Math Nation - 7th Grade - Volume 2 of 2

Table of Contents Unit 8 - Evaluating Numeric Expressions Lesson 8.1

Product Laws of Exponents

Page Number 3

8.2

Quotient Laws of Exponents

10

8.3

Exponential Expressions

16

8.4

Evaluating Numeric Expressions - Part 1

21

8.5

Evaluating Numeric Expressions - Part 2

27

8.6

Evaluating Numeric Expressions - Part 3

32

Unit 9 - Algebraic Expressions 9.1

Lesson Adding and Subtracting Linear Expressions With Integer Coefficients

Page Number

9.2

Adding Linear Expressions With Rational Coefficients

46

9.3

Subtracting Linear Expressions With Rational Coefficients

52

9.4

Adding and Subtracting Expressions With Rational Coefficients

57

9.5

Equivalent Numerical Expressions

61

9.6

Rewriting Equivalent Numerical Expressions

65

9.7

Equivalent Linear Expressions

70

9.8

Equivalent Expressions Using Properties of Operations

77

39

Unit 10 - Equations and Inequalities Lesson

Page Number

10.1

Exploring Inequalities - Part 1

85

10.2

Solving One-Step Inequalities Using Addition and Subtraction

91

10.3

Exploring Inequalities - Part 2

98

10.4

Solving One-Step Inequalities Using Multiplication and Division

104

10.5

Solving Two-Step Equations Using Properties of Equality

110

10.6

Solving Two-Step Equations

117

i

10.7

Solving Equations Involving the Distributive Property

122

10.8

Solving Real-World Problems With Two-Step Equations

128

10.9

Writing Two-Step Equations From Real-World Contexts

134

10.10 Writing and Solving Equations in Real-World Context

139

Unit 11 - Representing Proportional Relationships Lesson

Page Number

11.1 Graphing Proportional Relationships

147

11.2 Constant of Proportionality - Part 1

152

11.3 Constant of Proportionality - Part 2

159

11.4 Features of Proportional Relationships

164

11.5 Is It Proportional?

169

11.6 Representing Proportional Relationships With Equations

175

11.7 Translating Between Representations of Proportional Relationships

182

11.8 Putting It All Together

187

11.9 Revisiting Relationships in Circles

192

Unit 12 - Surface Area and Volume Lesson

Page Number

12.1 Finding Surface Area of Right Circular Cylinders Using Nets

199

12.2 Finding Surface Area of Right Circular Cylinders

204

12.3 Solving Problems Involving Surface Area of Right Circular Cylinders

208

12.4 Volume of Right Circular Cylinders - Part 1

212

12.5 Volume of Right Circular Cylinders - Part 2

217

12.6 Problem-Solving With Volume of Cylinders

223

Unit 13 - Numerical and Categorical Data Lesson

ii

Page Number

13.1 Effects of Outliers

233

13.2 Justifying Measures of Center and Spread

239

13.3 Determining Appropriate Measures of Center and Variation

245

13.4 Comparing Graphical Representations

251

13.5 Comparing Data Sets

257

13.6 Comparing Data Sets and Graphs

262

13.7 Representing Real-World Contexts With Graphs

268

13.8 Creating Displays for Numerical Data

275

13.9 Creating Displays for Categorical Data

281

Additional Resources Resource

Page Number

Benchmark Index

287

Appendix C: 6-12 Mathematics Glossary

288

Appendix D: Properties of Operations, Equality and Inequality

295

Appendix E: K-12 Formulas

297

iii

THIS PAGE WAS INTENTIONALLY LEFT BLANK.

iv

Unit 8 - Evaluating Numeric Expressions 8.1 8.2 8.3 8.4 8.5 8.6

Product Laws of Exponents Quotient Laws of Exponents Exponential Expressions Evaluating Numeric Expressions - Part 1 Evaluating Numeric Expressions - Part 2 Evaluating Numeric Expressions - Part 3

Lesson

Unit 8: Evaluating Numeric Expressions

Page Number 3 10 16 21 27 32

1

The following B.E.S.T. Standards will be covered in this unit: MA.7.NSO.1.1 Know and apply the Laws of Exponents to evaluate numerical expressions and generate equivalent numerical expressions, limited to whole-number exponents and rational number bases.

• Clarification 1: Instruction focuses on building the Laws of Exponents from specific examples. Refer to the K-12 Formulas (Appendix E) for the Laws of Exponents. !!

• Clarification 2: Problems in the form !" = 𝑎𝑎" must result in a whole-number value for 𝑝𝑝.

MA.7.NSO.2.1 Solve mathematical problems using multi-step order of operations with rational numbers including grouping symbols, whole-number exponents and absolute value.

• Clarification 1: Multi-step expressions are limited to 6 or fewer steps.

MA.7.NSO.2.2 Add, subtract, multiply and divide rational numbers with procedural fluency.

2

Unit 8: Evaluating Numeric Expressions

Unit 8, Lesson 1: Product Laws of Exponents

8.1.1 Warm-Up: Math Fail The picture is an example of a “math fail.”

1.

Benchmark

Explain the error shown in the image with your partner.

MA.7.NSO.1.1 Know and apply the Laws of Exponents to evaluate numerical expressions and generate equivalent numerical expressions, limited to whole-number exponents and rational number bases.

Learning Target 

I can explain and use the power of a product, power of a power, product of powers, and identity laws of exponents.

Unit 8: Evaluating Numeric Expressions

3

8.1.2 Exploration: Product of Powers Exponent Law

Product

Repeated Multiplication

22 ∙ 23

(2 ∙ 2) ∙ (2 ∙ 2 ∙ 2) = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2

Recall that repeated multiplication is a way of writing a number with an exponent.

35 ∙ 34

The number ʹ, called the base, is being multiplied repeatedly. base

25

51 ∙ 54 ∙ 53

23 = 2 × 2 × 2

2 is being multiplied repeatedly

b.

How do the exponents in the first column relate to the exponents in the last column?

c.

What do you notice about the bases in all three columns?

A base is the number used as a factor in exponential form.

Power of the Form 𝒂𝒂𝒃𝒃

Since the base, ʹ, is a factor ͵ times, the exponent is ͵. exponent

23 = 2 × 2 × 2

2 is a factor 3 times

An exponent is the number of times the base occurs as a factor.

1.

Write a rule for the product of two exponential expressions with the same base.

With your partner, use the definition of an exponent to complete the table. a.

4

2.

Write each product as a repeated multiplication expression, then write each expression with an exponent.

Product of Powers Exponent Law When multiplying two powers that have the same nonzero base, the exponents and keep the same base.

Unit 8: Evaluating Numeric Expressions

3.

With your partner, follow the steps.

• • • •

4.

Determine which column you will start in, and which column your partner will start in.

Using the directions for your column, evaluate the expression. Your partner should use the directions in their column to evaluate the same expression.

Verify if your answers are the same. If not, find the error(s).

After each problem, switch columns.

The first expression has been evaluated.

Apply the product of Evaluate each power powers exponent law, then multiply. then evaluate the power. 𝟐𝟐 𝟕𝟕 ∙ 𝟕𝟕𝟏𝟏 2 1 7 ∙7 72 ∙ 71 7(2 + 1)

49 ∙ 7

73

343

𝟔𝟔𝟗𝟗 ∙ 𝟔𝟔𝟑𝟑

Julian says that he can evaluate 32 ∙ 23 using the product of powers exponent law. a.

Explain to Julian if he is correct.

b.

Complete the statement about the product of powers exponent law.

When 𝑎𝑎 is a nonzero real number, and 𝑚𝑚 and 𝑛𝑛 are

integers, the exponential expression 𝑎𝑎𝑚𝑚 · 𝑎𝑎 𝑛𝑛 is equal to

o 𝑎𝑎𝑚𝑚𝑚𝑚 . o 𝑎𝑎𝑚𝑚−𝑛𝑛 . o 𝑎𝑎𝑚𝑚+𝑛𝑛 .

343

(−𝟖𝟖)𝟒𝟒 ∙ (−𝟖𝟖)𝟓𝟓

Unit 8: Evaluating Numeric Expressions

5

8.1.3 Exploration: Power of a Product Exponent Law 1.

With your partner, complete the table and answer the questions. a.

Write each power of a product as a repeated multiplication expression, and group repeated factors by using the commutative and associative properties. Then, write each expression with an exponent in the form 𝑎𝑎𝑚𝑚 ∙ 𝑏𝑏𝑛𝑛 .

Power of a Product (2 ∙ 3)3 (6 ∙ 5)4

Power Repeated of the Multiplication Form 𝒂𝒂𝒎𝒎 ∙ 𝒃𝒃𝒏𝒏 (2 ∙ 3) ∙ (2 ∙ 3) ∙ (2 ∙ 3) (2 ∙ 2 ∙ 2) ∙ (3 ∙ 3 ∙ 3) 23 ∙ 33 Group Repeated Factors

(10 ∙ 4)2 b.

c.

2.

6

How does the base in the first column relate to the bases in the last column?

How does the exponent in the first column relate to the exponents in the last column?

Power of a Product Exponent Law When raising a product to a power, distribute the power to each factor.

3.

With your partner, follow the steps.

• • • •

Determine which column you will start in, and which column your partner will start in.

Using the directions for your column, evaluate the expression. Your partner should use the directions in their column to evaluate the same expression.

Verify if your answers are the same. If not, find the error(s).

After each problem, switch columns.

The first expression has been evaluated.

Evaluate inside the parentheses, then apply the exponent.

Apply the power of a product exponent law, then evaluate the power.

( −5 ∙ 6)2

( −5 ∙ 6)2

( −30)2

( −30 ) ∙ ( −30) 900

( −𝟓𝟓 ∙ 𝟔𝟔)𝟐𝟐

−

𝟓𝟓

( 𝟔𝟔 ∙ 𝟒𝟒)

( −𝟑𝟑 ∙ −𝟐𝟐)𝟖𝟖

Write a rule for the power of a product.

Unit 8: Evaluating Numeric Expressions

( −5)2 ∙ 62 25 ∙ 36 900

4.

Aiyanna says that she can evaluate (10 + −2)3 using the power of a product exponent law. Explain to Aiyanna if she is correct.

8.1.4 Exploration: Power of a Power Exponent Law 1.

With your partner, complete the table. a.

5.

Complete the statement about the power of a product exponent law.

Power of a Power (42 )3

When 𝑎𝑎 and 𝑏𝑏 are nonzero real numbers, and 𝑚𝑚 is an integer, the exponential expression (𝑎𝑎 · 𝑏𝑏)𝑚𝑚 is equal to

o o o

Write each power of a power as a repeated multiplication expression using the outer exponent.

Then, using the Product of Powers Exponent Law, rewrite the multiplication expression in the form 𝑎𝑎𝑚𝑚 .

𝑎𝑎𝑏𝑏𝑏𝑏 . 𝑎𝑎𝑚𝑚 ∙ 𝑏𝑏𝑚𝑚 . 𝑎𝑎𝑚𝑚 + 𝑏𝑏𝑚𝑚 .

(104 )4

b.

2.

Repeated Multiplication 42 ∙ 42 ∙ 42

Power of the Form 𝒂𝒂𝒎𝒎 46

(31 )5

How do the exponents in the first column relate to the exponents in the last column?

Two different methods for evaluating expressions with exponents are illustrated in the table. With your partner, follow the steps.

• • • •

Select which column you will start in, and which column your partner will start in.

Review Method 1 and Method 2. When you are working in each column, make sure to follow the same process as in the example for that column.

After finding the value of each expression, verify your answers are the same. If not, find the error(s).

After each problem, swap columns so that you and your partner are practicing both methods.

Unit 8: Evaluating Numeric Expressions

7

Example a.

Method 1 [( −𝟐𝟐)𝟒𝟒 ]𝟐𝟐 (16)2

Method 2 [( −𝟐𝟐)𝟒𝟒 ]𝟐𝟐

b.

( −2)4 ∙ 2

1.

( −2)8

16 ∙ 16 256

[( −𝟐𝟐)𝟒𝟒 ]𝟐𝟐

8.1.5 Exploration: Identity Exponent Law

256

[( −𝟔𝟔)𝟏𝟏 ]𝟖𝟖

With your partner, use the meaning of an exponent to complete the table. Write each product as a repeated multiplication expression, then write the value of each expression.

Exponential Expression

Repeated Multiplication

Value

101

10

10

1001 111

[( −𝟏𝟏𝟏𝟏)𝟐𝟐 ]𝟔𝟔

c.

2.

Complete the statement about the identity exponent law.

When 𝑎𝑎 is a real number, the exponential expression 𝑎𝑎1 is equal to

3.

When 𝑎𝑎 is a nonzero real number, and 𝑚𝑚 and 𝑛𝑛 are

integers, the exponential expression (𝑎𝑎𝑚𝑚 )𝑛𝑛 is equal to

o o

o o

Complete the statement.

o

o

𝑎𝑎𝑚𝑚+𝑛𝑛 Ǥ 𝑎𝑎𝑚𝑚 ∙ 𝑎𝑎𝑚𝑚 Ǥ 𝑎𝑎𝑚𝑚×𝑛𝑛 Ǥ

Identity Exponent Law When raising a base to the power of 1, the base is a factor one time, thus, has the same value as the base.

Power of a Power Exponent Law When raising a base with a power to another power, multiply the exponents and keep the same base.

8

1Ǥ 𝑎𝑎 + 1Ǥ 𝑎𝑎Ǥ

Unit 8: Evaluating Numeric Expressions

8.1.6 Your Turn!

1.

8.1.7 Wrap-Up: Ticket Out the Door

Complete the following steps for each expression.

• • •

Rewrite each expression shown by using the laws of exponents.

Choose the exponent law you used from the list.

Evaluate the exponential expression.

(23 )4

(23 )4 = ________ 0.54 ∙ 0.53

0.54 ∙ 0.53 = ________ 24 1 ( ) 8

o o o o

Product of powers law Power of product law Power of power law Identity exponent law

o o o o

Product of powers law Power of product law Power of power law Identity exponent law

o o o o

Product of powers law Power of product law Power of power law Identity exponent law

o o o o

Product of powers law Power of product law Power of power law Identity exponent law

24 1

( 8 ) = ________ (0.5 ∙

(0.5 ∙

−

−

8)2

1.

Rewrite the expressions using laws of exponents. State the exponent law used.

Expression

Exponent Law

Equivalent Expression

12.755 ∙ 12.754 1 7 [( ) ] 4

2

(12 ∙ 0.75)3 285.6

−3 1 ( ) 5 106 ∙ (−1.5)6

8)2 = ________ Unit 8: Evaluating Numeric Expressions

9

Unit 8, Lesson 2: Quotient Laws of Exponents

8.2.1 Warm-Up: Use a Pattern Look at the pattern shown in the figures below. Work with your partner to answer the questions.

Benchmark MA.7.NSO.1.1 Know and apply the Laws of Exponents to evaluate numerical expressions and generate equivalent numerical expressions, limited to whole-number exponents and rational number bases.

Figure 1

Figure 2

Figure 3

Learning Target 

1.

Describe the pattern in words.

2.

If the pattern continued, how many squares would be in Figure 10?

I can explain and use the quotient of powers, power of a quotient, and zero exponent laws.

10

Unit 8: Evaluating Numeric Expressions

b.

8.2.2 Activity: Station 1: Quotient of Powers Exponent Law Work with your partner or a small group to complete three stations involving the laws of exponents.

Equivalent Expression

Student Work

Given

Task 1: 1. Review the steps Rosa, Victor, and Emma each used to generate an equivalent expression. Then, answer the questions that follow.

a.

Rosa

Victor

Emma

56 53

104 102

74 71

5 ∙5 ∙5 ∙5 ∙5 ∙5 5 ∙5 ∙5 5 ∙5 ∙5

10 ∙ 10 ∙ 10 ∙ 10 10 ∙ 10

7 10 10 10 10 7 ×7 × ×7 ∙ ∙ ∙ 7 10 10 1 1 1 ∙ 1 ∙ 10 ∙ 10 10 ∙ 10

53

7 ×7 ×7 ×7 7

102

Task 2: 1.

What do you notice about the bases of the numerator and denominator of each given exponential expression?

Fill in the blank to complete the rule for dividing exponential expressions with the same bases.

When dividing two powers that have the same nonzero base, keep the base and the exponents. This is called the quotient of powers exponent law.

2.

When 𝑎𝑎 is a nonzero real number, and 𝑚𝑚 and 𝑛𝑛 are integers, the exponential expression

7 ×7 ×7

73

Circle the exponents in the given exponential expression and circle the exponent in the equivalent expression each student created. How are they related?

o 𝑎𝑎𝑚𝑚𝑚𝑚 . o 𝑎𝑎𝑚𝑚−𝑛𝑛 . o 𝑎𝑎𝑚𝑚+𝑛𝑛 .

𝑎𝑎𝑚𝑚 𝑎𝑎𝑛𝑛

is equal to

Task 3: Show your steps of applying the quotient of powers exponent law to evaluate.

1.

(−2)6

2.

(3)6

3.

(−7)4

(−2)3

(3)2

(−7)1

Unit 8: Evaluating Numeric Expressions

11

8.2.3 Activity: Station 2: Power of a Quotient Exponent Law

Task 2: 1.

Task 1: Luis and Maya have shown their steps for using the power of a quotient exponent law to create equivalent expressions. Amir is confused. Help Amir complete his task by showing similar steps to Luis and Maya for his

Fill in the blank to complete the rule for raising a fraction to a power.

When raising a fraction to a power, the result is equal to the quotient of the numerator and denominator raised separately to the power. This is called the power of a quotient exponent law.

7 4

expression (3) .

Luis 8 3 ( ) 10

8 8 8 ∙ ∙ 10 10 10 8 ∙8 ∙8 10 ∙ 10 ∙ 10 83 103

12

Maya 3 5 ( ) 11

3 3 3 3 3 ∙ ∙ ∙ ∙ 11 11 11 11 11 3 ∙3 ∙3 ∙3 ∙3 11 ∙ 11 ∙ 11 ∙ 11 ∙ 11 35 115

7 ∙ 3

Amir 7 4 ( ) 3

2.

When 𝑎𝑎 and 𝑏𝑏 are nonzero real numbers, and 𝑚𝑚 is an 𝑎𝑎 𝑚𝑚

integer, the exponential expression (𝑏𝑏) o 𝑎𝑎𝑏𝑏 − 𝑚𝑚 . o

𝑎𝑎𝑚𝑚

is equal to

.

𝑏𝑏 𝑚𝑚 𝑚𝑚

o 𝑎𝑎 − 𝑏𝑏𝑚𝑚 .

Task 3: Apply the power of quotient exponent law. Do not simplify the fractions. −4 2

1.

(8)

2.

(−2)

3.

(−8)

−4 6

2

5

Unit 8: Evaluating Numeric Expressions

8.2.4 Activity: Station 3: Zero Exponent Law

Task 2: Use the tables to answer the questions. exponent form value

Task 1: Two students used their understanding of exponents to write an equivalent expression for the exponential expression 1.

23 23

.

2 ×2 ×2 2 ×2 ×2

2 2 2 × × 2 2 2

2.

32 9

21 2 31 3

20 30

Sebastian 23 23

2.

What pattern do you see in the bottom row of each table?

20

3.

How is the pattern in the bottom row of each table related to the value of the base in the top row of the table?

4.

Use the patterns in the tables to find the values of 20 and 30 . Fill in the empty cell in each table.

23−3

Using Hannah’s and Sebastian’s work, make a prediction about the value of any number raised to the power of 0.

33 27

22 4

What pattern do you see in the top row of each table?

1×1 ×1 1

34 81

23 8

1.

Discuss each step that the students completed.

Hannah 23 23

exponent form value

24 16

20 = ______ 30 = ______

Task 3: Fill in the blank to complete the rule for a base with a zero exponent.

When a nonzero real number is raised to the zero power, 𝑎𝑎0 , its value equals . This is called the zero exponent law.

Unit 8: Evaluating Numeric Expressions

13

8.2.5 Bring It Together: Exponential Expressions With Rational Bases

1.

Match each set of expressions with the Exponent Law that it demonstrates. Use arrows to indicate matches.

2.

Complete the following steps for each expression.

Rewrite each expression shown by using the laws of exponents.

Check off the exponent law used from the list.

Evaluate the exponential expression.

• • •

0.5 3 ( ) 1.9

0.5 0.5 0.5 ( ) ∙ ( ) ∙ ( ) 1.9 1.9 1.9 0.5 ∙ 0.5 ∙ 0.5 1.9 ∙ 1.9 ∙ 1.9 0.53 1.93

2 0 ( ) 3 1

10.16 10.14

10.1 ∙ 10.1 ∙ 10.1 ∙ 10.1 ∙ 10.1 ∙ 10.1 10.1 ∙ 10.1 ∙ 10.1 ∙ 10.1

10.1 10.1 10.1 10.1 10.1 10.1 ∙ ∙ ∙ ∙ ∙ 10.1 10.1 10.1 10.1 1 1 1 ∙ 1 ∙ 1 ∙ 1 ∙ 10.1 ∙ 10.1 10.12

14

0.58 0.55 Zero Exponent Law

Quotient of Powers Law

Power of a Quotient Law

o Quotient of powers law o Power of quotient law o Zero exponent law

0.58 = ________ 0.55

0.2 4 ( ) 10

o Quotient of powers law o Power of quotient law o Zero exponent law

0.2 4 ( ) = _____ 10

3 0 ( ) 4

3 0 ( ) = ________ 4

Unit 8: Evaluating Numeric Expressions

o Quotient of powers law o Power of quotient law o Zero exponent law

8.2.6 Your Turn!

1.

8.2.7 Wrap-Up: Reflection

Rewrite the expressions using laws of exponents. State the exponent law used.

Expression

Equivalent

1.

I’m so confident, I could explain this to someone else!

Exponent Law

8.66 0.26

I can get to the right answer, but I don’t understand well enough to explain it yet.

15.7 15.7

I understand some of this, but I don’t understand all of it yet.

7

6.75 6.753

I tried hard and I listened, but I am finding this challenging. I will make sure that I get help with this next lesson.

5 10 ( ) 20 (

I do not understand any of this yet. There are things I could do to be a better learner next lesson.

0.85 7 ) 2.64 8.450

Shade the arrow up to the statement that best describes your current level of understanding.

Unit 8: Evaluating Numeric Expressions

15

Unit 8, Lesson 3: Exponential Expressions Benchmark

8.3.1 Warm-Up: Make an Expression 1.

Create an expression with the largest possible value by filling in the blue boxes with the digits 1 through 5. You can only use each digit one time.

MA.7.NSO.1.1 Know and apply the Laws of Exponents to evaluate numerical expressions and generate equivalent numerical expressions, limited to whole-number exponents and rational number bases.

Learning Targets  

16

I can evaluate numeric expressions with the same base using the laws of exponents.

I can generate equivalent numeric expressions by applying the laws of exponents.

Unit 8: Evaluating Numeric Expressions

+

×

8.3.2 Collaboration: Applying More Than One Exponent Law An expression can be rewritten as an equivalent expression by applying one or more exponent laws.

26

3

Winston and Yvette were assigned ( 4 ) to rewrite as an 2

equivalent expression using multiple exponent laws. Their teacher told them to first make a plan and then apply the laws to the expression. Complete Yvette’s work by following her plan.

Winston I will first apply the quotient of powers exponent law and then apply the power of a power exponent law.

6 3

2 ( 4 ) = (22 )3 = 26 = 64 2 1.

Yvette I will first apply the power of a quotient exponent law, then the power of a power exponent law, and finally the quotient of powers exponent law.

6 3

(26 )3 2 ( 4) = 4 3 = (2 ) 2

8.3.3 Guided Practice: Writing Equivalent Expressions to Evaluate 1.

Two students evaluated the expression its value.

Caroline’s Work 45 ∙ 40

42

4(5+0)

42

45

42

4(5 −2)

43

64

identity

Ask your partner which method they prefer. Write their explanation here.

and found

Jamir’s Work 45 ∙ 40

42 45 ∙ 1

42 45

42

4(5 −2) 43

64

a.

Caroline used the __________ exponent law first by rewriting

product of powers

45 ∙ 40 as 45 by adding the

power of a product

_____________________

power of a power

2.

42

Complete the statements explaining each student’s work by using the phrases from the word bank. zero

Explain which method you prefer.

4 5 ∙ 40

quotient of powers power of a quotient

exponents. Next, she used the

exponent law by subtracting the exponents 5 and 2. Then,

Caroline found the value of the

Unit 8: Evaluating Numeric Expressions

exponential expression.

17

b.

zero

2.

Jamir used the

identity

________________ exponent

power of a power

law first by writing 40 as 1.

product of powers power of a product quotient of powers power of a quotient

Then, he multiplied 45 by 1.

Ruth was given the following problem to evaluate.

11.60 ∙

(−15)4 1.54

a.

Why would rewriting the following expression using laws of exponents be useful to Ruth before evaluating the expression?

b.

Rewrite the expression using as many laws of exponents as possible. Then, evaluate the expression. (−15)4 11.60 ∙ 1.54

Next, he used the

________________ exponent law by subtracting the exponents 5 and 2. Finally, he found the value of the

exponential expression.

c.

The students used the laws differently, but each expression they wrote is

18

o less than o equivalent to o greater than

4 5 ∙ 40 Ǥ 42

Unit 8: Evaluating Numeric Expressions

Team member 3: _________________ 3 Exponent law used 40 4 ( 2∙4 ) 4

8.3.4 Activity: Roundtable In teams of three, assign a number to each team member.

Each student completes the first step of their problem and writes the exponent law used. Then, pass your problem to the student on your right to check your work and write the next step to the problem. For each new step, continue passing the problems to the student on your right until all three problems are evaluated.

Team member 1: _________________ (30 ∙ 32 )3 Exponent law used

Team member 2: _________________ (0.53 )2 Exponent law used 1 2 (0.5 )

Unit 8: Evaluating Numeric Expressions

19

2.

8.3.5 Your Turn!

1.

Use the laws of exponents to evaluate the expressions.

Record the law used at each step.

Equivalent Expressions 1

2.55 ( 3) 2.5

2 0 2 2 [( ) ∙ ( ) ] 3 3

a.

0.42

(0.41 )

2

Laws of Exponents 6

b.

(22 ∙ 3.20 )

c.

(−4)2 ∙ (−4)1

d.

0.12 ∙ (0.11 )2

210

(−4)0

3

0.53 0.5 2 ∙ ( ) 2.53 2.5

20

Evaluate each expression showing each step of your work. Use a calculator to verify your solutions.

e.

2.1

3.20

∙ 2.12

Unit 8: Evaluating Numeric Expressions

Unit 8, Lesson 4: Evaluating Numeric Expressions – Part 1

8.3.6 Wrap-Up: Ticket Out the Door 1.

Use the laws of exponents to evaluate your work.

0.23 23

∙ 0.12 . Show

Benchmarks MA.7.NSO.2.1 Solve mathematical problems using multi-step order of operations with rational numbers including grouping symbols, whole-number exponents, and absolute value.

MA.7.NSO.2.2 Add, subtract, multiply and divide rational numbers with procedural fluency.

2.

Select all of the expressions equivalent to      

0.15

0.20 ∙ 0.12 0.13 ∙ 0.12 0.25 0.23

0.2 3

( 2 ) ∙ 0.12

0.23 23

∙ 0.12 .

Learning Target 

I can evaluate numeric expressions with up to three operations.

0.25

Unit 8: Evaluating Numeric Expressions

21

8.4.1 Warm-Up: Math Symbols

8.4.2 Refresher: Order of Operations

In the 18th century, Robert Recorde decided to use a symbol for the idea of “is equal to” so he would not have to write the words over and over. As more mathematicians also began to use the symbol, the “=” became accepted as the standard symbol for equality. Math is full of symbols that shorten lengthy words and calculations. But just like symbols you use when communicating with others, like “LOL” in texting, once you learn their meaning, understanding the mathematics is not so challenging.

1.

List math symbols that you are familiar with and what they represent.

Math symbols are used to provide instructions on how to apply number relationships and operations. Properties of operations and the order of operations work together to guide the order in which operations are performed.

1.

Recall that the order of operations is an agreed upon set of rules for how to evaluate an expression. Work with a partner to number each operation so that the order of operations is in the correct order.

Perform operations inside grouping symbols.

Perform multiplication and division in order from left to right. (→) Evaluate expressions with exponents.

2.

List other math symbols you have seen before, but you are unfamiliar with.

Perform addition and subtraction in order from left to right. (→)

22

Unit 8: Evaluating Numeric Expressions

8.4.3 Guided Instruction: Using Order of Operations 1.

4.

Explain if the power of a product exponent law should have been used to evaluate Expression A.

5.

Complete the statements about the expressions.

Use order of operations to evaluate one of the expressions shown in the table while your partner evaluates the other expression.

Expression A (6 + 0.5)2 ÷ 0.25

Expression B (62 + 0.52 ) ÷ 0.25

The two expressions are because they 2.

How are the problems similar?

o equivalent o not equivalent

o have the same value. o do not have the same value.

For Expression A, first o add inside the parentheses, then apply the exponent to the sum. o apply the exponent to each addend, then add to find the sum.

3.

How are the problems different?

For Expression B, first o add inside the parentheses, then apply the exponent to the sum. o apply the exponent to each addend, then add to find the sum.

Recall the power of a product exponent law states that when raising a product to a power, distribute the power to each factor.

For both expressions, addition was done before division because

o the addition is to the left of the division. o the addition is inside the grouping symbols.

Unit 8: Evaluating Numeric Expressions

23

6.

Anna evaluated Expression A and Expression B. Her work is shown.

Expression A |−10| ÷ 2 + 102 10 ÷ 2 + 102 10 ÷ 2 + 100 5 + 100 105

Expression B 2

−10 ÷ 2 + 10 −10 ÷ 2 + 100 −5 + 100 95

8.4.4 Collaboration: Determining Equivalent Expressions Answer the questions with your partner.

Tanisha says that when she evaluated an expression, the result was 7.25.

1.

Anna’s partner gave her a on both of her answers.

Why did Anna arrive at two different values when evaluating these similar looking expressions?

2.

Use mental math to estimate each value. Then, circle which of the following expressions could have been Tanisha’s expression.

(6 − 0.5)2 × 5

6 − 0.52 × 5

6 + (−0.5)2 × 5

5 × 6 − 0.52

Find the values of the expressions to verify your choice.

Expression

Absolute value notation should be treated the same as grouping symbols when following the order of operations.

(6 − 0.5)2 × 5 6 − 0.52 × 5 6 + (−0.5)2 × 5 5 × 6 − 0.52

24

Unit 8: Evaluating Numeric Expressions

Value

8.4.5 Guided Practice: Using the Order of Operations 1.

For each table, an expression is evaluated, but the steps are mixed up. Order the expressions by writing the numbers 1-4 next to the expressions shown.

Explain the steps in the column on the right. The tables have been started.

ʹ ͳ ͳ 2.

18.75

6.75 + 1.5 × 8

3

6.75 + 1.5 × 2 6.75 + 12

32.768 − 14.8

|(−3.2)3 |

− 14.8

Josiah evaluated 𝑎𝑎3 ÷ 4(6.4 + 𝑏𝑏) when 𝑎𝑎 = 4 and 𝑏𝑏 = −8.4. His work is shown below. a.

Circle Josiah’s error and choose which step of the order of operations he did not apply correctly.

43 ÷ 4(6.4 + −8.4) 43 ÷ 4( −2)

Evaluate the exponent

64 ÷ 4( −2)

Original expression

60 ÷ −8 −7.5

Original expression

17.968

|−32.768| − 14.8

3.

b.

• Evaluate expressions with exponents.

• Perform operations inside grouping symbols.

• Perform multiplication and division in order from left to right.

• Perform addition and subtraction in order from left to right.

Discuss your answer with your partner. Then, work together to evaluate the expression correctly.

Evaluate each expression. a.

(−8) ÷ 24 × 2

b.

4 × ( 14 )

c.

−5 × |3.6 + −4.2|

3.5 2

Unit 8: Evaluating Numeric Expressions

25

3.

8.4.6 Your Turn!

1.

Evaluate each expression without using a calculator.

Write an explanation using the order of operations next to each step. |𝟖𝟖 − 𝟏𝟏𝟏𝟏| ÷ (−𝟏𝟏𝟏𝟏)

(𝟏𝟏. 𝟓𝟓 × 𝟐𝟐)𝟐𝟐 ÷ 𝟐𝟐

2.

26

Evaluate each expression. a.

(−8×0.25)4

b.

4 × |10.6 × (−0.3)|

c.

(−95 × 0.7)2 + 8.75

Explanation

Explanation

Determine if the equations in the table are true or false.

Show work to support your answer.

Equation |7.3 − 8.5| × (−2.5) = 3

2 − 2 × 22 = −6

4.2 × 3.5 + (−3)3 = 41.7

5

Unit 8: Evaluating Numeric Expressions

True or False?

Unit 8, Lesson 5: Evaluating Numeric Expressions – Part 2

8.4.7 Wrap-Up: Cool Down For each problem, a student was asked to evaluate the expression. Their work is shown in the table. If there are mistakes in the student’s work, explain what the mistakes are and write the correct response. If there are no mistakes, leave the rest of the row blank.

Problem 1. 72 = 2. (85 )0 = 3.

(−3)3

=

4. (4 + 5)

2

Mistake(s)

Correct Response

Benchmarks MA.7.NSO.2.1 Solve mathematical problems using multi-step order of operations with rational numbers including grouping symbols, whole-number exponents, and absolute value.

MA.7.NSO.2.2 Add, subtract, multiply and divide rational numbers with procedural fluency.

Learning Target 

I can evaluate numeric expressions with up to three or four operations.

5. (−4)5 ∙ (4)0

6. [(8 + 32 ) × 4]2

Unit 8: Evaluating Numeric Expressions

27

8.5.2 Guided Instruction: Using the Order of Operations on Expressions With Nested Grouping Symbols

8.5.1 Warm-Up: Estimating Length The roll of bubble wrap on the left is 1 foot high and 100 feet long.

The order of operations can be used to evaluate expressions using grouping symbols, rational numbers, exponents, and absolute values.

1.

Two students evaluated the expression (

(3+0.5) 3 0.5

) . This

expression shows nested grouping symbols, or groupings inside of other groupings.

Complete the following to guess the length of the bubble wrap on the right.

1.

2.

What is a guess that is too low?

Give a guess that does not make sense and explain how you know it doesn’t make sense.

Devon 3 (3 + 0.5) ( ) 0.5

(7)3

3.53 0.53

3.5 3 ( ) 0.5 343

(3 + 0.5)3 0.53 42.875 0.125 343

How were Abby’s steps and Devon’s steps different from each other?

3.

What is a guess that is too high?

4.

What is your best guess of the length of the bubble wrap on the right when unrolled?

28

Abby 3 (3 + 0.5) ( ) 0.5

Unit 8: Evaluating Numeric Expressions

When using the order of operations to evaluate an expression with nested grouping symbols, apply the order of operations inside the inner grouping symbols first.

2.

8.5.3 Exploration: Grouping Symbols 1.

Zion says since multiplication is commutative and the order does not matter, he thinks he can evaluate the expression [0.05 × (3 × 2)4 ]2 by either multiplying 0.05 × 3 first or by multiplying 3 × 2 first.

Without working the problems, discuss with your partner which expression in the table you expect to have the largest value.

Expression A

Expression B 2

10 − (0.75 × 12) + 4

(10 − 0.75) × (12 + 42 )

Expression C

Expression D

10 − 0.75 × 12 + 42

10 − 0.75 × (12 + 4)2

Discuss with your partner if Zion is correct when he stated that both would work.

3.

Telena says the problem can be worked either by following the order of operations or by using the laws of exponents. Complete the table to show the work for both possibilities to determine if Telena is correct.

Order of Operations 3 2 [0.1 ∙ ( ∙ −3) ] 3

2

Laws of Exponents 3 2 [0.1 ∙ ( ∙ −3) ] 3

2

2.

Put an asterisk (*) next to the expression you expect will have the largest value. Write your explanation here.

3.

Evaluate each of the expressions in the table with your partner.

Unit 8: Evaluating Numeric Expressions

29

4.

With another set of partners, share if your guess was or was not correct. Discuss how the order of operations affected the value of your guesses. Summarize your discussion.

8.5.4 Collaboration: Using the Order of Operations to Evaluate Expressions With Nested Grouping Symbols

1.

The expressions in each row have the same value.

Evaluate the expressions in one column while your partner evaluates the expressions in the other column.

Compare values and if they are not equivalent, work together to find and correct the error(s).

Partner A

5.

For the two expressions in the table that resulted in the same value, explain why they have the same value.

30

Partner B

|(35.5 − 52 ) ÷ 3|

|4.3 × 5 − 52 |

32 ÷ [(−2)2 × 5 − 4]

[(−2)3 + 16] ÷ (−2)2

[(18 − 4 × 5) × 3]4

[(2.4 − 5.4) × (−2)1 ]4

11 + (0.30 + 3)2

Value of Both Expressions

Unit 8: Evaluating Numeric Expressions

[36 ÷ (23 + 4)]3

8.5.5 Your Turn!

1.

8.5.6 Wrap-Up: Reflection

Evaluate each expression without using a calculator.

1.

Complete each statement based on your learning today.

a.

[(10 − 15) × 2]3 + 50

•

A barrier to my learning was …

b.

(24 − 3 × 6)2

•

I will try to overcome this by …

•

Today I am still unsure about …

•

To fill in this gap I intend to …

c.

|18 + (−2) × 14|

5

d.

[0.8 × (1.2 + 2.8)2 ]2

e.

[(0.8 × 10)2 + 4.5] ÷ 0.5

f.

(10 × 0.4)2 + (28.4 ÷ (−4))

Unit 8: Evaluating Numeric Expressions

31

Unit 8, Lesson 6: Evaluating Numeric Expressions – Part 3 Benchmarks

8.6.1 Warm-Up: Bell Work Explain the steps needed to evaluate the expressions.

1.

(10 + 18 ∙ 2)5 ÷ 3

2.

[(7 − 10)2 ]4 × 2

MA.7.NSO.2.1 Solve mathematical problems using multistep order of operations with rational numbers including grouping symbols, whole-number exponents, and absolute value.

MA.7.NSO.2.2 Add, subtract, multiply and divide rational numbers with procedural fluency.

Learning Target 

I can evaluate numeric expressions with up to six operations.

32

Unit 8: Evaluating Numeric Expressions

8.6.2 Guided Instruction: Evaluating Expressions With Multiple Steps

Two students evaluated the expression below in two different ways. Review the work of each student. Then, answer the questions.

Diego [0.5 × (0.5 + 3.5)2 ]3 + 26 ÷ 13 [0.5 × 42 ]3 + 26 ÷ 13

[0.5 × 16]3 + 26 ÷ 13 3

8 + 26 ÷ 13

512 + 26 ÷ 13 512 + 2 514

Valeria [0.5 × (0.5 + 3.5)2 ]3 + 26 ÷ 13

0.53 × (0.5 + 3.5)2×3 + 26 ÷ 13 0.53 × (0.5 + 3.5)6 + 26 ÷ 13 3

6

0.5 × 4 + 26 ÷ 13

0.125 × 46 + 26 ÷ 13

0.125 × 4,096 + 26 ÷ 13 512 + 26 ÷ 13 512 + 2 514

1.

2.

8.6.3 Activity: Expression Game 1.

Each group will use a standard number cube with sides numbered 1 − 6 to play a game. One player rolls the number cube. Each player decides which box to place that number in the expression. Continue rolling the cube until all blanks in the expression are filled.

Evaluate your expression and record your score based on the point system in the table. If on any round two players’ expressions have the same value, both players lose 𝟏𝟏𝟏𝟏 points.

Round 1 Scoring – the player with the greatest value earns 8 points.

Round 2 Scoring – the player with the least value earns 5.5 points.

How were Diego’s steps and Valeria’s steps different from each other?

Round 3 Scoring – the player with the greatest absolute value loses 3.7 points.

Discuss with your partner whether you think one method was easier to use to evaluate the expression over the other.

Round 4 Scoring – the player with the absolute value closest to zero earns 11 points.

Unit 8: Evaluating Numeric Expressions

[( ______ × ______2 ) + ______] ÷ ______

My answer ______ Points earned ______

[(______ × ______)2 + ______] ÷ ______

My answer ______ Points earned ______

(______ × ______2 ) + (______ ÷ ______)

My answer ______ Points earned ______

(______ × ______)2 + (______ ÷ ______)

My answer ______ Points earned ______

33

My Game Score Round 1

2.

Round 2

Round 3

8.6.4 Try It: Evaluating Expressions With Multiple Steps

Round 4

My Score

Discuss with the other players what strategies they used to determine where to put the numbers.

Summarize your discussion.

1.

In the table an expression has been evaluated, but the steps are mixed up. Order the expressions using the numbers 1 − 7 on the left. Write an explanation for each step.

Step

Expression

Explain the Step

|−11.591| + 15 − (−3.1)2 11.591 + 15 − 9.61 11.591 + 15 − (−3.1)2 1

|18.2 + (−3.1)3 | + 15 − (−3.1)2 26.591 − 9.61 |18.2 + (−29.791)| + 15 − (−3.1)2 16.981

34

Unit 8: Evaluating Numeric Expressions

Original expression

2.

Evaluate each expression without using a calculator. a.

b.

c.

3 2

(4) × 32 + 32

[(10 + 4.5)2 − 10.25]2 ÷ 40

6.7 − (23 − 3)3 ÷ 10 + 4

8.6.5 Activity: Class Debate James wants to choose an expression that has a value of −778. He can choose between the expressions shown.

Expression A

Expression B

14 − (10 × 32 )2 + 6 × 3 Expression C

14 − 10 × (32 )2 + 6 × 3

(14 − 10) × (32 + 6)2 × 3

14 − [10 × (32 )2 ] + 6 × 3

Expression D

Which expression should James choose? _________ •

Each corner of the room is labeled A, B, C, or D. After selecting the expression that you think James should choose, go to the corner of the room that matches the label for that expression.

•

With your classmates who selected the same expression, discuss why your expression is the expression that James should choose. Discuss why James should not choose the other expressions.

One member of your group will act as a representative during a class debate.

•

For the debate, prepare an opening statement and potential rebuttals for arguments from other groups. Keep arguments and rebuttals centered on the mathematics.

•

After the debate, explain whether you would select the same expression again.

Unit 8: Evaluating Numeric Expressions

35

8.6.6 Wrap-Up: Ticket Out the Door 1.

THIS PAGE WAS INTENTIONALLY LEFT BLANK.

Evaluate each expression without a calculator. Show each step you use.

3 4 − |5 − | + (42 − 61) 5 15 Work

(1.5 + |−3.25 − 4.25|)3 12 Work

36

Unit 8: Evaluating Numeric Expressions

Unit 9 - Algebraic Expressions 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8

Lesson Adding and Subtracting Linear Expressions With Integer Coefficients Adding Linear Expressions With Rational Coefficients Subtracting Linear Expressions With Rational Coefficients Adding and Subtracting Expressions With Rational Coefficients Equivalent Numerical Expressions Rewriting Equivalent Numerical Expressions Equivalent Linear Expressions Equivalent Expressions Using Properties of Operations

Unit 9: Algebraic Expressions

Page Number 39 46 52 57 61 65 70 77

37

The following B.E.S.T. Standards will be covered in this unit: MA.7.AR.1.1 Apply properties of operations to add and subtract linear expressions with rational coefficients.

• Clarification 1: Instruction includes linear expressions in the form 𝑎𝑎𝑎𝑎 ± 𝑏𝑏 𝑜𝑜𝑜𝑜 𝑏𝑏 ± 𝑎𝑎𝑎𝑎 , where 𝑎𝑎 and 𝑏𝑏 are rational numbers.

• Clarification 2: Refer to Properties of Operations, Equality and Inequality (Appendix D).

MA.7.AR.1.2 Determine whether two linear expressions are equivalent.

• Clarification 1: Instruction includes using properties of operations accurately and efficiently.

• Clarification 2: Instruction includes linear expressions in any form with rational coefficients.

• Clarification 3: Refer to Properties of Operations, Equality and Inequality (Appendix D).

38

Unit 9: Algebraic Expressions

Unit 9, Lesson 1: Adding and Subtracting Linear Expressions With Integer Coefficients Benchmark MA.7.AR.1.1 Apply properties of operations to add and subtract linear expressions with rational coefficients.

Learning Target 

I can use algebra tiles to add and subtract linear expressions.

9.1.1 Warm-Up: Career in Robotics Tessa Lau is the Chief Robot Whisperer for a robotics company. The company creates robot helpers for the service industry. Some of their robots perform tasks like delivering items to guests at a hotel. Tessa has studied applied physics, computer science, artificial intelligence, and human-computer interaction. She loves hands-on experiments, building things, and the speed of creating with computers. Through programming and engineering, one of her primary jobs is to look at a robot in context and make sure the robots behave well in human environments.

1.

Have you ever considered a career as an engineer?

2.

How do you think a robot could be helpful to you in your everyday life?

3.

Tessa’s advice is “Don’t listen to someone who tells you that you can’t do something.” Describe a time someone told you that you were not able to do something.

Unit 9: Algebraic Expressions

39

9.1.2 Refresher: Adding and Subtracting Integers With Algebra Tiles Algebra tiles, like integer chips, can be used to model adding and subtracting integers. Each small square represents a positive 1 (blue) or a negative 1 (red).

Integer Chips

Algebra Tiles

Adding a positive 1 (blue) and a negative 1 (red) represents a zero pair because 1 + −1 = 0, as shown. =0

1.

Describe why this diagram represents the value 0.

2.

Complete the following problems. a.

Complete the table.

Numeric Expression

Value

−7 + 2 6−8 −1 − 5 4 − ( −3) b.

For some of the representations with algebra tiles, zero pairs had to be added to the representation.

Put an asterisk (*) next to the expressions where this was the case.

c.

Explain why zero pairs had to be added to the representation in these cases.

A zero pair is a pair of numbers whose sum is zero.

40

Algebra Tiles Representation

Unit 9: Algebraic Expressions

9.1.3 Guided Instruction: Algebra Vocabulary There are some fundamental definitions that are necessary to know for working with algebraic expressions.

A term is a single number or variable, or numbers and variables multiplied together. For example, the expression 0.5 − 4𝑦𝑦 has two terms.

The expression 0.5 − 4𝑦𝑦 has two terms.

0.5

term 1

In the expression 4𝑥𝑥𝑥𝑥, 4 is the coefficient.

In the expression 𝑥𝑥, 1 is the coefficient.

In the expression −8𝑦𝑦, −8 is the coefficient.

−4𝑦𝑦

term 2

One of the terms in this expression is a constant term.

1.

A coefficient is the number or constant that multiplies a variable in an algebraic expression. If no number is specified, the coefficient is 1.

Variable terms have coefficients while constant terms do not.

Which term is the constant term? Explain.

An expression is a mathematical statement containing numerals, operators, grouping symbols, and symbols or variables for unknown values. An expression does not contain an equal sign or inequality symbol.

For example, 4 + 5 , 3𝑥𝑥 − 2, and 6𝑥𝑥 − 7𝑦𝑦 + 11 are all expressions.

Unit 9: Algebraic Expressions

41

9.1.4 Exploration: What Are Like Terms?

1.

b.

Here is a collection of algebra tiles.

Use the definitions of coefficient and term to complete the expression by writing the coefficient or constant term that matches the number of tiles of each type.

______𝑥𝑥 + (−_____𝑥𝑥) + ____𝑦𝑦 + (−_____𝑦𝑦) + ______ + (−______)

a.

Complete the table below.

Type of Algebra Tile

Number of Tiles

c.

Draw the number of algebra tiles of each type under the term in the expression.

d.

Circle the zero pairs in your drawing.

e.

Complete the expression below after eliminating the zero pairs. ______𝑥𝑥 + ____𝑦𝑦 + ______

𝑥𝑥

(−𝑥𝑥) 𝑦𝑦

(−𝑦𝑦) ͳ

(−1) 42

Like terms are terms with the exact same variable(s) raised to the same exponent.

A constant is a fixed numerical value with no variables. All constants are like terms.

Unit 9: Algebraic Expressions

2.

Answer the following questions about like terms. a.

Classify the following terms.

Like Terms

Unlike Terms

𝑦𝑦 and 6

◯

◯

◯

◯

5𝑥𝑥 and −3𝑥𝑥

◯

◯

◯

◯

3𝑥𝑥 and 3

b.

9.1.5 Guided Instruction: Adding and Subtracting Expressions With Algebra Tiles

10, −2, 7.5

How did you determine whether the terms were like or unlike?

The algebra tiles shown represent 𝑥𝑥 and −𝑥𝑥 as rectangles, and integers as squares. Red represents the negative of that variable or constant, while the blue shades represent the positive.

Variables also form zero pairs, as shown. =0 1.

c.

Compare answers with your partner.

Complete the table by crossing out the zero pairs and recording how many there are. Write the resulting algebraic expression after eliminating the zero pairs.

Algebra Tiles Representation:

Unit 9: Algebraic Expressions

Number of Zero Pairs

Algebraic Expression

43

2.

Draw a representation of the following expressions with algebra tiles. Cross out any zero pairs. Then write an equivalent algebraic expression after eliminating any zero pairs.

Algebraic Expression and Representation 𝒙𝒙 − 𝟑𝟑𝟑𝟑 + 𝟏𝟏

Equivalent Algebraic Expression

9.1.6 Guided Practice: Adding and Subtracting Expressions With Algebra Tiles 1.

Consider the expressions 2𝑥𝑥 and 𝑥𝑥 + 2. a.

Draw a model of 2𝑥𝑥 and 𝑥𝑥 + 2 with algebra tiles.

Expression

Algebra Tiles Model

Representation:

2𝑥𝑥

−𝟒𝟒 + 𝟑𝟑𝟑𝟑 − 𝒚𝒚 + 𝟐𝟐

𝑥𝑥 + 2

Representation:

44

b.

How are they different?

c.

Are 2𝑥𝑥 and 𝑥𝑥 + 2 equivalent algebraic expressions?

Unit 9: Algebraic Expressions

9.1.7 Your Turn!

9.1.8 Wrap-Up: Cool Down

Write an equivalent expression by combining like terms.

1.

2𝑥𝑥 + 4𝑥𝑥 = _____________________

1.

For each key word, check the box in the column that describes your knowledge of the word. Then answer the questions or complete the statements in the last column.

Key Word 2.

−𝑦𝑦 + 7 + 4𝑦𝑦 − 1 = _______________ Expression

3.

4.

5.

𝑥𝑥 − 𝑥𝑥 + 𝑥𝑥 − 𝑥𝑥 + 1 = _______________

Term

8𝑥𝑥 − 4 − 4𝑥𝑥 = __________________

Coefficient

2𝑥𝑥 − 4𝑥𝑥 = _____________________

Equivalent

I already knew the meaning of this word.

I’ve heard this word before, but I was not sure what it meant.

Questions

An example of a numeric expression is _____________.

An example of an algebraic expression is _____________.

In the expression 8𝑥𝑥 − 9 + 2𝑦𝑦, how many terms are there? _____ In the expression 8𝑥𝑥 − 9 + 2𝑦𝑦, what are the coefficients? _______________ When working with algebraic expressions the word “equivalent” means _______________ _______________ _______________.

Unit 9: Algebraic Expressions

45

Unit 9, Lesson 2: Adding Linear Expressions With Rational Coefficients

9.2.1 Warm-Up: Visual Patterns Three figures are shown.

Benchmark MA.7.AR.1.1 Apply properties of operations to add and subtract linear expressions with rational coefficients.

Learning Target 

46

1.

What pattern do you see?

2.

Explain how many small purple squares you think will be in the next figure.

I can add linear expressions with rational coefficients.

Unit 9: Algebraic Expressions

9.2.3 Collaboration: Adding Expressions With Algebra Tiles

9.2.2 Exploration: Which One Doesn’t Belong?

1.

Which expression does not belong with the other three?

Explain your reasoning.

Expression A

Expression B

7 − 5𝑥𝑥 − 3 Expression C

3 1 𝑥𝑥 + 6 − 2 − 6 𝑥𝑥 2 2

−5𝑥𝑥 + 7.25 − 4.25

−2𝑥𝑥 − 3𝑥𝑥 + 4

Expression D

Algebra tile representations of an expression are given.

Work with your partner to find the sum by following the steps below.

1.

On the lines, write the algebraic expression for each representation.

Rearrange the tiles to group like tiles. Draw the representation.

_________ + _________

Eliminate any zero pairs to find the sum.

Write the sum.

Unit 9: Algebraic Expressions

47

2.

On the lines, write the algebraic expression for each representation.

Rearrange the tiles to group like tiles. Draw the representation.

Eliminate any zero pairs to find the sum.

Write the sum.

9.2.4 Guided Instruction: Adding Linear Expressions

_________ + _________

When adding two expressions, rearrange the terms so that like terms are grouped together. Then add the coefficients of the like variable terms together and add the constants.

Zero pairs are eliminated in this process.

1.

Subtracting an integer is the same as adding the opposite. For example, the expression 9 − 13 can be rewritten as 9 + −13.

Find the sum of the two expressions. a.

Original Expression

(2𝑥𝑥 + 6) + (5𝑥𝑥 + 4)

Regrouped Expression: Rearrange the terms so ______𝑥𝑥 + ______𝑥𝑥 + ______ + ______ that like terms are together.

Simplified Expression: Add the coefficients of the ______𝑥𝑥 + ______ like variable terms and add the constants. b.

Original Expression

48

(−2𝑥𝑥 + 6) + (5𝑥𝑥 − 4)

Regrouped Expression: Rearrange the terms so that like terms are together.

______𝑥𝑥 + ______𝑥𝑥 + ______ + ______

Simplified Expression: Add the coefficients of the like variable terms and add the constants.

______𝑥𝑥 + ______

Unit 9: Algebraic Expressions

9.2.5 Collaboration: Adding Linear Expressions 1.

Work with your partner to find the sum of the two expressions in simplest form.

c.

Original Expression

(−5𝑦𝑦 + 2) + (5𝑦𝑦 − 8)

a.

Regrouped Expression Original Expression

4 + (8 − 0.4𝑥𝑥)

Regrouped Expression

Simplified Expression

Simplified Expression d.

Original Expression

b.

Original Expression

1 3 1 1 (3 𝑥𝑥 − ) + (2 𝑥𝑥 + ) 2 5 4 3

(0.7 − 2.4𝑥𝑥) + (4.1𝑥𝑥 − 5.32)

Regrouped Expression

Regrouped Expression

Simplified Expression Simplified Expression

Unit 9: Algebraic Expressions

49

e.

Original Expression

9.2.6 Your Turn!

3 1 ( 𝑤𝑤 − 5.5) + (− 𝑤𝑤 + 4) 4 2

1.

Find the sum of each expression using the fewest terms possible. a.

(𝑥𝑥 + 7) + (𝑥𝑥 − 10)

c.

(1 7 − 𝑦𝑦) + (4𝑦𝑦 + 7)

Regrouped Expression

b.

(3𝑥𝑥 − 5) + (5 + 6𝑥𝑥)

d.

(4𝑤𝑤 − 6) + (−6𝑤𝑤 − 1)

Simplified Expression

50

6

Unit 9: Algebraic Expressions

1

e.

(9 − 4.6𝑥𝑥) + (−5.3𝑥𝑥 − 7)

f.

5

1

(− 2 𝑘𝑘 − 7) + (5 𝑘𝑘 + 4)

9.2.7 Wrap-Up: Error Analysis 1.

1

Your friend finds the sum of (−2 2 𝑥𝑥 + 7.1) + (4𝑥𝑥 − 5.2) to 3

be − 𝑥𝑥 + 1.9.

2

Write a detailed note to your friend explaining why their answer is correct or why it is incorrect.

Unit 9: Algebraic Expressions

51

Unit 9, Lesson 3: Subtracting Linear Expressions With Rational Coefficients

9.3.1 Warm-Up: Bell Work Combine like terms.

1.

Benchmark

(3𝑦𝑦 + 4.6) + (−5𝑦𝑦 − 3.25)

MA.7.AR.1.1 Apply properties of operations to add and subtract linear expressions with rational coefficients.

Learning Target 

52

I can subtract linear expressions with rational coefficients.

Unit 9: Algebraic Expressions

2.

3

5

(6 4 𝑥𝑥 − 7) + (− 2 𝑥𝑥 + 7)

9.3.2 Exploration: Subtracting Expressions With Algebra Tiles 1.

Algebra tile representations are given below. Use the algebra tiles to find the difference.

On the lines, write the algebraic expression for each representation.

9.3.3 Collaboration: Subtracting Expressions 1.

Noah, Lin, Jada, and Andre are rewriting the expression (𝑥𝑥 + 8) − (4 − 9𝑥𝑥) as an equivalent expression with fewer terms. Each student described their work.

Discuss with your partner which students made an error. Describe each student’s error and correct the error. a.

_________ − _________

Noah says, “I dropped the parentheses, combined like terms, and ended up with −8𝑥𝑥 + 4.” (𝑥𝑥 + 8) − (4 − 9𝑥𝑥) 𝑥𝑥 + 8 − 4 − 9𝑥𝑥 −8𝑥𝑥 + 4

Rearrange the tiles to show subtraction of like tiles together. Draw the representation. b.

If necessary, create any zero pairs needed for subtraction.

Find the difference by subtracting (taking away) like tiles. Write the algebraic representation of the difference.

Lin says, “I simplified inside the parentheses and ended up with 13𝑥𝑥.”

Unit 9: Algebraic Expressions

(𝑥𝑥 + 8) − (4 − 9𝑥𝑥) (8𝑥𝑥) − (−5𝑥𝑥) 8𝑥𝑥 + 5𝑥𝑥 13𝑥𝑥

53

c.

d.

54

Jada says, “I know that subtracting is adding the opposite, and ended up with 4 + 10𝑥𝑥.” (𝑥𝑥 + 8) − (4 − 9𝑥𝑥) (𝑥𝑥 + 8) + (−4 + 9𝑥𝑥) 𝑥𝑥 + 8 − 4 + 9𝑥𝑥 4 + 10𝑥𝑥

When subtracting expressions, use the distributive property to add the opposite. With the algebra tile representations below, model finding the difference by adding the opposite.

On the lines, write the algebraic expression for each representation.

Andre says, “I used the distributive property since there is an invisible (−1) in front of the second parentheses to distribute, but I ended up with 10𝑥𝑥 + 4.” (𝑥𝑥 + 8) − (4 − 9𝑥𝑥) (𝑥𝑥 + 8) − 1(4 − 9𝑥𝑥) 𝑥𝑥 + 8 − 4 + 9𝑥𝑥 10𝑥𝑥 + 4

2.

Rewrite subtracting to adding the opposite by changing (or flipping) the sign of the 2nd expression. Draw the representation.

Combine any zero pairs to find the sum. Write the sum.

Unit 9: Algebraic Expressions

_________ − _________

9.3.4 Guided Instruction: Subtracting Linear Expressions 1.

To subtract linear expressions, •

9.3.5 Guided Practice: Subtracting Linear Expressions With Rational Coefficients 1.

add the ______________, or additive inverse of each term in the expression.

Find the difference of each expression.

a.

(3𝑦𝑦 + 3.5) − (8.2𝑦𝑦 − 6)

c.

(−2 +

b.

OR •

2.

1

distribute a __________ to each term in the expression.

Complete the table to show both methods.

Additive Inverse Method

Applying the Distributive Property

(4𝑥𝑥 + 5) − (−2𝑥𝑥 + 6)

(4𝑥𝑥 + 5) − (−2𝑥𝑥 + 6)

3.

3

(𝑚𝑚 − 8) − (3 4 − 𝑚𝑚)

9

2

1

𝑦𝑦) − (5.3𝑦𝑦 − 5 ) d.

2

1

(−7.5𝑡𝑡 − 3) − (7 + 3 𝑡𝑡) 8

When adding and subtracting, parentheses are used to group expressions. Why is the use of parentheses especially important when subtracting?

Unit 9: Algebraic Expressions

55

2.

What is the result when −3𝑥𝑥 + 4.48 is subtracted from −2.2𝑥𝑥 + 3? a.

b.

c.

Write the algebraic expression.

9.3.6 Your Turn!

1.

Find the difference of each expression in simplest form.

a.

(−2ℎ + 8) − (ℎ + 10)

b.

(2 𝑥𝑥 + 3) − (3 −

c.

(−4.4𝑦𝑦 − 9) − (5 − 2.1𝑦𝑦)

d.

(2 3 𝑛𝑛 + 1) − (− 4 𝑛𝑛 − 6)

4

9

8

𝑥𝑥)

Rewrite the expression without parentheses.

Simplify the expression by rewriting it using the fewest terms possible.

2.

5

4

2

1

What is the result when 𝑥𝑥 + is subtracted from 5 4

56

7

𝑥𝑥 −

31 6

?

Unit 9: Algebraic Expressions

8

3

Unit 9, Lesson 4: Adding and Subtracting Expressions With Rational Coefficients

9.3.7 Wrap-Up: Ticket Out the Door

Write the following expressions using the fewest possible terms.

1.

1

1

Benchmark

(5𝑥𝑥 − 13 4) − (4 2 − 5𝑥𝑥)

MA.7.AR.1.1 Apply properties of operations to add and subtract linear expressions with rational coefficients.

Learning Target 

2.

(0.7 − 2.4𝑥𝑥) + (−5.32 + 4

1

10

I can apply properties of operations to add and subtract linear expressions with rational coefficients.

𝑥𝑥)

Unit 9: Algebraic Expressions

57

9.4.2 Collaboration: Adding and Subtracting Linear Expressions

9.4.1 Warm-Up: Bell Work 1.

Rewrite each expression by combining like terms. a.

5.7𝑥𝑥 − (3.5𝑥𝑥 + 6)

1.

Two students made errors when rewriting the expression (6𝑥𝑥 + 9) − (7 − 𝑥𝑥) + (8𝑥𝑥 + 11).

Zoe’s Work (6𝑥𝑥 + 9) − (7 − 𝑥𝑥) + (8𝑥𝑥 + 11)

Patricia’s Work (6𝑥𝑥 + 9) − (7 − 𝑥𝑥) + (8𝑥𝑥 + 11)

−𝑥𝑥 − 9

13𝑥𝑥 + 13

(6𝑥𝑥 + 9) − (7𝑥𝑥 + 18)

b.

58

5

1

(5𝑥𝑥 + 2) + (8𝑥𝑥 + 11)

a.

Working with your partner, write an explanation to Zoe that describes her error.

b.

Write an explanation to Patricia that describes her error.

2

(− 6 𝑦𝑦 + 5 2) − (3 𝑦𝑦 + 34)

Unit 9: Algebraic Expressions

c.

Simplify the expression (6𝑥𝑥 + 9) − (7 − 𝑥𝑥) + (8𝑥𝑥 + 11).

9.4.3 Guided Practice: Equivalent Expressions 1.

Circle the expressions that are equivalent to −5𝑥𝑥 − 3 and cross out those that are not.

1 3 (−4 𝑥𝑥 − 2) + (0.5𝑥𝑥 + 3) − (1 𝑥𝑥 + 4) 8 8

−2𝑥𝑥 − (8𝑥𝑥 + 5) + (𝑥𝑥 + 2 ) −5𝑥𝑥 − 3

(7𝑥𝑥 + 4) − (3𝑥𝑥 + 2) + (−9𝑥𝑥 − 9 )

1 3 − ( 𝑥𝑥 − 6) + (3𝑥𝑥 − 7.25) − (7.5𝑥𝑥 + 1 ) 2 4

Unit 9: Algebraic Expressions

59

9.4.4 Your Turn!

1.

Rewrite each expression by combining like terms. a.

b.

c.

60

9.4.5 Wrap-Up: Growth Mindset

(1 − 4𝑦𝑦) − (2.5𝑦𝑦 + 7) − (− 0.3𝑦𝑦 + 3)

1 (3 2 𝑥𝑥 − 4) + (𝑥𝑥 − 3.2) − (− 4 𝑥𝑥 + 9) 1

1.

Mistakes are an opportunity to learn. Complete the statement to highlight a mistake you learned from.

My favorite mistake from today was …

Means I Start To Acquire Knowledge Experience Skills

−(2𝑟𝑟 + 5) − (6 − 3𝑟𝑟) + (10𝑟𝑟 − 4)

Unit 9: Algebraic Expressions

Unit 9, Lesson 5: Equivalent Numerical Expressions

9.5.1 Warm-Up: Which One Doesn’t Belong?

1.

Benchmark MA.7.AR.1.2 Determine whether two linear expressions are equivalent.

Which one doesn’t belong? Justify your reasoning.

(13 − 12 ÷ 2) + 3 ∗ 2

12 + (9 ÷ 3)2

20 − 16 ÷ 8 + 5

38 − (2 + 4) + 20 − 3

Learning Target 

I can determine if numerical expressions are equivalent.

Unit 9: Algebraic Expressions

61

9.5.2 Exploration: Equivalent Numeric Expressions 1.

Is 14 −

3 4

(12) equivalent to

47 5

you determined your answer.

c.

Does

−3(12) 4

have the same value as the previous

expressions?

44

− ( ÷ 2)? Explain how 5 d.

3

Create another expression equivalent to − 4 (12).

Share your answer with your partner.

The expressions in the previous problem all have the same value, but they are written differently.

2.

62

3

Consider the expression − (12).

4

3.

What do you notice as the similarities and differences 3 −3(12) between − 4 (12), 0.75(−12), and 4 ?

4.

Explain which representation is easiest for you to evaluate using mental math.

3

a.

What is the value of − 4 (12)?

b.

Explain if − (12) can be rewritten as 0.75(−12)?

3 4

Unit 9: Algebraic Expressions

9.5.3 Guided Instruction: Equivalent Numerical Expressions

9.5.4 Try It: Equivalent Numerical Expressions

Numerical expressions are equivalent if their values, when simplified, are the same.

Algebraic expressions are equivalent if they have the same value no matter what number is substituted for each of the variables.

1.

1.

Evaluate expressions A-L. 1

B (−2.25 × 4) ∙ 2

A − (−24) + 3 3

1

Are the following expressions equivalent? Show your work or explain your thinking.

9

4

1

1

+ (3.5 − 7 2) and (− 2 ∙

3 4

7

+ 6.5) − 7 8

1

D 3(2.5 − 6) + 7 2 2.

15

C − 15 + 4 3

1

E 4(− 4)(1)(−12 )

F 3+

−24 −3

Work with your partner to create a third expression that 9 1 is also equivalent to 4 + (3.5 − 7 2).

Unit 9: Algebraic Expressions

63

G 5(−4 1 + 6.25) 4

2

9.5.5 Collaboration: Generating Equivalent Expressions

H −6 3 ∙ 2.4 ÷ (−4) I −7.5 + (6 1 ∙ 3) 6

3

1.

Work with a partner to choose a numerical value.

Individually, create a unique expression using rational numbers that equals the numerical value chosen when evaluated. Include a minimum of each of the following:

4

4

J (−4.2 + 9 5) + (1 + 4 10)

−18

K − ( −5 ) − 0.4

• • •

1

L |−3 8 − 0.875|

four operations one grouping symbol two rational numbers written as fractions

When you are finished creating your expression, trade and check each other’s numeric expressions.

Numerical Value _________ Your Expression

2.

Your Partner’s Expression

Sort the expressions into groups based on equivalence.

Record the letters of the equivalent expressions.

Equivalent Expressions 2.

64

Are they equivalent? If not, correct any errors.

Unit 9: Algebraic Expressions

Unit 9, Lesson 6: Rewriting Equivalent Numerical Expressions

9.5.6 Wrap-Up: Cool Down 1.

Are the following expressions equivalent? Show your work or explain your reasoning.

1

1

(−7.6 + 2 2) (−6) and (−8) (−3 10) +

2 3

(8.6)

Benchmark MA.7.AR.1.2 Determine whether two linear expressions are equivalent.

Learning Target 

I can determine if two numerical expressions are equivalent by rewriting each in the same form.

Unit 9: Algebraic Expressions

65

9.6.2 Guided Instruction: Rewriting Equivalent Expressions

9.6.1 Warm-Up: Two Truths and a Lie 1.

Students were asked to write two truths and a lie, or two true statements and one false statement, about the associative property of addition. One student’s work is shown.

1.

Consider the expressions 18 + 12(30 + 7) and 18 + 12(7 + 30). a.

Describe what is different about the expressions.

b.

Evaluate 18 + 12(30 + 7) and 18 + 12(7 + 30).

18 + 12(30 + 7)

Which statement is the lie? Justify your response.

66

18 + 12(7 + 30)

c.

Are the two expressions equivalent?

d.

Was it necessary to evaluate the two expressions to determine whether they were equivalent or not?

Unit 9: Algebraic Expressions

2.

The

o associative o commutative o distributive

property of addition

states that the order in which two numbers are added does not matter.

3.

4.

For what other mathematical operations does this property hold true?

Eric thinks the expressions shown are equivalent to each other.

18 + 12(30 + 7)

18 + 360 + 84

30(30 + 7)

Without evaluating each expression, explain if he is correct.

5.

The

o associative o commutative o distributive

property of operations can

be used to show that 18 + 12(30 + 7) = 18 + 360 + 84.

Equivalent expressions can be rewritten in different forms using the properties of operations.

The following properties of operations apply to all rational numbers.

The distributive property of multiplication over addition or subtraction states that a factor is “distributed” to all terms within a set of parentheses.

The commutative property states that for addition and multiplication, the order in which the operation is performed does not change the value of the expression.

The associative property states that for addition and multiplication, the way in which numbers are grouped does not change the value of the expression.

Unit 9: Algebraic Expressions

67

6.

Work with your partner to rewrite the expression 5 – 6.2 using the commutative property.

9.6.3 Try It: Rewriting Equivalent Expressions 1.

Maria was asked to rewrite the following expression in a variety of ways.

5

7.

Work with your partner to rewrite the following expression in two equivalent forms using the properties of operations.

2 1 ( + 4) + 6 − (5 − 6.2) 5 2

1 3 − 4(−3.5 + 7) + ( + 5) + 8.25 4 5

She came up with four different expressions, but she made some errors. Without evaluating, decide which expressions are equivalent using properties of operations. Then, identify the errors she made.

Expression

1 3 5 − 14 − 28 + ( + 5) + 8.25 4 5

1 3 5 − 4(−3.5 + 7) + + (8.25 + 5) 4 5

68

Unit 9: Algebraic Expressions

Is it equivalent?

Justify why or why not.

9.6.4 Your Turn!

1.

1 3 5 + 14 − 28 + + (5 + 8.25) 4 5

Rewrite the following expression in two ways using the properties of operations.

12 1 1 2 3 ∙ + + ( + ) 35 2 5 5 7

3 1.25(−3.5 + 7) + (5 + ) + 8.25 5

Unit 9: Algebraic Expressions

69

Unit 9, Lesson 7: Equivalent Linear Expressions

9.6.5 Wrap-Up: Error Analysis 1.

Juan was asked to rewrite the expression 1 2 (−6 + 2 2 + 3) + 4 − 2 (−3 3) using the properties of

Benchmark

operations.

1

1

Juan wrote −6 + (3 + 2 2 + 4) + 7 3.

Determine if Juan is correct. Show your work or explain your thinking.

MA.7.AR.1.2 Determine whether two linear expressions are equivalent.

Learning Target 

70

I can verify using substitution that linear expressions are equivalent.

Unit 9: Algebraic Expressions

9.7.2 Refresher: Evaluating Algebraic Expressions

9.7.1 Warm-Up: Origins of Algebra The word algebra comes from the Arabic word ‫الجبر‬, or al-jabr, which means, “the restoring of broken parts.” The word algebra comes from the title of a book written by a Persian mathematician named Muhammad ibn Musa al-Khwarizmi in the 9th century.

1.

Evaluate each expression for the given value of 𝑥𝑥.

Based on what you know about algebraic expressions and equations, explain why you think this meaning of the Arabic word makes sense to describe algebra.

1.

−3𝑥𝑥 − 8 for 𝑥𝑥 = 5

2.

4 − 2(𝑥𝑥 2 − 7) for 𝑥𝑥 = −1

3.

(4.5 + 𝑥𝑥) ∙ 3 for 𝑥𝑥 = −

Unit 9: Algebraic Expressions

5 2

71

9.7.3 Collaboration: Evaluating Equivalent Expressions Using Substitution 1.

2.

Depending on the value being substituted into the

With your partner, complete the table by choosing who will be Partner A or Partner B to determine if the expressions are equivalent.

Partner and Value

A 𝑥𝑥 = 2

B 𝑥𝑥 = 5

A 𝑥𝑥 = −5

B 𝑥𝑥 = (−2) 72

𝟖𝟖𝟖𝟖 + 𝟒𝟒𝟒𝟒 − 𝟐𝟐

𝟐𝟐(𝟔𝟔𝟔𝟔 − 𝟏𝟏)

Complete the statement to describe the results in the last column of the table.

o always expressions, the evaluated expressions o sometimes o never had the same value as each other.

Equivalent?

3.

What do the results tell you about the expressions 8𝑥𝑥 + 4𝑥𝑥 − 2 and 2(6𝑥𝑥 − 1)?

4.

Rewrite 8𝑥𝑥 + 4𝑥𝑥 − 2 by combining the like terms.

5.

Use the distributive property to rewrite 2(6𝑥𝑥 − 1).

6.

Discuss with your partner how the two expressions verify the results from your table.

o yes o no

o yes o no

o yes o no

o yes o no

Unit 9: Algebraic Expressions

9.7.4 Guided Instruction: Determining Equivalent Expressions Using Substitution The two expressions 8𝑥𝑥 + 4𝑥𝑥 − 2 and 2(6𝑥𝑥 − 1) are equivalent, but not equal. The expressions 12𝑥𝑥 − 2 and 12𝑥𝑥 − 2 are equal.

1.

What is the difference between equivalence and equality?

Algebraic expressions are equivalent if they have different terms but are equal when evaluated using the same value for the variables. Algebraic expressions are equal if they have the same terms and operations.

2.

Determine whether 5 − 3(2𝑥𝑥 + 1) and 4𝑥𝑥 + 2 are equivalent using substitution of numerical values. a.

Complete the table. 𝒙𝒙

𝟓𝟓 − 𝟑𝟑(𝟐𝟐𝟐𝟐 + 𝟏𝟏)

𝟒𝟒𝟒𝟒 + 𝟐𝟐

What is the value when 𝑥𝑥 = 0 ?

What is the value when 1 𝑥𝑥 = ?

2

Choose another value for 𝑥𝑥 to substitute. b.

Explain if the expressions are equivalent.

Unit 9: Algebraic Expressions

73

c.

Explain why the expressions were equal when “0” was substituted for 𝑥𝑥.

3.

Determine whether 3 + 𝑥𝑥(4 − 2) and 𝑥𝑥(3 − 7) + 5 are equivalent using substitution of numerical values. a.

Complete the table. 𝒙𝒙

𝟑𝟑 + 𝒙𝒙(𝟒𝟒 − 𝟐𝟐)

𝒙𝒙(𝟑𝟑 − 𝟕𝟕) + 𝟓𝟓

What is the value when 𝑥𝑥 = 1 ?

d.

Discuss with your partner the advantages and disadvantages of using “0” to substitute for a variable to check for equivalence.

What is the value when 𝑥𝑥 = 0 ?

Choose another value for 𝑥𝑥 to substitute. b.

74

How many values are best to use when substituting values in for the variable to check for equivalency?

Unit 9: Algebraic Expressions

b.

9.7.5 Guided Practice: Determining Equivalent Expressions Using Substitution 1.

What conclusion can you draw about the expressions from the information in the table?

Consider the three expressions below. Determine whether the expressions are equivalent by substituting the given value of 𝑥𝑥. a.

Complete the table.

𝒙𝒙

𝟏𝟏 𝟒𝟒𝟒𝟒 − (𝟔𝟔 + 𝟖𝟖𝟖𝟖) 𝟐𝟐

𝑥𝑥 = 3

𝑥𝑥 = −2.75

𝟖𝟖𝟖𝟖 − 𝟑𝟑

−(𝟒𝟒𝟒𝟒 + 𝟑𝟑) + 𝟒𝟒𝟒𝟒

2.

2

Determine whether the expressions 5 (15𝑥𝑥 − 30𝑦𝑦) + 10𝑥𝑥 and 4(4𝑥𝑥 − 2𝑦𝑦) are equivalent.

𝒙𝒙 and 𝒚𝒚

𝟐𝟐 (𝟏𝟏𝟏𝟏𝟏𝟏 − 𝟑𝟑𝟑𝟑𝟑𝟑) + 𝟏𝟏𝟏𝟏𝟏𝟏 𝟓𝟓

𝟒𝟒(𝟒𝟒𝟒𝟒 − 𝟐𝟐𝟐𝟐)

𝑥𝑥 = −2 𝑦𝑦 = 0

𝑥𝑥 = ____ 𝑥𝑥 = ____

𝑦𝑦 = ____

Unit 9: Algebraic Expressions

75

9.7.6 Your Turn!

1.

A

C

9.7.7 Wrap-Up: Reflection

Determine which expressions are equivalent by substituting the same values for the variable.

(5𝑥𝑥 − 3) −

3 (8 + 2𝑥𝑥) 4

−2 9 − 6 ( 𝑥𝑥 + 4) 3

B

D

1.

1 −5 (4𝑥𝑥 − 1) 4

I can verify using substitution that linear expressions are equivalent.

3 𝑥𝑥 − 9 + 2𝑥𝑥 2 2.

E

2.

30 ( ) (−1) + 4𝑥𝑥 2

F

Rate yourself using the scale below on today’s learning target.

Write an explanation of why you rated yourself where you did.

1 3 6 − 12 𝑥𝑥 − − 8.5𝑥𝑥 2 4

Complete each statement using the expressions from the table.

_________________ is equivalent to _________________ _________________ is equivalent to _________________ _________________ is equivalent to _________________ 76

Unit 9: Algebraic Expressions

Unit 9, Lesson 8: Equivalent Expressions Using Properties of Operations

9.8.1 Warm-Up: Fact or Fiction?

1.

Determine whether each equation below is fact (true) or fiction (false).

Equation

Benchmark MA.7.AR.1.2 Determine whether two linear expressions are equivalent.

Learning Targets  

I can use properties of operations to write equivalent forms of an algebraic expression.

I can determine whether two linear expressions are equivalent.

(30 + 8) ∙ 9 = 270 + 72

4 + 5(9 − 15) = 9(9 − 15) 2 ∙ (7 + 8) = (2 ∙ 7) + 8 (5 + 4) − 10 = 5 + (4 − 10)

Unit 9: Algebraic Expressions

Fact or Fiction o Fact o Fiction o Fact o Fiction o Fact o Fiction o Fact o Fiction

77

9.8.2 Refresher: Rewriting Expressions Using Properties of Operations 1.

Using the associative property of multiplication, 2 ∙ (7 ∙ 3) can be rewritten as _____________. Both products are equal to _____________.

h.

Using the associative property of multiplication, (−6)(4𝑦𝑦) can be rewritten as _____________. Both expressions will have the same product regardless of the value of 𝑦𝑦.

Fill in the missing blanks for each property. a.

Using the commutative property of addition, 3 + 12 can be rewritten as ________________.

Both expressions have a sum of ______________.

b.

Using the commutative property of addition, 4𝑥𝑥 + 5 can be rewritten as ______________. Both expressions will have the same sum regardless of the value of 𝑥𝑥.

c.

d.

e.

f.

78

g.

i.

Using the commutative property of multiplication, (4)(−2) can be rewritten as _____________. Both expressions have a product of ____________.

Using the commutative property of multiplication, (−3𝑦𝑦)(0.5) can be rewritten as _______________. Both expressions will have the same product regardless of the value of 𝑦𝑦.

Using the associative property of addition, −5 + (6 + 1) can be rewritten as _____________.

Both sums are equal to ______________.

j.

2.

Using the distributive property of multiplication over addition, 5(6 + 2) can be rewritten as _____________. Both expressions have a value of ______________.

Using the distributive property of multiplication over addition, 7(3𝑦𝑦 + 8) can be rewritten as _____________. Both expressions will have the same product regardless of the value of 𝑦𝑦.

Discuss with your partner why the associative property and the commutative property do not hold true for subtraction and division. Give an example of each to show mathematically that these properties do not apply to subtraction and division.

Using the associative property of addition, 9𝑥𝑥 + (−2𝑥𝑥 + 5) can be rewritten as _____________. Both expressions will have the same sum regardless of the value of 𝑥𝑥.

Unit 9: Algebraic Expressions

3.

Jay gave the example 6 + −11 is equal to −11 + 6 to show that subtraction is commutative. Explain to Jay why his example does not show that subtraction is commutative.

9.8.3 Guided Instruction: Equivalent Expressions Using Properties of Operations Algebraic expressions are considered equivalent if when evaluating the expression by substituting values in for variables, the resulting values are the same.

Algebraic expressions are also considered equivalent when one expression is rewritten using one or more of the properties of operations.

1.

Complete each statement. a.

The

o associative o commutative o distributive

property can be used to

state that 5(𝑥𝑥 + 2) − 3𝑥𝑥 is equivalent to 5𝑥𝑥 + 10 − 3𝑥𝑥. b.

The

o associative o commutative o distributive 2

3

5

property can be used to 2

5

3

state that 3 𝑥𝑥 + (8 + 6 𝑥𝑥) is equivalent to 3 𝑥𝑥 + (6 𝑥𝑥 + 8).

2.

Which properties can be used to show that 2(𝑥𝑥 − 4) + 5(3 + 6𝑥𝑥) is equivalent to 2𝑥𝑥 + 30𝑥𝑥 − 8 + 15?

Unit 9: Algebraic Expressions

79

9.8.4 Guided Practice: Rewriting Expressions Using Properties of Operations 1.

Write an equivalent expression using a property of operations. State the property used.

Equivalent Expression

Expression −6(2 − 3𝑥𝑥) + 1 −

Property Used

𝑥𝑥 2

12𝑥𝑥 + 10 + 4𝑥𝑥 − 17

2.

5𝑥𝑥 + (3𝑥𝑥 + 9) + 2

Several expressions are given below. Determine which expressions are equivalent using properties of operations.

35𝑥𝑥 + (7𝑥𝑥 + 56) (8𝑥𝑥 + 7) ∙ 5

5(8𝑥𝑥 + 7) 7(5𝑥𝑥 + 8)

35𝑥𝑥 + 56

(35𝑥𝑥 + 7𝑥𝑥) + 56

Complete the table by writing the expressions that are equivalent based on the property used.

Equivalent Expressions

Property Used Associative Property Commutative Property Distributive Property

80

9.8.5 Your Turn!

Write an equivalent expression using the associative, commutative, or distributive property. State the property used.

1.

2.

3.

2

10𝑥𝑥 + 5 (−30 + 15𝑥𝑥)

(18𝑥𝑥 + 15) + 4𝑥𝑥 − 17

1

(− 5 𝑥𝑥 + 6) ∙ 5

Unit 9: Algebraic Expressions

4.

5.

5 4

7

𝑛𝑛 + 3 −

11 2

1

𝑛𝑛 − 12 3

9.8.6 Wrap-Up: Ticket Out the Door Use properties of operations to determine if the expressions are equivalent.

1.

1

−3 (2 𝑥𝑥 + 4) −

3 5

4 5

1

(𝑥𝑥 − 1) − (2𝑥𝑥 + 1) and 8

11 20

𝑥𝑥 −

37 40

(−1 + 10𝑥𝑥)

2.

1

3

10

1

𝑥𝑥 + 𝑥𝑥 −

Unit 9: Algebraic Expressions

5

1

10

− 𝑥𝑥 −

1

10

1

4

and 1 10 − 5 𝑥𝑥

81

THIS PAGE WAS INTENTIONALLY LEFT BLANK.

82

THIS PAGE WAS INTENTIONALLY LEFT BLANK.

Unit 9: Algebraic Expressions

Unit 10 - Equations and Inequalities 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10

Lesson

Exploring Inequalities - Part 1 Solving One-Step Inequalities Using Addition and Subtraction Exploring Inequalities - Part 2 Solving One-Step Inequalities Using Multiplication and Division Solving Two-Step Equations Using Properties of Equality Solving Two-Step Equations Solving Equations Involving the Distributive Property Solving Real-World Problems With Two-Step Equations Writing Two-Step Equations From Real-World Contexts Writing and Solving Equations in Real-World Context

Unit 10: Equations and Inequalities

Page Number 85 91 98 104 110 117 122 128 134 139

83

The following B.E.S.T. Standards will be covered in this unit: MA.7.AR.2.1 Write and solve one-step inequalities in one variable within a mathematical context and represent solutions algebraically or graphically.

• Clarification 1: Instruction focuses on the properties of inequality. Refer to Properties of Operations, Equality and Inequality (Appendix D). !

• Clarification 2: Instruction includes inequalities in the forms 𝑝𝑝𝑝𝑝 > 𝑞𝑞; " > 𝑞𝑞 ; 𝑥𝑥 ± 𝑝𝑝 > 𝑞𝑞; 𝑎𝑎𝑎𝑎𝑎𝑎 𝑝𝑝 ± 𝑥𝑥 > 𝑞𝑞 , where 𝑝𝑝 and 𝑞𝑞 are •

specific rational numbers and any inequality symbol can be represented.

Clarification 3: Problems include inequalities where the variable may be on either side of the inequality symbol.

MA.7.AR.2.2 Write and solve two-step equations in one variable within a mathematical or real-world context, where all terms are rational numbers.

• Clarification 1: Instruction focuses the application of the properties of equality. Refer to Properties of Operations, Equality and Inequality (Appendix D).

• Clarification 2: Instruction includes equations in the forms 𝑝𝑝𝑝𝑝 ± 𝑞𝑞 = 𝑟𝑟 𝑎𝑎𝑎𝑎𝑎𝑎 𝑝𝑝(𝑥𝑥 ± 𝑞𝑞) = 𝑟𝑟 where 𝑝𝑝, 𝑞𝑞, and 𝑟𝑟 are specific rational numbers.

• Clarification 3: Problems include linear equations where the variable may be on either side of the equal sign.

84

Unit 10: Equations and Inequalities

Unit 10, Lesson 1: Exploring Inequalities – Part 1

10.1.1 Warm-Up: Which One Doesn’t Belong?

Four inequalities are shown.

Benchmark MA.7.AR.2.1 Write and solve one-step inequalities in one variable within a mathematical context and represent solutions algebraically or graphically.

Inequality A

Inequality B

Inequality C

Inequality D

𝑥𝑥 ≥ 5

12 ≤ 7

2 > 𝑥𝑥

𝑥𝑥 > 1

1.

Javier says that Inequality B does not belong with the others. What is one reason Javier might have said this?

2.

Select a different inequality and explain why it doesn’t belong.

Learning Targets   

I can relate the solution set of an inequality to its graph on a number line.

I can relate the solution set of an inequality to the context it represents.

I can determine the minimum or maximum value that makes an inequality true.

Unit 10: Equations and Inequalities

85

10.1.2 Refresher: Representing Inequalities on a Number Line

4.

Name two values of 𝑥𝑥 that are not solutions to the inequality.

5.

Explain how you know these values are not solutions.

6.

Alexis says the graph on the number line also represents the solutions to the inequality 1 < 𝑥𝑥. Discuss with your partner whether you agree with Alexis, then write a brief summary of your discussion.

Work with your partner to complete the following.

The values that make the inequality 𝑥𝑥 > 1 true are represented on the number line shown.

1.

Select the values of 𝑥𝑥 from this list that could make the inequality 𝑥𝑥 > 1 true.      

3

7 8

7

−3 8 1 700 1.05 −16.4

The 𝑥𝑥-values that make this inequality true are called solutions; however, there are many more possible solutions than those represented on the list.

2.

Name two more values of 𝑥𝑥 that are solutions to the inequality.

3.

Explain how you know the values are solutions.

86

Unit 10: Equations and Inequalities

10.1.3 Exploration: Representing Inequalities in Context

2.

Nick is happy to realize that he is tall enough to go on the Crush ‘n’ Gusher raft ride. Nick’s friend Matt is ʹ inches shorter than Nick. Explain whether Matt is tall enough to go on the ride with Nick.

3.

Complete the statements.

Height restrictions are often in place for roller coasters and rides at amusement parks to keep guests safe. At Disney’s Typhoon Lagoon Water Park, the following height restrictions are in place.

Attraction

Height Restriction

Bay Slides

Guests must be 60 inches or shorter

Crush ‘n’ Gusher

Guests must be 48 inches or taller

Ketchakiddee Creek Guests must be 48 inches or shorter

1.

Work with your partner to determine which graph represents the height restrictions of each attraction at Typhoon Lagoon. Then, draw an arrow from each attraction to its matching graph.

Attraction

Bay Slides

Crush ‘n’ Gusher

Graph Representing Safe Heights for the Attraction

If Nick is 𝑛𝑛 inches tall, then Matt is

o 𝑛𝑛 + 2 o 𝑛𝑛 − 2

inches tall.

For Matt to be tall enough to go on Crush ‘n’ Gusher, the value of equal to

o 𝑛𝑛 + 2 o 𝑛𝑛 − 2

o 48. o 60.

o inequality o o o

must be

o less than o greater than

or

Therefore, the solutions to the

𝑛𝑛 + 2 ≤ 𝑛𝑛 + 2 ≥ 𝑛𝑛 − 2 ≤ 𝑛𝑛 − 2 ≥

o 48 o 60

represent Nick’s height

that mean Matt could also ride the Crush ‘n’ Gusher.

Ketchakiddee Creek

Unit 10: Equations and Inequalities

87

4.

On the number line, show all the possible heights that Nick could be if Matt is also tall enough to go on the ride.

10.1.4 Bring It Together: Inequalities in Mathematical and Real-World Contexts Inequalities have an infinite number of solutions. The set of all the solutions of an inequality is called the solution set.

5.

What is the minimum height Nick must be in order for Matt to be able to go on the ride as well?

The solution set of an inequality includes the values that, when substituted for the variable, make the inequality true.

One way to represent the solution set of an inequality is by shading the values on the number line, beginning with the minimum or maximum value, and including all the values to the right or left of it that are included in the solution set.

1.

Complete each statement.

A minimum or maximum value represented by an open

Julia noticed the graph also represents the solutions to the inequality 𝑛𝑛 ≥ 50.

6.

Work with your partner to explain why the solutions to the inequality 𝑛𝑛 − 2 ≥ 48 can be represented the same way on a number line as 𝑛𝑛 ≥ 50.

circle means the value is o included in o excluded from solution set.

the

A minimum or maximum value represented by a closed circle means the value is o included in o excluded from solution set.

the

Graphs representing the solution set of inequalities are helpful when interpreting them in real-world contexts.

However, sometimes there are parts of the solution set that may not make sense in a given context.

88

Unit 10: Equations and Inequalities

For example, to ride the Bay Slides at Typhoon Lagoon, guests must be 60 inches or shorter. The graph below represents the graph of ℎ ≤ 60 in this situation, where ℎ is the height of guests that can ride the Bay Slides.

2.

What does the arrow pointing to the left on the number line represent?

4.

Sometimes there are values that are part of the mathematical solution set that do not make sense in the context. In those cases, the variable may have a constraint (restriction or limitation) placed on it so that the mathematical model is realistic.

What does 60 represent in this situation?

3.

The Exploration demonstrated the inequalities 𝑛𝑛 − 2 ≥ 48 and 𝑛𝑛 ≥ 50 are equivalent because all of the same values of 𝑛𝑛 make both inequalities true. Both inequalities were also represented using the same graph.

Inequalities that have the same solution set are called equivalent inequalities.

Describe at least two values that are part of the solution set of ℎ ≤ 60 that do not make sense in this context.

Unit 10: Equations and Inequalities

89

10.1.5 Collaboration: Exploring Inequalities

4.

The graph of an inequality is shown.

Work with your partner to complete the following.

1.

Circle the inequalities that can be represented by the graph.

Match each inequality with the graph that represents all the values of 𝑦𝑦 that make it true.

Inequality

𝑚𝑚 + 1.2 < 2.2

Graph

𝑦𝑦 − 4 ≥ −1

5.

𝑦𝑦 + 4 ≥ −1

90

4 1 4

> 𝑚𝑚 +

3 4 3

> 𝑚𝑚 − 4

6.

Suppose the solution set shown on the number line represents the possible height of Krista’s feet in relation to the surface of the water in her pool when she jumps off the pool’s deck. Describe any possible constraints on the solution set in this context.

7.

Describe the minimum or maximum value that makes this inequality true given the context.

2 < 𝑦𝑦 + 7

3.

Complete the statement.

1 < 𝑚𝑚

1

The inequalities circled above are _________________ inequalities.

−2 < 𝑦𝑦 − 5

2.

1 > 𝑚𝑚

𝑚𝑚 − 1.2 < 2.2

Choose one inequality and explain how you determined which graph represents its solutions.

What is the minimum value that makes the inequality 𝑦𝑦 − 4 ≥ −1 true?

Unit 10: Equations and Inequalities

Unit 10, Lesson 2: Solving One-Step Inequalities Using Addition and Subtraction

10.1.6 Wrap-Up: Cool Down 1.

Complete the table.

The terms and information used in today’s lesson that I knew were…

Something I wondered during today’s lesson was…

Something I learned during today’s lesson was…

Benchmark MA.7.AR.2.1 Write and solve one-step inequalities in one variable within a mathematical context and represent solutions algebraically or graphically.

Learning Targets  

I can solve one-step inequalities in one variable using addition or subtraction.

I can relate properties of inequality to solving one-step inequalities algebraically.

Unit 10: Equations and Inequalities

91

10.2.1 Warm-Up: Cost Estimators

10.2.2 Exploration: Properties of Inequality

Cost estimators use data analytics to estimate the cost of manufacturing a product, constructing a building, or providing a service. They predict the amount of time, money, materials, and labor that will be needed to complete a project. This includes factoring in variables that may arise during the course of a project such as bad weather or shipping delays.

1.

How do you think cost estimators use the minimum and maximum values of inequalities in their work?

Three inequalities are shown. 𝑥𝑥 > −2.3

−

−

2.3 > 𝑥𝑥

2.3 < 𝑥𝑥

1.

Circle the inequalities that represent the same solution set.

2.

Complete the statement.

For both inequalities, all the values of 𝑥𝑥 that are ____________ than _____ make both inequalities true.

These two inequalities are equivalent by the asymmetric property of inequality.

2.

Cost estimators usually specialize in a particular product or industry. Why do you think this is helpful?

3.

The asymmetric property of inequality states that if 𝑎𝑎 > 𝑏𝑏, then 𝑏𝑏 < 𝑎𝑎.

Use the asymmetric property of inequality to write an 2 7 inequality equivalent to 𝑞𝑞 + ≤ .

5

92

Unit 10: Equations and Inequalities

10

4.

Four true inequalities are given in the table. Work with your partner to complete the table by following the steps indicated for each inequality and then answering the questions.

Given Inequality −6 < −2 −5 <

3 4

7.45 > 7 0>− 5.

2 3

Is the What is new the new inequality inequality? true?

Step Add 8.1 to both sides

10.2.3 Guided Instruction: Solving One-Step Inequalities With Addition and Subtraction 1.

𝒙𝒙

𝒙𝒙 − 𝟑𝟑

1

sides

Subtract

5 6

−4 −7

−3.2

−2 −5

−

1 2

0

1

2

−1

3

6 7

4 1

2.

For which value of 𝑥𝑥 is the equation 𝑥𝑥 − 3 = −2 true?

3.

Represent the solution to the equation 𝑥𝑥 − 3 = −2 on the number line.

4.

For which values of 𝑥𝑥 in the table is the inequality 𝑥𝑥 − 3 > −2 true?

Subtract 2 from both sides Add 3 5 to both

Complete the table to determine the value of 𝑥𝑥 − 3 for each value of 𝑥𝑥.

from

both sides

Use your work on the previous problem to complete the description of each property.

Property of Inequality Addition property of inequality

Subtraction property of inequality

Description When the same value is added to both sides of a true inequality, …

When the same value is subtracted from both sides of a true inequality, …

Unit 10: Equations and Inequalities

93

5.

Represent the values from the table that are part of the solution set of 𝑥𝑥 − 3 > −2 on the number line.

This solution set can be determined algebraically by adding the same value to both sides of the inequality.

8.

What value can be added to both sides of the inequality 𝑥𝑥 − 3 > −2 to isolate 𝑥𝑥?

The solution set of 𝑥𝑥 − 3 > −2 includes more values than shown on the previous number line.

Adding the same value to both sides of an inequality can be used to solve an inequality by the addition property of inequality.

6.

Graph the solution set for the inequality.

Addition Property of Inequality If 𝑎𝑎 > 𝑏𝑏, then 𝑎𝑎 + 𝑐𝑐 > 𝑏𝑏 + 𝑐𝑐.

The addition property of inequality states that if the same value is added to both sides of an inequality, the result will be an equivalent inequality that is also true.

7.

Write a value on the line to show an equivalent inequality represented by the graph above. 𝑥𝑥 > ______

The solution set of the inequality 𝑥𝑥 − 3 > −2 is all the values of 𝑥𝑥 that are greater than 1, or 𝑥𝑥 > 1.

94

The subtraction property of inequality can also be used to solve an inequality.

Subtraction Property of Inequality If 𝑎𝑎 > 𝑏𝑏, then 𝑎𝑎 − 𝑐𝑐 > 𝑏𝑏 − 𝑐𝑐.

The subtraction property of inequality states that if the same value is subtracted from both sides of an inequality, the result will be an ______________ inequality that is also _________.

Unit 10: Equations and Inequalities

These properties can be used to solve one-step inequalities algebraically.

9.

Use the inequality −12.25 > 𝑦𝑦 − 2.4 to complete the following. a.

Find the solution set by isolating the variable. Show your work.

b.

Graph the solution set on the number line.

c.

Use substitution to test a value or values in order to verify that the solution and the graph are correct.

10.2.4 Guided Practice: Solving One-Step Inequalities With Addition and Subtraction 1.

3

Use the inequality −4 + 𝑐𝑐 ≤ −7.4 to complete the following.

4

a.

Solve the inequality. Show your work.

b.

Graph the solution set on the number line.

c.

Use substitution to test a value or values to verify that the solution and the graph are correct.

Unit 10: Equations and Inequalities

95

2.

Greg and Bianca both solved the inequality −5.2 + 𝑥𝑥 > −8.4. Their solutions are shown.

Greg’s Solution

Bianca’s Solution

−3.2 < 𝑥𝑥

𝑥𝑥 < −3.2

Explain who is correct. Show your work.

3.

b.

Graph the solution set on the number line below.

c.

Which solutions on the number line do not make sense for the context?

Michael got an estimate on the cost to repair his mountain bike. The parts will cost $16.38 and he was told that the repair will be no more than $40.50, including labor.

The inequality 16.38 + 𝑥𝑥 ≤ 40.50 represents the situation where 𝑥𝑥 is the cost of labor. a.

96

Solve the inequality to determine the maximum amount the labor will cost.

Unit 10: Equations and Inequalities

10.2.5 Your Turn!

10.2.6 Wrap-Up: Cool Down

Solve each inequality. Then, graph its solution set.

1.

2.

𝑦𝑦 − 11 > −5 a.

Solve the inequality and show your work.

b.

Graph the solution set.

a.

b.

2

−2.25 = 𝑥𝑥 + 2

−2.25 ≤ 𝑥𝑥 + 2

1.

Describe how the processes of solving the equation and the inequality are similar.

2.

Complete the table to describe how the solutions to the equation and inequality are similar and different.

How are the solutions similar?

1

−8 ≥ 𝑥𝑥 + 7.25

An equation and an inequality are shown.

How are the solutions different?

Solve the inequality and show your work.

Graph the solution set.

Unit 10: Equations and Inequalities

97

Unit 10, Lesson 3: Exploring Inequalities – Part 2

10.3.1 Warm-Up: Visual Models An unbalanced hanger diagram is shown.

It shows that the weight of one circle is greater than the weight of one pentagon.

Benchmark MA.7.AR.2.1 Write and solve one-step inequalities in one variable within a mathematical context and represent solutions algebraically or graphically.

1.

Complete the table.

Hanger Diagram

Learning Targets   

I can determine the minimum or maximum value that makes an inequality true.

I can relate the solution set of an inequality to the context it represents.

I can explore the relationship between the coefficients of the terms in an inequality and solution set.

Description

Inequality

The weight of one circle, 𝑐𝑐, is greater than the weight of one pentagon, 𝑝𝑝. 𝑠𝑠 < 𝑝𝑝 The weight of one square, 𝑠𝑠, is __________ than the weight of a circle and a pentagon.

𝑐𝑐 + 𝑝𝑝 > ____

𝑠𝑠 + 𝑐𝑐 > 2𝑡𝑡

98

Unit 10: Equations and Inequalities

10.3.2 Exploration: Determining the Minimum or Maximum Value That Makes an Inequality True

c.

6

lunches the stage manager can order from the cafeteria?

Work with your partner to complete the following.

1.

The mathematical solution to the inequality 6𝑛𝑛 ≤ 83 5 is 𝑛𝑛 ≤ 13 . What is the maximum number of boxed

Bak Middle School of the Arts in West Palm Beach, FL is hosting a rehearsal for its upcoming musical. The stage manager has $83 from a recent fundraiser that can be used to order lunch for the cast and crew. The school’s cafeteria can make boxed lunches for a cost of $6 each.

The inequality 6𝑛𝑛 ≤ 83 represents the situation, where 𝑛𝑛 represents the number of boxed lunches that can be ordered from the cafeteria. a.

Select the values that represent a valid number of boxed lunches the stage manager can order from the cafeteria.      

b.

14 13

d.

5

What is the minimum number of boxed lunches the stage manager can order from the cafeteria?

6

12 9.5 2 − 4

Explain why some values in the previous question are not valid in this context.

Unit 10: Equations and Inequalities

99

e.

The graph of the solution set for 6𝑛𝑛 ≤ 83 is shown.

b.

a.

100

≤ 0.75 can be used to represent

Use your answer to Part A to determine the solution 𝑑𝑑 set for ≤ 0.75.

3.7

c.

Hiromi walks around his neighborhood each afternoon, tracking his pace each day. The fastest rate Hiromi can walk for an extended period of time is 3.7 miles per hour.

𝑑𝑑

3.7

the distance Hiromi walks, 𝑑𝑑, in 0.75 hours given his fastest walking rate of 3.7 miles per hour.

Discuss with your partner how the graph that mathematically represents the solution set is not the best representation of the real-world context.

Then, represent the solutions in the real-world context on the number line shown.

2.

The inequality

The graph of the solution set for

𝑑𝑑

3.7

≤ 0.75 is shown.

Discuss with your partner how the graph that mathematically represents the solution set is not the best representation of the real-world context.

Then, represent the solutions in the real-world context on the number line.

What is the maximum distance Hiromi can walk in 0.75 hours?

Unit 10: Equations and Inequalities

10.3.3 Bring It Together: Solutions and Mathematical Models in Context Each of the inequalities in the Exploration has a mathematical solution set that can be represented on a number line. These graphs are examples of mathematical models.

Mathematical models are helpful tools used to understand a situation and make predictions.

However, sometimes the mathematical model can include parts that do not make sense in a real-world context. When interpreting models to solve problems, it is important to analyze whether aspects of the model make sense in a given context.

10.3.4 Exploration: Matching Inequalities to Solutions 1.

Complete the tables by substituting each value of 𝑥𝑥 to determine if it is a solution to each inequality.

Value of 𝒙𝒙 −10

What is the value of 𝟑𝟑𝟑𝟑?

−7.4 1

28 6

Value of What is the 𝒙𝒙 value of −𝟑𝟑𝟑𝟑? −10

−7.4 2

1 8

6

Unit 10: Equations and Inequalities

Is 𝟑𝟑𝟑𝟑 ≥ −𝟏𝟏𝟏𝟏 Solution? true or false? o Yes o No o Yes o No o Yes o No o Yes o No Is −𝟑𝟑𝟑𝟑 ≥ 𝟏𝟏𝟏𝟏 Solution? true or false? o Yes o No o Yes o No o Yes o No o Yes o No

101

2.

The solution sets for the inequalities 3𝑥𝑥 ≥ −15 and − 3𝑥𝑥 ≥ 15 are given in the table. Write the inequality that matches each graph of the solution set.

Inequality

3.

Graph of the Values of 𝒙𝒙

−8.2

What is the 𝒙𝒙 value of ? −𝟏𝟏

Is

𝒙𝒙

−𝟏𝟏

≤ 𝟓𝟓 true

or false?

−6 0

14 3

Value of 𝒙𝒙 −10

−7.4 2

1 8

6

102

What is the 𝒙𝒙 value of ? 𝟐𝟐

Is

𝒙𝒙

𝟐𝟐

≤ −𝟐𝟐. 𝟓𝟓

true or false?

Yes No Yes No Yes No Yes No

Solution? o o o o o o o o

𝑥𝑥

−1

≤ 5 and

𝑥𝑥

2

≤ −2.5

are given in the table. Write the inequality that matches each graph of the solution set.

Graph of the Values of 𝒙𝒙

5.

Look back at your answers to questions 2 and 4.

What do you notice about the original inequality symbol and the direction of the arrows on the graphs of the solution sets?

6.

Compare your work on questions 1-5 with your partner.

Discuss any differences and revise your work if necessary.

Solution? o o o o o o o o

The solution sets for the inequalities

Inequality

Complete the tables by substituting each value of 𝑥𝑥 to determine if it is a solution to each inequality.

Value of 𝒙𝒙

4.

Yes No Yes No Yes No Yes No

Unit 10: Equations and Inequalities

7.

The graphs of two solution sets are shown.

Graph A Graph B

10.3.5 Wrap-Up: Cool Down 1.

The solution set for each inequality in the table matches one of the graphs. Based on your work and observations thus far, work with your partner to determine which graph matches each inequality.

Inequality

Make a conjecture about the relationship between the inequality symbol and the sign of the coefficient of the variable term when determining the solution set for an inequality. Use examples from the lesson if they’re helpful.

Which Graph?

−2𝑝𝑝 > 4

3.8𝑝𝑝 > −7.6 𝑝𝑝 3

>−

𝑝𝑝

2 3

− 5 > 0.4

8.

Explain how you and your partner determined which graphs match each inequality.

Unit 10: Equations and Inequalities

103

Unit 10, Lesson 4: Solving One-Step Inequalities Using Multiplication and Division

10.4.1 Warm-Up: Bell Work 1.

Benchmark

Using substitution, determine whether the given numerical value is part of the solution set of the inequality. Show your work. a.

Is 𝑛𝑛 = −5.5 part of the solution set for −4𝑛𝑛 > 20?

b.

Is 𝑦𝑦 = 8 part of the solution set for −6 ≤ − 𝑦𝑦?

c.

Is 𝑏𝑏 = 2.7 part of the solution set for −9.6 > −12.3 + 𝑏𝑏?

MA.7.AR.2.1 Write and solve one-step inequalities in one variable within a mathematical context and represent solutions algebraically or graphically.

Learning Targets  

I can solve one-step inequalities in one variable using multiplication or division.

I can relate properties of inequality to solving one-step inequalities algebraically.

2.

3 4

Explain how to determine whether a value is part of the solution set for an inequality.

104

Unit 10: Equations and Inequalities

10.4.2 Collaboration: The Effect of a Negative Coefficient 1.

2.

Work with your partner to complete the following. a.

Two simple inequalities are shown.

a.

b.

1>0

and

−1 > 0

Circle the inequality that is true and cross out the

inequality that is false.

d.

0 < −𝑞𝑞

0 ____ 𝑞𝑞

0 ____ 𝑚𝑚 𝑤𝑤 ____ 0

In each equivalent inequality, the sign of the variable changed to its opposite and the value 0 remained the same.

_____________ than 0.

b.

Describe which operation(s) could be applied to both sides of the given inequalities above to result in the signs of the variables changing and the value 0 remaining the same.

c.

Describe what happens to the comparison symbol in the inequality when either of these operations are applied.

Write the correct inequalities by filling in a comparison symbol on each line. and

Equivalent Inequality

−𝑤𝑤 ≤ 0

Complete the statement.

0 _______ 1

Given Inequality

0 > −𝑚𝑚

If 1 is greater than 0, then the opposite of 1 is

c.

Complete the inequalities in the table based on your understanding of opposites.

0 ______ − 1

Verify your answers with your partner on parts A-C and, if necessary, resolve any differences.

Unit 10: Equations and Inequalities

105

10.4.4 Guided Instruction: Solving One-Step Inequalities With Multiplication and Division

10.4.3 Bring It Together: The Effect of a Negative Coefficient 1.

Complete the statements using words from the bank shown. Some words may not be used.

adding

subtracting multiplying dividing opposite

When the variable term in an inequality is negative, the inequality can be rewritten by _________________ or _________________ the inequality by −1. In the

resulting inequality, each term and the comparison symbol will change to its _________________.

2.

Use the inequality −𝑛𝑛 ≥ 2 to complete the following. a.

b.

Complete the inequality to create an equivalent inequality. 𝑛𝑛 ____

As with solving one-step inequalities with addition and subtraction, properties of inequality can be used to solve inequalities with multiplication and division. As discovered in the Collaboration, though, when the value that is being multiplied or divided is negative, this changes both the values in the inequality and the comparison symbol to their opposites.

Multiplying both sides of an inequality by the same value can be used to solve an inequality by the multiplication property of inequality.

1.

Multiplication Property of Inequality If 𝑎𝑎 > 𝑏𝑏 and 𝑐𝑐 > 0, then 𝑎𝑎 × 𝑐𝑐 _____ 𝑏𝑏 × 𝑐𝑐.

If 𝑎𝑎 > 𝑏𝑏 and 𝑐𝑐 < 0, then 𝑎𝑎 × 𝑐𝑐 _____ 𝑏𝑏 × 𝑐𝑐.

Find the solution to both inequalities using the multiplication property of inequality. 𝑦𝑦 4

≤ 2.1

Graph the solution set for −𝑛𝑛 ≥ 2.

106

Unit 10: Equations and Inequalities

𝑦𝑦

−4

≤ 2.1

Dividing both sides of an inequality by the same value can be used to solve an inequality by the division property of inequality.

10.4.5 Guided Practice: Solving One-Step Inequalities With Multiplication and Division 1.

Division Property of Inequality

3

Use the inequality 7 ≥ − 𝑦𝑦 to complete the following.

2

a.

Solve the inequality. Show your work.

b.

Graph the solution set on the number line.

c.

Use substitution to test a value or values to verify that the solution and the graph are correct.

If 𝑎𝑎 > 𝑏𝑏 and 𝑐𝑐 > 0, then 𝑎𝑎 ÷ 𝑐𝑐 _____ 𝑏𝑏 ÷ 𝑐𝑐.

If 𝑎𝑎 > 𝑏𝑏 and 𝑐𝑐 < 0, then 𝑎𝑎 ÷ 𝑐𝑐 _____ 𝑏𝑏 ÷ 𝑐𝑐.

2.

Find the solution to both inequalities using the division property of inequality. −25 > 5𝑐𝑐

−25 > −5𝑐𝑐

Unit 10: Equations and Inequalities

107

2.

Use the inequality a.

𝑘𝑘

3

< −5 to complete the following.

3.

Solve the inequality. Show your work.

Gabriel and Lauren made errors when solving the inequality −4𝑥𝑥 ≤ 12. Their work is shown.

Gabriel’s Work

a.

Explain which mistake(s) each student made.

Gabriel’s Error

b.

Graph the solution set on the number line.

c.

Use substitution to test a value or values to verify that the solution and the graph are correct.

b.

108

Lauren’s Work

Lauren’s Error

Find the correct solution to −4𝑥𝑥 ≤ 12.

Unit 10: Equations and Inequalities

10.4.6 Your Turn!

1.

2.

𝑛𝑛

b.

5.5 ≤ −

c.

−3.6𝑦𝑦 ≤ −17.1

8

Frankie got a summer job as a lifeguard at the local pool, where he will earn $9.50 per hour. He wants to make at least $299 to buy the newest gaming system.

The inequality 299 ≤ 9.50𝑥𝑥 mathematically represents this situation where 𝑥𝑥 is the number of hours Frankie works. Find the solution to the inequality to determine the minimum number of hours Frankie must work to save enough money for the gaming system. Round your answer to the nearest hundredth.

Solve each inequality. Then, graph its solution set. a.

9𝑥𝑥 > −

3 4

Unit 10: Equations and Inequalities

109

10.4.7 Wrap-Up: Cool Down

Two inequalities are shown.

2.5𝑥𝑥 ≥ −10 1.

−2.5𝑥𝑥 ≥ −10

Complete the table to describe how the process of solving each inequality is similar and different.

How is the process of How is the process of solving each inequality solving each inequality similar? different?

Benchmark MA.7.AR.2.2 Write and solve two-step equations in one variable within a mathematical context or real-world context, where all terms are rational numbers.

Learning Targets  

110

Unit 10, Lesson 5: Solving Two-Step Equations Using Properties of Equality

I can relate properties of equality to solving two-step equations.

I can solve two-step equations in one variable.

Unit 10: Equations and Inequalities

10.5.1 Warm-Up: Notice and Wonder

2.

Explain what is happening in this situation using the words “equal,” “greater than,” and/or “less than.”

A math teacher and her daughter experimented with a seesaw at her local park. Their goal was to determine how the daughter could play on the seesaw on her own. She placed a crate on the other side of the seesaw and began adding bricks to it one at a time. Each time she added a brick, she tried to seesaw with the crate. A series of screenshots of her experiment are shown.

Seesaw with 𝟎𝟎 Bricks

Seesaw after adding 𝟏𝟏 Brick and a Crate

Seesaw after adding 𝟏𝟏𝟏𝟏 Bricks and a Crate

1.

Complete the table to indicate one thing you notice and one thing you wonder when looking at this situation.

One thing I notice is…

One thing I wonder is…

Unit 10: Equations and Inequalities

111

10.5.2 Exploration: Properties of Equality 1.

b.

What do you notice about each of the resulting equations?

c.

With your partner, compare the results of multiplying each term in the original equation, 1 − 6 + 10 = 4, by and dividing each term by 2.

The equation −6 + 10 = 4 is used in the table to show different operations. a.

Work with your partner to complete the table.

Operation

+ − ×

÷ 112

Step Add 5 to both sides of the equal sign

Show Step

−

Resulting Equation

2

Summarize your observations.

6 + 10 + 5 = 4 + 5 _____ = _____

Subtract 8 from both sides of the equal sign 2.

Multiply the terms on both sides of the equal sign by ½

Divide the terms on both sides of the equal sign by 2

Complete the statement.

Dividing by 4 is the same as multiplying by

3.

.

1

Use the equation 5 + 8 = 13.5 to complete the following. a.

2

Discuss with your partner why the equation could 1 also be written 13.5 = 5 2 + 8. Summarize your discussion.

Unit 10: Equations and Inequalities

b.

Work with your partner to complete the table for 1 the equation 5 + 8 = 13.5.

2

Operation

+

−

×

÷

Step

Show Step

Prisha said it is not possible to complete the last row of the table. Explain why Prisha said it is not possible.

d.

What do you notice about the resulting equations in the first three rows of the table?

Resulting Equation

Add 23 to the left side of the equation and 14 to the right side Subtract 5 from the left of the equation and 10 from the right side

c.

4.

Complete the statements.

An equation compares two equivalent expressions using

Multiply the terms on the left side of the equation by 3 and those on the right by −1

Divide the left side of the equal sign by 0 and the right side of the equal sign by 0.5

the mathematical symbol = to show the values of both quantities are

o equal. o unequal.

When the same operation

is performed on both sides of an equation, the quantities

o remain o are no longer

equivalent. When the

operations performed on both sides of an equation are not the same, the resulting equation is

o still o no longer

true because the quantities being compared are o still equivalent. o no longer Unit 10: Equations and Inequalities

113

10.5.3 Bring It Together: Properties of Equality

Properties of equality can be used to solve equations.

1.

Complete the table with a numeric example of each property of equality.

Property Symmetric Property of Equality Addition Property of Equality Subtraction property of Equality Multiplication Property of Equality Division Property of Equality

Algebraic Statement

Numeric Example

If 𝑎𝑎 = 𝑏𝑏, then 𝑏𝑏 = 𝑎𝑎.

If 𝑎𝑎 = 𝑏𝑏, then 𝑎𝑎 + 𝑐𝑐 = 𝑏𝑏 + 𝑐𝑐.

When solving equations, the goal is to isolate the variable to find the value(s) of the variable that make the equation true.

The operations being performed on the variable can be “undone” using inverse operations and opposites in order to isolate the variable. Properties of equality can be applied to create equivalent equations using steps to isolate the variable.

1. Consider the equation 5𝑥𝑥 a.

If 𝑎𝑎 = 𝑏𝑏, then 𝑎𝑎 − 𝑐𝑐 = 𝑏𝑏 − 𝑐𝑐.

− 9.6 = 21.4.

Complete the statement to describe the equation.

The variable, 𝑥𝑥 , is first being

________________ by 5, and then 9.6 is being

If 𝑎𝑎 = 𝑏𝑏, then 𝑎𝑎 ∙ 𝑐𝑐 = 𝑏𝑏 ∙ 𝑐𝑐.

________________ from this product. The value of the expression, 5𝑥𝑥

If 𝑎𝑎 = 𝑏𝑏 and 𝑐𝑐 ≠ 0, then 𝑎𝑎 ÷ 𝑐𝑐 = 𝑏𝑏 ÷ 𝑐𝑐.

2.

Explain why 𝑐𝑐 ≠ 0 for the division property of equality.

3.

Which property justifies the equation 2𝑚𝑚 − 7.3 = 24.45 is equivalent to 24.45 = 2𝑚𝑚 − 7.3?

114

10.5.4 Guided Instruction: Solving Two-Step Equations Using Properties of Equality

b.

− 9.6, is _________ to 21.4.

Describe the operation that could be performed to isolate 5𝑥𝑥 on the left side of the equation.

Unit 10: Equations and Inequalities

c.

Show this step to create an equivalent equation.

h.

Which properties of equality were applied to solve this equation?

i.

Cameron said the equation 5𝑥𝑥 = 31 could have been solved by applying the multiplication property of equality. What value can be used as the factor that is multiplied to both sides of the equation to isolate 𝑥𝑥?

5𝑥𝑥 − 9.6 + _______ = 21.4 + _______ _________ = _________

d.

e.

Describe the operation that could be performed to isolate the 𝑥𝑥 on the left side of the equation.

Show this step to create an equivalent equation.

5𝑥𝑥 ÷ _____ = 31 ÷ _____ 𝑥𝑥 = _______

f.

What is the coefficient of the 𝑥𝑥 after dividing both sides?

g.

Use substitution in the original equation to verify the solution.

5𝑥𝑥 − 9.6 = 21.4

Unit 10: Equations and Inequalities

115

2.

Solve −6.7 = a.

𝑦𝑦

−4

1

Fill in each box with an operation and each blank with a value. 𝑦𝑦 1 −6.7 = − − 4

−6.7

_______

b.

_______ =

_______ =

𝑦𝑦

−4

𝑦𝑦

−

4

116

2

−

1 2

_______ = _______

Work with your partner to complete the following.

1.

_______

Use the equation 4.76 = −5 + 3.2𝑛𝑛 to complete the following. a.

Apply properties of equality to solve the equation.

b.

Use substitution to verify the solution.

𝑦𝑦

−4

_______ = 𝑦𝑦

Use substitution in the original equation to verify the solution. 𝑦𝑦 1 −6.7 = − − 4

c.

10.5.5 Try It: Solving Two-Step Equations Using Properties of Equality

− 2 by applying properties of equality.

2

Which properties of equality were applied to solve this equation?

2.

5

3

Use the equation 7 𝑟𝑟 + 6 7 = −8 to complete the following. a.

Apply properties of equality to solve the equation.

b.

Use substitution to verify the solution.

Unit 10: Equations and Inequalities

10.5.6 Wrap-Up: Reflection Complete the statements as you reflect on today’s lesson.

1.

Today I am still unsure about ____________________ ____________________________________________.

2.

To fill in this gap I intend to _____________________

Unit 10, Lesson 6: Solving Two-Step Equations Benchmark

MA.7.AR.2.2 Write and solve two-step equations in one variable within a mathematical context or real-world context, where all terms are rational numbers.

____________________________________________.

Learning Targets  

I can solve two-step equations in one variable.

I can relate the solution of an equation representing a real-world context to the context.

Unit 10: Equations and Inequalities

117

10.6.1 Warm-Up: Bell Work

10.6.2 Collaboration: Examining Student Work

Michaela solved the equation 2.9 − 𝑟𝑟 = −5.11. Her work is shown.

Wilma and Maurio solved the equation 42 = −3𝑥𝑥 + 15.6 by applying the subtraction property of equality and the division property of equality. Their work is shown.

1.

Wilma’s Work

1.

Use substitution to verify if Michaela’s solution is correct.

42 = −3𝑥𝑥 + 15.6

42 = −3𝑥𝑥 + 15.6

26.4 = −3𝑥𝑥

−

42 − 15.6 = −3𝑥𝑥 + 15.6 − 15.6 26.4 −3

=

−

3𝑥𝑥

−3

____ = 𝑥𝑥

2.

If Michaela found the correct solution, describe the properties of equality applied. If Michaela did not find the correct solution, describe the error(s) she made.

42

−3 −

=

−

3𝑥𝑥

−3

+

15.6 −3

14 = 𝑥𝑥 − 5.2

14 + 5.2 = 𝑥𝑥 − 5.2 + 5.2

____ = 𝑥𝑥

a.

Complete the last step in each student’s work.

Then, check your work with your partner.

b.

Discuss the similarities and differences between each students’ work with your partner. Summarize your discussion in the table.

How are the students’ work similar?

118

Maurio’s Work

Unit 10: Equations and Inequalities

How are the students’ work different?

Notice that when division property of equality is applied first, each term in the equation must be divided by the same value to maintain equivalence.

2.

Several students made errors when solving equations.

Their work is shown in the table.

Work with your partner to find the error in each student’s work. Circle the step where the student made the error. Then, work together to complete the table.

Student Work

Describe the Error

Correct Solution

10.6.3 Guided Instruction: Relating Equations and Their Solutions to Real-World Contexts Two-step equations can be used to represent real-world situations when solving problems.

Dan’s family recently adopted a dog, so they are putting in a new fence in their backyard. Dan found fencing on sale at a local hardware store, but the store only had 80 linear feet (ft) of the fencing left in stock. Dan knows the backyard is 48.6 ft long and wants to determine how wide the fenced-in portion of the yard would be if he used 80 linear ft of fencing. The dotted line in the figure shows how the fence will be constructed to enclose the backyard.

48.6 ft 𝑤𝑤 ft

1.

𝑤𝑤 ft

The equation 48.6 + 2𝑤𝑤 = 80 can be used to determine how wide the fenced-in portion of the yard will be. a.

Solve 48.6 + 2𝑤𝑤 = 80. Show your work.

Unit 10: Equations and Inequalities

119

b.

Verify your solution is correct.

c.

Explain what the solution means in this context.

In most countries, temperatures are measured in degrees Celsius, while in the United States, the Fahrenheit scale is 𝐹𝐹−32 used. The equation 𝐶𝐶 = 1.8 can be used to convert between

b.

Verify your solution is correct.

c.

Explain what the solution means in this context.

these scales, where 𝐶𝐶 is the temperature in degrees Celsius (°C) and 𝐹𝐹 is the temperature in degrees Fahrenheit (°F).

2.

The equation 25 =

𝐹𝐹−32 1.8

can be used to find the

temperature in °F when it is 25°C. a.

120

Solve the equation to find the value of 𝐹𝐹.

Unit 10: Equations and Inequalities

2.

10.6.4 Your Turn!

1.

1

Solve −10 3 +

𝑥𝑥

6

8

= − 3 using the properties of equality.

Then, verify your solution is correct.

The basketball team at C.H. Price Middle School in Putnam County, Florida sold candy bars to raise money for new Razorbacks uniforms. The coach spent $78.95 on supplies for the fundraiser and team members sold each candy bar for $1.50. When the fundraiser ended, the team had $285.55 to spend on new uniforms after paying the coach back for the supplies she bought. The equation 1.5𝑐𝑐 − 78.95 = 285.55 can be used to represent this situation. a.

Solve the equation to find the value of 𝑐𝑐.

3.

b.

Verify your solution is correct.

c.

Explain what the solution means in this context.

Solve −9.2 =

𝑚𝑚+7.22 4

using the properties of equality.

Then, verify your solution is correct.

Unit 10: Equations and Inequalities

121

10.6.5 Wrap-Up: Growth Mindset 1.

Mistakes are an opportunity to learn. Describe your favorite mistake from today’s lesson which could be a mistake you made or observed that helped you learn.

I

Start

Acquire

Knowledge Experience Skills

122

Unit 10, Lesson 7: Solving Equations Involving the Distributive Property Benchmark

MA.7.AR.2.2 Write and solve two-step equations in one variable within a mathematical context or real-world context, where all terms are rational numbers.

Means

To

Learning Target



I can solve two-step equations involving the distributive property.

Unit 10: Equations and Inequalities

10.7.1 Warm-Up: Two-Step Equations 1.

Create an equation with the largest possible value for 𝑥𝑥 by filling in the blue boxes using the digits 1 through 9.

Each digit can only be used one time, at most. This may take several attempts. Show your work for each attempt you make.

𝑥𝑥 −

10.7.2 Refresher: The Distributive Property 1.

Several expressions are shown. (−2 ∙ 𝑥𝑥) + 3

(𝑥𝑥 + 3)(−2)

(−2) ∙ 𝑥𝑥 + (−2) ∙ 3

=

a. b.

2.

−2𝑥𝑥 − 6

(−2 + 𝑥𝑥)3

Circle the expressions that are equivalent to −2(𝑥𝑥 + 3).

Use substitution to show that each of the expressions chosen is equivalent to −2(𝑥𝑥 + 3).

Rewrite each expression by applying the distributive property. a.

(−𝑥𝑥 + 6)4

b.

−3.5(−7 + 2𝑦𝑦)

c.

7

4

(16𝑚𝑚 − 5) 8

Unit 10: Equations and Inequalities

123

10.7.3 Collaboration: Equivalent Equations 1.

All of the equations in the table below are equivalent.

Work with your partner to explain how each equation is equivalent to the previous equation.

Equation

𝑥𝑥 = −6 𝑥𝑥 − 3 = −9 −9 = 𝑥𝑥 − 3

How is this equation equivalent to the equation above it?

2.

Two students wrote equations equivalent to 1 2 (𝑚𝑚 + 1 4) = 3.5.

Simon’s Equation 1

2𝑚𝑚 + 2 = 3.5 2

1

𝑚𝑚 + 1 = 1.75 4

a.

Describe what Simon did to generate his equivalent equation.

b.

Describe what Daphne did to generate her equivalent equation.

This is the given equation.

900 = −100(𝑥𝑥 − 3) 900 = −100𝑥𝑥 + 300

124

Daphne’s Equation

Unit 10: Equations and Inequalities

10.7.4 Guided Instruction: Solving Equations Involving the Distributive Property

b.

Verify the solution is correct using substitution.

c.

Discuss each method of solving the equation with your partner. Then, explain which method you preferred for solving this equation.

Equations can be solved in more than one way. When an equation is written in the form 𝑎𝑎(𝑏𝑏 + 𝑐𝑐) = 𝑑𝑑, it can be solved by either distributing 𝑎𝑎 first or by dividing both sides the equation by 𝑎𝑎 first.

1.

The equation 65 = 8(9 − 𝑝𝑝) can be solved using either method. a.

Solve the equation 65 = 8(9 − 𝑝𝑝) using each method.

Method 1 Apply the distributive property first

Method 2 Divide both sides by the factor being distributed first

65 = 8(9 − 𝑝𝑝)

65 = 8(9 − 𝑝𝑝)

Unit 10: Equations and Inequalities

125

2.

With your partner, decide who is Partner A and who is Partner B. a.

Partner A Apply the distributive property first 1

2

8 = (𝑥𝑥 + 6) 2

c.

126

2

Solve the equation 8 = (𝑥𝑥 + 6) using the method indicated.

b.

1

3.

3

2

3

The perimeter of an equilateral triangle is 41.46 inches.

If the length of each side is (𝑞𝑞 − 5.7), the equation 3(𝑞𝑞 − 5.7) = 41.46 can be used to find the value of 𝑞𝑞. a.

What is the value of 𝑞𝑞?

b.

The length of each side of the triangle is 13.82 inches because 41.46 ÷ 3 = 13.82. Explain how the solution to the equation 3(𝑞𝑞 − 5.7) = 41.46 can be used to show this as well.

Partner B Divide both sides by the factor being distributed first 1 2 8 = (𝑥𝑥 + 6) 2

3

Share your solution and your work with your partner. Correct any errors, if necessary. Then, write your partner’s method in the space provided once both partners agree the work is correct.

Explain which method you preferred for solving this equation.

Unit 10: Equations and Inequalities

10.7.5 Guided Practice: Solving Equations Involving the Distributive Property Solve each equation using your preferred method. Then, verify that your solution is correct using substitution.

1.

2.

3 4

1

5

10.7.6 Collaboration: Error Analysis

Work with your partner to complete the following.

1.

(𝑘𝑘 − 5 2) = 8

Francesca was asked to solve the equation 1 1 −4 = (𝑥𝑥 − 1.5). Her work is shown.

2

3

1

1

−4 = (𝑥𝑥 − 1.5) 2

3

1

−4.5 + 1.5 = (𝑥𝑥 − 1.5) + 1.5 1

−2 = 𝑥𝑥 −2 1 3

−3 = (𝑤𝑤 − 1.7)(−10)

=

3

3

1 𝑥𝑥 3 1 3

𝑥𝑥 = −6

a.

Draw an arrow next to the step(s) where Francesca made a mistake.

b.

Describe the mistake(s) Francesca made.

c.

Find the solution to −4 = (𝑥𝑥 − 1.5).

Unit 10: Equations and Inequalities

1 2

1 3

127

10.7.7 Wrap-Up: Ticket Out the Door 1.

2

Solve 5 (𝑥𝑥 + ) = 5

−

13.

Unit 10, Lesson 8: Solving Real-World Problems with Two-Step Equations

Benchmark

MA.7.AR.2.2 Write and solve two-step equations in one variable within a mathematical context or real-world context, where all terms are rational numbers.

Learning Targets  

I can solve two-step equations within a real-world context.

I can relate the solution of an equation representing a real-world context to the context.

128

Unit 10: Equations and Inequalities

10.8.2 Collaboration: Matching Equations to Diagrams and Contexts

10.8.1 Warm-Up: Number Patterns A pattern is shown.

Equations can be represented using bar diagrams.

1.

A bar diagram is shown.

18 1.

𝑥𝑥 + 5

Describe how the pattern is changing in each step. a.

How many circles will be in the next step?

𝑥𝑥 + 5

Select all the equations that match the diagram.     

2.

𝑥𝑥 + 5

𝑥𝑥 + 5 = 18 18 ÷ 3 = 𝑥𝑥 + 5 3(𝑥𝑥 + 5) = 18 1 𝑥𝑥 + 5 = ⋅ 18 3 3𝑥𝑥 + 5 = 18

b.

Compare your selections with your partner. Resolve any differences if necessary.

c.

With your partner, choose one of the equations that can be modeled by the bar diagram and explain why it matches the diagram.

Unit 10: Equations and Inequalities

129

d.

Explain why 𝑥𝑥 + 5 = 18 does not match the bar diagram.

b.

With your partner, discuss what the variable 𝑥𝑥 represents in each of the equations. Then, write the meaning of 𝑥𝑥 in each equation based on the context it matches.

Equation

Equations can be used to represent real-world contexts.

2.

1

A: + 4𝑥𝑥 = 8 4

Three equations are shown.

Equation A 1 4

a.

+ 4𝑥𝑥 = 8

Equation B 1

4 + 𝑥𝑥 = 8 4

Equation C 1

8𝑥𝑥 + = 4 4

1

B: 4 + 𝑥𝑥 = 8

Work with your partner to determine which equation matches each context. Then, write each equation in the appropriate row.

Context A stack of nested paper cups is 8 inches tall. The first cup is 4 inches tall and each of the rest 1 of the cups in the stack adds

4

Equation

1

C: 8𝑥𝑥 + 4 = 4

4

inch to the height of the stack.

A baker uses 4 cups of flour.

1 She uses 4 cup to cover the

counter with flour so the dough doesn’t stick, and the rest to make 8 identical loaves of bread.

Elena has an 8-foot piece of ribbon. She cuts off a piece that 1 is of a foot long and cuts the

4

remainder into four pieces of equal length.

130

Unit 10: Equations and Inequalities

Meaning of 𝑥𝑥

10.8.3 Guided Instruction: Relating Equations to Their Contexts Olivia had her birthday party at a trampoline park. It cost $48.50 to reserve a private jumping area for their group and $15.72 per person for food, drinks, and unlimited jump time.

Olivia’s parents paid $315.74 for Olivia and her friends to attend the party.

1.

2.

The total cost for Olivia’s birthday party can be represented by the equation 315.74 = 15.72𝑛𝑛 + 48.50.

3.

Work with your partner to answer the following questions. a.

What does 𝑛𝑛 represent in the equation?

b.

What does the variable term, 15.72𝑛𝑛, represent in the equation?

c.

What does the constant term, 48.50, represent in the equation?

What was the total amount of money spent on the party?

One of the costs depends on the number of people attending the party, while the other does not. a.

b.

Which cost depends on the number of people who attend the party?

Which cost is the same regardless of how many people attend the party?

Unit 10: Equations and Inequalities

131

4.

Solve the equation 315.74 = 15.72𝑛𝑛 + 48.50.

10.8.4 Guided Practice: Relating Equations to Their Contexts 1.

Franco is saving money to take a trip to Cocoa Beach, Florida with his friends. Every week, he sets aside the same portion of his paycheck after taking out the $325 he needs for his weekly expenses. After 11 weeks, Franco has $489.94 saved for his trip. a.

( 5.

Number of weeks

If Olivia was included as one of the people attending the party, determine how many of her friends attended.

)(

Amount of weekly paycheck

Amount of − weekly expenses

)

Which of these quantities is unknown?

Total

= Saved

b.

Write the equation that represents this situation using the variable 𝑝𝑝 for the unknown quantity.

c.

What does (𝑝𝑝 − 325) represent in the equation?

132

An equation can be created relating these quantities as described below.

Unit 10: Equations and Inequalities

d.

Solve the equation.

10.8.5 Try It: Relating Equations to Their Contexts Work with your partner to complete the following.

1.

Two equations are shown. a.

Solve both equations. Show your work.

Equation A 46.92 = 6(𝑦𝑦 − 1.75)

e.

Equation B 6𝑦𝑦 − 1.75 = 46.92

Explain what the solution means in this context.

b.

A family of 6 is going to the local fair. They have a coupon for $1.75 off of each ticket. They pay $46.92 for all their tickets. Explain which equation represents this situation.

c.

Explain what the solution to the equation that correctly models this situation means.

Unit 10: Equations and Inequalities

133

10.8.6 Wrap-Up: Reflection 1.

Circle the value on the emoji scale that best represents your understanding of today’s lesson.

Briefly explain your choice.

Learning Targets  

134

Benchmark

MA.7.AR.2.2 Write and solve two-step equations in one variable within a mathematical context or real-world context, where all terms are rational numbers.

2.

Unit 10, Lesson 9: Writing Two-Step Equations from Real-World Contexts

I can write two-step equations to represent real-world contexts.

I can solve two-step equations within a real-world context.

Unit 10: Equations and Inequalities

10.9.2 Guided Instruction: Writing Two-Step Equations From Real-World Contexts

10.9.1 Warm-Up: Bell Work

Tremaine solved the equation −10 = −8.5 + 2𝑥𝑥 and checked his solution. His work and check are shown.

Work

Check

When writing equations to represent a real-world context, it is helpful to start by considering the known and unknown quantities involved in the situation.

1.

Amanda is draining her pool for resurfacing. The pool began with 12,340 gallons of water and it is draining at a rate of 780 gallons per hour. How long has the pool been draining if currently there are 5,125 gallons in the pool? a.

Complete the table to describe the known and unknown quantities in this situation.

Known Quantities

Unknown Quantities

Tremaine knows his solution is incorrect, but he is not sure where he went wrong.

1.

Explain how Tremaine knows his solution is incorrect.

b.

2.

Describe the error he made.

3.

Find the correct solution to the equation −10 = −8.5 + 2𝑥𝑥. Show your work.

With your partner, discuss how the quantities are related in this situation.

When writing equations to represent real-world contexts, any variables used must be defined to identify which quantity each variable in the equation represents.

Unit 10: Equations and Inequalities

135

c.

Define the unknown quantity using a variable.

2.

Six friends met for a book club at a local coffee shop.

Each person ordered a small cup of coffee and a muffin.

The total for the order, including taxes, was $37.56. If the cost of a small coffee is $3.79 after taxes, find the cost of one muffin after taxes. a.

d.

Write an equation that could be used to find the unknown quantity using the variable you defined.

e.

Solve the equation.

f.

136

Explain what the solution means in this context.

Complete the table to describe the known and unknown quantities in this situation.

Known Quantities

Unknown Quantities

b.

Define the unknown quantity using a variable.

c.

Write an equation that could be used to find the unknown quantity using the variable defined.

Unit 10: Equations and Inequalities

d.

Solve the equation.

10.9.3 Try It: Writing Two-Step Equations From Real-World Contexts 1.

e.

The average toddler’s height increases by 0.2 inches per month. Maurice’s youngest brother James, who is a toddler, is 35.45 inches tall. If he grows at the average rate, how long will it take James to reach a height of 37 inches? a.

Define the unknown quantity using a variable.

b.

Write an equation that could be used to find the unknown quantity using the variable defined.

c.

Solve the equation.

d.

Explain what the solution means in this context.

Explain what the solution means in this context.

Unit 10: Equations and Inequalities

137

1.

10.9.4 Guided Practice: Writing Two-Step Equations From Real-World Contexts Involving Percentages

Victoria works as a salesperson. She earns $350 per week, plus 10% commission on her weekly sales. If Victoria needs to earn $800 this week, determine how much she needs to sell this week. a.

Define the unknown quantity using a variable.

b.

Write an equation that could be used to find the unknown quantity using the variable defined.

c.

d.

138

2.

Maddox has a coupon for $10 off his purchase at Foot Zone. He found a pair of shoes he wants that is marked 20% off. If Maddox paid $36.20 before taxes were applied, what was the original price of the shoes? a.

Define the unknown quantity using a variable.

b.

What percentage of the original price of the shoes did Maddox pay?

c.

Write an equation that can be used to find the unknown quantity using the variable defined.

d.

Solve the equation.

e.

Explain what the solution means in this context.

Solve the equation.

Explain what the solution means in this context.

Unit 10: Equations and Inequalities

Unit 10, Lesson 10: Writing and Solving Equations in Real-World Context

10.9.5 Wrap-Up: Error Analysis Mr. Humphries told his class he has some quarters and four dimes in his pocket, worth a total of $2.40.

Devin wrote the equation 25𝑥𝑥 + 40 = 2.40 to represent the situation.

1.

Mr. Humphries told Devin his equation did not correctly represent the situation. Write an equation that correctly represents the situation.

Benchmark MA.7.AR.2.2 Write and solve two-step equations in one variable within a mathematical context or real-world context, where all terms are rational numbers.

Learning Targets  

2.

I can write two-step equations to represent real-world contexts.

I can solve two-step equations within a real-world context.

If 𝑥𝑥 represents the number of quarters, solve the equation you created in Question 1 to determine how many quarters Mr. Humphries has.

Unit 10: Equations and Inequalities

139

10.10.1 Warm-Up: Bell Work Mary McLeod Bethune was an educator, civil rights activist, and advisor to five U.S. presidents. She turned her passion for the U.S. making racial progress into founding the Daytona Literacy and Industrial Training School for Negro Girls in 1904. Twenty years later, the school would merge with Cookman Institute of Jacksonville to create what is known today as Bethune-Cookman University, a Historically Black College University (HBCU).

Mary McLeod Bethune was president of the university until 1942.

1.

In 2021, there were 3,773 students enrolled at BethuneCookman University, which is 3,648 more than 25 times the number of students enrolled in 1904. a.

Write an equation that could be used to find the number of students enrolled in 1904, 𝑛𝑛.

b.

Solve your equation.

10.10.2 Collaboration: Writing and Solving Equations in Real-World Contexts

Work with your partner or small group to complete the table for each real-world context.

1.

Peyton was given a $40 gift card to her local coffee shop. She went to the coffee shop four times and purchased the same thing each time. If she has $7.08 left on her gift card, what was the cost of her purchase each time she went to the coffee shop?

Define the variable.

Write an equation that can be used to answer the question.

Solve the equation.

Explain the meaning of the solution.

140

Unit 10: Equations and Inequalities

2.

Florida Power and Light (FPL) charges customers 22.358 cents per kilowatt-hour of electricity used each month. Monthly charges to customers also include $18.75 per month for service. If a customer receives a bill for $237.86, how many kilowatt-hours were used during the month? Round your answer to nearest tenth.

3.

Kai is making ham and cheese sandwiches for an 1 upcoming camping trip with his family. He uses pound 8

of cheese for each sandwich. He has a 2-pound block of cheese and needs to use 0.75 pounds of it for a different recipe. How many ham and cheese sandwiches will he be able to make?

Define the variable.

Define the variable.

Write an equation that can be used to answer the question.

Write an equation that can be used to answer the question.

Solve the equation.

Solve the equation.

Explain the meaning of the solution.

Explain the meaning of the solution.

Unit 10: Equations and Inequalities

141

4.

Fatima and Luna are members of the running club at school. Fatima ran 7 fewer miles this week than she did last week. She ran 9 times as far as Luna ran this week.

Luna only had time to run 4 miles this week. How many miles did Fatima run last week?

Define the variable.

Write an equation that can be used to answer the question.

Solve the equation.

Explain the meaning of the solution.

142

5.

Shirley gives each of her 8 grandchildren a $15 gift card to their favorite restaurant and some cash for their birthday. She spends the same amount of money on each grandchild. If she spends $280 each year on her grandchildren’s birthday gifts, how much cash does she give each grandchild on their birthday?

Define the variable.

Write an equation that can be used to answer the question.

Solve the equation.

Explain the meaning of the solution.

Unit 10: Equations and Inequalities

10.10.3 Bring It Together: Writing and Solving Equations in Real-World Contexts Consider some of the problems you’ve solved and errors you’ve made over the last few lessons.

1.

In the first column of the table, write yourself a note or reminder that you can use when writing an equation for a real-world context so you can avoid similar errors in the future.

My Note or Reminders

My Partner’s Reminders

10.10.4 Your Turn!

1.

Complete the table for the real-world situation.

Kiara is baking cookies for an upcoming bake sale. Each 2 batch of cookies requires 1 cups of sugar. She has 12 3

cups of sugar, but her mom asked her to save

1 3

cup for

her to use in her coffee until she can go to the grocery store in a few days. How many batches of cookies can Kiara make?

Define the variable.

Write an equation that can be used to answer the question.

Solve the equation.

2.

Share your list with your partner. Add any helpful reminders your partner shares in the second column.

Explain the meaning of the solution.

Unit 10: Equations and Inequalities

143

1.

THIS PAGE WAS INTENTIONALLY LEFT BLANK.

10.10.5 Wrap-Up: Reflection

Circle the icon that best describes your understanding of each item.

I need help.

I got it.

I could help someone.

Solving two-step equations Writing two-step equations to represent real-world contexts Explaining the meaning of solutions to equations in context

144

Unit 10: Equations and Inequalities

Unit 11 - Representing Proportional Relationships Lesson

Page Number

11.1 Graphing Proportional Relationships

147

11.2 Constant of Proportionality - Part 1

152

11.3 Constant of Proportionality - Part 2

159

11.4 Features of Proportional Relationships

164

11.5 Is It Proportional?

169

11.6 Representing Proportional Relationships With Equations

175

11.7 Translating Between Representations of Proportional Relationships

182

11.8 Putting It All Together

187

11.9 Revisiting Relationships in Circles

192

Unit 11: Representing Proportional Relationships

145

The following B.E.S.T. Standards will be covered in this unit: MA.7.AR.4.1 Determine whether two quantities have a proportional relationship by examining a table, graph or written description.

• Clarification 1: Instruction focuses on the connection to ratios and on the constant of proportionality, which is the ratio between two quantities in a proportional relationship.

MA.7.AR.4.2 Determine the constant of proportionality within a mathematical or real-world context given a table, graph or written description of a proportional relationship.

MA.7.AR.4.3 Given a mathematical or real-world context, graph proportional relationships from a table, equation or a written description.

• Clarification 1: Instruction includes equations of proportional relationships in the form of 𝑦𝑦 = 𝑝𝑝𝑝𝑝, where 𝑝𝑝 is the constant of proportionality.

MA.7.AR.4.4 Given any representation of a proportional relationship, translate the representation to a written description, table or equation.

• Clarification 1: Given representations are limited to a written description, graph, table or equation.

• Clarification 2: Instruction includes equations of proportional relationships in the form of 𝑦𝑦 = 𝑝𝑝𝑝𝑝, where 𝑝𝑝 is the constant of proportionality.

MA.7.GR.1.3 Explore the proportional relationship between circumferences and diameters of circles. Apply a formula for the circumference of a circle to solve mathematical and real-world problems.

• Clarification 1: Instruction includes the exploration and analysis of circular objects to examine the proportional relationship between circumference and diameter and arrive at an approximation of pi (𝜋𝜋) as the constant of proportionality.

• Clarification 2: Solutions may be represented in terms of pi (𝜋𝜋) or approximately.

MA.7.GR.1.4 Explore and apply a formula to find the area of a circle to solve mathematical and real-world problems.

• Clarification 1: Instruction focuses on the connection between formulas for the area of a rectangle and the area of a circle.

• Clarification 2: Problem types include finding areas of fractional parts of a circle.

• Clarification 3: Solutions may be represented in terms of pi (𝜋𝜋) or approximately.

146

Unit 11: Representing Proportional Relationships

Unit 11, Lesson 1: Graphing Proportional Relationships Benchmark MA.7.AR.4.3 Given a mathematical or real-world context, graph proportional relationships from a table, equation, or a written description.

11.1.1 Warm-Up: Dancing With Math Karin Ditchfield lived in three different countries growing up – Brazil, England, and Argentina – where three different languages are spoken. Learning new languages can be challenging, but she explains despite these challenges the language of math was always the same.

1.

What are some reasons why math could be considered a global language?

Learning Targets  

Karin says, “Math is not just about working with numbers. I see it as a global language that has a positive impact on the world.”

I can graph a proportional relationship from a table, an equation, or a written description.

I can describe common features of the graphs of proportional relationships.

2.

Karin mentioned that growing up in Brazil meant growing up dancing. She says, “When you’re learning a new dance, practice makes perfect. It’s exactly the same with math.”

Explain whether you agree with this explanation.

Unit 11: Representing Proportional Relationships

147

11.1.2 Exploration: Graphs of Proportional Relationships

b.

Explain how you determined the unit price.

c.

Graph the relationship by plotting a point to represent each family member’s bag of blueberries.

In this course, proportional relationships have been used to make predictions about populations from sample data, create scale drawings, convert between units of measure, and solve problems involving percentages.

A proportional relationship is a collection of pairs of numbers that are in equivalent ratios.

Proportional relationships can be represented by graphs, tables of values, equations, and written descriptions.

Work with your partner to complete the following.

Madelyn and her family went to a U-Pick farm in Lithia, FL to pick blueberries. The table shows the weight, in pounds (lb), and price, in dollars, of the bag of blueberries each family member picked.

Weight (lb)

Price ($)

0.75

2.85

2

7.60

2.6

a.

1.2

9.88

Price ($)

1.

4.56

What is the unit price per pound of blueberries?

Weight (lb)

148

Unit 11: Representing Proportional Relationships

d.

Plot a point on the graph to represent the unit price per pound of blueberries.

e.

Plot a point on the graph to represent the price for 0 pounds of blueberries.

f.

2.

There are approximately 2.54 centimeters (cm) in every 1 inch (in.). This proportional relationship can be represented by the equation 𝑐𝑐 = 2.54𝑛𝑛, where 𝑐𝑐 represents the length of an object in centimeters, and 𝑛𝑛 represents the length of the same object in inches. a.

Use the equation to complete the table of values. 𝒏𝒏 length (in.)

Connect the points plotted on the graph. What do you notice?

1

𝒄𝒄 length (cm)

6

h.

Discuss with your partner whether the graph of the relationship between the number of pounds of blueberries picked and price extends beyond the points plotted on the graph in either direction.

b.

1 2

2.25

Graph the ordered pairs from the table of values.

Complete the statements. o does extend below the point (0, 0) o does not o positive because it is not possible to pick a o negative The graph

number of pounds of blueberries.

The graph

o does o does not

Length (cm)

g.

3

extend above the point

(2.6, 9.88) because it is possible to pick any o positive o negative

number of pounds of blueberries.

Length (in.)

Unit 11: Representing Proportional Relationships

149

Connect the points plotted on the graph to create a ray that represents all possible equivalent lengths.

d.

Discuss with your partner whether there is a point on the line that represents the approximate number of inches that are equivalent to 5 centimeters. If there is such a point, draw an X to mark this point on the line. If there is not such a point, explain why not.

d.

An empty bathtub is filling at a rate of 4.5 gallons per minute. a.

How many gallons of water are in the tub after 0 minutes?

b.

How many gallons of water are in the tub after 3 minutes?

c.

Time (minutes) e.

Connect the points plotted on the graph to create a ray.

f.

Complete the statement.

How many minutes does it take for the tub to fill to 31.5 gallons of water?

The bathtub is filling up at a constant unit rate of ______

g.

150

Using the values determined in Parts A-C, represent the relationship of the amount of water in the tub, in gallons, with time, in minutes, on the coordinate grid.

Amount of water in the tub (gallons)

3.

c.

o gallons o minutes

per o gallon. o minute.

Place an X on the graph at the point that represents the unit rate described in Part F.

Unit 11: Representing Proportional Relationships

11.1.3 Bring It Together: Graphs of Proportional Relationships

11.1.4 Wrap-Up: Ticket Out the Door

Each graph created in the Exploration represents a proportional relationship. Graphs representing proportional relationships have certain features in common.

Complete each statement to describe the common features of the graphs of proportional relationships. a.

Graphs of proportional relationships o include o do not include

the origin, (0,0), because when

the value of one quantity equals 0, the value of the other quantity

b.

equal 0.

o line. o curve.

A unit rate for the proportional relationship can be identified on the graph using the the point where the

o 𝑥𝑥-value o 𝑦𝑦-value

o 𝑥𝑥-value o 𝑦𝑦-value

is 1.

The table of values represents the proportional relationship between the amount of garbage produced in the U.S., in pounds, per person, with time, in days.

Number of Days 1 3 4.5 5

Garbage per Person (lb) 4.4 13.2 19.8 22

Graph this proportional relationship.

Graphs of proportional relationships form a

c.

o will also o will not

The Environmental Protection Agency (EPA) estimated that in 2013, the average amount of garbage produced in the U.S. was 4.4 pounds (lb) per person per day.

of

Garbage Produced per Person (lb)

1.

1.

Time (days) Unit 11: Representing Proportional Relationships

151

Unit 11, Lesson 2: Constant of Proportionality - Part 1 Benchmarks MA.7.AR.4.2 Determine the constant of proportionality within a mathematical or real-world context given a table, graph, or written description of a proportional relationship.

11.2.1 Warm-Up: Graphing Story A student is pouring water from a container into an empty beaker. The beaker measures volume in milliliters (mL).

The screenshots show the volume of water in the beaker at different points in time. A line is sketched at the approximate water level in each screenshot.

Volume at 𝟒𝟒 Seconds Volume at 𝟏𝟏𝟏𝟏 Seconds

MA.7.AR.4.3 Given a mathematical or real-world context, graph proportional relationships from a table, equation, or a written description.

Learning Targets  

1.

Sketch the graph of the relationship.

Volume (mL)



I can graph a proportional relationship from a table, an equation, or a written description.

I can determine the constant of proportionality from a graph.

I can determine the constant of proportionality from a table.

Time (seconds) 152

Unit 11: Representing Proportional Relationships

11.2.2 Exploration: Unit Rates and the Graphs of Proportional Relationships

c.

Use either unit rate

$2.74

1 gallon

or

$1

gallons

to

complete the table of values to represent this

Work with your partner to complete the following.

situation. Round to the nearest hundredth when

1.

necessary.

Malcolm went to the gas station to fill up his tank with gasoline. The current price for regular unleaded gasoline at the station is $2.74 per gallon. a.

Determine an approximately equivalent ratio to represent the same relationship per dollar. Round to the nearest thousandth. $2.74 ≈ 1 gallon

b.

Both

$2.74

1 gallon

and

$1

gallons

Gallons of Gasoline Cost in Dollars

12

22.74

2.5

15

$1 gallons are unit rates. Complete

each statement to interpret both unit rates.

•

Each gallon of gasoline will cost Malcolm $_______.

•

For every $1 Malcolm spends, he can add _______ gallons to his tank.

Unit 11: Representing Proportional Relationships

153

Two different graphs can be used to represent the proportional relationship between gasoline, in gallons, and cost, in dollars. Both graphs are shown.

d.

Label the axes of both graphs based on your work in Parts A-C.

e.

Discuss with your partner which point on each graph represents a unit rate for this relationship.

Then, place an X on each graph at this point.

Graph A

____________________________

Graph B

________________________

154

Unit 11: Representing Proportional Relationships

c.

11.2.3 Guided Instruction: Constant of Proportionality 1.

Ana’s refrigerator uses an average of 1.6 kilowatt-hours (kWh) of energy each day. a.

0

1

2

3

o increases o decreases

time the 𝑥𝑥-value d.

by

o increases o decreases

o 1 unit every o 1.6 units by

o 1 unit. o 1.6 units.

Use this information to determine two additional points on the graph of this relationship.

Plot the ordered pairs from the table on the coordinate grid shown.

The graph of the relationship can be completed by using a straightedge to draw a ray with an endpoint at (0,0), going through each point that represents the relationship between the two quantities, with an arrow extending beyond the grid shown.

Energy Used by the Refrigerator (kWh)

b.

The 𝑦𝑦-value

Complete the table of values to represent this relationship.

Number of Days Energy Used (kWh)

Complete the statements to describe the relationship between each point plotted on the graph in Part B.

Number of Days

e.

Complete the graph to represent the relationship by drawing the ray.

f.

Discuss with your partner why the graph of the relationship representing this context has an endpoint at (0,0) and a ray that extends beyond the grid shown.

Unit 11: Representing Proportional Relationships

155

The graph of a proportional relationship represents an infinite number of points. The point (3.75,6) is included on the graph. g.

h.

Explain what this point means in terms of the context.

156

Two points on the graph and in the table of values representing the proportional relationship between the number of days and the amount of energy Ana’s refrigerator uses are (3, 4.8) and (2, 3.2). a.

The constant of proportionality is the constant value of the ratio of two proportional quantities.

The constant of proportionality is the unit 𝑦𝑦 rate. It can be determined using the ratio 𝑥𝑥 or 𝑥𝑥

𝑦𝑦

Complete each ratio of the energy used, 𝑦𝑦, to the number of days, 𝑥𝑥, for each ordered pair. 𝑦𝑦 = 𝑥𝑥

Place an X at this point on the graph.

From a graph or a table of values representing a proportional relationship, the constant of proportionality can be determined by expressing any ordered pair as a ratio.

2.

=

b.

Determine the value of each ratio using a calculator.

c.

Complete the statement.

The constant of proportionality is the value of each ratio in this relationship, which is _____.

d.

Find the reciprocal of the constant of proportionality from Part C.

. When two quantities are in a proportional

relationship, there are two constants of proportionality, and they are always reciprocals of each other.

Unit 11: Representing Proportional Relationships

e.

Complete the statements.

11.2.4 Guided Practice: Constant of Proportionality

The other constant of proportionality for the same relationship is _______. In this context, this means as the

1.

The graph of a proportional relationship is shown.

o number of days the refrigerator runs o amount of energy the refrigerator uses

increases by _______ units for every 1 unit increase in the

o number of days. o amount of energy used.

𝑦𝑦

a.

Determine the constant of proportionality, 𝑥𝑥 .

b.

Complete the statement to describe the constant of proportionality.

The 𝑦𝑦-value

o increases o decreases

by ______ every time

the 𝑥𝑥-value increases by 1.

Unit 11: Representing Proportional Relationships

157

The table of values shows how many tablespoons (tbsp) of chocolate milk Dax adds to different quantities of milk to make his preferred flavor of chocolate milk.

Cups of Milk, 𝒄𝒄

Tablespoons of Chocolate Syrup, 𝑻𝑻

3

1

1 6 1

1.

2

3

1

4.5

2

2

6

𝑇𝑇

a.

Determine the constant of proportionality, 𝑐𝑐 .

b.

Complete the statement to interpret this constant of proportionality.

Dax

o increases o decreases

by ______ amount of

c.

the amount of

o cups o tablespoons o syrup o milk

At a local frozen yogurt restaurant in Lake Mary, FL, the price of frozen yogurt is determined using its weight, in ounces (oz). The graph shows this relationship.

1

1

1

11.2.5 Your Turn!

Price ($)

2.

o syrup o milk

o 1 cup. o 1 tbsp. 𝑐𝑐

Determine the constant of proportionality, 𝑇𝑇.

Determine the constant of proportionality, .

b.

Complete the statement to describe the constant of proportionality.

𝑥𝑥

The

o price o weight

o dollars o ounces by

158

𝑦𝑦

a. used

every time the

used increases by

Weight (oz)

o increases o decreases

by _________

o price every time the o weight

o 1 dollar. o 1 ounce.

Unit 11: Representing Proportional Relationships

increases

11.2.6 Wrap-Up: Reflection 1.

Unit 11, Lesson 3: Constant of Proportionality – Part 2

Circle the icon that best describes your understanding of each concept after today’s lesson.

I need help.

I got it.

Understand what the constant of proportionality is in a proportional relationship.

Determine the constant of proportionality from a graph.

Determine the constant of proportionality from a table.

I could help someone.

Benchmark MA.7.AR.4.2 Determine the constant of proportionality within a mathematical or real-world context given a table, graph, or written description of a proportional relationship.

Learning Targets  

I can determine the constant of proportionality from a written description.

I can relate the constant of proportionality to an equation representing a proportional relationship.

Unit 11: Representing Proportional Relationships

159

11.3.2 Guided Instruction: Relating the Constant of Proportionality to Written Descriptions

11.3.1 Warm-Up: Bell Work The graph represents a proportional relationship.

1.

The volume of sand in a rectangular sandbox, 𝑦𝑦, changes proportionally with relationship to the height of the sand in the sandbox, 𝑥𝑥. The table of values and graph represent this relationship for a rectangular sandbox measuring 10 feet (ft) long and 8 ft wide with the capacity to fill to a height of 2 ft.

Height Volume of of Sand Sand (ft) (ft3) 1 2 3

1.

Describe a situation that could be represented by this graph.

4

1

14 2

a.

2.

Label the axes with the quantities in your situation.

3.

There is a point marked at (16, 12) on the graph. What does this point represent in your situation?

160

40 60

100 160

Graph

Volume of Sand (ft3)

Table of Values

Height of Sand (ft)

𝑦𝑦

Determine the constant of proportionality, 𝑥𝑥 , using the table of values. Show your work.

Unit 11: Representing Proportional Relationships

b.

Explain how to determine the constant of proportionality using the graph.

a.

Complete the table of values to represent the same relationship. 𝒙𝒙

𝒚𝒚 c.

Describe the meaning of the constant of proportionality in this context.

−5

−3

0

5

6.2

b.

Verify the values in your table with your partner and resolve any differences, if necessary.

c.

What is the constant of proportionality, , in this

𝑦𝑦 𝑥𝑥

relationship?

2.

1

The graph of a proportional relationship is shown.

d.

Verify the ratio of

𝑦𝑦 𝑥𝑥

is constant for each coordinate

pair represented in your table using a calculator.

Make edits to your work as needed.

e.

Discuss with your partner how the constant of proportionality can be used to determine the 𝑦𝑦values corresponding to specific 𝑥𝑥-values in the table in cases where it is difficult to determine from looking at the graph.

Unit 11: Representing Proportional Relationships

161

f.

This proportional relationship can be used to represent the relationship between the original price, 𝑥𝑥, and the change in price due to a storewide sale, 𝑦𝑦. Complete the statement to interpret the constant of proportionality found in Part C in this context.

11.3.3 Guided Practice: Determining the Constant of Proportionality From a Written Description 1.

Kyla is training for a half-marathon. She sets her treadmill to a constant speed to build up her endurance on longer runs. The relationship between the time she runs, 𝑥𝑥, and the distance she runs, 𝑦𝑦, is proportional.

She runs 10.5 miles in 1.75 hours at this pace.

The constant of proportionality is _____, so the change in price due to the sale is a ____% o reduction o mark up

from the original price. This means

𝑦𝑦

a.

Determine the constant of proportionality, 𝑥𝑥 .

b.

Complete the statement to interpret the constant of proportionality.

the store is having a ____% off sale.

g.

Explain which part(s) of the graph representing this proportional relationship would not make sense mathematically for this real-world context.

At this rate, Kyla runs _____ o 1 mile o 1 hour

162

o miles o hours

for every

she trains on the treadmill.

In a real-world context, the constant of proportionality is the amount one quantity increases or decreases each time the other quantity increases by 1 unit.

Unit 11: Representing Proportional Relationships

Proportional relationships can also be represented by equations.

11.3.4 Collaboration: Equations Representing Proportional Relationships

2.

On January 14, 2021, 1 U.S. dollar was worth 73.08 Indian rupees. This is a written description of a proportional relationship. Below are two other representations of the same relationship.

Graph

Value in rupees, 𝒓𝒓

Use the quantities determined in the table of values to complete each equation that represents this relationship. 𝑑𝑑 =

3.

∙ 𝑟𝑟

Table of Values

Value in Rupees, 𝒓𝒓 Value in Dollars, 𝒅𝒅

73.08 1

10

1

10

5.

∙ 𝑑𝑑

𝑑𝑑

𝑟𝑟

What is the constant of proportionality, 𝑑𝑑, in this context?

Value in Dollars, 𝒅𝒅

𝑟𝑟 =

What is the constant of proportionality, 𝑟𝑟 , in this context?

4.

or

Discuss with your partner how the constants of proportionality relate to the equations of the proportional relationship.

Work with your partner to complete the following.

1.

Complete the table of values by filling in the missing values. Round values to the nearest thousandth, where necessary.

Unit 11: Representing Proportional Relationships

163

Unit 11, Lesson 4: Features of Proportional Relationships

11.3.5 Wrap-Up: Reflection 1.

Color in the arrow up to the statement that best describes your current understanding of the benchmark below.

Determine the constant of proportionality within a mathematical or real-world context given one of these forms: a table, graph, or written description of a proportional relationship.

Benchmark MA.7.AR.4.1 Determine whether two quantities have a proportional relationship by examining a table, graph, or written description.

I’m so confident, I could explain how to determine the constant of proportionality for proportional relationships to someone else!

I can get to the right answer, but I don’t understand well enough to explain how to determine the constant of proportionality yet.

I understand how to find the constant of proportionality for some representations of proportional relationships, but I don’t understand how to for all forms yet.

Learning Targets  

I can determine if two quantities have a proportional relationship.

I can relate tables and graphs that represent the same relationship between two quantities in a real-world context.

I have been trying hard, but I am not yet able to determine the constant of proportionality for proportional relationships.

I do not understand how to determine the constant of proportionality for proportional relationships yet. There are things I could do to be a better learner next lesson.

164

Unit 11: Representing Proportional Relationships

11.4.1 Warm-Up: Notice and Wonder The graphs below show the cost, 𝑦𝑦, of buying 𝑥𝑥 pounds of fruit. One graph shows the cost of buying 𝑥𝑥 pounds of peaches, and the other shows the cost of buying 𝑥𝑥 pounds of plums.

1.

After looking at the relationships graphed, complete the table with one thing you notice and one thing you wonder.

One thing I notice is…

One thing I wonder is…

11.4.2 Activity: Card Sort Several scenarios are described in the chart. Your teacher will give you two sets of cards showing tables and graphs.

1.

Work with your partner to match a table of values and graph to each scenario. Write the matching table number and letter of the graph in the chart.

Table Written Description of Graph Values a. T-shirts are priced $10 each at a store. There is a sale posted that says, “Buy two T-shirts, get the second one half-price.” The table and graph show the total cost, 𝑦𝑦, of 𝑥𝑥 T-shirts. b. A movie streaming service costs $8.49 each month. The table and graph show the total cost, 𝑦𝑦, for 𝑥𝑥 months of the subscription. c.

d.

e.

2.

Explain which fruit you think costs more per pound.

f.

A recipe uses 2 tablespoons of honey, 𝑥𝑥, for every 8 cups of flour, 𝑦𝑦.

Camping fees, 𝑦𝑦, at Blue Springs State Park in Volusia, FL include a $6.70 reservation fee and $24 per night, 𝑥𝑥.

Rachel ran 6 laps in 18 minutes around the track at Riverview High School in Sarasota, FL. She notes the number of minutes, 𝑦𝑦, that have passed after each lap, 𝑥𝑥.

Kenneth is filling his fish tank with water at a flow rate of 1.6 gallons per minute. The table and graph show the number of gallons of water, 𝑦𝑦, after 𝑥𝑥 minutes.

Unit 11: Representing Proportional Relationships

165

2.

Complete the statements using the words from the bank. constant

equivalent

line

ordered

origin

table

4.

When two quantities are in a proportional relationship,

Use Graph D and the context it represents to complete the following parts with your partner. a.

Discuss why, for this context, it makes sense that every point on the line shown is possible.

b.

Explain whether you think the graph representing this relationship will continue increasing every minute that passes.

the graph will include the ___________________, (0, 0) and the points on the graph will form a straight

___________________. The ___________________ pairs of the points on the graph match the pairs of values in the ___________________ representing the same relationship. The ordered pairs in a proportional relationship are ___________________ ratios. The value of the ratio is called the ___________________ 5.

of proportionality.

3.

Use Graph E and the table it matches to complete the following parts with your partner. a.

The points on Graph E form a straight line, but the situation does not represent a proportional relationship. Explain why it does not represent a proportional relationship.

b.

Use at least two ordered pairs from the table of values that matches Graph E to show why this relationship is not proportional.

Work with your partner to determine which scenarios from the card sort represent proportional relationships using the matching tables and graphs. Write the letter P next to the written description in the chart for each relationship that is proportional.

166

Unit 11: Representing Proportional Relationships

11.4.3 Bring It Together: Determining if a Relationship Is Proportional

Graphs representing proportional relationships have the following in common.

•

The points that satisfy the relationship lie on a straight line.

•

The line they lie on passes through the origin, (0, 0).

11.4.4 Your Turn!

1.

A graph is shown.

Tables representing proportional relationships have the following in common.

•

Each row in the table shows a ratio of the two quantities in the relationship.

•

All of the ratios in the relationship are equivalent. a.

Complete the statement.

The graph

o represents o does not represent

a proportional

relationship between two quantities.

b.

Explain your selection for Part A.

Unit 11: Representing Proportional Relationships

167

2.

Fountain drinks at Wawa are priced by size. The volume in fluid ounces (fl. oz.) and price in dollars ($), before tax, for each size is shown in the table.

Volume (fl. oz.)

Cost ($) a.

16

1.49

Complete the statement.

The table

21

30

1.59

o represents o does not represent

11.4.5 Wrap-Up: Cool Down 1.

1.89 a proportional

Your friend is absent today and will need the notes from class. Write a quick note to help them understand how to determine if two quantities have a proportional relationship.

relationship between two quantities.

b.

Explain your selection for Part A.

168

Unit 11: Representing Proportional Relationships

Unit 11, Lesson 5: Is It Proportional?

11.5.1 Warm-Up: Visual Pattern 1.

Benchmarks

The first 3 steps in a pattern created using Froot Loops are shown.

MA.7.AR.4.1 Determine whether two quantities have a proportional relationship by examining a table, graph, or written description.

MA.7.AR.4.2 Determine the constant of proportionality within a mathematical or real-world context given a table, graph, or written description of a proportional relationship.

a.

Describe how the pattern is changing in each step.

b.

Draw what you think the 6th step in this pattern will look like using small circles to represent each Froot Loop.

Learning Targets  

I can determine if two quantities have a proportional relationship.

I can determine the constant of proportionality of a proportional relationship.

Unit 11: Representing Proportional Relationships

169

11.5.2 Guided Practice: Determine if a Relationship Is Proportional From Tables and Graphs 1.

For a proportional relationship, the constant of proportionality can be determined from a graph in two ways.

Four graphs are shown in the chart. Indicate whether each graph represents a proportional relationship.

• •

Graph A

Graph B

Use the point (1, 𝑘𝑘), where 𝑘𝑘 is the constant of proportionality. 𝑦𝑦 Find the value of the ratio using any point on the graph.

2.

𝑥𝑥

Two of the graphs in the previous question represent proportional relationships. Complete the statements to state the value of the ratio between the two quantities in each of those two graphs. a.

Graph _____ represents a proportional relationship where the constant of proportionality is _________.

o proportional o not proportional

o proportional o not proportional

o proportional o not proportional

o proportional o not proportional

Graph C

170

Graph D

b.

Graph _____ represents a proportional relationship where the constant of proportionality is _________.

Unit 11: Representing Proportional Relationships

3.

Dylan and Maya were running laps around the football field at Hernando High School. The coach recorded their times at the end of laps 2, 4, 6, and 8. a.

c.

The table of values representing

Complete the table to find the ratio of minutes to laps for Dylan throughout his run.

Time (minutes)

2

4

4

10

8

23

6

b.

run is

proportionality is _________.

Minutes per Lap

15

Complete the table to find the ratio of minutes to laps for Maya throughout her run.

o Dylan’s o Maya’s

a proportional relationship. The constant of

Dylan’s Run Distance (laps)

Complete the statement.

When determining whether two quantities have a proportional relationship from a table, check every pair of values in the table to determine if the ratios between them are all equivalent.

Maya’s Run Distance (laps)

Time (minutes)

2

5

4

10

8

20

6

Minutes per Lap

15

Unit 11: Representing Proportional Relationships

171

11.5.3 Guided Instruction: Determining if a Relationship Is Proportional From a Written Description To determine if a relationship is proportional from a written description, identify pairs of values that are part of the relationship based on the information given. It is helpful to consider at least 3 or 4 pairs of values, if possible.

1.

The cooking time needed when using a pressure cooker changes depending on the elevation above sea level.

Cooking time should be increased by 5 percent for every 1,000 feet (ft) after 2,000 ft above sea level. a.

b.

c.

172

By what percentage should the cooking time be increased when cooking at an elevation of 1,000 ft?

By what percentage should the cooking time be increased when cooking at an elevation of 2,000 ft?

By what percentage should the cooking time be increased when cooking at an elevation of 3,000 ft?

2.

d.

Write the ratio of percentage increase to elevation in cooking time from Parts A-C.

e.

Find the value of each ratio using a calculator.

f.

Explain whether the relationship between the elevation and cooking time is proportional.

The actual distance between Palm Beach, FL and Melbourne, FL is 113 miles (mi). The actual distance between Melbourne, FL and Kissimmee, FL is 55 mi. On a map of Florida, the distance between Palm Beach and Melbourne is 28.25 centimeters (cm) and the distance between Melbourne and Kissimmee is 13.75 cm. a.

Which two quantities are being compared?

Unit 11: Representing Proportional Relationships

b.

Identify pairs of values of the two quantities being compared.

11.5.4 Try It: Determining if a Relationship Is Proportional 1.

The table shows the distance from which a lighthouse is visible to a boat at sea. a.

c.

Determine whether the relationship between the two quantities is proportional. If it is proportional, state the constant of proportionality. If it is not proportional, explain why not.

Determine whether the table represents a proportional relationship.

Height of a Lighthouse (feet)

Distance from which it can be Seen (miles)

20

6

70

11

150

16

45

9

95

b.

13

Complete the statements.

The table

o does o does not

represent a proportional

relationship. o It does not have a constant of proportionality. o The constant of proportionality is ________.

Unit 11: Representing Proportional Relationships

173

2.

The Koreshan State Park in Estero, FL charges the following admission fees for vehicles entering the park.

• $4 for vehicles occupied by 1 person • $5 for vehicles occupied 2 − 8 people • $2 additional per passenger over 8 people a.

Identify pairs of values that are comparing the cost of admission into the park for a vehicle to the number of people in the vehicle.

b.

Determine whether the relationship between the two quantities is proportional. If it is proportional, state the constant of proportionality. If it is not proportional, explain why not.

174

3.

Aluminum cans can be recycled instead of being thrown in the garbage. Some states pay people for returning cans so the aluminum can be reused for other products.

The weight of 10 aluminum cans is 0.16 kilograms (kg).

The weight of 25 aluminum cans is 0.4 kg. a.

Identify pairs of values of the two quantities being compared.

b.

Determine whether the relationship between the two quantities is proportional. If it is proportional, state the constant of proportionality. If it is not proportional, explain why not.

Unit 11: Representing Proportional Relationships

Unit 11, Lesson 6: Representing Proportional Relationships With Equations

11.5.5 Wrap-Up: Ticket Out the Door 1.

The table shows how loud a sound is, in decibels (dB), depending on how far away the listener is from it, in feet (ft).

Benchmarks Distance to Listener (ft) 5 10 20 40

Sound Level (dB) 85 79 73 67

MA.7.AR.4.2 Determine the constant of proportionality within a mathematical or real-world context given a table, graph, or written description of a proportional relationship.

MA.7.AR.4.4 Given any representation of a proportional relationship, translate the representation to a written description, table or equation.

Use the table to complete the statements.

Learning Targets The table

o does o does not

represent a proportional 

relationship.



o It does not have a constant of proportionality.

I can determine the constant of proportionality of a proportional relationship.

I can translate a given representation of a proportional relationship between two quantities to an equation.

o The constant of proportionality is ________.

Unit 11: Representing Proportional Relationships

175

11.6.2 Exploration: Proportional Relationships and Equations

11.6.1 Warm-Up: Fighting for Equal Pay Megan Rapinoe is an American professional soccer player and captain of the OL Reign, a team in the National Women’s Soccer League (NWSL), as well as co-captain of the U.S. women’s national team. She has won many awards and championships, including the 2019 FIFA Women’s World Cup. In March of 2019, she and a group of 27 of her teammates filed a lawsuit against the U.S. Soccer Federation over gender discrimination as part of the team’s fight for equal pay. The U.S. women’s national team has been largely more successful and popular than the U.S. men’s national team. Yet, as of 2019, the top women’s players earned only 38% of the pay their top male counterparts earned.

1.

Work with your partner to complete the following.

1.

A recipe says that 2 cups of dry rice will produce 6 servings. a.

How many servings will 1 cup of dry rice produce?

b.

Complete the table.

Number of Servings, 𝒚𝒚

2

6

1

In 2019, players from the men’s national team earned an average of $13,166 per game. How much did the average women’s national team player earn per game at the time?

176

Cups of Dry Rice, 𝒙𝒙

3

12 43

c.

Discuss with your partner how you determined the number of servings 43 cups of dry rice would produce.

d.

How many servings does 𝑥𝑥 cups of dry rice produce?

Unit 11: Representing Proportional Relationships

e.

Write an equation that could be used to find the number of servings, 𝑦𝑦, that any number of cups of dry rice, 𝑥𝑥, could produce.

c.

With your partner, discuss what was similar and what was different about completing this table when compared with the table in Problem 1b.

Summarize your discussion below.

How was completing the tables similar?

2.

How was completing the tables different?

On their party-planning menu, a catering company says that 6 spring rolls are considered 3 servings. a.

How many servings will 1 spring roll produce?

b.

Complete the table.

Number of Spring Rolls, 𝒙𝒙

Number of Servings, 𝒚𝒚

6

3

1

10 16 25

d.

How many servings would 𝑥𝑥 spring rolls be?

e.

Write an equation that could be used to find the number of servings, 𝑦𝑦, that any number of spring rolls, 𝑥𝑥, could produce.

Unit 11: Representing Proportional Relationships

177

11.6.3 Guided Instruction: Writing Equations to Represent Proportional Relationships

a.

𝑦𝑦 = 𝑥𝑥

In each situation from the Exploration, an equation can be written to represent the relationship.

When 𝑦𝑦 is proportional to 𝑥𝑥, 𝑥𝑥 can be multiplied by the same number to get 𝑦𝑦. The number multiplied by 𝑥𝑥 is the constant of proportionality, 𝑘𝑘.

b.

The equation representing this proportional relationship is 𝒚𝒚 = 𝒌𝒌𝒌𝒌.

More than half of the sugarcane produced in the United States comes from farms in Florida. Sugarcane is grown in stalks. The graph and table of values show the relationship between the amount of sugar produced in pounds (lb), 𝑦𝑦, and the number of stalks harvested, 𝑥𝑥.

Graph

Sugarcane Produced (lb)

1.

Complete the steps to determine the constant of 𝑦𝑦 proportionality, 𝑥𝑥 , using the point on the graph.

𝑦𝑦 = 𝑥𝑥 c.

178

𝒙𝒙

𝒚𝒚

𝑥𝑥

=

Complete an equation that represents this relationship. 𝑦𝑦 =

d.

=

Select any ordered pair from the table of values and 𝑦𝑦 use it to find the constant of proportionality, .

𝑥𝑥

Complete the written description of this proportional relationship. o sugarcane stalk The average o pound of sugar o sugarcane stalks. o pounds of sugar.

Sugarcane Stalks Harvested Table of Values

12

10 3

15

4.5

30 9

100 30

Unit 11: Representing Proportional Relationships

produces ______

When given a table of values or a graph, an equation representing the proportional relationship can be written by finding the constant of proportionality. e.

h.

Write another equation to represent the same relationship in terms of 𝑥𝑥. 𝑥𝑥 =

𝑦𝑦

Consider the equation 𝑦𝑦 = 𝑥𝑥 ∙ 𝑥𝑥. Work with your

partner to explain why both sides of this equation are equivalent.

f.

An equation for a proportional relationship can also be written in terms of 𝑥𝑥. Complete the equation using a ratio that would make both sides of the equation equivalent. 𝑥𝑥 =

g.

∙ 𝑦𝑦

While two equations can be written to represent any proportional relationship, equations are most commonly written 𝑦𝑦 = 𝑘𝑘𝑘𝑘, where 𝑥𝑥 is the independent variable, 𝑦𝑦 is the dependent variable, and 𝑘𝑘 is the constant of 𝑦𝑦 proportionality, 𝑥𝑥 .

Any two variables can be used in the equation of a proportional relationship.

𝑥𝑥

Determine the constant of proportionality, 𝑦𝑦, for the relationship between the amount of sugar produced in pounds (lb), 𝑦𝑦, and the number of stalks harvested, 𝑥𝑥.

Unit 11: Representing Proportional Relationships

179

2.

In 2018, there were 481,000 tons of sugar produced, 𝑠𝑠, from the 14,800 acres of sugarcane harvested, 𝑐𝑐, in Martin County, FL. a.

this written description.

b.

𝑠𝑠

Determine the constant of proportionality, 𝑐𝑐, from

11.6.4 Guided Practice: Writing Equations to Represent Proportional Relationships 1.

The graph of a proportional relationship is shown.

Write an equation to represent this relationship.

Write an equation to represent the relationship.

2.

The aluminum from recycled cans can be used to make other products. The aluminum from 32 cans that are recycled in Tampa, FL has a value of $0.40. If 𝑎𝑎 is the number of aluminum cans recycled and 𝑣𝑣 is their value, in dollars, complete each equation to represent this relationship. 𝑎𝑎 =

180

Unit 11: Representing Proportional Relationships

𝑣𝑣

𝑣𝑣 =

𝑎𝑎

1.

A bakery uses 8 tablespoons of honey for every 10 cups of flour to make bread dough. If 𝑓𝑓 is the number of cups of flour used and ℎ is the number of tablespoons of honey, complete each equation to represent this relationship. 𝑓𝑓 =

2.

11.6.6 Wrap-Up: Ticket Out the Door

ℎ

ℎ =

𝑓𝑓

1.

The graph shows the amounts of almonds, in grams, for different amounts of oats, in cups, in a granola mix.

Almonds (grams)

11.6.5 Your Turn!

The table shows the amount of red paint and blue paint needed to make a certain shade of purple paint called Venusian Sunset.

Red Paint (parts)

Blue Paint (parts)

3

12

1.5

6

7

Oats (cups) Write an equation that represents the relationship between the number of cups of oats, 𝑥𝑥, and the number of grams of almonds, 𝑦𝑦, in the granola mix.

28

If 𝑟𝑟 is the amount of red paint used and 𝑏𝑏 is the amount of blue paint used, complete each equation to represent this relationship. 𝑏𝑏 =

𝑟𝑟

𝑟𝑟 =

𝑦𝑦 =

𝑥𝑥

𝑏𝑏

Unit 11: Representing Proportional Relationships

181

Unit 11, Lesson 7: Translating Between Representations of Proportional Relationships

11.7.1 Warm-Up: Equivalent Ratios 1.

Benchmarks

Fill in the blue boxes using the digits 1 to 9 to make two ratios that are equivalent to 20 ∶ 30.

:

MA.7.AR.4.2 Determine the constant of proportionality within a mathematical or real-world context given a table, graph, or written description of a proportional relationship.

:

MA.7.AR.4.4 Given any representation of a proportional relationship, translate the representation to a written description, table, or equation.

Learning Targets  

I can determine the constant of proportionality of a proportional relationship.

I can translate a given representation of a proportional relationship to a written description, table, or equation.

182

Unit 11: Representing Proportional Relationships

11.7.2 Guided Instruction: Translating Between Different Representations of Proportional Relationships

2.

The graph of a proportional relationship is shown.

Suppose in the previous relationship, 𝑥𝑥 represents the price, in dollars, a store owner pays for merchandise from their manufacturer and 𝑦𝑦 represents the price, in dollars, for which the store owner sells the merchandise. 𝑦𝑦

a.

What is the constant of proportionality, ?

b.

Explain what the constant of proportionality means in this context.

c.

Complete the written description.

𝑥𝑥

As the value of 𝑥𝑥 increases by ____ dollars, the 1.

Table of Values 𝒙𝒙 −4 0 4 8

d.

Complete the representations of the proportional relationship by filling in the blanks in the chart.

Equation

𝒚𝒚 𝑦𝑦 =

𝑥𝑥

Written Description

As the value of 𝑥𝑥 increases by ____, the value of 𝑦𝑦 increases by ____.

value of 𝑦𝑦 increases by ____dollars.

Discuss with your partner what is different about the written description given the context compared to the written description in Problem 1.

e.

Explain which parts of the graph of this relationship do not make sense in the context.

f.

Use the equation to determine the sale price of an item the store owner purchases for $32.60.

Unit 11: Representing Proportional Relationships

183

g.

11.7.3 Guided Practice: Translating Between Different Representations of Proportional Relationships

Write the coordinate pair of the point on the graph of this relationship that represents the item described in Part F. 1.

h.

Add a row to the table of values to represent this point.

i.

Show that the ratio

𝑦𝑦 𝑥𝑥

Each table represents a proportional relationship. a.

Find the constant of proportionality and write an equation that represents the relationship.

Table of Values

for this point is equivalent to

the constant of proportionality.

Constant of Proportionality

𝒔𝒔 𝑷𝑷

Equation

b.

Constant of Proportionality Equation

184

3 12

5 20

10 40

𝑃𝑃 = ___________

Find the constant of proportionality and write an equation that represents the relationship.

Table of Values

2 8

Unit 11: Representing Proportional Relationships

𝒅𝒅 𝒄𝒄

2 6.28

3 9.42

5 15.7

𝑐𝑐 = ___________

10 31.4

2.

It took Petra 5 minutes to fill a cooler with 8 gallons of water from a faucet flowing at a steady rate. Let 𝑤𝑤 be the number of gallons of water in the cooler after 𝑡𝑡 minutes. a.

b.

3.

In Citrus County, FL, a 6% sales tax is added to the price of any item sold. a.

Circle the two equations that represent the relationship between 𝑤𝑤 and 𝑡𝑡. 𝑤𝑤 = 1.6𝑡𝑡

𝑤𝑤 = 0.625𝑡𝑡

𝑡𝑡 = 1.6𝑤𝑤

𝑡𝑡 = 0.625𝑤𝑤

𝒑𝒑 ($) 𝒕𝒕 ($)

What does 1.6 represent in the context of this situation?

b.

c.

Complete the table of values to represent the relationship between an item’s price, 𝑝𝑝, in dollars, and the total cost to a customer, 𝑡𝑡, in dollars, after sales tax is applied.

1.50

4

20

Write an equation that could be used to find the total cost, 𝑡𝑡, in dollars, of any item sold in Citrus County, after taxes.

What does 0.625 represent in the context of this situation?

Unit 11: Representing Proportional Relationships

185

11.7.4 Collaboration: Matching Equations to Graphs

Graph E

Graph F

Several equations are shown.

1.

𝑦𝑦 = 2𝑥𝑥

4

𝑦𝑦 = 𝑥𝑥 5

4

2

𝑦𝑦 = 𝑥𝑥 3

4

𝑦𝑦 = 𝑥𝑥 3

3

𝑦𝑦 = 2 𝑥𝑥

Work with your partner to match each equation to the graph representing the same relationship. Write the equation in the space provided below each graph.

Graph A

Graph C

1

𝑦𝑦 = 𝑥𝑥

186

Graph B

Graph D

2.

Choose one graph and explain how you determined the equation that matched the graph.

3.

Share your explanation with your partner. If necessary, make edits to your explanation based on feedback from your partner.

4.

A store is having a 20% off sale. With your partner, discuss which graph represents the relationship between the original price, in dollars, of an item, 𝑥𝑥, and its sale price, 𝑦𝑦, in dollars. Then, write a brief summary of your discussion.

Unit 11: Representing Proportional Relationships

Unit 11, Lesson 8: Putting It All Together

11.7.5 Wrap-Up: Reflection 1.

What was the muddiest (most difficult or confusing) point in today’s lesson on translating proportional relationships to written descriptions, tables, or equations?

Benchmarks MA.7.AR.4.1 Determine whether two quantities have a proportional relationship by examining a table, graph or written description.

MA.7.AR.4.2 Determine the constant of proportionality within a mathematical or real-world context given a table, graph, or written description of a proportional relationship.

MA.7.AR.4.3 Given a mathematical or real-world context, graph proportional relationships from a table, equation or a written description.

MA.7.AR.4.4 Given any representation of a proportional relationship, translate the representation to a written description, table or equation.

Learning Targets    

I can determine if two quantities have a proportional relationship given a real-world context.

I can graph a proportional relationship from a table, an equation, or a written description.

I can determine the constant of proportionality of a proportional relationship.

I can translate a given representation of a proportional relationship to a written description, table, or equation.

Unit 11: Representing Proportional Relationships

187

11.8.1 Warm-Up: Critical Thinking – Shower or Bath?

Some people say that taking a bath uses more water than a shower. Others disagree.

1.

What information would you collect in order to answer the question of which uses more water?

11.8.2 Activity: Station 1 – Gym Membership Plans In January, Gigi signed up for a membership at Total Fitness. The plan she chose costs $65 in start-up fees and then $20 per month starting in February. Julien also signed up at Total Fitness in January. His plan costs $32 per month starting in February, and his start-up fees and January costs were waived.

1.

2.

What are some proportional relationships that may be involved in helping to answer this question?

Complete the tables to represent the total costs of their gym memberships for both Gigi and Julien for the first 8 months of their memberships.

Gigi’s Membership Costs Months Total Since Cost January 0 1

2 3 4 5 6 7

188

Unit 11: Representing Proportional Relationships

Julien’s Membership Costs Months Total Since Cost January 0 1 2 3 4 5 6 7

2.

Determine whether either or both gym memberships are a proportional relationship. Then, complete the statements.

Gigi’s membership costs

o represent o do not represent

a

proportional relationship because _________________

11.8.3 Activity: Station 2 – Who Earns More Money?

Tia works at the Boys and Girls Club after-school program at Lockhart Middle School in Orlando, FL. The table shows how much money she earned each day she worked last week.

____________________________________________

Time Worked (hours) Money Earned ($)

____________________________________________.

Julien’s membership costs

o represent o do not represent

a

proportional relationship because _________________ ____________________________________________

4.

For the relationship(s) that is proportional, write an equation to represent the total cost, 𝑐𝑐, in dollars, for 𝑚𝑚 months of the membership.

Discuss with your partner which membership seems to be a better deal. Then, summarize your discussion.

18.55

2.5

26.50

4.25

45.05

Kahlil has a job bagging groceries at Publix where he earns $9.75 per hour.

1.

Who would make more money if they both work 10 hours? Explain or show work.

2.

On the coordinate grid shown, draw a graph that represents 𝑦𝑦, the amount of money, in dollars, Tia makes for working 𝑥𝑥 hours. Label both axes appropriately.

____________________________________________.

3.

1.75

Unit 11: Representing Proportional Relationships

189

4.

On the same grid, draw a graph that represents 𝑦𝑦, the amount of money, in dollars, Khalil makes for working 𝑥𝑥 hours.

Label each graph with the name of the person whose earnings it represents.

5.

Explain how the graph shows which person has the greatest hourly rate of pay.

6.

For each person, determine the constant of proportionality and write an equation to represent the relationship between their total earnings, 𝑦𝑦, in dollars, and the hours worked, 𝑥𝑥.

Tia’s Earnings ($)

Constant of Proportionality Equation

Constant of Khalil’s Earnings ($) Proportionality Equation

190

𝑦𝑦 = ___________

𝑦𝑦 = ___________

11.8.4 Activity: Station 3 – Robot Races The students in the STEM Club at Umatilla Middle School built solar-powered robots which they raced in the gym at their STEM Showcase.

The graphs shown represent the distance 𝑑𝑑, in meters, that each of the three robots A, B, and C traveled after 𝑡𝑡 seconds.

Distance (in meters)

3.

Time (in seconds) 1.

Determine whether each statement in the table is true or false.

True or Statement False?

Robot B traveled in a different direction o true than the other robots. o false o true Robot B traveled 5 meters in 7.5 seconds. o false o true Robot B traveled 7.5 meters in 5 seconds. o false 2 o true Robot B traveled 3 meters per second. o false 3 o true Robot B traveled 2 meters per second. o false

Unit 11: Representing Proportional Relationships

2.

𝑑𝑑

Determine the constant of proportionality, 𝑡𝑡 , for each robot.

Robot

Constant of Proportionality

A

11.8.5 Wrap-Up: Reflection So far in this unit, you have learned multiple concepts about proportional relationships including those listed below.

• Determining if two quantities have a proportional relationship • Determining the constant of proportionality of a proportional relationship given a table, graph, or written description • Graphing proportional relationships from a table, equation, or written description • Translating between different representations of proportional relationships

B C

3.

Explain what the constant of proportionality determined for each robot represents in this context.

4.

Write an equation to represent the relationship between distance, 𝑑𝑑, and time, 𝑡𝑡, for each robot.

Robot

1.

Use the stoplight reflection below to share how you are feeling about each of these concepts at this point.

Equation

A

I still feel stuck about how to…

B C 5.

I’m starting to get the hang of…

Assume the robots traveled at the same rate for 10 seconds. How far would each robot have traveled in that time?

Robot A B

Distance Traveled in 𝟏𝟏𝟏𝟏 seconds (meters)

I feel confident and ready to go with…

C

Unit 11: Representing Proportional Relationships

191

Unit 11, Lesson 9: Revisiting Relationships in Circles Benchmarks

MA.7.GR.1.4 Explore and apply a formula to find the area of a circle to solve mathematical and real-world problems.

Learning Targets I can find the circumference of a circle in terms of 𝜋𝜋.

I can find the area of a circle in terms of 𝜋𝜋.

I can relate proportional relationships to circles.

A series of screenshots of Circle 𝑀𝑀 are shown. In each screenshot, the line segment and arc that are the same color have the same length.

Description

MA.7.GR.1.3 Explore the proportional relationship between circumferences and diameters of circles. Apply a formula for the circumference of a circle to solve mathematical and real-world problems.

  

11.9.1 Warm-Up: Bell Work

The diameter, 𝐾𝐾𝐾𝐾, is wrapped around the circle starting at point 𝐸𝐸 and ending at point 𝐷𝐷.

The diameter, 𝐾𝐾𝐾𝐾, is wrapped around the circle starting at point 𝐷𝐷 and ending at point 𝐹𝐹.

The diameter, 𝐾𝐾𝐾𝐾, is wrapped around the circle starting at point 𝐹𝐹 and ending at point 𝐻𝐻.

A small part of Circle 𝑀𝑀’s circumference is shown between points 𝐻𝐻 and 𝐸𝐸.

Its equivalent length is shown on the diameter.

192

Unit 11: Representing Proportional Relationships

Screenshot

11.9.2 Exploration: Revisiting Relationships in Circles

All 3 diameters are shown wrapped around Circle 𝑀𝑀’s circumference, as well as the small amount of the circumference that is not a complete diameter.

1.

The formulas for finding the circumference and area of a circle are shown.

Complete the statements to describe the relationship shown in the series of screenshots of Circle 𝑀𝑀.

The o perimeter o circumference

of any circle is equal to the

length of the o radius o diameter

times 𝜋𝜋. The value of 𝜋𝜋 is

approximately ______. The screenshots show the length of the

o perimeter o circumference

o2 length of slightly more than o 3 o4

2.

is the same as the o radii. o diameters.

The part of the circumference that represents the small

1.

Area of a Circle, 𝑨𝑨

𝐴𝐴 = 𝜋𝜋𝑟𝑟 2 , where 𝑟𝑟 is the radius Circumference of a Circle, 𝑪𝑪 𝐶𝐶 = 𝜋𝜋𝜋𝜋 𝐶𝐶 = 2𝜋𝜋𝜋𝜋ǡ

where 𝑑𝑑 is the diameter and 𝑟𝑟 is the radius

Work with your partner to complete each table of values in terms of 𝜋𝜋.

Diameter (inches) Circumference (inches) Radius (inches) Circumference (inches)

Table A

1

Radius (inches) Area (sq. inches)

Unit 11: Representing Proportional Relationships

3

4

5

3

4

5

3

4

5

Table B 1

amount of the circumference that is not a complete diameter has an approximate length of ______.

2

2 Table C

1

2

193

2.

Three graphs are shown.

On your own, mark the table that matches each graph and label the 𝑥𝑥- and 𝑦𝑦-axes appropriately.

Graph

Table

o Table A o Table B o Table C o Table A o Table B o Table C

3.

Check your work with your partner and if necessary, resolve any differences.

4.

Discuss with your partner which relationships appear to be proportional, including what evidence there is to support your thinking. Then, summarize your discussion.

o Table A o Table B o Table C

194

Unit 11: Representing Proportional Relationships

11.9.3 Bring It Together: Revisiting Relationships in Circles Complete the statements.

1.

There

o is o is not

a proportional relationship between

11.9.4 Collaboration: Diameter and Circumference Circle 𝑊𝑊 is shown with diameter 𝐹𝐹𝐹𝐹.

the diameter and circumference of a circle. o Therefore, there is no constant of proportionality. o The constant of proportionality is ______.

1.

2.

There

o is o is not

a proportional relationship between

Miranda uses a drawing program to change the size of Circle 𝑊𝑊. Find the circumference, in terms of 𝜋𝜋, for each diameter she uses.

Diameter

the radius and circumference of a circle.

Circumference

12

o Therefore, there is no constant of proportionality. o The constant of proportionality is ______.

19 45

3.

There

o is o is not

a proportional relationship between

the radius and area of a circle. o Therefore, there is no constant of proportionality. o The constant of proportionality is ______.

2.

146

Suppose 𝑥𝑥 represents the diameter and 𝑦𝑦 represents the circumference. Write the constant of 𝑦𝑦 proportionality, . 𝑥𝑥

Unit 11: Representing Proportional Relationships

195

3.

Complete the statement.

As the value of the diameter increases by 1, the value of circumference increases by ______.

4.

Write an equation that represents the relationship between 𝑥𝑥, the diameter, and 𝑦𝑦, the circumference.

196

11.9.5 Wrap-Up: Reflection 1.

Take a SELFIE by checking the box that best matches what you did today.

Showed my work

Explained my answers

Lots of math vocabulary used

Found connections to other content

I persevered through the problem

Eliminated careless errors

Unit 11: Representing Proportional Relationships

Unit 12 - Surface Area and Volume 12.1 12.2 12.3 12.4 12.5 12.6

Lesson Finding Surface Area of Right Circular Cylinders Using Nets Finding Surface Area of Right Circular Cylinders Solving Problems Involving Surface Area of Right Circular Cylinders Volume of Right Circular Cylinders - Part 1 Volume of Right Circular Cylinders - Part 2 Problem-Solving With Volume of Cylinders

Unit 12: Surface Area and Volume

Page Number 199 204 208 212 217 223

197

The following B.E.S.T. Standards will be covered in this unit: MA.7.GR.2.1 Given a mathematical or real-world context, find the surface area of a right circular cylinder using the figure’s net.

• Clarification 1: Instruction focuses on representing a right circular cylinder with its net and on the connection between surface area of a figure and its net.

• Clarification 2: Within this benchmark, the expectation is to find the surface area when given a net or when given a three-dimensional figure.

• Clarification 3: Within this benchmark, the expectation is not to memorize the surface area formula for a right circular cylinder.

• Clarification 4: Solutions may be represented in terms of pi (𝜋𝜋) or approximately.

MA.7.GR.2.2 Solve real-world problems involving surface area of right circular cylinders.

• Clarification 1: Within this benchmark, the expectation is not to memorize the surface area formula for a right circular cylinder or to find radius as a missing dimension.

• Clarification 2: Solutions may be represented in terms of pi (𝜋𝜋) or approximately.

MA.7.GR.2.3 Solve mathematical and real-world problems involving volume of right circular cylinders.

• Clarification 1: Within this benchmark, the expectation is not to memorize the volume formula for a right circular cylinder or to find radius as a missing dimension.

• Clarification 2: Solutions may be represented in terms of pi (𝜋𝜋) or approximately.

198

Unit 12: Surface Area and Volume

Unit 12, Lesson 1: Finding Surface Area of Right Circular Cylinders Using Nets

12.1.1 Warm-Up: Which One Doesn’t Belong?

1.

Circle the figure that does not belong.

2.

Explain why that figure does not belong.

Benchmark MA.7.GR.2.1 Given a mathematical or real-world context, find the surface area of a right circular cylinder using the figure’s net.

Learning Targets  

I can draw the net of a right circular cylinder.

I can find the surface area of a right circular cylinder using the figure’s net.

Unit 12: Surface Area and Volume

199

12.1.2 Guided Instruction: Drawing the Net of a Right Circular Cylinder

2.

Draw the net of the right circular cylinder shown. Label the net with its dimensions. Then, using the terms in the word bank, label its parts.

A cylinder is a figure containing two congruent, parallel, circular bases whose edges are connected by a perpendicular curved surface. The net of a cylinder consists of a parallelogram and two circles.

𝑟𝑟 = 4 centimeters (cm.)

Parts of a Right Cylinder Circular base Curved surface

ℎ = 9 cm.

A net is a two-dimensional representation of a threedimensional figure that can be folded to form the threedimensional shape.

1.

Discuss with a partner if these nets can be folded into a right circular cylinder. Circle the nets that represent a net of a right circular cylinder. 𝑟𝑟 ℎ

ℎ

Tin cans are used for packaging many things, including various types of foods. The tin can shown has a diameter of 4 inches (in.) and a height of 6 in. Draw the net of the tin can. Label the net with its dimensions. 𝑑𝑑 = 4 in.

𝑟𝑟

𝑟𝑟 ℎ

200

3.

𝑟𝑟

ℎ

Unit 12: Surface Area and Volume

ℎ = 6 in.

12.1.3 Exploration: Relationship Between Circumference and Lateral Area of a Right Circular Cylinder

If the circle represents a circular base of a right circular cylinder, discuss with your partner how the diagrams show the relationship between the circumference of the circle and the length of the lateral area of the cylinder.

Write a summary of the discussion.

The perpendicular curved surface that connects the circular bases of the right circular cylinder is also called the lateral area of the cylinder.

1.

The diagrams show a circle with a diameter of 4.9 units.

The circle is then rolled along the ruler so that the circumference (represented in green) is “peeled” off.

Diagram 1 2.

Willard’s grandmother collects vintage labels from canned goods. The label shown is from a can of squash.

Diagram 2

a.

Discuss with your partner how the label relates to the net of a cylinder. Draw a net of the can, labeling the part of the net that represents the vintage label.

Diagram 3

Unit 12: Surface Area and Volume

201

b.

Complete the statement to describe how the length of the label is related to the circular base when the label is wrapped around the can.

12.1.4 Guided Instruction: Surface Area of a Right Circular Cylinder The surface area of a right cylinder is the total area of the two congruent, parallel circular bases and the rectangular curved surface.

Surface area is measured in square units.

The length of the label is equal to the

of the circular base.

1.

A net of a right circular cylinder is shown with its labeled dimensions.

Height, ℎ

𝑟𝑟

Radius,

o area o circumference o diameter o radius

Diameter, 𝑑𝑑

Length, 𝜋𝜋𝜋𝜋

a.

Discuss with your partner why there are two labels for the length of the rectangle.

b.

Select all of the formulas that can be used to find the area of 1 of the circular bases and the area of the rectangle.

Area of 𝟏𝟏 of the circular bases 2  2𝜋𝜋𝜋𝜋 ∙ ℎ  𝜋𝜋𝑑𝑑 2 1  𝜋𝜋𝑟𝑟 2  𝜋𝜋 (2 𝑑𝑑)  𝜋𝜋𝜋𝜋 ∙ ℎ  2𝜋𝜋𝜋𝜋

202

Length, 2𝜋𝜋𝜋𝜋

Unit 12: Surface Area and Volume

Area of the rectangle   

2 ∙ 𝜋𝜋 ∙ 𝑑𝑑 2𝜋𝜋𝜋𝜋 ∙ ℎ 𝜋𝜋𝜋𝜋

  

𝜋𝜋𝑟𝑟 2 2𝜋𝜋𝜋𝜋 ℎ ∙ 𝜋𝜋 ∙ 𝑑𝑑

2.

For an art project, Caleb will use a new dowel that measures 2 inches (in.) in diameter and 19 in. in height.

To find the surface area of the dowel, Caleb drew the net below. Use 3.14 as 𝜋𝜋.

12.1.5 Wrap-Up: Ticket Out the Door 1.

The net of a right circular cylinder is shown.

ℎ = 19 in. 𝑙𝑙 = ______ Complete the following for the given context. Represent your answers as an approximation by using ͵ǤͳͶ for 𝜋𝜋. a.

Find the missing value in the net Caleb drew.

Include units.

b.

What is the area of one of the circular bases?

Include units.

c.

What is the circumference of one of the circular bases? Include units.

d.

Before Caleb can use the new dowel, he must first remove the tight plastic casing the entire dowel is enclosed in. How much plastic is the dowel enclosed in? Include units.

Use the net to find the surface area of the right circular 22 cylinder. Use for 𝜋𝜋 and include units.

Unit 12: Surface Area and Volume

7

203

Unit 12, Lesson 2: Finding Surface Area of Right Circular Cylinders Benchmarks

12.2.1 Warm-Up: Manufacturing Engineer Joenel Johnson, a senior manufacturing engineer, mentions that one of his responsibilities is to interpret and explain, in layman’s terms (common language), the drawings of the design engineers to the technicians he works with.

1.

MA.7.GR.2.1 Given a mathematical or real-world context, find the surface area of a right circular cylinder using the figure’s net.

Describe a time when you had to interpret for yourself or explain a math problem to a partner. What did you do to ensure that you or your partner understood the problem?

MA.7.GR.2.2 Solve real-world problems involving surface area of right circular cylinders.

Learning Targets  

I can draw the net of a right circular cylinder.

I can find the surface area of a right circular cylinder.

204

Unit 12: Surface Area and Volume

12.2.2 Guided Instruction: Finding the Surface Area of Right Circular Cylinder 1.

b.

What is the surface area of a can of soda whose diameter is ͺ centimeters (cm.) with a height of ͳ͵ cm?

c.

Your partner is absent from class today. Write a summary describing to your partner how to find the surface area of a right circular cylinder when given the cylinder’s height and diameter.

Assume that a can of soda is in the shape of right circular cylinder. a.

Complete the table to find the surface area of a can of soda whose diameter is ͺ centimeters (cm.) with 22 a height of ͳ͵ cm. Use 7 for 𝜋𝜋. Round your answer to the nearest thousandth.

Draw a net and label its dimensions.

(Include units.)

2.

Determine the area of one circular base.

(Include units.)

Find the sum of the areas of both circular bases.

(Include units.)

Evelyn stores all her gift wrap paper in a plastic storage bag that is shaped like a right cylinder. The storage bag is Ͷͳ inches ሺin.ሻ tall, and its radius measures ͷǤ͸ in.

What is the surface area of Evelyn’s storage bag? Leave your answer in terms of 𝜋𝜋 and include units.

Find the lateral area.

(Include units.)

Find the sum of the areas for all the parts.

(Include units.)

Sum of the Areas of the Circular Bases

Lateral Area

≈___________

Unit 12: Surface Area and Volume

205

3.

Charlie calculated the surface area of a mailing tube whose diameter is ͷ inches (in.) and height is ʹ͸ in.

His calculation is shown. Work with your partner to determine if Charlie is correct. If not, identify his error(s) and correct his work. (𝜋𝜋 ∙ 52 ) + 2 ∙ 𝜋𝜋 ∙ 5 ∙ 26 25𝜋𝜋 + 10𝜋𝜋 ∙ 26 25𝜋𝜋 + 260𝜋𝜋 285𝜋𝜋

12.2.3 Guided Practice: Finding Surface Areas of Right Circular Cylinders 1.

There are three different Tom-tom types of bass drums: tomtoms, floor toms, and kick drum. Each drum is the shape of a right circular cylinder.

Suppose you intend to place each drum individually in a heavy-duty nylon bag when traveling locally. Find the Kick Drum amount of nylon needed for Floor tom each drum bag. Use 3.14 for 𝜋𝜋 and include units in your answer.

Bass Drum

Tom-tom 𝑑𝑑 = 7 in. ℎ = 10 in.

Floor tom 𝑑𝑑 = 15 in. ℎ = 16 in.

Kick drum 𝑟𝑟 = 8 in. ℎ = 22 in.

206

Unit 12: Surface Area and Volume

Work

Amount of Nylon Needed

12.2.4 Your Turn!

1.

12.2.5 Wrap-Up: Reflection

A small business on Etsy sells confetti cannons for buyers to celebrate special occasions. The confetti cannons come in an enclosed tube in the shape of a right circular cylinder. Each cannon is 6 inches (in.) tall and 1.5 in. wide. How much material is used to make the enclosed tube of each cannon? Leave your answer in terms of 𝜋𝜋 and include units.

1.

Take a SELFIE by checking the box that best matches what you did today.

Showed my work Explained my answers Lots of math vocabulary used Found multiple solutions I persevered through the problem Eliminated careless errors

Unit 12: Surface Area and Volume

207

Unit 12, Lesson 3: Solving Problems Involving Surface Area of Right Circular Cylinders

Benchmarks

12.3.1 Warm-Up: Photographs 1. A photographer wants to print a photograph and two smaller copies on the same rectangular sheet of paper.

The smaller picture is proportional to the larger picture.

Dimensions are given in inches (in.). The diagram is not drawn to scale. ? in.

MA.7.GR.2.1 Given a mathematical or real-world context, find the surface area of a right circular cylinder using the figure’s net.

MA.7.GR.2.2 Solve real-world problems involving surface area of right circular cylinders.

Learning Targets  

? in.

6 in. ? in.

I can draw the net of a right circular cylinder.

I can solve problems involving the surface area of a right circular cylinder.

? in.

What are the dimensions of the small photographs?

208

4 in.

Unit 12: Surface Area and Volume

12.3.2 Guided Instruction: Finding the Surface Area of Right Circular Cylinders 1.

For her final art project, Courtney uses an oldworld map to decorate a right-cylindrical pedestal.

She wraps the map around the lateral area of the pedestal. a.

2.

With your partner, write a formula that could be used to find the surface area of a right circular cylinder.

3.

Ask two classmates to share and explain the formula they wrote. Record their formula in the table. You are the only person who should write in the first column of the table. Have the person verify that you correctly recorded their formula by writing their initials.

Formula

Draw a net of the pedestal and label the part of the net where the map will be located.

4.

b.

Initials

Review the formulas from your classmates with your partner. Determine if there is more than 1 formula that can be used to find the surface area of a right circular cylinder. If necessary, revise the formula you and your partner wrote.

The length of the map is 16 feet (ft), and its height is 6.5 ft. The radius of the pedestal is 2.55 ft. Find the surface area of the pedestal. Use 3.14 for 𝜋𝜋 and round your answer to the nearest hundredth.

Unit 12: Surface Area and Volume

209

12.3.3 Guided Practice: Solving Problems Involving Surface Area of Right Circular Cylinders

Surface Area of a Cylinder 𝑆𝑆𝑆𝑆 = 2𝐵𝐵 + 𝑃𝑃ℎ, where 𝐵𝐵 is the area of the base, 𝑃𝑃 is the perimeter of the base and ℎ is the height Or 2 𝑆𝑆𝑆𝑆 = 2𝜋𝜋𝜋𝜋ℎ + 2𝜋𝜋𝜋𝜋 , where 𝜋𝜋 is the radius of the circular base and ℎ is the height

5.

The Push Pop candy, a creative spin on the lollipop, was very popular in the 1990s. It is cylindrical in shape, with a diameter of 2 centimeters (cm) and a height of 8 cm.

Find the surface area of the Push Pop candy. Use

1.

Cheese is often made into a cylindrical shape. Some cheeses are encased in paraffin wax after being formed into a cylindrical shape. The wax specially for coating cheeses aides in preventing unwanted mold growth and retains moisture. An example of what the wax coating looks like when removed from around the cheese is shown in the picture.

The circumference of a type of Gouda cheese shown is 9.5 in. and it has a height of 5 in.

22 7

for 𝜋𝜋 and round to the nearest hundredth.

How much wax coating will be needed to encase the 22 cheese? Use for 𝜋𝜋 and round your answer to the 7

nearest hundredth.

210

Unit 12: Surface Area and Volume

2.

A checker piece is in the shape of a right circular cylinder and has a diameter of 1.75 in. Each checker piece that is sold in a vintage game set is individually wrapped snug in 6.9 inʹ of material. What is the height, in inches, of each individual checker piece? Use 3.14 for 𝜋𝜋 and round your answer to the nearest hundredth.

3.

Tim plans to give his wife a roll of toilet paper as a gag gift for their first wedding anniversary since paper is the traditional 1 year wedding anniversary gift. If the roll of toilet paper is 10 cm tall by 12 cm wide, how much wrapping paper, in cmʹ will he need to cover the gift? Use

22 7

4.

Amoré gave her homeroom teacher a cylindrical jar with a hot chocolate mix inside. The jar she used was 4.2 in. wide by 6 in. tall. The gift was such a hit that she wants to give her 5 other teachers the same hot chocolate mix and tightly wrap each jar in plastic to preserve the ingredients. How much plastic will she need in total to wrap the 5 jars? Leave your answer in terms of 𝜋𝜋 and include units.

for 𝜋𝜋 and round

your answer to the nearest hundredth.

Unit 12: Surface Area and Volume

211

12.3.4 Wrap-Up: Ticket Out the Door Consider the formula, 𝑆𝑆𝑆𝑆 = 2𝜋𝜋𝜋𝜋ℎ + 2𝜋𝜋𝑟𝑟 2 , to complete the following.

1.

A hydration lotion is sold in a cylindrical container. The lotion ships in a box, but for quality control purposes the jar is wrapped in vacuum sealed plastic before being placed into a shipping box. The dimensions of the jar are 7.8 in. tall by 3.39 in. wide. How much plastic material is used to wrap around 60 individual jars of lotion? Use 3.14 for 𝜋𝜋 and round your answer to the nearest whole number.

212

Unit 12, Lesson 4: Volume of Right Circular Cylinders – Part 1 Benchmark

MA.7.GR.2.3 Solve mathematical and real-world problems involving volume of right circular cylinders.

Learning Target 

I can find the volume of a right circular cylinder.

Unit 12: Surface Area and Volume

12.4.2 Activity: Penny Stacks – Volume of a Right Circular Cylinder

12.4.1 Warm-Up: Would You Rather?

Gold is worth a significant amount of money. Its value changes with the market. For example, on January 28, 2021 the value of 1 ounce of gold was nearly $2000.

1.

Would you rather have one cube made of solid gold measuring 14 meters (m) on each side, or two cubes made of solid gold, one measuring 12 m on each side and one measuring 8 m on each side?

1.

Stack the given pennies to form a right circular cylinder. a.

How many pennies high is the stack?

b.

Compare your stack of pennies with your partner’s stack. Suppose both stacks are hollow such that liquid could be poured into them. Discuss with your partner which stack would hold more liquid. Write a summary of your discussion.

14 m

2.

Explain your choice.

12 m

8m

Capacity is the amount of space that can be filled in a container. Both capacity and volume are used to measure three-dimensional spaces.

Capacity usually refers to fluid measures, whereas volume is measured in cubic units.

c.

Complete the statement.

The capacity, or volume, of the empty cylinder depends on o the radius of each coin. o the height of the coin stack. o both the height and radius of the coin stack.

Unit 12: Surface Area and Volume

213

d.

Circle the penny stack that has the greatest volume.

12.4.3 Exploration: Discovering the Volume of a Right Circular Cylinder 1.

A penny stack with radius, 𝑟𝑟, and height, ℎ, and an empty cylinder with the same dimensions are shown.

Complete the statement.

The volume of the penny stack is the volume of the empty cylinder.

214

Unit 12: Surface Area and Volume

o equal to o not equal to

2.

A cylinder can be sliced to create a cross section.

b. Complete the statements.

Area measures the two-dimensional space an object

takes up. The

o radius o height

of the empty cylinder

makes it a three-dimensional space, as multiple layers with the same area are stacked upon each other.

Complete the statements.

To find the volume of a cylinder, multiply the

The cross section of the cylinder shown is the same size

o circular base as the o lateral area o height

_________ of the base by the ___________ of the of the cylinder.

cylinder.

The expression o 𝜋𝜋𝑟𝑟 2 represents the area of the o 2𝜋𝜋𝜋𝜋 cross section of the cylinder.

3.

4.

Which formula represents the volume of a right cylinder?

With your partner, discuss the relationship between one of the cross sections of the cylinder and the capacity of the entire cylinder.

o o

a.

o

Summarize your discussion.

𝑉𝑉 = 𝑟𝑟 ∙ ℎ, where 𝑟𝑟 is the radius and ℎ is the height. 𝑉𝑉 = 𝐶𝐶ℎ, where 𝐶𝐶 is the circumference of the base and ℎ is the height. 𝑉𝑉 = 𝐵𝐵ℎ, where 𝐵𝐵 is the area of the base and ℎ is the height.

Unit 12: Surface Area and Volume

215

1.

12.4.4 Guided Practice: Calculating the Volume of a Right Circular Cylinder 𝑟𝑟 = 5 m.

A diagram of a right circular cylinder is shown with the radius, 𝑟𝑟, of 5 meters (m.) and height, ℎ, of 9 m. ℎ = 9 m. a.

Shade in the base of the cylinder.

b.

Calculate the area of the base. Leave terms of 𝜋𝜋 and use the appropriate units.

c.

2.

Calculate the volume of the cylinder. Leave in terms of 𝜋𝜋 and use the appropriate units.

Consider a cylinder with a radius 5 inches (in.) and a height of ͳͲ in. a. Sketch the cylinder and label its dimensions.

b.

Calculate the volume of the cylinder using

22 7

ͳǤ

Suppose three cylinders each have the same height of 8 centimeters (cm.). Cylinder 1 has a radius of 1 cm.

Cylinder 2 has a radius of 2 cm. Cylinder 3 has a radius of 3 cm. Label the cylinders with their heights and radii, then find the volume of each cylinder. Use 3.14 to approximate 𝜋𝜋. Round your answers to the nearest tenth and include appropriate units.

Cylinder Number

1

2

to

approximate 𝜋𝜋. Round to the nearest thousandth and use appropriate units.

216

12.4.5 Your Turn!

3

Unit 12: Surface Area and Volume

Label with Height and Radius

Volume

12.4.6 Wrap-Up: Reflection 1.

Complete the table.

The terms and information used in the Exploration and Guided Practice that I knew were …

Something I wondered during the Exploration and Guided Practice was …

Something I learned during the Exploration and Guided Practice was …

Unit 12, Lesson 5: Volume of Right Circular Cylinders – Part 2 Benchmark

MA.7.GR.2.3 Solve mathematical and real-world problems involving volume of right circular cylinders.

Learning Target 

I can find the volume of a right circular cylinder.

Unit 12: Surface Area and Volume

217

12.5.2 Guided Instruction: Finding the Volume of a Right Circular Cylinder

12.5.1 Warm-Up: What Do You Wonder?

1.

A silo is a cylindrical container that is used on farms to hold large amounts of goods, such as grain. The artist Cam Scale painted a mural across 7 adjacent silos. The painting is of a young girl standing in an open meadow looking up into the open sky. The mural took about 4 weeks to paint.

1.

To find the volume of a cylinder,

o area o circumference

Describe at least two steps you think the artist needed to take before painting this mural.

b.

What is something you wonder about this artist’s project?

218

o add o multiply

of the base of the cylinder

```````` the height of the cylinder.

2. a.

Complete the statements.

𝑑𝑑 2

the o to o by

Landon says he can use the formula 𝑉𝑉 = 𝜋𝜋 ( 2 ) ℎ to find the volume of a cylinder. Shayna disagrees and says the formula for finding the volume of a cylinder is 𝑉𝑉 = 𝜋𝜋𝑟𝑟 2 ℎ. Discuss with your partner who you think is correct.

Unit 12: Surface Area and Volume

3.

The cylinder shown has a diameter of 6 meters (m.) and a height of 4.8 m.

𝑑𝑑 = 6 m.

With your partner, decide who is partner A and who is partner B. Then calculate the volume of the cylinder using the formula indicated. Leave your answer in terms of 𝜋𝜋.

Partner A Use Landon’s formula, 𝒅𝒅 𝟐𝟐

𝑽𝑽 = 𝝅𝝅 (𝟐𝟐) 𝒉𝒉

b.

c.

Four cylinders are shown and specific measurements are given. The measurements include the radius, 𝑟𝑟, diameter, 𝑑𝑑, height, ℎ, or circumference, 𝐶𝐶. a.

ℎ = 4.8 m. a.

4.

Partner B Use Shayna’s formula, 𝑽𝑽 = 𝝅𝝅𝒓𝒓𝟐𝟐 𝒉𝒉

Both formulas should result in the same answer.

Check your answer with your partner. Then, discuss the similarities and differences in your calculations.

Write a summary of your discussion.

For each cylinder, select which formula you would use to find the volume.

Cylinder Formula 𝑟𝑟 = 3 in.

ℎ = 5 in. 𝑑𝑑 = 3 cm. ℎ = 11 cm.

ℎ = 4.1 m.

o 𝑉𝑉 = 𝜋𝜋𝑟𝑟 2 ℎ

𝑑𝑑 2

o 𝑉𝑉 = 𝜋𝜋 ( ) ℎ 2 o 𝑉𝑉 = 𝜋𝜋𝑟𝑟 2 ℎ

𝑑𝑑 2

o 𝑉𝑉 = 𝜋𝜋 ( 2 ) ℎ o 𝑉𝑉 = 𝜋𝜋𝑟𝑟 2 ℎ

𝑑𝑑 2

o 𝑉𝑉 = 𝜋𝜋 ( 2 ) ℎ

Complete the statement.

The volume of the cylinder is _____ 𝜋𝜋

𝑟𝑟 = 2 m.

𝑑𝑑 2

o 𝑉𝑉 = 𝜋𝜋 ( 2 ) ℎ

o m. o m2. o m3.

𝐶𝐶 = 6𝜋𝜋 units ℎ = 10 units

Unit 12: Surface Area and Volume

o 𝑉𝑉 = 𝜋𝜋𝑟𝑟 2 ℎ

219

b.

Either formula can be used to calculate the volume of a cylinder. Compare your selections above with your partner. Discuss your reasoning for each selection.

12.5.3 Guided Practice: Using Diameter or Radius to Find the Volume of a Right Circular Cylinder 1.

Calculate the volume of each right circular cylinder.

Leave your answers in terms of 𝜋𝜋. 𝑟𝑟 = 3 in.

ℎ = 5 in.

𝑑𝑑 = 7 cm. ℎ = 11 cm.

220

Unit 12: Surface Area and Volume

2.

On Mrs. Scarham’s farm, there is a silo with a height of 28 feet (ft.) and diameter of 12 ft. a.

Draw a sketch of the silo and label the dimensions.

12.5.4 Collaboration: Error Analysis 1.

Simone calculated the volume of a cylinder with a diameter of ͸ inches (in.) and height of ͳ͸ in. Her calculations are shown. 𝑑𝑑 = 6 in. (𝜋𝜋 ∙ 62 )(16) 36𝜋𝜋 ∙ 16 576𝜋𝜋 in²

b.

How many cubic feet of grain can this silo hold? Use 3.14 for 𝜋𝜋.

ℎ = 16 in.

Simone made errors in her work. Work with your partner to determine what errors were made. Write a note to Simone identifying her errors and explaining how she can correct them.

Unit 12: Surface Area and Volume

221

3.

12.5.5 Your Turn!

1.

The dimensions of four cylinders are given in the column on the left. Match each set of dimensions with the corresponding volume of the cylinder.

1.___ 𝑟𝑟 = 2 units; ℎ = 5 units A.

36𝜋𝜋 units³

3.̴​̴​̴​̴​̴ 𝑑𝑑 = 6 units; ℎ = 4 units C.

20𝜋𝜋 units³

2.̴​̴​̴​̴​̴ 𝑟𝑟 = 9 units; ℎ = 3 units B.

4.̴​̴​̴​̴​̴ 𝑑𝑑 = 7 units; ℎ = 8 units D.

2.

98𝜋𝜋 units³

The figure shows two grain storage silos. The diameter of each measures 24 feet (ft.) and the height of each cylinder measures 51 ft. Find the volume of the cylindrical portion of one of the silos.

Use 3.14 for 𝜋𝜋.

approximately 763 units³

Find the volume of the cylinder shown. Use

22 7

for 𝜋𝜋.

Round your answer to the nearest thousandth.

𝑟𝑟 = 2 units

222

ℎ = 5 units

Unit 12: Surface Area and Volume

12.5.6 Wrap-Up: Ticket Out the Door 1.

At a farm, animals are fed from buckets of grain. Each bucket of grain is a cylinder with a diameter of 12 inches.

The height of the bucket is 14 inches. What is the volume of grain the bucket can hold?

22 Use 7 for 𝜋𝜋.

Unit 12, Lesson 6: Problem-Solving with Volume of Cylinders Benchmark

MA.7.GR.2.3 Solve mathematical and real-world problems involving volume of right circular cylinders.

Learning Target 

I can solve real-world problems involving the volume of right circular cylinders.

Unit 12: Surface Area and Volume

223

12.6.1 Warm-Up: Notice or Wonder 1.

Two cylindrical cans are shown.

b.

Describe how the dimensions of the two cans compare to each other.

Dimension radius

height A close-up picture of each can shows that both cans contain 15 ounces of food.

a.

224

Explain how it could be possible for two differently shaped cans to have the same volume.

Unit 12: Surface Area and Volume

How do they compare?

12.6.2 Exploration: Popcorn Containers 1.

The dimensions of each piece of paper prior to rolling it to form a cylinder are shown.

Container A

Dan is using paper to create a popcorn container. He can make two differently shaped cylindrical containers by rolling two identical pieces of paper. Container A is taller and narrower, while Container B is shorter and wider.

Container A

a.

Container B

Container B

Which container do you think will hold more popcorn?

2.

The radius of each resulting cylinder is given in the table. Complete the statements.

Container A

Container B

• The height of the paper is the b.

Without having popcorn to test, what information would be helpful in determining which container holds more popcorn?

o height o circumference

of

the cylinder.

the

of

the cylinder.

• The radius of the cylinder is 1.354 inches (in.).

Unit 12: Surface Area and Volume

the o height o circumference of the cylinder.

• The width of the paper is o height o circumference

• The height of the paper is

• The width of the paper is the

o height o circumference

of the cylinder.

• The radius of the cylinder is 1.752 in.

225

3.

Calculate the volume of each container. Use 3.14 for 𝜋𝜋 and round to the nearest thousandth.

Volume of Container A

Volume of Container B

12.6.3 Guided Instruction: Using the Volume Formula to Problem-Solve 1.

Cylinder C and Cylinder D have the same volume in cubic centimeters (cm³).

Cylinder C

Cylinder D 6 cm. ℎ cm.

4.

a.

Find the volume of Cylinder C. Leave it in terms of 𝜋𝜋.

b.

Write an expression for the volume of Cylinder D.

Complete the statement.

Container

o A o B

can hold more popcorn.

𝑉𝑉 = __________________ c.

Write an equation to show the volumes are equal to each other. _______________ = _______________

226

Unit 12: Surface Area and Volume

d.

Solve the equation.

12.6.4 Try It: Finding the Height 1.

e.

What does the solution to the equation represent?

Julius wants to build an aquarium for his pet goldfish in the shape of a right circular cylinder. He read that his fish needs to live in 1538.6 cubic inches (in.³) of water.

If his aquarium has a diameter of 14 in., what is the height of the aquarium? Use 3.14 for 𝜋𝜋.

Unit 12: Surface Area and Volume

227

3.

12.6.5 Your Turn!

1.

Find the volume of the cylinder shown. Write your answer in terms of 𝜋𝜋.

2.

The ceramic cylinder shown is advertised as holding 71.5 cubic inches. How tall must the ceramic cylinder be? Use 3.14 for 𝜋𝜋 and round your answer to the nearest thousandth.

Ruby is building a new above-ground pool for her backyard. The manual says the volume of the pool is 21.613 cubic meters (m³) and has a diameter of 4.6 m.

22 What is the height of the new pool? Use 7 for 𝜋𝜋 and round to the nearest tenth.

𝑑𝑑 = 4.5 in. ℎ =?

228

Unit 12: Surface Area and Volume

THIS PAGE WAS INTENTIONALLY LEFT BLANK.

12.6.6 Wrap-Up: Reflection 1.

Place an X on the line to indicate your understanding of each learning target. Then, provide a brief explanation or evidence to support your self-assessment.

Learning Target I can find the volume of a cylinder.

Self-Assessment I need help.

I can’t get started.

I’m getting there, but I need more practice.

I understand this, and I feel confident.

I need help.

I can’t get started.

I’m getting there, but I need more practice.

I understand this, and I feel confident.

Evidence

I can apply the volume formula to solve realworld problems.

Evidence

Unit 12: Surface Area and Volume

229

THIS PAGE WAS INTENTIONALLY LEFT BLANK.

230

THIS PAGE WAS INTENTIONALLY LEFT BLANK.

Unit 12: Surface Area and Volume

Unit 13 - Numerical and Categorical Data 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9

Lesson

Effects of Outliers Justifying Measures of Center and Spread Determining Appropriate Measures of Center and Variation Comparing Graphical Representations Comparing Data Sets Comparing Data Sets and Graphs Representing Real-World Contexts With Graphs Creating Displays for Numerical Data Creating Displays for Categorical Data

Unit 13: Numerical and Categorical Data

Page Number 233 239 245 251 257 262 268 275 281

231

The following B.E.S.T. Standards will be covered in this unit: MA.7.DP.1.1 Determine an appropriate measure of center or measure of variation to summarize numerical data, represented numerically or graphically, taking into consideration the context and any outliers.

• Clarification 1: Instruction includes recognizing whether a measure of center or measure of variation is appropriate and can be justified based on the given context or the statistical purpose.

• Clarification 2: Graphical representations are limited to histograms, line plots, box plots and stem-and-leaf plots.

• Clarification 3: The measure of center is limited to mean and median. The measure of variation is limited to range and interquartile range.

MA.7.DP.1.2 Given two numerical or graphical representations of data, use the measure(s) of center and measure(s) of variability to make comparisons, interpret results and draw conclusions about the two populations.

• Clarification 1: Graphical representations are limited to histograms, line plots, box plots and stem-and-leaf plots.

• Clarification 2: The measure of center is limited to mean and median. The measure of variation is limited to range and interquartile range.

MA.7.DP.1.5 Given a real-world numerical or categorical data set, choose and create an appropriate graphical representation.

• Clarification 1: Graphical representations are limited to histograms, bar charts, circle graphs, line plots, box plots and stem-and-leaf plots.

232

Unit 13: Numerical and Categorical Data

Unit 13, Lesson 1: Effects of Outliers Benchmark MA.7.DP.1.1 Determine an appropriate measure of center or measure of variation to summarize numerical data, represented numerically or graphically, taking into consideration the context and any outliers.

13.1.1 Warm-Up: Precision Is Key Becky Miller is a quality control engineer with GE Aviation.

Like many other careers, manufacturing engineering has many moving parts that make up an entire process. Becky says that manufacturing is problem-solving.

1.

List at least two steps that you take when faced with a difficult task.

Learning Target 

I can determine the implications an outlier has in a given data set.

Becky mentions that an error “about the size of a human hair” can make a difference in manufacturing jet engines.

2.

What is a time that you have made a small error that caused a big impact in your next steps?

Unit 13: Numerical and Categorical Data

233

13.1.2 Refresher: Measures of Center and Variation

Measures of center, such as mean and median, are used to describe a typical value in the data set.

Measures of variation, such as range and interquartile range, are used to describe how “spread out” the data is.

1.

Suppose an office tracks the number of phone calls received on business days, Monday through Friday, over two weeks. The data set is shown below.

183 152

165 153

58

152

135

124

179 112

2.

Discuss with your partner what you notice about the data set.

3.

Match the measures of center and variation with their calculations for the data set above.

Fill in the blanks with the terms provided.

mean

234

range

median

interquartile range(IQR)

The ___________ is the arithmetic average of a set of numbers. It is a measure of center, or central tendency.

A. Mean

The ___________ is the difference between the highest data value and the lowest data value in a data set. It is a measure of variation.

A measure of variation in a set of numerical data, the __________________ is the distance between the first and third quartiles of the data set.

B. Median

C. Range

152+152 2

D. IQR

= 152

165 − 124 = 41

183 + 165 + 58 + 152 + 179 + 152 + 153 + 135 + 124 + 112 = 141.3 10

The ___________ is the middle of an ordered list of the values. If the list has an odd number of values, it is the middle value of that list. If the list has an even number of values, it is the average of the two middle values. It is a measure of center, or central tendency.

Unit 13: Numerical and Categorical Data

183 − 58 = 125

4.

A histogram, line plot, and box plot representing the data are shown.

b.

5.

a.

Label the display types by filling in the box above each display.

Complete the table by identifying which measures of center and variation can be determined from each display.

Mean

Median

Range

IQR

Histogram









Line Plot









Box Plot









Are there any data values that are far away from the rest?

An outlier is a value that is much higher or much lower than the other values in a set of data.

Unit 13: Numerical and Categorical Data

235

13.1.3 Guided Instruction: Stem-and-Leaf Plots

2.

What similarities do What differences do you notice between the you notice between the displays? displays?

Recall the data collected on the number of phone calls an office received on business days over two weeks.

183 152

165 153

58

135

152

Compare the stem-and-leaf plot with the line plot from the refresher.

179

124

112

In addition to displaying the data graphically with a histogram, line plot, or box plot, the data can also be displayed in a stem-and-leaf plot.

3.

1.

A stem-and-leaf plot is a table that organizes data by place value to compare data frequencies.

What do you notice about the data in the stem-and-leaf plot?

236

Stem 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Leaf 8

2 4 5

2, 2, 3 5 9 3 Key: ͷȁͺ α ͷͺ

What value does the entry 13|5 represent?

4.

Which measure(s) of center can be determined from the stem-and-leaf plot? o mean o median o both the mean and median

5.

Which measure(s) of variation can be determined from the stem-and-leaf plot? o range o interquartile range o both the range and the interquartile range

Unit 13: Numerical and Categorical Data

13.1.4 Collaboration: Determining, Describing, and Comparing Numerical Data Represented Graphically

c.

Discuss with your partner what you notice about the mean and the median of the data. Summarize your discussion.

d.

What is the range of the data?

e.

Find the interquartile range (IQR).

Work with your partner to complete the following.

1.

Twenty students timed how long it took each of them to solve a puzzle. Below is a list of their solution times, in minutes, and a line plot displaying the data.

2, 6, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9 ,9, 9, 10, 10, 11, 11, 11, 12

Q1

Puzzle Solving Time (minutes)

Q3

IQR

a.

Determine the mean of the data set. Then, draw an arrow (↑) below the number line that points to this value.

b.

Determine the median of the data set. Then, circle the value on the number line that represents the median.

2.

Imagine that the fastest time of 2 minutes was removed from the data set. Discuss with your partner what effect, if any, removing this value has on the values of center and variation (mean, median, range, IQR). Summarize your discussion.

3.

Describe how the shape of the graph would change if 2 is removed.

Unit 13: Numerical and Categorical Data

237

4.

Complete the table to compare the values of each measure before and after 2 is removed.

Original Value Mean

>, <, =

New Value After ʹ is Removed

The following data set shows the number of tickets a movie theater sold in one day for each movie that is playing at the theater.

1.

162 265

2.

Complete the statement.

Median Range

13.1.5 Your Turn!

623 82

76 263

277 124

181 167

64 150

105 245

One of the values in the data set is an outlier. Explain which value you think is the outlier.

IQR

5.

Which measure(s) of center will be affected if the outlier is removed? o mean o median o both the mean and median

6.

Removing the outlier, 623, causes the mean to

Which measure(s) of variation will be affected if the outlier is removed? o range o interquartile range o both the range and the interquartile range

238

o increase o decrease o remain the same

o increase o decrease o remain the same

and the range to

because it is much

than the other values in the data set.

Unit 13: Numerical and Categorical Data

o larger o smaller

Unit 13, Lesson 2: Justifying Measures of Center and Spread

13.1.6 Wrap-Up: Reflection 1.

Rate yourself on today’s learning target using the scale below.

Benchmark

I can determine the implications an outlier has in a given data set.

MA.7.DP.1.1 Determine an appropriate measure of center or measure of variation to summarize numerical data, represented numerically or graphically, taking into consideration the context and any outliers.

Learning Target

2.



I can justify whether or not a measure of center or variation appropriately summarizes data.

Write an explanation of why you rated yourself where you did.

Unit 13: Numerical and Categorical Data

239

13.2.2 Collaboration: Center and Variation in Context

13.2.1 Warm-Up: Income The diagram below shows bar charts for the median and mean income between 2012 and 2014 in selected countries. The income is given in U.S dollars ($).

1.

In a study of wild bears, researchers measured the weights, in pounds (lb), of 143 wild bears that ranged in age from newborn to 15 years old. The data was used to make the histogram shown.

Frequency

Wild Bear Weight

1.

What do you notice about the values for median and mean income?

Weight (pounds) a.

Complete the statements.

Based on the data, it appears wild bears typically

2.

Why do you think the mean income for the U.S. was so much higher than the median income?

weigh between

o 50 and 150 o 100 and 200 o 150 and 250

pounds. A good

estimate for the typical weight for wild bears is __________ pounds.

240

Unit 13: Numerical and Categorical Data

b.

c.

Compare your estimate with your partner. Discuss whether it is possible for both of your estimates to be reasonable, even if they are not the same.

2.

The two histograms, labeled A and B, show height distributions of 50 male professional basketball players and 50 male professional baseball players.

Histogram A: Heights of Professional Basketball Players

Discuss with your partner whether your estimate from Part A is closer to the median or the mean.

Summarize your discussion.

Height (in inches)

d.

Histogram B: Heights of Professional Baseball Players

Angelica and Raymond were asked to describe the variation in weight of the wild bears.

Angelica said, “The bear weights range from less than 50 to over 500 lbs, so the range best describes the spread of the weights of the wild bears.”

Raymond said, “It looks like most bears in the sample weigh between 100 and 300 lbs, with few weighing less and few weighing more. The small weights could be because there were babies in the sample. So, the interquartile range is the best measure of spread to describe this data.”

Explain whether you agree with Angelica or Raymond. Justify your reasoning.

Height (in inches)

Work with your partner to complete each statement. a.

In

o Histogram A, o Histogram B,

the average height is around

72 inches. The values of the mean and median are similar. b.

To explain typical height, the variation in o Histogram A o Histogram B

is best described by the

interquartile range because the range may be misleading due to outliers. The values of the mean and median are similar.

Unit 13: Numerical and Categorical Data

241

3.

An earthworm farmer set up several containers of a certain species of earthworm so he could learn about their lengths. Lengths of earthworms provide information about their ages. The farmer measured the lengths of the earthworms in one container.

The farmer displayed the earthworms’ lengths, in millimeters (mm), in a stem-and-leaf plot.

Stem 0 1 2

Use the display to answer the following questions. a.

3 4 5 6 7 8 9

Leaf 6 1, 8, 9 0, 3, 3, 5, 5, 6, 7, 7, 8, 9 2, 3 1, 2, 8 2, 4, 9 0 7

3 Key: 2|8 = 28

Half of the earthworms sampled were less than ________ mm long.

c.

Compare your answers to Parts A and B with your partner, and if necessary, resolve any differences.

d.

The farmer reported that the lengths varied between 6 mm and 93 mm. The farmer’s assistant said that most earthworms had lengths between 23 and 48 mm.

Work with your partner to complete the statements.

The farmer is using the

o IQR o range

to summarize

the variation in worm length, while the farmer’s assistant is using the

o IQR. o range.

If a researcher is interested in how long earthworms are, the

o IQR o range

is an

appropriate measure of variation for the farmer b.

242

The farmer reported that the average length is 35.92 mm. Why might this measure of center be misleading about the typical length of the earthworms in this sample?

to report because

Unit 13: Numerical and Categorical Data

.

13.2.4 Guided Practice: Comparing Distributions

13.2.3 Bring It Together: Outlier Effects 1.

Complete each statement to describe the effect outliers have on each measure of center or variation. ●

The presence of an outlier does not have a significant effect on the

●

o mean. o median.

2019-2020 NCAA Women’s Division 1 Soccer Top Scorers

o mean. o median.

Number of Goals

The presence of an outlier does not have a significant effect on the

●

Below is a line plot of the number of goals scored by the top 15 women goal scorers during the 2019-2020 NCAA Women’s Division 1 soccer season.

The presence of an outlier will change, often significantly, the value of the

●

1.

o IQR. o range.

The presence of an outlier will change, often significantly, the value of the

o IQR. o range.

When outliers are present or a distribution is skewed, the _______________ is the preferred measure of center and the ____________ is the preferred measure of variation to describe the distribution. However, the context must be considered because outliers may be important in understanding the data. .

Rank

Player, School

1 2 3 4 5 5 7 8 9 10 10 10 13 13 13

Evelyne Viens, South Fla. Catarina Macario, Stanford Raimee Sherle, Boise St. Ally Watt, Texas A&M Tziarra King, NC State Deyna Castellanos, Florida St. Alexa Genas, Campbell Morgan Weaver, Washington St. Peyton DePriest, Middle Tenn.

Elise Flake, BYU Samantha Dewey, Xavier Hayley Younginer, Wake Forest Dakota Mills, Saint Joseph's Katie McClure, Kansas Taylor Kornieck, Colorado

Unit 13: Numerical and Categorical Data

Number of Goals 73 63 61 49 48 48 46 43 42 40 40 40 39 39 39

243

a.

b.

Without doing any calculations, explain which measure of central tendency you expect to be larger, the mean or the median.

Demetrica described the variation by saying that most of the players in the top 15 had around the same number of goals scored, with only a few scoring much higher. a.

Given her description, discuss with your partner whether Demetrica should use the range or the interquartile range to numerically summarize the spread of the number of goals scored.

b.

Explain which measure of variation is most appropriate to summarize the data.

Find the mean number of goals.

c.

Find the median number of goals.

d.

Check your work with your partner, and if necessary, resolve any differences.

e.

2.

Demetrica stated that the typical number of goals in the season for a top-15 scorer was about 48.

Explain whether you agree with Demetrica using your work from Parts A-C.

3.

At times, a statistical purpose may justify using a particular measure to describe the data.

Complete the statement.

If the statistical purpose of collecting the data is to determine how high the very top scorers score, the o IQR o range

would be the most appropriate measure

to use because this information is not captured in the o IQR. o range.

244

Unit 13: Numerical and Categorical Data

Unit 13, Lesson 3: Determining Appropriate Measures of Center and Variation

13.2.5 Wrap-Up: Reflection Draw an X on the line to indicate where your current learning is in relation to the following learning targets.

1.

I can justify whether or not a measure of center appropriately summarizes data.

I need help.

I can’t get started.

I’m getting there, but I need more practice.

I understand this and I feel confident.

Benchmark MA.7.DP.1.1 Determine an appropriate measure of center or measure of variation to summarize numerical data, represented numerically or graphically, while taking into consideration the context and any outliers.

Learning Target

2.

I can justify whether or not a measure of variation appropriately summarizes data.

I need help.

I can’t get started.

I’m getting there, but I need more practice.



I can determine an appropriate measure of center and variation to summarize numerical data.

I understand this and I feel confident.

Unit 13: Numerical and Categorical Data

245

13.3.2 Guided Practice: Measures of Center and Variation

13.3.1 Warm-Up: Between Two Numbers The temperature of the sun’s core is 15 million degrees Celsius.

The temperature of Earth’s core is 6000 degrees Celsius.

1.

What is the ratio of core temperatures of the sun to Earth?

1.

The U.S. Energy Information Administration reports weekly average gas prices for the United States. The box plot represents the weekly average gas prices in 2019.

2019 Weekly Gas Prices

Price (in dollars)

246

a.

Explain which measure of center, the mean or the median, best describes the center of the data.

b.

If possible, find the value of the measure you chose in Part A.

Unit 13: Numerical and Categorical Data

c.

d.

Describe the variation of the weekly average price of gasoline.

2.

Robin, a small business owner, is trying to determine the cost of providing a small group health care plan to her 30 employees. Her insurance agent asks her for a summary of the employees’ ages, since age is a factor that affects the premium cost of insurance. The employees’ ages are displayed in the stem-and-leaf plot and box plot.

Stem Leaf 1 ͺǡ ͻ 2 ͷǡ ͺ 3 Ͳǡ Ͳǡ ͳǡ ʹǡ ʹǡ ͸ǡ ͸ǡ ͸ǡ ͹ǡ ͹ǡ ͹ǡ ͺǡ ͻ 4 Ͳǡ ͳǡ ʹǡ ͺ 5 ͳǡ ʹǡ ͷǡ ͺǡ ͺǡ ͻǡ ͻ 6 Ͳǡ Ͷ Key: 1|8 α ͳͺ

Given your description, does the interquartile range or range better represent the variation in price?

Employee Ages (in years) Complete the statements to explain which measure of center and variation Robin should provide to her insurance agent to accurately summarize the ages of her employees. a.

The

o mean o median

best summarizes the center of

the data for the agent because …

Unit 13: Numerical and Categorical Data

247

b.

The

o IQR o range

13.3.3 Try It: Measures of Center and Variation

best summarizes the variation of

the data for the agent because …

1.

Tom Brady is considered by most to be one of the best NFL quarterbacks of all time. His pass-completion percentages from 2009 to 2019 are shown in the box plot and stem-and-leaf plot.

Tom Brady’s Completion Percentages 2009-2019

a.

The

Stem Leaf 60 ͷǡ ͺ 61 62 63 Ͳ 64 ͳǡ Ͷ 65 ͸ǡ ͹ǡ ͺǡ ͻ 66 ͵ ͸7 Ͷ Key: 60ȁͷ α ͸ͲǤͷΨ

o mean o median

the data. The

best summarizes the center of o mean o median

is not the best measure

of center because it is influenced by the values o 60.5% and 60.8% o 66.3% and 67.4%

that are far from the rest of

the data values. b.

The

o IQR o range

the data. The

best summarizes the variation of o IQR o range

is not the best measure

of variation because it is affected by the value o 60.5% o 67.4% 248

Unit 13: Numerical and Categorical Data

that is far from the rest of the data.

The Rachards family is moving to Gainesville, FL and want to purchase a home in a specific area. They have a budget of $160,000. The prices of homes in the area are shown below, in thousands of dollars.

150, 302, 201, 212, 150, 160, 140, 145

13.3.4 Your Turn!

Zane and 12 of his friends listed how far, in miles, they live from the school and created a histogram.

0.4, 0.5, 1, 1, 2.4, 2.7, 3, 3.2, 4, 4.2, 4.4, 5, 10.3

The Rachards are concerned that the homes may be out of their budget, so they ask the realtor to summarize the home prices. a.

Distance from School

Which measure of center should the realtor report to the Rachards, the mean or the median? Explain your reasoning.

Frequency

2.

b.

Which measure of variation should the realtor report to the Rachards, the interquartile range or the range? Explain your reasoning.

1.

0.4 1.4

2.4 3.4

4.4 5.4

6.4

7.4

Distance (in miles)

8.4 9.4 10.4

Zane concluded that the best description of the typical distance that a student in his class lives from school is 3 miles, the value of the median.

Do you agree with Zane? Explain your reasoning.

Unit 13: Numerical and Categorical Data

249

2.

Zane explained to his classmate, Shelby, that the students lived anywhere from 0.4 to 10.3 miles from school, so the range of 9.9 miles was the best summary for the spread of the data.

Shelby argued, “That’s misleading, only one person lived more than 5.4 miles from school, so the interquartile range is a more accurate depiction of the spread of the distances.” a.

b.

250

Do you agree with Zane or Shelby? Explain your reasoning.

Under what circumstances would you agree with the other person’s chosen measure of variation?

3.

Noelle is Zane’s friend who used to live 10.3 miles away from the school. She recently moved closer, and now only lives 5.5 miles away from the school.

The data set with Noelle’s new distance from school, in green, is shown.

0.4, 0.5, 1, 1, 2.4, 2.7, 3, 3.2, 4, 4.2, 4.4, 5, 𝟓𝟓. 𝟓𝟓

a.

What measure of center best describes the typical distance Zane’s friends live from school? Explain your reasoning.

b.

What measure of variation best describes the spread of the data distribution? Explain your reasoning.

Unit 13: Numerical and Categorical Data

Unit 13, Lesson 4: Comparing Graphical Representations

13.3.5 Wrap-Up: Ticket Out the Door The dot plot summarizes the hours of battery life for students’ cell phones when streaming videos.

Benchmark MA.7.DP.1.2 Given two numerical or graphical representations of data, use the measure(s) of center and measure(s) of variability to make comparisons, interpret results, and draw conclusions about the two populations.

Learning Target

Battery Life (in hours)

Mr. Funk’s world history class has an upcoming field trip to Tallahassee, FL that involves a five-hour drive. The students want to be able to watch videos on their phones during the drive without charging their cell phones.

1.



I can use two graphical representations of data to make comparisons and draw conclusions about the populations.

Determine the appropriate measure of center and variation for the battery life of the students’ cell phones when streaming videos.

Unit 13: Numerical and Categorical Data

251

13.4.2 Guided Practice: Comparing Line Plots

13.4.1 Warm-Up: Defrost Time 1.

A microwave shows that it takes 9 minutes and 45 seconds to defrost 1.5 pounds of meat and 26 minutes to defrost 4 pounds of meat. a.

Without performing any calculations, estimate a range of minutes you think it will take to defrost 3.5 pounds of meat.

b.

Explain how the given information could be used to find the exact amount of time it takes to defrost 3.5 pounds of meat.

c.

Determine how long it would take for the microwave to defrost 3.5 pounds of meat. Show your work.

252

1.

Clare recorded the amount of time a sample of students in the sixth, eighth, and tenth grades spent doing homework, in hours per week. She made a line plot of the data for each grade and provided the following summary: ● Students in sixth grade tend to spend less time on homework than students in eighth and tenth grades. ● The homework times for the tenth-grade students are more alike than the homework times for the eighth-grade students.

Use Clare's summary to match each line plot to the correct grade (sixth, eighth, or tenth).

Grade Line Plot Level

Unit 13: Numerical and Categorical Data

Time (in hours)

13.4.3 Activity: Station 1 – Histograms and Line Plots 1.

a.

The two histograms show height distributions of 50 male professional baseball players and 50 male professional basketball players, in inches (in.).

Complete the statements that draw a conclusion about the typical height of professional basketball players and baseball players.

The sample data in the histograms shows the center of the data for basketball players is

Heights of Basketball Players

o less than the center of the data for baseball o greater than

players. It can be concluded that professional basketball players are typically professional baseball players. b.

Heights of Baseball Players

o shorter than o taller than

Complete the statement.

The histograms show that in the samples, the range of heights of basketball players is o less than the range of the heights of o greater than

baseball players. We can conclude that professional basketball players have

o more o less

variability in

their heights than professional baseball players.

Height (in.)

One possible reason for this is …

Unit 13: Numerical and Categorical Data

253

2.

Mr. Logan surveyed a sample of his junior and senior students about the time they spend studying math per day. He then tabulated the results and created a line plot displaying the data for each group.

Time Spent Studying (Juniors)

Time Spent Studying (Seniors)

13.4.4 Activity: Station 2 – Box Plots 1.

The mile times for a sample of track team members at East Bay High School and Leto High School are shown in the two box plots.

East Bay High

Leto High Minutes

a.

b.

Minutes

The value of the larger median study time for the two groups is ______.

The value of the larger mean study time for the two groups is ______.

Mile Time (minutes) a.

Using the box plots above, calculate and describe each of the features for Leto High in the table.

Feature

c.

254

In one to two sentences, describe the difference between the number of minutes Mr. Logan’s juniors and seniors study by comparing the center and shapes for the groups.

East Bay High School The shape of the data is Shape asymmetrical.

The median mile time for the sample of East Center Bay High team members is 5 minutes.

The range of the sample of East Bay High team members’ mile times is 1.5 minutes.

Variability The IQR of the sample of East Bay High team members’ mile times is 0.75 minutes.

Unit 13: Numerical and Categorical Data

Leto High School

b.

Based on the data, explain which team is more likely to win at a track meet.

13.4.5 Activity: Station 3 – Stem-andLeaf Plots and Box Plots 1.

A sample of two stores’ guitar prices are shown in the stem-and-leaf plots.

Store A’s Guitar Prices Stem

c.

Suppose a runner at a track meet runs a mile in 3 minutes and 45 seconds. Explain which track team you would assume they were on, using information from the data displays.

Leaf 2 1, 2, 8 3 3, 6 4 5, 8, 8 5 1, 6 6 7 2, 2 8 1, 6 9 1 Key: ʹȁͳ α ̈́ʹͳͲǤͲͲ

Store B’s Guitar Prices Stem

4 5 6 7 8 9 10

Leaf 5, 8 6, 9 1, 1, 3, 5 5, 5, 6 0, 6 1, 8 0

Key: ʹȁͳ α ̈́ʹͳͲǤͲͲ

Zach wants to purchase a guitar and is trying to determine which store to purchase from. Using the displays he makes the following conclusion. d.

Based on your conclusions and answers to Parts A through C, use some of the terms from the word bank below to write a 4-5 sentence summary comparing the population of runners at East Bay High School to the population of runners at Leto High School.

mean

median

center lower quartile

range

variability sample

IQR

upper quartile population

“Store B is better for more experienced guitar players because it typically sells more expensive guitars than Store A. The median price of Store B’s guitars is $700, while the median price of Store A’s guitars is $480.

Prices of guitars at Store A tend to vary more, with an IQR of $390, compared to the prices of guitars at Store B, with a much smaller IQR.” Do you agree or disagree with Zach’s conclusion?

Explain your reasoning.

Unit 13: Numerical and Categorical Data

255

2.

Suppose a sample of test scores is taken from Mr. Spellman’s morning math classes. The data is displayed in the top box plot. Another sample of test scores is taken from Mr. Spellman’s afternoon math classes. The data is displayed in the bottom box plot.

13.4.6 Wrap-Up: Reflection Complete each statement based on your current understanding.

1.

Morning Test Scores

I am still unsure about _________________________ ____________________________________________ ____________________________________________

Afternoon Test Scores

2.

One question I have that will help me better understand is __________________________________________ ____________________________________________ ___________________________________________?

Explain the difference between taking Mr. Spellman’s math test in the morning compared to the afternoon.

Use some of the terms from the word bank and be sure to include both a measure of center and variability in your explanation. mean

median

center lower quartile

256

range

variability sample

IQR

upper quartile population

Unit 13: Numerical and Categorical Data

Unit 13, Lesson 5: Comparing Data Sets

13.5.1 Warm-Up: Notice and Wonder The line plots represent the distribution of the tip amounts, in dollars, left at two different restaurants on the same night.

Benchmark MA.7.DP.1.2 Given two numerical or graphical representations of data, use the measure(s) of center and measure(s) of variability to make comparisons, interpret results, and draw conclusions about the two populations.

Learning Target 

I can compare and interpret two numerical representations of data for two populations.

Tip Amount (in dollars) 1.

Complete each statement. a.

When looking at the data displays, one thing I notice is …

b.

One thing I wonder is …

Unit 13: Numerical and Categorical Data

257

13.5.2 Exploration: Calculating Measures of Center and Variation

2.

Complete the statements. a.

On a college entrance exam, the possible scores on the math section range from ʹͲͲǦͺͲͲ. Below are the scores for ʹͲ randomly selected female high school seniors.

ͶͷͶ ͶͲͻ Ͷ͵͹ Ͷͳͷ Ͷͷʹ ͶͶͲ ͶͷͶ Ͷ͹ͺ ͶͲͳ ͶͲ͵ Ͷʹ͸ Ͷʹ͸ Ͷ͵͹ Ͷ͸ͷ Ͷ͸ͷ Ͷʹͻ Ͷ͹Ͳ ͶͶͺ ͷͳͳ Ͷͷ͵

1.

o lower o higher

scores

on the math section of the college entrance exam, with a median of _______, than males, who have a median of _______. b.

Scores for males tend to have

o less o more

variability,

with an IQR of _______, than scores for females,

The scores for ʹͲ randomly selected male high school seniors are shown below. ͶͲͺ ͶͶͲ ͶͳͶ ͵ͻͳ ͶͺͲ Ͷͳͷ Ͷʹ͸ ͵ͻʹ Ͷ͸ͻ ͶͶ͸ ͶͲͻ Ͷ͵ʹ ͵͹ͷ ͵ͺͺ Ͷͳ͵ Ͷͷͳ ͵͹Ͳ Ͷͳͷ Ͷͺ͹ ͶͲͶ

Female students tend to have

with an IQR of _______. c.

Explain why the measures median and IQR were chosen to describe the data sets above, instead of mean and range.

d.

Check your work on Parts A-C with your partner and if necessary, resolve any differences.

Work with your partner to complete the table by calculating the measures of center and variability for each set of data.

Measure Female Students Male Students Mean Median Range IQR Potential Outliers

258

Unit 13: Numerical and Categorical Data

13.5.3 Try It: Drawing Conclusions Shawn wants to determine the best way to get to school.

Over several weeks, he took turns riding in a car and riding his bicycle and recorded the time, in minutes.

Car ͳͻ ͳ͹ ͳͷ ͳͺ ͳʹ ʹʹ ʹͻ ͳͶ

13.5.4 Activity: Guess My Data Sets Suppose an English teacher takes five random samples of students and records their most recent test scores.

Data Set A: {ʹʹǡ Ͳǡ ͶͶǡ ͶͲǡ ͷ͸ǡ ͵ʹǡ ͸Ͷǡ ͶͶǡ ͷʹǡ ͸Ͳ}

Bicycle ʹͶ ʹͷ ʹ͹ ʹ͸ ʹͳ ʹͺ ͵ʹ ʹ͵

Data Set B: {͸Ͳǡ ͸ʹǡ ͸ʹǡ ͸Ͳǡ ͸Ͷǡ ͸ͺǡ ͸Ͷǡ ͸ʹǡ ͸ͺǡ ͸Ͳ} Data Set C: {͹Ͳǡ ͺʹǡ ͹Ͷǡ ͹͸ǡ ͺͲǡ ͹ʹǡ ͺͶǡ ͹Ͳǡ ͹ͺǡ ͺʹ} Data Set D: {͹ʹǡ ͺ͸ǡ ͳͲͲǡ ͹͸ǡ ͻʹǡ ͻͲǡ ͳͲͲǡ ͹ͺǡ ͻͺǡ ͺ͸}

Shawn calculated the measures of center and spread for each transportation method. They are shown in the table.

Mean Median Range IQR

Car

Bicycle

18.25

25.75

6.0

4.0

17.5 17.0

Data Set E: {͸Ͷǡ ͵͸ǡ ͹Ͷǡ ͺʹǡ ͻ͸ǡ ͺͲǡ ͸͸ǡ ͹ͺǡ ͸ͺǡ Ͳ} Without telling your partner, choose two data sets from above.

25.5

1.

11.0

1.

Make an argument as to why Shawn should ride in a car to get to school.

2.

Make an argument as to why Shawn should ride his bicycle to school.

Write a paragraph comparing the two data sets you have chosen using measures of center and variability.

Make sure to refer to the data sets as “Data Set 1” and “Data Set 2” in the paragraph you write.

Unit 13: Numerical and Categorical Data

259

2.

Determine with your partner who is partner A and who is partner B. Write your names on the lines for each partner in the table.

3.

Partner A should read their paragraph from Number 1 to partner B, and partner B will guess which two data sets were used. Then, partner B will explain their reasoning for their guess. Both partners will summarize the guess and reasoning in the first row of the table.

4.

Then, switch roles so that partner B reads their paragraph while partner A makes a guess and explains their reasoning.

Partner

Guess

Reasoning for Guess

13.5.5 Your Turn!

1.

A company is trying to determine which method of communication is best to tell their employees important information. They send text messages to nine employees and emails to another nine employees. They record how many minutes it takes for the employees to reply.

Text Email

a.

Partner A:

ͷ ʹͷ

b.

o mean o median

o range o IQR

_________

5.

Reveal to your partner which data sets were used and discuss any errors that may have led to incorrect conclusions.

260

ͻ ʹͶ

ͳ͵ ͵Ͳ

ͳʹ ͳͻ

Ͷ ʹͻ

ͳ ͵ͺ

ͳͳ ͳʹ

Email:

o mean o median

Which measure of variability best summarizes the data sets?

Text:

Partner B:

Ͷͷ ͵ͷ

Which measure of center best summarizes the data sets?

Text:

_________

ʹͲ ͳͷ

Unit 13: Numerical and Categorical Data

Email: o range o IQR

c.

Calculate the measures of center and variability.

Mean Median Range IQR

d.

Text

Email

13.5.6 Wrap-Up: Cool Down 1.

Use the data from Your Turn to complete the prompt below.

Lucas recommends that the company emails its employees because of the smaller range in the data set.

Explain why you agree or disagree with Lucas.

Use the most appropriate measures of center and variability to compare the effectiveness of the two methods of communicating with employees.

Unit 13: Numerical and Categorical Data

261

Unit 13, Lesson 6: Comparing Data Sets and Graphs

Benchmark

MA.7.DP.1.2 Given two numerical or graphical representations of data, use the measure(s) of center and measure(s) of variability to make comparisons, interpret results, and draw conclusions about the two populations.

13.6.1 Warm-Up: Would You Rather?

1.

Would you rather…

A. Pay for ͵ gifts and get ͳ of equal or lesser value free?

B. Pay for 4 gifts using a coupon for ʹͲ% off?

Explain your reasoning.

Learning Target 

I can compare and interpret a numerical representation of data of one population and a graphical representation of another population.

262

Unit 13: Numerical and Categorical Data

c.

13.6.2 Guided Practice: Summarizing Data 1. ͻ ʹͷ

The commute time, in minutes, for 24 randomly selected people in Duval County, FL is shown. ͳͲ ʹͷ

ͳͷ ʹͷ

ͳ͹ ʹ͸

ͳͺ ʹ͸

ʹͲ ʹͻ

ʹͲ ʹͻ

ʹͳ ʹͻ

ʹʹ ͵ʹ

ʹ͵ ͵Ͷ

ʹͶ ͵ͺ

ʹͷ ͶͲ

The box plot displays the commute time, in minutes, for 24 randomly selected people in Union County, FL.

Use at least three terms from the term bank below to write a three-sentence summary comparing travel times to work in Duval County to travel times to work in Union County.

mean

median

center lower quartile

range

variability sample

IQR

upper quartile population

Travel Time in Union County, FL

Time (in minutes)

a.

Explain why it would be beneficial to create a box plot of the Duval County data when comparing these data sets.

b.

What are some measures used to describe data that cannot be determined from a box plot?

Unit 13: Numerical and Categorical Data

263

13.6.3 Try It: Comparing Data Sets and Graphs 1.

b.

Which team do you think is more likely to win the next race? Explain your reasoning.

The Deltona High School Wolves and Wildwood High School Wildcats cross-country teams ran an obstacle course. Each team’s times are summarized.

Deltona HS Wolves’ Obstacle Course Times 4: 25 5: 21 5: 54 6: 48

4: 43 5: 31 6: 08 6: 53

4: 49 5: 32 6: 20 7: 16

5: 02 5: 37 6: 26 7: 23

5: 12 5: 52 6: 33 8: 05

Wildwood HS Wildcats’ Obstacle Course Times

Course Times a.

Which statements are true about the data for the Deltona Wolves and the Wildwood Wildcats? Select all that apply.  





264

The median time of the Deltona Wolves is less than the median time of the Wildwood Wildcats.

The fastest 25% of athletes on both teams complete the obstacle course in about the same amount of time.

The interquartile range of the Deltona Wolves is less than the interquartile range of the Wildwood Wildcats.

Approximately 50% of Wildwood Wildcats have times between 5 and 6 minutes.

Unit 13: Numerical and Categorical Data

2.

13.6.4 Activity: Roll the Dice With your partner, decide which of you will be “Partner 1” and “Partner 2.” Partner 1 will be creating a line plot and Partner 2 will be recording a data set.

Mean Median

Using a standard six-sided die, each person will roll one die ʹͷ times. After each roll, record the results in either a line plot or numerical data set below, depending on if you are Partner 1 or Partner 2.

1.

When both partners are finished, copy their data below.

Line plot (Partner 1’s rolls):

Complete the table by calculating the measures of center and spread for your data. Share your data with your partner and add their data to the table.

Partner ͳ

Partner ʹ

Range IQR 3.

With your partner, discuss similarities and differences in your data. Summarize your discussion in the table.

Similarities

Differences

Data set (Partner ʹ’s rolls):

Unit 13: Numerical and Categorical Data

265

For each statement, state whether it is true or false.

For any statements that are true, explain why.

13.6.5 Your Turn!

1.

A group of ten rowers that practice outdoors on the water (Group 1) are challenging a group of ten who use indoor rowing machines (Group 2). They want to see who can row faster using rowing machines.

The time, in minutes, it took each of the ten rowers in Group 1 to row ͷͲͲ meters on the rowing machine is represented in the stem-and-leaf plot.

Group

Statement

The center of the Group 1 times is similar to the center of the Group 2 times.

1: Outdoor Rower Times (in minutes)

Stem

Leaf 15 2, 8 16 8 17 0, 5, 8 18 0, 2, 3, 7 Key: ͳͷȁʹ α ͳǤͷʹ

The variability of the Group 1 times is similar to the variability of the Group 2 times.

The time, in minutes, it took each of the ten rowers in Group 2 to row ͷͲͲ meters on the rowing machine is represented in the data set. {ͳǤ͸͵ǡ ͳǤͺʹǡ ͳǤͻͷǡ ͳǤͻ͹ ʹǤͲǡ ʹǤͲʹǡ ʹǤͳʹǡ ʹǤʹͲǡ ʹǤʹͷǡ ʹǤͺͺ}

The Group 1 rowers that practice on the water are typically faster than the Group 2 rowers that row on machines.

The Group 1 rowers that practice on the water are more consistent than the Group 2 rowers that row on machines.

266

Unit 13: Numerical and Categorical Data

True or False?

Explanation

2.

The dot plots display the arrival times of 30 flights for two different airlines.

Arrival Times of Airline P

Based on the data, from which airline would you choose to buy your ticket? Use measure(s) of center and variability to justify your choice.

Arrival Times of Airline Q

•

A negative number represents the number of minutes the flight arrived before its scheduled time.

•

A positive number represents the number of minutes the flight arrived after its scheduled time.

•

A zero indicates that the flight arrived at its scheduled time.

Unit 13: Numerical and Categorical Data

267

13.6.6 Wrap-Up: Cool Down 1.

Write a short note to a classmate who missed class summarizing today’s lesson.

Unit 13, Lesson 7: Representing Real-World Contexts With Graphs

Benchmark MA.7.DP.1.5 Given a real-world numerical or categorical data set, choose and create an appropriate graphical representation.

Learning Target 

268

I can choose an appropriate graphical representation for a real-world context.

Unit 13: Numerical and Categorical Data

13.7.2 Exploration: Choosing the Best Graphical Representation for Numerical Data

13.7.1 Warm-Up: Bell Work To choose an appropriate graphical representation for data, the first step is to determine if the data is numerical (data points are numbers) or categorical (data points fall within categories).

1.

Complete the table. The first two rows have been completed.

Possible Question

Possible Responses

What is your favorite color?

red, orange, yellow, green

How many people are in your family?

How old are you?

ʹ, ͵, Ͷ, ͷ…

11, 12, 13…

Numerical or Categorical

Four graphical representations that can be used to display numerical data are box plots, histograms, line plots, and stem-and-leaf plots. Each graphical representation is shown in the table.

Examples of graphical representations that can be used to display numerical data Box Plot

o categorical o numerical

Line Plot

o categorical o numerical o categorical o numerical

How tall are you?

o categorical o numerical

What is your favorite sport?

o categorical o numerical

Histogram

watermelon, apple, o categorical pear, orange o numerical

Unit 13: Numerical and Categorical Data

Stem-and-Leaf Plot

Stem Leaf 5 8 6

7

8

9

10

11 2 12 4 13 5 14

15 2, 2, 3 16 5 17 9 18 3 Key: ͷȁͺ α ͷͺ

269

Work with your partner to complete the following.

1.

Camilla wants to create a graphical display that represents the typing speeds of the 425 students at her school. a.

Which representation(s) will best display the data?    

Explain your reasoning.

c.

Camilla says, “Making a line plot of this data would take a lot of time and it would be hard to make sure each dot is in the right location on the number line, since speed can be given in fractions of seconds. I’d have to make 425 dots to represent each student!” Explain whether you agree with Camilla that a line plot would not best display the data.

270

Dallas records the highest temperature from each month over the past year in Immokalee, FL.

a.

{76, 78, 81, 85, 90, 92, 92, 93, 91, 87, 82, 78}

Which representation(s) will best display the data?    

box plot line plot histogram stem-and-leaf plot

b.

d.

2.

box plot line plot histogram stem-and-leaf plot

b.

Explain your reasoning.

c.

Dallas decides to change his data set to the average weekly high temperature. Explain whether this changes the display he should use.

Which other display should not be considered based on the size of the data set?

Unit 13: Numerical and Categorical Data

3.

Basil recorded the weights, rounded to the nearest pound, of 15 dogs at a pet show. a.

7, 10, 12, 22, 28, 59, 62, 65, 72, 79, 85, 98, 100, 150, 163

Explain which display would be better for the data, a line plot or a stem-and-leaf plot.

4.

Ms. Rallen asked her students how many pets they each have, and wants to use a graphical representation that will allow her students to see the overall shape of the data and give them enough information to calculate the ͷǦnumber summary. a.

Which graphical representations should Ms. Rallen consider?    

b.

Basil also recorded the weights, rounded to the nearest pound, of 15 cats at the pet show.

4, 4, 5, 7, 7, 8, 8, 8, 9, 10, 11, 12, 13, 13, 14

b.

box plot line plot histogram stem-and-leaf plot

Explain your reasoning.

Explain which display would be better for the data, a line plot or a histogram.

c.

Basil updated his data to show the weights of the cats to the nearest tenth.

4.1, 4.3, 5.2, 7.6, 7.9, 8.0, 8.2, 8.7, 9.3, 10.5, 11.2, 12.8, 13.4, 13.5, 14.6

Explain which display would be better for the data, a line plot or a histogram. If your recommendation is different from Part B, explain why.

Unit 13: Numerical and Categorical Data

271

13.7.3 Bring It Together: Choosing the Best Graphical Representation for Numerical Data

Line plots are best for smaller sets of data and when the goal is to see each value over a scale, making it easier to calculate measures of center and spread.

Histograms are best for larger sets of data, data over a large range, or continuous data.

One disadvantage is that the display cannot be used to calculate measures of center and spread since individual values cannot be determined from the display.

13.7.4 Guided Instruction: Graphical Representations for Categorical Data Two graphical representations that can be used to display categorical data are bar graphs and circle graphs.

The vertical bar graph shows data collected from a survey on how students get to school.

Stem-and-leaf plots are helpful in visualizing the shape of a distribution, while still being able to see the individual values.

Box plots are helpful in visualizing the shape of a distribution and quickly identifying the fivenumber summary. This display cannot be used to find the mean.

140

100 80 60 40 20 0

272

Mode of Transportation to School

120

Number of Students

A bar graph is a visual display of categorical data values where each category is represented by a bar whose height represents the number in that category. Bar graphs can be represented vertically or horizontally.

Bus

Unit 13: Numerical and Categorical Data

Car Bike Transportation Type

Walk

The same data is also displayed in a horizontal bar graph.

Mode of Transportation to School

1.

Complete the statements.

The bar graph visually shows

Walk

o how each part relates to the whole. o how each part compares to the other parts.

Bike

The circle graph shows

Car

o how each part relates to the whole. o how each part compares to the other parts.

Bus 0

20

40

60

80

100

Number of Students

120

140

A circle graph is a visual display of categorical data. The whole set of data is represented by the circle and its interior. The categories are represented by fractional parts of the circle.

This is also called a pie chart.

The circle graph below shows the same data.

Mode of Transportation to School 11% 43%

22%

24% Walk

Bike

Car

Bus

Unit 13: Numerical and Categorical Data

273

2.

Greer wants to convince the head chef at his school to include oat milk as an option at meals, so he asks a random sample of students from his school what their favorite kind of milk is: dairy, oat, or soy. He wants to use a graphical representation that makes it easy to compare the number of students who prefer oat milk to the total number of students surveyed.

Jordan asks a random sample of students from his middle school which of these apps is their favorite: TikTok, Snapchat, Instagram, Pinterest, YouTube, WhatsApp, Kik, or Discord. He wants to display the data so it is easy to compare the different categories to each other.

13.7.5 Your Turn!

1.

{ͳǡ ͳǡ ͵ǡ ͷǡ ͷǡ ͸ǡ ͸ǡ ͸ǡ ͸ǡ ͹ǡ ͺǡ ͻǡ ͳͲǡ ͳͲǡ ͳͳǡ ͳʹǡ ͳʹǡ ͳͶǡ ͳͶǡ ͳͷǡ ͳͷǡ ͳͷǡ ͳ͹ǡ ͳ͹ǡ ʹͲǡ ʹ͵ǡ ʹͷǡ ʹ͹}

He wants to make a graph that shows the number of books read by each student. Which graphical representation would you recommend for this situation?

Explain your reasoning.

Complete each statement. o Greer o Jordan

should use a bar graph. A bar graph is

the best option for this student because it o visually shows how each part relates to the whole. o visually shows how each part compares to the other parts.

o Greer o Jordan

Mr. Shapiro’s seventh grade class read more books than any other class in the middle school over the summer. The number of books read by the students in his class is shown below.

2.

Kendrick wants to make a graph that will show what proportion of students in his homeroom like science class best when compared to language arts, math, and social studies. Explain which graphical representation would best show this comparison.

should use a circle graph. A circle graph

is the best option for this student because it o visually shows how each part relates to the whole. o visually shows how each part compares to the other parts.

274

Unit 13: Numerical and Categorical Data

Unit 13, Lesson 8: Creating Displays for Numerical Data

13.7.6 Wrap-Up: Growth Mindset 1.

Every mistake a person makes, their brain actually grows! Take a moment to celebrate your mistakes by summarizing your favorite mistake from today.

My favorite mistake from today was …

Benchmark MA.7.DP.1.5 Given a real-world numerical or categorical data set, choose and create an appropriate graphical representation.

Learning Target



I can create an appropriate graphical representation for a numerical data set.

Unit 13: Numerical and Categorical Data

275

13.8.2 Collaboration: Who Made It Best?

13.8.1 Warm-Up: Mystery Boxes The data set below contains 14 numbers, listed in order from least to greatest. Six of the values have been removed. Your job is to use the clues given to determine the missing values. Write your answers in the boxes.

The The The The The

• • • • • ͳ

Ͷ

mean is ʹ͸. median is ʹʹ. interquartile range is ʹͲ. mode is ͳͺ. range is ͸Ͳ.

ͳͷ ͳͺ

1.

Dante and Lorraine are both managers of the seventhgrade basektball team. They are trying to decide which graphical representation to use to show the number of turnovers the team has had per game in the first ͳʹ games of the season, shown in the data set below. {ʹǡ ʹǡ ͵ǡ Ͷǡ Ͷǡ ͷǡ ͷǡ ͸ǡ ͹ǡ ͺǡ ͳͲǡ ͳʹ}

Lorraine made a histogram, as shown.

Turnovers per Game

ʹͻ ͵Ͳ ͵ʹ

ͷͺ

Frequency

1.

Number of Turnovers per Game

Dante made a line plot, as shown.

Turnovers in the First Twelve Games of the Season

276

Unit 13: Numerical and Categorical Data

a.

13.8.3 Try It: Creating Graphical Representations

Which of the two graphical representations shown is best to display the data?

1.

b.

Explain your reasoning.

Laverne and Rocco planned to travel the greatest distance between any two points in the United States of America — from Log Point, Elliot Key, Florida to Kure Atoll, Hawaii — a distance of ͷǡͺͷͻ miles.

They took a ferry from Log Point to Miami, then drove to Los Angeles, California, where they took a plane to Hawaii. They drove anywhere from ͷǦͺ hours a day, and kept track of how many miles they traveled per day on their trip. This information is shown below. {ʹʹͶǡ ʹͻͶǡ ͵Ͳͻǡ ͵ͳ͹ǡ ͵ʹͳǡ ͵͵ͻǡ ͵ͷͷǡ ͵͸ͳǡ ͵ͺ͵ǡ Ͷͷ͸ǡ ʹͷͲͲ}

c.

Compare your work with your partner and if necessary, resolve any differences.

d.

Discuss with your partner whether a box plot or stem-and-leaf plot would be useful for displaying this data set. Summarize your discussion.

a.

Create an appropriate graphical representation for the data.

Unit 13: Numerical and Categorical Data

277

b.

If the value 2500 were removed from the data set, which display would you create to represent the data? Explain your reasoning.

2.

An earthworm farmer set up several containers of a certain species of earthworm so he could learn about their lengths. Lengths of earthworms provide information about their ages. The farmer measured the lengths of the earthworms in one container.

The farmer recorded the earthworms’ lengths, in millimeters (mm), in the table shown.

Earthworm Lengths in Millimeters (mm)

c.

278

Create an appropriate graphical representation for the data without the value 2500.

6.0

11.2 18.3 19.4 20.2 23.1 23.3 25.2 25.7 26.0

54.1

59.7 60.3 77.6 93.2

27.8 a.

27.9 28.4 29.9 32.5 33.2 41.1 42.3 48.5 52.1

Create an appropriate graphical representation for the data.

Unit 13: Numerical and Categorical Data

b.

Explain your reasoning for choosing your display type.

13.8.4 Your Turn!

1.

The birth weights, in ounces, of golden retriever puppies born at a kennel were recorded for the past month. The data is recorded in the table.

13 14 15 15 16 16 16 16 17 17 17 17 17 17 17 18 18 18 18 18 18 18 18 19 20

c.

a.

Create an appropriate graphical display for the data.

b.

Explain your reasoning for choosing that display type.

If you were asked to create an additional display, which would you make and why?

Unit 13: Numerical and Categorical Data

279

2.

Tom Brady’s pass-completion percentages from 2009 to 2020 are listed in the table.

Year % Year %

2009

2010

2011

2012

2013

2014

2015

2016

2017

2018

2019

2020

65.7 64.4

65.9 67.4

65.6 66.3

63

65.8

60.5 60.8

Compare your graph with your partner. If you created the same type of graph, explain why this graph is the best choice to represent the data. If you created different displays, discuss which graph is a better choice.

64.1 65.7

a.

Create an appropriate graphical display for the data.

b.

Explain your reasoning for choosing that display type.

280

c.

Unit 13: Numerical and Categorical Data

Unit 13, Lesson 9: Creating Displays for Categorical Data

13.8.5 Wrap-Up: Cool Down 1.

What is your favorite type of graphical representation to create when displaying numerical data?

Benchmark 2.

Write a note to your teacher explaining why this is your favorite type of display to create. Be specific in your explanation. You can use illustrations and examples from the lesson to help in your explanation.

MA.7.DP.1.5 Given a real-world numerical or categorical data set, choose and create an appropriate graphical representation.

Learning Target 

I can create an appropriate graphical representation for a categorical data set.

Unit 13: Numerical and Categorical Data

281

13.9.2 Guided Instruction: Graphical Representations for Categorical Data

13.9.1 Warm-Up: Visual Representations Various number systems have developed over time. In some of the original systems, symbols representing values were written one after another and then added together to determine the value represented. The Roman Numeral system added the ability to represent values with fewer symbols. Even so, to represent Ͷͻͻͻ in Roman Numerals is MMMMCMXCIX, which is still quite long. The base-10 system used today allows for positional notation where the value of a number depends on its placement.

1.

Lupe is reviewing her bills to determine if there is a way to save money by using less energy. In 2020, she paid a total of $1477 in electricity costs to Florida Power and Light (FPL). The table shows the total cost associated with each area of electricity use in her household for the year.

Area of Electricity Use Air conditioning Space heating Water heating Appliances, electronics, and lighting

What are some reasons it is better to use the base-ͳͲ representation of the number instead of the Roman Numerals?

1.

2.

133 207 738

Percentage

Use the data to create a bar graph for Lupe’s data set.

The symbols used for the base-ͳͲ system varied regionally for a long time. The current digits used worldwide, Ͳͳʹ͵Ͷͷ͸͹ͺͻǡ evolved from the North African region of the Arab empire. Why do you think having a standard worldwide numbering system is helpful?

282

Annual Cost, in Dollars 399

Unit 13: Numerical and Categorical Data

2.

Complete the table by calculating the percentage of Lupe’s electricity costs for each area of electricity use.

3.

Create a circle graph for Lupe’s data set.

Lupe can use the o bar graph o circle graph

to analyze whether

there are any areas of electricity use that make up too large of a portion of her bill to determine where she should try to reduce her electricity consumption. She can use the o bar graph o circle graph

to show her family the

actual cost of each area of use as they set goals for using less energy, such as taking shorter showers or running the air conditioner at a higher temperature.

4.

5.

Discuss with your partner how each data display can be useful to Lupe as she considers ways to reduce her energy bills.

Complete the statements. o actual amount The bar graph displays the o proportion of the total cost Lupe pays for each area of electricity use, while the o actual amount circle graph displays the o proportion of the total cost Lupe pays for each.

Bar graphs and circle graphs are both useful for visually representing categorical data.

Bar graphs are best when the goal is to compare categories to each other or to compare the actual values. They visually show how each part compares to other parts and display the exact counts or values for each category.

Circle graphs are best when the goal is to compare parts to a whole. They visually show each category as a proportion related to the whole.

The purpose of displaying the data can also be used to determine which graph is best.

Unit 13: Numerical and Categorical Data

283

13.9.3 Try It: Graphical Representations for Categorical Data

2.

Use your calculated percentages to create a circle graph displaying Terrance’s survey data.

3.

With your partner, discuss some conclusions that can be drawn about Terrance’s customer service based on the circle graph. Then, write three statements summarizing the conclusions discussed.

Terrance is a server at a local restaurant and is being evaluated by his manager on his customer service skills.

After their meals, 50 customers sitting at his tables were given a survey asking to rate their satisfaction with their service on the scale shown.

Amazing Good

Okay

Poor

Terrible

The results are shown in the frequency table.

Rating Frequency

1.

ͳʹ

ʹͷ

ͺ

͵

ʹ

Complete the table to determine the percentage of customers who responded with each rating. The first row has been completed for you.

Rating

Percentage 12 50

284

= 0.24

0.24 × 100 = 24%

Unit 13: Numerical and Categorical Data

4.

Complete the statement.

7.

The best graphical representation for Terrance’s data is a

o bar graph o circle graph

because it helps us visualize how

each rating category compares to the whole.

5.

If Terrance’s manager wants to compare two servers’ customer service skills, explain whether it would be better to create two circle graphs or two bar graphs.

Complete the statements.

If the same number of customers responded to each o bar graph server’s survey, a o circle graph

is a better choice

for comparing these two data sets because the o height of the bars o size of each section

in each server’s graph can be

easily compared side-by-side.

If a different number of customers responded to each o bar graph server’s survey, a o circle graph

would be better

because it would compare the o actual number of customers o proportion of customers

who gave each

response rather than the raw numbers that could be misleading in this case.

6.

Compare your recommendation with your partner’s.

Discuss any differences in your reasoning.

Unit 13: Numerical and Categorical Data

285

THIS PAGE WAS INTENTIONALLY LEFT BLANK.

13.9.4 Wrap-Up: Ticket Out the Door The manager at a new restaurant surveys a sample of customers to identify where their marketing efforts have been successful. The results are shown in the frequency table.

Source New Customers

Facebook Ad

Friend

Google Ad

Newspaper

ʹ

ͻ

ͳ͵

ͺ

1.

The restaurant manager wants to make a recommendation to the owner about where to spend more money on advertisements using a data display.

What graphical display is best for the manager to use to display the data?

2.

Create the data display you chose in Question 1 to display the data collected by the restaurant manager.

286

Unit 13: Numerical and Categorical Data

Benchmark Index MA.7.GR.1.5

Lesson 5.5, Lesson 5.6, Lesson 5.7, Lesson 5.8, Lesson 5.9, Lesson 5.10

MA.7.GR.2.1

Lesson 12.1, Lesson 12.2, Lesson 12.3

MA.7.GR.2.2

Lesson 12.2, Lesson 12.3

MA.7.GR.2.3

Lesson 12.4, Lesson 12.5, Lesson 12.6

MA.7.NSO.1.1

Lesson 8.1, Lesson 8.2, Lesson 8.3

MA.7.NSO.1.2

Lesson 2.3, Lesson 2.4, Lesson 2.5

MA.7.NSO.2.1

Lesson 8.4, Lesson 8.5, Lesson 8.6

MA.7.NSO.2.2

Lesson 3.1, Lesson 3.2, Lesson 3.3, Lesson 3.4, Lesson 8.4, Lesson 8.5, Lesson 8.6

Lesson 11.4, Lesson 11.5, Lesson 11.8

MA.7.NSO.2.3

Lesson 3.5, Lesson 3.6, Lesson 3.7, Lesson 3.8

MA.7.AR.4.2

Lesson 11.2, Lesson 11.3, Lesson 11.5, Lesson 11.6, Lesson 11.7, Lesson 11.8

MA.K12.MTR.1.1

embedded throughout

MA.7.AR.4.3

Lesson 11.1, Lesson 11.2, Lesson 11.8

MA.K12.MTR.2.1

embedded throughout

MA.7.AR.4.4

Lesson 11.6, Lesson 11.7, Lesson 11.8

MA.K12.MTR.3.1

embedded throughout

MA.7.AR.4.5

Lesson 6.9, Lesson 6.10, Lesson 6.11

MA.K12.MTR.4.1

embedded throughout

MA.7.DP.1.1

Lesson 13.1, Lesson 13.2, Lesson 13.3

MA.K12.MTR.5.1

embedded throughout

MA.7.DP.1.2

Lesson 13.4, Lesson 13.5, Lesson 13.6

MA.K12.MTR.6.1

embedded throughout

MA.7.DP.1.3

Lesson 5.1, Lesson 5.2, Lesson 5.3, Lesson 5.4

MA.K12.MTR.7.1

embedded throughout

MA.7.DP.1.4

Lesson 7.1, Lesson 7.2, Lesson 7.3, Lesson 7.4

ELA.K12.EE.1.1

embedded throughout

MA.7.DP.1.5

Lesson 13.7, Lesson 13.8, Lesson 13.9

ELA.K12.EE.2.1

embedded throughout

MA.7.DP.2.1

Lesson 2.1, Lesson 2.2

ELA.K12.EE.3.1

embedded throughout

MA.7.DP.2.2

Lesson 2.5, Lesson 2.6

ELA.K12.EE.4.1

embedded throughout

MA.7.DP.2.3

Lesson 2.7, Lesson 2.9

ELA.K12.EE.5.1

embedded throughout

MA.7.DP.2.4

Lesson 2.7, Lesson 2.8, Lesson 2.9, Lesson 5.2

ELA.K12.EE.6.1

embedded throughout

MA.7.GR.1.1

Lesson 4.1, Lesson 4.2

MA.7.GR.1.2

Lesson 4.3, Lesson 4.4

MA.7.GR.1.3

Lesson 4.5, Lesson 4.6, Lesson 4.7, Lesson 4.10, Lesson 11.9

MA.7.GR.1.4

Lesson 4.8, Lesson 4.9, Lesson 4.10, Lesson 11.9

MA.7.AR.1.1

Lesson 9.1, Lesson 9.2, Lesson 9.3, Lesson 9.4

MA.7.AR.1.2

Lesson 9.5, Lesson 9.6, Lesson 9.7, Lesson 9.8

MA.7.AR.2.1

Lesson 10.1, Lesson 10.2, Lesson 10.3, Lesson 10.4

MA.7.AR.2.2

Lesson 10.5, Lesson 10.6, Lesson 10.7, Lesson 10.8, Lesson 10.9, Lesson 10.10

MA.7.AR.3.1

Lesson 7.5, Lesson 7.6, Lesson 7.7, Lesson 7.8, Lesson 7.9, Lesson 7.10

MA.7.AR.3.2

Lesson 6.6, Lesson 6.7, Lesson 6.8, Lesson 6.9, Lesson 6.11

MA.7.AR.3.3

Lesson 6.1, Lesson 6.2, Lesson 6.3, Lesson 6.4, Lesson 6.5, Lesson 6.9

MA.7.AR.4.1

ELD.K12.ELL.MA.1 embedded throughout ELD.K12.ELL.SI.1 embedded throughout

287

Appendix C: 6-12 Mathematics Glossary absolute value

bar graph

A visual display of categorical data values where each category is represented by a bar whose height represents the number in that category. Bar graphs can be represented vertically or horizontally.

base (of an exponent)

The number used as a factor in exponential form.

bivariate data

Data that measures two characteristics of a population.

box plot

A plot displaying the spread or distribution of a data set using a five number summary, the minimum, lower quartile, median, upper quartile and maximum. It is also called a boxand-whisker plot.

capacity

Angles are formed wherever two lines, segments or rays intersect. Angles are measured in degrees.

The amount of space that can be filled in a container. Both capacity and volume are used to measure three-dimensional spaces; however, capacity usually refers to fluid measures, whereas volume is measured in cubic units.

categorical data

The measure, in square units, of the inside region of a closed two-dimensional figure.

A type of data which is divided into groups.

Categorical data are qualitative.

central angle

An angle that has its vertex at the center of a circle with radii as its sides.

circle graph

A visual display of categorical data. The whole set of data is represented by the circle and its interior. The categories are represented by fractional parts of the circle. Also called a pie chart.

circumference

The distance around a circle.

circumscribed circle

The smallest circle that includes a plane figure.

If the figure is a polygon, then the circle must contain all of the vertices of the polygon. Not every polygon has a circumscribed circle, but all triangles and all regular polygons have circumscribed circles.

additive identity property

Refer to Properties of Operations, Equality and Inequality (Appendix D).

additive inverse property

Refer to Properties of Operations, Equality and Inequality (Appendix D).

addition property of equality

Refer to Properties of Operations, Equality and Inequality (Appendix D).

addition property of inequality

Refer to Properties of Operations, Equality and Inequality (Appendix D).

algorithm

A step-by-step way to solve a problem.

analytic geometry

The branch of mathematics that uses functions and relations to study geometric phenomena.

angle (∠) area arithmetic sequence

association

288

A number’s distance from zero (0) on a number line. Distance is expressed as a positive value.

FLORIDA’S B.E.S.T. STANDARDS: MATHEMATICS

A sequence of numbers in which each consecutive pair of numbers has a common difference.

A way to describe the form, direction or strength of the relationship between the two variables in a bivariate data set. For numerical data, descriptions include linear or nonlinear; positive or negative; strong or weak. For categorical data, descriptions include strong or weak.

associative property

Refer to Properties of Operations, Equality and Inequality (Appendix D).

axes (of a graph)

The horizontal and vertical number lines used in a coordinate plane system.

Appendix C: 6-12 Mathematics Glossary cluster (data)

Data that are in a close group on a scatter plot or univariate numerical data that have similar values.

coefficient

The number or constant that multiplies a variable in an algebraic expression. If no number is specified, the coefficient is 1.

commutative property (of addition or multiplication)

Refer to Properties of Operations, Equality and Inequality (Appendix D).

complementary angles

Two angles whose measures sum to 90°.

composite number

A whole number greater than 1 that has at least one whole-number factor other than one and itself.

compound inequality

A conjunction of two or more inequalities.

concave polygon

A polygon with one or more diagonals that have points outside the polygon. See convex polygon.

conditional relative frequency

The ratio of a joint relative frequency and a marginal relative frequency. Equivalently, the ratio of a relative frequency and a marginal frequency.

cone

A three-dimensional figure with a circular base and an apex that is connected to the base by a collection of line segments that form a curved surface.

congruent

Having exactly the same shape and size.

Equivalently, two figures are congruent if one can be mapped to the other using a rigid transformation.

constant of proportionality

The constant value of the ratio of two proportional quantities.

converse of Pythagorean Theorem

If the lengths 𝑎𝑎, 𝑏𝑏 and 𝑐𝑐 of the three sides of a triangle satisfy the relationship 𝑎𝑎! + 𝑏𝑏! = 𝑐𝑐 ! , then the triangle is a right triangle.

FLORIDA’S B.E.S.T. STANDARDS: MATHEMATICS convex polygon

A polygon with each interior angle measuring less than 180°. All diagonals of a convex polygon lie inside the polygon. See concave polygon.

coordinate

A number used to locate a point on a number line. One of the numbers in an ordered pair, or triple, that locates a point on a coordinate plane or in coordinate space, respectively.

coordinate plane

A plane determined by two perpendicular number lines called axes. The axes intersect at the origin. Each point in the coordinate plane is represented by a pair of coordinates that represent the direction and distance from each axis. The origin is represented by the coordinate pair (0,0).

corresponding angles

Angles that are in the same position on two parallel lines in relation to a transversal.

cube

A rectangular prism with six congruent square faces.

customary units

The units of measure used in the United States.

• Customary units for length include inches, feet, yards, and miles.

• Customary units for weight include ounces, pounds, and tons.

• Customary units for volume include cubic inches, cubic feet, and cubic yards.

• Customary units for capacity include fluid ounces, cups, pints, quarts, and gallons.

A figure containing two congruent, parallel, circular bases whose edges are connected by a cylinder (circular) curved surface. The net of the cylinder consists of a parallelogram and two circles. data

Values that are collected together for reference or analysis.

diameter

A line segment from any point on the circle passing through the center to another point on the circle.

289

Appendix C: 6-12 Mathematics Glossary dilation

A proportional increase or decrease in size in all directions.

distributive property

Refer to Properties of Operations, Equality and Inequality (Appendix D).

domain

The complete set of possible values of the input of a function or relation. The domain may vary depending on the context. See range (of a relation or function).

equilateral triangle

A triangle with three equal-length sides and three 60° interior angles. Also known as an equiangular triangle.

estimation

The use of methods to determine a reasonably accurate approximation, without calculating an exact answer.

geometric sequence

A sequence of numbers in which each consecutive pair of numbers has a common ratio.

greatest common factor (GCF) of two or more whole numbers

The largest whole number that evenly divides the given whole numbers.

histogram

A visual display of numerical data using bars along a number line with no spaces between the bars. The height of each bar represents either the frequency or relative frequency of data within that interval.

hypotenuse

The longest side of a right triangle; the side opposite the right angle.

identity property of addition

Refer to Properties of Operations, Equality and Inequality (Appendix D).

event

A set of possible outcomes resulting from an experiment. In general, an event is any subset of a sample space.

exponent (exponential form)

The number of times the base occurs as a factor.

identity property of multiplication

Refer to Properties of Operations, Equality and Inequality (Appendix D).

exponential function

An exponential function is a function with a constant percent rate of change.

inscribed angle in a circle

An angle which is formed in the interior of a circle when two chords share an endpoint.

experimental probability

The ratio of the number of times an event occurs to the total number of trials or times the activity is performed. Also called empirical probability.

inscribed circle

frequency table

A table that shows how often each item, number, or range of numbers occurs in a set of data.

The largest possible circle that can be contained in a plane figure. If the plane figure is a polygon, then the circle must be tangent to all of the sides of the polygon. Not every polygon has an inscribed circle, but all triangles and all regular polygons have inscribed circles.

inscribed polygon in a circle

A polygon which has all of its vertices on a circle.

integers

Whole numbers and their opposites.

function

function notation

290

FLORIDA’S B.E.S.T. STANDARDS: MATHEMATICS

A mathematical relation for which each element of the domain corresponds to exactly one element of the range.

A notation that describes a function. For a function 𝑓𝑓 when 𝑥𝑥 is a member of the domain, the symbol 𝑓𝑓(𝑥𝑥) denotes the corresponding member of the range.

Appendix C: 6-12 Mathematics Glossary intercept

The value of a variable when all other variables in the equation equal 0. On a graph, the values where a function crosses an axis.

interquartile range (IQR)

A measure of variation in a set of numerical data, the interquartile range is the distance between the first and third quartiles of the data set. See quartile and box plot.

inverse functions irrational number joint frequency

Two functions, 𝑦𝑦 = ℎ(𝑥𝑥) and 𝑥𝑥 = 𝑔𝑔(𝑦𝑦), are said to be inverses when 𝑔𝑔(ℎ(𝑥𝑥)) = 𝑥𝑥 and ℎ(𝑔𝑔(𝑦𝑦)) = 𝑦𝑦. The function inverse to 𝑓𝑓(𝑥𝑥) is denoted 𝑓𝑓 "# (𝑥𝑥).

A real number that cannot be expressed as a ratio of two integers.

In a two-way table, joint frequency is the number of times a combination of two conditions occurs.

joint relative frequency

Joint relative frequency is the ratio of the joint frequency and the total number of data points.

least common multiple (LCM)

The lowest number that is a multiple of two or more given numbers.

line of fit

A line drawn on a scatter plot to estimate the relationship between two sets of data. Also known as a trend line. See scatter plot.

line graph

A graph that displays numerical data using connected line segments.

line plot

A visual display of data values where each data value is shown as a dot or mark above a number line. Also known as a dot plot.

linear expression (or linear equation)

A polynomial expression or equation that contains a term of degree 1, but no term of higher degree.

linear function

A function that has a constant rate of change.

FLORIDA’S B.E.S.T. STANDARDS: MATHEMATICS

line of symmetry

A line that divides a figure into two congruent parts, so that the reflection of either part across the line maps precisely onto the other part.

matrix

A rectangular array of numbers or variables.

mean

The arithmetic average of a set of numbers. It is a measure of central tendency.

measures of center

A numerical value used to describe the overall clustering of data in a set, or the overall central value of a set of data. The three most common measures of central tendency are the mean, median, and mode.

measures of variability

A numerical value that measures how much a data set varies from a central value.

median

The middle of an ordered list of values. If the list has an odd number of values, it is the middle value of that list. If the list has an even number of values, it is the average of the two middle values. It is a measure of central tendency.

metric units

The units of measure used in most of the world. Like the decimal system, the metric system uses the base 10.

• Metric units for length include millimeters, centimeters, meters, and kilometers.

• Metric units for mass include milligrams, grams, and kilograms.

• Metric units for volume include cubic millimeters, cubic centimeters, and cubic meters.

• Metric units for capacity include milliliters, centiliters, liters, and kiloliters.

mode

The value found most often in a set of numbers. There may be no mode, one mode, or more than one mode in a set of numbers. It is a measure of central tendency.

291

Appendix C: 6-12 Mathematics Glossary

piecewise function

A function defined by multiple sub functions, each of which applies to a certain interval of the main function’s domain.

polynomials

The sum or difference of terms which have variables raised to non-negative integer powers and which have coefficients that may be real or complex.

population (in data analysis)

The entire set of cases or individuals under consideration in a statistical analysis.

prime factorization

The expression of a number as the product of prime factors.

prime number

A whole number greater than 1 that is not divisible by any whole number other than 1 and itself.

principal square roots

The principal square root is the positive square root of a positive real number.

proportional relationships

A collection of pairs of numbers that are in equivalent ratios.

A value that is much higher or much lower than the other values in a set of data.

quadrant

Any of the four regions separated by the axes in a coordinate plane.

percent of change

The difference between a final value and an initial value, expressed as a percentage of the initial value.

quadratic expression (or quadratic equation)

A polynomial expression or equation that contains a term of degree 2, but no term of higher degree.

percent error

The difference between the estimated number and the actual number as a percentage of the actual value.

quadratic function

A polynomial function with degree of 2.

quartiles

For a data set with median M, the first quartile is the median of the data values less than M and the third quartile is the median of the data values greater than M. The second quartile is the median M.

radius

A line segment extending from the center of a circle or sphere to a point on the circle or sphere.

monomial

A polynomial with one term.

multiplicative identity

Refer to Properties of Operations, Equality and Inequality (Appendix D).

multiplicative inverse (reciprocal)

Refer to Properties of Operations, Equality and Inequality (Appendix D).

net

A two-dimensional diagram that can be folded or made into a three-dimensional figure.

order of operations

The order of performing computations is to first work within grouping symbols using the order of operations. Then simplify terms with exponents. Next, while reading from left to right, perform multiplication and division in the order in which it appears. Finally, while reading from left to right, perform addition and subtraction in the order in which it appears.

origin

outlier

pi (𝜋𝜋)

292

FLORIDA’S B.E.S.T. STANDARDS: MATHEMATICS

The point of intersection of the 𝑥𝑥- and 𝑦𝑦-axes in a rectangular coordinate system, where the 𝑥𝑥coordinate and 𝑦𝑦-coordinate are both 0.

The symbol designating the ratio of the circumference of a circle to its diameter. It is an irrational number. Common approximations are 3.14,

!! $

or

%&& ##%

.

Appendix C: 6-12 Mathematics Glossary

random sampling

A smaller group of people or objects chosen from a larger group or population by a process giving equal chance of selection to all possible people or objects, and all possible subsets of the same size.

random variable

An assignment of a numerical value to each outcome in a sample space.

range (of a data set)

The difference between the highest data value and the lowest data value.

range (of a relation or function)

The complete set of possible values of the output of a relation or function. See domain.

rate

A ratio that compares two quantities of different units.

rate of change

FLORIDA’S B.E.S.T. STANDARDS: MATHEMATICS

rigid transformation

A transformation of points in space consisting of a sequence of one or more translations, reflections, or rotations. Rigid transformations preserve distances and angle measures (congruency).

rotation

A transformation of a figure by turning it about a center point or axis. The amount of rotation can be expressed in the number of degrees.

The direction of the rotation for twodimensional figures can be expressed as clockwise or counterclockwise.

sample space

In a probability model for a random process, a list of the individual outcomes that are to be considered.

The ratio of change in one quantity to the corresponding change in another quantity.

scale

The numeric values, set at fixed intervals, assigned to the axes of a graph.

rational expression

A quotient of two polynomials with a non-zero denominator.

scale factor

rational number

A real number that can be expressed as the ratio of two integers.

The constant that is multiplied by the length of each side of a figure to produce an image that is the same shape as the original figure.

scale model

A model or drawing based on a ratio of the dimensions for the model and the actual object it represents.

scatter plot

A graph in the coordinate plane representing a set of bivariate numerical data that is used to observe the relationship between two variables. See line of fit.

scientific notation

A method of writing very large or very small numbers using exponents in which a number is expressed as the product of a power of 10 and a number that is at least 1 and less than 10.

real numbers

The set of all rational and irrational numbers.

reflection

A transformation that produces the mirror image of a geometric figure across a line of reflection.

regular polygon

A polygon that is both equilateral (all sides congruent) and equiangular (all angles congruent).

relation

A set of input-output pairs.

repeated experiment

A random experiment done with the same conditions and parameters as a previous one.

293

Appendix C: 6-12 Mathematics Glossary

set-builder notation

significant digits

similarity

A shorthand used to write sets, often sets with an infinite number of elements. The set {𝑥𝑥: 𝑥𝑥 > 0} is read aloud, "the set of all 𝑥𝑥 such that 𝑥𝑥 is greater than 0." It is read aloud exactly the same way when the colon : is replaced by the vertical line | as in {𝑥𝑥|𝑥𝑥 > 0}.

The nonzero digits of a number and the zeros that are included between them or any trailing zeros that are considered to be precise.

Having exactly the same shape but not necessarily the same size. Equivalently, two figures are similar if one can be mapped to the other using a rigid transformation combined with a dilation, including cases with a scale factor of 1.

simple interest

A method of computing interest. Interest is computed from the (original) principal alone no matter how much money has accrued so far.

simulation

A simulation is an approximate imitation of a statistical experiment, often done with a computer program to examine the statistics of a large quantity of trials.

slope

294

FLORIDA’S B.E.S.T. STANDARDS: MATHEMATICS

The ratio of the change in the vertical direction (𝑦𝑦 direction) to change in the horizontal direction (𝑥𝑥 direction), often expressed as

∆( ∆)

theoretical probability

A number between 0 and 1 representing the likelihood of an event in a theoretical model based on a sample space. If all outcomes in the sample space are equally likely, then theoretical probability of an event is the ratio of the number of outcomes in the event to the number of outcomes in the sample space.

translation

A transformation in which every point in a figure is moved in the same direction and by the same distance.

transversal

A line that intersects two or more lines in the same plane at different points.

trigonometric function

Any of the six functions (sine, cosine, tangent, cotangent, secant, cosecant) that, for an acute angle of a right triangle, may be expressed in terms of ratios of sides of the right triangle.

unit rates

A ratio comparing a number of units of one quantity to one unit of a second quantity.

vertical angles

The opposite angles formed when two lines intersect.

𝑥𝑥-intercept

The value of 𝑥𝑥 at the point where a line or graph intersects the 𝑥𝑥-axis. The value of 𝑦𝑦 is 0 at this point.

𝒚𝒚-intercept

The value of 𝑦𝑦 at the point where a line or graph intersects the 𝑦𝑦-axis. The value of 𝑥𝑥 is 0 at this point.

.

statistical question

A question that can be answered by collecting data. Often there will be variability in the data.

stem-and-leaf plot

A table that organizes data by place value to compare data frequencies.

supplementary angles

Two angles with measures the sum of which is exactly 180°.

Appendix D: Properties of Operations, Equality and Inequality

FLORI DA’ S B. E. S. T. STANDARDS : MATHEMATI CS

Properties of Operations

Properties of Equality

The table below illustrates the properties of operations. For each property, the variables 𝑎𝑎, 𝑏𝑏, and 𝑐𝑐 stand for arbitrary numbers in a given number system. The properties of operations apply to the rational number system, the real number system and the complex number system.

Property of Operation Associative property of addition Commutative property of addition Additive identity property of zero Existence of additive inverses Associative property of multiplication Commutative property of multiplication Multiplicative identity property of one Existence of multiplicative inverses Distributive property of multiplication over addition

Property of Equality

Example

Reflexive property of equality

(𝑎𝑎 + 𝑏𝑏) + 𝑐𝑐 = 𝑎𝑎 + (𝑏𝑏 + 𝑐𝑐)

Symmetric property of equality

𝑎𝑎 + 𝑏𝑏 = 𝑏𝑏 + 𝑎𝑎

Transitive property of equality

𝑎𝑎 + 0 = 𝑎𝑎 0 + 𝑎𝑎 = 𝑎𝑎

For every 𝑎𝑎 there exists −𝑎𝑎 so that 𝑎𝑎 + (−𝑎𝑎) = 0 and (−𝑎𝑎) + 𝑎𝑎 = 0. (𝑎𝑎 × 𝑏𝑏) × 𝑐𝑐 = 𝑎𝑎 × (𝑏𝑏 × 𝑐𝑐)

𝑎𝑎 × " = 1 and

Subtraction property of equality

Division property of equality

For every 𝑎𝑎 ≠ 0 there exists !

Addition property of equality

Multiplication property of equality

𝑎𝑎 × 𝑏𝑏 = 𝑏𝑏 × 𝑎𝑎 𝑎𝑎 × 1 = 𝑎𝑎 1 × 𝑎𝑎 = 𝑎𝑎

The table below illustrates the properties of equality. For each property, the variables 𝑎𝑎, 𝑏𝑏, and 𝑐𝑐 stand for arbitrary numbers in a given number system. The properties of equality apply to the rational number system, the real number system and the complex number system.

!

"

× 𝑎𝑎 = 1.

𝑎𝑎 × (𝑏𝑏 + 𝑐𝑐) = (𝑎𝑎 × 𝑏𝑏) + (𝑎𝑎 × 𝑐𝑐)

!

"

so that

Substitution property of equality

Example 𝑎𝑎 = 𝑎𝑎

If 𝑎𝑎 = 𝑏𝑏, then 𝑏𝑏 = 𝑎𝑎.

If 𝑎𝑎 = 𝑏𝑏 and 𝑏𝑏 = 𝑐𝑐, then 𝑎𝑎 = 𝑐𝑐.

If 𝑎𝑎 = 𝑏𝑏, then 𝑎𝑎 + 𝑐𝑐 = 𝑏𝑏 + 𝑐𝑐.

If 𝑎𝑎 = 𝑏𝑏, then 𝑎𝑎 − 𝑐𝑐 = 𝑏𝑏 − 𝑐𝑐.

If 𝑎𝑎 = 𝑏𝑏, then 𝑎𝑎 × 𝑐𝑐 = 𝑏𝑏 × 𝑐𝑐.

If 𝑎𝑎 = 𝑏𝑏 and 𝑐𝑐 ≠ 0, then 𝑎𝑎 ÷ 𝑐𝑐 = 𝑏𝑏 ÷ 𝑐𝑐.

If 𝑎𝑎 = 𝑏𝑏, then 𝑏𝑏 may be substituted for 𝑎𝑎 in any expression containing 𝑎𝑎.

295

Appendix D: Properties of Operations, Equality and Inequality

FLORI DA’ S B. E. S. T. STANDARDS : MATHEMATI CS

Properties of Inequality The table below illustrates the properties of inequality. For each property, the variables 𝑎𝑎, 𝑏𝑏, and 𝑐𝑐 stand for arbitrary numbers in a given number system. In addition, exactly one of the following is true: 𝑎𝑎 < 𝑏𝑏, 𝑎𝑎 = 𝑏𝑏, or 𝑎𝑎 > 𝑏𝑏. The properties of inequality apply to the rational number system and the real number system.

Property of Inequality Asymmetric property of inequality Transitive property of inequality Addition property of inequality Subtraction property of inequality Multiplication property of inequality Division property of inequality

296

Example If 𝑎𝑎 > 𝑏𝑏, then 𝑏𝑏 < 𝑎𝑎.

If 𝑎𝑎 > 𝑏𝑏 and 𝑏𝑏 > 𝑐𝑐, then 𝑎𝑎 > 𝑐𝑐.

If 𝑎𝑎 > 𝑏𝑏, then 𝑎𝑎 + 𝑐𝑐 > 𝑏𝑏 + 𝑐𝑐.

If 𝑎𝑎 > 𝑏𝑏, then 𝑎𝑎 − 𝑐𝑐 > 𝑏𝑏 − 𝑐𝑐.

If 𝑎𝑎 > 𝑏𝑏 and 𝑐𝑐 > 0, then 𝑎𝑎 × 𝑐𝑐 > 𝑏𝑏 × 𝑐𝑐.

If 𝑎𝑎 > 𝑏𝑏 and 𝑐𝑐 < 0, then 𝑎𝑎 × 𝑐𝑐 < 𝑏𝑏 × 𝑐𝑐.

If 𝑎𝑎 > 𝑏𝑏 and 𝑐𝑐 > 0, then 𝑎𝑎 ÷ 𝑐𝑐 > 𝑏𝑏 ÷ 𝑐𝑐.

If 𝑎𝑎 > 𝑏𝑏 and 𝑐𝑐 < 0, then 𝑎𝑎 ÷ 𝑐𝑐 < 𝑏𝑏 ÷ 𝑐𝑐.

Appendix E: K-12 Formulas

FLORIDA’S B.E.S.T. STANDARDS: MATHEMATICS

Area of a two-dimensional figure 𝐴𝐴 = 𝑙𝑙𝑙𝑙, where 𝑙𝑙 is the length and 𝑤𝑤 is the width 𝐴𝐴 = 𝑏𝑏ℎ, where 𝑏𝑏 is the base and ℎ is the height

Rectangle

𝐴𝐴 = 𝑙𝑙𝑙𝑙, where 𝑙𝑙 is the length and 𝑤𝑤 is the width 𝐴𝐴 = 𝑏𝑏ℎ, where 𝑏𝑏 is the base and ℎ is the height 𝐴𝐴 = 𝑠𝑠 ! , where 𝑠𝑠 is the side length

Square

"

𝐴𝐴 = ! 𝑏𝑏ℎ, where 𝑏𝑏 is the base and ℎ is the height

Triangle

"

Trapezoid Parallelogram

𝐴𝐴 = ! (𝑏𝑏" + 𝑏𝑏! )ℎ, where 𝑏𝑏" and 𝑏𝑏! are the bases and ℎ is the height

𝐴𝐴 = 𝑏𝑏ℎ, where 𝑏𝑏 is the base and ℎ is the height 𝐴𝐴 = 𝑏𝑏ℎ, where 𝑏𝑏 is the base and ℎ is the height " 𝐴𝐴 = 𝑑𝑑" 𝑑𝑑! , where 𝑑𝑑" and 𝑑𝑑! are the diagonals

Rhombus

!

𝐴𝐴 = 𝜋𝜋𝑟𝑟2 , where 𝑟𝑟 is the radius

Circle Equilateral triangle Regular polygon

𝐴𝐴 =

√% ! 𝑠𝑠 , & "

where 𝑠𝑠 is the side length

𝐴𝐴 = ! 𝑃𝑃𝑃𝑃, where 𝑃𝑃 is the perimeter and 𝑎𝑎 is the apothem

Surface Area of a three-dimensional figure Cube Prism Cylinder Cone Pyramid Regular pyramid Sphere

𝑆𝑆𝑆𝑆 = 6𝑠𝑠 ! , where 𝑠𝑠 is the side length

𝑆𝑆𝑆𝑆 = 2𝐵𝐵 + 𝑃𝑃ℎ, where 𝐵𝐵 is the area of the base, 𝑃𝑃 is the perimeter of the base and ℎ is the height 𝑆𝑆𝑆𝑆 = 2𝐵𝐵 + 𝑃𝑃ℎ, where 𝐵𝐵 is the area of the base, 𝑃𝑃 is the perimeter of the base and ℎ is the height

𝑆𝑆𝑆𝑆 = 𝐵𝐵 + 𝜋𝜋𝜋𝜋ℎ' , where 𝐵𝐵 is the area of the base, 𝑟𝑟 is the radius and ℎ' is the slant height 𝑆𝑆𝑆𝑆 = 𝐵𝐵 + 𝐴𝐴(𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓), where 𝐵𝐵 is the area of the base and 𝐴𝐴(𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓) is the area of each face "

𝑆𝑆𝑆𝑆 = 𝐵𝐵 + ! 𝑃𝑃ℎ' , where 𝐵𝐵 is the area of the base, 𝑃𝑃 is the perimeter of the base and ℎ' is the slant height 𝑆𝑆𝑆𝑆 = 4𝜋𝜋𝑟𝑟 , where 𝑟𝑟 is the radius !

𝑆𝑆𝑆𝑆 = 3𝜋𝜋𝑟𝑟 ! , where 𝑟𝑟 is the radius and the area of the flat side is Hemisphere included

Volume of a three-dimensional figure Cube Prism Cylinder Cone Pyramid Sphere

𝑉𝑉 = 𝑠𝑠 ! , where 𝑠𝑠 is the side length

𝑉𝑉 = 𝐵𝐵ℎ, where 𝐵𝐵 is the area of the base and ℎ is the height 𝑉𝑉 = 𝐵𝐵ℎ, where 𝐵𝐵 is the area of the base and ℎ is the height "

𝑉𝑉 = ! 𝐵𝐵ℎ, where 𝐵𝐵 is the area of the base and ℎ is the height " 𝑉𝑉 = 𝐵𝐵ℎ, where 𝐵𝐵 is the area of the base and ℎ is the ! height #

𝑉𝑉 = ! 𝜋𝜋𝑟𝑟 ! , where 𝑟𝑟 is the radius

Laws of Exponents (where 𝑚𝑚 and 𝑛𝑛 are integers and 𝑎𝑎 and 𝑏𝑏 are nonzero real numbers)

Product of powers Quotient of powers Power of a power Power of a product Power of a quotient Negative exponent Identity exponent Zero exponent

𝑎𝑎! ⋅ 𝑎𝑎" = 𝑎𝑎!#" and conversely 𝑎𝑎()* = 𝑎𝑎( ⋅ 𝑎𝑎* !! !"

!!

= 𝑎𝑎"#$ and conversely 𝑎𝑎"#$ =

!"

(𝑎𝑎" )$ = 𝑎𝑎"⋅$ and conversely 𝑎𝑎"⋅$ = (𝑎𝑎" )$

(𝑎𝑎𝑎𝑎)( = 𝑎𝑎( ⋅ 𝑏𝑏( and conversely 𝑎𝑎( ⋅ 𝑏𝑏( = (𝑎𝑎𝑎𝑎)( ! "

& ' &

= "

!!

and conversely

&!

𝑎𝑎+" = , and conversely ! #'

& ' & "

𝑎𝑎 = 𝑎𝑎

=

𝑎𝑎- = 1 !

"

&

!

!

and conversely

"

𝑎𝑎 " = ( √𝑎𝑎*

Rational, fractional where 𝑎𝑎 > 0 ! " exponent " "

𝑎𝑎

'

!! &!

= 𝑎𝑎#' &

!

! "

= & ' &

! #'

= & ' &

"

"

and conversely ( √𝑎𝑎* "

!

= 𝑎𝑎 " ,

!

= ( √𝑎𝑎 * and conversely ( √𝑎𝑎" * = 𝑎𝑎 " ,

where 𝑎𝑎 > 0

297

NOTES

298

NOTES

299

230

7211