh?

How far do you have to travel from home to school each day?

How old are Olympic gold medal winners when they win their medal?

How far do students have to travel to get from their home to school each day?

How far do you have to travel from home to school each day?

How many calories do people burn per day?

Determine which data collection method best describes each of these scenarios. a

Asking people on the street about the age of their oldest living relative and recording their answers.

Using a batter bowl and a scale to determine the density of dough before and after it rises to see if there is a relationship.

A long term study that provides different levels of subsidies for childcare and looks to see if this affects the income and mental well-being of those children when they reach adulthood.

Watching a variety of gardens to see how many pollinators visit per hour.

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Is each question leading or not? a

Do you take a multi-vitamin?

How much time do you waste on social media per day?

How much time do you spend reading every week?

Do you think the government should be allowed to cut down some of the oldest trees in the area to construct a metro railway line in the city?

Do you think bike helmets should be mandatory for all bike riders?

Do you eat at least the recommended number of servings of fruits and vegetables to ensure a healthy and long life?

How much time do you spend sitting every day?

Determine whether the scenario represents collecting data from a population or sample: a

Oscar has determined the cost of 5% of houses from each suburb in Richmond.

Ainsley tests every lamp that the factory produces.

Habib scans every carry-on bag for a flight Roanoke.

Drake does a checkup on all children brought to his doctor’s office to assess the health of all children in the city.

In which type of sampling does every individual in the population have an equal chance of being selected? A

Random sampling

Systematic sampling

Stratified sampling

What is the main difference between random and systematic sampling? A

Random sampling involves selecting individuals at regular intervals, while systematic sampling involves selecting individuals randomly.

Random sampling involves selecting individuals randomly, while systematic sampling involves selecting individuals at regular intervals.

There is no difference between random and systematic sampling.

John is conducting a research study on the satisfaction levels of employees in a company. He decides to use random sampling. a

Which option best describes the process of random sampling? A Random sampling involves dividing the population into distinct subgroups based on specific characteristics and then randomly selecting participants from each subgroup. B Random sampling involves selecting participants from a population in a completely random manner, ensuring each individual has an equal chance of being chosen. C Random sampling involves selecting participants from a population at regular intervals, using a predetermined starting point.

Which option best describes one advantage of using this random sampling? A One advantage of using this method is that it reduces the chances of bias and provides a representative sample of the population. B One advantage of using this method is that it ensures representation from each subgroup, allowing for more accurate estimates for each group.

A school principal wants to estimate the number of students who ride a bicycle to school. Is each sample biased? a

All students who are in the school band.

Eight students in the hallway after school.

Ten students from each grade, chosen at random.

130 randomly selected students during the lunch periods.

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For each statistical question, identify the independent and dependent variables. a

Is there a relationship between the number of website visitors and the percentage of visitors who make a purchase?

Is there a relationship between the number of words in a child’s vocabulary and the number of books that are read to them per week?

Is there likely to be a relationship between each pair of variables? a

Time spent on phone per day and number of apps on phone

Amount spent on pet food and amount spent on candy

Price of lemons and average SAT scores

Car speed and gas mileage

Let’s practice 15

For each scenario: i

Identify possible independent and dependent variables.

Write a statistical question related to the scenario.

Gertrude notices the eggs with more Omega-3 tend to cost more and wonders if these are related.

Latisha has started baking bread from scratch. As she gets more experienced she finds it takes less time. She wonders if there is a link between experience and time it takes to bake bread.

Theo likes running and is working on his hill sprints on different hills. He is curious about his maximum speed on different slopes.

Change the following questions to make them statistical questions. a

How many books does your teacher have?

How many points did the grade school basketball team score in its last game?

What is your grade in Algebra 1 during the first term?

Jeremiah formulated the statistical question “Is there a relationship between spending on advertisement and business revenue?” Design a simple study that could be used to collect data to answer her question.

Natalie notices that finding coats that are long enough for her is difficult. She is curious what other body measurements are the best predictor of torso length. She formulates the question “Is there a relationship between arm length and torso length in teenagers?”

Should she use measurement, observation, or acquire data to collect the data?

What type of data would she be collecting?

What type of sample should she use to ensure it is representative of the population?

Rewrite each question so it is not leading and it could be used to accurately collect data using a sample survey. a

A new study said that using more than two bottles of shampoo per year is wasteful. How many bottles of shampoo do you use per year?

Most people with nice hair use some kind of oil product, what is your favorite hair product?

Would you describe your hair as beautiful?

How strong is the relationship between hair length and intelligence?

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The owner of a movie theater wants to use stratified sampling in their survey of people who come to their theater. Are these methods considered to be stratified sampling?

Interview 10% of the people who used the concessions and 10% of people who didn’t.

Interview every person that sees a romantic movie.

Interview 10% of the people from each movie.

Interview every 10th person that purchases a ticket.

For each scenario, determine the type of sampling method used: a

Drawing out the winning ticket in a lottery

Choosing every 50th person on the class roll to take part in a survey

Choosing 5% of the of the students in each grade for grades 7−12

David is conducting a research study to investigate the eating habits of people in a particular city. He opts for stratified sampling. What is an advantage of using this method?

Sarah is conducting a survey to gather data about the shopping preferences of customers in a large retail store. She chooses to use systematic sampling. What is a disadvantage of using this method?

Explain why the following samples are biased: a

Hannah is surveying customers at a shopping mall. She wants to know which stores customers shop at the most. She walks around an entertainment store and chooses 30 customers from the store for the survey.

A TV station wants to know what the most popular type of music is, so they ask listeners to contact them and vote for their favorite type of music.

The community health nurse wants to survey the students in a school about their eating habits. At lunchtime, she stands by a vending machine and surveys every student who purchases something from the machine.

Let’s extend our thinking 25

Imagine you are investigating the relationship between a student’s SAT score and their college GPA. Identify potential confounding variables that could influence this relationship and how you might account for them in your investigation.

Patricia surveys her class about their favorite music. Her results are shown in the table: Genre No. students

Country 15

Pop 2

According to the survey, which genre is most popular among her class?

This was her survey question: “The coolest kids like country music, and nobody likes pop. Do you like country music or pop music?” Do you trust the results of Patricia’s survey? Explain your answer.

You want to survey a group of n students about their favorite sports, but you only have time to survey 20 students. How would you use systematic sampling to select the 20 students for the survey?

You want to conduct a survey of the reading habits of students at your school. How would you use stratified sampling to ensure a representative sample?

Discuss the importance of formulating appropriate statistical questions in the data cycle. How does this impact the subsequent steps in the cycle such as data collection, analysis, and interpretation?

For a topic that interests you, consider a relationship that might exist. a

Formulate a statistical question that could be used to explore the relationship.

Describe a sampling and data collection method that could be used.

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Answers

b i I ndependent variable: Month or years of experience baking

9.01 Data and sampling

Dependent variable: Time to bake bread

What do you remember?

ii Is there a relationship between the experience of a baker and the time it takes them to bake bread?

1 Univariate data involves a single variable while bivariate data involves two variables. Bivariate data is used to ﬁnd the relationship or association between the two variables. 2 An example could be investigating the relationship between a baby’s age and weight. 3 a Univariate data

b Bivariate data

c Univariate data

d Bivariate data

e Bivariate data

Bivariate data

g Univariate data 4 a No e No 5 a No e No i

b Yes

c i Independent variable: Steepness of the hill Dependent variable: Maximum sprinting speed ii Is there a relationship between the steepness of a hill and maximum sprinting speed on the hill? 16 a A nswers will vary. A possible answer is: How many nooks do teachers in our school have? b A nswers will vary. A possible answer is: What is a typical number of points scored by the grade school basketball team in its games this season? c A nswers will vary. A possible answer is: What are the grades Algebra 1 students during the ﬁrst term?

c Yes

d No

b No

c Yes

d Yes

1. Determine the population:

Yes

g Yes

h No

17 Many possible designs. For example:

6 a Survey

b Measurement

c Experiment

• Sampling Method: Use stratified sampling to ensure diverse representation from various types of businesses like photographers, manufacturers, publishers, cleaners, transportation, etc.

• Sample Size: Depending on the resources available, he should choose the largest number of possible businesses

d Observation

7 a Not leading

b Leading

c Not leading

d Leading

e Not leading

Leading

g Not leading 8 a Sample

3. Collect the data:

b Population

c Population

He would probably need to do a survey or acquire data from a secondary source.

d Sample

9 Option A

10 Option B 11 a Option B

b Option A

12 a Biased

b Biased

c Not biased

d Not biased

13 a Independent variable: Number of website visitors Dependent variable: Percentage of visitors who make a purchase b I ndependent variable: Number of books read to the child per week Dependent variable: Number of words in the child’s vocabulary 14 a Yes

• Target Population: All businesses in Jeremiahs’s town

2. Develop a sampling strategy:

Yes

b No

c No

d Yes

Let’s practice 15 a i Independent variable: Amount of Omega-3 Dependent variable: Cost of eggs ii Is there a relationship between the amount of Omega-3 in eggs and their cost per dozen?

• Distribution method: He could distribute the survey electronically through school email or giving a link to an online form to make data collection easy and efficient. Alternatively, he could provide paper surveys with pre-paid return envelopes for those who don’t like using the internet.

18 a Measurement b W e would be collecting two lengths which are measureable numerical data. c U sing a stratified sample would be appropriate in this case, because it would ensure that people with a wide variety of body types, ages, and backgrounds would be included. She should make the survey anonymous to ensure that people will feel comfortable to respond honestly. 19 a How many bottles of shampoo do you use per year? b What is your favorite hair product? c How would you describe your hair in one word? d I s there a relationship between hair length and intelligence?

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20 a Yes

b No

21 a Random sampling

c Yes

d No

b Systematic sampling

c Stratified sampling 22 Stratified sampling ensures representation from different segments or groups within the population, allowing for more accurate analysis and conclusions about specific subgroups. 23 One disadvantage of using systematic sampling is that it may introduce a potential bias if there is a systematic pattern or periodicity in the population’s characteristics, leading to an underrepresentation or overrepresentation of certain individuals or groups in the sample. 24 a T he sample is not representative of the target population. b T he sample is from self-selecting participants, i.e. only those that made an effort to respond. c T he sample is not representative of the target population. Let’s extend our thinking 25 Several variables could influence the relationship between a student’s SAT score and their college GPA, including the student’s study habits, course load, major, and involvement in extracurricular activities. To account for these variables, we could collect data on these additional factors and include them in our analysis as control variables. Alternatively, we could focus our investigation on a more specific population (e.g., students within the same major) to reduce the impact of these variables. 26 a Country b N o, because the question uses emotive or leading language and suggests that country music is cool. It also asks two questions rather than one, as it’s possible to like both types of music (or neither).

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27 Example answer: You would first create a list of all the students, and then choose every nth student on the list, where n is the total number of students divided by 20. 28 Example answer: You would ﬁrst divide the students into different groups based on reading level, grade level, or other relevant factors. Then, you would randomly select students from each group to ensure representation from each subgroup. 29 Formulating appropriate statistical questions is a critical ﬁrst step in the data cycle because it guides the subsequent steps. The question determines what kind of data needs to be collected, how it should be collected, and what kind of analysis methods are appropriate. A poorly formulated question can lead to irrelevant or inaccurate data, inappropriate analysis methods, and misleading or invalid interpretations. Conversely, a well-formulated question can help ensure that the data collected is relevant and accurate, that the analysis is appropriate and meaningful, and that the interpretations and conclusions are valid and useful. 30 a A nswers vary. Sample answer: In Ultimate Frisbee, is there a relationship between the height of a player and number of goals they score in the season? Does it vary by league? b F or teams in the UFA, we could use secondary data and use the website to acquire the data for each player. This might be time consuming without writing a script or program to source all the data. We could do a stratified sample and select 5 players from each team with a mix of positions if we were short on time. For local teams, we could do a survey and ask 10 players from each team that were selected using a systematic sample two questions: How tall are you? and How many goals have you scored this season?

9.02 Scatterplots Subtopic overview Lesson narrative In this lesson, students will be representing the relationship between two variables on a scatterplot. Students will be analyzing the relationship between bivariate data using concepts such as correlation and causation, as well as determining best-fit models for the data. Students should be able to justify whether or not two variables are correlated, as well as identify the strength of the correlation. By the end of the lesson, students should be comfortable plotting data with a linear pattern on a scatterplot and estimating a best-fit line. This is essential as in future lessons students will be required to fit functions to data.

Learning objectives Students: Page 514

Key vocabulary 

bivariate data



correlation



dependent variable



independent variable



negative linear relationship



positive linear relationship



quantitative variable



scatterplot

Essential understanding Correlation does not imply causation.

Standards This subtopic addresses the following Virginia Standards of Learning for Mathematics standards.

Mathematical process goals MPG1 — Mathematical Problem Solving Teachers can encourage mathematical problem solving by guiding students to apply the concepts of bivariate data and scatterplots to solve real-world problems. For example, students can be asked to create their own scatterplots from real-world data, such as tracking the relationship between temperature and ice cream sales over a certain period. Teachers can also engage students in identifying potential problems in scatterplot representations, like outliers or unaccounted variables, thereby promoting critical thinking. 9.02 Scatterplots 1085 mathspace.co

MPG2 — Mathematical Communication

MPG5 — Mathematical Representations

Teachers can foster mathematical communication by having students explain their reasoning when identifying the type of relationship in a scatterplot. For instance, students can be asked to justify their reasoning when classifying scatterplots as having positive linear relationships, negative linear relationships, or no relationship. Additionally, students can share their observations and interpretations of scatterplots in group discussions or in written form.

Teachers can integrate the goal of mathematical representations by having students represent bivariate data using scatterplots, both manually and using technology. Students can be taught to understand that each point on the scatterplot represents a pair of values for the two variables. Teachers can also encourage students to use different representations, such as tables or graphs, to further reinforce the concept of scatterplots. They can also have students explain the meaning of the scatterplots they’ve created in their own words, reinforcing the concept that representation is both a process and a product.

A.ST.1d — Given a table of ordered pairs or a scatterplot representing no more than 30 data points, use available technology to determine whether a linear or quadratic function would represent the relationship, and if so, determine the equation of the curve of best fit.

A.ST.1a — Formulate investigative questions that require the collection or acquisition of bivariate data.

A.ST.1h — Analyze relationships between two quantitative variables revealed in a scatterplot.

A.ST.1b — Determine what variables could be used to explain a given contextual problem or situation or answer investigative questions.

A.ST.1i — Make conclusions based on the analysis of a set of bivariate data and communicate the results.

A.ST.1c — Determine an appropriate method to collect a representative sample, which could include a simple random sample, to answer an investigative question.

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Engage Activity Happiness scores

60 mins

Students will investigate a sample of bivariate data and determine possible reasons as to why the two variables have an association.

Understanding and skills

Will use

Will develop

Creating scatterplots.

Understanding correlation.

Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: None.

Support students with disabilities Support language - write explanations of mathematical thinking When students are describing possible reasons for association in the data, use sentence starters such as: • I think there is an association between ⬚ and ⬚ because... • Countries with high sugar consumption have ⬚ happiness ratings. • Countries with sugar consumption have ⬚ happiness ratings. • I think the relationship between sugar consumption and happiness is ⬚ because...

Support for English language learners Three reads Have students read the titles on the graph aloud. On the ﬁrst read, ask students to describe the graph. Prompt: Students read the problem. Students think/write: Answer the question “What is the problem about?” Answers may look like: • Sugar consumption vs. happiness • Different countries sugar consumption and happiness • Investigating data and making inferences Share: Students are called upon to discuss their answers with the class. On the second read, ask students to interpret the question. Prompt: Students read the problem. Students think/write: Answer the question “What does an answer look like?” Answers may look like: • A claim and evidence to support or refute the claim • An explanation of whether the two variables in the problem have any causation On the third read, have students identify important information. Prompt: Students read the problem. Students think/write: Answer the question “What are the important pieces of information given in the question?” Answers may look like: • The individual data points • The units and labels on the axes of the graph

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Classroom guide Hook

Notice and wonder

Students write observations about bivariate data presented as a scatterplot. Students look for conclusions that can be made from the data, or questions they have about the data that would require further research or information provided.

Nobel Laureates per 10 Million Population

r = 0.791 P < 0.0001

US NL FR BE CA AU PT GR IT

10 5 0

CN 0

5 10 Chocolate Consumption (kg/yr/capita)

Slide 1 from Student Engage Activity

Launch Ask students if they have ever heard of the happiness score. You may wish to share a summary of what the score and how it is reported, which can be found here: https://worldhappiness. report/faq/. You may also allow students to quickly research the score and summarize it in with their partner before having a brief share-out to the class. Important mathematical concepts: Scatterplot, two-way frequency table, relative frequency table, bivariate data Important contextual information: Happiness score Suggested grouping: Form pairs

5 mins The following graph shows data for various country’s sugar consumption compared to their national happiness score ranking: 90

Happiness Score

80 70 60 50 40 30 20 10

Sugar consumption

g day

20 40 60 80 100 120 140 160 180

Slide 2 from Student Engage Activity

Continue when Students have read the Launch and understand the context of the problem.

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5 mins

What do you notice? What do you wonder?

Implementation details Ask students to look at individual flags and the data as a whole, and encourage them to notice and wonder about individual data points as well as the data set as a whole. Since correlation and causation are often conflated with one another, it may be challenging for students to articulate why one statement is a notice while the other must be a wonder. For example, a student might wonder, “Does more chocolate consumption produce more Nobel laureates?” while another student might think they can say “chocolate consumption influences the number of Nobel laureates” which is not necessarily true.

•

Explore

Think-pair-share

•

35 mins

Anticipated strategies Interpret data to make claim Students may wish to interpret the given data to make a claim. Students will produce their own claim based on the data. One such claim that students might make would be, “The higher a country’s sugar consumption, the higher their happiness score.” After researching their claim, they may be able to recognize that both sugar consumption and happiness scores are typically higher in countries with higher GDPs and thus a causal relationship is unlikely. During the second stage of the Explore, the goal is to have students recognize that there is a positive association between sugar consumption and happiness scores, but that there are likely other variables inﬂuencing both happiness and sugar consumption simultaneously, namely the wealth of the nation and its level of development. Whether or not they are actually valid claims, it is theoretically possible for the two variables to have an association for other reasons as well, such as a coincidence or perhaps sugar consumption is just one of many factors inﬂuencing happiness and not the sole factor.

Misconceptions Correlation is the same as causation Does it mean that simply owning books leads to better grades? What about income levels or parent’s education level? Might those be inﬂuencing both the number of books and the students’ grades?

Purposeful questions Use the following questions to check for understanding and encourage critical thinking: • Is there any information in the graph to imply that sugar consumption causes happiness? • Is it possible for two variables to have an association without causation? How might that be possible? • What are some reasons that two variables may have an association?

Continue when Students have written a claim about the data and justiﬁed whether or not their claim is supported by any reputable sources they found during their investigation.

Discuss

15 mins

Start with a class discussion. Consider making connections from the discussion to how to determine if a source is credible or not.

Discussion guide Ask several pairs to share their claim with the class. Allow the class to decide whether they think each claim is true before asking the pair whether or not they found evidence to support their claim. The discussion is a great opportunity to have students practice determining whether a source is credible evidence of a particular claim. Feel free to allow groups to determine which sources used are credible or not credible. After several pairs have presented, ask several other pairs to share the possible factors for why the two variables have an association. As pairs are sharing their answers with the class there may be themes in what kinds of answers are being shared. Use this as an opportunity to have students generalize the types of claims into different categories such as “A is just one cause of B” or “C causes both A and B.” If the claims produced in class seem to fall heavily into one or two categories, you may wish to provide additional examples of claims for students to try and generalize.

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Tools You may ﬁnd these tools helpful: • Graphing calculator • Graph paper

Lesson supports The following support may be useful for this lesson. More speciﬁc supports may appear throughout the lesson:

Concrete-Representational-Abstract (CRA) Approach Targeted instructional strategies Concrete: Begin by engaging students in a hands-on activity to explore the relationship between two variables. Have them collect real-world data, such as measuring each other’s heights and arm spans, or recording the time spent walking and the distance covered. Provide tools like measuring tapes or stopwatches for data collection. This physical activity helps students understand how variables can be related and gives them concrete data to work with. Representational: Guide students to organize the collected data into a table. Then, teach them how to create a scatterplot using graph paper or a digital graphing tool. Show them how to plot each data pair accurately, with one variable on the x-axis and the other on the y-axis. Encourage them to observe any patterns or trends in the plotted data, such as clusters or a general direction. Abstract: Introduce the concepts of correlation and causation using mathematical terminology. Explain how to describe the relationship shown in the scatterplot as positive correlation, negative correlation, or no correlation. Teach students how to estimate and draw a line of best fit through the data points. Discuss how this line can be represented with an equation and how the slope relates to the strength of the correlation. Emphasize that correlation does not imply causation, and provide examples to illustrate this concept Connecting the stages: Help students connect their hands-on data collection with the scatterplots and abstract concepts. Ask guiding questions like: • “How does the pattern in our scatterplot relate to the measurements we took?” • “What does the line of best fit tell us about the relationship between these variables?” Encourage them to consider whether one variable causes the other or if they are simply correlated. By linking the concrete activity, visual representation, and abstract ideas, students will develop a deeper understanding of analyzing relationships in scatterplots.

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Student lesson & teacher guide Scatterplots Students are guided through key features of the analysis of bivariate data. These features include the form, strength, and direction of the relationship. Additionally, the inclusion of categorical variables in scatterplots is discussed to enhance understanding Examples of data sets with different strength and direction are given.

Students: Pages 514–515

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7 5 2

3 1

x 1

10 20 30 40 50 60 70 80 90

Strong negative relationship since points are tightly clustered along the model

2 3 4 5 6 7 8 9

Moderate negative relationship since points are relatively clustered along the model

260

259

258

257

5 4

256

255

254

x 0.2

x 1

0.3

Weak positive relationship since points are loosely clustered along the model

2 3 4 5 6 7 8 9

No relationship since there is no evident clustering of the data

When comparing bivariate data, it may be necessary to separate the data into categories. For example, when comparing the weights of dogs during their first year after birth, the data might not show a relationship because large dogs (like Boxers) will grow much more than small dogs (like Yorkies). We can compare categorical variables in scatterplots by using different colors or symbols. The weights of small, medium, and large dogs over time are shown in the scatterplot. Weight of dogs over time small dogs

weight (lb)

60 50

medium dogs

large dogs

30 20 10 0

9 10 11 12

Age (in months)

Different colored dots represent the different categories or sizes of dogs. For each category, there is a strong, positive linear relationship between the dogs’ age and weight. It is important to note that the existence of a relationship between two variables in a scatterplot, regardless of strength, does not necessarily imply that one causes the other. Causation can only be determined from an appropriately designed statistical experiment. 9.02 Scatterplots mathspace.co

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515

Provide a list of guiding questions for student analysis Targeted instructional strategies When asked to describe the relationship between variables represented in a scatterplot, there are several factors that students need to consider. For example, the variables, form, direction, and strength of the relationship between those variables. Provide this list of guiding questions for students to ask themselves during their analysis: 1. What are the independent and dependent variables? 2. Is there a relationship between the variables? If so, it is linear or nonlinear? 3. If the relationship is linear, is it positive or negative? 4. How strong is the relationship? After answering these questions, students must describe the relationship in context. Provide the following sentence frame: The relationship between (independent variable) and (dependent variable) is (strength), (direction if linear), and (form). This means that as (independent variable) increases, (dependent variable) tends to (describe the behavior of the y-values). Consider providing various examples to help students feel comfortable describing various types of relationships in context. • The relationship between age and height is strong, positive, and linear. This means that as age increases, height tends to increase. • The relationship between the number of items produced and the cost of producing those items is moderate and nonlinear. This means that as the number of items produced increases, the cost of production tends to increase at first, then decrease. • The relationship between time and savings is strong and nonlinear. This means that as time increases, savings tends to increase at an increasing rate.

Relationship of data forming vertical and horizontal lines Address student misconceptions Students might think that data forming vertical or horizontal lines have a strong, linear relationship since it forms a line. The strength of the relationship measures how well we can predict the value of one variable given the other. With a vertical line, knowing the value of x does not give a good indication for what the value of y should be since there are y-values along the entire range. Similarly, with horizontal lines, knowing the value of y does not help predict the value of x. We can conclude that there is no relationship between the quantities. 9 8 7 6 5 4 3 2 1

9 8 7 6 5 4 3 2 1

1 2 3 4 5 6 7 8 9

Horizontal trend, no relationship

Vertical trend, no relationship

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Information gap English language learner support Prepare a set of cards where half contain various scatterplots, and the other half have descriptions of the form, strength, and direction of relationships in scatterplots. Half of the class will be given a scatterplot card, and the other half will be given a description card. Inform students that they are not allowed to show each other their cards or use the words found on their cards. The students with the description cards must ask questions to the students with the graph cards and find the person that has the graph that matches their description. To support the discussions, provide example questions like the following: • How tightly clustered are the points in your graph? • Do the points in your graph roughly follow a straight line or a curve? • As the x-values of the points in your graph increase, do the y-values also increase?

Examples Students: Page 516 Example 1 For each scatterplot, determine whether the variables have a linear relationship, a nonlinear relationship, or no relationship. If there is a relationship, describe its strength. If the relationship is linear, describe the direction as positive or negative. a

55 50 45 40 35 30 25 20 15 10 5

x 1

Create a strategy A relationship between two variables exists if the points follow a similar trend. The points will roughly form a line (linear) or a curve (nonlinear) if there is a relationship. To describe the strength of the relationship, we can analyze how tightly the data points are clustered or grouped together.

Apply the idea The y-values are decreasing at a slower and slower rate, causing the point to form a curve. This shows there is a nonlinear relationship between the variables. Because the points are tightly clustered, the relationship is strong.

b 90

Purpose 80 70 Check if students can visually determine whether variables have a relationship, estimate the strength, and 60 identify the direction of the relationship. 50 40 30 20 10

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Create a strategy

together.

Apply the idea The y-values are decreasing at a slower and slower rate, causing the point to form a curve. This shows there is a nonlinear relationship between the variables.

Students:Because Pagesthe516–517 points are tightly clustered, the relationship is strong. b 90

80 70 60 50 40 30 20 10

x 1

8 9

Create a strategy A relationship between two variables exists if the points follow a similar trend. If there is no trend or no shape to the data, then there is no relationship between the variables.

Apply the idea There is no trend in this data, meaning there is no relationship between the variables.

Reflect and check 516 Mathspace Virginiaa SOL We could try to sketch lineAlgebra of fit for1 the data, like the one shown, but the points are far from the line. A negative, linear mathspace.co trend would suggest that y decreases as x increases, which we cannot conclude for this data set.

80 70 60 50 40 30 20 10

x 1

y 50 Purpose 45 40 Check if students can visually determine whether variables have a relationship, estimate the strength, and 35 identify the 30 direction of the relationship. 25 Expected mistakes 20 15 try to draw a line from the top left corner of the scatterplot to the bottom right side of Students might 10 the scatterplot to indicate a weak, negative linear relationship. Remind them that a weak, negative linear 5 x

relationship implies that the y-values decrease as the x-values increase. 2 4 6 8 10 12 14 16 18 Then, ask them if they think the previous statement is valid. Highlight some of the points on the graph, such as (8.2, 92). Help them see that x is relatively large, but y is also large, meaning that the relationship is not Create a strategy necessarily negative. First, we must determine if a relationship between the variables exists. If a relationship exists, we can describe the strength by analyzing how tightly the data points are clustered. If the relationship between the variables is linear, the direction of the relationship can be described as positive or negative. • Positive relationship: as the independent variable increases, the dependent variable increases • Negative relationship: as the independent variable increases, the dependent variable decreases

Apply the idea As the x-values increase, the y-values also increase. This indicates there is a positive, linear relationship between the variables. 9.02 Scatterplots 1095 However, the points are not tightly clustered, so the relationship between the variables is moderate. mathspace.co

40 30 20 10

Students: Page 517 c

50 45 40 35 30 25 20 15 10 5

x 1

x 2 4 6 8 10 12 14 16 18

Create a strategy First, we must determine if a relationship between the variables exists. If a relationship exists, we can describe the strength by analyzing how tightly the data points are clustered. If the relationship between the variables is linear, the direction of the relationship can be described as positive or negative. • Positive relationship: as the independent variable increases, the dependent variable increases • Negative relationship: as the independent variable increases, the dependent variable decreases

Apply the idea As the x-values increase, the y-values also increase. This indicates there is a positive, linear relationship between the variables. However, the points are not tightly clustered, so the relationship between the variables is moderate.

Purpose Check if students can visually determine whether variables have a relationship, estimate the strength, and identify the direction of the relationship.

Outline the points to visualize the relationship Student with disabilities support

use with Example 1 9.02 Scatterplots

517

mathspace.co If students struggle to visualize whether there is a linear relationship, nonlinear relationship, or no relationship in the scatterplot, encourage them to outline the points. Then, have them draw a path down the middle of the outline, from one end to the other. • If the outline has an oval shape, draw a path that connects the two ends of the oval, dividing the length of the oval in half. This indicates a linear relationship. • The thinner the oval, the stronger the relationship.

• If the outline looks like a worm or snake, draw a path through the middle, connecting the two ends of the outline. This indicates a nonlinear relationship. • The thinner the worm, the stronger the relationship. • If the outline is a square or circle, there is no relationship.

1096 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

For the scatterplots in this example, the drawing may look like the ones shown. 55 50 45 40 35 30 25 20 15 10 5

50 45 40 35 30 25 20 15 10 5

x 1

x 2 4 6 8 10 12 14 16 18

The outline looks like a worm, so the path connecting the ends is a curve. 90

The outline is an oval, so the path connecting the ends is a straight line.

For the scatterplot in part (b), the outline would form a rough square shape.

Students might want to connect the corners of the square. If they do, point out that there are multiple lines we could draw (such as a vertical line down the middle or a horizontal line across the middle).

70 60 50 40 30 20 10

x 1

Show them that the points are far from each of the lines, so no line is better than the other. This indicates that there is no relationship between the variables.

8 9

9.02 Scatterplots 1097 mathspace.co

Students: Page 518

Purpose Assess whether students can identify questions that explore the relationship between two variables.

Students: Pages 518–519

1098 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

The data should be separated into two categories: • Patients that take medication only • Patients that take medication and attend physical therapy sessions

Reflect and check Since the data is bivariate and numerical, it can be represented by a scatterplot. The independent variable is time, and the dependent variable is the patients’ pain level. An example scatterplot is shown: The data should be separated into two categories: Pain level changes over time • Patients that take medication only • Patients that take medication and attend physical therapy sessions 8 Group A (Physical therapy and medication)

Reflect and check 6

Group B (Medication only)

Pain level

Since the data is bivariate and numerical, it can be represented by a scatterplot. The independent variable is time, 5 and the dependent variable is the patients’ pain level. An example scatterplot is shown: 4

Pain level changes over time

3 2 8 71

Pain level

0 6 5

Group A (Physical therapy and medication) 0

3 4 5 6 Time (weeks)

Group B (Medication only)

4 3 2

PurposeExample 3 1 Check ifAstudents can identify the datastates needed tothe answer a statistical question. surfing company is 0 located in bivariate various coastal across U.S. When analyzing their data, they separate the 0

store locations into two regions: the Western region and the Eastern region. The scatterplot shows data collected to Time (weeks) Students:answer Pages the 519–520 question, “How have the sales of our product changed over time in each of the sales regions?” Product sales over time 720

Example 3

640

Western region

Sales ($)

560 A surfing company is located in various coastal states across the U.S. When analyzing their data, they separate the Eastern region shows data collected to 480 store locations into two regions: the Western region and the Eastern region. The scatterplot 400the sales of our product changed over time in each of the sales regions?” answer the question, “How have 320 240 720 160 640 80 560 0 480 0 2 400

Product sales over time

Western region 4

8 10 12 14 16 18 20 Time (months)

Eastern region

320

a Identify the independent and dependent variables in this context.

Create a strategy

240 160 80

Recall that the independent variable 0 is not affected by the other variable, while the dependent variable may be 0 2 4 6 8 10 12 14 16 18 20 affected or changed by the other variable. Time (months) On a scatterplot, the independent variable is placed on the horizontal axis, and the dependent variable is placed on the vertical axis. a Identify the independent and dependent variables in this context.

Create a strategy 9.02 Scatterplots Recall that the independent variable is not affected by the other variable, while the dependent variable may be mathspace.co affected or changed by the other variable.

519

On a scatterplot, the independent variable is placed on the horizontal axis, and the dependent variable is placed on the vertical axis.

9.02 Scatterplots mathspace.co

519

9.02 Scatterplots 1099 mathspace.co

Apply the idea The independent variable is time (measured in months), and the dependent variable is the amount of sales (measured in dollars).

b The owner of the company makes this conclusion: “The sales of the product are improving with time.” Which sales region was the owner analyzing?

Purpose Check ifApply students can identify the independent and dependent variables in a given context. the idea Create a strategy

The independent variable is time (measured and thevariable, dependent is the amount ofvariable. sales (measured In part (a), we found that the amount of sales in is months), the dependent and variable time is the independent

Students:in Page 520 dollars). This means the owner concluded that the dependent variable increases as the independent variable increases. Apply the idea

Reflect and check

Apply the idea

Reflect and check

b The owner of the company makes this conclusion: “The sales of the product are improving with time.” Which sales According to the owner’s statement, both variables are If the owner was analyzing the Eastern region (the black region was the owner analyzing? increasing which indicates a positive relationship. Both points), the conclusion would have been, “The sales of sets of data values show a linear relationship, but only the product are decreasing over time.” Create strategy the blue adots show a positive relationship. According to In found the thatblue the amount of sales isdata the from dependent variable, and time is the independent variable. thepart key(a), (orwe legend), points represent This means the owner concluded that the dependent variable increases as the independent variable increases. the Western region.

According If the owner was analyzing the Eastern region (the black Exampleto4the owner’s statement, both variables are increasing which indicates a positive relationship. Both points), the conclusion would have been, “The sales of Apply the idea show a linear relationship, but only sets data values are decreasing Adriaofheard that children who learn to speak at a young agethe areproduct more likely to be giftedover and time.” talented in later stages The independent variable is time (measured in months), the blue dots show a positive relationship. to and the dependent variable is the amount of sales (measured of life. She decides to investigate this usingAccording the data cycle. in dollars). the key (or legend), the blue points represent data from a Formulate a statistical question for Adria that would lead to the collection of data that can be represented in a the Western region. scatterplot. b The owner of the company makes this conclusion: “The sales of the product are improving with time.” Which sales region was the owner analyzing? Create a strategy

Example 4

First, we need to identify the variables of interest. Then, we need to write a question such that the answer to the

strategy both variables. PurposeCreate questionaaddresses Adria heard that children who learn to speak at a young age are more likely to be gifted and talented in later stages part (a), found that theconclusions amount sales is the dependent variable, and time is the independent variable. Check ifIn students can connect and plotted of life. Shewe decides to investigate thisofusing the data cycle.data. Apply the idea This means the owner concluded that the dependent variable increases as the independent variable increases.

a Formulate a statistical question for Adria that would lead to the collection of data that can be represented in a Reflecting with students From the given information, we gather that Adria is interested in two variables: scatterplot. Apply the Reflect and check Extend this problem byaprompting advanced learners to consider other factors that might explain the different 1. The ageidea when child first spoke According to the owner’s statement, both variables are If the owner was analyzing the Eastern region (theinvestigation black sales patterns in each region, such as marketing efforts, weather patterns, or cultural trends. This 2. Their intelligence level later in life Create a strategy increasing which indicates a positive relationship. Both points), the conclusion would have been, “The sales of will alsoFirst, remind them the scatterplot shows a correlation between time and sales, itsuch does not The information not while specific about the stages of life. We can any stage of life after birth, wedata need toisthat identify the variables of later interest. Then, we need to choose write a question such that the answer to as thethe sets of values show a linear relationship, but only the product are decreasing over time.” teenage years. necessarily imply that time is the cause of the change in sales. question addresses variables. the blue dots show aboth positive relationship. According to Onekey possible statistical question is, represent “What is the relationship between the age at which a child first spoke and their the (or legend), the blue points data from

level ofthe intelligence idea Students:Apply Page 520 the Western region. as teenagers?”

From the given information, we gather that Adria is interested in two variables:

Reflect and check

1. The age when a child first spoke Other possible questions are: 2. Their intelligence level later in life Example 4 • How does the age at which a child first spoke influence their level of intelligence as adults? The information is not specific about thewhat later level stages life. We can any stagetoofhave life after birth, such as the • If aheard child that first spoke at who 6 months of of intelligence arechoose they a teenager? Adria children learn old, to speak at a young age are more likelyexpected to be gifted and as talented in later stages teenage years. Which of ages for when this a child firstthe spoke to the highest levels of intelligence? of• life. Sherange decides to investigate using data correspond cycle. One possible question “What is the relationship between age at which a child first spoke and their This could alsostatistical be separated intois, multiple categories: age when a childthe first spoke versus intelligence level after a Formulate a statistical question for Adria that would lead to the collection of data that can be represented in a level of intelligence as teenagers?” middle school, after high school, and after university. scatterplot.

Reflect and check Create a strategy Virginia SOL Algebra 1 520 Mathspace Other possible questions are: mathspace.co First, we need the to identify the variables of interest. Then, we their needlevel to write a question as such that the answer to the • How does age at which a child first spoke influence of intelligence adults? question addresses both • If a child first spoke at variables. 6 months old, what level of intelligence are they expected to have as a teenager? • Which range of ages for when a child first spoke correspond to the highest levels of intelligence?

Apply thealso ideabe separated into multiple categories: age when a child first spoke versus intelligence level after This could From givenafter information, we gather thatuniversity. Adria is interested in two variables: middlethe school, high school, and after 1. The age when a child first spoke 2. Their intelligence level later in life 520

Mathspace

Virginia SOL Algebra 1

mathspace.co The information is not specific about the later stages of life. We can choose any stage of life after birth, such as the teenage years. 1100 Mathspace Virginia SOL Algebra 1 Teacher Edition One possible statistical question is, “What is the relationship between the age at which a child first spoke and their mathspace.co level of intelligence as teenagers?”

Reflect and check

a Formulate a statistical question for Adria that would lead to the collection of data that can be represented in a scatterplot.

Create a strategy First, we need to identify the variables of interest. Then, we need to write a question such that the answer to the question addresses both variables.

Apply the idea From the given information, we gather that Adria is interested in two variables: 1. The age when a child first spoke 2. Their intelligence level later in life The information is not specific about the later stages of life. We can choose any stage of life after birth, such as the teenage years. One possible statistical question is, “What is the relationship between the age at which a child first spoke and their level of intelligence as teenagers?”

Reflect and check Other possible questions are: • How does the age at which a child first spoke influence their level of intelligence as adults? • If a child first spoke at 6 months old, what level of intelligence are they expected to have as a teenager? • Which range of ages for when a child first spoke correspond to the highest levels of intelligence? This could also be separated into multiple categories: age when a child first spoke versus intelligence level after middle school, after high school, and after university.

520

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Purpose Check if students can formulate statistical questions that would lead to the collection of data that can be represented in a scatterplot.

Students: Page 521 b The table shows the ages of some teenagers when they first spoke and their results in an aptitude test: Age when ﬁrst spoke (months) Aptitude test results

14 96

27 69

9 93

16 101

21 87

17 92

10 99

7 104

19 93

24 97

Create a scatterplot to model the data.

Create a strategy Let x represent the age when the child first spoke and y represent the aptitude test results as a teenager. The minimum value for x is 7 and the maximum is 27, so we can use a scale of 5 to label the x-axis. The minimum value for y is 69 and the maximum is 104, so we can use a scale of 20 to label the x-axis.

Apply the idea Aptitude score 100 80 60 40 20 Age (months) 5

c Draw a conclusion about the data by answering the statistical question from part (a).

Purpose Create a strategy Check if students can recognize the independent and dependent variables and create a scatterplot with To describe the relationship between the age at which a child first spoke and their level of intelligence as teenagers, appropriate scale and labels. we can analyze the following features of the data: • Form: linear or nonlinear • Strength: strong or weak If the data follows a linear trend, we can describe the direction as positive or negative.

Apply the idea

Reﬂect and check

The points are relatively close together, indicating a strong relationship. As the age increases, the aptitude score decreases slightly, indicating a negative, linear relationship.

The closer the points are to forming a line or curve, the stronger their relationship will be. A strong relationship between two quantities suggests 9.02 that Scatterplots the value of one quantity can be predicted with some mathspace.co accuracy given the other quantity, but is not enough evidence to suggest that changes in one quantity directly cause changes in the other.

The relationship between the age when a child first spoke and their aptitude test score as a teenager has a strong, negative, linear relationship.

1101

Reﬂecting with students Ask students whether they think the relationship is affected by the scale of the graph. The choice of scale may affect how strong the relationship appears, but it does not affect the actual relationship. The following graphs show the same data as the example, but by eye, a different interpretation of the strength of the relationship may be given. b The table shows the ages of some teenagers when they first spoke and their results in an aptitude test:

Aptitude score

Age when ﬁrst spoke (months) 110 Aptitude test results

Aptitude score 14 96

27 69

9 93

100

Create a scatterplot to model the data.

90 Create a strategy Let x80 represent the age when the child first spoke and y represent the aptitude test results as a teenager. The 70 minimum value for x is 7 and the maximum is 27, so we can use a scale of 5 to labelAge the x-axis. The minimum (months) value for y is 69 and the maximum is 104, so we can use 5 10 15 20 25 a scale of 20 to label the x-axis.

Points appear more widely spread

16 101

21 17 87 160 92

10 99

7 104

19 93

24 97

120

Apply the idea 80 Aptitude score 100 40

Age (months)

Points appear more closely clustered 40

To remove subjectivity, it is important to consider the best scale for20 the data set. For this lesson, students should be aware that an inappropriately chosen scale may distort the strength of the relationship. Age (months)

Students: Page 521

c Draw a conclusion about the data by answering the statistical question from part (a).

Create a strategy To describe the relationship between the age at which a child first spoke and their level of intelligence as teenagers, we can analyze the following features of the data: • Form: linear or nonlinear • Strength: strong or weak If the data follows a linear trend, we can describe the direction as positive or negative.

Apply the idea

Reﬂect and check

The points are relatively close together, indicating a strong relationship. As the age increases, the aptitude score decreases slightly, indicating a negative, linear relationship.

The closer the points are to forming a line or curve, the stronger their relationship will be. A strong relationship between two quantities suggests that the value of one quantity can be predicted with some accuracy given the other quantity, but is not enough evidence to suggest that changes in one quantity directly cause changes in the other.

The relationship between the age when a child first spoke and their aptitude test score as a teenager has a strong, negative, linear relationship. This suggests that as the age at which a child first spoke increases, their intelligence level as a teenager tends to decrease.

Purpose Check if students can derive a conclusion from a scatterplot to answer a statistical question. Reflecting with students Encourage students to use an iterative approach and consider what another cycle of the data cycle could look like to further refine the question and conclusion. 9.02 Scatterplots mathspace.co

1102 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

521

Creating scatterplots with technology

use with Example 4

Student with disabilities support For students who struggle to draw scatterplots by hand, allow them to create scatterplots with technology. To create a scatterplot using the Desmos Graphing calculator, click the plus sign in the top left corner and add a table. Then, enter the x-values into the first column and enter the y-values in the second column of the table

Students: Page 522

Idea summary The analysis of bivariate data should include: • •

Form, usually described as a linear relationship or a nonlinear relationship Strength, describing how closely the data points match the model line or curve

If the relationship between the variables is linear, the direction of the relationship can be described as positive or negative. • •

Positive relationship: as the independent variable increases, the dependent variable increases Negative relationship: as the independent variable increases, the dependent variable decreases

Practice What do you remember? 1

Create a scatterplot that models each set of data. Include labels and scales on each axis. a

The heights and weights of the female Olympic “All around champions” in gymnastics.

Suni Lee Simone Biles Gabby Douglas Nastia Liukin Carly Patterson Simona Amanar b

Height (inches) 60 57 59 62 59 62

Weight (lbs) 112 104 90 99 97 97

The test scores on the midterm and ﬁnal exam for a sample of students.

9.02 Scatterplots 1103 mathspace.co

Practice Students: Pages 522–528

What do you remember? 1

Create a scatterplot that models each set of data. Include labels and scales on each axis. a

The heights and weights of the female Olympic “All around champions” in gymnastics.

Suni Lee Simone Biles Gabby Douglas Nastia Liukin Carly Patterson Simona Amanar b

Weight (lbs) 112 104 90 99 97 97

The test scores on the midterm and ﬁnal exam for a sample of students. Midterm Exam Final Exam

Height (inches) 60 57 59 62 59 62

95 90

90 92

88 83

84 80

75 62

77 80

65 60

70 74

99 100

85 85

Scientists conducted a study where each person was asked to read a paragraph then recount as much information as they could remember. They found that the longer the paragraph, the less information each person could retain. If the length of the paragraph were plotted (on the horizontal axis) against the amount of information retained (on the vertical axis), would the relationship be positive or negative?

For each pair, identify the independent and dependent variable: a

Amount of fertilizer and plant height

Length of stride and height

Number of family members and time (in hours) spent cooking

Time spent traveling and distance to destination

Time spent practicing and performance in piano lessons

Which question would lead to data that could be represented by a scatterplot? A

On which days of the week do most teenagers play videos?

How many hours does an average teenager spend playing video games each day?

What is the difference in the number of hours teenagers play video games?

How does the amount of time spent outside impact the amount of time spent playing video games?

1104 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Does each scatterplot show a linear or nonlinear relationship? a

5 x 5

9.02 Scatterplots 1105 mathspace.co

Let’s practice 6

Describe the strength and direction for each linear relationship. a

5 x 5

Four different classes with four different professors had the same ﬁnal exam. The exam results and number of classes attended by each student is displayed for each class. Does each graph suggest that there is a relationship between number of classes attended and ﬁnal exam grade? a

b 100

Final exam grade

100

x 0

Number of class attended

1106 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

x 0

Number of class attended

d 100

Final exam grade

100

x 10

Number of class attended

This scatterplot shows the relationship between air and sea temperature. a

Which is the best description of the relationship between the variables?

35 Sea temperature

A Strong, positive, linear B Moderate, negative, linear C Weak, nonlinear b

Describe the relationship between the variables in context.

30 25

Air temperature

This table shows the scores of 12 students in math and P.E. class. a

Select the question that could be answered by the data. A Do students prefer Math or P.E.? B If a student does well in Math, do they also do well in P.E.? C Does a student’s Math grade impact their P.E. grade? D How many students are enrolled in Math and P.E.?

Construct a scatterplot for the students’ scores in math versus their scores in P.E. class.

Is the relationship between students’ grades in math and P.E. linear or nonlinear? If it is linear, describe the direction as positive or negative.

Describe the strength of the relationship between students’ grades in math and P.E.

Student 1 2 3 4 5 6 7 8 9 10 11 12

Maths 63 82 60 79 88 81 61 91 72 62 66 92

P.E. 44 94 52 70 67 60 73 86 84 93 57 92

A shop owner in Morocco collected data to answer the statistical question, “How does the temperature outside impact the number of fans sold?” The data is shown in the table: Temperature (°C) Number of fans sold

6 12

8 13

10 14

12 17

14 18

Identify the independent and dependent variables.

Which method was most likely used to collect the data? A Measurement

Observation

16 19

18 21

20 23

Survey

Construct a scatterplot using the data from the table.

Describe the relationship between the temperature and the number of fans sold.

Experiment

9.02 Scatterplots 1107 mathspace.co

Data was collected to answer the statistical question, “What is the relationship between the age at which a child first walked and the age at which they first spoke?” The data collected is shown in the table: Age first walked Age first spoke

12 12

18 21

15 16

13 20

17 19

9 13

16 22

11 15

20 20

Construct a scatterplot for the data.

What is the relationship between the age at which a child first walked and the age at which they first spoke?

A researcher is studying the relationship between the number of passers-by in an emergency, and the time taken (in seconds) before a passer-by helps a stranger during an emergency. The data is recorded in this table. 1 8

2 19

3 26

4 37

5 51

6 65

Was the data most likely collected through measurement, observation, a survey or an experiment?

Construct a scatterplot using the data from the table.

Describe the relationship between the number of passers-by and the time until assistance is offered.

As more passers-by are present, what happens to the time taken until help is offered?

Each point on the scatterplot shows the time (in weeks) Sumon spent training for a half marathon and the corresponding miles they were able to run. Using the scatterplot, are these statements true or false? a

The number of weeks that Sumon trained for the half marathon is the independent variable.

The y-coordinates of the points represent the time spent by Sumon training.

There is evidence to suggest that the longer Sumon trains, the further they can run.

14 17

Number of passers-by (n) Time until help is offered (t)

10 12

The relationship between the number of weeks training and the number of miles Sumon is able to run is positive.

As preparation for a science test, a group of 10 students was given a practice worksheet containing 60 questions. The table shows the number of questions from the worksheet successfully completed by each student and the score out of 100 of that student on the test. a

Formulate a question that could be answered by the data.

Was the data most likely collected through measurement, observation, a survey or an experiment?

Which variable is independent and which variable is dependent?

Construct a scatterplot of the data.

Draw a conclusion about the data by answering the statistical question from part (a).

1108 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Running distance (mi) 20 15 10 5

Number of questions 11 14 36 60 57 42 20 27 50 59

No. of weeks 15 20

Test result 20 23 62 97 100 66 35 52 87 99

A chemical company is testing the effect of different chemicals on slowing the melting of snow. They are currently testing four different chemicals with the results showing how much ice is remaining after 3 hours at a certain temperature. A

B 100 Snow remaining (%)

Snow remaining (%)

100

x 0

Temperature outside (°C)

D 100 Snow remaining (%)

100

Snow remaining (%)

Temperature outside (°C)

x 0

Temperature outside (°C)

A snow sculpture company wants a chemical that will keep snow from melting. Which chemical should they choose to ensure that at least half of the snow is still there after 3 hours?

If the company operates at a temperature of 3 °C to 4 °C, which chemical should they choose?

Mona has a checking account and a savings account with her bank. Her savings accout accrues interest from the bank, and she tries to deposit and withdraw from both accounts equally. She has been tracking the balance of each account over the past year.

Balance ($)

1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 300 200 100 0

Checking account Savings account

Formulate a question that could be answered by the scatterplot.

Describe a method that Mona could have used to collect the data.

Identify the independent and dependent variables.

Which account had a higher balance at the beginning of the year?

Which account had a higher balance at the end of the year?

Draw a conclusion about the data by answering the statistical question from part (a).

0 1 2 3 4 5 6 7 8 9 10 11 12 Time (months)

9.02 Scatterplots 1109 mathspace.co

Let’s extend our thinking 17

Determine whether each statement is true or false. Provide an example to support your claim. a

If the value of variable A increases as the value of variable B increases, there is a relationship between variable A and variable B.

If there is a relationship between variable A and variable B, then changes in variable A directly cause changes in variable B.

If variable A and variable B move in opposite directions (as one increases, the other decreases), then they have a negative relationship.

Consider the scatterplot: a

Explain why the relationship between the variables is weak.

Determine whether the following pairs of variables could be represented by the data set:

Scores in an English test and distance traveled from home to school.

Cost of cars and cost of gasoline.

iii Distance traveled in a car and the cost of a driver’s license. x

Brody wants to take his dog on a hike, then stop to pick up groceries on the way home from the hike. However, he is worried about leaving his dog in the car for half an hour while he is inside the store because he has heard it is unsafe. Temperature over time in a closed car Brody wants to use the data cycle to investigate the temperature inside the car over time. a

Formulate a question which could be investigated using a scatterplot.

Determine what variables could be used to answer your investigative question.

The current temperatures where Brody lives are between 70–80 °F. He acquired data on the temperature inside a car on a 70 °F day and an 80 °F day, shown in the table. Plot each set of data on the same scatterplot.

Draw a conclusion about the data by answering the statistical question from part (a).

Brody learns that at 103 °F, dogs lose their ability to regulate their body temperature. Determine an approximate range of time that it takes for the inside of a car to reach 103°F when outside temperatures are between 70°F and 80°F.

Time in minutes 0 5 10 15 20 25 30 35 40 45 50 55 60

Temperature in car on 70°F day 70°F 83°F 89°F 94°F 99°F 102°F 104°F 106°F 108°F 110°F 111°F 112°F 113°F

Temperature in car on 80°F day 80°F 94°F 99°F 105°F 109°F 111°F 114°F 117°F 118°F 119°F 121°F 122°F 123°F

When Sherrie was trying on shoes, the sales attendant told her to always try on both shoes because, for most people, one foot is longer than the other. Go through the whole data cycle at least once to investigate the relationship between the lengths of a person’s feet.

1110

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Answers

5 a Linear relationship

9.02 Scatterplots What do you remember? 1 a T he scale on the axes should count by equal amounts. The values on the x-axis must begin at or below the minimum value of 57 and end at or above the maximum value of 62. The values on the y-axis must begin at or below the minimum of 90 and end at or above the maximum of 112. The chosen scale should display the data clearly.

For example: 120

4 a D

d Nonlinear relationship

e Linear relationship

Linear relationship

Let’s practice 6 a Positive and strong b Positive and weak c Positive and moderate-weak d Negative and strong 7 a No

b Yes

c Yes

d Yes

8 a A

Weight (lbs)

b A s air temperature increases, sea temperature increases.

115 110 105

9 a C

100

P.E. 90

85 80

Height (inches) 65

b T he scale on the axes should count by equal amounts. The values on the x-axis must begin at or below the minimum value of 65 and end at or above the maximum value of 99. The values on the y-axis must begin at or below the minimum of 60 and end at or above the maximum of 100. The chosen scale should display the data clearly.

For example:

Final Exam

75 65 55 Midterm Exam 65

Math 40 50 60 70 80 90

c Positive, linear

d Weak

10 a T he independent variable is the temperature, and the dependent variable is the number of fans sold. c

b B

2 Negative 3 a Independent variable: amount of fertilizer

b Nonlinear relationship

c Nonlinear relationship

Dependent variable: plant height

24 22 20 18 16 14 12 10 8 6 4 2

Number of fans sold

Temperature (°C) 2 4 6 8 10 12 14 16 18 20 22 24

d S trong, positive, linear. As the temperature increased, the number of fans sold increased.

b Independent variable: height

Dependent variable: length of stride

c Independent variable: number of family members

Dependent variable: time (in hours) spent cooking

d Independent variable: distance to destination

Dependent variable: time spent traveling

e Independent variable: time spent practicing

Dependent variable: performance in piano lessons

Answers mathspace.co

1111

11 a

24 22 20 18 16 14 12 10 8 6 4 2

16 a M any possible answers, for example, “How does the balance of each account change over time?”

Age first spoke

b A bank issues a statement each month that provides the transactions and ending balance of the account. Mona could have acquired this data from those statements. c T he independent variable is time in months, and the dependent variable is the balance of each account in dollars. Age first walked

d Checking

2 4 6 8 10 12 14 16 18 20 22 24

e Savings

b T here is a moderate, positive, linear relationship between the age at which a child first walked and the age at which they first spoke. This means that as the age a child first walked increased, the age at which they first spoke also increased. 12 a Observation b 70

17 a T rue. For example, if as the temperature increases, the sale of ice cream also increases, then there is a positive relationship between temperature and ice cream sales. This does not imply causation but indicates a relationship where both variables move in the same direction.

50 40 30 20 10

n 1

c Strong, positive, linear 13 a True

b False

d Increases c True

d True

14 a M any possible answers, for example, “As the number of questions completed increases, how does a students’ test result change?” b Survey c T he independent variable is the number of questions and the dependent variable is the test result. d 100 Test result

c T rue. For example, as the amount of rainfall increases, the amount of time spent outdoors may decrease, indicating a negative relationship between rainfall and time spent outdoors. This shows an inverse relationship but does not imply one causes the other. 18 a I t appears that y can change signiﬁcantly without much corresponding change in x. ii Yes

iii Yes

19 a M any possible answers, for example, “How does the temperature inside a car change over time?” b A nswers will vary based on the question in part (a). Using the example question, “How does the temperature inside a car change over time?”:

Number of questions

e T here is a strong, positive, linear relationship between the variables which shows that as the number of completed questions increases, the students’ test score also increased.

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b F alse. For example, if there is a relationship between the number of ice cream sales and the number of sunglasses sold, that does not mean increasing ice cream sales directly cause an increase in sunglasses sales. There could be a third factor, such as warm weather, influencing both.

b i Yes

10 15 20 25 30 35 40 45 50 55

15 a D

oth accounts are growing over time. The checking B account increases at a linear rate, while the savings account increases at a nonlinear rate. The savings account is growing faster than the checking account over time.

Let’s extend our thinking

95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15

b B

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The independent variable could be time in minutes, and the dependent variable could be the temperature (in degrees Fahrenheit) inside a closed car. There could be categorical variables, such as the specific temperature outside of the car.

3. Create a data display using a scatterplot.

Temperature in a closed car over time

This scatterplot represents the sample data from the previous part.

Temperature (°F)

130 120 110

100

Left foot (cm)

5 10 15 20 25 30 35 40 45 50 55 60

Time (minutes)

80°F day

70°F day

d T here is a strong, nonlinear relationship between the time and the temperature of a closed car. The temperature increases fastest during the first 10 minutes, then the temperature increases at a slower rate. The temperature increases in the same pattern regardless of the temperature outside. The warmer it is outside, the warmer it is inside the car, and the temperature inside the car is always hotter than the temperature outside the car. e I t would take about 13–27 minutes for the car to reach 103 °F when outside temperatures are between 70 °F and 80°F. 20 1. Formulate questions

Possible questions:

• What is the relationship between the length of a person’s left and right feet?

• How does the length of the right foot differ from the length of the left foot?

• Which foot tends to be longer?

Right foot (cm) 23 24 25 26 27 28 29

4. Analyze and explain the results. From the scatterplot, we can observe that there is a strong, positive linear relationship between the length of a person’s left and right feet. The relationship shows that as the length of one foot increases, the length of the other foot also increases. However, the relationship between the lengths of each foot is not 1 : 1. Instead, the scatterplot shows that the left foot tends to be slightly longer than the right foot. This could lead us to formulate a new question like “How does the relationship between a person’s foot length compare between people who are right-hand dominant versus left hand dominant?” or “How does the difference in people’s feet lengths affect how shoes are made?”

2. Collect data using the questions above. We can collect data by measuring a person’s left foot and right foot in centimeters. The table shows an example sample of 30 students. Right foot 22.9 23.1 23.2 23.8 23.9 24.3 24.5 24.6 25.1 25.2 (cm) Left foot 23.4 24.4 23.9 24.3 24.3 25.3 24.5 25.2 25.1 25.7 (cm) Right foot 25.3 25.3 25.4 25.2 25.6 25.6 25.9 25.9 (cm)

26.1

Left foot 25.2 25.6 25.3 25.5 25.8 26.6 25.6 26.9 26.7 26.8 (cm) Right foot 26.2 26.3 26.4 26.8 26.8 27.2 27.8 (cm) Left foot (cm)

27.3 26.6 26.4 26.8

28.4 28.6

27.2 28.2 28.2 29.3 28.4

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9.03 Linear regression Subtopic overview Lesson narrative In this lesson, students will develop their knowledge of fitted functions from the previous lesson as they learn how to fit functions to data using technology, and use the correlation coefficient to assess the strength of the model’s fit to the data. By the end of the lesson, students will be able to find linear regression models for a set of data, interpret the coefficient of determination to determine the strength of the fit, interpret how the model relates to the context, and use the model to make predictions.

Learning objectives Students: Page 529

Key vocabulary 

extrapolation



interpolation



line of best fit

Essential understanding The relationship between the variables in a set of bivariate data reveals the type of function that best models the data.

Standards This subtopic addresses the following Virginia Standards of Learning for Mathematics standards.

Mathematical process goals MPG1 — Mathematical Problem Solving

MPG4 — Mathematical Connections

Teachers can challenge students to solve real-world problems using scatterplots and linear regression. For example, they can provide a dataset related to a realworld scenario and ask students to use their knowledge of scatterplots, line of best fit, and linear regression to analyze the data and solve the problem.

Teachers can help students make mathematical connections by relating the concept of scatterplots and linear regression to other mathematical topics they have learned. For example, they can discuss how the slope and y-intercept in the linear regression model relate to the concepts they learned in the linear functions unit.

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MPG5 — Mathematical Representations Teachers can encourage students to use various representations to express their understanding of linear regression. For instance, they can ask students to represent the relationship between two variables in a scatterplot visually, then use a line of best fit to model this relationship. They can also ask students to express this relationship algebraically by writing the equation of the line of best fit. It’s also important to show how these representations relate to real-world contexts, such as predicting outcomes based on the linear model.

A.ST.1e — Use linear and quadratic regression methods available through technology to write a linear or quadratic function that represents the data where appropriate and describe the strengths and weaknesses of the model.

A.ST.1a — Formulate investigative questions that require the collection or acquisition of bivariate data.

A.ST.1f — Use a linear model to predict outcomes and evaluate the strength and validity of these predictions, including through the use of technology.

A.ST.1b — Determine what variables could be used to explain a given contextual problem or situation or answer investigative questions.

A.ST.1g — Investigate and explain the meaning of the rate of change (slope) and y-intercept (constant term) of a linear model in context.

A.ST.1c — Determine an appropriate method to collect a representative sample, which could include a simple random sample, to answer an investigative question.

A.ST.1h — Analyze relationships between two quantitative variables revealed in a scatterplot.

A.ST.1i — Make conclusions based on the analysis of a set of bivariate data and communicate the results.

Prior connections 8.PS.3 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on scatterplots. A.F.1 — The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.

Future connections A2.ST.2 — The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on representing bivariate data in scatterplots and determining the curve of best fit using linear, quadratic, exponential, or a combination of these functions.

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Engage Activity Nutrition facts

60 mins

Students will create a nutrition fact label by using the line of ﬁt and the context of the data for calories and macronutrients.

Understanding and skills

Will use Writing and interpreting linear functions from a graph.

Will develop Solving real-world problems using the line of ﬁt and the context of the data. Fitting a linear function to a data set displayed in a scatterplot that suggests a linear association with and without technology. Interpreting key features of the line of ﬁt in context, such as slope and y-intercept.

Preparation and materials • Open and complete the student preview, anticipating classroom responses. • Materials: Paper, pencil • Download and print copies of the students graphic organizer from the student Launch slide.

Support students with disabilities Support conceptual processing - make generalizations Provide students a list of potential macronutrient values to test against the scatterplot. Choose the most reasonable value for carbohydrates (carbs), fats, and proteins from the table provided. Choices do not need to come from the same row. For the 1000 calorie item:

For the 600 calorie item: Carbs 10 20 30 40

Fats 40 50 60 70

Protein 30 35 40 45

Carbs 55 65 75 85

Fats 100 125 150 175

Protein 40 60 80 100

Once students have chosen their values, encourage them to justify mathematically and reﬁne their estimations.

Support for English language learners Compare and connect Invite students to share their representations for estimating the protein content in both the 600 calorie menu item and the 1000 calorie item. With their partners, ask students to ﬁnd the similarities and differences in the strategies used to estimate the macronutrient contents of each item. Consider asking: “For which item does the strategy appear to be more accurate? What other strategies might help with the other menu item?” Listen for and amplify observations which include the advantages and disadvantages to different approaches, such as drawing an extending a line for the 1000 calorie item.

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Classroom guide Hook

Co-craft questions

•

5 mins

Students create questions about a nutrition facts label.

Implementation details The objective of the Explore is to create a nutrition fact label. The Hook allows students to interact with a label and ask questions related to the categories and speciﬁc values displayed. Some questions may include: • What is the ratio of fat to protein in the chicken nuggets? • What is the conversion factor between kJ and calories? • What is a healthy ratio of saturated fat to calories? • Are the chicken nuggets a nutrient dense food? How can we interpret the given nutritional values to determine that?

What mathematical questions could we ask about this? Nutrition Facts

Calories 275

(1149 kJ)

Chicken Nuggets (6 piece) % Daily Value1 Total Fat Sat. Fat

17.3 g 3g

27% 15%

Cholesterol

42 mg

14%

Sodium

597 mg

25%

Total Carbs. Dietary Fiber Sugar

16.1 g 0g 0g

5% 0%

Protein

14.3 g

Calcium

11.9 mg

Potassium

0 mg

Slide 1 from Student Engage Activity

Launch Give students time to explore the scatterplots in the applet and familiarize themselves with the context terms unique to this problem. Engage students in a discussion on macronutrients and calories as needed.

5 mins

Consider the following applet:

800

Important mathematical concepts: Line of ﬁt, estimate, linear function, scatterplot, slope, intercepts

600

Important contextual information: Macronutrient, carbohydrates, fats, proteins, calories

400

Suggested grouping: Form pairs

200

Carbohydrates

40 Protein

50 Fat

Slide 2 from Student Engage Activity

Continue when Students have read the Launch and understand the context of the problem.

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Explore

Think-pair-share

•

35 mins

Anticipated strategies Draw a line of ﬁt on the graph Students may draw a line on the graph and estimate the values for carbohydrates, protein, and fat for a 600 and 1000 calorie item. Use prompts to encourage students to recognize informally strong correlation and weak correlation: • Which macronutrient was easiest to work with? Why? • Which macronutrient do you think might have the most error in the estimation? Why? • Do you think other pairs could have different answers? Then, push students to formalize their process and use mathematics as a tool to explain their reasoning: • How did you come up with this value? How would you convince someone else your solution is valid? • How can you use mathematics to support your reasoning and solution?

Write an equation for line on graph Students may write equation for line drawn on graph or line that best models data and use equation to ﬁnd the values for carbohydrates, protein, and fat for a 600 and 1000 calorie item.

Visually estimate Students may visually estimate the values for carbohydrates, protein, and fat for a 600 and 1000 calorie item. Students who have successfully completed this activity will submit a nutrition facts label representing the carbohydrates, protein, and fat in the 600 calorie and 1000 calorie menu item. Answers will vary signiﬁcantly. The values below are from the line of best ﬁt for each. 600 calorie item: • Carbohydrates: 38g • Protein: 36g • Fat: 71g 1000 calorie item: • Carbohydrates: 84g • Protein: 73g • Fat: 143g The accompanying descriptions will include an analysis of the meaning for both the slope and y-intercept for each macronutrient.

Misconceptions Thinking the y-intercept needs to pass through the origin for line of best ﬁt What is the y-intercept of the line you drew on the graph? What does this mean in context? How do you know?

Purposeful questions Use the following questions to check for understanding and encourage critical thinking: • What are the macronutrient values for your 600 calorie/1000 item? • Can you use the points on the graph to make a prediction about the macronutrients in your 600 calorie/1000 item? • How is a line of best ﬁt useful for determining each of the macronutrient values in your item?

Continue when Students have created and submitted two nutrition facts labels.

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Discuss

15 mins

Pairs of students will present to the whole class before facilitating a class discussion. Consider sequencing the strategies presented from visually estimating, to drawing a line of best ﬁt, to writing an equation for the line.

Discussion guide Select several pairs to present their nutrition facts labels. You should choose groups who made a wide range of predictions on the macronutrient values for each menu item, including one group who underestimated, one group who overestimated, and 2 − 3 groups who had estimates around the same values as one another. As each pair shares their nutrition label, focus the discussion around how each scatterplot was used to inform their estimation. If any pairs used a line of best ﬁt, have them present last. Questions to ask pairs as they present: • What strategy did you use to estimate the macronutrients for the 600 calorie menu item? Did you use the same strategy for the 1000 calorie menu item? • What do the scatterplots tell us about the nutrition of the menu overall? Were there any trends in the data that you noticed? After having pairs present you may provide the whole class an opportunity to revisit their work and make a revised estimation for each of the macronutrient values after seeing presentations on how different groups produced their estimates. You may have students reﬂect on the following prompt: If you were to estimate the macronutrient values for a 100 calorie menu item, what method would you use to determine the macronutrient values? Is this the same method you used in the Explore? Why or why not? If time permits students can apply their method to solve for the macronutrients in the 100 calorie item.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Grade 8 — 4.05 Scatterplots and lines of best fit Algebra 1 — 3.01 Slope Algebra 1 — 3.03 Slope-intercept form Algebra 1 — 9.02 Scatterplots

Tools You may ﬁnd these tools helpful: • Graphing calculator • Graph paper

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Student lesson & teacher guide Linear regression Students will start by engaging in an exploration to determine the relationship between variables and the line of best fit.

Students: Page 529

Collect and display English language learner support As students are working, note how students describe the concepts of “regression”, “interpolation” and “extrapolation”. Collect the different ways that students find to understand these concepts and display them in a common place for the students to access. If students do not come up with alternative ways to describe these concepts and are confused by them, suggest some of your own. For example: • Regression • Creating a model to estimate the relationship between variables • Finding the line of best fit for data points • Predicting future values based on past data • Interpolation • Predicting points between other known points • A way of using the trend of the actual data to make a reliable prediction • Using an x-value that is between the other x-values to make a prediction for y • Extrapolation • Predicting a point that is to the left or right of all other data points • A less reliable form of prediction because we do not know if the trend of the actual data continues • Using an x-value that is outside the domain of the other x-values to make a prediction for y Take care to address any rewordings that contradict or are too similar to other concepts that the student will learn in the future, such as ensuring students know that it is not limited to linear relationships.

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Interpreting mathematics in context Student with disabilities support Provide some tips and sentence frames to support with interpreting the meaning of the slope and y-intercept. Slope: (y is the dependent variable, x is the independent variable) • The slope is the • Use the units to help, they will be “unit of y” per “unit of x” • Sentence frame: The slope means that for each increase by 1 in (insert independent variable), the (insert dependent variable) (increases/decreases) by (insert rate of change). y-intercept: ( y is the dependent variable) • y-intercept occurs when x = 0 • Use the units of y to help • Sentence frame: The y-intercept means that the initial value is (insert y-intercept). • Sentence frame: The y-intercept means that when (insert independent variable) is 0, the (insert dependent variable) is (insert y-intercept).

Reliability of predictions Address student misconceptions Students might assume that any prediction made with the line of best fit is reliable. Remind them that when doing pattern analysis the reliability of the predictions decrease if: • The relationship between the variables is not strong (the points are not tightly clustered around the line), or • The prediction was made using extrapolation. To help students understand why these two factors decrease the reliability of the predictions, use specific examples that show the difference between actual data values and predicted data values. 24 22 20 18 16 14 12 10 8 6 4 2

Age first spoke

Age first walked

For example, this graph shows data collected on the age (in months) a child first walked and the age they first spoke (in months). If we use the line of best fit to predict the age a child will speak if they start walking at 13 months, we might expect them to start talking at 16 months old. However, according to the actual data value, a child that talked at 13 months started talking at 20 months. There is a 4-month difference between the predicted age and the actual age.

2 4 6 8 10 12 14 16 18 20 22 24

24 22 20 18 16 14 12 10 8 6 4 2

In addition, explain to students that when we make predictions outside the range of known data values, we are assuming that the trend of the data will continue. However, the trend may not necessarily continue in the same way, as shown in this example.

Age first spoke

Age first walked 2 4 6 8 10 12 14 16 18 20 22 24

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Exploration Students: Page 529

Suggested student grouping: In pairs Students are asked to determine if there is a relationship between the variables shown in a scatterplot and to select the line of best fit. The aim is to connect the concept of correlation from the previous lesson with identifying a line that appears to best fit the trend of the data. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Is there a relationship between the years since purchased and the value in thousands of dollars? Explain. Yes, there is a strong, negative linear trend between the value of the car and the time since the car was purchased. That is, as the time since purchase increases, the value of the car decreases. 2. Which of the lines on the graph is the line of best ﬁt? The dashed blue line better represents the line of best fit. The blue line has a balance in the number of points above and below the line. Purposeful questions • How do we know when there is a relationship between two variables in a scatterplot? • Can you relate the relationship observed to the context in real-life? • Which line would generally give closer predictions for the value of a car given its age? • Would you consider predictions made from the line reliable? Possible misunderstandings • Students may come up with different criteria for what determines the line of best fit, such as passing through the most data points or joining the first and last data points. Remind them that in 8th grade, the goal was to draw a line with a similar number of points above and below the line.

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Following the exploration, students are shown that the line of best fit is best approximated by technology. They will learn how to interpret the key features of a line of best fit, including the slope and y-intercept, and use it to predict values not represented in the data. Students are also introduced to the terms interploation and extrapolation to help them evaluate the validity and strength of their predictions.

Students: Pages 530–531 A line of best fit (or trend line) is a straight line that best represents the data on a scatterplot. We can use lines of best fit to help us make predictions or conclusions about the data. We previously approximated a line of best fit by trying to balance the number of points above the line with the number of points below the line. This can result in multiple different models. Height (cm)

14 12

8 y = 1.4x + 1.6

6 4

4 2

Weekly growth 2

y=x+3

2 1

8 9

3 points above, 3 points below 14

Height (cm)

Weekly growth 6 7 8 9

5 points above, 4 points below We get a more accurate line of best fit when we use technology, referred to as linear regression analysis.

Height (cm)

Once we have found the line of best fit for a scatterplot, we can interpret the key features and use the line to predict values that don’t appear in the data set.

10 8 y = 1.21x + 2.14

6 4 2

Weekly growth 1

8 9

In the context of a line of best fit, the slope-intercept form represents

y = mx + b m

the rate of change for y with respect to x

the starting value of y when x is 0

For example, this graph models a plant’s growth over several weeks. 14

The slope of the line y = 1.21x + 2.14 means that the plant is growing at a rate of 1.21 centimeters per week.

Height (cm)

The y-intercept of 2.14 means the plant was 2.14 centimeters tall at week 0. This is feasible if the plant was not a seed when measurements began.

12 10 8 y = 1.21x + 2.14

6 4 2 1

Weekly growth 6 7 8 9

These terms describe the range in which we make predictions: • Interpolation: Prediction within the range of x-values in the data • Extrapolation: Prediction outside the range of x-values in the data

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8 y = 1.21x + 2.14

6 4 2

Weekly growth 1

8 9

Interpolations

Extrapolations

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Using the previous example of the plant height over time: 14

Interpolating which week the plant was 9 centimeters tall, we will solve 9 = 1.21x + 2.14. The plant was 9 centimeters tall at 5.67 weeks.

Height (cm)

Extrapolating the plant’s height at 10 weeks, we will evaluate y = 1.21(10) + 2.14. The plant will be 14.24 centimeters tall at 10 weeks.

10 8 y = 1.21x + 2.14

6 4 2

Weekly growth 1

8 9

The reliability of predictions depends on the strength of the relationship, whether the data is interpolated or extrapolated, and the number of points in the data set. • A larger sample size increases reliability. • Interpolation with a strong correlation implies a reliable prediction. • Interpolation with a moderate or weak correlation leads to a less reliable prediction. • Extrapolation generally leads to an unreliable prediction. The further outside the range of known values, the less reliable it is.

Example 1 Natalia collected data to answer the question, “What is the relationship between the years since purchasing a car and Examples its value?” Her data is shown in the table.

The followingTime support be useful the examples since may purchase (years) for 0.5 0.8 1.2 in this 1.3 section. 1.5 1.7 Value (thousands of dollars)

28.5 28.5

27.4

28.5

2.1 1.8 25.9 25.9

Calculator Desmos graphing calculators Time since steps purchasefor (years) 2.6 and 2.8 TI 3.1 3.4 3.6 3.9 4.05 Targeted instructional strategies Value (thousands of dollars) 24.6 23.5 24.6 23.3

4.6 21

2 24.7

2.5 26.4

4.8 20.1

The examples students with the fit. steps to finding the line of best fit using the Desmos graphing a Find provide the equation of the line of best calculator. Steps for finding the line of best fit on TI-83 or TI-84 are shown. TI-83 orCreate TI-84: a strategy

To find the equation using technology, we can follow these steps:

1. Select the STAT button.

1. Click the plus sign in the top left corner of the screen, and select table.

2. Select 2. EDIT. Enter the x-values and y-values in the respective columns of the table. the enter table. 3. Enter3.the under L In ax-values new line beneath the table, the equation y1 ∼ mx1 + b. 1 in 4. Enter the y-values under L2 in the table. 5. Press the STAT button again. 6. Press the right arrow to select CALC at the top of the screen. 7. Select LinReg(ax+b).

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531

Using the previous example of the plant height over time: Interpolating which week the plant was 9 centimeters tall, we will solve 9 = 1.21x + 2.14. The plant was 9 centimeters tall at 5.67 weeks.

Height (cm)

14 buttons in this order: 8. Press these 12 button in the top left) • 2ND (blue • 1 (has L110in blue above it) 8 • , (the comma above 7) • 2ND 6 y = 1.21x + 2.14 • 2 (has L42 in blue above it)

Extrapolating the plant’s height at 10 weeks, we will evaluate y = 1.21(10) + 2.14. The plant will be 14.24 centimeters tall at 10 weeks.

2 should show LinReg(ax+b)(L ,L ) Your screen 1 2 Weekly growth

9. Select ENTER.1

8 9

The reliability of predictions depends on the strength of the relationship, whether the data is interpolated or These steps will produce the values of a and b. The value of a is the slope of the line, and the value of b is the extrapolated, and the number of points in the data set. y-intercept. • A larger sample size increases reliability. • Interpolation with a strong correlation implies a reliable prediction. • Interpolation with a moderate or weak correlation leads to a less reliable prediction. Extrapolation generally leads to an unreliable prediction. The further outside the range of known values, the less Students:•Pages 531–533 reliable it is.

Example 1 Natalia collected data to answer the question, “What is the relationship between the years since purchasing a car and its value?” Her data is shown in the table. Time since purchase (years) Value (thousands of dollars)

0.5 29

0.8 1.2 28.5 28.5

1.3 27.4

1.5 28.5

1.7 27

1.8 2.1 25.9 25.9

2 24.7

Time since purchase (years) Value (thousands of dollars)

2.6 24.6

2.8 23.5

3.4 23.3

3.6 21

3.9 21

4.05 22

4.8 20.1

3.1 24.6

4.6 21

2.5 26.4

a Find the equation of the line of best fit.

Create a strategy To find the equation using technology, we can follow these steps: 1. Click the plus sign in the top left corner of the screen, and select table. 2. Enter the x-values and y-values in the respective columns of the table. 3. In a new line beneath the table, enter the equation y1 ∼ mx1 + b.

Apply the idea 1. Click the plus sign in the top left corner of the screen, and select table. 9.03 Linear regression mathspace.co

531

2. Enter the x-values and y-values in the respective columns of the table.

9.03 Linear regression 1125 mathspace.co 3. In a new line beneath the table, enter the equation y1 ∼ mx1 + b.

2. Enter the x-values and y-values in the respective columns of the table.

3. In a new line beneath the table, enter the equation y1 ∼ mx1 + b.

The parameters of the equation are m = −2.19507 (which is the coefficient of x) and b = 30.4638 (which is the constant, or y-intercept). If we round the parameters to two decimal places, the equation of the line of best fit is y = −2.2x + 30.46.

Reflect and check 532

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The points are tightly clustered around the line, indicating that the relationship between the years since the car was mathspace.co purchased and the value of the car is strong. This means the line of best fit can be used to make relatively reliable predictions. Remember, a strong relationship does not imply that one variable causes changes in the other. We cannot say that the year since the car was purchased causes the value of the car to decrease, as there may be other factors that affect the value of the car.

b Interpret the slope and y-intercept of the line.

Purpose Createhow a strategy Demonstrate to find the line of best fit for a set of data using technology. Use the independent and dependent variables to determine the units of the slope and y-intercept. Car value over time Value (thousands of dollars) 30

To help us visualize the relationship better, we can sketch the scatterplot and line of best fit, and add labels on the axes of the graph. Remember that the y-values are in thousands of dollars. This means we will need to multiply the y-value of the slope and y-intercept by 1000 when interpreting them in context.

25 20

1126 Mathspace 15Virginia SOL Algebra 1 Teacher Edition mathspace.co 10 5 Time since purchase (years)

The points are tightly clustered around the line, indicating that the relationship between the years since the car was purchased and the value of the car is strong. This means the line of best fit can be used to make relatively reliable predictions. Remember, a strong relationship does not imply that one variable causes changes in the other. We cannot say that the year since the car was purchased causes the value of the car to decrease, as there may be other factors that the 533 value of the car. Students:affect Page b Interpret the slope and y-intercept of the line.

Create a strategy Use the independent and dependent variables to determine the units of the slope and y-intercept. Car value over time Value (thousands of dollars) 30

25 20 15 10 5 Time since purchase (years) 1

Apply the idea The y-intercept of (0, 30.46) means that at the time of purchasing the car, it would have a value of $30 460. The slope of −2.2 means that each year, the car’s value would decrease by $2200.

c Make a prediction about the value of a car after 3 years.

PurposeCreate a strategy Check ifWe students can interpet the slope and y-intercept in context. are given the years since the car was purchased, which is the indpendent variable (x), and we are looking for the value of the car, which is the dependent variable (y).

Expected mistakes We can use the graph to estimate the y-value at x = 3 or use the line of best fit to get a more accurate prediction. Students might not consider the units of the dependent variable, causing them to state the y-values as is rather than converting Apply thethem idea to thousands. Remind them that the y-values are in thousands of dollars, which means each value should be multiplied When we substitute x = 3 into by the 1000. equation, we get For example, y = 30.46 in context is “30.46 thousand” which is y = −2.2(3) + 30.46 30.46= 23.86 ⋅ 1000 = 30 460 Based on the equation of the line of best fit, a car that is initially valued at $30 460 will be worth $23 860 three years

Reflecting afterwith it wasstudents purchased. Encourage students to evaluate the reasonableness of these values. Is it realistic for a car to initally be worth $30 460? Is it reasonable to assume the car’s value drops by $2200 each year? If students are not familiar with the cost of cars or how they depreciate, encourage them to research the current prices and average depreciation of cars. 9.03 Linear regression mathspace.co

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Time since purchase (years) 1

Apply the idea The y-intercept of (0, 30.46) means that at the time of purchasing the car, it would have a value of $30 460. slope of533–534 −2.2 means that each year, the car’s value would decrease by $2200. Students:The Pages

c Make a prediction about the value of a car after 3 years.

Create a strategy We are given the years since the car was purchased, which is the indpendent variable (x), and we are looking for the value of the car, which is the dependent variable (y). We can use the graph to estimate the y-value at x = 3 or use the line of best fit to get a more accurate prediction.

Apply the idea When we substitute x = 3 into the equation, we get y = −2.2(3) + 30.46 = 23.86 Based on the equation of the line of best fit, a car that is initially valued at $30 460 will be worth $23 860 three years after it was purchased.

Reflect and check When using technology to evaluate x = 3, we will get a slightly different answer. This is because the coefficients were rounded in our line of best fit. The calculator does not round the coefficients, making its result more accurate. 9.03 Linear regression mathspace.co

533

d Make a prediction about the value of a car after 10 years.

PurposeCreate a strategy Check students’ ability use theisequation line to make Since 10 years aftertopurchase not shown of on a the graph, we canpredictions. use the equation of the line of best fit to determine the value of a car at that time.

Students: Page 534

Apply the idea We can use technology to find the value of y when x = 10 by tracing the line of best fit. d Make a prediction about the value of a car after 10 years.

Create a strategy Since 10 years after purchase is not shown on the graph, we can use the equation of the line of best fit to determine the value of a car at that time.

Apply the idea We can use technology to find the value of y when x = 10 by tracing the line of best fit.

1128 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

A car that is initially valued at $30 460 will be worth $8513 ten years after it was purchased.

d Make a prediction about the value of a car after 10 years.

Create a strategy Since 10 years after purchase is not shown on the graph, we can use the equation of the line of best fit to determine the value of a car at that time.

Apply the idea We can use technology to find the value of y when x = 10 by tracing the line of best fit.

A car that is initially valued at $30 460 will be worth $8513 ten years after it was purchased.

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Purpose Check students’ ability to use the equation of a line to make predictions outside the range of the given data.

Students: Page 535 e Is the prediction for the car’s value after 3 years or after 10 years more reliable?

Create a strategy To determine the reliability of the predictions, consider whether interpolation or extrapolation was used to make the prediction. Interpolation leads to a more reliable outcome than extrapolation.

Apply the idea

Reflect and check

The given data ranges between x = 0.5 and x = 4.8. This means the prediction of the car’s value after 3 years falls within the range of known data, while the prediction after 10 years falls outside of that range. The prediction of the car’s value after 3 years is more reliable.

Interpolation is more reliable than extrapolation because the predictions follows the same pattern as the known data values. With extrapolation, we assume that the trend continues beyond the known data values. Realistically, the trend may not continue which makes extrapolation less reliable.

Example 2 Purpose A teacher recorded the number of days since a student last studied for an exam and their score out of a possible To assess students’ understanding of interpolation and extrapolation. 80 points on the exam. Reflecting with students Days since studying 3 2 6 4 4 1 6 3 4 2 Exam score or all students 64 59 to determine 42 57 58 appropriate 72 33 domain 63 55 62linear model and to Prompt advanced learners an for the defend their choices. Students should consider when the model is most reliable. Then, have them find the a Formulate an investigative question that can be answered by the data. corresponding range for their domain choice. Create a strategy The question should be focused on the relationship between the variables represented by the data. The independent variable is the number of days since studying, and the dependent variable is the score on the exam.

Apply the idea

Reflect and check

One possible question is, “How does the number of days since a student last studied impact their exam score?”

Other possible questions are: Linear regression 1129 • What is the relationship between the9.03 number of days mathspace.co since a student last and their exam score? • How many days prior to the exam should a student study to increase their exam score?

TheIsgiven data ranges = 0.5after and3x years = 4.8.or This Interpolation more reliable than extrapolation because e the prediction for between the car’s xvalue after 10 years moreisreliable? means the prediction of the car’s value after 3 years falls the predictions follows the same pattern as the known within the range of known data, while the prediction after data values. With extrapolation, we assume that the trend Create a strategy 10 years falls outside of that range. The prediction of the continues beyond the known data values. Realistically, To determine the3reliability of thereliable. predictions, consider whether interpolation extrapolation used to make the car’s value after years is more the trend may notor continue which was makes extrapolation prediction. Interpolation leads to a more reliable outcome than lessextrapolation. reliable.

Students: Page 535

Apply the idea

Reflect and check

The given data ranges between x = 0.5 and x = 4.8. This Interpolation is more reliable than extrapolation because Example 2 means the prediction of the car’s value after 3 years falls the predictions follows the same pattern as the known within the recorded range of known data, while thesince prediction after values. With extrapolation, we assume the trend A teacher the number of days a student last data studied for an exam and their score out of athat possible 10 years falls outside of that range. The prediction of the continues beyond the known data values. Realistically, 80 points on the exam. car’s value after 3 years is more reliable. the trend may not continue which makes extrapolation Days since studying 3 2 6 4 4less reliable. 1 6 3 4 2 Exam score 64 59 42 57 58 72 33 63 55 62 a Formulate Example 2 an investigative question that can be answered by the data. A teacher Create a recorded strategy the number of days since a student last studied for an exam and their score out of a possible 80 on the exam. Thepoints question should be focused on the relationship between the variables represented by the data. The independent variable is the number of days since studying, and the dependent variable is the score on the exam. Days since studying 3 2 6 4 4 1 6 3 4 2 Exam score 64 59 42 57 58 72 33 63 55 62

Apply the idea

Reflect and check

OneFormulate possible question is, “Howquestion does thethat number ofanswered days Other possible a an investigative can be by the data. questions are: since a student last studied impact their exam score?” • What is the relationship between the number of days since a student last and their exam score? Create a strategy • How many days prior to the exam should a student The question should be focused on the relationship between the variables represented by the data. The independent study to increase their exam score? variable is the number of days since studying, and the dependent variable is the score on the exam. • If a student studies on the same day as the exam, what is their expected score on the exam?

Apply the idea

Reflect and check

One possible question is, “How does the number of days Other possible questions are: b Was the datalast most likely impact collected through measurement,•observation, survey or an experiment? since a student studied their exam score?” What is the arelationship between the number of days since a student last and their exam score? PurposeApply the idea • How and manycheck days prior to the exam should a student Reflect Check students’ ability to formulate investigative questions given a data set. study to score? The teacher did not measure, observe, or control the time Although theincrease teachertheir may exam have had access to the • If a student studies on the same as thewas exam, since a student studied. Instead, it is more likely that the students’ exam scores (assuming theday teacher the one is their expected score on the exam? Students:teacher Pageasked 535the students how many days it has been thatwhat assigned the exam), they could have still included a since they last studied.

survey question about the exam score to keep the data organized. The data was most likely collected through a survey. b Was the data most likely collected through measurement, observation, a survey or an experiment? For example, their survey questions could have been, “How many days has it been since you last studied for Apply the idea Reflect and check this subject?” and “What was your score on the exam?” The teacher did not measure, observe, or control the time Although the teacher may have had access to the since a student studied. Instead, it is more likely that the students’ exam scores (assuming the teacher was the one teacher asked the students how many days it has been that assigned the exam), they could have still included a 9.03 Linear regression 535 since they last studied. survey question about the exam score tomathspace.co keep the data organized. The data was most likely collected through a survey. For example, their survey questions could have been, “How many days has it been since you last studied for this subject?” and “What was your score on the exam?”

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535

Purpose Assess the students’ understanding of different data collection methods and their ability to determine which method is applicable in a given scenario.

1130 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Students: Page 536 c Describe the relationship between the number of days since studying and the exam score.

Create a strategy To describe the relationship, we should construct a scatterplot to get a visual of the data. Then, we will consider the form (linear or nonlinear), strength (strong, moderate or weak), and direction (positive or negative).

Apply the idea Score

c Describe the relationship between the80 number of days since studying and the exam score. 70

Create a strategy

50 To describe the relationship, we should construct a scatterplot to get a visual of the data. Then, we will consider the form (linear or nonlinear), strength (strong,40moderate or weak), and direction (positive or negative). 30

Apply the idea

20 10 Score 80 70

Days since studying 2

60 linear relationship. The data appears to have a strong, negative, 50

Relating this back to the context, we can say that as the number of days since a student last studied increases, and 40 their score on the exam tends to decrease. 30 20

d Calculate the line of best ﬁt using technology. 10

Days since studying

PurposeCreate a strategy 1 2 3 4 5 6 Check the students’ ability to create and analyze arelationship. scatterplot. The data to using have atechnology, strong, negative, To find theappears equation we canlinear follow these steps:

Relating backsign to the context, wecorner can say the number of days since a student last studied increases, and 1. Click this the plus in the top left ofthat the as screen, and select table.

Students:their Pages 536–537 score on the exam tends to decrease.

2. Enter the x-values and y-values in the respective columns of the table. 3. In a new line beneath the table, enter the equation y1 ∼ mx1 + b. d Calculate the line of best ﬁt using technology.

Apply the idea Create a strategy

1. Click the plus sign in the top left corner of the screen, and select table. To find the equation using technology, we can follow these steps: 1. Click the plus sign in the top left corner of the screen, and select table. 2. Enter the x-values and y-values in the respective columns of the table. 3. In a new line beneath the table, enter the equation y1 ∼ mx1 + b.

Apply the idea 1. Click the plus sign in the top left corner of the screen, and select table.

536

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536

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1131

2. Enter the x- and y-values in the respective columns of the table.

3. In a new line beneath the table, enter the equation y1 ∼ mx1 + b.

The equation of the line of best fit is y = −6.2245x + 78.2857

Reflect and check If the instructions do not specify to round the coefficients, it is best to include all the digits given by the calculator. This increases the accuracy of the model and the predictions.

Purpose Check that students can find the line of best fit for a set of data using technology.

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537

Students: Page 538 e Answer the question formulated in part (a).

Create a strategy To answer the question, “How does the number of days since a student last studied impact their exam score?”, we can describe the direction of the linear relationship. To be more specific, we can interpret the slope of the line in context. In the previous part, we found the equation of the line of best fit to be y = −6.2245x + 78.2857, which tells us the slope is −6.2245.

Apply the idea As the number of days since a student last studied increases, their exam score decreases. More specifically, for each additional day since a student last studied, their exam score is expected to decrease by about 6 points.

Reflect and check Matching the rise and run of the slope to their respective units can help us interpret its meaning in context.

e AnswerScore the question formulated in part (a). The y-values represent the exam score, which is the “rise” of the 80 slope. The x-values represent the number of days since studying, 70a strategy Create which is the “run” of the slope. 60 the question, “How does the number Since To answer of days a student last itstudied impact their exam thesince slope is negative, represents a decrease ofscore?”, 6.2245 in the we can50describe the direction of the linear relationship. To beformore specific, we can interpret the slope of the line exam score every 1 day since studying. in context. 40

In the previous part, we found the equation of the line of best fit to be y = −6.2245x + 78.2857, which tells us the 30 slope is −6.2245. 20 10

Days since studying

Apply the idea 1

4 5 6 7 As the number of days since a student last studied increases, their exam score decreases. More specifically, for each additional day since a student last studied, their exam score is expected to decrease by about 6 points.

fReflect If a student studied the same day as the exam, what would we expect their score to be? and check

Purpose Matching the rise and run of the slope to their respective units can help us interpret its meaning in context. Create a strategy Determine if students can interpret the slope of the line in context.

If the number of days since a student last studied is 0, then their exam score is the y-value of the y-intercept.

Reflecting with students The y-values represent the exam score, which is the “rise” of the Apply theScore idea 80 The x-values represent causes the number of scores days since Discuss with students whether or not the numberslope. of days since studying test tostudying, decrease. Remind The y-intercept tells us that a student who has studied on the day of the exam has a predicted score of 78.2857, 70 which is the “run” of the slope. students that a strong according to therelationship linear model. does not imply that one variable causes a change in the other variable. the slope negative, it represents a decrease of 6.2245 in the Ask students to60list other factors that might haveSince an impact onis test grades. Examples may include the number of 50 exam score for every 1 day since studying. and check absencesReflect from school recently, hours of sleep the night before the test, average class participation amount, etc. 40 Although this value was x = 0studying is not very causes far outside of test the range of known values. as there In other words, we cannot sayfound the through numberextrapolation, of days since the scores to decrease 30 relationship is strong, this prediction is relatively reliable. Since the may be other factors at play. If an experiment was conducted to control the outside factors and the test scores 20 still decreased,10then we can say there is a causal relationship there. Days since studying

Students: Page 538 f

If a student studied the same day as the exam, what would we expect their score to be?

Create a strategy Mathspace Virginia SOLaAlgebra 1 last studied is 0, then their exam score is the y-value of the y-intercept. 538 If the number of days since student mathspace.co

Apply the idea The y-intercept tells us that a student who has studied on the day of the exam has a predicted score of 78.2857, according to the linear model.

Reflect and check Although this value was found through extrapolation, x = 0 is not very far outside of the range of known values. Since the relationship is strong, this prediction is relatively reliable. 9.03 Linear regression 1133 mathspace.co

If the number of days since a student last studied is 0, then their exam score is the y-value of the y-intercept.

Apply the idea The y-intercept tells us that a student who has studied on the day of the exam has a predicted score of 78.2857, according to the linear model.

Purpose Check the students’ ability to interpret the y-intercept of a linear regression model.

Students: Page 539 538

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Idea summary A line of best fit for a set of data can be used to interpret a given situation and make predictions about values not represented by the data. A line of best ﬁt has an equation of the form y = mx + b. We can use technology to perform the linear regression analysis. In the context of a line of best ﬁt, the slope-intercept form represents

y = mx + b m

the rate of change for y with respect to x

The starting value of y when x is 0

These terms describe the range in which we make predictions: • •

Interpolation: Prediction within the range of x-values in the data Extrapolation: Prediction outside the range of x-values in the data

The reliability of predictions depends on the strength of the relationship, whether the data is interpolated or extrapolated, and the number of points in the data set. In general, interpolation is more reliable than extrapolation.

Practice What do you remember?

Practice 1

Choose the line of best ﬁt for this scatterplot.

Students: Pages 539–545

8 7 Line 3

What do you remember?

4 3

Choose the line of best ﬁt for this scatterplot.

Line 2

5 Line 1

2 1

9 1

y x

82 3 4 5 6 7 8 9 7 Line 3

Line 2

5 4 3

Line 1

2 1

x 1

2 3 4 5 6 7 8 9

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539

Sketch the line of best fit for each scatterplot: a

5 x 5

The scatterplot represents the given data set: {(36, 114), (20, 164), (22, 154), (24, 150), (26, 140), (28,138), (30,134), (32, 122), (34,118)} a

Create a table of values for the data set.

Find the equation of the line of best fit.

180

170 160 150 140 130 120 110 100

x 20 22 24 26 28 30 32 34 36

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Does each graph show interpolation or extrapolation? a

10 20 30 40 50 60 70 80 90

x 10 20 30 40 50 60 70 80 90

Is each statement true or false? a

The line of best fit can only be used for interpolation, not extrapolation.

Using the line of best fit for extrapolation is generally more reliable than interpolation.

Interpolation is more accurate when the data points are tightly clustered around the line of best fit.

A line of best fit may not accurately represent the relationship between variables if the relationship is non-linear.

It is not important to consider the validity of predictions when using interpolation or extrapolation.

Considering the strengths and weaknesses of a regression model helps ensure that the conclusions are accurate and reliable.

Let’s practice 6

Find the equation of the line of best fit for each data set. a

x y

28 3926

x y

17 16

Speed Time

Age Accidents

30 6482 19 19

20 85

21 21 25 87

20 41

32 10 589 23 16 30 75

25 44

34 17 098 25 17

35 82 30 39

36 28 236

27 27 40 69

35 34

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29 19 45 73

40 30

38 46 985

50 60 45 25

55 57 50 22

60 45 55 18

65 49 60 19

65 17

For the data set shown: {(93, 51.2), (57, 25.4), (86, 38.9), (97, 58.6), (78, 38.2), (96, 60.8), (68, 26.3), (69, 28.5), (54, 5.4), (92, 92)}

Find the equation of the line of best fit.

Predict the value of y when x = 3.49. Round your answer to two decimal places.

Is the prediction in part (b) an example of interpolation or extrapolation?

A cafe manager collected data about sales of hot cocoa during a winter weekend. The data included the outside temperature and the number of hot cocoa sold every two hours. Outside temperature in °F, t Number of hot cocoas sold, n

0 20

2 16

5 3

2 16

1 18

3 17

6 8

4 12

Is the relationship between the outside temperature and the number of hot cocoa sold linear or nonlinear?

Calculate the regression model for this data set. Round all values to the nearest tenth.

The amount of money households spend on dining out each week, D, is measured against their weekly income, I. This linear model D = 0.3I + 27 is ﬁt to the data.

Explain the meaning of the y-intercept.

State the slope of the line.

If the weekly income of a family increases by $200, by how much can we expect their spending on dining out to increase?

The life expectancy (E), in years, of individuals at different annual incomes (I), per $1000, is shown: The equation of the line of best ﬁt is E = 0.09I + 72.55. a

By how much does average life expectancy change for each $1000 of annual income?

Find the average life expectancy of someone who earns no income.

E 100 75 50 25 I 50

Scientists collect data to answer the statistical question, “What is the relationship between the number of aphids and the number of ladybugs in various areas?” The data is shown in the scatterplot.

100

150

A 3200

A = −3.82L + 3865.21 represents the line of best ﬁt. a

Describe the variables the scientists used to answer their statistical question.

How much does the average aphid population change by with each extra ladybug? Round your answer to the nearest aphid.

Find the average aphid population of a region with no ladybugs. Round your answer to the nearest aphid.

2400 1600 800 L 200 400 600 800 1000

9.03 Linear regression 1137 mathspace.co

The average monthly temperature and the average wind speed in a particular location was plotted over several months. The graph shows the points for each month’s data and their line of best fit.

Identify the independent and dependent variables used in this study.

Use the line of best fit to approximate the wind speed on a day when the temperature is 41 °F.

How reliable is this prediction? Explain your answer.

Wind speed (knots)

6 5 4 3 2 1

Temperature (°F ) 39

The scatterplot shows data collected on the amount of caffeine consumed, in milligrams, in a day and the number of hours of sleep for 30 adults. Formulate a question that can be answered by the scatterplot.

Which equation is most likely the line of best fit?

B y = 0.015x − 9.58

C y = −0.015x + 9.58

D y = −0.015x − 9.58

4 Caffeine (mg)

Describe the strength of the relationship. Explain what this relationship implies in context.

100 150 200 250 300

Scientists conducted a study to analyze the time it took people to perform a simple matching activity after they’ve had different amounts of sleep. The participants were placed in similar rooms under the same conditions, and they were all given the same matching activity. The data is shown in the table and scatterplot, along with the line of best fit. Number of hours sleep (x) Completion time in seconds (y)

1.1 4.66

1.5 4.1

2.1 4.66

2.5 3.7

3.5 3.6

Formulate a question that could be answered by the data.

Which data collection method was used? A Measurement

Hours of sleep

A y = 0.015x + 9.58

Observation

4 3.4

Survey

Experiment

Use technology to find the equation for the line of best fit.

Use the line of best fit to predict the task completion time for someone who has slept 5 hours.

Predict the number of hours someone has slept if they complete the matching task in 4 seconds.

A student collected data to answer the statistical question, “How does the temperature outside impact the number of people at the beach?” Their data is shown in the table. Temperature (°F) Number of people

69 80

72 88

74 120

77 134

79 162

80 177

82 180

83 188

85 220

87 230

Did the student collect the data through measurement, observation, a survey or an experiment?

About how many people might be at the beach when the temperature is 80°F?

What is the temperature when there are 81 people at the beach?

About how many people might be at the beach when the temperature is 90°F?

Is the prediction for 80°F or 90°F more reliable?

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One liter of gas is raised to various temperatures, and its pressure is measured. The results are shown in the table. Temperature (K) Pressure (Pa)

302 2416

304 2434

308 2462

310 2478

312 2496

314 2512

316 2526

318 2546

Use technology to calculate the equation for the line of best fit.

Use the line of best fit to predict the pressure when the temperature is 306 K.

Is the prediction in part (a) an example of interpolation or extrapolation?

Is the prediction in part (a) reliable?

Will using the line of best fit to predict pressure within each of these ranges of temperatures result in a reliable prediction? 300 ≤ Temp ≤ 320

300 ≤ Temp ≤ 600

iii 0 ≤ Temp ≤ 320

280 ≤ Temp ≤ 340

300 2400

Concern over student use of the social media app SnappyChatty leads to a study of student grades in Mathematics versus minutes spent using the app. The results are shown in the table: Minutes, M Grade, P %

292 26

153 63

354 13

253 37

11 97

42 89

195 51

7 98

162 59

254 36

Predict the grade of a student who spends no time on the SnappyChatty app. Use a model to justify your response.

Use your model to explain and interpret the relationship between minutes spent on the SnappyChatty app and mathematics grades.

Let’s extend our thinking 18

Based on the given scatterplot and line of best fit, could the model be used to make reliable predictions? Explain.

8 7 6 5 4 3 2 1

x 1

Lorena and Frasier each draw a possible line of best ﬁt. a

Frasier said he noticed that there was a point that was far away, so he moved his line closer to it. Does Frasier’s line represent a line of best ﬁt? Explain. Lorena said she noticed that there was a point that was far away, but decided to ignore it. Would Lorena’s line better represent a line of best ﬁt? Explain.

2 3 4 5 6 7 8 9

y 7

Frasier

6 5 4 3

Lorena

2 1

x 2 4 6 8 10 12 14 16 18

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Each week, a school counselor helps students who are struggling in Math and English organize tutoring sessions. He hopes that the tutoring sessions will have a positive effect on students’ grades. a

Formulate a question which could be investigated using a scatterplot.

Determine what variables could be used to answer the statistical question from part (a).

Describe a method the school counselor could use to collect the data.

The data the school counselor collected on the students who receive Math and English tutoring is shown in the tables. Create a scatterplot of the data. Math students Hours of tutoring per week Math grade

1.5 63

3.5 75

1 60

4 79

2 68

2.5 69

3 71

Hours of tutoring per week Math grade

3.25 72

1.75 66

2.25 68

3.75 74

1.25 62

2.75 70

4.5 79

English students

Hours of tutoring per week English grade

1 69

2.5 80

0.75 64

1.5 72

4 91

1.25 71

2 77

Hours of tutoring per week English grade

3.5 89

1.25 67

2.75 83

1.75 73

3 85

2.25 79

3.75 87

Find the regression model for each set of data.

Draw a conclusion about the data by answering the statistical question from part (a).

Use the regression models to predict the grades of a math student and an English student who each receive 3 hours and 15 minutes of tutoring each week. Explain whether these predictions are reliable.

Go through the whole data cycle at least once to investigate whether a relationship between a person’s height and the length of their stride exists.

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Answers 9.03 Linear regression

b Extrapolation

c Extrapolation

d Interpolation

5 a False e False

What do you remember? 1 Line 2 2 a

4 a Interpolation b False f

d True

True

Let’s practice y

6 a y = 4100.9x − 1 16 445

b y = 0.375x + 10.661

c y = −0.93x + 107.87

d y = −0.65x + 56.56

7 a y = 1.26x − 57.01

b y = −52.61

c Extrapolation b n = −2.5t + 20.9

8 a Linear

9 a W hen a family has no income, their average spending on dining out is $27.

x 5

b 0.3

c $60

10 a 0.09 years per $1000 b

b −4

b 3865

12 a T he independent variable is the average monthly temperature, and the dependent variable is the average wind speed. Each point represents one month, so data was collected over 8 months.

10 5 x 5

b 72.55 years

11 a T he independent variable is the number of ladybugs, and the dependent variable is the number of aphids.

b 4 knots

c T his prediction is not very reliable because the sample size is small and the relationship between the variables is weak.

y 20

13 a M any possible questions, for example, “Does higher caffeine consumption correlate with fewer hours of sleep among adults?”

15 10

b C

c T here is a strong relationship between the variables. We expect adults who consume more caffeine to sleep less.

x 5

14 a M any possible questions, for example, “What is the relationship between the hours of sleep someone gets and their reaction time?”

b D

c y = −0.406294x + 5.01542

d 3.8 seconds e 2.5 hours

15 a Observation

c 70°F

3 a

c True

e 80°F

114

164 154 150 140 138 134 122

118

b 168 people d 256 people

16 a y = 7.98333x + 4.93333

b 2448 Pa

c Interpolation

d Yes

e i Yes

iii No

ii No

iv No

b y = −3.08x + 223.44

Answers mathspace.co

1141

17 a A scatterplot can help visualize the relationship between the quantities, but is not necessary for a complete response:

c I f there is a relatively small number of students receiving tutoring, the school counselor could collect data on all students receiving tutoring. Since he is already tracking the tutoring hours each student receives, he only needs to collect data on the students’ grades. He could acquire this data from the school’s online system or from the students’ Math and English teachers.

Grades

80 70

Grade 90

20 10

minutes 50 100 150 200 250 300 350 400 450

There is a strong, negative, linear relationship between minutes spent on SnappyChatty and math grades. The linear regression model is y = −0.25x + 99.6 A student who spends 0 minutes on SnappyChatty is predicted to have a 99.6% grade in mathematics. b A ccording to the linear model, for every additional minute spent on SnappyChatty, a student’s math grade is expected to drop 0.25%. Or, for every additional 100 minutes spent on SnappyChatty, a student’s grade is predicted to drop 25%. Let’s extend our thinking 18 No, the line does not follow the pattern in the data. A nonlinear model would be more appropriate for making predictions. 19 a T here are only 2 points above the line and 18 points below it, so this line does not go through the middle of the data. It is too heavily impacted by the outlier, so is not a line of best fit. b A n outlier should not be completely ignored, but the line should go through the majority of the data. Lorena’s line has 9 points above it and 10 points below it, and goes through the middle of the data, so it is a line of best fit. 20 a M any possible questions, for example, “How have the weekly tutoring hours impacted grades for math and English students?” b A nswers will vary based on question from part (a). Using the example, “How have the weekly tutoring hours impacted grades for math and English students?”: The independent variable is the hours of tutoring per week, and the dependent variable is the students’ class grades. The categorical variables are students who receive tutoring for Math and students who receive tutoring for English.

1142 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Weekly tutoring hours

2 English

3 4 Math

e Math students: y = 5.3396x + 55.6024 f

English students: y = 8.0136x + 59.7553 nswers will vary based on question from part (a). A Using the example, “How have the weekly tutoring hours impacted grades for math and English students?”:

There is a strong, positive, linear relationship between the weekly tutoring hours and grades of both math and English students. This implies that the weekly tutoring sessions have had a positive impact on students’ grades. As the weekly tutoring hours increase, the students’ grades also increase. More specifically, an increase in 1 hour of tutoring each week can increase a math student’s grade by about 5% and an English student’s grade by about 8%. g 3 hours and 15 minutes corresponds to x = 3.25.

For math students: y ≈ 73%

For English students: y ≈ 86%

Since there is a strong relationship between the variables and both predictions are interpolations, these predictions are reliable. 21 1. Formulate questions

Possible questions:

• What is the relationship between a person’s height and their stride length?

• How does a person’s stride length change with their height?

• What are the stride lengths of people between 5 and 6 feet tall?

2. Collect data using the questions above.

4. Analyze and explain the results.

We can collect data by measuring a person’s height in inches and the length of their stride in inches. The table shows an example sample of 30 students.

From the scatterplot, we can observe that there is a moderate, positive linear relationship between a person’s height and their stride length. The relationship shows that as height increases, stride length tends to increase.

Height

Stride 29.4 28.1 28.2 32.1 29.4 28.2 29.1 28.5 27.9 29.6 Height

Stride 26.5 28.7 29.4 29.7 28.4 27.7 26.4 28.3 27.6 25.3 Height

Stride 28.4 28.6 25.3 28.7

30.6

27.2 30.5 25.6

This could lead us to formulate a new question like “How does the relationship between height and stride length compare between boys and girls?” or “What is the expected stride length of someone who is 6 feet tall?”

3. Create a data display using a scatterplot. This scatterplot represents the sample data from the previous part. Since it shows an approximate linear relationship, a line of best fit has been drawn. The equation of the line of best fit is y = 0.364x + 5.44. 38

Stride length (in)

36 34 32 30 28 26 24 22

Height (in) 52 54 56 58 60 62 64 66 68 70 72 74 76 78

Answers 1143 mathspace.co

9.04 Quadratic regression Subtopic overview Lesson narrative In this lesson, students will use their knowledge of graphs to determine if a graph is quadratic. Given a set of data points, students will graph the points to determine the type of equation needed. Students will find the quadratic regression equation when the graph is quadratic. This equation will be used to make predictions about other values not in the problem. By the end of this lesson, students will be able to find the quadratic regression and apply it to find additional values.

Learning objectives

9.04 Quadratic regression

Students: Page 546

After this lesson, you will be able to… • determine if a table of values represents a quadratic model. • determine the equation of the curve of best fit given a table of values. • predict values using a quadratic model.

Quadratic regression Functions can be used to model real-world events and interpret data from those events. Data that measures or Key vocabulary

compares two characteristics of a population is known as bivariate data.  domain constraint bivariate data When analyzing data, we previously described the relationship between two variables as linear or nonlinear. In this  scatterplot  standard form (of a quadratic function) lesson, we will focus on nonlinear relationships that can be modeled by a quadratic function. 

Exploration

Essential understanding Each table shown represents a different set of of bivariate data. The relationship between the variables in a set data reveals the type of function that best models the data. Table 1 x y

13 Standards

0.2 7

0.4 4

0.6 3

0.8 1

1 0

1.2 2

1.4 3

1.6 6

1.8 9

This subtopic addresses the following Virginia of Learning for Mathematics standards. Table Standards 2 x 3 process 3.5 4 goals 4.5 Mathematical y

5 104

5.5 100

6 112

6.5 120

7 114

7.5 127

8 127

MPG2 — Mathematical Communication

MPG4 — Mathematical Connections

Table 3 by Teachers can foster mathematical communication

Teachers can make mathematical connections by linking the linear regression, and 7 concepts 8 9of scatterplots, 10 quadratic regression with previous lessons, thereby 55 53 52 50 helping students see the relevance and applicability of their prior knowledge. For instance, connecting the concept of regression to lessons on the 6.8 7.2 7.4quadratic 8 standard form of a quadratic equation or the meaning of 6 5.5 4 2 coefficients in this context.

encouraging students their x 0 1 to explain 2 3 reasoning 4 5 when6 decidingythe type of relationship between variables, 63 65 61 59 58 59 54 calculating the curve of best fit, or evaluating the strength Table 4 should and weaknesses of a quadratic model. Students be encouraged their understanding x 1 to articulate 2 2.5 3 4 5 of the 6.3 2 coefficients in a quadratic equation, the meaning y 1.5 3 4.8 5 7.4 8 of R 7, and the limitations of their models. Without creating a scatterplot: 1.

Does the data in Table 1 have a linear or quadratic relationship? Explain your answer.

Does the data in Table 3 have a linear or quadratic relationship? Explain your answer.

Does the data in Table 4 have a linear or quadratic relationship? Explain your answer.

1144 Mathspace Virginia SOL Algebra 1 Teacher Edition 2. Does the data in Table 2 have a linear or quadratic relationship? Explain your answer. mathspace.co

MPG5 — Mathematical Representations Teachers can incorporate mathematical representations into their lessons by having students use technology to create scatterplots and perform quadratic regression. Students can then write the equation of their quadratic function and use it to predict outcomes. They can also visually represent the curve of best fit and interpret the meaning of the coefficients in this context. Teachers should encourage students to make connections among different representations — such as the scatterplot (visual), the equation (symbolic), and the real-world context (contextual).

A.ST.1a — Formulate investigative questions that require the collection or acquisition of bivariate data.

A.ST.1h — Analyze relationships between two quantitative variables revealed in a scatterplot. A.ST.1i — Make conclusions based on the analysis of a set of bivariate data and communicate the results.

A.F.2 — The student will investigate, analyze, and compare characteristics of functions, including quadratic and exponential functions, and model quadratic and exponential relationships.

Lesson Preparation Suggested review Depending on your students’ level of prior knowledge, consider revisiting the following lessons: Algebra 1 — 7.03 Quadratic functions in vertex form Algebra 1 — 7.04 Quadratic functions in standard form

9.04 Quadratic regression 1145 mathspace.co

Tools You may ﬁnd this tool helpful: • Graphing calculator

Student lesson & teacher guide Quadratic regression Students are introduced to the idea of using quadratic models to represent nonlinear data. The lesson describes situations in which the different forms of a quadratic equation could be most efficient, based on the information given.

Students: Page 546

9.04 Quadratic regression After this lesson, you will be able to… • determine if a table of values represents a quadratic model. • determine the equation of the curve of best fit given a table of values. • predict values using a quadratic model.

Quadratic regression Functions can be used to model real-world events and interpret data from those events. Data that measures or compares two characteristics of a population is known as bivariate data. When analyzing data, we previously described the relationship between two variables as linear or nonlinear. In this lesson, we will focus on nonlinear relationships that can be modeled by a quadratic function.

Exploration Each tableand shown represents a different Stronger clearer each timeset of data. Table 1 English language learner support

500 450 400 350 x 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 300 After a few minutes, pair students to share their explanations and receive y 63 68 77 90 104 100 112 120 114 127 127 250 feedback. In these pairs, encourage students to ask their partners for 200 further clarification of their response. They can3 ask questions such as: Table 150 • Why did/didn’t of of8the 9 x 0you include 1 2 an explanation 3 4 5 the6strength 7 10 100 relationship? y 63 65 61 59 58 59 54 55 53 52 50 50 x

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

Profit($)

Provide students with model and y 13 the 7 following 4 3scatterplot 1 0and quadratic 2 3 6 9 ask, “What is the relationship between days and profit?” Have students Table 2 relationship shown. individually write an explanation of the quadratic

• Should the relationship be described as nonlinear or quadratic? Table 4 • Do we need to describe what happens to the y-values? x

2.5

6.3

6.8

7.2

Days 1 2 3 4 5 6 7 8 9 10

7.4

Then, mix pairs for1.5further refinement of 7.4 their explanations. Finally, students should be given time to individually y 3 4.8 5 8 7 6 5.5 4 2 refine their original explanations, incorporating feedback and new ideas. Encourage students to share their final explanations withcreating the class. Without a scatterplot: 1. explanations Does the data inorTable 1 havestudents a linear or with quadratic relationship? answer. Highlight good provide an example of Explain a goodyour explanation, such as, “There is a 2. Does the data in between Table 2 have a linear quadratic relationship? yourthe answer. strong, quadratic relationship days andorprofit. Between 1 andExplain 6 days, profit increases. After 6 days, 3. Does the data in Table 3 have a linear or quadratic relationship? Explain your answer. the profit decreases.” 4.

Does the data in Table 4 have a linear or quadratic relationship? Explain your answer.

1146 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Quadratic relationships are not the only type of nonlinear relationship Address student misconceptions In Algebra 1, students only focus on linear and quadratic relationships. However, this might create the misconception that quadratic relationships are the only type of nonlinear relationship between two variables. Students may benefit from, and advanced learners may enjoy, seeing examples of data that have other types of nonlinear relationships, like the ones shown: Population 60000 Rainfall in inches 4

50000

9.04 Quadratic regression

40000

30000

20000

After this lesson, you will be able to…

1 1

• determine if a table of values represents a quadratic model. 10000 • determine the equation of the curve of best fit given a table of values. Month • predict values using a quadratic model.

Time 0.1

0.2 0.3 0.4 0.5

Quadratic regression

Exploration Functions can be used to model real-world events and interpret data from those events. Data that measures or compares two characteristics of a population is known as bivariate data.

Students: Page 546

When analyzing data, we previously described the relationship between two variables as linear or nonlinear. In this lesson, we will focus on nonlinear relationships that can be modeled by a quadratic function.

Exploration Each table shown represents a different set of data. Table 1 x y

0 13

0.2 7

0.4 4

0.6 3

0.8 1

x y

3 63

3.5 68

4 77

4.5 90

5 104

1 0

1.2 2

1.4 3

1.6 6

1.8 9

6 112

6.5 120

7 114

7.5 127

8 127

6 54

7 55

8 53

9 52

10 50

6.3 7

6.8 6

7.2 5.5

7.4 4

8 2

Table 2 5.5 100

Table 3 x y

0 63

1 65

2 61

3 59

4 58

5 59

Table 4 x y

1 1.5

2 3

2.5 4.8

3 5

4 7.4

5 8

Without creating a scatterplot: 1.

Does the data in Table 1 have a linear or quadratic relationship? Explain your answer.

Does the data in Table 2 have a linear or quadratic relationship? Explain your answer.

Does the data in Table 3 have a linear or quadratic relationship? Explain your answer.

Does the data in Table 4 have a linear or quadratic relationship? Explain your answer.

9.04 Quadratic regression 1147 mathspace.co 546

Mathspace Virginia SOL Algebra 1 mathspace.co

Suggested student grouping: Small groups In this exploration, students are examining four tables of data and looking at the relationship between x and y in each. Without creating a scatterplot, students will determine if the data represents a linear or quadratic relationship, explaining their reasoning. Ideal student responses These ideal responses may differ from other correct student responses. Less formal responses can be connected with the more precise mathematical language presented here. 1. Does the data in Table 1 have a linear or quadratic relationship? Explain your answer. The data in Table 1 seems to have a quadratic relationship. The y-values decrease, reach a minimum value, then increase, which is typical of a quadratic function. 2. Does the data in Table 2 have a linear or quadratic relationship? Explain your answer. The data in Table 2 seems to have a linear relationship because the y-values are only increasing for each half unit increase in x. 3. Does the data in Table 3 have a linear or quadratic relationship? Explain your answer. The data in Table 3 seems to have a linear relationship. The y-values are only decreasing for each unit increase in x. 4. Does the data in Table 4 have a linear or quadratic relationship? Explain your answer. The data in Table 4 seems to have a quadratic relationship. The y-values increase, reach a maximum value, then decrease, which is typical of a quadratic function. Purposeful questions • As the x-values increase, what happens to the y-values? • Choose a y-value in the center of the table and compare it to the first and last y-values in the table. Does it appear that the values have only increased, only decreased, or have increased and decreased? Possible misunderstandings • Students might think that the y-values should change by the same amount for linear functions, or that the values should only increase/decrease. Remind them that real data is messy, and encourage them to instead consider the overall trend of the data. • Students might not consider the entire domain of the data, instead only looking for what happens to the first few outputs. Emphasize the importance of considering the entire domain, as the y-values of quadratic functions will change direction at some point in the domain.

Students: Page 547

To more easily analyze a set of data and determine if there is a quadratic relationship between the variables, we often construct a scatterplot.

8 7

Data presents a quadratic relationship if it forms a symmetric curve or parabolic shape.

6 5 4

The quadratic curve of best fit that approximately models the data can be calculated using technology. Most calculators will write the model in standard form (of a quadratic function), y = ax2 + bx + c.

3 2 1 1

2 3 4 5 6 7 8 9

If points are more tightly clustered along the model, it represents a stronger relationship between the variables. The curve of best fit can help us make predictions or conclusions about the data. If we are given an x-value, we can predict the y-value by substituting x into the equation and solving for y. We can also use the graph of the model to approximate x and y-values. 9

1148 Mathspace Virginia 7 SOL Algebra 1 Teacher Edition mathspace.co 6

9 8 7 6

1 1

2 3 4 5 6 7 8 9

x 1

2 3 4 5 6 7 8 9

When x = 8, y ≈ 3

When y = 7, x ≈ 4 and 6

When anayzing the data, it is often helpful to interpret the x-intercepts or the vertex in context. For example, if the equation models a company’s sales over time, the x-intercepts represent the times the company made no sales, and the vertex represents the time the highest amount of sales were made. It is important to consider the context of the data when communicating results as the model may only be appropriate over a part of the domain. Domain constraint A limitation or restriction of the possible x-values, usually written as an equation, inequality, or in set-builder notation

Examples The following support may be useful for the examples in this section.

Calculator steps for Desmos, TI graphing calculators, and Google Sheets Targeted instructional strategies The examples provide students with the steps to finding the line of best fit using the GeoGebra Statistics calculator. Steps for finding the line of best fit with other technologies are shown. TI-83 or TI-84:

9.04 Quadratic regression mathspace.co

547

1. Select the STAT button. 2. Select EDIT. 3. Enter the x-values under L1 in the table. 4. Enter the y-values under L2 in the table. 5. Press the STAT button again. 6. Press the right arrow to select CALC at the top of the screen. 7. Select QuadReg. 8. Press these buttons in this order: • 2ND (blue button in the top left) • 1 (has L1 in blue above it) • , (the comma above 7) • 2ND • 2 (has L2 in blue above it) Your screen should show QuadReg(L1, L2) 9. Select ENTER.

9.04 Quadratic regression 1149 mathspace.co

These steps will produce the values of a, b and c. The value of a is the leading coefficient, the value of b is the coefficient of the middle term, and the value of c is the y-intercept. Google Sheets: 1. Enter the data from the table into two rows or columns with labels 2. Highlight the data and navigate to “Insert” and click “Chart” 3. By default it should insert a scatterplot, but if not change the “Chart type” to “Scatter chart” 4. Under the Chart editor click on “Customize” and under “Horizontal axis” and “Vertical axis” set an appropriate view 5. Under the Chart editor click on “Customize” and under “Series” tick the box for “Trendline” and change the “Type” to “Polynomial” and “Label” to “Use Equation” Notice that the equation is displayed by default to 3 significant figures, so the rounding may vary from a given solution.

Students: Page 548 Example 1 For each scatterplot, determine whether the variables have a linear relationship or a quadratic relationship. If there is a relationship, describe its strength. a

y 35 30 25 20 15 10 5

x 1

Create a strategy A relationship between two variables exists if the points follow a similar trend. The points will roughly form a line if there is a linear relationship or a parabola if there is a quadratic relationship. To describe the strength of the relationship, we can analyze how tightly the data points are clustered or grouped together.

Apply the idea As the x-values increase, the y-values decrease then increase, causing the points to form a U-shaped curve. This shows there is a quadratic relationship between the variables. Because the points are tightly clustered, the relationship is strong.

b 90

Purpose 80 Show students 70 how to determine the type and strength of a relationship between variables from a scatterplot. 60 50 40 30 20 10

x 1

2 3

4 5

Apply the idea

1150 Mathspace Virginia SOL Algebra 1 Teacher Edition As the x-values increase, the y-values decrease. This indicates there is a linear relationship between the variables. mathspace.co However, the points are not tightly clustered, so the relationship between the variables is moderate.

Reflect and check

relationship, we can analyze how tightly the data points are clustered or grouped together.

b 90

80 70 60 50 40 30 20 10

x 1

2 3

4 5

Apply the idea As the x-values increase, the y-values decrease. This indicates there is a linear relationship between the variables. However, the points are not tightly clustered, so the relationship between the variables is moderate.

Reflect and check Recall that we can describe a linear relationship as positive or negative. For this data set, the relationship is negative since one variable increases and the other decreases. This implies that the equation of the line of best fit would have a negative slope.

548

Virginia SOL Algebra 1

Mathspace

Purpose mathspace.co Test students’ understanding of how scatterplots can be used to identify and describe the strength of a relationship between variables.

Students: Page 549 c

50 45 40 35 30 25 20 15 10 5

x 2 4 6 8 10 12 14 16 18

Apply the idea A relationship between two variables exists if the points follow a similar trend. If the y-values increase and decrease over the domain, the relationship can be modeled by a quadratic function.

Reflect and check As the x-values increase, the y-values increase then decrease, causing the points to form an upside down, U-shaped curve. This shows there is a quadratic relationship between the variables. Because the points are not tightly clustered, the relationship is moderate.

PurposeExample 2 Test students’ ability to identify quadratic relationships from scatterplots and assess the strength of the A conservationist tracks the population, y, of manatees that regularly visit a river over a number of years, x, (starting at relationship. zero). The data is displayed in the table: x y

0 65

1 61

2 58

3 60

4 66

5 74

6 90

9.04 Quadratic regression mathspace.co

a Was the data most likely collected through measurement, observation, a survey or an experiment?

Create a strategy

1151

Students: Page 549 Example 2

A conservationist tracks the population, y, of manatees that regularly visit a river over a number of years, x, (starting at zero). The data is displayed in the table: x y

0 65

1 61

2 58

3 60

4 66

5 74

6 90

a Was the data most likely collected through measurement, observation, a survey or an experiment?

Create a strategy Consider whether the population was measured (with a measurement tool such as a rule or protactor) or observed. Also consider whether anyone was surveyed or whether any variables were controlled.

Apply the idea The population of manatees was not measured, and the conservationist did not survey anyone to collect the data. The information does not specify whether any other variables were controlled, so we can assume that an experiment was not used. The data was most likely collected by observation.

Reﬂect and check Many times, populations of species are tracked using tracking devices. It is possible that the manatees each have a tracking device, and a conservationist collects data from those devices each year.

Purpose Show students how to deduce the method used to collect data based on the nature of the data and context given. 9.04 Quadratic regression mathspace.co

Students: Page 550

549

b Determine if the manatee population over time has a quadratic relationship.

Create a strategy Construct a scatterplot to visually determine if a linear or quadratic model is a better ﬁt.

Apply the idea After plotting the data on a graph, we get the following scatterplot: y 90 85 80 75 70 65 60

x 1

There is a clear curve in the pattern of the data, so a quadratic function would better ﬁt the data.

c Using technology, determine an appropriate equation to model the data set. Round all values to two decimal places.

Purpose Show students to use graphical analysis to determine whether the relationship between two variables is Create ahow strategy quadratic. To ﬁnd the equation using technology, we can follow these steps: 1. Click the plus sign in the top left corner of the screen, and select table. 1152 Mathspace Virginia SOL Algebra 1 Teacher Edition 2. Enter the x-values and y-values in the respective columns of the table. mathspace.co 3. In a new line beneath the table, enter the equation y1 ∼ ax21 + bx1 + c.

85 80 75 70

Reflecting with students 65 Ask students to describe the strength of the quadratic relationship between the variables. Point out that 60 although the relationship is strong, it is not perfectly quadratic. While xit is possible for relationships to be 1 2 data. 3 4 5 6 7 perfectly quadratic, it is rare to see that in real-world is a clear curve in the pattern of the data, so a quadratic function would better ﬁt the data. Students:There Pages 550–551 c Using technology, determine an appropriate equation to model the data set. Round all values to two decimal places.

Create a strategy To ﬁnd the equation using technology, we can follow these steps: 1. Click the plus sign in the top left corner of the screen, and select table. 2. Enter the x-values and y-values in the respective columns of the table. 3. In a new line beneath the table, enter the equation y1 ∼ ax21 + bx1 + c.

Apply the idea 1. Click the plus sign in the top left corner of the screen, and select table.

2. Enter the x-values and y-values in the respective columns of the table. Then, click the magnifying glass with the plus sign on the bottom left side of the table to see the points.

550

Mathspace Virginia SOL Algebra 1 mathspace.co

3. In a new line beneath the table, enter the equation y1 ∼ ax21 + bx1 + c.

9.04 Quadratic regression 1153 mathspace.co

3. In a new line beneath the table, enter the equation y1 ∼ ax21 + bx1 + c.

Rounding the values to two decimal places, we find the approximate curve of best fit is y = 1.94x2 − 7.75x + 65.74.

Purpose Show students how to use Algorithmic thinking to follow a set of step to perform regression analysis with technology to find an equation that best fits a set of data.

Students: Page 552 Quadratic regression d Using the model in part (b), determine the population 10 years afer the numbers were 9.04 ﬁrst recorded.

551

mathspace.co

Create a strategy We can ﬁnd the population, y, after 10 years by substituting x = 10 into the equation of the curve of best ﬁt.

Apply the idea y = 1.94x2 − 7.74x + 65.74

State the equation

y = 1.94 (10)2 − 7.74 (10) + 65.74

Substitute x = 10

y = 182.34

Evaluate

We can see that after 10 years, the population will have grown to about 182 manatees.

Reflect and check Remember that the coefficients in the equation for the curve of best fit have been rounded. Rounding values reduces the accuracy of the prediction. If we had used technology to make this prediction, we would have gotten a slightly different answer.

1154 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Apply the idea y = 1.94x2 − 7.74x + 65.74

State the equation

y = 1.94 (10) − 7.74 (10) + 65.74

Substitute x = 10

y = 182.34

Evaluate

We can see that after 10 years, the population will have grown to about 182 manatees.

The calculator’s answer is more accurate because it includes more decimal values in the coeﬃcients and does not round them to only four place values. However, the differences between these values are small and does not change our ﬁnal, rounded answer.

Purpose Show students how to use a quadratic model to predict future values of a variable. Expected mistakes Students might not consider the context of the problem and state the population as a decimal value. Remind students that we cannot have part of a manatee, so we need to round to the nearest whole manatee.

Converting data to vertical tables

use with Example 2

Student with disabilities support When working with regressions and curves of best fit, students may benefit from additional practice converting from ordered pairs to tabular data. Since most technology requires entering the data into columns, creating a 552 Mathspace Virginia SOL Algebra 1 vertical tablemathspace.co can help them avoid entry mistakes. For example, data sets given as a set of coordinate pairs or a horizontal table: • {(0, 65), (1, 61), (2, 58), (3, 60), (4, 66), (5, 74), (6, 90)} •

x y

0 65

1 61

2 58

3 60

4 66

5 74

6 90

9.04 Quadratic regression 1155 mathspace.co

can be written as a table with columns instead: x 0 1 2 3 4 5 6

y 65 61 58 60 66 74 90

This allows students to enter the data into their calculators exactly as shown on their paper.

Students: Page 553 Example 3 Carlos is a goalie on the school soccer team. When he kicks a soccer ball dropped from his hands, he notices that the angle of trajectory for each kick is different. He also notices that there are times when the ball does not travel as far as other times. He wants to investigate this further using the data cycle. a Formulate a statistical question that Carlos can use for his investigation.

Create a strategy

Example 3

We can assume that Carlos is interested in determining the optimum angle at which he should kick a soccer ball dropped from his on hands achieve the team. maximum distance. are many statistical questions we he cannotices ask, but wethe Carlos is a goalie the to school soccer When he kicksThere a soccer ball dropped from his hands, that should the question purpose thenotices investigation. angle offocus trajectory for eacharound kick is the different. He of also that there are times when the ball does not travel as far as other times. He wants to investigate this further using the data cycle.

Apply the idea a Formulate a statistical question that Carlos can use for his investigation.

One possible statistical question is, “At what angle should Carlos kick the soccer ball for it to travel farthest?”

Create a strategy Reﬂect and check

We can assume that Carlos is interested in determining the optimum angle at which he should kick a soccer ball Other possible questions dropped from his hands toare: achieve the maximum distance. There are many statistical questions we can ask, but we • How does the distancearound the ballthe travels change with the angle of trajectory? should focus the question purpose of the investigation. • If Carlos kicked the ball and it traveled 130 feet, what was the ball’s angle of trajectory? • If the ballidea is kicked at the optimum angle, what is the farthest distance the ball will travel? Apply the One possible statistical question is, “At what angle should Carlos kick the soccer ball for it to travel farthest?” b Determine what variables could be used to answer the statistical question formulated in part (a).

Reﬂect and check

PurposeApply the idea Other possible questions are: Make students awarethat that theythe can formulate a statistical question based on a given scenario. • How distance ballneed travels change withon the of the trajectory? The two does thingsthe Carlos would to collect data toangle answer question are the angle of trajectory for each • Ifand Carlos thethe ballball and it traveled 130 feet, what was the ball’s angle of trajectory? kick thekicked distance travels.

Students:•Page 553 If the ball is kicked at the optimum angle, what is the farthest distance the ball will travel? Reﬂect and check The angle of trajectory can impact the ballthe travels, but the distance the ball travels impact the b Determine what variables couldthe be distance used to answer statistical question formulated in partcannot (a). angle of trajectory. This means the angle of trajectory is the independent variable, and the distance the ball travels is the dependent Apply the ideavariable. The two things that Carlos would need to collect data on to answer the question are the angle of trajectory for each and the distance the ball ckickCarlos records 10 kicks andtravels. analyzes them to determine the angle of trajectory and also the distance traveled. His results are recorded in the table:

Reﬂect and check

Angle (degrees) 24 30 33 37 43 48 51 56 60 64 The angle of trajectory impact the distance the ball the distance travels cannot impact the Distance (feet) can112 129 138 155 161travels, 164but 158 148 the 134 ball124 angle of trajectory. This means the angle of trajectory is the independent variable, and the distance the ball travels is the data suggests a linear or quadratic relationship. Explain your answer. the Determine dependentifvariable.

Create a strategy c Carlos records 10 kicks and analyzes them to determine the angle of trajectory and also the distance traveled.

We His canresults determine if the datainsuggests are recorded the table:a linear or quadratic relationship by plotting the points on a coordinate plane and determining if the data resembles a line or a parabola. (degrees) 24 1 Teacher 30 33 43 48 51 56 60 64 Virginia SOL Algebra Edition37 1156 MathspaceAngle To do this using technology, we can follow these steps: mathspace.co Distance (feet) 112 129 138 155 161 164 158 148 134 124 1. Click the plus sign in the top left corner of the screen, and select table. Determine if the data suggests a linear or quadratic relationship. Explain your answer. 2. Enter the x-values and y-values in the respective columns of the table.

Apply the idea The two things that Carlos would need to collect data on to answer the question are the angle of trajectory for each kick and the distance the ball travels.

PurposeReﬂect and check Show students how to identify the independent and dependent variables in a statistical investigation. The angle of trajectory can impact the distance the ball travels, but the distance the ball travels cannot impact the angle of trajectory. This means the angle of trajectory is the independent variable, and the distance the ball travels is

Students:thePages 553–554 dependent variable.

c Carlos records 10 kicks and analyzes them to determine the angle of trajectory and also the distance traveled. His results are recorded in the table: Angle (degrees) Distance (feet)

24 112

30 129

33 138

37 155

43 161

48 164

51 158

56 148

60 134

64 124

Determine if the data suggests a linear or quadratic relationship. Explain your answer.

Create a strategy We can determine if the data suggests a linear or quadratic relationship by plotting the points on a coordinate plane and determining if the data resembles a line or a parabola. To do this using technology, we can follow these steps: 1. Click the plus sign in the top left corner of the screen, and select table. 2. Enter the x-values and y-values in the respective columns of the table. 3. To adjust the scales of the axes to see the data, click the magnifying glass with the plus sign on the bottom left side of the table.

Apply the idea 1. Click the plus sign in the top left corner of the screen, and select table.

9.04 Quadratic regression mathspace.co

553

2. Enter the x-values and y-values in the respective columns of the table.

3. To adjust the scales of the axes to see the data, click the magnifying glass with the plus sign on the bottom left side of the table.

9.04 Quadratic regression 1157 mathspace.co

3. To adjust the scales of the axes to see the data, click the magnifying glass with the plus sign on the bottom left side of the table.

The data has a parabolic shape which is symmetric. The y-values begin increasing, then reach a maximum value, then decrease after. This means the data has a quadratic relationship.

554

Mathspace

Virginia SOL Algebra 1

mathspace.co Purpose Challenge students to identify the type of relationship between two variables by examining a scatterplot.

Students: Page 555 d Using technology, determine an appropriate equation to model the data set.

Create a strategy We can use technology to calculate the quadratic regression equation. In a new line beneath the table, enter the equation y1 ∼ ax21 + bx1 + c.

Apply the idea

y = −0.1132x2 + 10.3245x − 74.5885, where x is the angle of trajectory (in degrees) and y is the distance traveled (in feet).

1158 Mathspace Virginia SOL Algebra 1 Teacher Edition Reﬂect and check mathspace.co If the instructions do not specify to round the coeﬃcients, it is best to include all the digits given by the calculator. This increases the accuracy of the model and the predictions.

d Using technology, determine an appropriate equation to model the data set.

Create a strategy We can use technology to calculate the quadratic regression equation. In a new line beneath the table, enter the equation y1 ∼ ax21 + bx1 + c.

Apply the idea

d Using technology, determine an appropriate equation to model the data set.

Create a strategy We can use technology to calculate the quadratic regression equation. In a new line beneath the table, enter the equation y1 ∼ ax21 + bx1 + c.

Apply the idea

y = −0.1132x2 + 10.3245x − 74.5885, where x is the angle of trajectory (in degrees) and y is the distance traveled (in feet).

Reﬂect and check If the instructions do not specify to round the coeﬃcients, it is best to include all the digits given by the calculator. This increases the accuracy of the model and the predictions.

e Draw a conclusion about the data by answering the statistical question from part (a).

Purpose Create a strategy Show students how to use technology to find a quadratic regression equation for a given set of data.

The statistical question from part (a) is, “At what angle should Carlos kick the soccer ball for it to travel farthest?” When considering the quadratic regression model, the largest y-value represents the farthest distance traveled by Reflecting with students the ball. If students state the 2 equation without describing what the variables in the equation represent, encourage them yThe = −0.1132x 10.3245x − 74.5885, where x is the angle of trajectory (in degrees) and y(x) is that the distance traveled vertex is + the maximum point of the parabola and represents the angle of trajectory Carlos should kick the to describe what x and y represent. If they struggle to do this, ask them which variable is the independent (in feet). ball for it to travel farthest (y). We can ﬁnd this angle (x-value) using the equation .

variable and which is the dependent variable. Then, they can connect the variables in the equation back to the Reﬂect and check contextual variables they identified in part (b). Apply the idea

If the instructions do not specify to round the coeﬃcients, it is best to include all the digits given by the calculator. The vertex represents the optimum angle and to kick ball to achieve the maximum distance traveled. This increases the accuracy of the model thethe predictions.

Students: Pages 555–556

e Draw a conclusion about the data by answering the statistical question from part (a).

Create a strategy

9.04 Quadratic regression mathspace.co

555

The statistical question from part (a) is, “At what angle should Carlos kick the soccer ball for it to travel farthest?” When considering the quadratic regression model, the largest y-value represents the farthest distance traveled by the ball. The vertex is the maximum point of the parabola and represents the angle of trajectory (x) that Carlos should kick the ball for it to travel farthest (y). We can ﬁnd this angle (x-value) using the equation

Apply the idea The vertex represents the optimum angle to kick the ball to achieve the maximum distance traveled.

9.04 Quadratic regression mathspace.co

555

9.04 Quadratic regression 1159 mathspace.co

The statistical question from part (a) is, “At what angle should Carlos kick the soccer ball for it to travel farthest?” When considering the quadratic regression model, the largest y-value represents the farthest distance traveled by the ball. The vertex is the maximum point of the parabola and represents the angle of trajectory (x) that Carlos should kick the ball for it to travel farthest (y). We can ﬁnd this angle (x-value) using the equation

Apply the idea The vertex represents the optimum angle to kick the ball to achieve the maximum distance traveled. The equation of the curve of best ﬁt is y = −0.1132x2 + 10.3245x − 74.5885, where a = −0.1132 and b = 10.3245. Equation of the x-value of the vertex Substitute a = −0.1132, b = 10.3245

9.04 Quadratic regression mathspace.co

555

Simplify For the ball to travel farthest, Carlos would need to kick the ball at an angle of about 45.6°.

Reﬂect and check To ﬁnd the farthest distance the ball is expected to travel, we can substitute x = 45.6 into the equation and solve for y. y = −0.1132x2 + 10.3245x − 74.5885

State the equation

y = −0.1132(45.6)2 + 10.3245(45.6) − 74.5885

Substitute x = 45.6

y ≈ 160.8

Simplify

The vertex occurs at about (45.6, 160.8) which means that the maximum distance of 160.8 feet is achieved by kicking the ball at an angle of 45.6°. When looking at the raw data, we see that Carlos actually kicked the ball farther than this. One of his kicks traveled 164 feet when it was kicked at an angle of 48°. This implies that there are other factors that affect the distance the ball travels, such as the force Carlos uses to kick the ball.

Idea summary

Purpose Data presents a quadratic relationship if it forms a symmetric curve or parabolic shape. Challenge students to draw conclusions from a statistical investigation by using the results of a quadratic If points are more tightly clustered along the model, it represents a stronger relationship between regression analysis. the variables. Expected mistakes If students do not consider the independent and dependent variables in this problem, they might think the x-valuesPractice represent the distance the ball travels horizontally, and therefore think the rightmost x-intercept represents the farthest distance. To help What students the problem, draw a few examples of Carlos’ kicks using the angle of trajectory do understand you remember? and the distance the ball travels. 1

Masturah is using an app on her phone to learn French. She uses the app to learn and practice her French each day, and the following day, the app quizzes her on how much she remembered from the previous day. a

Which statistical question would lead to data that can be represented by a scatterplot? A What is the average amount of time Masturah spends learning French each day? B What day of the week does Masturah practice French the longest? C How many times does Masturah practice French in a week? D What amount of time should Masturah practice French each day to maximize the amount she for the following day? 24°remembers 33° 48°

112 ft

138 ft

Determine the variables that could be used to answer the statistical question.

y = 4x2 + 5

164 ft

Then, help students see that the angles are the x-values of the points, and the distances are the y-values of the 2 Determine or balls not thethat following quadratic points. Then, point outwhether that the travelare farther arefunctions: the ones with higher y-values. b

y2 = x2 − 5x + 6

e y the = 10xdata +9 Scaffold cycle f

y = 4(x − 7)2 + 8

y=x+2

y = (x − 5) (x − 8)

use with Example 3

Student with disabilities support To help students with disabilities navigate the data cycle in Carlos’s soccer kick investigation, provide explicit support at each stage of the cycle. Start by introducing the data cycle visually—display the diagram of the data Mathspace Virginia SOL Algebra 1 cycle in 556 the classroom and refer to it frequently. Break down each step: mathspace.co 1. Formulate questions: Guide students in crafting the statistical question by providing sentence starters or question templates. For example, “What is the relationship between ⬚ and ⬚?” Encourage them to identify what Carlos wants to find out about his kicks.

1160 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

2. Collect or acquire data: Model how data can be collected in this context. Use a sample data table similar 2 to Carlos’s and demonstrate howﬁtto angles and distances. a blank data table The equation of the curve of best is yrecord = −0.1132x + 10.3245x − 74.5885, Provide where a =students −0.1132 andwith b = 10.3245. template to fill in. Equation of the x-value of the vertex

3. Organize and represent data: Assist students in organizing the data by creating graphs or charts. Offer a = −0.1132, b = 10.3245 graph paper with labeled axes orSubstitute use graphing software with preset parameters. Walk them through plotting the data points step-by-step. Simplify 4. Analyze and Teach students how graphs For the ballcommunicate to travel farthest,results: Carlos would need to kick the balltoatinterpret an angle ofthe about 45.6°. by identifying patterns or trends. Use guiding questions like, “What do you notice about the shape of the data?” Provide sentence Reﬂect andthem checkarticulate their conclusions, such as “The data suggests a ⬚ relationship because ⬚.” frames to help To ﬁnd the farthest distance the ball is expected to travel, we can substitute x = 45.6 into the equation and solve for y.

By scaffolding each phase2 of the data− 74.5885 cycle, you enable students toequation focus on one component at a time, + 10.3245x State the y = −0.1132x 2 reducing cognitive Incorporate periodic− 74.5885 check-ins afterSubstitute each step to ensure understanding before moving + 10.3245(45.6) x = 45.6 y = load. −0.1132(45.6) on. This structured approach makes the statistical investigation more accessible and helps students build y ≈ 160.8 Simplify confidence in working data. The vertex occurs with at about (45.6, 160.8) which means that the maximum distance of 160.8 feet is achieved by kicking the ball at an angle of 45.6°. When looking at the raw data, we see that Carlos actually kicked the ball farther than this. One of his kicks traveled feet when it was kicked at an angle of 48°. This implies that there are other factors that affect the distance the ball Students:164 Page 556 travels, such as the force Carlos uses to kick the ball.

Idea summary Data presents a quadratic relationship if it forms a symmetric curve or parabolic shape. If points are more tightly clustered along the model, it represents a stronger relationship between the variables.

Practice What do you remember? Practice 1

Masturah is using an app on her phone to learn French. She uses the app to learn and practice her French

each 556–561 day, and the following day, the app quizzes her on how much she remembered from the previous day. Students: Pages a

Which statistical question would lead to data that can be represented by a scatterplot? A What is the average amount of time Masturah spends learning French each day?

B What day of the week does Masturah practice French the longest? What do you remember? C How many times does Masturah practice French in a week?

D What amount of time should Masturah practice French each day to maximize the amount she

Masturah is using an app on her phone to learn French. She uses the app to learn and practice her French remembers for the following day? each day, and the following day, the app quizzes her on how much she remembered from the previous day. a

Determine the variables that could be used to answer the statistical question.

Which statistical question would lead to data that can be represented by a scatterplot? Determine or not the following quadratic functions: A2 What is thewhether average amount of timeareMasturah spends learning French each day? a

y = 4x2 + 5

y2 = x2 − 5x + 6

y = 10x + 9

y = 4(x − 7) + 8

y=x+2

B What day of the week does Masturah2practice French the longest?

y = (x − 5) (x − 8)

C How many times does Masturah practice French in a week? D What amount of time should Masturah practice French each day to maximize the amount she remembers for the following day? b 2

SOL Algebra 1 556 Mathspace Determine the Virginia variables that could be used to answer the statistical question. mathspace.co

Determine whether or not the following are quadratic functions: a

y = 4x2 + 5

y2 = x2 − 5x + 6

y = 10x + 9

y = 4(x − 7)2 + 8

y=x+2

y = (x − 5) (x − 8)

9.04 Quadratic regression mathspace.co

1161

Determine whether or not the following graphs could represent a quadratic relation: a

b 9

−3 −2

2 x

−1

1 1

x 2

x 1

Determine whether or not the following tables could represent a quadratic function: a

x −2 −1 0 y 40 24 10

1 8

2 18

x y

1 −6

2 −16

3 −24

4 −29

5 −26

x 12 13 14 15 y 6 2 1 0

16 −6

x y

0 9

1 9

2 9

3 9

4 9

Consider the scatterplot:

1.2

Select a quadratic function that ﬁts the data the best. A

y = 0.3(x − 9)2 + 1.2

y = −0.03(x − 20)2

y = −0.003(x − 19) + 1

y = 3(x − 25)2 + 0.8

1 0.8 0.6 0.4 0.2 0

15 20 25 30 35

Let’s practice SOL

Consider the data set shown: {(−8, 15.3), (−6, 25.1), (−4, 31.5), (−2, 35.2), (0, 37.8), (2, 35.6), (4, 30.1), (6, 21.7), (8, 10.4)} Select the equation of the curve of best fit. A C

y = 0.38x2 − 0.3x + 37 2

y = −1.3x − 0.3 + 37.1

1162 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

y = 1.3x2 + 0.27x + 37

y = −0.38x2 − 0.27x + 37.1

Calculate the quadratic curve of best fit for each data set: a

x −3 −2 −1 0 1 y −18 −20 −26 −23 −24

2 −19

3 −14

x y

0 17

1 10

2 3.5

3 2

4 0

5 2

6 5.5

x −3 −2 −1 0 y 3.1 8.8 15.9 19.5

2 10.7

3 5.1

x y

−7 −12

−6 −4

−5 1

−4 6

−3 5

−2 −2

−1 −9

1 17.2

Match each regression model to the set of data it fits best. i

y = 0.4x2 − 5.67x + 41

y = −0.22x2 + 2.72x + 8.52

iii

y = −0.32x2 + 0.48x + 19.7

y = 0.22x2 − 1.6x + 2.27

x 0.9 1.8 6.2 7.3 y 1 0.1 1 2

8.5 4.7

7.6 2.9

y 8

7 20

2.8 27

16 52

0 42

19 75

13 6.8

15 0.6

x 1

2 4 6 8 10 12 14 16 18 20

x 3.7 8.3 0.5 2.8 y 15 6 19 18

6.1 12

9 3

y 22 20 18 16 14 12 10 8 6 4 2

x y 18

4 16

11 12

0.6 9.6

1.7 13

16 14 12 10 8 6 4 2

x 1 2 3 4 5 6 7 8 9 10

13 35

x y

x 2 4 6 8 10 12 14 16 18

The creators of the online game Nomad’s Horizon formulated the statistical question, “How has the number of people playing our game changed over time?” They collected data on the number of people, y, playing the game in the years after its release, x. x y

0 419

1 112

2 13

3 148

4 397

5 855

6 1602

Describe the independent and dependent variables.

Is the relationship between the variables linear or quadratic?

Find the equation of the curve of best fit. Round all values to two decimal places. 9.04 Quadratic regression 1163 mathspace.co

The population, y, of a particular species of bird is tracked over a number of years, x, (starting at zero), with the data displayed in the table: x y

0 64

1 63

2 65

6 96

7 113

8 127

Formulate a question that could be answered by the data. Which data collection method was most likely used? B

Observation

9 149

10 161

11 180

12 208

Experiment

Survey

Determine an appropriate equation to model the data. Round all values to two decimal places.

Using the model in part (c), predict what the population will be 20 years after the species was first recorded.

Ten pregnant women at various weeks of pregnancy were asked at their medical appointments to rate their level of discomfort on a scale of 0 to 10 where 0 is completely comfortable and 10 is in severe discomfort or pain. The results are displayed in the given graph table. 8 5

12 3

16 2

20 1

22 1

24 2

28 3

32 6

36 7

40 10

Was the data collected through measurement, observation, a survey or an experiment?

Determine if the data suggests a quadratic relationship. Explain your answer.

Determine an appropriate equation to model the data set. Round all values to four decimal places.

Interpret the meaning of the vertex of the model.

Explain the significance of there being no x-intercepts.

Jiang is helping his mom to determine the best price for a dozen eggs for new contracts. Some experimentation and research provided the results shown for different expected profits based on the price. Price per dozen Proﬁt per month($)

5 86

Week Discomfort Level

4 82

A Measurement

3 75

1.3 3200

1.35 3230

1.4 3250

1.45 3210

1.5 3100

1.55 3000

1.6 2800

Formulate a question that could be answered by the data.

Determine if the data suggests a quadratic relationship. Explain your answer.

Determine an appropriate equation to model the data set with integer coefficients.

Determine the price which the model predicts would result in the highest profit.

Interpret the y-intercept.

Interpret the x-intercepts.

Researchers collected data to answer the statistical question, “How does the electricity consumption (in kilowatt-hours) of households change throughout the afternoon and evening hours?” a

Describe the variables that could be used to answer the statistical question.

The data the researchers collected is shown in the table. Describe the relationship between the variables. Hours after noon Usage (kWh)

0.5 1.2

2.5 2.3

3.5 2.7

4.5 3

5 3.4

6.5 3

7.5 3.2

1 1.5

8.5 2.6

Hours after noon Usage

2 1.9

9.5 2.3

10.5 2

0 1

11.5 1.4

1.5 1.7

3 2

4 2.2

12 0.9

Hours after noon Usage (kWh)

5 2.8

6 3.5

7 3.8

8 3.6

9 3

10 2.5

11 1.8

12 1.2

1164 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

Calculate the regression model for this data.

Answer the researchers’ statistical question.

Use the curve of best fit to predict a household’s electricity consumption at 5 p.m.

What time(s) of the day is the model’s predictions most reliable? A From noon to 2:30 p.m. and from 10:30 p.m. to midnight B From 2:30 p.m. to 10:30 p.m. C From noon to 7 p.m. D From 7 p.m. to midnight

Let’s extend our thinking 14

At the beginning of the school year, Shirah and her friends decided that they want to take a trip during spring break in March. They researched average prices of flights and found the following data. Months until trip Average price

6 $540

5 $503

4 $461

3 $432

2 $450

1 $520

0 $637

Formulate a question Shirah and her friends can use for their investigation.

Calculate the regression model for this data.

Determine when Shirah should purchase her flight and accommodation. Explain your reasoning.

The table shows the average weekly wage (in dollars) of an American resident from 1996 to 2006, where x is the number of years since 1996. Year x f (x)

7 $547

1996 0 800

2000 4 961.76

2003 7 1065.86

2006 10 1155.20

Determine whether a linear or quadratic function would accurately model this situation. Explain your reasoning.

Predict the average weekly wage of an American in 2010.

Would the model from part (a) make sense for long term analysis? Explain your answer.

Write a report about the changes in the average weekly wage of an American resident from 1996–2016.

The populations of U.S. cities are constantly changing. Some cities see large increases in population, while others face large decreases in population size. a

Formulate a question about the population of Pittsburgh, Pennsylvania that would require the collection of bivariate data.

Describe the variables that could be used to answer the question from part (a).

Collect the census data on the population of Pittsburgh, Pennsylvania from 1870 to 2000.

Use technology to create a scatterplot and describe the form or shape of the data.

Use technology to find an appropriate equation to model the data set.

Draw a conclusion about the data by answering the question formulated in part (a).

Could the model from part (e) be used to make reasonable predictions after 2000? Explain your answer.

9.04 Quadratic regression 1165 mathspace.co

Answers

12 a M any possible questions, for example, “What price of eggs will yield the highest profit?”

9.04 Quadratic regression

b Y es, a quadratic model is appropriate because when drawn on a graph, it has a parabolic shape which is symmetric.

What do you remember? 1 a D b T he independent variable is the amount of time Masturah practices French daily, and the dependent variable is the amount she remembers the following day. 2 a Yes e No

b No f

c No

d Yes

c y = −9000x2 + 24 807x − 13 845, where x is the price per dozen of eggs and y is the proﬁt. y 3400 3200

Yes

3000

3 a No

b Yes

c No

d Yes

4 a Yes

b Yes

c No

d No

2800 2600 x

5 C 0.5

Let’s practice 6 C 7 a y = 0.976x2 + 0.571x − 24.5 b y = 1.12x2 − 8.64x + 17.1 c y = −1.62x2 + 0.396x + 18 d y = −1.75x2 − 13.4x − 20.7

c iii: y = −0.32x2 + 0.48x + 19.7

b Quadratic

13 a T he independent variable is the time of day between noon and midnight, and the dependent variable is the electricity consumption in kilowatt-hours. b B y creating a scatterplot, we can see that there is a quadratic relationship between the variables. 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5

10 a M any possible questions, for example, “How has the population of this species of bird changed over time?” b B c y = 0.98x2 + 0.14x + 63.02 d 458 birds 11 a Survey

c y = 0.0223x2 − 0.8950x + 10.5078, where x is the number of weeks and y is level of discomfort d T he vertex is a minimum and occurs at about (20.07, 1.53) which means that women are generally the most comfortable around 20 weeks and are fairly comfortable then. e T his means that there is no point in pregnancy where these women felt completely comfortable.

1166 Mathspace Virginia SOL Algebra 1 Teacher Edition mathspace.co

2.5

The x-intercepts are the two prices that will result in breaking even, so the cost of the hens is equal to the amount of money earned by selling the eggs. This will occur at the prices of $0.78 and $1.98.

c y = 98.612 − 398.12x + 419

b Y es, a quadratic model is appropriate because when drawn on a graph it has a parabolic shape which is symmetric.

e I f they were to give their eggs away for free (a price of $0, then they would lose $13 845 per month with the cost of the hens.

b i: y = 0.4x2 − 5.67x + 41

9 a T he independent variable is the years after the game’s release, and the dependent variable is the number of people playing the game.

1.5

d T he vertex is a maximum and occurs at about (1.378, 3249) which means that the model predicts $1.38 to be the price that would result in the highest proﬁt.

8 a iv: y = 0.22x2 − 1.6x + 2.27

d ii: y = −0.22x2 + 2.72x + 8.52

Usage (kWh)

Hours after 12 p.m. 1 2 3 4 5 6 7 8 9 10 11 12

c y = −0.064x2 + 0.8072x + 0.6719 where x is the hours after 12 p.m. and y is the electricity consumption in kilowatt-hours d T he electricity consumption of households increases from noon to about 7 p.m., then it decreases from 7 p.m. to midnight. e T he approximate electricty consumption of a household at 5 p.m. (x = 5) is 3.1 kWh. f

Let’s extend our thinking 14 a M any possible answers, for example, “In what month will the price of flights be lowest?” b If x is deﬁned as the number of the month with the current month being x = 0, next month is x = 1, etc., then the month of the trip is month 7. An example of a scatterplot using this deﬁnition is shown: Price 600

500 450 Month 1

A quadratic model can be used to analyze the data. Using technology to find the quadratic regression function for the data gives f (x) = 12.05x2 − 88.40x + 609.83 where f (x) represents the price of the flight. b E xample answer: Shirah should purchase her flight between the middle of November and beginning of December. According to the model, flights are cheapest in the middle of November. Based on past data, flights are cheapest at the beginning of December. Regardless, flights are the cheapest within that range of dates compared to the rest of the year. 15 a When graphed, the data values appear as shown: 1250 1200 1150 1100 1050 1000 950 900 850 800

This study only analyzed how weekly wages increased from 1996–2016. No research was conducted on why the weekly wages were increasing more each year before 2006.

b A nswers will vary based on question from part (a). Using the example, “How has Pittsburgh’s population changed over time?”: The independent variable is the time in years, and the dependent variable is the population. c

A quadratic model ﬁts the data best because the increase in average weekly wage is decreasing over time, and the curve slopes upward more steeply for smaller values of x. Using technology to determine a regression model, a quadratic function would model the data perfectly. b E xample of a quadratic regression model: f (x) = −0.82x2 + 43.72x + 800 Using the quadratic model: $1251.36

c I f the long term implications are assessed, a linear model would be more accurate because wages will only increase. It would not make sense for wages to eventually decrease, as a quadratic model suggests. Using technology to determine a regression model, a linear function could model relatively well. Example of a linear regression model: f (x) = 36x + 809

Year

1870

1880

1890

1900

1910

1920

1930

Population (in thousands)

86.1

156.4 238.6 321.6 533.9 588.3 669.8

Census

Year

1940

1950

1960

1970

1980

1990 2000

Population 671.7 676.8 604.3 520.1 (in thousands)

424

369.9 334.6

650 600 550 500 450 400 350 300 250 200 150 100 50

x 4

Census

d Let x represent the census row of the table, where the 1870 census is represented by x = 1, the 1880 census is x = 2, etc. Let y represent the population in thousands.

f (x)

Weekly wages increased by an average of $35.52 each year from 1996–2006, but only increased by an average of $19.12 each year from 2006–2016.

16 a M any possible questions, for example, “How has Pittsburgh’s population changed over time?”

550

d F rom 1996–2016, weekly wages of an American resident have increased by almost $546.40

x 1 2 3 4 5 6 7 8 9 10 11 12 13 14

The data has a quadratic relationship because it has a parabolic shape. e y = −11.3212x2 + 190.162x − 162.85 f

nswers will vary based on question from part (a). A Using the example, “How has Pittsburgh’s population changed over time?”:

Pittsburgh’s population increased at a slower and slower rate from 1870 (when x = 1) to 1950 (when x = 9), then decreased at an increasing rate from 1950 to 2000 (when x = 14). g N o, it cannot be used to make reasonable predictions after 2000. According to the model, Pittsburgh’s population will continue decreasing, and there will be no one living there by 2020. According the actual data, Pittsburgh’s population was about 305 400 in 2010 and about 302 800 in 2020. This shows that the model does not follow the trend in the actual data after 2020.

Answers 1167 mathspace.co

9.05 Analyze bivariate data Subtopic overview Lesson narrative In this lesson, students will use their knowledge of regression to determine the model needed given a set of data. Given a set of data points, students will graph the points to determine the type of equation needed. Students will find the appropriate regression model depending if it is linear or quadratic. Students will interpret the correlation coefficient or coefficient of determination to determine if the model represents a good fit of the data. By the end of the lesson, students will be able to determine the appropriate regression model, calculate the regression model for the data set, draw conclusions about the bivariate data, and use the correlation coefficient or coefficient of determination to evaluate if the model is a good fit.

Learning objectives

9.05 Analyze bivariate data

Students: Page 562

After this lesson, you will be able to… • analyze relationships of variables given a set of bivariate data. • make conclusions given a set of bivariate data.

Analyze bivariate data The process of analyzing bivariate data involves a two-step process. First, we plot the data on a scatterplot. Key vocabulary

This allows us to visually inspect the relationship between the two variables. Then, we use mathematical models to describe this relationship. Two common models that we have used are the linear regression model and the  linear  quadratic bivariate data model model quadratic regression model.  regression equation  scatterplot 

Interactive exploration Explore online to answer the question Essential understanding

mathspace.co The relationship between the variables in a set of bivariate data reveals the type of function that best models the data. Use the interactive exploration in 9.05 to answer this question. 1.

Which function fits the data better? How do you know?

Standards

This addresses following Standards of we Learning fortechnology Mathematics Tosubtopic determine the curvethe of best fit for aVirginia set of bivariate data, can use suchstandards. as graphing calculators or software. These tools allow us to perform both linear and quadratic regression on the same set of data and compare Mathematical process goals the results. MPG1 — Mathematical Solving To decide which curveProblem best models the data, we can visually assess whether the curves follow the trend in the data and how close the points are to each curve. We can also use thestudents context to determine if a model is a good Teachers can integrate this goal into their instruction by having solve real-world problems wherefit.they determine which model, linear or quadratic, best represents the relationship between two variables. Through these Distance from ground (ft.) problems, students can apply mathematical concepts and skills and the relationships among them to solve problem situations of varying complexities. This can be especially effective with the use of technology, such as graphing 48 and quadratic regression on the same set of data and compare the calculators or software, to perform both linear results. 32 1168 Mathspace Virginia SOL Algebra 1 Teacher Edition 16 mathspace.co

Time

MPG2 — Mathematical Communication

MPG4 — Mathematical Connections

To incorporate this goal, teachers can emphasize the importance of communicating the results of the analysis, including a clear explanation of the chosen model and the reasoning behind the choice. Teachers can encourage students to use the language of mathematics, including graphs, equations, and descriptive language, to effectively communicate their results.

To incorporate this goal, teachers can build on students’ prior knowledge of scatterplots, linear regression, and quadratic regression from previous lessons. By making connections between these previous concepts and the current lesson on analyzing bivariate data, students can see mathematics as an integrated field of study. Real-world examples can be used to reinforce the relevance of these mathematical connections.

MPG5 — Mathematical Representations Teachers can integrate this goal into their instruction by encouraging students to represent and describe mathematical ideas, generalizations, and relationships using a variety of methods in the context of analyzing bivariate data. This can include physical, visual, symbolic, verbal, and contextual representations. For example, students could use scatterplots to visually represent data, equations for symbolic representation, and written explanations for verbal representation.