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Grade 8 Student Notebook

GEORGIA

Student Notebook – Grade 8 ISBN: 978-1-64861-275-6 Published by Accelerate Learning Inc., 5177 Richmond Ave, Suite 800, Houston, TX 77056. Copyright © 2023 by Accelerate Learning Inc. All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without prior written consent of Accelerate Learning Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning.

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GEORGIA

Student Notebook - Grade 8

Table of Contents Scope Name

Page Number

Square Roots and Cube Roots

Irrational Numbers

Integer Exponents

Scientiﬁc Notation

Operations with Scientiﬁc Notation

Solve Equations

Solve Inequalities

103

Create Non-Proportional Relationships from Proportional Relationships

129

Functions

137

Rate of Change and Initial Value

161

Linear Forms

179

Bivariate Data

199

Parallel and Perpendicular Lines

217

Solving Pairs of Linear Equations

233

iii

GEORGIA

Student Notebook - Grade 8

Table of Contents (Cont.) Scope Name

Page Number

Pythagorean Theorem

255

Volume

275

Skills Quizzes

289

Glossary of Terms

365

Workspace

401

Square Roots and Cube Roots

Explore 1

Name: _______________________ Date: ___________

Square Roots and Perfect Squares Part I Use linking cubes to complete the table of side lengths and areas. Side Length

Area

Equation (s2 = A)

Square Roots and Cube Roots | 3

Explore 1

Square Roots and Cube Roots

Reflect 1. What patterns do you notice in the table of side lengths and area measurements?

2. When given the side length, how can you determine the area?

3. When you were given that the area of a square was 16, how did you know the side length?

4. What equation represents ﬁnding the area of a square with a side length of 15?

5. What equation could you write to ﬁnd the side length of a square with an area of 225?

6. Consider the equation y2 = 441. What would the solutions be? Explain.

4 | Square Roots and Cube Roots

Explore 1

Square Roots and Cube Roots

Part II Use your knowledge of perfect squares and square roots to determine the side length of each option. Option 1

Option 2

Option 3

Option 4

Square Roots and Cube Roots | 5

Square Roots and Cube Roots

Explore 1 Reflect

1. We can use square roots to ﬁnd the solutions to the equation x² = p. Explain what happens in each step of the problem below. x2 = 36 √ x2 = √ 36 x = ±6

2. What is (√ 25) ? Explain how you know.

3. Do you think all solutions to whole-number square roots are integers? Use the number line below to help explain your answer.

6 | Square Roots and Cube Roots

Square Roots and Cube Roots

Explore 2

Name: _______________________ Date: ___________

Cube Roots and Perfect Cubes Part I Use linking cubes to complete the table of side lengths and volumes of the memory boxes. Side Length

Volume

Equation (s³ = V) V

1 2 3 4 125 6 343 512 9 1,000

Square Roots and Cube Roots | 7

Explore 2

Square Roots and Cube Roots

Reflect 1. What patterns do you notice in the table of side lengths and volume measurements?

2. When given the side length, how can you determine the volume?

3. When you were given the volume of 125, how did you know the side length?

4. What equation represents ﬁnding the volume of a cube with a side length of 11?

5. Using the equation x³ = p, where p is the volume of a cube, write an equation to represent a cube with a volume of 125.

6. Reﬂect on square roots and the number of solutions they have. Will the same be true for perfect cubes and cube roots? Explain.

8 | Square Roots and Cube Roots

Explore 2

Square Roots and Cube Roots

Part II Use your knowledge of perfect cubes and cube roots to determine the missing side lengths. Option 1

Option 2

Option 3

Option 4

Square Roots and Cube Roots | 9

Square Roots and Cube Roots

Explore 2 Reflect

1. We can use cube roots to ﬁnd the solution to the equation x³ = −p. Explain what happens in each step of the problem below: x3 = −216 ∛ x3 = ∛ −216 x = −6

2. Jeanna sees that her box needs to be reduced by 64. How much does each side need to be reduced? Explain.

3. What is (∛ 27) ? Explain how you know.

4. Do you think all solutions to whole-number cube roots are integers? Use the number line below to help explain your answer.

10 | Square Roots and Cube Roots

Irrational Numbers

Explore 1

Name: _______________________ Date: ___________

Irrational Numbers vs. Rational Numbers Part I Complete the Venn diagram with words and examples to clearly show what each classiﬁcation represents.

Real Numbers

Rational Numbers Integers Whole Numbers

Irrational Numbers

Natural Numbers

Irrational Numbers | 13

Irrational Numbers

Explore 1 Part II

Work cooperatively with your group to classify each of the numbers as rational or irrational. Explain the classiﬁcation using mathematical proof. Number

√49 √49

Rational or Irrational?

Proof

Rational

√ 49 = 7 = 7 1

Rational numbers can be written as fractions.

0.5

√3

1.275

Nonrepeating decimals

14 | Irrational Numbers

Irrational Numbers

Explore 1 Number

Rational or Irrational?

Proof

5 2

−1

Repeating decimals

Reflect 1. Based on the Real Numbers Venn Diagram, is zero a rational number or an irrational number?

2. Is a terminating decimal rational or irrational? Explain using an example.

Irrational Numbers | 15

Irrational Numbers

Explore 2

Name: _______________________ Date: ___________

Decimal Expansion Part I: Converting Fractions to Decimals with Decimal Expansion Your boss at Team Builders has given you an example to show you two ways to determine the decimal expansion for a fraction. Method 1

1 3

0.333 3 1.000 –9 10 –9

0.333…

10 –9 1

Method 2 2 = 4 = 0.4 5 10

1. Which number is a terminating decimal? Explain how you know.

2. Which number is a repeating decimal? Explain how you know.

3. What other ways can I show that the digit is repeating?

Irrational Numbers | 17

Irrational Numbers

Explore 2

Use the Land Survey Cards to ﬁnd the three measurements for Riverside and Rolling Hills. Convert the fractions to decimals using decimal expansion. Use bar notation for repeating decimals. Riverside Measurement 1

Measurement 2

Measurement 3

Fraction Representation

Workspace

Decimal Representation

Rolling Hills Measurement 1

Measurement 2

Measurement 3

Fraction Representation

Workspace

Decimal Representation 18 | Irrational Numbers

Explore 2

Irrational Numbers

Reflect 1. What two types of decimals did you get when you converted the fractions to decimals?

2. Can you determine the decimal expansion for any fractions without having to do any work?

3. Can you convert an irrational number to a fraction?

4. What is the decimal form for √ 4 ?

Irrational Numbers | 19

Irrational Numbers

Explore 2 Part II: Converting Decimals to Fractions

Your boss shows you the method they use to convert a repeating decimal to a fraction using an algorithm by powers of 10. Description of Steps

Work Shown Let x = 0.333…

Multiply both sides by 10.

10x x = 3.333… 10x x = 3.333…

Subtract the original equation from the new equation.

−x x = 0.333… 9x x=3 9x x=3

Divide both sides by 9 to get x by itself.

9=9 1

x= 3 1. Why did your boss multiply by 10?

2. Why do you think multiplying by 10 helps to ﬁnd the fraction for a repeating decimal?

3. Could you multiply by a different number and have the same effect?

20 | Irrational Numbers

Irrational Numbers

Explore 2

Use your boss’s method to determine the fraction for each of the measurements at the downtown and rural locations. Downtown Youth Sports Area

Measurement 1 _______ Let x =

Measurement 2 _______ Let x =

Could you use the same strategy if the measurement was 1.8? How would that change your answer?

Irrational Numbers | 21

Irrational Numbers

Explore 2 Rural Youth Sports Area

Measurement 1 _______ Let x =

22 | Irrational Numbers

Measurement 2 _______ Let x =

Explore 2

Irrational Numbers

Reflect 1. Should you convert a fraction to a decimal to determine whether it is rational? Explain.

2. Why do you use powers of ten when converting a repeating decimal to a fraction?

3. Does this method work for 0.123? Show your work to prove if this method works.

Irrational Numbers | 23

Irrational Numbers

Explore 3

Name: _______________________ Date: ___________

Locate and Compare Irrational Numbers on a Number Line Part II Using the special Team Builders Measuring Tape, determine which two perfect squares each of the given measurements falls between. Then, estimate the decimal value of the given measurement.

Measurements

Square Below

Estimate (to the nearest tenth)

Square Above

√3 √7 √10

Reflect

√12

1. How does ﬁnding the square below and the square above the square root help to determine the approximate decimal value?

Irrational Numbers | 25

Irrational Numbers

Explore 3 Part III

Using the Team Builders Measuring Tape and estimation, complete each of the comparison statements. Measurement

Comparison Symbol

Measurement

√2

1.2

√ 18

4.36

𝜋

3.1

√3

𝜋

Reflect

(√ 2 )

√ 25 5

√100 6

1. Use the number line above to compare 7.2 and √ 70. 2. What strategy, other than using a number line, can you use to compare an irrational number and a rational number?

26 | Irrational Numbers

Irrational Numbers

Explore 4

Name: _______________________ Date: ___________

Estimate and Compare Irrational Number Expressions Look at the image of the concrete slab that was poured for a new section of the youth activity center. Participate in the class discussion to determine the length of one side of the square slab. Question

Solution

What is the perimeter of the entire square slab?

What is the expression that Team Builders provided us with?

Why is the expression divided by 4?

What is the value of √ 410? What is the length of one side?

Irrational Numbers | 27

Irrational Numbers

Explore 4

Use the Construction Cards to determine the value of each irrational number expression. Approximate irrational numbers to the nearest tenth. Fence Measurement given: Length = Width =

Expression to solve: feet

feet

Workspace:

Solution:

Garden Measurement given:

Expression to solve:

Workspace:

Solution:

28 | Irrational Numbers

Irrational Numbers

Explore 4 Basketball Court Measurement given:

Expression to solve:

Workspace:

Solution:

Square Plate Measurement given:

Expression to solve:

Workspace:

Solution:

Irrational Numbers | 29

Irrational Numbers

Explore 4 Reflect

1. Explain why when you square a square root, you get the number inside the square root symbol.

2. Is √ 2

140

the same as √ 70?

3. Place the numbers in their approximate locations on the number line. √9

√49

2𝜋

√5

4. Answer the comparison statements below. a.

√5

_____ √ 9

2𝜋

_____ √ 9

√49

_____ √ 9

30 | Irrational Numbers

Integer Exponents

125

Evaluate

5·5·5 25

5·5

Expanded

Evaluate

4·4

4·4·4

Expanded

Evaluate

2·2

2·2·2

Expanded

5−1

4−1

2−1

5−2

4−2

2−2

Integer Exponents | 33

1 1 · · 1 5 5 5

5−3

1 1 1 · · 4 4 4

4−3

1 1 · · 1 2 2 2

2−3

Name: _______________________ Date: ___________

Properties of Integer Exponents

Look at each table. Fill in the missing sections.

Part I: Finding the Pattern

Explore 1

Integer Exponents

34 | Integer Exponents

3. Looking at the chart, what could a negative exponent mean for the integer?

2. Why is 4−1 equal to 4 ?

1. Why is 41 equal to 4?

Reflect

Explore 1

Integer Exponents

Property

= 56

64 · 63 =

Then

58 ·

42 · 33 ≠ 45

But

And

42 · 43 = 45

−3

(54)2 = 58 =

74 = 73

Integer Exponents | 35

= 4−12 = (43)−4

(33)2 ≠ 31 ≠ 35

85 ≠ 83 72

= 914

(42)3 = 46 = (43)2

85 = 83 82

Look at the If and But statements below. Use them to ﬁll in the Then and And statements. Glue the Property Deﬁnition Card that matches all the statements in the Property row.

Part II: Finding the Property

Explore 1

Integer Exponents

40 =

Then

36 | Integer Exponents

Property

30 ≠ 3

But

And

30 = 1

Explore 1

71 =

81 ≠ 1

81 = 8

1 3−3

= 34

88 =

3−3 ≠

63 = 1−3

Integer Exponents

2. How did the But section help you identify the property?

1. Why is it important to know the properties of integer exponents?

Reflect

Explore 1

Integer Exponents | 37

Integer Exponents

Explore 2

Name: _______________________ Date: ___________

Multiplying with Exponents Take turns ﬂipping the Matching Cards. When you ﬁnd a match, record the expressions on the table below. Use the Generate column to prove your cards are equivalent.

Expression 1

Generate

Expression 2

Integer Exponents | 39

Integer Exponents

Explore 2 Reflect 1. Why is it important to be able to generate equivalent expressions?

2. Given the expression 32 · 32, generate an equivalent expression.

3. For the expression 6(64 · 64), 6 has no exponent. Explain how you would get 69 and not 68.

40 | Integer Exponents

Integer Exponents

Explore 3

Name: _______________________ Date: ___________

Dividing with Exponents Record the expression that matches the True or False Card letter in the box below. Decide whether it’s true or false, and prove your answer. Expression

True/False

Proof

Integer Exponents | 41

Integer Exponents

Explore 3 Expression

True/False

Proof

42 | Integer Exponents

Integer Exponents

Explore 3 Expression

True/False

Proof

Reflect 1. How is creating an equivalent expression when dividing numbers with like bases different from when dividing numbers with like exponents?

2. What happens when neither the bases nor the exponents are the same?

3. How is dividing numbers with like bases different from multiplying numbers with like bases?

Integer Exponents | 43

Integer Exponents

Explore 4

Name: _______________________ Date: ___________

Exponential Powers Determine equivalent expressions for each of Ms. Taylor’s review expressions. Show your steps in the workspace provided, and explain the process of each step. Ms. Taylor’s Expression

(32)4

Workspace with Explanation

Equivalent Expression Ms. Taylor’s Expression

(6x3)3

Workspace with Explanation

Equivalent Expression Ms. Taylor’s Expression

(52 · 32)7

Workspace with Explanation

Integer Exponents | 45

Integer Exponents

Explore 4 Ms. Taylor’s Expression

(27 · 22)3

Workspace with Explanation

Equivalent Expression

Ms. Taylor’s Expression

( 42 ) 3

Workspace with Explanation

Equivalent Expression

Ms. Taylor’s Expression

( 88 ) 3

Workspace with Explanation

Equivalent Expression 46 | Integer Exponents

Explore 4

Integer Exponents

Reflect 1. What does it mean when an exponential term has an exponent?

2. What strategies could you use to check your work with equivalent expressions?

3. Why is it important to know the properties of integer exponents?

Integer Exponents | 47

Scientific Notation

Scientiﬁc Notation

Explore 1

Name: _______________________ Date: ___________

Writing in Scientific Notation Part I: Writing Large Numbers in Scientific Notation Write the numbers in the table in standard form and scientiﬁc notation. Penelope’s New Movies Penelope is working on several projects for the movies Lions and Tigers and Underwater Adventures. She has been given the task of representing the number of frames per second in standard form and the number of frames in scientiﬁc notation.

Type and Title of Project

Total Number of Frames/Second (standard form)

Movie trailer Lions and Tigers

2,000

Short ﬁlm Lions and Tigers Full-length ﬁlm Lions and Tigers

Number of Frames (scientiﬁc notation)

2.5 × 104

125,000

Full-length ﬁlm, extended version Underwater Adventures

7 × 105

Short ﬁlm Underwater Adventures

5.5 × 104

Movie trailer Underwater Adventures

5,000

Full-length ﬁlm Underwater Adventures

550,000

Scientiﬁc Notation | 51

Scientiﬁc Notation

Explore 1 Use the information in Part I to answer the following questions.

1. How do you convert a large number written in standard form to scientiﬁc notation?

2. What is 754,000,000,000 written in scientiﬁc notation?

3. What is 9.4 × 109 written in standard form?

4. Is 41 × 107 written in scientiﬁc notation? Why or why not?

Reflect 1. What are the advantages and beneﬁts of writing a very large number in scientiﬁc notation?

2. Why do you think this method of rewriting very large numbers is called scientiﬁc notation?

3. What is the relationship between the positive exponent in scientiﬁc notation and the number of decimal places in standard form?

52 | Scientiﬁc Notation

Scientiﬁc Notation

Explore 1 Part II: Writing Small Numbers in Scientific Notation Write the numbers in the table in standard form and in scientiﬁc notation.

Representing Penelope’s New Movies (Frame/Total Number of Frames) Penelope’s next task is to represent the 1 frame/total number of frames in standard form as a ratio expressed as a decimal and as a decimal in scientiﬁc notation. 1 Frame Total Number of Frames

Standard Form (ratio expressed as a decimal)

Movie trailer Lions and Tigers

1 2,000

0.0005

Short ﬁlm Lions and Tigers

1 25,000

Full-length ﬁlm Lions and Tigers

1 125,000

Short ﬁlm Underwater Adventures

1 55,000

Movie trailer Underwater Adventures

1 5,000

0.0002

Full-length ﬁlm Underwater Adventures

1 550,000

0.00000182

Type and Title of Project

Number of Frames (scientiﬁc notation)

4 × 10−5

0.000008

1.818 × 10−5

Scientiﬁc Notation | 53

Scientiﬁc Notation

Explore 1 Use the information in Part II to answer the following questions.

1. How do you convert a small number written in standard form to scientiﬁc notation?

2. What is 0.000000728 written in scientiﬁc notation?

3. What is 6.1 × 10−9 written in standard form?

4. Is 0.25 × 10−7 written in scientiﬁc notation? Why or why not?

Reflect 1. What are some examples of very small measurements that might be better written in scientiﬁc notation?

2. What are the similarities and differences between writing really big numbers in scientiﬁc notation and writing really small numbers in scientiﬁc notation?

3. What is the relationship between the negative exponent in scientiﬁc notation and the number of decimal places in standard form?

54 | Scientiﬁc Notation

Scientiﬁc Notation

Explore 2

Name: _______________________ Date: ___________

Estimating Numbers and Scientific Notation Estimate the numbers that are in standard form to the nearest tenth, and then write them in scientiﬁc notation. Complete the missing values using the Estimation Card Match cards. Pierre’s Top Movie Report Pierre is working on a movie report and needs to estimate the ticket sales from the top movies that were shown over the weekend. Film Title

Actual Ticket Sales ($)

Aquarium

7,938,876

Classroom Chaos

239,871

The Furry Dragon

37,183

The Powerful 4

81,759,482

Out of Ofﬁce

519,275

The Adventures of Boomer and Missy

4,824,827

Estimated Ticket Sales ($)

Estimate 45,817,826 to the nearest million. Write your answer in scientiﬁc notation.

Scientiﬁc Notation | 55

Scientiﬁc Notation

Explore 2 Use the matching activity to help answer the following questions. 1. How can you estimate numbers using scientiﬁc notation?

2. The Furry Dragon makes $3,780,924 more money in ticket sales the following week. Estimate this value to the nearest whole number, and write it in scientiﬁc notation.

3. The Adventures of Boomer and Missy makes less money the following week. It makes –7.43%, or 0.0743, less in ticket sales. Estimate this value to the nearest whole number, and write it in scientiﬁc notation.

Reflect 1. Is it easier to estimate a very small number or a very large number? Use an example written in scientiﬁc notation to explain your reasoning.

2. Provide an example of when the estimation of a very large or very small number might be appropriate to use instead of using the actual value.

56 | Scientiﬁc Notation

Scientiﬁc Notation

Explore 3

Name: _______________________ Date: ___________

Comparing Numbers in Scientific Notation Use the Comparing Trivia Task Cards to compare two values written in scientiﬁc notation. Begin by randomly selecting two cards, card 1 and card 2, one at a time. Then, use the table below to compare the values expressed in scientiﬁc notation. Repeat this process two more times for cards 3–6.

Letters on Cards

Value on Each Card Value on card 1 Scientiﬁc notation

Standard notation

(2)

Value on card 2 Scientiﬁc notation

Standard notation

Value on card 3 Scientiﬁc notation

Standard notation

(4)

Value on card 4 Scientiﬁc notation

Standard notation

Value on card 5 Scientiﬁc notation

Standard notation

(6)

Value on card 6 Scientiﬁc notation

Standard notation

By How Much? Larger value Smaller value

_______ is ____ (larger value written in scientiﬁc notation)

times greater than

___.

_______.

The value on card

_______ is ____

the value on card

(smaller value written in scientiﬁc notation)

(larger value written in scientiﬁc notation)

times greater than

___.

_______.

The value on card

_______ is ____

___ is less than

and (5)

the value on card

___ is greater than

and (3)

The value on card ___ is greater than

and (1)

Inequality Statement

the value on card ___.

(smaller value written in scientiﬁc notation)

times less than _______.

(larger value written in scientiﬁc notation)

Scientiﬁc Notation | 57

Scientiﬁc Notation

Explore 3 Use the comparing activity to help answer the following questions. 1. How can you compare two values written in scientiﬁc notation?

2. Which value is smaller: 5 × 10−5 or 2.5 × 10−5?

larger value

3. Using the ratio smaller value , how many times smaller is 6 × 10−5 than 3 × 10−4?

4. Which value is larger: 6 × 10−2 or 5 × 10−3?

larger value

5. Using the ratio smaller value , how many times larger is 4.9 × 105 than 7 × 104?

Reflect 1. Do you think it’s easier to compare two very large numbers written in scientiﬁc notation or two very small numbers written in scientiﬁc notation?

2. Between two large numbers written in scientiﬁc notation, how can you generally tell which number is smaller by looking at just the exponents of these values?

58 | Scientiﬁc Notation

Operations with Scientific Notation

Name: _______________________ Date: ___________

Work Order Letter Combined Data

Operations with Scientiﬁc Notation | 61

Free HDD Space

Use the Work Order Cards to ﬁnd the amount of data there will be once it has been combined as well as the amount of free space that will be available on the new hard drives. Fill in the information on the table below.

Adding and Subtracting with Scientific Notation

Explore 1

Operations with Scientiﬁc Notation

62 | Operations with Scientiﬁc Notation

Work Order Letter

Explore 1

Combined Data

Free HDD Space

Operations with Scientiﬁc Notation

Work Order Letter

Explore 1

Combined Data

Operations with Scientiﬁc Notation | 63

Free HDD Space

Operations with Scientiﬁc Notation

64 | Operations with Scientiﬁc Notation

3. What happens when you add or subtract numbers written in scientiﬁc notation but the answer is not written in correct scientiﬁc notation?

2. Do you prefer making the powers of ten equal by making the exponent larger or smaller? Explain.

1. If you’re trying to add two numbers that are raised to a different power of ten, how can you make it so you can add those numbers together?

Reflect

Explore 1

Operations with Scientiﬁc Notation

Multiplying with Scientific Notation

Name: _______________________ Date: ___________

Legends Course

Course Name Model

Workspace

Operations with Scientiﬁc Notation | 65

Square Footage

Look at the Course Model Cards. Use the space below to draw a model of each course and ﬁnd the total area that Greyson’s Lawn Service will need to mow. Show your work using scientiﬁc notation.

Explore 2

Operations with Scientiﬁc Notation

66 | Operations with Scientiﬁc Notation

Royal National Course

Tiger Tees

Course Name

Explore 2 Model

Workspace

Square Footage

Operations with Scientiﬁc Notation

Meadow Valley Course

Bricker Fire Course

Course Name

Explore 2 Model

Workspace

Operations with Scientiﬁc Notation | 67

Square Mileage

Operations with Scientiﬁc Notation

68 | Operations with Scientiﬁc Notation

3. Why is it important to be able to multiply in scientiﬁc notation?

2. Why do you need to change the numbers written in decimal notation to be in scientiﬁc notation?

1. What happens when you get an answer that is not written in correct scientiﬁc notation?

Reflect

Explore 2

Operations with Scientiﬁc Notation

Dividing with Scientific Notation

Name: _______________________ Date: ___________

Operations with Scientiﬁc Notation | 69

(building or room)

___________________.

___________________

(building or room)

times greater than

is ________

(building or room)

___________________

(building or room)

vs.

(building or room)

___________________

(building or room)

___________________.

___________________

is ________

(building or room)

___________________

By How Much?

times greater than

Larger value Smaller value

Workspace

vs.

(building or room)

___________________

Building or Room Name

Begin by randomly selecting two Building Volume Cards, one at a time. Then, use the table below to compare the values expressed in scientiﬁc notation. Repeat this process three more times for a total of four comparisons.

Explore 3

Operations with Scientiﬁc Notation

70 | Operations with Scientiﬁc Notation

(building or room)

___________________.

___________________

(building or room)

times greater than

is ________

(building or room)

___________________

(building or room)

vs.

(building or room)

___________________

(building or room)

___________________.

___________________

is ________

(building or room)

___________________

By How Much?

times greater than

Larger value Smaller value

Workspace

vs.

(building or room)

___________________

Building or Room Name

Explore 3

Operations with Scientiﬁc Notation

Operations with Scientiﬁc Notation | 71

2. Why do you need to change the numbers written in decimal notation to be in scientiﬁc notation?

1. Do you think it’s easier to divide numbers written in decimal notation or scientiﬁc notation?

Reflect

3. Using the ratio smaller value , how many times smaller is 6 × 10−3 than 3 × 10−2?

larger value

2. Which value is smaller: 3 × 10−7 or 2.5 × 10−4?

1. How can you divide two values written in scientiﬁc notation?

Use the table to help answer the following questions.

Explore 3

Operations with Scientiﬁc Notation

Solve Equations

One-Variable Equations

Name: _______________________ Date: ___________

Substitute x = 0

6x x + 3 = 6x + 2

x + 6 = 3x – 4

5x x + 15 = 15 + 5x

Equation

Substitute x = 5

Solve Equations | 75

Substitute x = 10

Look at the one-variable equations below. Substitute each value in for x. Then, shade each box where the substituted value is a possible solution for the equation.

Part I

Explore 1

Solve Equations

76 | Solve Equations

2. Why do you think the equation 5x x + 15 = 15 + 5x was true for all three values?

1. What do you notice about the equations?

Reflect

Explore 1

Solve Equations

10x x – 5 = 2x + 11

x + 6 + 2x = 3x + 6

4x x – 6 = 4x

Equation

Solve for x.

One Solution?

Solve for x. Use the solution to predict the number of solutions for each equation.

Part II

Explore 1

Many Solutions?

Solve Equations | 77

No Solution?

Solve Equations

78 | Solve Equations

7x x – 2x + 5 = 5x + 5

−12x x + 15 = 3x – 30

3x x + 7 + 4x = 7x + 14

Equation

Explore 1 Solve for x.

Many Solutions?

No Solution?

One Solution?

Solve Equations

5. Were there any clues to help you determine the type of equation before solving it?

4. Tell the steps you took to solve the equations and determine the type of equation.

3. What was similar in the solutions to the equations that you predicted have many solutions?

2. What was similar in the solutions to the equations that you predicted have one solution?

1. What were the solutions to the equations that you predicted have no solutions?

Reflect

Explore 1

Solve Equations | 79

Solve Equations

Explore 2

Name: _______________________ Date: ___________

Write, Model, and Solve Two-Step Equations Part I Carefully read and analyze each question to help Lily prepare. Day 1 – Brownies • Lily received a $4 tip. • She sold all of her brownies for a dollar each. • She made a total of $12 on the first day. • How many brownies, b, did Lily sell on day 1? 1. Define the variable to represent the unknown value, and then write an equation to represent the situation. a. Define the variable. b represents ______________________________________________________. b. Write an equation.

2. Model the problem using algebra tiles and the Algebra Equation Mat. 3. Record your model below.

4. Solve the equation, show your work and solution, and give your answer in a complete sentence.

Solve Equations | 81

Solve Equations

Explore 2 Day 2 – Cookies • Lily spent $7 on supplies. • She sold all of her bags of cookies for 2 dollars each. • She made a proﬁt of $17 (proﬁt is revenue minus expenses). • How many bags of cookies, c, did Lily sell on day 2?

5. Deﬁne the variable to represent the unknown value, and then write an equation to represent the situation. a. Deﬁne the variable. c represents ______________________________________________________. b. Write an equation.

6. Model the problem using algebra tiles and the Algebra Equation Mat. 7. Record your model below.

8. Solve the equation, show your work and solution, and give your answer in a complete sentence.

82 | Solve Equations

Explore 2

Solve Equations

Day 3 – Banana Bread • Lily spent $3 on supplies and $5 on a new customized sign for her stall. • She sold all of the loaves of banana bread for $4 each. • She made a profit of $20. • How many loaves of banana bread, b, did Lily sell on day 3? 9. Define the variable to represent the unknown value, and then write an equation to represent the situation. a. Define the variable. b represents ______________________________________________________. b. Write an equation.

10. Model the problem using algebra tiles and an Algebra Equation Mat. 11. Record your model below.

12. Solve the equation, show your work and solution, and give your answer in a complete sentence.

Solve Equations | 83

Solve Equations

Explore 2 Part II

1. Read each scenario, and then analyze Lily’s equations and strategies for solving. Day 4: Lily spent $25.75 on supplies and sold cookies at $1.50 each. At the end of the day, she made a proﬁt of $37.25. How many cookies did she sell on day 4?

Day 5: Lily rented space at a local plaza 3 for $5.50 plus 5 of her total sales. At the end of day 5, she had to pay $45.10 for her rental space. How much did she make in total sales on day 5?

Lily’s Work

Step 1: −25.75 + 1.50x x = 37.25

Step 1: 5.50 + 3 x = 45.10

Step 2: 1.50x x = 63

Step 2: 3 x = 39.60

Step 3: x = 42 cookies sold

Step 3: 3x x = 198

Step 4: x = $66 total proﬁt 2. For day 4, in which step did Lily use additive inverses to solve?

3. For day 4, in which step did Lily use the division property of equality to solve?

4. For day 5, explain what Lily did going from step 2 to step 3 of her work.

5. Use substitution to verify Lily’s solutions for days 4 and 5. Day 4 Workspace:

84 | Solve Equations

Day 5 Workspace:

Solve Equations

Explore 2 Reflect 1. What are the rules of solving a two-step equation?

2. In the equation 3x x + 4 = 13, why can’t you subtract 3 from both sides?

3. In the equation 6 − 3 x = 8, why is adding 3 on both sides not helpful?

4. In the equation 7 w − 3 = 1, why is adding 7 on both sides not helpful?

5. How will you verify your answer to check if it makes the equation true?

Solve Equations | 85

Solve Equations

Explore 3

Name: _______________________ Date: ___________

Write, Model, and Solve Multistep Equations Part I Use the questions to help you identify the 3-digit combination of the new vault. William’s Findings – The Right Wall • William counts 8 bands of ten-dollar bills (each band is $1,000) plus a bag of gold coins arranged on the top shelf of the right wall of the vault. • There are 5 bags of gold coins placed on the bottom shelf of the right wall. • The total value of the top shelf is equal to the total value of the bottom shelf. The value of 1 bag of gold coins, c, is the ﬁrst digit of the 3-digit combination. Answer the questions below to help you solve for c. 1. Deﬁne the variable. c represents _________________________________________________________.

Key Key

2. Write an equation.

= 3. Complete the key provided. 4. Model the problem using the balanced hanger.

= One band of

= ten-dollar bills =

5. Show your work and solution below.

6. Describe what the solution means in context.

7. What is the ﬁrst digit of the 3-digit combination?

Solve Equations | 87

Solve Equations

Explore 3

Henry’s Findings – The Center Wall • Henry counts two bags of gold coins plus 9 bands of ten-dollar bills kept on the middle shelf of the center wall of the vault. • He also counts three bags of gold coins plus 4 bands of ten-dollar bills kept on the bottom shelf of the center wall. • The total value of the middle shelf is equal to the total value of the bottom shelf. The value of 1 bag of gold coins, c, is the second digit of the 3-digit combination. Use the questions to help you determine c. 8. Deﬁne the variable. c represents _________________________________________________________.

Key Key

9. Write an equation.

= 10. Complete the key provided.

= One band of

= ten-dollar bills =

11. Model the problem using the balanced hanger.

12. Show your work and solution below.

13. Describe what the solution means in context.

14. What is the second digit of the 3-digit combination?

? 88 | Solve Equations

Solve Equations

Explore 3

Ethan’s Findings – The Left Wall • Ethan counts three piles of two bags of gold coins kept on the middle shelf on the left wall of the vault. • He also counts two piles that each have 2 bags of gold coins plus 3 bands of tendollar bills on the top shelf of the left wall. • The total value of the middle shelf is equal to the total value of the top shelf. The value of 1 bag of gold coins, c, is the third digit of the 3-digit combination. Answer the questions below to help you determine the third digit of the combination. 15. Deﬁne the variable. c represents ________________________________________________________.

Key Key

16. Write an equation.

= 17. Complete the key provided. 18. Model the problem using the balanced hanger.

= =

19. Show your work and solution below.

20. Describe what the solution means in context.

21. What is the third digit of the 3-digit combination?

? ?

22. What is the 3-digit combination of the new vault?

Solve Equations | 89

Solve Equations

Explore 3 Part II Analyze the hanger model, and answer the questions that follow.

Key Key The value of a bag of

= gold = coins

One band of ten-dollar

= bills, = or $1,000

1. Use the key to write the equation represented by the hanger model.

2. Solve the equation to ﬁnd the value of one bag of gold coins that is represented by one green rectangle.

3. Use substitution to verify your solution.

90 | Solve Equations

Solve Equations

Explore 3 Reflect 1. Write one observation that is true about this diagram as it relates to equations.

2. Write one observation that is true about this diagram as it relates to equations.

3. How will you ﬁnd the weight of one triangle if each square weighs 3 kg?

4. What are the rules of solving a multistep equation with variables on both sides?

5. In the equation 5x x + 3 = 6x, would subtracting 3 from each side be the most efﬁcient ﬁrst step? Why or why not?

6. In the equation 5 – 2 x = 3x – 9, would multiplying both sides by 2 help solve the equation? Why or why not?

7. How can you verify your solution?

Solve Equations | 91

Solve Multistep Equations

Name: _______________________ Date: ___________

Model:

Explanation:

Solve Equations | 93

8x x – 4 = 7x x+8

Algebraic steps:

Week 1: Diego walked 8 dogs and spent $4 on dog treats, and Andrew walked 7 dogs and received $8 in tips. At the end of the week, the brothers had the exact same amount of money. Let x represent how much Diego and Andrew charged for each dog walked during week 1.

Use the information given for each week to determine how much Diego and Andrew charged for each dog they walked. Sketch a model with algebra tiles, explain it in words, and write out the algebraic steps to justify your answer.

Part I

Explore 4

Solve Equations

Explanation:

2(4x x – 2) = 6x + 14

Algebraic steps:

What does the solution mean in terms of the situation?

94 | Solve Equations

Write and solve an equation to represent this situation.

Week 3: Diego was hired by the Cortinez family to walk their dogs for a week. He was paid a ﬂat fee of $100 from the Cortinez family and also walked 5 additional dogs. Andrew worked overtime to walk 30 dogs. At the end of the week, the brothers had the exact same amount of money. Let x represent how much Diego and Andrew charged for each dog walked during week 3.

Model:

Week 2: Diego was only able to work 2 days. Each day, he walked 4 dogs and spent $2 on treats. Andrew worked for the whole week, walked 6 dogs, and received $14 in tips. At the end of the week, the brothers had the exact same amount of money. Let x represent how much Diego and Andrew charged for each dog walked during week 2.

Explore 4

Solve Equations

Step 3: 6(0.25x x – 22.71) = x Step 4: 1.5x x – 136.26 = x

Step 3: 1 x = 22.71

Step 4: x = 272.52

Step 6: 272.52 = x

Step 5: −136.26 = −0.5x

4. What does the solution mean in terms of the situation?

Step 2: 0.25x x – 22.71 = 6

Step 1: 0.25x x + 24.75 = 6 + 47.46

Step 2: 1 x = 22.71 + 1 x

Andrew’s Work

Step 1: 24.75 + 1 x = 47.46 + 1 x

Diego’s Work

Week 4: The boys tried waiting tables at different restaurants. Diego earned 1 $24.75 per shift plus 4 of all combined tips. Andrew earned $47.46 per shift 1 plus 6 of all combined tips.

Solve Equations | 95

3. Which brother’s work was more efﬁcient? Explain.

2. What are the differences between the strategies each brother used to solve?

1. In which step did each brother use additive inverses to solve?

Each brother wrote and solved an equation to represent their different jobs for the rest of the summer. Read each scenario, and then analyze Diego’s and Andrew’s equations and strategies for solving.

Part II

Explore 4

Solve Equations

3(3x x – 1.25) – 15 = 2(5x) – 33.75 – 20 9x x – 3.75 – 15 = 10x – 33.75 – 20 9x x – 18.75 = 10x – 53.75 −18.75 = 1x x – 53.75 35 = 1x 35 = x

3(3x x – 1.25) – 15 = 2(5x) – 33.75 – 20

9x x – 3.75 – 15 = 10x – 33.75 – 20

9x x – 3.75 + 5 = 10x – 33.75

9x x – 3.75 + 38.75 = 10x

9x x + 35 = 10x

9x x = 10x – 35

96 | Solve Equations

4. What does the solution mean in terms of the situation?

3. Which brother’s work was more efﬁcient? Explain.

x = 35

−1x x = −35

Andrew’s Work

Diego’s Work

Week 5: Diego and Andrew decided to tutor some students from the elementary school. Diego held 3 sessions per day. At each session, he tutored 3 students and spent $1.25 on coffee. Diego was also charged $15 per day to rent a workspace. Andrew held 2 sessions per day, where he tutored 5 students at each session. Andrew spent $33.75 on workspace rental and $20 on snacks for his students.

Explore 4

2. What are the differences between the strategies each brother used to solve?

1. In which step did each brother use the division property of equality to solve?

Solve Equations

B. 2x – 12 = 2x + 12

4. How many strategies are there for solving equations? How do you know which strategy is best?

Explanation:

Step 2: 2x x – 12 = 72 + 5x

Step 1: 3 (2 (2x x – 12) = 3 (72 + 5x)

Andrew’s Work

Solve Equations | 97

3. Andrew is solving an equation with fractions on both sides. Explain what he did to go from step 1 to step 2 and why it was helpful.

2. Diego says that the equation 2x x + 10 = 6x + 10 has no solutions. Do you agree with him?

A. 2x x – 12 = 2(x – 6)

1. One of the equations below is true for all values of x. Which one is it, and how do you know?

Reflect

Explore 4

Solve Equations

Explore 5

Name: _______________________ Date: ___________

Solve Literal Equations Part I Solve each equation for the ,⃝ and then ﬁnd the matching Code Card and place it on the level 1 Puzzle Dial. Use the workspace as needed. Once all of the codes are found, present the completed level 1 Puzzle Dial to your teacher. Puzzle

Workspace

Code

= C

(

−

Solve Equations | 99

Solve Equations

Explore 5 Reflect

1. What are the similarities and differences between solving these puzzles and solving algebraic equations?

2. Let’s replace the shapes in puzzle A with the variables a, b, and c. What would the solution be? How did your process for solving change?

+ a

= +

−

a = ____________

3. How would the code for puzzle C change if you were to solve for the triangle instead?

(

100 | Solve Equations

Solve Equations

Explore 5 Part II

Solve each literal equation for the speciﬁed variable, and then ﬁnd the Code Card that matches your solution. Place the cards on the level 2 Puzzle Dial. Use the workspace provided. Once all of the codes are found, present the completed level 2 Puzzle Dial to your teacher. Puzzle

Workspace

Code

E P = 2l + 2w Solve for w.

w = ____________

F y = mx + b Solve for x.

x = ____________

G 1

A = 2 bh Solve for b.

b = ____________

H I = prt Solve for p.

p = ____________

Solve Equations | 101

Solve Equations

Explore 5 Reflect 1. Consider the formula P = 2l + 2w. a. What can be calculated using this formula?

b. What variables need to be given for the formula to be solved as is?

c. What could be found if you were given the perimeter and the length?

2. Why would it be beneﬁcial to know how to manipulate literal equations?

102 | Solve Equations

Solve Inequalities

103

Write Inequalities

Inequality:

Let x represent the number of hours worked on day 2.

Tamara earned $20 an hour and spent $7.50 on snacks. Devi earned $13 an hour and received a $5 tip. At the end of the day, Tamara had earned the same or more than Devi.

Tamara earned $15.25 per hour and spent $2 on a slushie. Devi earned $14.75 per hour and received a tip of $4. At the end of the day, Tamara earned more than Devi.

Let x represent the number of hours worked on day 1.

Day 2

Day 1

Inequality:

Solve Inequalities | 105

Let x represent the number of hours worked on day 3.

Tamara earned $13.50 an hour and $10 in tips. Devi earned $14 an hour and $7 in tips. At the end of the day, Tamara earned less than Devi.

Day 3

Name: _______________________ Date: ___________

Read each scenario. Determine an inequality to represent the girls’ earnings for the day.

Part I

Explore 1

Solve Inequalities

106 | Solve Inequalities

Scenario:

11.50x x + 5 ≥ 12.25x – 2

10x x – 6 ≤ 12x – 4

Scenario:

Day 5

Day 4

Scenario:

12x x + 3 < 14x – 4

Day 6

Read each inequality with your group. Determine a scenario to represent the girls’ earnings for the day.

Part II

Explore 1

Solve Inequalities

4. How did you write a scenario from an inequality?

3. How can you determine the inequality from a scenario?

2. How are these inequalities different from ones we have seen previously?

1. Does the variable always have to be x?

Reflect

Explore 1

Solve Inequalities | 107

Solve Inequalities

Name: _______________________ Date: ___________

Model:

Explanation: Given

Solve Inequalities | 109

___________________________.

She earned __________________

3x x – 1 < 2x + 2

Algebraic steps:

Day 1: Erika worked for 3 hours and spent $1 on a soda. Tammi worked for 2 hours and earned $2 in tips. At the end of the day, Tammi had more money than Erika. Let x represent the amount of money per hour each girl was paid on day 1. How much money did Tammi get paid per hour on day 1?

With the class, read the scenario from day 1. Determine an inequality to represent the girls’ earnings for the day. Then, solve using algebra tiles, and solve algebraically.

Part I

Write, Model, and Solve Inequalities with Variables on Both Sides

Explore 2

Solve Inequalities

110 | Solve Inequalities

2. How are these inequalities different from ones we have seen previously?

1. What do you notice about the steps using algebra tiles and the steps solving the inequality algebraically?

Reflect

Explore 2

Solve Inequalities

Model:

Explanation: Given

Solve Inequalities | 111

___________________________.

She earned __________________

5x x – 2 ≥ 4x + 1

Algebraic steps:

Day 2: Erika worked for 5 hours and spent $2 on snacks. Tammi worked for 4 hours and earned $1 in tips. At the end of day 2, Erika earned at least as much as Tammi on day 2. Let x equal the amount of money per hour each girl earned on day 2. How much money did Erika get paid per hour on day 2?

Read the scenario with your group. Determine an inequality to represent the girls’ earnings for the day. Then, solve using algebra tiles, and solve algebraically.

Part II

Explore 2

Solve Inequalities

112 | Solve Inequalities

Model:

Explanation: Given

___________________________.

She earned __________________

3x x + 4 > 4x – 12

Algebraic steps:

Day 3: Erika worked 3 hours and earned $4 in tips. Tammi worked 4 hours and spent $12 on food. Tammi earned less than Erika. Let x represent the dollars per hour that each girl earned on this day. How much money did Tammi get paid per hour on day 3?

Explore 2

Solve Inequalities

Model:

Explanation: Given

Solve Inequalities | 113

___________________________.

She earned __________________

6x x – 1 ≤ 7x – 3

Algebraic steps:

Day 4: Erika worked 6 hours and spent $1 on a soda. Tammi worked 7 hours and spent $3 on food. Erika earned the same as or less than Tammi on day 4. Let x represent the dollars per hour that each girl earned on this day. How much money did Erika get paid per hour on day 4?

Explore 2

Solve Inequalities

114 | Solve Inequalities

Model:

Explanation: Given

___________________________.

She earned __________________

8x x – 4 ≤ 7x + 8

Algebraic steps:

Day 5: Erika worked 8 hours and spent $4 on food. Tammi worked 7 hours and earned $8 in tips. Tammi earned at least as much as Erika on day 5. Let x represent the dollars per hour that each girl earned on this day. How much money did Tammi get paid per hour on day 5?

Explore 2

Solve Inequalities

3. What does the solution represent in terms of the situation?

2. How can you prove a solution is correct for an inequality?

1. When might you rather solve algebraically than solve using algebra tiles?

Reflect

Explore 2

Solve Inequalities | 115

Solve Inequalities

3. Explain your solution in Zahra’s context.

2. Solve the inequality. Start by adding 4x x to each side. Use the related equation to solve the inequality.

1. Zahra created the inequality 24 – 4x x ≥ 5 to model the number of bargain games she can purchase. Explain the coefﬁcients and constants in the inequality.

Part I

Name: _______________________ Date: ___________

Solve Inequalities | 117

6. Why are the solutions to −x x < 8 different from the solutions to x < −8?

5. When do you need to ﬂip the sign of an inequality when you solve?

4. Solve the inequality again. Start by subtracting 24 from each side. Do your answers match?

Solve Inequalities with Variables on Both Sides

Explore 3

Solve Inequalities

Solution Set

55 + 10x x < 62 + 8x

Inequality

118 | Solve Inequalities

4. Which of these stores would you prefer for buying games, and why?

3. How can your number line be used to determine when Game Shop will be cheaper than Retro Games?

b. 1 new game and 4 bargain games:

a. 1 new game and 3 bargain games:

2. Which store would be cheaper if Zahra buys the following number of games? Show your work.

1. Solve the inequality, and then test a point to make sure your inequality symbol faces the correct direction. Write your answer, and complete the number line in the table.

Advertisement A

Part II

Explore 3

Solve Inequalities

Solution Set

Inequality

9. Which store would you prefer, and why?

games at ________.

Solve Inequalities | 119

Games. Since the bargain games are the ________ price at both stores, it will ________ be cheaper to buy the

8. Game Shop’s membership promotion plus one new game costs ________ than the cost of one game at Retro

7. Why can’t the solutions to this inequality be graphed on a number line? Explain your answer in the context of the two game companies.

6. Solve the inequality. Write the solution in the table.

5. Write an inequality in the table to represent the situation.

Advertisement B

Explore 3

Solve Inequalities

All real numbers

Solution Set

Inequality

120 | Solve Inequalities

13. Which of these subscription services would you prefer, and why?

12. What would the solution set to this inequality look like on a number line? Explain your answer in the context of the two game companies.

11. Solve the inequality.

10. In the table, write an inequality to represent this situation, where x represents the number of months Zahra uses each subscription.

Advertisement C

Explore 3

Solve Inequalities

18. Which of these subscription services would you prefer, and why?

b. 15 months:

a. 5 months:

Solve Inequalities | 121

Solution Set

Inequality

17. Which subscription would cost more if you subscribed for the following number of months?

16. Describe the meaning of your solution in terms of the situation.

15. Solve the inequality, and then test a point to make sure your inequality symbol faces the correct direction. Write your answer, and complete the number line in the table.

14. In the table, write an inequality to represent this situation, where x represents the number of months Zahra uses each subscription.

Advertisement D

Explore 3

Solve Inequalities

122 | Solve Inequalities

4. Which game purchasing option would you choose? Use mathematics to justify your answer.

3. How can substitution be used to verify the solution set?

2. How would you describe solving inequalities with a negative coefﬁcient to a friend who missed class today?

1. How does the solution look different when the result is all real numbers or no solutions?

Reflect

Explore 3

Solve Inequalities

Name: _______________________ Date: ___________

Expression to Determine 5-Week Average

Pari’s 5th Grade

2. Complete the table.

5-Week Average Yes

Exactly a 90?

Solve Inequalities | 123

5. Write the inequality to represent the scenario in question 4.

4. How would this be different if you were solving to ﬁgure out what grade she needs to earn a 90 or better?

3. Write and solve an equation to determine the grade Pari needs to earn a 90 exactly.

1. Write an expression to determine Pari’s current average in gym class. Then, ﬁnd her current average for weeks 1–4.

Part I

Solve Inequalities with the Distributive Property and Fractions

po Re

Explore 4

Solve Inequalities

Equation to Determine Grade Needed to Average a 90 Exactly

Inequality to Determine Grade Needed to Average a 90 or Better

124 | Solve Inequalities

What grade does Elizabeth need in week 5 to earn a 90 average or better? Show all work.

Average of Weeks 1–4

Prediction of Grade Needed for Week 5

Elizabeth

Complete the table, and determine what possible grades each student needs to earn in week 5 to average a 90 or better on their progress report.

Part II

Explore 4

Solve Inequalities

Prediction of Grade Needed for Week 5

Equation to Determine Grade Needed to Average a 90 Exactly

Tomas

Solve Inequalities | 125

Inequality to Determine Grade Needed to Average a 90 or Better

What grade does Tomas need in week 5 to earn a 90 average or better? Show all work.

Average of Weeks 1–4

Explore 4

Solve Inequalities

342 + 5x

126 | Solve Inequalities

4. Solve the inequality from question 3.

3. Explain why the inequality = ≥ 80 would help Tomas determine what 10 10 average he needed during the last ﬁve weeks of the term to bring his grade up to a B after he received his ﬁve-week progress report average.

63 + 69 + 60 + 2(75) + 5x

2. What grades could Tomas earn on his ﬁnal week assessment in English to end the term with at least an 80 so he has a B? Set up an inequality, and solve.

1. Tomas wants to earn a B for his final grade in English. If he earns an 80 on the final week’s assessment, will his final grade be a B?

At the end of the term, Tomas remembered calculating his gym grade for progress reports, and he wanted to do the same thing to determine his ﬁnal term grade in English.

Explore 4

Solve Inequalities

3. Are all 3 students able to earn an A in gym class? Which student needs the highest grade?

Solve Inequalities | 127

2. How did you approach solving an inequality for Tomas’s gym grade compared to his English grade?

1. How did you approach solving an inequality that was written as a fraction? Is there another way?

Reflect

Explore 4

Solve Inequalities

Create Non-Proportional Relationships from Proportional Relationships

129

Explore 1

Create Non-Proportional Relationships from Proportional Relationships

Name: _______________________ Date: ___________

Create Non-Proportional Relationships Part I Use the table below to help Jake graph the salary he could earn per hours worked. Offer 1 200

Number of Hours

Salary ($)

0 20

180 160 140 Salary ($)

120 100 80 60 40 20

3 4 5 6 7 8 Number of hours

9 10

What is the equation that represents the relationship between the hours worked and the salary above?

Offer 2 Offer 2 has the same salary as offer 1 but has a signing bonus of $100. Use that information to complete the table and graph below. Salary ($)

0 1

180 160 140 Salary ($)

Number of Hours

200

120 100 80 60

x 1

3 4 5 6 7 8 Number of hours

9 10

What is the equation that represents the relationship between the hours worked and the salary above?

Create Non-Proportional Relationships from Proportional Relationships | 131

Explore 1

Create Non-Proportional Relationships from Proportional Relationships

Use the table below to help Jake graph the salary he could earn per hours worked. Offer 3 Number of Hours

150 135

Salary ($)

120

Salary ($)

105 90 75 60 45 30 15

3 4 5 6 7 8 Number of hours

9 10

What is the equation that represents the relationship between the hours worked and the salary above?

Offer 4 Offer 4 has the same salary as offer 3 but has a signing bonus of $75. Use that y information to complete the table and graph below. 225

Number of Hours

210

Salary ($)

195 180

Salary ($)

165 150 135 120 105 90 75

3 4 5 6 7 8 Number of hours

9 10

What is the equation that represents the relationship between the hours worked and the salary above?

Which job offer will allow Jake to make the most money?

132 | Create Non-Proportional Relationships from Proportional Relationships

Explore 1

Create Non-Proportional Relationships from Proportional Relationships

Part II Use the job listings to determine an equation. Graph that equation, and answer the questions below. Job Listing 1 Determine the equation.

200

What are 3 points that show how much money Jania will make at this job?

180 160

How do you know those points are amounts of money she will make?

Salary ($)

140 120 100 80 60 40 20

3 4 5 6 7 8 Number of hours

9 10

Job Listing 2 Determine the equation. 250

What are 3 points that show how much money Jania will make at this job?

225 200

How do you know those points are amounts of money she will make?

Salary ($)

175 150 125 100 75 50 25 0

x 1

3 4 5 6 7 8 Number of hours

9 10

Create Non-Proportional Relationships from Proportional Relationships | 133

Explore 1

Create Non-Proportional Relationships from Proportional Relationships

Job Listing 3 Determine the equation.

200

What are 3 points that show how much money Jania will make at this job?

180 160

Is this graph proportional? Why or why not?

Salary ($)

140 120 100 80 60 40 20

3 4 5 6 7 8 Number of hours

9 10

Job Listing 4 Determine the equation.

200

What are 3 points that show how much money Jania will make at this job?

180 160

Is this graph proportional? Why or why not?

Salary ($)

140 120 100 80 60 40 20 0

x 1

3 4 5 6 7 8 Number of hours

9 10

134 | Create Non-Proportional Relationships from Proportional Relationships

Explore 1

Create Non-Proportional Relationships from Proportional Relationships

Reflect 1. What did you notice about the graphs of the offers with proportional relationships?

2. If all of the other solutions to this equation were graphed, where would they appear?

3. What did you notice about scenarios with proportional relationships?

4. What did you notice about equations with proportional relationships?

5. When the x value is 0, what does the point on the y-axis represent?

6. What does it mean when the starting rate is at (0, 0)?

Create Non-Proportional Relationships from Proportional Relationships | 135

Functions

137

Month

Functions | 139

Which variable represents the y-coordinate?

2nd Half of the Year

Which variable represents the x-coordinate?

Month

1st Half of the Year

What do you notice about November and December?

Understand Functions on a Graph

Name: _______________________ Date: ___________

Functions

What do you notice about March and April?

Deposit (dollars)

Plot the points by using the information from the Monthly Deposits Cards.

Part I

Explore 1

Deposit (dollars)

$2,000 Output

Input

Output

Diagram 2

Output

$1,100 $1,200 $1,400 $1,450 $1,650 $1,900 $2,000

Input

What would the y value of an ordered pair represent? (Circle one.)

140 | Functions

Input

Diagram 1

Input

July August September October November December

Diagram 2

What would the x value of an ordered pair represent? (Circle one.)

Do any output values repeat (have more than one arrow)?

Do any input values repeat (have more than one arrow)?

Fill in the following table by writing yes or no answers.

$1,700

$1,550

$1,050

January February March April May June

Diagram 1

Draw an arrow from each input to its corresponding output. The ﬁrst one has been done for you.

Explore 1

Functions

What is the input when the output is 350?

Do any of the inputs have more than one output? Yes / No Is this a function?

What is the input when the output is 150?

Do any of the inputs have more than one output?

Yes / No

Is this a function?

Do any of the inputs have more than one output? Yes / No Is this a function?

Do any of the inputs have more than one output?

Yes / No

Is this a function?

Functions | 141

What is the input when the output is 300?

Yes / No

What is(are) the output(s) when the input is 4?

Yes / No

April

March

Yes / No

What is(are) the output(s) when the input is 2?

What is(are) the output(s) when the input is 3?

Yes / No

February

January

Part II

Explore 1

Functions

Do any of the inputs have more than one output?

142 | Functions

What point can you remove in order to make this a function?

Is this a function? Why or why not?

Yes / No

Do any of the outputs have more than one input?

$50

$100

$150

$200

Weeks

Place points on the graph so that it is not a function.

July

August

What point can you remove in order to make this a function?

$250

C. (3, 100)

Is this a function?

August

D. (4, 100)

A. (1, 100)

List the coordinates of the 6 points on the graph.

Yes / No

Which additional point can be plotted so the graph continues to represent a function?

B. (2, 100)

June

Functions

May

Explore 1

Revenue

List the outputs. Write each pair as a coordinate.

Is this a function? Why or why not?

December Write each coordinate. List two coordinates that could be added to keep this a function.

List the outputs.

Write each pair as a coordinate.

Is this a function? Why or why not?

November

List the inputs.

List the outputs.

Is this a function? Why or why not?

Functions | 143

List two coordinates that could be added that would not make this a function.

List the inputs.

Write each pair as a coordinate.

October

September

Explore 1

Functions

144 | Functions

3. Do duplicate output values affect whether a graph is a function or not? Why or why not?

2. Deﬁne a function in your own words.

1. When Rosie counts her money more than once a week in Part II, does this represent a function? Why or why not?

Reflect

Explore 1

Functions

Explore 2

Name: _______________________ Date: ___________

Understand Functions on a Table Week 1

Week 2

What is the output when the input is dresses?

What is the input when the output is 12?

What is the input when the output is 16?

What is the output when the input is Wednesday?

List all of the inputs.

Are there any repeating inputs?

Is this a function?

Which inputs do you need to remove in order to make this a function?

Week 3

Week 4

List the inputs.

What is the output when the input is Friday?

What input can be placed in the last row to make this table a function?

What input can be placed in the last row to make this table not a function?

What are the inputs when the output is 36?

Are there any repeating inputs?

Are there any repeating outputs?

Is this a function?

Functions | 145

Functions

Explore 2 Week 5

Week 6

Use the information to create a table.

Fill in the table to create a function.

Does the item determine the amount?

Item

Amount

Does the input determine the output?

Style

Amount Sold

Pants

35 28

Is this a function?

Dresses

Week 7

Week 8

Write the information as coordinates.

What is one input that can be added to the table so that the relationship continues to represent a function?

(_________, ___), (_________, ___), (_________, ___), (_________, ___),

a. Monday

(_________, ___), (_________, ___),

b. Thursday c. Sunday

List the inputs.

List the outputs. Are there any repeating input values?

What ordered pair could be added to the table so the relationship does not represent a function? a. (Tuesday, 67) b. (Thursday, 75)

Is this a function?

c. (Friday, 83) d. (Sunday, 80)

146 | Functions

Functions

Explore 2 Week 9

Week 10

List the inputs.

What is the input? (Circle one.) Color

List the outputs.

Amount

Use the information to create a table. Color

Amount

Is this a function?

Why or why not?

Week 11

Week 12

What is the input when the output is 20?

Use the information to create a table.

What is the output when the input is orange dress?

Item Amount

Cost

Orange dresses sold the same amount as green dresses. Does this affect whether this is a function? Explain your answer.

Is this a function?

If the cost were $40, what item amount would make this relationship not a function?

Functions | 147

Functions

Explore 2 Reflect 1. Do functions always have to be number values for their inputs or outputs?

2. Does it matter where the inputs and outputs are placed in the table? Why or why not?

3. Which coordinate does the input represent? Which coordinate does the output represent?

4. What do duplicate inputs create on a graph?

148 | Functions

Functions

Explore 3

Name: _______________________ Date: ___________

Analyzing Graphs

Use the Road Trip Cards to analyze the graphs, and describe each card as linear or nonlinear and increasing or decreasing. Match the description to the correct graph. Glue the description in the space on the table. Distance from Hotel 1. How can the descriptions linear, nonlinear, constant, increasing, and decreasing be used to describe the graph?

Description:

2. What is the y-intercept of the graph?

Functions | 149

Functions

Explore 3 Distance from the Museum 1. How can the descriptions linear, nonlinear, constant, increasing, and decreasing be used to describe the graph?

Description:

2. What is the y-intercept of the graph?

Distance from Zoo Entrance 1. How can the descriptions linear, nonlinear, constant, increasing, and decreasing be used to describe the graph?

Description:

2. What is the y-intercept of the graph?

150 | Functions

Functions

Explore 3 Distance from Home 1. How can the descriptions linear, nonlinear, constant, increasing, and decreasing be used to describe the graph?

Description:

2. What is the y-intercept of the graph?

Distance from Rest Stop 1. How can the descriptions linear, nonlinear, constant, increasing, and decreasing be used to describe the graph?

Description:

2. What is the y-intercept of the graph?

Functions | 151

Functions

Explore 3 Distance from Park 1. How can the descriptions linear, nonlinear, constant, increasing, and decreasing be used to describe the graph?

Description:

2. What is the y-intercept of the graph?

152 | Functions

Functions

Explore 3 Reflect

1. Compare and contrast graphs that show linear relationships and graphs that show nonlinear relationships.

2. What is an example of a real-world situation where the function is linear?

3. What is an example of a real-world situation where the function is nonlinear?

Functions | 153

Functions

Explore 4

Name: _______________________ Date: ___________

Sketching Graphs

Use the Beach Cards to read the story, and then sketch the graph. Describe each card as linear or nonlinear and constant, increasing, or decreasing. Seaplane Tour Graph:

Use the descriptions linear, nonlinear, constant, increasing, and decreasing to describe the graph.

Functions | 155

Functions

Explore 4 Bike Ride Graph:

Use the descriptions linear, nonlinear, constant, increasing, and decreasing to describe the graph.

156 | Functions

Functions

Explore 4 Swimming at the Beach Graph:

Use the descriptions linear, nonlinear, constant, increasing, and decreasing to describe the graph.

Functions | 157

Functions

Explore 4 Create Your Own Story Graph:

Use the descriptions linear, nonlinear, constant, increasing, and decreasing to describe the graph.

158 | Functions

Functions

Explore 4 Reflect 1. How did you determine whether the verbal description was describing a linear relationship?

2. How did you determine whether the verbal description was describing a nonlinear relationship?

3. What is an example of a real-world situation where the function is both linear and nonlinear?

Functions | 159

Rate of Change and Initial Value

161

Rate of Change and Initial Value

Explore 1

Name: _______________________ Date: ___________

Determine the y-intercept and Rate of Change Part I: Determine the y-intercept and Rate of Change in Tables and Graphs Use the Bike Rental Cards to answer the questions and determine the y-intercept. Bikes N’ More Weekday Rates 1. Identify the independent and dependent variables.

2. What is the hourly rate for a bike rental? Explain how you know.

3. What is the y value when the x value is 0? Write the ordered pair. What does the y value represent when the x value is 0 in this situation?

Bikes N’ More Weekend Rates 1. Identify the independent and dependent variables.

2. What is the daily rate for a bike rental on the weekend? Explain.

3. What is the y value when the x value is 0? Write the ordered pair. What does the y value represent when the x value is 0 in this situation?

Rate of Change and Initial Value | 163

Rate of Change and Initial Value

Explore 1

Two Wheels Only Weekday Rates 1. Identify the independent and dependent variables.

2. What is the y value when the x value is 0? Write the ordered pair. What does the y value represent when the x value is 0 in this situation?

3. What would be the rate to rent for one day? Find the missing costs to help you determine the cost.

Day

Cost

3 61

Two Wheels Only Weekend Rates 1. Identify the independent and dependent variables.

2. What is the y value when the x value is 0?

3. What does the y value represent when the x value is 0 in this situation?

4. What is the hourly weekend rate? What is the cost for an 8-hour rental? How would Javier ﬁnd that cost?

164 | Rate of Change and Initial Value

Rate of Change and Initial Value

Explore 1 Wheels R Us Weekday Rates 1. Identify the independent and dependent variables.

2. What is the cost per hour? Complete the table to help you determine the cost. Hour

Cost

2 79

4 96

3. What is the y value when the x value is 0? Write the ordered pair. What does the y value represent when the x value is 0 in this situation?

Wheels R Us Weekend Rates 1. Identify the independent and dependent variables.

2. What is the y value when the x value is 0? What does the y value represent when the x value is 0 in this situation?

3. What is the daily cost to rent a bike from Wheels R Us? Explain how you calculated the cost.

Rate of Change and Initial Value | 165

Explore 1

Rate of Change and Initial Value

1. What are the domain and range that are represented in the weekend graph for Two Wheels Only?

2. What are the domain and range that are represented in the weekend graph for Bikes N’ More?

Reflect 1. How do you determine the y-intercept from a table?

2. How do you determine the rate from a table?

3. How do you determine the y-intercept from a graph?

4. How do you determine the rate from a graph?

166 | Rate of Change and Initial Value

Rate of Change and Initial Value

Explore 1

Part II: Determine the y-intercept in Equations and Verbal Descriptions Use the Bike Rental Cards Part II to answer the questions and determine the y-intercept and rate of change. Card 1 1. Identify the independent and dependent variables.

2. What is the y-intercept? Write the ordered pair.

3. What is the rate of change? What does it represent in this situation?

Card 2 1. Identify the independent and dependent variables.

2. What is the y-intercept? Write the ordered pair.

3. What is the rate of change? What key words helped you identify it?

4. How can this situation be represented as an equation?

Rate of Change and Initial Value | 167

Rate of Change and Initial Value

Explore 1 Card 3 1. Identify the independent and dependent variables.

2. What is the y-intercept? Write the ordered pair. What does it represent in this situation?

3. What is the rate of change? What does it represent in this situation?

Card 4 1. Identify the independent and dependent variables.

2. What is the y-intercept? Write the ordered pair. What does it represent in this situation?

3. What is the rate of change? What key words helped you identify it?

4. How can this situation be represented as an equation?

168 | Rate of Change and Initial Value

Explore 1

Rate of Change and Initial Value

Reflect 1. How can you describe the y-intercept in real-world situations?

2. What is an example of y-intercept used in a real-world situation?

3. How can you describe the rate of change in real-world situations?

4. What is an example of rate of change used in a real-world situation?

Rate of Change and Initial Value | 169

Rate of Change and Initial Value

Explore 2

Name: _______________________ Date: ___________

Comparing Key Features Part I 1. The ﬁrst step in identifying the best allowance for you is to sort the Allowance Options Graphs into possible appropriate domains. If f(x) represents the number of weeks you are collecting an allowance, you want to know which graphs would be appropriate. Complete the table below to help you sort the graphs based on the x-axis, or domain. Domain

Implication for Allowance

Allowance Option

Takes into account future weeks Negative integers 2. Which allowance options would be the least desirable considering an appropriate domain? Why?

3. Your cousin said you should consider all of the allowance options as viable options. Justify why you agree or disagree.

You decide to also investigate the key features of intercepts and slope. Compare and contrast the key features to answer the following questions. 4. Consider A and G. Using key features, what implications would this have on your decision?

5. Consider two graphs with the same starting amount. Using key features, what implications would this have on your decision?

Rate of Change and Initial Value | 171

Rate of Change and Initial Value

Explore 2 Part II

Use the following questions to help you analyze all of the Allowance Option Cards. 1. The ﬁrst step in identifying the best allowance for you is to analyze the Allowance Options Cards. Complete the table below to help you sort the graphs based on certain characteristics. Description in Context

Word for This Key Feature

Allowance Option

List the options in order from least to greatest based on initial value. List the options in order from least to greatest based on allowance rate. 2. Which allowance options have a negative y-intercept? What does that mean in terms of this situation?

3. Your cousin offers to help you look over your options. Looking at I and N, he believes N is the better choice because the y-intercept is greater. Do you agree or disagree? Why?

4. Looking over your texts with your parents, you see that you do owe them some money already. Which options should you consider? Which is the better option and why?

5. If the cards all demonstrate f(x) as the length of time for the allowance, what is the appropriate domain for all cards? Would this affect your choice of cards? Explain.

172 | Rate of Change and Initial Value

Explore 2

Rate of Change and Initial Value

Reflect 1. Which function from the graphs would you pick for your allowance? Why?

2. Which function from the cards would you pick for your allowance if you were collecting your allowance for only 1 year? Why?

3. When answering questions involving graphs, tables, words, and equations, what key features can help you compare them accurately?

4. When comparing the slope of a graph of a function to the slope represented by a table of a different function, what would you do?

5. What can be determined about an allowance option given the equation? How does this compare to the information that can be determined given a graph or table?

Rate of Change and Initial Value | 173

Rate of Change and Initial Value

Explore 3

Name: _______________________ Date: ___________

Create Equations Part I Record the correct equations with the ﬂyer advertisements in the table below. Bikes N’ More Weekday Special Bikes N’ More Weekend Special Wheels R Us Weekday Special Wheels R Us Weekend Special Two Wheels Only Weekday Special 1. Your cousin ﬁnds some extra equations. Write possible advertisements that would match the equations. a.

y = 35x + 62

y = 99x – 15

Rate of Change and Initial Value | 175

Explore 3

Rate of Change and Initial Value

Part II 1. Using the Bike Shop Cards, create the equations for the bike shop specials. Wheels R Us Friday Special! Bikes N’ More Friday Special! Two Wheels Only Friday Special! Pedaled Out Friday Special! 2. Which bike shop has the least expensive hourly rate to work on the bike, and which has the most expensive hourly rate?

3. What does the y-intercept indicate for the bike tune-up equations? Which is the best in the scenarios, and which is the worst?

4. Your bikes will only need minor tune-ups that should take about 2 hours. Put the bike shops in order of best to worst choice. Explain.

176 | Rate of Change and Initial Value

Explore 3

Rate of Change and Initial Value

Reflect 1. In these scenarios, what do the slope and y-intercepts represent?

2. Which weekday special would you choose if you wanted to rent bikes for 2 days of your vacation during the week? Explain.

3. Which weekend special would you choose if you were going to rent bikes for a 2-day weekend? Explain.

4. Would you change the bike tune-up choice from Part II if your bike had an estimated 5-hour tune-up? Why or why not?

5. Would you change the bike tune-up choice from Part II if your bike had an estimated 30-minute tune-up? Why or why not?

Rate of Change and Initial Value | 177

Linear Forms

179

Linear Forms

Explore 1

Name: _______________________ Date: ___________

Point-Slope Form Part I Use the Jewel Island Treasure Map to answer the following questions. 1. You begin your hunt for treasure at the origin and quickly ﬁnd a pink circle jewel in quadrant IV where the shovel is depicted. What is the slope of the line you traveled along to reach this point?

2. Had there been another jewel at the point (2, −3), would you have picked it up on your way from the origin to the pink circle jewel?

3. From the pink circle jewel, you now start digging toward the third quadrant along a line that has a slope of − 1 . Will you pick up another jewel? Use slope to justify why 4 or why not.

4. If you continue digging along this path past the purple circle jewel, would you pick up a jewel at the point (−12, 0)?

5. After traveling from one point to another, how can you predict if you would pick up another jewel along the same path?

Linear Forms | 181

Linear Forms

Explore 1 Part II

1. Use your chenille stem to create a line from the yellow triangular jewel to the blue teardrop jewel. Write an equation of the line connecting the yellow triangular jewel and the blue teardrop jewel in the form y = mx + b. Which of the values is easy to ﬁnd, and which is difﬁcult by looking at the graph?

2. Since we don’t have an equation that is easy to create, we will have to use the slope to check if other potential coordinates with jewels will also lie on the line. Complete the table below.

Starting Point

End Point

Slope Calculation

Yellow triangle (−9, 7)

Blue teardrop (6, −2)

−2 − 7 6 − (−9)

Slope Value

Is the end point on the line?

− 3

Yes

Yellow triangle (−9, 7) Yellow triangle (−9, 7) Yellow triangle (−9, 7) Yellow triangle (−9, 7) 3. What is true about every point that ends up on the line?

4. What equation would have to be true in order to make the point (x, y) also end up on the line?

182 | Linear Forms

Linear Forms

Explore 1

5. Multiply both sides of the equation you created in question 4 by the denominator containing x to reveal a new equation for the line passing through the yellow triangle jewel and blue teardrop jewel.

6. How do the values in this equation connect to the work you did in the table in question 2?

7. Use your equation from question 5 to determine if (21, −11) is on the line.

8. You now want to tell the machine an equation so it will dig up the orange heart and green triangle jewels in the ﬁrst quadrant, as well as any other jewels along that path. What is the slope between the two jewels in quadrant I?

9. Write an equation for the slope between the point (x, y) and the green triangular jewel if you know that (x, y) is also on the line. Then, multiply both sides by the denominator that includes x.

10. Write an equation for the slope between the point (x, y) and the orange heart jewel if you know that (x, y) is also on the line. Then, multiply both sides by the denominator that includes x.

11. Isolate y in both of your equations from questions 9 and 10. What do you notice?

Linear Forms | 183

Linear Forms

Explore 1 Part III

1. Use the Jewel Island Treasure Map and the equations below to help determine a path to treasure! Graph the new line, and list the color and shape of the jewels you reach. Be sure to graph the entire line to see if you get any jewels! Complete the table. GPS Equation

Color of Jewel at Starting Point

y – 4 = −2(x + 7)

Light blue diamond

y + 8 = 2(x – 2)

Any Other Jewels?

Pink circle Blue teardrop

Orange heart

Red oval

Purple circle

2. Substitute the coordinates of the red oval jewel into the equation you created. What do you notice?

3. Your friend creates a GPS equation as y = 2x – 12. Which GPS equation above is this the same equation for? How do you know?

184 | Linear Forms

Linear Forms

Explore 1 Reflect

1. Create an equation of a line that passes through the point (5.2, −8.3) and has a slope of −10. Is this the same equation as y = −10x + 43.7? Why or why not?

2. How can you write the equation of a line when given two points?

3. What key information can you extract from the equation y – k = m(x x – h)?

4. A line with a slope of 2 and a y-intercept at (0, −6) would have an equation in slopeintercept form of y = 2x – 6. Write the equation using point-slope form, and show that the two are equivalent.

5. Why might it be more efﬁcient to write the equation of a line in point-slope form instead of using slope-intercept form? When might slope-intercept form be a better choice?

6. Find the slope, m, between the points (h, k) and (x, y). How does this connect to the point-slope linear form?

Linear Forms | 185

Linear Forms

Explore 2

Name: _______________________ Date: ___________

Standard Form Part I Use the Jewel Island Treasure Map to answer the following questions. 1. You begin your hunt for treasure at the point (0, −8) and quickly ﬁnd a pink circle jewel in quadrant IV where the shovel is depicted. What is the slope of the line you traveled along to reach this point?

2. You continue digging at the same slope. Will you pick up another jewel? Use slope to justify why or why not.

3. You didn’t record your other initial intercept and want to write it down so you don’t repeat the same dig. What intercept besides (0, −8) would have led you to the pink circle jewel and blue teardrop jewel? Justify your answer.

4. You lucked out with your last attempt and found all the jewels in the ﬁrst quadrant, but you forgot to write down your data that led you to the jewels. a. What is the slope that has all the jewels in the ﬁrst quadrant? b. What is the y-intercept of the line? c. Does the x-intercept fall on a whole number point? If not, between which points does it lie?

5. What do you notice about all points that cross the x-axis? What do you notice about all points that cross the y-axis?

Linear Forms | 187

Explore 2

Linear Forms

Part II 1. Use your chenille stem or straightedge to create a line between the red oval jewel and the orange heart jewel. Write an equation of the line connecting the orange heart jewel and the red oval jewel in the form y = mx + b. Can you easily determine the intercepts from the equation and chenille stem or straightedge? Why or why not?

2. Write the equation for the red oval jewel and orange heart jewel in point-slope form and in the form y = mx + b.

3. What do you notice about both of your equations for the red oval jewel and orange heart jewel?

4. Rewrite your y = mx + b equation so that b is left alone on the right side of the equation.

5. Consider the y-intercept of the equation (use question 2 for reference). What value could you substitute into the equation from question 4 in order to get that resulting y-intercept? Where would you substitute it? Explain.

6. What can you conclude about ﬁnding the y-intercept?

7. Make a prediction about your equation that you found in question 4 and about calculating the x-intercept.

188 | Linear Forms

Linear Forms

Explore 2

8. Use your chenille stem or straightedge to estimate the x-intercept. What value could you substitute into the equation from question 4 in order to get about your estimated x-intercept? Where would you substitute it? Explain.

9. Use your chenille stem or straightedge to determine what intercepts would be needed to collect the purple circle jewel and jewels and coins treasure. Record the data in the table below. Slope

y-intercept

x-intercept

10. Write the equation for your data from question 9 in y = mx + b form.

11. From the equation you just created in question 10, rewrite the equation so the variables are on one side of the equal sign and the constant is on the other side.

12. Check the equation you just made by solving for the x- and y-intercepts. Does your equation work? Justify your answer.

13. What do you notice about equations when they are in the form you made from questions 4 and 11 (when variables are isolated on one side of the equal sign and constants are on the other)?

Linear Forms | 189

Linear Forms

Explore 2 Part III

1. Use the Jewel Island Treasure Map and the equations below to help determine a path to treasure! Graph the new line using intercepts, and list the color and shape of treasures you reach. Be sure to graph the entire line to see if you get any jewels! Complete the table. GPS Equation

Color of Jewel at Starting Point

−x + y = −8

Blue teardrop

4x x + 3y = −16

Any Other Jewels?

Jewels and coins Yellow triangle

Orange heart

Pink circle

Purple circle

2. Substitute the coordinates of the purple circle jewel into the equation you created. What do you notice?

190 | Linear Forms

Explore 2

Linear Forms

Reflect 1. Create three forms of equations of a line that passes through the point (5.2, −8.3) and has a slope of −10 in the form used in Part III.

2. How can you write the equation of a line when given two points to the form used in Part III?

3. What key information can you extract from the standard form equation Ax + By = C?

4. A line with a slope of 2 and a y-intercept at (0, −6) would have an equation in slopeintercept form of y = 2x – 6. Write the equation using standard form, and show that the two are equivalent.

5. Why might it be more efﬁcient to write the equation of a line in standard form instead of using slope-intercept form? When might slope-intercept form be a better choice?

Linear Forms | 191

Linear Forms

Explore 3

Name: _______________________ Date: ___________

Compare Equation Forms Part I You are evaluating the data from your machines by checking the production documents, which specify the distance in feet along the beach, s, as a function of the time in seconds, t, for each dig.

Machine 1

Machine 2

1. Using the tables, determine the rates of change of the distances with respect to time and the y-intercept. Machine 1 rate of change:

Machine 1 y-intercept:

Machine 2 rate of change:

Machine 2 y-intercept:

2. Write equations that will allow you to compare the rate of change and y-intercept for machine 1 and machine 2. Machine 1: Machine 2: 3. Why did you choose that form of equation to be the best form to compare the equations?

Linear Forms | 193

Linear Forms

Explore 3

The machines get very hot from working outside in the heat and have different rates at which they return back to nonworking temperature. Use the data given by the machine to write equations. The machine manual speciﬁes the relationship between temperature, y, in degrees Fahrenheit and the time, x, in minutes as shown below.

Machine 1

Machine 2

Rate of cooling: 4 degrees per minute

Rate of cooling: 3.2 degrees per minute

Recorded temperature: 82°F when the cooling had occurred for 2 minutes.

Recorded temperature: 79°F when the cooling had occurred for 5 minutes.

4. Write equations that will allow you to include all the information in the tables above for both machines. Machine 1:

Machine 2:

5. Explain your decision to use the linear form you used for each equation above.

194 | Linear Forms

Linear Forms

Explore 3

6. If you needed to know the starting temperature for the machines, what form would you change the equations to? Explain. Then, ﬁnd the starting temperature for both from these equations.

Equation

Starting Temperature

Machine 1 Machine 2

7. If you needed to know the starting temperature for the machines and how long it took them to return to their original temperatures, what form would you change the equations to? Explain. Then, ﬁnd how long it would take for each machine to return to its original temperature. Round to the nearest tenth as necessary.

Equation

Minutes to Return to Original Temperature

Original Temperature

Machine 1 Machine 2

Linear Forms | 195

Linear Forms

Explore 3 Part II

After having worked with your machine and the different linear forms, you want to create a reference note card for yourself to know when to use each equation.

Linear Form

How It Is Written

What Key Features Are Included

When to Use This Linear Form When graphing using the y-intercept and slope When given two points and one is the y-intercept

y = mx + b

(h, k): a point on the line m: slope

Point-slope form

Ax + By = C

None directly, but can calculate the x- and y-intercepts

Determine which form you need for the following scenarios: a. You need to know the starting temperature and how long it took to return to its cooled temperature.

b. You know how quickly the machine is cooling and the starting temperature.

c. You know how hot the machine was some time into cooling and how quickly it is cooling.

196 | Linear Forms

Linear Forms

Explore 3 Reflect

1. Give an example of a linear function written in slope-intercept form. Identify the rate of change (slope).

2. Give an example of a linear function with a slope of zero.

3. Convert y – 4 = −2(x + 1) into slope-intercept form, and then identify the slope.

4. Which of the equations below have the same slope? Show how you know. A. y = 4x – 6 B. 3x x–y=6 C. y + 5 = 4(x – 4) D. x + y = 4

Linear Forms | 197

Bivariate Data

199

Bivariate Data

Explore 1

Name: _______________________ Date: ___________

Bivariate Data Part I Sketch the scatterplot created by each group. Use the scatterplots to complete General Zane’s report. 1.

Age vs. Eye Color

Height vs. Shoe Size

x 3.

x 4.

Minutes Studied vs. Exam Grade

Amount of Money Spent vs. Amount of Money Saved y

x 5. Time Spent on Phone Before Bed

vs. Amount of Sleep

Age vs. Favorite Sport y

x Bivariate Data | 201

Bivariate Data

Explore 1 General Zane’s Report 1. Which scatterplots could be described as having a positive relationship?

2. Which scatterplots could be described as having a negative relationship?

3. Which scatterplots could be described as having no relationship?

4. Which scatterplots could be described as having a linear relationship?

5. Which scatterplots could be described as having a nonlinear relationship?

202 | Bivariate Data

Bivariate Data

Explore 1 Part II

Analyze the scatterplot to determine the association between the number of absences and test scores. Then, complete Principal Zed’s report.

100

Absences vs. Test Scores

Test scores

80 60 40 20 0

x 2

Number of absences Principal Zed’s Report 1. Circle all of the terms that correctly describe the association between the number of absences and test scores. Positive

Negative Linear

Nonlinear

2. Sometimes scatterplots show data points that are considered outliers. Draw a box around any point that could be considered an outlier.

3. Are there any data points that are clustered together? If so, circle the cluster(s).

4. Is the coordinate (13, 64) a good representation of the relationship between the number of absences and test scores? Explain.

Bivariate Data | 203

Bivariate Data

Explore 1 Reflect

1. The association between students’ averages in math and averages in science is shown on the scatterplots below.

Class A

x Average in math

Class B

Average in science

x Average in math

Which class has a stronger relationship between the average in math and the average in science? Explain your reasoning.

2. Sketch a scatterplot that has a negative linear association with 2 outliers and no clusters.

x 204 | Bivariate Data

Bivariate Data

Explore 2

Name: _______________________ Date: ___________

Lines of Best Fit Use the Class Data Table to construct a scatterplot. Analyze the scatterplot to complete the Woodville Hospital glove order. Length of Hand vs. Width of Hand

y 16

Width of hand (cm)

12 10 8 6

2 0

Length of hand (cm)

1. How would you describe the association between the length of a hand and the width of a hand?

2. What does this graph tell you about the relationship between the length and width of a hand?

3. Circle the point or points that appear to be outliers. 4. Place your hard spaghetti noodle on the graph to show a line that best represents the data. Trace along the spaghetti noodle with your pencil. Make sure your line goes across the entire graph, not just through your data points. © Accelerate Learning Inc. – All Rights Reserved

Bivariate Data | 205

Bivariate Data

Explore 2 5. How well does your line ﬁt the data points?

6. Is there another line, different from what you drew, that could also represent the data well? Explain.

7. Why would it be useful to draw a line through the data points?

Use your scatterplot to predict the missing dimensions for each doctor’s glove order.

Woodville Hospital Glove Order

Doctor

Length of Hand (cm)

Santo

20.5

Chen Hawkins Vygotsky

206 | Bivariate Data

Width of Hand (cm)

11 22.25 15

Bivariate Data

Explore 2 Reflect 1. Draw a line that would be a good representation of the data in the scatterplot.

2. Draw a line that would NOT be a good representation of the data in the scatterplot.

x 0

3. How do outliers affect the line that best represents the data?

4. What are some strategies you can use to draw the line that best ﬁts the data?

5. Would it be useful to draw a line that best represents data with no correlation? Explain.

Bivariate Data | 207

Bivariate Data

Explore 3

Name: _______________________ Date: ___________

Linear Equations Part I Use the data in the table to construct a scatterplot representing the 9th-grade class’s cookie sales. Draw a line of best ﬁt, and analyze the scatterplot to answer the questions that follow. Week

Funds Raised ($)

175

225

250

275

300

250

400

1. What is the slope of the line of best ﬁt?

9th-Grade Cookie Sales

375

2. What does the slope tell you about the proﬁt?

350 325 300

Funds raised ($)

275

3. What is the y-intercept of the line of best ﬁt?

250 225 200

4. What does the y-intercept tell you about the proﬁt?

175 150 125

5. Write an equation to represent the line of best ﬁt for cookie sales.

100 75 50 25 0

9 10

6. Use your equation to predict the proﬁt at the end of the 20-week fundraiser.

Bivariate Data | 209

Bivariate Data

Explore 3 Part II

Analyze the scatterplots and lines of best fit for each class to determine an equation to represent the lines of best fit. Use the equations to predict the profit after 20 weeks.

300

1. What is the slope of the line of best ﬁt, and what does it represent in the scenario?

10th-Grade Bracelet Sales

275 250

Funds raised ($)

225

2. What is the y-intercept of the line of best ﬁt, and what does it represent in the scenario?

200 175 150 125

3. Write an equation to represent the line of best ﬁt.

100 75 50 25 0

9 10

4. Use your equation to predict the proﬁt at the end of the 20-week fundraiser.

Week

Funds raised ($)

300

1. What is the slope of the line of best ﬁt, and what does it represent in the scenario?

11th-Grade Ice Pop Sales

2. What is the y-intercept of the line of best ﬁt, and what does it represent in the scenario?

200

3. Write an equation to represent the line of best ﬁt.

100

9 10

4. Use your equation to predict the proﬁt at the end of the 20-week fundraiser.

Week 210 | Bivariate Data

Bivariate Data

Explore 3

Funds raised ($)

300

1. What is the slope of the line of best ﬁt, and what does it represent in the scenario?

12th-Grade T-Shirt Sales

2. What is the y-intercept of the line of best ﬁt, and what does it represent in the scenario?

200

3. Write an equation to represent the line of best ﬁt.

100

9 10

4. Use your equation to predict the proﬁt at the end of the 20-week fundraiser.

Week

Reflect 1. Which classes should meet the goal of raising at least $500?

2. The teachers held their own fundraiser selling school supplies. The line of best fit for their profit over the first 8 weeks can be represented by the equation y = 20x + 150.

a. What does the 150 represent in this scenario?

b. What does the 20 represent in this scenario?

c. If the teachers run their fundraiser for 35 weeks, what will their approximate proﬁt be?

Bivariate Data | 211

Bivariate Data

Explore 4

Name: _______________________ Date: ___________

Interpret Data Use the data from each graph to answer the questions below.

Sleeping Habits 100

Test scores

90 80 70 60 50

x 7

Hours of sleep 1. What test score would you expect from someone who gets 10 hours of sleep?

2. How many hours of sleep would you expect from someone who got a 75 on their test?

3. What inference can you draw from the data?

Bivariate Data | 213

Bivariate Data

Explore 4

100

1. Why does this graph not have a line of best ﬁt?

Students’ Ages

Test scores

90 80 70

2. What inference can you draw from the data?

60 50

x 11

Age

100

1. What test score would you expect from a student who studies 4 hours a week?

Study Habits

Test scores

2. What inference can you draw from the data?

80 70 60 50

x 2

3. What can you infer based on the location of the y-intercept?

Hours studying per week

214 | Bivariate Data

Bivariate Data

Explore 4

1. What test score would you expect from a student who is involved in extracurricular activities for 12 hours?

Extracurricular Activities

100

Test scores

2. What inference can you draw from the data?

80 70 60 50

x 2

3. What can you infer based on the location of the y-intercept?

Hours in extracurricular activities

Reflect 1. Is there an association between the extracurricular activities and the study habits data?

2. How can you use a line of best ﬁt to make predictions?

Bivariate Data | 215

Parallel and Perpendicular Lines

217

Parallel and Perpendicular Lines

Explore 1

Name: _______________________ Date: ___________

Parallel and Perpendicular Slopes Part I 1. The ﬁrst box the box factory wanted made was perfectly square and had a bottom and side as shown. Box 1

a. The top of this image should be equal in length and parallel to the bottom side. Draw the top of the square.

y 5

b. What equation would represent the line for the top of the box?

4 3

c. What is the slope of the line for the top of the box?

2 1

d. How does this compare to the slope of the line for the bottom of the box?

e. The left side of this image should be equal in length and parallel to the right side. Draw the left side of the square.

f. What equation would represent the line for the left side of the box?

g. What is the slope of the line for the left side of the box?

h. How does this compare to the slope of the line for the right side of the box?

2. What does it mean for the lines to be parallel? How can you tell on a graph that lines are parallel?

Parallel and Perpendicular Lines | 219

Parallel and Perpendicular Lines

Explore 1

Brandon begins a table to analyze a square box that is partially programmed into the computer. Help Brandon ﬁnish the box. 3. Draw the rest of the box, and determine the slope for each side. Box 2 y 10

ft s ide

Slope

sid

Top

7 6

Left

5 4

Right

3 2

Bottom

1 0

4. How does the slope of the top compare to the slope of the bottom?

5. How does the slope of the right compare to the slope of the left?

6. What generalization can be made about the slopes of parallel lines? Give an example to support your generalization.

220 | Parallel and Perpendicular Lines

Parallel and Perpendicular Lines

Explore 1 Part II

Brandon begins a table to analyze a rectangular box that is partially programmed into the computer. Help Brandon ﬁnish the box. Box 3

1. Determine the slope and equation for each side of the box.

y 10

Slope

Left

t gh

− 3 2

e sid

Top

Right

ide

s ft Le

4 3

Bottom

2 1 0

2. Instead of looking at the parallel sides, Brandon wants to focus on the slopes of the sides forming the corners of each box. What relationship do you see when looking at the slopes of the top side and left side of box 2 and box 3? Complete the table, and record your thoughts below.

Left Side Slope Box 2

Box 3

− 3

Top Side Slope

Product

3. How could you write the slope of the left side of box 2 as a fraction? How does this representation compare to the top side slope?

4. What generalization can you make about the slopes of perpendicular lines? Justify your answer.

Parallel and Perpendicular Lines | 221

Parallel and Perpendicular Lines

Explore 1

Brandon decides to investigate an additional box plan to determine whether he has correctly identiﬁed how to plan perpendicular sides. 5. Draw the rest of the box, and determine the slope for each side. Box 4

Slope

Top

Top s

ide

Left

Right

Bottom 1

6. Do these slopes follow the generalization made in question 4? Support your answer by multiplying the slopes to show whether or not they follow the generalization.

222 | Parallel and Perpendicular Lines

Parallel and Perpendicular Lines

Explore 1 Reflect

1. Summarize your ﬁndings by completing the table below using words like opposite reciprocal, same, and different. Graph of Lines

Slopes

Parallel Perpendicular 2. Brandon’s robotics class begins to practice writing plans for their projects. For each box, Brandon needs to explain whether the slopes are correct. If not, he needs to provide a suggestion of a slope that would be correct. a. Brandon is analyzing a box; the slopes of the top and bottom are 3, and the slopes of the left and right sides are −3. Are they perpendicular?

b. Brandon is analyzing a box; the slope of the left side is −5, and the slope of the right side is −5. Are they parallel?

c. Brandon is analyzing a box; the slopes of the top and bottom are 5 , and the slopes of the right and left sides are −5. Are the corners perpendicular? Explain.

3. What are the parallel and perpendicular slopes of a line that has a slope of b ?

Parallel and Perpendicular Lines | 223

Parallel and Perpendicular Lines

Explore 2

Name: _______________________ Date: ___________

Parallel Lines Part I: Matching Parallel Equations 1. Brandon ﬁnds the notes left with the machine to program the parallel sides of boxes. Somehow, the notes were ripped apart before Saanvi and Brandon arrived. Find pairs of parallel lines, and record them in the table below. Parallel Lines Set A Parallel Lines Set B Parallel Lines Set C Parallel Lines Set D 2. Saanvi notices some notes that are scribbled on another piece of paper near the machine. Some are crossed off, but there are some that are still readable. Decide whether the notes could be used or if they have errors and should be discarded. Explain how you know. a.

The side parallel to y = 4x + 8 must go through the point (−2, 0).

The side parallel to 3y y – x = −6 must go through the point (0, 3).

Parallel and Perpendicular Lines | 225

Parallel and Perpendicular Lines

Explore 2 Part II: Creating Parallel Equations

The students decide to work on carnation boxes. The boxes need to be long and skinny. y

1. The top of the box is shown on the graph. What is the equation for the top of the box in slope-intercept form?

5 4

(4, 4)

2. The students decide that the bottom of the box needs to pass through the point (4, 4) in order to be the correct width. Write the equation they need for the bottom of the box in point-slope form, and sketch the line on the graph.

2 1

3. Rewrite the equation for the bottom of the box in slope-intercept form.

4. One of the machines is already set to start cutting the other side of the box at (0, 3). If it is parallel to the line given for the top of the box, write an equation in slope-intercept form for the line that is parallel and goes through (0, 3).

5. The machine is malfunctioning and will only accept equations in standard form. Rewrite the equation from question 4 in standard form.

6. Another group believes that the parallel line for question 5 is 3y y + 4x = 12. Would this be a valid equation to input into the machine to create the desired line? Explain.

226 | Parallel and Perpendicular Lines

Explore 2

Parallel and Perpendicular Lines

Reflect 1. An equation for a side of a box is 2x x + 3y = 24. a. What information do we need to extract from this equation to write the equation of a parallel side? Find the information.

b. If the parallel side must pass through the point (3, −1), write an equation for this new side.

2. Line k passes through (3, −8) and (9, −8). a. What is the slope of line k? Explain how you know.

b. What is an equation of a line that would be parallel to line k? Justify your answer.

3. Given any line graphed on the coordinate plane, list one or more methods to create new lines that are parallel.

Parallel and Perpendicular Lines | 227

Parallel and Perpendicular Lines

Explore 3

Name: _______________________ Date: ___________

Perpendicular Lines Part I The staircase program uses pivot points in order to complete the cuts. The computer starts at the solid circle and ends at the X. When it reaches an open-circle pivot coordinate, it will cut in the direction of the next equation. 1. Continue the graph until the point (4, 0). Complete the table with equations to match each segment of the graph.

y Line

Equation

x = −2

9 8 A

B C D E F G

6 5 C 4

D E

3 2

1 –3 –2 –1 0

G 4

2. What patterns occur in the slopes and equations for this staircase program?

Parallel and Perpendicular Lines | 229

Parallel and Perpendicular Lines

Explore 3 Saanvi decides to start working on the equation for line segment A. 3. What is an equation she could write?

y 10 9

4. Draw line segment B on the graph.

5. Write an equation in point-slope form for line segment B.

7 6 5 4

6. Write another equation that would create the equation for line segment B in slope-intercept form.

3 2 1

7. Complete the equations in point-slope form or slope-intercept form for lines C and D in the staircase program. Line C:

–5 –4 –3 –2 –1 0 –1 C –2

–3 D

–4 –5 –6 –7

Line D:

230 | Parallel and Perpendicular Lines

Parallel and Perpendicular Lines

Explore 3 Part II

Saanvi tries to help Brandon by providing some tips and strategies for writing equations of perpendicular lines. Saanvi’s Strategies A. Use the slope formula to determine the slope.

D. Create an equation in point-slope form.

B. Convert an equation into slopeintercept form.

E. Identify the opposite reciprocal slope.

C. Write the equation of a horizontal or vertical line.

F. Determine the slope of a line from the equation.

Decide which strategies, if any, would work best for each question, and then ﬁnd the perpendicular equation. Best Perpendicular Strategies Equation 1. Line p contains the points (4, 4) and (8, 7). Write an equation for line r that is perpendicular to line p and goes through (3, 4). 2. Write an equation of a line that is perpendicular to y = 3 and goes through the point (−8, 0). 3. What is the equation of a line perpendicular to 5x x + 3y = 9 that passes through (−7, −1)? 4. Write an equation of a line that is perpendicular to x = 10 and goes through the point (−2, 2). 5. Write the equation of a line that is perpendicular to the line y = 3x – 5 and has the same y-intercept. © Accelerate Learning Inc. – All Rights Reserved

Parallel and Perpendicular Lines | 231

Parallel and Perpendicular Lines

Explore 3 Reflect

1. What is true about the slope of perpendicular lines? Select all that apply. They are the same. They are opposite signs. They are reciprocals.

2. Is the following statement true or false? There are an inﬁnite number of perpendicular equations passing through a single point perpendicular to a given line.

3. An equation for a side of the box is y = 5 x + 3. If the perpendicular side must have a y-intercept of (0, 7), which equation could be used to program the machine? Select all that apply. 1

y= 5 x+7 y = 5x + 7 y = −5x + 7 5x + y = 7

4. When transforming an ordered pair called point P, (x, y), performing a rotation of 90 degrees clockwise about the origin will result in a new ordered pair called point P’, (y, −x). Explain how a rotation of 90 degrees clockwise about the origin is similar to lines with perpendicular slopes.

232 | Parallel and Perpendicular Lines

Solving Pairs of Linear Equations

233

Solving Pairs of Linear Equations

Explore 1

Name: _______________________ Date: ___________

Graph Pairs of Linear Equations Part I The equation f = 25 + 10t shows Saanvi’s followers over time and is graphed below. y

Number of followers

100 80 60

Time (minutes)

Graph a line to represent Gabriella’s followers over time using the following equation: f = 50 + 5t Who has more followers after 3 minutes? Explain how you can tell from the graph.

Will Saanvi ever have more followers? Explain how you can tell from the graph.

Solving Pairs of Linear Equations | 235

Solving Pairs of Linear Equations

Explore 1 Reflect

1. How many equations did we plot on the same graph for this scenario?

2. What do we call the point where two or more lines cross?

3. What does the point (5, 75) mean in the context of this scenario?

4. What does the intersection of two lines represent?

236 | Solving Pairs of Linear Equations

Solving Pairs of Linear Equations

Explore 1 Part II

The graph of Saanvi’s followers over time is graphed below. Graph a line to represent Daisy’s followers on the same grid, and use the pair of linear equations to answer the questions. y

Number of followers

100 80 60

Time (minutes)

Graph a line to represent Daisy’s followers over time using the following equation: f = 10t Daisy, who just started her photo-sharing page, is excited that she’s gaining followers at the same rate as Saanvi, 10 followers per minute! She jokes that at 10 minutes, she will have more followers than Saanvi. Based on the graph, do you agree with Daisy? Why or why not?

At what point will Saanvi and Daisy have the same number of followers? Explain your answer.

Solving Pairs of Linear Equations | 237

Solving Pairs of Linear Equations

Explore 1

The graph of Saanvi’s followers over time is graphed below. Graph a line to represent Debra’s followers on the same grid, and use the pair of linear equations to answer the questions. y

Number of followers

100 80 60

Time (minutes)

Graph a line to represent Debra’s followers over time using the following equation: f = 20 + 15t When do Saanvi and Debra have the same number of followers? How do you know?

Between Saanvi and Debra, who is winning the competition? Why?

238 | Solving Pairs of Linear Equations

Solving Pairs of Linear Equations

Explore 1

The graph of Saanvi’s followers over time is graphed below. Graph a line to represent Devon’s followers on the same grid, and use the pair of linear equations to answer the questions. y

Number of followers

100 80 60

Time (minutes)

Graph a line to represent Devon’s followers over time using the following equation: 2ff – 20t = 50 What do you notice when you graph the equations of both lines?

What does the observation you noted in the previous question mean in the context of the scenario?

Of the ﬁve friends, Saanvi, Gabriella, Daisy, Debra, and Devon, who won the competition after 10 minutes? Why?

Solving Pairs of Linear Equations | 239

Solving Pairs of Linear Equations

Explore 1 Reflect

1. By looking at the graph for a system of equations, how can you tell if a pair of linear equations has one solution, no solution, or an inﬁnite number of solutions?

2. Consider this scenario: A dog is chasing you. Which graph shows that the dog caught up with you? Why? y

3. The system of linear equations y = 2x + 4 and y = 3x + 2 is solved graphically below. a. Based on the graph, is (1, 6) the solution to this system of equations? Why or why not?

y 14 12 10 8

b. How could you tell if (1, 6) were a solution to the pair of linear equations without using a graph?

6 4 2

x 1

240 | Solving Pairs of Linear Equations

Solving Pairs of Linear Equations

Explore 2

Name: _______________________ Date: ___________

Analyze Systems of Linear Equations Part I Round 1 Saanvi’s distance versus time on the mini-race car course is graphed below. y 10

Distance (feet)

9 8

Driver

Distance vs. Time Equation

Saanvi

d = 5t

Gabriella

d = 5t + 2

Devon

d = 5t – 5

7 6 5 4 3 2 1 0

x 1

Time (seconds)

Graph lines to represent Gabriella’s and Devon’s distances versus time on the graph above. Then, label each line on the graph. 1. Looking at the equations, what is the same about each equation?

2. Looking at the equations, what is different about each equation?

3. In round 1, did Saanvi pass any of her friends? If so, at what point?

Solving Pairs of Linear Equations | 241

Solving Pairs of Linear Equations

Explore 2

Round 2 Saanvi’s distance versus time on the mini-race car course is graphed below. y 10

Distance (feet)

9 8

Driver

Distance vs. Time Equation

Saanvi

d = 5t

Gabriella

2d d – 10t = 0

Devon

3d d = 15t

7 6 5 4 3 2 1 0

x 1

Time (seconds)

Graph lines to represent Gabriella’s and Devon’s distances versus time on the graph above. Then, label each line on the graph. 1. Looking at the equations, what is the same about each equation?

2. Looking at the equations, what is different about each equation?

3. In round 2, did Saanvi pass any of her friends? How do you know?

242 | Solving Pairs of Linear Equations

Solving Pairs of Linear Equations

Explore 2

Round 3 Saanvi’s distance versus time on the mini-race car course is graphed below. y 10

Distance (feet)

9 8

Driver

Distance vs. Time Equation

Saanvi

d = 5t

Gabriella

d=t+4

Devon

d = 3t

7 6 5 4 3 2 1 0

x 1

Time (seconds)

Graph lines to represent Gabriella’s and Devon’s distances versus time on the graph above. Then, label each line on the graph. 1. Looking at the equations. What is the same about each equation?

2. Looking at the equations. What is different about each equation?

3. In round 3, did Saanvi pass any of her friends? If so, when?

Solving Pairs of Linear Equations | 243

Solving Pairs of Linear Equations

Explore 2 Part II

For each of the rounds of the mini-race car course, determine if Bart and Saanvi passed each other by analyzing the equations and give your reasoning. Round 1

Equation Saanvi: d = 3t Bart: d = t + 2

Did anyone pass another player?

Explain why or why not.

Round 2

Equation Saanvi: d = 2t + 1 Bart: −4tt + 2d = 2

Did anyone pass another player?

Explain why or why not.

244 | Solving Pairs of Linear Equations

Solving Pairs of Linear Equations

Explore 2 Round 3

Equation Saanvi: d = 2t + 1 Bart: d – 5 = 2t

Did anyone pass another player?

Explain why or why not.

Round 4

Equation Saanvi: d = 2t + 1 Bart: d – 5 = 3t

Did anyone pass another player?

Explain why or why not.

Solving Pairs of Linear Equations | 245

Solving Pairs of Linear Equations

Explore 2 Reflect 1. Summarize your ﬁndings by completing the chart below. Equation Features y-intercepts

Slopes

Different

Type of Solution

Graph of Lines

No solution Coincide

2. If two equations of a system have the same y-intercept but different slopes, how many solutions would there be? Explain your answer.

Distance (feet)

a. Sketch an example on the graph provided to justify your answer.

10 8 7 6 5 4 3 2 1 0

3. Consider the following equations.

x 1

Time (seconds)

y = 5x + 7 y – 7 = mx a. What would m need to be in order for this system of equations to have inﬁnitely many solutions?

b. How could you change the second equation so the system of equations would have no solution?

246 | Solving Pairs of Linear Equations

Solving Pairs of Linear Equations

Explore 3

Name: _______________________ Date: ___________

Solving with Substitution Use the information below to determine how many adults, a, and how many children, c, were on each ride by solving with substitution. Roller Coaster Equation 1: a + c = 30

Equation 2: c = 18

How many adults were on the roller coaster?

Use substitution to ﬁnd the value of a. a + _____ = 30

Number of adults:

Number of children:

Solving Pairs of Linear Equations | 247

Solving Pairs of Linear Equations

Explore 3 Ferris Wheel Equation 1: 2a + c = 50

Equation 2: c = 8

Use substitution to ﬁnd the value of a. 2a + _____ = 50

Number of adults:

Number of children:

Dungeon Drop Equation 1: 2a + c = 50

Equation 2: c = a + 2

Use substitution to ﬁnd the value of a.

Number of adults:

248 | Solving Pairs of Linear Equations

Number of children:

Solving Pairs of Linear Equations

Explore 3 Bumper Boats Equation 1: a + c = 25

Equation 2: c = 4a

Use substitution to ﬁnd the value of a.

Number of adults:

Number of children:

Reflect 1. Which ride has the least adults?

2. What would you look for in your equations to determine whether substitution can be used rather than graphing?

3. Daniel wants to solve the following system of equations through graphing to ﬁnd the point of intersection. 4d d + h = 27 h=d+2 Pillar thinks that solving using substitution would be faster. Which method do you think would be faster to use for these equations?

Solving Pairs of Linear Equations | 249

Solving Pairs of Linear Equations

Explore 4

Name: _______________________ Date: ___________

Solving with Elimination Part I Use the receipts below to ﬁgure out the cost of each item ordered. Saanvi

Gabriella

1 Grilled cheese sandwich 1 Order of tots

1 Grilled cheese sandwich 2 Order of tots

Total: $3.50

Total: $5.00

Saanvi’s equation: 1s + 1tt = 3.50

Gabriella’s equation: 1s + 2tt = 5.00

How much does one order of tots cost?

How much does one grilled cheese sandwich cost?

Analyze, and ﬁll in the chart below. Add System

1s + 2tt = 5.00 + 1s + 1tt = 3.50

Subtract 1s + 2tt = 5.00 – (1s + 1tt = 3.50)

Result Was this useful to help us solve? Why or why not?

Solving Pairs of Linear Equations | 251

Solving Pairs of Linear Equations

Explore 4 Reflect 1. Why would it be useful to subtract the equations?

2. Which equation should you write ﬁrst when subtracting?

3. What happened to the s terms when you subtracted the equations?

4. At what point on a graph would these two lines intersect?

Part II Use the Order Cards to determine the cost of each item on the snack menu. Order 1 equation:

Order 2 equation:

Algebraic steps:

Cost of one hamburger:

252 | Solving Pairs of Linear Equations

Cost of one fries order:

Explore 4 Order 3 equation:

Solving Pairs of Linear Equations

Order 4 equation:

Algebraic steps:

Cost of one chicken basket:

Cost of one drink:

Order 5 equation:

Order 6 equation:

Algebraic steps:

Cost of one cheeseburger:

Cost of one fries order:

Solving Pairs of Linear Equations | 253

Explore 4 Order 7 equation:

Solving Pairs of Linear Equations

Order 8 equation:

Algebraic steps:

Cost of one cheeseburger:

Cost of one drink:

Reflect 1. What did you do when you solved the system of equations for orders 5 and 6 and had a coefﬁcient of 2 left for fries?

2. When is using the elimination method the best way to solve?

254 | Solving Pairs of Linear Equations

Pythagorean Theorem

255

Pythagorean Theorem

Explore 1

Name: _______________________ Date: ___________

Modeling the Pythagorean Theorem and the Converse of the Pythagorean Theorem Part I: Modeling the Pythagorean Theorem Use the Modeling the Pythagorean Theorem Work Mat and the Demonstrating the Pythagorean Theorem Triangles handout to complete the table. Write the formula that applies the Pythagorean theorem to each triangle and proves that the triangle is a right triangle.

Side Length

Area

Pythagorean Theorem Formula 2 a + b2 = c 2

Pythagorean Theorem | 257

Pythagorean Theorem

Explore 1 Reflect 1. How were models used to demonstrate the Pythagorean theorem?

2. How can you determine the area of a square given one of its side lengths?

3. How can you determine the side length of a square given its area?

4. The hypotenuse is the longest side of a right triangle. How can you identify the hypotenuse when given an image of a right triangle?

5. How might applying the Pythagorean theorem be useful?

258 | Pythagorean Theorem

Pythagorean Theorem

Explore 1 Part II: The Converse of the Pythagorean Theorem Determine which triangles form a right triangle. Is it a right triangle?

Use the Pythagorean theorem to help Mason decide which of the following two holes’ side lengths form a right triangle. Hole 7

6 yards

8 yards 10 yards

Pythagorean theorem: ___ + ___ = ___ Right triangle? Yes or No Hole 12

12 yards

9 yards

10 yards

Pythagorean Theorem | 259

Explore 1

Pythagorean Theorem

Use the space provided below to sort the Converse of the Pythagorean Theorem Cards into two categories: right triangle and not a right triangle.

Right Triangle

Not a Right Triangle

260 | Pythagorean Theorem

Explore 1

Pythagorean Theorem

Reflect 1. How can you use the Pythagorean theorem to prove whether a triangle is a right triangle?

2. Given the three side lengths of a triangle, how can you identify the hypotenuse?

3. When applying the Pythagorean theorem to verify whether a triangle is a right triangle, does it matter which of the triangle’s two legs is a and which is b? Why or why not?

4. When might it be helpful to use the Pythagorean theorem in identifying right triangles?

5. Side lengths 3, 4, and 5 can be substituted into the Pythagorean theorem to form a true equation. This set of positive integers is also known as a Pythagorean triple. What are some other examples of Pythagorean triples?

Pythagorean Theorem | 261

Pythagorean Theorem

Explore 2

Name: _______________________ Date: ___________

Finding an Unknown Side Length in a Right Triangle Part I: Finding the Hypotenuse of a Right Triangle Gemma has begun designing hole 1; however, she needs your help ﬁnishing this hole, along with designing 3 more holes. Use the Geoboard to create 3 other right-triangle designs. Then, use the Pythagorean theorem to help Gemma determine the lengths of the hypotenuses. Round your answer to the nearest hundredth, if necessary.

Sketch of hole 1

Hole

Side Length a

Side Length b

Pythagorean Theorem 2 a + b2 = c 2

Hypotenuse c (√ c2 = c)

Pythagorean Theorem | 263

Pythagorean Theorem

Explore 2 Use the information in Part I to answer the following questions. 1. How was the Geoboard useful in creating right triangles?

2. How did you determine the length of the hypotenuse in a right triangle given the measurements of both legs?

3. Triangle A has two legs with lengths of 5 units and 12 units. Triangle B has two legs with lengths of 7 units and 9 units. What are the differences between both hypotenuses?

Reflect 1. Given two side lengths, how can you determine whether the length of a hypotenuse will be an integer or an irrational number?

2. Outside of the math classroom, when might you need to determine the length of a hypotenuse given the measurements of the two legs of a right triangle?

264 | Pythagorean Theorem

Pythagorean Theorem

Explore 2 Part II: Finding the Missing Leg of a Right Triangle

Gemma has space for 3 more holes on the course. She creates 3 different righttriangle-shaped holes and measures each of their side lengths. In recording these measurements, she forgets to write down one of the leg lengths from each of the holes. Apply the Pythagorean theorem and use the Finding the Missing Leg of a Right Triangle Card Sort to help Gemma calculate the lengths of the missing legs. Glue the cards in the columns Leg a, Hypotenuse c, and Leg b. Use the table below to record your calculations in the column Applying the Pythagorean Theorem.

Hole Design

Leg a

Hypotenuse c

Applying the Pythagorean Theorem

Leg b (√ b2 = b)

10 yards

7.5 yards

b yards

√120 yards

b yards

√25 yards

b yards 13 yards

20 yards

Pythagorean Theorem | 265

Pythagorean Theorem

Explore 2 Use the information in Part II to answer the following questions.

1. How can you determine the length of a leg given the measure of the other leg and the hypotenuse of a right triangle?

2. Using the Pythagorean theorem, what operations are used to determine the length of a leg given the measures of the other two sides of a right triangle?

3. A right triangle’s leg measures 10 inches, and the hypotenuse measures 15 inches. Use the Pythagorean theorem to determine the length of the other leg. Round your answer to the nearest hundredth, if necessary.

Reflect 1. When solving for the missing side length in a right triangle, do you think it’s easier to solve for a missing leg or a missing hypotenuse? Use the Pythagorean theorem to explain your reasoning.

2. When checking your work, how do you know your answer is reasonable when solving for the measure of a missing leg in a right triangle?

266 | Pythagorean Theorem

Pythagorean Theorem

Explore 3

Name: _______________________ Date: ___________

The Pythagorean Theorem in Rectangular Prisms Kyle packages his latest shipment using a cardboard box. The dimensions of the rectangular prism-shaped box are provided below.

2 ft

3 ft

4 ft What is x, the maximum length of a putter that is able to ﬁt inside this box when placed diagonally from the bottom-most left front corner to the top-most right back corner?

Use step 1 and step 2 below to calculate the values of d and x by ﬁlling in the blanks. Step 1: Calculate the value of d.

Step 2: Calculate the value of x.

Use the Pythagorean theorem to calculate the length of d, the diagonal of the bottom rectangular base. Round your answer to the nearest hundredth, if necessary.

Using the diagonal length of d, apply the Pythagorean theorem to calculate the length of x, the diagonal of the rectangular prism. Round your answer to the nearest hundredth, if necessary.

x= d=

Pythagorean Theorem | 267

Pythagorean Theorem

Explore 3

Kyle uses two other-sized boxes to send his shipments. Use the Putter Boxes to ﬁnd the maximum length of a putter that is able to ﬁt inside each box when placed diagonally from the bottom-most left front corner to the top-most right back corner. Use the table below to calculate each diagonal length using the Pythagorean theorem. Putter Box A

Putter Box B

Diagonal length of rectangular base, d:

Maximum length of putter that is able to ﬁt diagonally:

268 | Pythagorean Theorem

Pythagorean Theorem

Explore 3 1. How can you determine the diagonal length of a rectangle?

2. How can you determine the diagonal length of a rectangular prism?

3. A box is shaped like a rectangular prism. The rectangular base measures 18 inches by 24 inches. The height of the box is 20 inches. What is the diagonal distance from the bottom-most left front corner to the top-most right back corner?

Reflect 1. Outside of the math classroom, how is using the Pythagorean theorem to determine the diagonal of a rectangular prism useful?

2. Do you think rotating a rectangular prism on a lateral side instead of its base changes the length of the prism’s diagonal? Use Putter Box A or B to justify your response.

Pythagorean Theorem | 269

Pythagorean Theorem

Explore 4

Name: _______________________ Date: ___________

The Pythagorean Theorem on a Coordinate Grid Use a number cube and the coordinate grid below to help Yvette create an outline of the ﬁrst new addition. N W

E S

Addition 1 Begin this addition by placing point A at (1, 1). Roll the number cube once. Move this many spaces north. Record this new point below. Point B ( _____ , _____ ) Connect point A to point B. Roll the number cube again. Move this many spaces east. Record this new point below. Point C ( _____ , _____ ) Connect point B to point C. Then, connect point C to point A. Use the information above to complete the table. Round your answer to the nearest hundredth, if necessary. Distance between Point A and Point B

Distance between Point B and Point C

Use the Pythagorean theorem to determine the distance between point A and point C.

Distance between Point A and Point C

Pythagorean Theorem | 271

Pythagorean Theorem

Explore 4

Use a number cube and the coordinate grid below to help Yvette create an outline of the second new addition. N W

E S

Addition 2 Begin this addition by placing point D at (7, 1). Roll the number cube once. Move this many spaces west. Record this new point below. Point E ( _____ , _____ ) Connect point D to point E. Roll the number cube again. Move this many spaces north. Record this new point below. Point F ( _____ , _____ ) Connect point E to point F. Then, connect point F to point D. Use the information above to ﬁll in the following table. Round your answer to the nearest hundredth, if necessary. Distance between Point D and Point E

272 | Pythagorean Theorem

Distance between Point E and Point F

Use the Pythagorean theorem to determine the distance between point D and point F.

Distance between Point D and Point F

Pythagorean Theorem

Explore 4

Use both tables and the addition 1 and addition 2 coordinate grids to answer the following questions. 1. With both new additions, how can you identify both legs of the right triangle when drawn on the coordinate grid?

2. How is the hypotenuse in addition 1 similar to the hypotenuse in addition 2?

3. Yvette decides to design a third new addition to the course. She places the following points along the perimeter of this right-triangle-shaped design: Point H (2, 0)

Point I (2, 7)

Point J (6, 0)

Determine the distance between point I and point J. Round your answer to the nearest hundredth, if necessary.

Reflect 1. How is using the coordinate grid to create right triangles similar to using a geoboard to make right triangles?

2. How are the four directions on the compass beneﬁcial in creating a right triangle on the coordinate grid?

Pythagorean Theorem | 273

Volume

275

Volume

Explore 1

Name: _______________________ Date: ___________

Cylinders Part II: The Cannery Use the Can It Cards to complete the table. Draw a model of each cylinder, and label the dimensions. Use the workspace to represent the area of the base expressed in terms of 𝜋, and then ﬁnd the volume. Use 3.14 as an approximation for 𝜋. Round to the nearest hundredth, if necessary. Card A Model:

Workspace:

Volume: Card B Model:

Workspace:

Volume | 277

Volume

Explore 1 Card C Model:

Workspace:

Volume:

Card D Model:

Workspace:

Volume:

278 | Volume

Volume

Explore 1 Card E Model:

Workspace:

Height:

Card F Model:

Workspace:

Volume:

Volume | 279

Explore 1

Volume

Reflect 1. Why are the units of volume in cubic units?

2. If a cylinder were on its side, how would you know which measurements to use for the dimensions to ﬁnd the volume of the cylinder?

3. Compare the following two cylinders, and determine which cylinder has the greater volume. o

Cylinder 1: diameter of 20 cm, height of 5 cm

Cylinder 2: radius of 5 cm, height of 10 cm

280 | Volume

Volume

Explore 2

Name: _______________________ Date: ___________

Cones Part II: Solving Volume of a Cone Problems Use the Cone It Cards to complete the table. Draw a model of each cone, and label the dimensions. Use the workspace to represent the area of the base expressed in terms of 𝜋, and then ﬁnd the volume. Use 3.14 as an approximation for 𝜋. Round to the nearest hundredth, if necessary. Sample Cone Model:

Workspace:

Volume: Medium Cone Model:

Workspace:

Volume | 281

Volume

Explore 2 Kiddie Cone Model:

Workspace:

Height:

Small Cone Model:

Workspace:

Volume:

282 | Volume

Volume

Explore 2 Wafﬂe Cone Model:

Workspace:

Volume:

Teeny Tiny Sundae Cone Model:

Workspace:

Volume:

Volume | 283

Explore 2

Volume

Reflect 1. What did you notice about the relationship between the height and the volume of a cone?

2. What did you notice about the relationship between the radius and the volume of a cone?

3. Do you think the relationship between the volume of a cylinder and the volume of a cone would work if the radii and heights were not the same? Explain.

284 | Volume

Volume

Explore 3

Name: _______________________ Date: ___________

Spheres Part II: Solving Volume of a Sphere Problems Use the Juice It Cards to complete the table. Draw a model of each sphere, and label the dimensions. Use the workspace to ﬁnd the volume. Use 3.14 as an approximation for 𝜋. Round to the nearest hundredth, if necessary. Navel Orange Model:

Workspace:

Volume: Personal Watermelon Model:

Workspace:

Volume | 285

Volume

Explore 3 Cantaloupe Model:

Workspace:

Volume:

Grape Model:

Workspace:

Volume:

286 | Volume

Volume

Explore 3 Clementine Model:

Workspace:

Volume:

Blueberry Model:

Workspace:

Radius:

Volume | 287

Volume

Explore 3 Reflect 1. How do you determine the height of a sphere?

2. What is the relationship between the volume of a sphere and the volume of a cylinder?

3. What is an example of a real-world scenario where you would need to ﬁnd the volume of a sphere?

288 | Volume

Skills Quizzes

289

Square Roots and Cube Roots

Skills Quiz

Name: _______________________ Date: ___________

Square Roots and Cube Roots Directions: Solve each problem, and show the steps you took to get your answer.

1. √225

2. ∛ – 27 64

3. Find the side length of the cube below.

Volume = 343 yd.3

Square Roots and Cube Roots | 291

Skills Quiz

Square Roots and Cube Roots

4. (−12)2 A.

−144

144

−24

5. 43

6. √ 49

7. 92

292 | Square Roots and Cube Roots

Skills Quiz

Square Roots and Cube Roots

8. ∛ 125 512 A.

5 64

5 8

25 64

5 512

9. ∛ 216

10. In a square with an area of 121 in.2, what is the side length?

Square Roots and Cube Roots | 293

Irrational Numbers

Skills Quiz

Name: _______________________ Date: ___________

Irrational Numbers Directions: Solve each problem. Show or explain your mathematical thinking.

1. √130 is in between which two integers? A.

10 and 11

11 and 12

12 and 13

13 and 14

2. Approximate the value of 3 . A.

0.222

0.333

0.666

0.999

3. What is the approximate value of √10?

Irrational Numbers | 295

Irrational Numbers

Skills Quiz 𝜋 4. Classify the number 2 . A.

Rational

Irrational

Neither

5. Write the decimal equivalent of the fraction 6 .

6. Which number line best represents the approximate value of √50? A. B. C. D.

7. Find the approximate value of 6𝜋.

296 | Irrational Numbers

Skills Quiz

Irrational Numbers

8. Classify the number √ 30. A.

Rational

Irrational

Neither

9. Classify the number 8 . A.

Rational

Irrational

Neither

10. The value of √113 is in between which two integers?

Irrational Numbers | 297

Integer Exponents

Skills Quiz

Name: _______________________ Date: ___________

Integer Exponents Directions: Solve each problem, and show the steps you took to get your answer.

1. Simplify the following expression: (96)4. A.

910

924

96+ 94

2. Which of the answer choices is equivalent to this expression? 3–8 A.

1 – 3–8

–38

1 38

–83

Integer Exponents | 299

Integer Exponents

Skills Quiz

3. Simplify the following expression. Express your answer in exponent form. 59 ÷ 54

4. Given the following expression, write an equivalent expression. (4 · 7)6

5. Simplify the expression 90.

300 | Integer Exponents

Integer Exponents

Skills Quiz

6. Simplify the following expression. Express your answer in exponent form. (124)4

7. Calculate the value of the following expression. 47 ÷ 44

8. Determine the value of the following expression. 120 A.

–12

Integer Exponents | 301

Integer Exponents

Skills Quiz

9. Simplify the following expression. Express your answer in exponent form. 63 · 65

10. Simplify the following expression. Express your answer in exponent form.

( 11 ) 3

302 | Integer Exponents

Scientiﬁc Notation

Skills Quiz

Name: _______________________ Date: ___________

Scientific Notation Directions: Solve each problem, and show the steps you took to get your answer.

1. Convert the following number to scientiﬁc notation. 5,400,000

2. How many times larger is 1.4 × 106 than 3.5 × 105? A.

1 4

1 8

Scientiﬁc Notation | 303

Scientiﬁc Notation

Skills Quiz 3. Convert the following number to scientiﬁc notation. 0.00000827

4. Convert the following number to scientiﬁc notation. 0.000477 A.

47.7 × 10–4

4.77 × 10–4

0.477 × 10–4

4.77 × 104

5. Which number written in scientiﬁc notation is larger? 2.3 × 1010 or 5.8 × 109 A.

2.3 × 1010

5.8 × 109

304 | Scientiﬁc Notation

Scientiﬁc Notation

Skills Quiz 6. Convert the following number to scientiﬁc notation. 798,400,000,000

7. Convert the following number to scientiﬁc notation. 0.0000000004

8. Which number written in scientiﬁc notation is larger? 6.2 × 10–7 or 4.9 × 10–8 A.

6.2 × 10–7

4.9 × 10–8

Scientiﬁc Notation | 305

Scientiﬁc Notation

Skills Quiz 9. How many times smaller is 3 × 10–6 than 2.1 × 10–5?

10. Convert the following number to scientiﬁc notation. 0.00157

306 | Scientiﬁc Notation

Skills Quiz

Operations with Scientiﬁc Notation

Name: _______________________ Date: ___________

Operations with Scientific Notation Directions: Solve each problem, and show the steps you took to get your answer.

1. (3.6 × 105) + (2.7 × 104)

2. (9.3 × 10–5) – (4.5 × 10–7)

Operations with Scientiﬁc Notation | 307

Skills Quiz

Operations with Scientiﬁc Notation

3. (8 × 105)(7 × 106)

4.5 × 1010 2 × 1015

[(9.2 × 108) – (2 × 107)] 4.5 × 104

308 | Operations with Scientiﬁc Notation

Skills Quiz

Operations with Scientiﬁc Notation

6. (4.4 × 10–5)(6 × 1012) + (5 × 108)

6.4 × 103 + 1.4 × 104 + 7.5 × 103

Operations with Scientiﬁc Notation | 309

Skills Quiz

Operations with Scientiﬁc Notation

8. What is 4.4 × 1013 + 3.02 × 1013 + 1.04 × 1013 in calculator notation?

9. Solve the problem in question 8, and write the answer in calculator notation.

10. Write the answer in question 9 in scientiﬁc notation.

310 | Operations with Scientiﬁc Notation

Solve Equations

Skills Quiz

Name: _______________________ Date: ___________

Solve Equations Directions: Solve each problem, and show the steps you took to get your answer.

1. The distance a jet travels, d, varies directly with the number of hours ﬂown, h. A jet travels 1,290 miles in three hours. Which equation below represents this direct variation? A.

d = 43(h)

d = 430(h)

h = 430(d)

h = 43(d)

2. Fill in the blank so each statement is true.

a. ____ + 3 = 4x x – 9 has no solution.

b. −x x + 4 = ____ + 4 has one solution.

Solve Equations | 311

Skills Quiz

Solve Equations

3. Tell whether each equation has one solution, no solution, or inﬁnite solutions. a. −2x x + 1 = 2x – 5

b. 4x x + 2 = 4x + 2

c. 3(4x x – 5) = 2(6x + 4)

d. 3x x – 10 + 4x = 7x – 10

312 | Solve Equations

Solve Equations

Skills Quiz 4. Rami solved the equation 3(d d + 2) = 6. These are his steps. Step 1: d + 2 = 2 Step 2: d = 0 Which choice explains step 1 in Rami’s process? A.

Multiply both sides by 2.

Divide both sides by 3.

Subtract 2 from both sides.

Subtract 4 from both sides.

Use your knowledge about solving equations to answer questions 5–8. 1

5. 3 x + 6 = 9

6. 3(y y – 5) = 5y – (3 + y)

7. 2(3 – w) = 4(w w – 3)

8. 6x x + 9 = 3x + 21

Solve Equations | 313

Solve Equations

Skills Quiz Solve for x for questions 9 and 10. 9. ax + 3x = bx + 4

314 | Solve Equations

10. −13x x + mx = 34x + w

Solve Inequalities

Skills Quiz

Name: _______________________ Date: ___________

Solve Inequalities Directions: Solve each problem, and show the steps you took to get your answer. 1. What is the solution to the equation modeled below? Key 1

–1 x

–x

2. Two after-school clubs are raising money. • The math club has already raised $500 and is selling candy for $2 each. • The drama club has already raised $350 and is selling candy for $3 each. Write an equation that can be used to determine c, the amount of candy each club must sell so that the math club raises more money than the drama club.

Solve Inequalities | 315

Solve Inequalities

Skills Quiz 3. Which situation best represents the inequality? 0.10x x + 8 > 0.05x + 10 A.

Super Cab charges a standard fee of $8 plus $0.05 per mile driven. Ride Time charges a standard fee of $10 plus $0.10 per mile driven. How many miles must be driven for the cost of using Super Cab to be more than the cost of using Ride Time?

Super Cab charges a standard fee of $8 plus $0.10 per mile driven. Ride Time charges a standard fee of $10 plus $0.05 per mile driven. How many miles must be driven for the cost of using Super Cab to be more than the cost of using Ride Time?

Super Cab charges a standard fee of $0.10 and $8 per mile driven. Ride Time charges a standard fee of $0.05 and $10 per mile driven. How many miles must be driven for the cost of using Super Cab to be less than the cost of using Ride Time?

Super Cab charges a standard fee of $8 plus a $0.10 per mile driven. Ride Time charges a standard fee of $10 plus $0.05 per mile driven. How many miles must be driven for the cost of using Super Cab to be less than the cost of using Ride Time?

4. Daria and Alicia are reading the same book for class. Daria has completed reading 100 pages of the book and plans to read an additional 5 pages every night until the book is ﬁnished. Alicia has completed reading 75 pages of the book and plans to read 10 pages per night until the book is ﬁnished. Write an inequality that could be used to determine the number of pages Alicia must read to have more of the book completed than Daria.

316 | Solve Inequalities

Solve Inequalities

Skills Quiz 5. Which situation best represents the following inequality? 4b + 3 ≤ 6b A.

Four binders and a $3 pack of markers cost less than six binders.

Six binders cost more than four binders and a $3 pack of markers.

Four binders and a $3 pack of markers cost the same as six binders.

Four binders and a $3 pack of markers cost no more than six binders.

Use the following information for questions 6–8. A bakery sells a box of 6 cookies for $5.25 and sells additional cookies for $0.48 each. If Maria has $20.00 to buy the box of cookies and some additional cookies, what is the maximum number of additional cookies she can buy? Be sure to deﬁne any variables you use. 6. Write an inequality that represents the situation above.

7. Solve the inequality that you wrote for question 6 to see how many additional cookies Maria can buy.

8. What steps did you use to solve the inequality?

Solve Inequalities | 317

Solve Inequalities

Skills Quiz Use the following inequality for questions 9 and 10. 3(−2x x – 6) < −30 9. Explain the steps you would take to solve this inequality.

10. Graph the solution set you found in question 9.

318 | Solve Inequalities

Skills Quiz

Create Non-Proportional Relationships from Proportional Relationships

Name: _______________________ Date: ___________

Create Non-Proportional Relationships from Proportional Relationships Directions: Solve each problem. Use the tables to answer questions 1–3. A.

3.5

5.25

22.5

8.75

37.5

Determine the equation that represents each table, and answer each question. 1. Table A equation: What would the equation be if the table changed to have the point (0, 4)?

2. Table B equation: What would the equation be if the table changed to have the point (0, 3)?

3. Table C equation: What would the equation be if the table changed to have the point (0, 12)?

Create Non-Proportional Relationships from Proportional Relationships | 319

Skills Quiz

Create Non-Proportional Relationships from Proportional Relationships

4. A bag of grapes costs $2.35 per pound. Which equation represents that scenario but includes a $5 entry fee to enter the farmers’ market? A.

y = 2.35x

y = 5x

y = 2.35x + 5

y = 5x + 2.35

320 | Create Non-Proportional Relationships from Proportional Relationships

Skills Quiz

Create Non-Proportional Relationships from Proportional Relationships

Use the graphs to answer questions 5–7. B.

C. 200

200

180

160

140

120

Salary ($)

16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Salary ($)

100 80 60 40

3 4 5 6 7

8 9 10 11 12

Number of hours

80 60 40

20 0 1 2

100

x 1

3 4 5 6 7 8 Number of hours

9 10

20 0

Number of hours

Determine the equation that represents each graph, and answer each question. 5. Graph A equation: What would the equation be if the graph changed to have the point (0, 5)?

6. Graph B equation: What would the equation be if the graph changed to have the point (0, 30)?

7. Graph C equation: What would the equation be if the graph changed to have the point (0, 80)?

Create Non-Proportional Relationships from Proportional Relationships | 321

Create Non-Proportional Relationships from Proportional Relationships

Skills Quiz

The graph below shows the amount of money Seffron will make per hour. Use the graph below to answer questions 8–10.

400

375 350 325

Salary ($)

300 275 250 225 200 175 150 125 100 0

x 1

Number of hours 8. Determine the equation that represents the graph.

9. Which 3 points are represented on the graph? A.

(0, 100), (3, 125), (5, 150)

(0, 0), (3, 100), (5, 125)

(0, 125), (3, 150), (5, 175)

(0, 100), (3, 175), (5, 225)

322 | Create Non-Proportional Relationships from Proportional Relationships

Skills Quiz

Create Non-Proportional Relationships from Proportional Relationships

10. Will Seffron ever make exactly $250 during her time working? Explain.

Create Non-Proportional Relationships from Proportional Relationships | 323

Functions

Skills Quiz

Name: _______________________ Date: ___________

Functions Directions: Solve each problem.

Use the graphs to answer questions 1–3. A.

6 5

–6 –5 –4 –3 –2 –1 –1

x 1

–6 –5 –4 –3 –2 –1 –1

x 1

1 –6 –5 –4 –3 –2 –1 –1

–2

–3

–4

–5

–6

x 1

Tell whether each graph represents a function. 1. Graph A:

2. Graph B:

3. Which graph is a linear function?

Functions | 325

Functions

Skills Quiz 4. Which graph shown below does not represent a function?

B. 6

1 -6

-5

-4

-3

-2

-1

-6

-5

-4

-3

-2

-1

0 -1

-2

-3

-4

-5

-6

D. 6

1 -6

-5

326 | Functions

-4

-3

-2

-1

0 -1

x 1

-6

-5

-4

-3

-2

-1

0 -1

-2

-3

-4

-5

-6

Functions

Skills Quiz Use the tables to answer questions 5 and 6. A.

3 4

–2

–4

5. Tell whether each table represents a function. Table A: Table B: Table C:

6. What input and output values make the table a nonfunction?

Functions | 327

Functions

Skills Quiz Use ﬁgure 1 to answer question 7.

50 40 30 20 10 0

7. Name one interval where the function is increasing and one where the function is decreasing.

328 | Functions

Functions

Skills Quiz Use the graph below to answer questions 8 and 9.

The graph below shows two channel packages for a video streaming service: standard member and premium member.

Cost

Premium Standard n Number of channels subscribed

8. Choose the answer that best describes the graph. A.

The standard package is always cheaper.

The premium package is always cheaper.

The premium package becomes free where the line is horizontal.

The standard package becomes more expensive than the premium package.

Functions | 329

Functions

Skills Quiz

9. Give an argument for why someone might choose the standard option over the premium option.

10. Sketch a graph that accurately models the following scenario. Jenny rides her skateboard on a half-pipe ramp (U-shaped ramp).

330 | Functions

Rate of Change and Initial Value

Skills Quiz

Name: _______________________ Date: ___________

Rate of Change and Initial Value Directions: Solve each problem, and show the steps you took to get your answer. Determine the rate of change and initial value of each situation presented. Make sure to label your answers based on each situation.

Cost ($)

The graph and table show how much you and your cousin make cutting grass.

100 90 80 70 60 50 40 30 20 10

0 1 2 3 4 5 6 7 8 9 10

Time (h) 1. Complete the following table by using the graph and table above.

Graph

Table

Rate of Change and Initial Value | 331

Skills Quiz

Rate of Change and Initial Value

Use the table you created in question 1 to answer questions 2 and 3.

2. Which has the greatest rate of change? What does it represent?

3. Which has the greatest initial value? What does it represent?

Using the coordinates (–3, 3) and (–4, 5), answer questions 4 and 5.

4. Rate of change:

5. Initial value:

332 | Rate of Change and Initial Value

Skills Quiz

Rate of Change and Initial Value

Use the following scenario for questions 6 and 7.

6. A gym charges a $35 membership fee to start plus $45 every month. Rate of change:

7. Initial value:

Rate of Change and Initial Value | 333

Rate of Change and Initial Value

Skills Quiz Use the following information for questions 8–10.

Marcos orders a chocolate milkshake at his favorite restaurant. The milkshake is 300 mL. He ﬁnishes his drink in 10 minutes by drinking it at a constant rate.

8. Graph the function representing the volume of the milkshake over time. 350

300 250 200 150 100 200 50 x -2

10 19

9. Write the equation for the function.

10. What is a reasonable domain for this graph?

The domain is __________.

334 | Rate of Change and Initial Value

Linear Forms

Skills Quiz

Name: _______________________ Date: ___________

Linear Forms Directions: Solve each problem, and show the steps you took to get your answer. Use this graph of h(x) for questions 1–3. y 8 6 4 2

x -8

-6

-4

-2

-2 -4

h(x)

-6 -8

1. Write a function in slope-intercept form that matches the graph.

2. Using a point other than an intercept, write a function in point-slope form that matches the graph.

Linear Forms | 335

Linear Forms

Skills Quiz

3. Prove your equations are equivalent. Compare and contrast the key features provided by each form of the equation.

4. Fabian lives 4 miles from his school and rides his bike to school each day. He rides past one block every 50 seconds. If each block is 400 feet long, what is the rate in feet per second at which Fabian rides his bike? Write a linear equation to reﬂect a one-block ride.

5. What is the slope of the equation y = 9 − 5x?

6. Point-slope formula is written as y − y1 = m(x x − x1). Which of the following expressions below can be used to determine the rate of change? A.

y1 – x1 y–x x – x1 y – y1

336 | Linear Forms

y – y1

x1 – x

x – x1

Linear Forms

Skills Quiz Use the following scenario for questions 7–9.

You are upgrading your current gaming system. You are selling games for $20 a piece and accessories for $5. You are hoping to make the $260 you still need to purchase the new gaming system.

7. Write an equation for the scenario where x represents the accessories and y represents the games.

8. What are the intercepts for your equation? What do they represent in the scenario?

9. Assume that you wanted to only sell 8 games and 4 accessories. Rewrite the equation to include that point. Then, prove it is equivalent to your original equation.

Linear Forms | 337

Linear Forms

Skills Quiz 10. Compare the two functions, f( f x) and g(x).

g(x) = −4(x x − 2)

f x) f(

f(x)

–9

–7

–3

4.5

a. Which function has a greater slope?

b. Which function has a greater x-intercept?

c. Which function has a greater y-intercept?

338 | Linear Forms

Bivariate Data

Skills Quiz

Name: _______________________ Date: ___________

Bivariate Data Directions: Answer the following questions by interpreting the graph for each set of problems.

Figure 1

Money spent on fast food this month, y

100 80 60 40 20

100 x

Cost of most recent concert tickets purchased, x

1. Interpret the data pictured in ﬁgure 1. Describe the relationship between the bivariate data.

Bivariate Data | 339

Bivariate Data

Skills Quiz Use the table in ﬁgure 2 to answer questions 2–4. Figure 2 Hours of Tablet Use

Battery Remaining

94%

86%

82%

79%

70%

67%

62%

61%

53%

45%

2. The table suggests an association between two variables. What are the two variables in this situation?

340 | Bivariate Data

Bivariate Data

Skills Quiz

3. Sketch a scatterplot for the data in the space below. Include labels for the x- and y-axes. y

100

Battery %

80 60 40 20

Time (hours)

100 x

4. Graph the equation y = −5x + 100 on top of your scatterplot. Does this line ﬁt our data?

Bivariate Data | 341

Skills Quiz

Bivariate Data

Use the scatterplot in ﬁgure 3 to answer questions 5–8.

5. What is the y-intercept of the estimated line of best ﬁt? What does the y-intercept represent?

6. Find the approximate slope of the estimated line of best ﬁt. What does the slope represent?

342 | Bivariate Data

Skills Quiz

Bivariate Data

7. Write an equation for the estimated line of best ﬁt using your y-intercept and slope.

8. Using your equation for the estimated line of best ﬁt, predict how many schooling options a population of 600,000 people may have.

Bivariate Data | 343

Bivariate Data

Skills Quiz Use the graph in ﬁgure 4 to answer questions 9 and 10.

Figure 4

Unit test grades, y

100 80 60 40 20

Mid-unit quiz grades, x

100 x

9. According to the line of best ﬁt, what would someone who scored a 40 on the mid-unit quiz expect to score on the unit test?

10. Based on the data, what conclusion can be drawn about the relationship between the unit test grades and the mid-unit quiz grades?

344 | Bivariate Data

Parallel and Perpendicular Lines

Skills Quiz

Name: _______________________ Date: ___________

Parallel and Perpendicular Lines Directions: Solve each problem. Show or explain your mathematical thinking. Use the following information to answer questions 1 and 2. The line y = 5 x + 11 that passes through (−1, 3) and (1, 8) is represented on the 2

graph below.

y 10

-10

-5

-10

1. Write the equation of the line that is parallel to the line above and passes through the point (2, 8).

2. Write the equation of the line that is perpendicular to the line above and passes through the point (10, −8).

Parallel and Perpendicular Lines | 345

Parallel and Perpendicular Lines

Skills Quiz

3. Line B passes through the point (4, 2) and is perpendicular to the x-axis. What is the slope of line B?

4. Write the equation of the line that is perpendicular to the y-axis and passes through the point (3, 12).

5. Use the graph below to draw the line that passes through the point (0, 3) and is perpendicular to the y-axis.

346 | Parallel and Perpendicular Lines

Skills Quiz

Parallel and Perpendicular Lines

6. Line A passes through the points (2, −1) and (0, 5). Which of the following functions is parallel to line A and passes through the point (9, 3)? A.

f x) = −3x f( x + 21

f x)) = − 3 x + 30 f(

f x) = −3x f( x + 30

f x)) = 1 x + 28 f(

7. Write the equation of a line in slope-intercept form that is perpendicular to x – 2y = 4 and contains the point (−1, 2).

8. What is the slope of a line that passes through the point (−5, 5) and is parallel to the y-axis? A.

−1

−5

Undeﬁned

Parallel and Perpendicular Lines | 347

Parallel and Perpendicular Lines

Skills Quiz Use the following information to answer questions 9 and 10. The line y = − 1 x − 1 is graphed below. 2

y 10

-10

-5

-10

9. Which equation represents a line that is perpendicular to y = − 1 x − 1? 2

y=− 1x+3

y = 2x + 3

y = −2x + 3

y= 1 x+3

10. Which equation represents a line that is parallel to y = − 1 x – 1? 2

y=− 1x+3

y = 2x + 3

y = −2x + 3

y= 1 x+3

348 | Parallel and Perpendicular Lines

Solving Pairs of Linear Equations

Skills Quiz

Name: _______________________ Date: ___________

Solving Pairs of Linear Equations Directions: Analyze the graphs, and solve the linear equation questions. Show all steps when solving. Use ﬁgure 1 for questions 1 and 2. 30

Figure 1

x 0

1. Write a system of equations represented by these two lines.

Solving Pairs of Linear Equations | 349

Skills Quiz

Solving Pairs of Linear Equations

2. Write the solution to the system of equations as an ordered pair. Label the solution on the graph.

350 | Solving Pairs of Linear Equations

Solving Pairs of Linear Equations

Skills Quiz Use ﬁgure 2 for questions 3 and 4. 5

–5

Figure 2

x 5

–5

–10

3. Write a system of equations represented by these two lines.

4. Write the solution to the system of equations as an ordered pair. Label the solution on the graph.

Solving Pairs of Linear Equations | 351

Solving Pairs of Linear Equations

Skills Quiz

5. Choose the correct answer for the system of two equations. 2x x + 5y = 1 2x x + 5y = 4 A.

The system has inﬁnitely many solutions.

The system has no solutions.

The solution to the system is x = 1, y = 1.

6. A system of two linear equations is graphed in the coordinate plane and has inﬁnitely many solutions. Use words to describe what this means for the two equations.

352 | Solving Pairs of Linear Equations

Solving Pairs of Linear Equations

Skills Quiz 7. Solve the system of two linear equations shown below. y = 6x −2x x+y=8

8. Solve the system of two linear equations shown below. 2y y = 10x + 40 y = 5x + 20

Solving Pairs of Linear Equations | 353

Skills Quiz

Solving Pairs of Linear Equations

Use the following scenario for questions 9 and 10. Roberto and Muhammad scored a total of 18 points in their basketball game. Roberto scored 4 more than Muhammad.

9. Write a system of equations representing the points scored by Roberto (y) and Muhammad (x).

10. Solve the system of equations to ﬁnd how many points were scored by Roberto.

354 | Solving Pairs of Linear Equations

Pythagorean Theorem

Skills Quiz

Name: _______________________ Date: ___________

Pythagorean Theorem Directions: Solve each problem, and show the steps you took to get your answer. For questions 1 and 2, using the Pythagorean theorem, tell whether the given side lengths are a Pythagorean triple.

1. a = 9

b = 11

c = 14

2. a = 3

b=4

c=5

Pythagorean Theorem | 355

Pythagorean Theorem

Skills Quiz

3. Is the triangle shown a right-angled triangle? Prove by using the Pythagorean theorem.

4. Find the length of the hypotenuse of a right triangle with legs of 50.25 and √1,200.

356 | Pythagorean Theorem

Pythagorean Theorem

Skills Quiz 5. Find the missing hypotenuse in the triangle shown.

6. Find the missing leg in the triangle shown.

Pythagorean Theorem | 357

Skills Quiz

Pythagorean Theorem

7. Find the length of the hypotenuse of a right triangle whose legs have lengths 75 and 225 using a triangle with leg lengths 1 and 3.

8. Create a right triangle that is similar to a right triangle with legs of 3 and 4 and a hypotenuse of 5.

358 | Pythagorean Theorem

Pythagorean Theorem

Skills Quiz

9. Find the distance between the two points on the coordinate plane by using the Pythagorean theorem. 6

5 4 3 2 1 –6 –5 –4 –3 –2 –1 0 –1

x 1

–2 –3 –4 –5 –6

10. Find the distance between the points (1, 2) and (10, 14).

Pythagorean Theorem | 359

Volume

Skills Quiz

Name: _______________________ Date: ___________

Volume Directions: Solve each problem, and show the steps you took to get your answer. Identify the formula for each 3-D solid listed in questions 1–3.

1. Cylinder:

2. Cone:

3. Sphere:

Find the volume of each ﬁgure shown by using 3.14 for pi. Round to the nearest hundredth if necessary.

4. 5 12

Volume | 361

Skills Quiz

Volume

7 3

7. Rearrange the formula of a cylinder to ﬁnd the height when given volume and radius.

362 | Volume

Skills Quiz

Volume

Find the volume of each 3-D ﬁgure. Use 3.14 for pi.

8. Cylinder with a diameter of 10 and a height of 4

9. Cone with a diameter of 15 and a height of 8

10. Sphere with a diameter of 7

Volume | 363

GLOSSARY OF TERMS absolute value

angle-angle criterion

absolute value: the distance a number

additive inverse: what must be added to

is from zero on a number line; also

a number in order for the sum of the two

called the magnitude of a number; never

numbers to be zero

negative adjacent angles: two angles that have absolute value equation: an equation

the same vertex and a common ray but

in which x is c units from b in either

no interior common points

direction algebraic expression: numbers, absolute value function: a function that

variables, and symbols grouped together

contains an algebraic expression within

without an equal sign to show a

absolute value symbols

relationship

absolute value inequality: an inequality

algorithm: a step-by-step method for a

in which the distance from x to b is less

solution

than/greater than c altitude: the height of a polygon acute angle: an angle that measures less than 90°

amplitude: the height from the center line to the peak (or to the trough)

acute triangle: a triangle where every angle measures less than 90°

angle: a geometric figure formed by two rays with the same endpoint (vertex)

addends: the numbers added together to form a sum; any numbers being added

angle-angle criterion: the criterion which states that if two triangles have

addition property of equality: the

two pairs of congruent angles, then the

mathematical property which states that

triangles are similar

365

GLOSSARY OF TERMS angle measure

bar graph

angle measure: the measure of the

association: the form (linear/nonlinear),

angle formed by the two rays from a

direction (positive/negative/none), and

common vertex

strength (weak/moderate/strong) seen between two variables in a scatterplot

angle sum theorem: the theorem which states that the sum of the three interior

associative property of addition: the

angles of a triangle is equal to 180°

mathematical property which states that when adding three or more numbers, the

approximate: to find a number that is

placement of the grouping symbols does

close to the given number on a number line

not affect the sum, e.g., (a + b) + c = a + (b + c)

arc: a part of the circumference of a circle or a section of a curve

associative property of multiplication: the mathematical property which states

area: the number of square units it takes

that when multiplying three or more

to cover the two-dimensional surface of

numbers, the placement of the grouping

an object

symbols does not affect the product, e.g.,

area model: a model where the length

(a × b) × c = a × (b × c)

and width represent the factors and

asymptote: a line that a graph

are configured through the operation of

approaches but never crosses as the

multiplication

value of a variable becomes extremely

arithmetic pattern: a number pattern

large or small

that changes at the same rate, either

axis of symmetry: the line that divides

increasing or decreasing

a figure into two identical parts that are

arithmetic sequence: a sequence where

mirror images of each other

the successive terms differ by the same

bar graph: a graph that uses horizontal

number d, called the common difference,

or vertical rectangular bars to show each

where d ≠ 0

category of qualitative data

366

GLOSSARY OF TERMS base

chance

base: (1) the lower number of an

budget: a financial plan that estimates

exponent that is multiplied by itself;

expenditure for a certain period of time

(2) the surface that a solid object stands on categorical data: a type of data that can base of a polygon: the polygon side that

be divided into groups

is perpendicular to the altitude categorical variable: nonnumerical data base of a triangle: the triangle side that

represented by a letter or symbol

is perpendicular to the altitude category: a collection of objects with benchmark fraction: a familiar fraction

shared attributes

used as a reference point in order to measure, compare, and assess the

causation: the action of one event

reasonableness of a fractional value

causing another event to occur

binomial: a polynomial expression

center: referring to measures of center in

containing two terms

data collection

bivariate categorical data: data for two

center of a circle: the point that is an

nonnumerical variables

equal distance from any point on the circle

bivariate data: data for two variables that are paired to each other

center of a data set: a value in the middle of a distribution that represents a

boundary line: a line that corresponds

typical value of the data set

to the function that divides the coordinate plane into two halves

central angle: an angle in a circle with its corner in the circle’s center

box plot: a diagram that shows the fivenumber summary of a distribution

chance: the possibility of something happening

367

GLOSSARY OF TERMS circle

commutative property of addition

circle: a closed round figure in which

commission: money earned for selling a

every point on the boundary is equidistant

product, usually earned as a percentage

from the center

of the sales

circumference: the distance around a

common coefficient: when a variable

circle

has the same coefficient in two or more equations regardless of the sign

classify: to arrange into groups according to shared characteristics

common denominator: a denominator that is the same in two or more fractions

clockwise rotation: rotating in the direction in which the hands of a clock

common difference: the nonzero

normally move

constant difference, d, of any term and the previous term in an arithmetic

cluster: a group of data occurring closely

sequence

together on a graph common factor: a factor that two or coefficient: the number placed directly

more numbers share

before a variable that tells you to multiply that number by the variable

common multiple: a multiple that two or more numbers share

coinciding lines: lines that lie one on top of the other; the same line with the

common ratio: the ratio of each term

equations expressed in different forms

of a geometric progression to the term preceding it

combine like terms: to add together terms that have the same variable(s),

commutative property of addition: the

including their exponent

mathematical property which states that when adding two or more numbers, the order of the addends does not affect the sum; a + b = b + a

368

GLOSSARY OF TERMS commutative property of multiplication

constant

commutative property of

compound event: a combination of two

multiplication: the mathematical

or more simple events (with two or more

property which states that when

outcomes)

multiplying two or more numbers, the order of the factors does not affect the

compound interest: interest calculated

product; a × b = b × a

multiple times in a given time period so that interest is calculated on the original

complementary angles: two acute

amount and previous interest

angles that, when added, make 90°; two angles whose sum is 90°

conditional relative frequency: the fraction used to express the ratio of the

complete the square: the process used

number of participants in a group that

to form a perfect square trinomial for

meet a certain qualification

the purpose of finding the solution(s) by taking the square root

cone: a solid (three-dimensional) shape that has a flat, circular base joined to a

complex fraction: a fraction where the

point (vertex) by a curved side

numerator and/or the denominator are fractions

congruent: having exactly the same shape and size; being identical; congruent

complex solutions of a quadratic

objects coincide when they overlap.

equation: in the form a + bi; solutions that occur when the value under the

congruent angles: angles that have the

radical of the quadratic formula is less

same measure

than zero congruent figures: figures with the composite figure: a figure that consists

same size and shape

of two or more geometric shapes constant: a fixed number that stands composite number: a number with more

alone in an equation or expression

369

GLOSSARY OF TERMS constant of proportionality

correlation coefficient

constant of proportionality: the

convert: to change the form of a

positive constant, usually denoted k, that

measurement using different units

relates two quantities in the form y = kx

without changing the size or amount of the quantity being measured

constant of variation: the constant (unchanged) ratio of two quantities; in

coordinates: a pair of numbers that

direct variation, it is usually denoted as k.

provides the location of a point along the coordinate plane using the values of the

constant rate of change: a rate of

x-axis and y-axis

change that does not vary coordinate pair: the location of a single constant speed: the rate of fixed speed

point on a coordinate plane where the

per time

first and second values represent the position relative to the x-axis and y-axis,

constraint: a condition that the solution

respectively (x, y); also known as ordered

must satisfy

pair

continuous: data that can contain any

coordinate plane: two perpendicular

real number value between data points;

number lines, called the x-axis and the

data points can be connected.

y-axis, that intersect at the point (0, 0)

converse of the Pythagorean theorem: the theorem which states that

and create four quadrants; also called a graph, coordinate grid, or Cartesian plane

if the square of the length of the longest

correlation: the relationship between two

side of a triangle is equal to the sum of

variables that vary together

the squares of the other two sides, then the triangle is a right triangle; if c² = a²

correlation coefficient: a number r

+ b², then it is a right triangle.

that describes how closely the points in a scatterplot are related, where −1 ≤ r ≤ 1

370

GLOSSARY OF TERMS corresponding angles

data point

corresponding angles: angles in the

cube root: a number that, when

same position in different plane figures

multiplied by itself three times, produces the given number

corresponding congruent angles: angles in identical positions formed by a

cube root function: a function of the

transversal line cutting through two lines

form f( f x) =

corresponding sides: two sides that are

cubic number: a number to the power

in the same position in different plane

of three, i.e., 2³ represents the cubic

figures; in scale drawings, these sides will

number 8 and can be read as two cubed

have a proportional relationship.

or two to the power of three.

corresponding similar sides: sides in

cylinder: a solid (three-dimensional)

matching positions of similar figures that

shape that has two flat, circular, parallel

have a proportional relationship

bases joined by a curved surface at a

fixed distance counterclockwise rotation: rotating in the opposite direction in which hands of a

data: a collection of organized facts,

clock normally move

usually in numerical form, words, measurements, or descriptions

credit: a positive money value data distribution: a function or a listing cross-section: a two-dimensional shape

that shows all the possible values (or

that is created when a three-dimensional

intervals) of the data

shape is sliced data point: a point on a scatterplot that cube: a solid figure with six congruent

represents the data

square faces

371

GLOSSARY OF TERMS data set

difference

data set: a collection of organized facts,

degree (°): the unit of measure for an

usually in numerical form, but can also

angle

be given in words, measurements, or descriptions

degree of a polynomial: the largest exponent or the largest sum of exponents

debt: money that is owed; describes a

of a term within a polynomial

person’s bank account balance when it is less than zero

denominator: the bottom number within a fraction; the number that represents

decimal: a number that uses a decimal

the whole and how many parts total are

point followed by digits that show a value

in the whole

smaller than one, in powers of ten that decrease; a number with one or more

dependent variable: a variable,

digits to the right of the decimal point

often y, that relies on the value of the independent variable

decimal expansion: the decimal form of a number

deposit: a sum of money that is put into a bank account

decimal notation/decimal form: a number that uses a decimal point followed

deviation: the amount by which a single

by digits showing values less than one

measurement differs from a fixed value

decompose: to separate into parts or

diameter: any straight line segment that

elements (e.g., geometric figures or

passes through the center of the circle

numbers)

and has endpoints that lie on the circle

decreasing: the measure of the

difference: a number that is the result of

steepness of a line that shows the slant

subtraction

downward from left to right

372

GLOSSARY OF TERMS difference of two squares

dividend

difference of two squares: the

distance: a measurement of the length

difference of two squares, such as a² – b²

between two points

being factored into (a + b)(a – b) distance formula: the formula used to digit: any one of the numbers 0–9 dilation: a type of transformation where

find the distance, d, between two points (x1, y1) and (x2, y2) on the coordinate plane;

a scale factor is used to enlarge or reduce the distances in the original image

distance-time graph: a graph that shows the distance traveled by an object

dimension: something measurable (such

against the time it takes; any given

as length, width, and height)

point represents the speed of the object

direct variation: a relationship between two variables including a constant (k) discount: the amount subtracted from the original cost of an item discrepancy: a lack of compatibility or similarity between two or more things discrete: data that cannot contain the real number values between data points; data points are not connected. discriminant: the expression under the square root of the quadratic formula that determines the types of solutions of a quadratic equation

(distance per time). distribution: a list of all the possible values of the data and how often they occur distributive property: the mathematical property which states that multiplying the sum or difference of a group of terms by a number or variable is the same as multiplying each term by a number or variable and then adding or subtracting the products dividend: the number you divide into; a quantity that is to be divided by another quantity; a number that shows the amount of equal parts of a whole; the numerator (top number) that tells the number or quantity; a quantity to be divided

373

GLOSSARY OF TERMS division property of equality

equivalent

division property of equality: the

elimination method: a method of

mathematical property that states that

solving systems by adding or subtracting

dividing both sides of an equation by

equations to eliminate a variable

the same number gives us an equivalent equation

end behavior: the trend the graph follows as x approaches infinity in the

divisor: the quantity by which another

negative and positive directions

quantity is to be divided endpoint: the point at the end of a line domain: the set of all possible input

segment or ray

(x x values) of a function enlarge: to create a similar image that is dot plot: a method of visually displaying

now larger than the original image

a distribution of data values where each data value is shown as a dot or mark

equal sign: the symbol used to show

above a number line

that two quantities or expressions are the same

double number line diagram: a pair of parallel number lines used to represent

equal to (=): having exactly the same

equivalent ratios

amount or value

downward: the direction a parabola

equation: a mathematical statement that

opens when the value of a < 0

shows that two expressions are equal to each other

edge: a line at which a space or shape terminates, where two faces of a 3-D solid

equilateral triangle: a triangle with

intersect

three congruent sides and three congruent angles

element: any distinct number or value that is part of a set

374

equivalent: equal in value or amount

GLOSSARY OF TERMS equivalent expressions

expression

equivalent expressions: expressions

exponent: a mathematical notation that

that name the same number no matter

indicates the number of times the base

what value is substituted for the variable

number is multiplied by itself; also called power

equivalent ratios: two or more ratios that are equal; two different ratios

exponential decay: the process of

representing the same value

reducing an amount by a consistent percentage rate over a period of time

estimate: an approximation of an overall amount or value

exponential expression: an expression involving a term with a variable as an

evaluate: to determine or calculate the

exponent; 2x for example

numerical value of something exponential function: a function in the even function: when x is replaced with

form of f( f x) = abx where a and b are real

−x x in a function and the function is

numbers and a ≠ 0, b ≠ 1, and b > 0

simplified, the resulting function will be identical to the original function.

exponential growth: the change that occurs when an original amount is

event: one (or more) outcome(s) of an

increased by a consistent rate over a

experiment

period of time

experimental probability: the ratio that

exponential notation: an expression

compares the number of occurrences to

that takes the form aⁿ, where a is

the number of trials

multiplied by itself n times

explicit formula: a formula to find the

expression: numbers, variables, and

nth term of a sequence

symbols grouped together without an equal sign to show a relationship

375

GLOSSARY OF TERMS exterior angle of triangles theorem

gap

exterior angle of triangles theorem: the

force of gravity: the universal force of

mathematical theorem which states that

attraction acting between all matter

an exterior angle is equal to the sum of the two opposite interior angles of a triangle

formula: a mathematical statement or rule written with symbols

factor: A number or algebraic expression that another number or algebraic

fraction: a number that shows a part of a

expression can be divided by without

whole or part of a set

having a remainder frequency: how often a number occurs in factors: expressions that are multiplied

a data set

together to get a polynomial; factors that appear in the form of ax + b and cannot

frequency table: a table that lists

be factored further

outcomes and the number of times that they occur

factor pair: a set of two factors that multiply to give a particular product;

function: a special relationship between

listing factor pairs is a strategy used to

values; each input value gives back

determine all the factors of a number.

exactly one output value.

factor tree: a mathematical tool to help

function notation: a way of representing

break down a number into its prime

y, the dependent value in a relationship,

factorization

as f( f x), read “ff of x” where f names the function

figure: a two-dimensional shape function rule: the dependent variable five-number summary: the five values

(range, output, y value) expressed

used to make a box plot, including the

in terms of the independent variable

lowest value, lower quartile, median,

(domain, input, x value)

upper quartile, and highest value gap: a missing range of values in a data set 376

GLOSSARY OF TERMS geometric sequence

horizontal reflection

geometric sequence: a sequence in

half-plane: a planar region consisting of

which the ratio of successive terms is

all points on one side of an infinite straight

a constant r, called the common ratio,

line, and no points on the other side

where r ≠ 0 and r ≠ 1 height: the perpendicular distance from a graph: a visual representation of data

vertex to the opposite side of a figure

graph of a quadratic function:

height (3-D figure): the vertical

the attributes of a quadratic function

distance from the top of an object or

including the vertex, the y-intercept, the

figure to its base

x-intercepts, and the axis of symmetry histogram: a special type of bar graph graphing method: a method of solving

with numerical intervals as its labels

systems by graphing horizontal: describes the direction of gratuity: money given above the amount

a line that travels from left to right,

charged for a service; tip

perpendicular to a corresponding vertical line; from left to right; parallel to the

greater than (>): more than another

horizon

(e.g., 49 > 12) horizontal dilation: the act of expanding greater than or equal to (≥): more

or contracting in the horizontal direction

than or the same as another horizontal number line: describes the greatest common factor: the largest

direction of a horizontal number line that

same factor of two or more numbers

travels from left to right, perpendicular to a corresponding vertical line; from left to

grouping symbols: symbols that help to

right: parallel to the horizon

organize mathematical expressions; braces { }, brackets [ ], and parentheses ( )

horizontal reflection: a reflection over a vertical line such as the y-axis

377

GLOSSARY OF TERMS horizontal shift

inequality phrase

horizontal shift: a change in a function

increasing slope: the measure of the

that moves the function left or right

steepness of a line that shows the slant upward from left to right

horizontal translation: a shift in the base of the graph to the left or right

increasing/decreasing: a function is increasing if f( f b) > f( f a) and decreasing

hundredths: the second digit to the right

if f( f b) < f( f a) for any two input values a

of the decimal point; a hundredth is one

and b.

out of 100 equal parts of a whole. increments: the evenly spaced and scaled hypotenuse: the longest side of the right

markings used to locate and plot points

triangle, the side opposite of the right angle independent variable: a variable, often identity property of addition: the

x, that does not rely on the value of

mathematical property which states that

another variable

adding zero to a number does not change the value

index: a number indicating how many of a kind you need to put together to be able

identity property of multiplication: the

to move that number or variable from

mathematical property which states that

inside the radical to outside the radical

multiplying 1 by any number does not change the value

inequality: a mathematical sentence that uses symbols such as <, ≤, >, or ≥ to

image: the new figure in a transformation

compare two quantities

improper fraction: a fraction that has a

inequality notation: notation in

numerator that is greater than or equal to

which the solution is represented by an

the denominator

inequality statement

increasing: when the y value increases

inequality phrase: phrase representing

as the x value increases

each of the inequalities

378

GLOSSARY OF TERMS inference

inverse function

inference: a conclusion based on the

integer exponent: a positive or negative

given data

whole number or zero that tells the number of times a base is multiplied by

infinite: having an unlimited number of

itself

values intercept: the point where the line on a infinite number: the concept of

graph crosses the x-axis or y-axis

something that is unlimited, endless, without bound

interest: money that is a percentage of an original amount typically owed as part

infinite solutions: in systems of

of a debt

equations, coinciding lines have infinite solutions.

interquartile range (IQR): the difference between the upper quartile

input: the set of values supplied to a

(Q3) and the lower quartile (Q1)

function intersecting lines: lines that cross at a input-output pair: an ordered pair

point

in which the input corresponds to the independent variable in the left column

intersection: the point at which two lines

of a function table and the output

cross

corresponds to the right column of a function table; an ordered pair is

interval: the set of continuous input

determined by evaluating the function

values on which a function’s outputs could

using the input.

be increasing, decreasing, or constant

integer: any one of the positive whole

inverse: the opposite number or

numbers, negative whole numbers, and

operation

zero; any member of the set of all whole numbers and their opposites

inverse function: a function that undoes the action of another function

379

GLOSSARY OF TERMS inverse operation

less than or equal to (≤)

inverse operation: the operation that

laws of exponents – multiplication

reverses the effect of another operation

of same bases: can be rewritten as the base raised to the sum of the powers

inverse property of addition: the mathematical property that states that

laws of exponents – negative

when you add a number to its opposite,

exponents: can be rewritten as the

you will always get zero as the sum

multiplicative inverse of the base raised to the positive opposite of the power

inverse property of multiplication: the mathematical property that states

laws of exponents – zero exponents:

that when you multiply a number by its

the mathematical law which states that

reciprocal, you will always get 1

any number raised to the power of zero equals one

irrational number: a decimal number that cannot be expressed as a fraction,

least common multiple: the smallest

is not imaginary, and does not repeat or

multiple that is the same in a set of two

terminate

or more numbers

isosceles triangle: a triangle with two

leg: either of the two sides in a right

or more congruent sides where angles

triangle that form the right angle and are

opposite of the congruent sides are

opposite of acute angles

congruent angles length: the measure of an object from joint frequency: the ratio of the

end to end; the distance from one end to

frequency in a particular category and the

the other end of an object

total number of data values less than (<): smaller than another laws of exponents – division of same

(e.g., 432 < 501)

bases: can be rewritten as the base raised to the difference of the powers

less than or equal to (≤): smaller than or the same as another

380

GLOSSARY OF TERMS like terms

magnitude

like terms: terms that have the same

linear expression: an expression in

variables, including their exponents

which all terms have an exponent of one

likelihood: the probability that an event

linear function: a relationship that when

will occur; also called chance

graphed is a straight line

line: a straight geometric element with

linear graph: a series of points

no thickness, extending endlessly in both

connected on the coordinate plane,

directions; the shortest distance between

forming a straight line that shows a

two points

relationship or rate of change

line of best fit (trend line): a line that

linear inequality: an inequality that

best represents the data on a scatterplot

involves a linear function

line plot: a graph that displays data as

linear parent function: the simplest

points above a number line, to show the

equation of the linear function, y = x or

frequency of each value

f x) = x f(

line segment: a section of a line with two

linear relationship: having a constant

distinct endpoints

rate of change between two quantities/ variables and making a straight line when

linear association: a proportional

graphed; a relationship that creates a

relationship that creates a straight line on

straight line

a graph long division: an algorithm used to find linear equation: an equation in which no

the quotient of two numbers

variable has a power greater than 1; the general form is y = mx + b, where m =

magnitude: the absolute value or

slope and b = y-intercept.

distance to zero

381

GLOSSARY OF TERMS mapping

measures of variability

mapping: a function represented by

measure: a number of units that shows

two sets of objects with arrows drawn

the amount or size of something

between them to show relationships between the objects or data

measure of center: a single value used to represent/summarize a collection

marginal frequency: the ratio of the

of data; three commonly used types

sum of the joint relative frequency in a

are mode, median, and mean; also

row or column and the total number of

called measures of central tendency or

data values

measures of average

markdown: a decrease in the cost of an

measurement: a number that shows the

item; a discount

size or amount of something

markup: an increase in the cost of an

measure of variation: a measure of

item to make a profit

how data is spread out, usually including range, interquartile range, variance, and

maximum: the greatest or highest

standard deviation

amount possible or attained measurement system: one of two main maximum value: the place where a

systems of measurement—the metric

function reaches its highest point, or

system and the standard or customary

vertex, on a graph

system, each of which uses different units to measure distance, mass, and volume

mean: the average of a set of numbers calculated by finding the sum of all data

measures of variability: measures of

and dividing by the number of data values

how data is spread out, usually including range, interquartile range, variance, and

mean absolute deviation: the average

standard deviation

difference between the mean and each data point

382

GLOSSARY OF TERMS median

multiplication property of equality

median: the middle number of a set of

mode: the number or value that appears

numbers when the numbers are arranged

the most frequently in a data set

from least to greatest, or the mean of the two middle numbers when the set has two

monomial: an expression containing only

middle numbers

one term

midpoint formula: the formula used to

multi-digit: a number that has more

calculate the point on a line segment that

than one digit

is equidistant from the endpoints (x1, y1) and (x2, y2) on the coordinate plane;

multiple: a product of two integers; one of the numbers that result from multiplying a whole number by the set of whole numbers

minimum: the least or smallest amount or

multiple representations: different

quantity possible, attainable, or required

mathematical ways to represent a relation or a function

minimum value: the place where a function reaches its lowest point, or

multiplicand: the number that is

vertex, on a graph

multiplied by another number; a quantity that is to be multiplied by another quantity

minuend: a number or quantity from which another number is to be

multiplication: a mathematical operation

subtracted; for example, in the equation

consisting of obtaining a product or

7 – 4 = 3, the number 7 is the minuend,

result by joining equal groups, repeated

the number you subtract from.

addition, or forming arrays

mixed number: a whole number and a

multiplication property of equality:

fraction combined; a number made up of

the mathematical property that states

a whole number and a fraction

that multiplying the same number by each side of an equation gives us an equivalent equation

383

GLOSSARY OF TERMS multiplicative comparison

non-proportional relationship

multiplicative comparison: shows the

negative exponent law: the

relationship between two amounts, where

mathematical law which states that any

one quantity is a certain number of times

nonzero number raised to a negative

as large as another quantity; a number is

exponent is equivalent to the reciprocal

multiplied by another number to result in

of the base raised to the opposite of the

a greater or lesser quantity.

negative exponent

multiplicative identity property: the

negative number: a number that is less

mathematical property which states that

than zero

the resulting product of any number and 1 is equal to the original number

negative reciprocal: the result of multiplying the reciprocal by −1

multiplicative inverse: one of two numbers whose product is 1; also called

negative slope: the measure of the

the reciprocal

steepness of a line that shows the slant downward from left to right

multiplier: the number you multiply by; the quantity that the multiplicand

net: a two-dimensional shape that when

is multiplied by; the number being

folded represents a three-dimensional

multiplied

figure

multistep problem: a mathematical

nonlinear association: a relationship

problem involving more than one

that does not create a straight line

operation nonlinear function: a relationship that negative association: a relationship

when graphed does not make a straight

between two variables that move in

line; a relationship that does not create a

opposite directions

straight line; nonlinear association non-proportional relationship: two quantities that do not have equal ratios

384

GLOSSARY OF TERMS nonvertical line

ordered pair

nonvertical line: a line that is horizontal

obtuse angle: an angle that measures

or diagonal

greater than 90°

no solution: in systems of equations,

obtuse triangle: a triangle that contains

parallel lines have no solution.

one obtuse angle and two acute angles

number line/number line diagram:

odd function: when x is replaced with

a line on which numbers are marked at

−x x in a function and the function is

intervals

simplified, the terms in the resulting function have the opposite signs of those

numerator: the top number within a

in the original function.

fraction, which represents the part of the whole

one solution: in systems of equations, intersecting lines have one solution (x, y).

numeric expression: a mathematical sentence that uses numbers and one or

opposites: numbers the same distance

more operation symbols

away from zero, located on different sides of zero

numerical data: data comprised of numbers, measurements, or quantities

order of operations: a set of rules that dictate which mathematical operation to

numerical radical expression: any

perform first, second, and so on when

numerical expression that contains a

evaluating a mathematical expression

radical ordered pair: the location of a single numerical reasoning: a process

point on a coordinate plane where

using numbers and quantities to draw

the first and second values represent

conclusions

the position relative to the x-axis and y-axis, respectively (x, y); also known as

observation: the value of what is being

coordinate pair

385

GLOSSARY OF TERMS organized data list

percent decrease

organized data list: elements listed in a

part-to-part ratio (comparison): a

particular sequence or order

relationship between one part of a whole and another part of a whole

origin: the center point of a coordinate plane, where the x-axis and y-axis

part-to-whole ratio (comparison): a

intersect, located at (0, 0)

relationship between one part of a whole and the total number of parts in the

outcome: the result of an event

whole

outlier: a number in a set of data that

partial product: the product of the

is much larger or smaller than other

multiplicand and one digit of the multiplier

numbers in the set pattern: a repeating arrangement of output: the result of the input placed in

numbers or shapes

the function pattern of association: a relationship parabola: the shape that a quadratic

between data sets

equation takes when graphed peak: the highest value(s) in a set of data parallel: existing in the same plane and equidistant and not intersecting

per (unit rate): a ratio for an amount for one unit of the other quantity

parallel lines: lines in the same plane that are equidistant and do not intersect

percent: a special ratio that compares a number to 100 using the percent symbol,

parallelogram: a quadrilateral with two

%; a rate per 100

sets of parallel sides percent decrease: the amount by which parameter: a quantity that influences

the cost decreased from the initial value,

the output or behavior of a mathematical

expressed as a percent

object but is viewed as being held constant 386

GLOSSARY OF TERMS percent error

plot

percent error: the measure of how far

periodicity: the tendency of a function

off an estimated value is from the true

to repeat itself in a regular pattern at

value, expressed as a percent

established intervals

percent increase: the amount by which

perpendicular: having the position of

the cost increased from the initial value,

two lines that intersect at a right angle;

expressed as a percent

intersecting at a 90° angle

percent rate of change: the percentage

perpendicular lines: two lines that

increase or decrease of an amount over a

intersect at a 90° angle

unit of time, denoted by r pi: a constant which is found by dividing percentage: a special ratio that

the circumference of a circle by its

compares a number to 100 using the

diameter; approximately 3.142

percent symbol, %; a rate per 100 piecewise function: a function that is perfect cube: an integer that is the result

defined by different formulas at different

of another integer times itself three times

inputs

perfect square: an integer that is the

place value: the numerical value that a

result of another integer times itself

digit has, based on its position within a number

perfect square trinomial: a trinomial whose factored form is the square of a

plane: a flat, two-dimensional surface

binomial; takes the form ax² + bx + c

that continues indefinitely

and satisfies the condition b² = 4ac plot: to indicate the position a number perimeter: the distance around the

is relative to zero on a number line or

outside of a figure or shape

relative to the origin on a coordinate plane

387

GLOSSARY OF TERMS point

power of a power law

point: a dot that represents a specific

positive association: a relationship in

spot on a number line or coordinate

which the values of one variable tend

plane; a geometric object with no

to increase as the values of the other

dimension used to indicate a location

variable increase

point of intersection: the point where

positive number: a number that is

two or more lines cross each other

greater than zero

point-slope form: an equation written

positive rational number: a number to

in the form of y – y1 = m(x x – x1), where

the right of (or greater than) zero that

m is the slope and (x1, y1) is any point

can be expressed as a fraction of two

contained in the line

integers

polygon: a closed figure that has three

positive slope: the measure of the

or more sides, no curved lines, and no

steepness of a line that shows the slant

intersections; a closed figure formed by

upward from left to right

line segments that meet at their endpoints power: a mathematical notation that polynomial: a mathematical expression

indicates the number of times the base

consisting of several terms

number is multiplied by itself; also called an exponent

population: a discrete group for the purposes of data collection and analysis

power law: the distribution of an exponent through multiplication to all

positive/negative interval: positive

parts of the base

intervals are those above the x-axis; negative intervals are those below the

power of a power law: the

x-axis.

mathematical law that states that when raising a base with an exponent to another exponent, the exponents are multiplied and the base stays the same

388

GLOSSARY OF TERMS power of one law

protractor

power of one law: the mathematical

probability model: a mathematical

law which states that any number to the

description of an experiment that lists

power of one is equal to that number

all of the possible outcomes and their probabilities

power of zero law: the mathematical law that states that any number to the

product: the solution when multiplying

power of 0 is equal to 1

two or more numbers; the answer to a multiplication problem

prediction: a reasonable guess as to what will happen

proof: evidence or argument establishing a fact or the truth of a statement

preimage: the original figure in a transformation

product of powers law: the mathematical law which states that when

prime number: a number with exactly

multiplying two exponents with the same

two factors—one and itself

base, the exponents are added together

prime factorization: a given set of prime

and the base stays the same

numbers that when multiplied together

proportion: two fractions or ratios that

equals the original number

are equal in value; a type of equation that

prism: a three-dimensional figure that

shows that two ratios are equal

has at least one set of congruent, parallel

proportional corresponding sides:

faces (bases) that are polygons with

sides in the same position in two similar

parallelograms as the remaining faces

polygons that are proportional

probability: the likelihood that something

proportional relationship: when two

will happen

quantities have the same ratio protractor: a mathematical tool for measuring and drawing angles

389

GLOSSARY OF TERMS pyramid

radius

pyramid: a three-dimensional figure in

quantitative data: numerical or

which the base is any polygon and the

measured data that is analyzed for

other faces are triangles that share a

statistical purposes

common vertex quantitative relationship: the Pythagorean theorem: a theorem that

relationship between magnitudes

states that the square of the hypotenuse is equal to the sum of the squares of the

quantity: a number or amount; an

other two sides of a right triangle; a² +

amount that tells how much

b² = c² quotient: the solution when dividing quadrant: one of four sections of

two numbers; the answer to a division

the coordinate plane, formed by the

problem; the result of the division of one

intersection of the x-axis and y-axis

quantity by another quantity

quadratic formula: the formula

quotient of powers law: the

, which gives the

mathematical law that states that when

solutions of equations in the form of ax²

dividing two exponents with the same

+ bx + c = 0, where a ≠ 0

base, one subtracts the exponents and keeps the base the same

quadratic function: a function that can be written in the form f( f x) = ax2 + bx +

radical: a symbol that indicates the root

c, where a, b, and c are real numbers and

of a quantity

a≠0 radicand: the value inside the radical quadratic parent function: the simplest

symbol

equation of the quadratic function, y = x² or f( f x) = x²

radius: the distance from the center of a circle or a sphere to any point that lies on

quadrilateral: a polygon with four sides

the circle or the sphere

and four angles 390

GLOSSARY OF TERMS random sampling/random sample

real-world problem

random sampling/random sample: a

ratio table: a list of pairs of equivalent

selection chosen by chance and which has

ratios used to determine the relationship

no predictability

between the ratios

range: (1) the difference between the

rational exponent: an exponent that can

maximum and minimum values within a

be expressed as

data set; (2) the set of all possible output,

a radical expression where m and n are

or y values, of a relation or function

integers and m represents the power

as a way to rewrite

of the base and n represents the root; rate: a type of ratio where the quantities have two different units rational number: a number that can rate of change: the rate that shows

be written as a fraction of integers a/b,

how one quantity changes in relation to

where b ≠ 0; a number that can be

another quantity

written as a ratio using two integers

ratio: a comparison of two quantities

ray: part of a line with a fixed starting

that shows their sizes in relation to one

point and no endpoint

another real number: any one of the set of all ratio language: language used to

rational and irrational numbers

mathematically describe the relationship between any two units that are being

real solution: a value that satisfies the

compared in a ratio using the phrase for

equation; called roots, x-intercepts, or

every… there are… or the word to

zeros

ratio relationship: equivalent ratios

real-world problem: a contextual-

form a ratio relationship between the two

based problem that can be interpreted,

quantities being compared

represented, and analyzed through the application of mathematics

391

GLOSSARY OF TERMS reciprocal

right angle

reciprocal: one of two numbers whose

relative frequency: how often a number

product is 1; also called the multiplicative

occurs in a data set divided by the total

inverse

number of outcomes

rectangle: a parallelogram with opposite

relative maximum: a point that is higher

equal sides and four right angles

than the points directly beside it on both sides

recursive formula: a formula that defines each term of a sequence using

relative minimum: a point that is lower

preceding term(s)

than the points directly beside it on both sides

recursive process: the calculation of the next number in a sequence by repeated

remainder: a leftover quantity resulting

application of a rule

from the quotient of 2 integers

reduction: the creation of a similar image

repeating decimal: a decimal number

that is now smaller than the original image

in which a digit or group of digits is repeated indefinitely, as in 0.333… or

reflect: to transform a point so that it is

1.851851851…

equidistant on opposite sides of the x- or y-axis

representative sample: a sample that matches or reflects a population

reflection: the mirror image of a figure; the flipping of a figure

residual: the difference between the observed y value (from the scatterplot)

regression: the process of drawing a line

and the predicted y value (from the

through data in a scatterplot

regression equation line)

relationship: the rule in a pattern

right angle: an angle that measures 90°

392

GLOSSARY OF TERMS right polygon

scientific notation

right polygon: a polygon with at least

scale: the representation of the

one right angle

relationship between a measurement on a model and the corresponding

right prism: a solid composed of a

measurement on the actual object

polygon as its base and vertical sides perpendicular to the base

scale drawing: a smaller or larger representation of an object that is

right rectangular prism: a prism with six

proportional to the original object

rectangular faces where the lateral edge is perpendicular to the plane of the base

scale factor: the ratio of corresponding side lengths in a scale drawing to those of

right triangle: a triangle with one 90º

the original figure

angle scaled interval: a measurement scale rotation: the turning of a figure around a

used on a graph with the distance

fixed point

between marks being equal and the marks counting by a constant value

rounding: the process of raising or lowering a number to a specific

scalene triangle: a triangle with no

place value position; representing an

congruent sides

approximate worth scatterplot: a series of plotted points ruler: a tool used to measure length and

that show the relationship between two

to draw straight lines

sets of data

sample: one part of the given population

scientific notation: a method of expression used to write very small and

sample space: all possible outcomes of

very large numbers by representing them

an experiment

with decimal numbers between 1 and 10, with each decimal being multiplied to a power of 10

393

GLOSSARY OF TERMS sequence

solution

sequence: an ordered arrangement of

simplest form: (1) the smallest possible

numbers or objects

way to write an equivalent fraction for the fraction given; (2) the smallest way to

set: (1) a collection of objects or things;

write an equivalent expression

(2) a group of unique numbers or objects called members or elements

simplify: to replace a numerical expression with the simplest name for its

shape: a description of the type of graph

value by using the substitution principle

seen, as symmetrical, peaks, skewed, or uniform

simulation: a model of random events

side: the line segment that connects two

skewed data: when data on a graph is

vertices in a figure

not symmetrical; when the graphed data shows a tail on one side or the other

signed number: a positive or negative number; a number that has the sign + for

slope: how steep a line is; represented as

positive or − for negative

m in the slope-intercept equation

similar figures: two or more figures that

slope formula: the formula used to find

are the same shape but different sizes

the slope between two points (x1, y1) and

similar triangles: two or more triangles

(x2, y2) ,

that have congruent angles and

slope-intercept form: a way to write

proportional sides

the equation of a line so that it is easy to view the slope and y-intercept of the line;

simple event: one event at a time with

y = mx + b

one single outcome solution: any number that makes an simple interest: a way to calculate

equation true

interest accrued using the formula I = Prt

394

GLOSSARY OF TERMS solution of a system of inequalities

stem-and-leaf plot

solution of a system of inequalities:

square unit: a unit of area, specifically

the overlapping region that makes both

square centimeters, inches, feet, and

inequalities true

meters

solution set: a set of numbers that

standard deviation: a measure of how

makes an inequality statement true

spread out numbers are; calculated by finding the square root of the variance

sphere: a three-dimensional round figure where every surface point is equidistant

standard form: a way to write numbers

from the center of the figure

by using the digits 0–9, with each digit having a place value

spread: a measure of how far the numbers in a data set are from the mean

standard form (linear): Ax + By = C,

or median; including the commonly used

where A, B, and C are constants and A

types range and quartiles; also known as

and B are not both 0

measures of variation or dispersion standard form (quadratic): y = ax² + square: any number or variable times itself

bx + c or ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0

square number: a number to the power of 2, i.e., 3² represents the square

statistical question: a question that

number 9 and can be read as “three

anticipates differences in data

squared” or “three to the power of two.” statistics: the study of data and square root: a number that, when

collecting, organizing, representing, and

multiplied by itself, produces the given

interpreting data

number stem-and-leaf plot: a plot where each square root function: a function of the

data value is split into a “leaf” (usually

form f( f x) =

the last digit) and a “stem” (the other

, where x is greater than

digits) 395

GLOSSARY OF TERMS step function

system of equations

step function: a piecewise-defined

sum: the solution when adding two or

function where each piece’s formula is a

more numbers; the answer to an addition

constant

problem

straight angle: an angle that measures

supplementary angles: two adjacent

exactly 180°

angles that, when added, make 180°; two angles whose sum is 180°

strict inequality: an inequality that has no equality conditions; the strict inequality

surface area: the total area of each of the

is either greater than or less than.

faces and curved surfaces of a solid figure

subcategory: a category within a category;

survey: a data collection tool or list of

a collection of objects with even more

questions used to gather information

specific characteristics than a category

about individuals or groups of people

substitution: replacing letters in an

symbol: a mark or character used as a

algebraic expression with known values

representation of an object, function, or process

substitution method: a method of solving systems by substituting equations

symmetrical: the relationship between

within one another

objects that are the same size and shape after a flip, slide, or turn

subtraction property of equality: the mathematical property that states that

symmetrical distribution: data that is

subtracting the same number from each

in the shape of a bell; it can be equally

side of an equation gives us an equivalent

divided in half.

equation system of equations: two or more subtrahend: a quantity or number to

equations with two or more variables

be subtracted from another; the number being subtracted 396

GLOSSARY OF TERMS system of inequalities

triangle

system of inequalities: two or more

thousandths: the third digit to the right

inequalities with two or more variables

of the decimal point; a thousandth is one out of 1,000 equal parts of a whole.

table: a chart that uses rows and columns to organize information

three-dimensional figure: a solid having three measurable dimensions

tape diagram: a rectangular visual model that represents equal parts, used to

transformation: changing a shape

model word problems involving part-part-

through movement on a coordinate plane

whole relationships translation: moving a figure along a line tax: a fee added to a good or service,

for a specific distance

usually a percentage of the total transversal: a line that cuts through two tenths: the first digit to the right of the

or more lines in the same plane

decimal point; a tenth is one out of 10 equal parts of a whole.

trapezoid: a quadrilateral with one set of parallel sides

term: (1) a number, a variable, or a product of numbers and variables in

trend: the general direction that data

an expression separated by addition,

points seem to follow

subtraction, or sometimes division; (2) in an algebraic expression, a number

tree diagram: a diagram with connecting

or variable, or a product or quotient of

lines to calculate the number of possible

numbers and variables

outcomes of an event

terminating decimal: a decimal number

triangle: a polygon with exactly three

that has a finite number of digits

straight sides and three angles

397

GLOSSARY OF TERMS triangle angle sum property

triangle angle sum property: the

variation

union: a combination of two or more things

mathematical property of a triangle which states that the angles of a triangle always

unit: a type of measurement such as an

add up to 180°

inch, a pound, or a second

triangle inequality theorem: the

unit cube: a cube in which all sides have

theorem that states that the sum of any

a length of one unit

2 sides of a triangle must be greater than the measure of the third side

unit of measurement: a standard amount that is used to measure

trinomial: a polynomial expression containing three terms

unit price: the price of goods per one unit of measure

truncated decimal: a decimal number where some digits are left off and the

unit rate: a rate with a denominator of 1

number is approximated at a certain point

that shows how many units of the first type

without rounding

correspond to one unit of the second type

two-dimensional figure: a flat figure

upward: the direction a parabola opens

with two measurable dimensions

when the value of a > 0

two-way relative frequency table: a

variability: how spread out data is

two-way table that displays percentages or ratios, called relative frequencies

variable: a letter or symbol that takes

two-way table: a chart used to show

a letter that can stand for an unknown

the relationship between two categorical

number or a set of numbers

variables

the place of a number that can change;

variation: how spread out data is

undefined slope: the slope of a vertical line

398

GLOSSARY OF TERMS vertex/vertices

x-axis

vertex/vertices: the common point of

vertical number line: a number line that

two rays that form an angle; the common

travels up and down, perpendicular to a

point to any two sides of a polygon

corresponding horizontal line; from top to bottom; perpendicular to the horizon

vertex: the minimum or maximum point in a quadratic function; identified as (h, k)

vertical reflection: a reflection over a horizontal line such as the x-axis

vertex form: y = a(x – h)² + k, where a, h, and k are constants and a ≠ 0

vertical shift: a change in a function that moves the function up or down

vertical: describes the direction of a line that travels up and down, perpendicular

vertical translation: a shift in the base

to a corresponding horizontal line; from

of the graph up or down

top to bottom; perpendicular to the horizon

volume: the amount of space an object occupies; the measured amount of cubic

vertical angles: angles opposite from

units that fit inside a solid figure

one another when two lines cross; opposite congruent angles that are

whole number: a number zero or above

formed on either side of intersecting lines

that contains no fractional or decimal part; a positive number without a

vertical dilation: expansion or

fractional piece

contraction in the vertical direction width: how many units wide something is vertical line test: a visual way to tell whether a line is a function; if any vertical

withdrawal: a sum of money that is

line intersects the graph more than once,

taken out of a bank account

then the graph is not a function. x-axis: a horizontal number line on a coordinate plane

399

GLOSSARY OF TERMS x-coordinate

x-coordinate: the first term in an

zero slope

zero slope: the slope of a horizontal line

ordered pair; provides the location along the x-axis within the coordinate plane x-intercept: the x-coordinate or coordinates where a graph intersects the x-axis, identified as (x, 0) y-axis: a vertical number line on a coordinate plane y-coordinate: the second term in an ordered pair; provides the location along the y-axis within the coordinate plane y-intercept: the point on a graph of an equation where the line crosses the y-axis zero: (1) the only integer that is neither negative nor positive and is its own opposite; (2) the value of x where an expression is equal to zero; this is the x-coordinate of the x-intercept of the expression’s graph. zero product property: the mathematical property which states that when multiplying two numbers together results in zero, then either a, b, or both a and b are zero; if ab = 0, then either a = 0 or b = 0 or both 400

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