lesson 17 - amazon s3 · practice lesson 17 equivalent expressions 188 lesson 17 equivalent...

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©Curriculum Associates, LLC Copying is not permitted. 183Lesson 17 Equivalent Expressions

Name: Equivalent Expressions

Lesson 17

Vocabularyevaluate to find the

value of an algebraic

expression

Prerequisite: Write and Evaluate Expressions with Variables

Study the example showing how to write and evaluate expressions with variables. Then solve problems 1–7.

1 What does 4r 1 6 represent in the expression?

2 Does the expression (4r 1 6) ·· 2 also represent the

number of females on the team this year? Explain

3 If there were only 9 runners on the team last year, how many female runners are on the team this year? Explain how you found the answer

Example

The number of runners on a marathon team this year is 6 more than 4 times the number of runners on last year’s team Half of the runners this year are female and half are male What expression represents the number of female runners on the team this year?

You can draw a model to represent the situation

Last year’srunners

r 6

This year’s runners

Female

Malerr rr 1

The model shows that the number of female runners can

be represented by the expression 1 ·· 2 (4r 1 6)

183183

the number of runners on the marathon team this year

Yes; Dividing by 2 is the same as multiplying by 1 ·· 2 .

21; Possible explanation: Evaluate the

expression 1 ·· 2 (4r 1 6) for r 5 9; 1 ·· 2 (4r 1 6) 5

1 ·· 2 (4 • 9 1 6) 5 1 ·· 2 (36 1 6) 5 1 ·· 2 (42) 5 21

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B

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©Curriculum Associates, LLC Copying is not permitted.184 Lesson 17 Equivalent Expressions

Solve.

Use the situation below to solve problems 4–6.

The temperature increased 12°F between 9 AM and noon It decreased 9°F between noon and 6 PM

4 Write an expression with three terms to show the change in temperature Let the fi rst term represent the temperature at 9 AM

5 If the temperature was 45°F at 9 AM, what was the temperature at 6 PM?

6 Suppose the temperature at 6 PM was 30°F What would the temperature have been at 9 AM? Explain how you can use the expression you wrote in problem 4 to fi nd the answer

7 Jill makes purses and backpacks To make each purse,

she uses 1 foot less than 1 ·· 2 the amount of fabric she

uses to make a backpack Write an expression for the

amount of fabric that Jill needs to make a purse If she

uses 6 feet of fabric to make a backpack, how many

feet of fabric will she use to make a purse?

Show your work.

Solution:

184

t 1 12 2 9

48°F

27°F; Possible explanation: I can guess and check by trying different numbers for t in

the expression until I find the number that gives a solution of 30.

Let f be the amount of fabric that Jill needs to make a backpack.

f ·· 2 2 1; f ·· 2 2 1 5 6 ·· 2 2 1 5 3 2 1 5 2

Jill uses 2 feet of fabric to make a purse.

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Solving

Unit 3 Exp

ressions and Eq

uations Unit 3

Practice Lesson 17 Equivalent Expressions


Name: Lesson 17

Properties of Operations

Study the example showing how to use properties of operations to write equivalent expressions with variables. Then solve problems 1–9.

1 What does b represent in the expressions?

2 What does 2b 1 5b represent?

3 Does the expression 2b 1 5b have like terms? Explain

4 What is the common factor of each term in the expression 2b 1 5b?

5 Explain how to use the distributive property to create an expression that is equivalent to 2b 1 5b

Example

Sam bought 2 granola bars and Hayley bought 5 granola bars Each granola bar was the same price

Write an expression for the total price of the granola bars Then simplify the expression to create an equivalent expression Use the model to help you

bbHayley

bbSam

b bb

2b 1 5b 5 b(2 1 5) 5 7b

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the unknown price of 1 granola bar

The sum of the prices of Sam’s granola bars and Hayley’s granola bars.

Yes; 2b and 5b are like terms because they have the same variable as a factor.

Possible explanation: Factor out the common factor, b: 2b 1 5b 5 b(2 1 5) 5 7b

b

B

B

B

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Solve.

Use the situation below to solve problems 6–7.

Larry bought 12 containers of pasta salad for a school picnic Each container held the same number of ounces of salad Students finished the pasta salad in 8 of the containers

6 Let p equal the number of ounces of pasta salad in one container Write an expression with two terms to represent how many ounces of pasta salad are left

7 Simplify the expression you wrote in problem 6 to create an equivalent expression Use the distributive property

8 A soccer coach bought 16 medium T-shirts and 9 large T-shirts Each T-shirt was the same price Onaje and Paula tried to write equivalent expressions to represent the total price of the T-shirts The expressions they wrote are shown below

Onaje: 16t 1 9t 5 t(16 1 9) 5 25tPaula: 16t 1 9t 5 16 1 9 1 2t 5 25 1 2t

Whose expression is correct? Why is the other expression incorrect?

9 Adem writes 18y to simplify an expression with three like terms

a. What could the expression be?

b. Simplify the expression you wrote for part (a) to check your answer

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12p 2 8p

12p 2 8p 5 p(12 2 8) 5 4p

Onaje; Possible explanation: Paula did not distribute the common factor.

Possible answer: 4y 1 6y 1 8y

Possible answer: 4y 1 6y 1 8y 5 y(4 1 6 1 8) 5 18y

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Solving

Unit 3 Exp

ressions and Eq

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Unit 3



Solve.

6 Use the distributive property to fi nd two expressions that are equivalent to 7(3x 2 4)

7 A rectangular play area is 8 yards long The expression 56 1 8x represents the area of the play area in square yards What expression represents the width of the play area in yards? Draw a picture to model the problem

Show your work.

Solution:

8 Use the distributive property to write two expressions that are equivalent to 12 1 30x Describe the steps you follow to fi nd the expressions

9 Are 9(4 2 x) and 36 2 9x equivalent expressions? Explain how you know

188

Possible answer: (7 • 3x) 2 (7 • 4)

21x 2 28

Possible student work:

56 1 8x 5 (8 • 7) 1 (8 • x) 5 8(7 1 x)

The width of the play area is (7 1 x) yards.

Possible answer: (6 • 2) 1 (6 • 5x); 6(2 1 5x)

Factor out the common factor, 6: 12 1 30x 5 (6 • 2) 1 (6 • 5x)

Simplify (6 • 2) 1 (6 • 5x): (6 • 2) 1 (6 • 5x) 5 6(2 1 5x)

Yes; Possible explanation: When you distribute the factor 9 to 4 and x, you get

(9 • 4) 2 (9 • x). When you simplify (9 • 4) 2 (9 • x), you get 36 2 9x.

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Name: Lesson 17

Properties and Equivalent Expressions

Study the example showing how to use properties of operations to write equivalent expressions with variables. Then solve problems 1–9.

1 Explain how the model shows 4(5 1 x)

2 Simplify the expression in the example

3 What are the factors in the expression 4(5 1 x)?

4 Show how to use the distributive property to simplify 4(5 1 x)

5 Are the expressions (5 1 5 1 5 1 5) 1 (x 1 x 1 x 1 x) and 4(5 1 x) equivalent? If so, write another expression that is equivalent to both of them If not, explain why not

Example

Four students are buying tickets to a play The tickets cost $5 each plus a service fee The expression 4(5 1 x) represents the total cost Write an expression that is equivalent to 4(5 1 x)

You can use math tiles to model 4(5 1 x)

From the math tiles, you can see that the expression 4(5 1 x) 5 (5 1 5 1 5 1 5) 1 (x 1 x 1 x 1 x)

1 1 111

1 1 111

1 1 111

1 1 1

x

x

x

x11

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There are four groups of five 1-tiles and one x-tile.

20 1 4x

4 and (5 1 x)

I can multiply 5 and x by 4 in the first expression to get (4 • 5) 1 (4 • x),

which equals 20 1 4x.

Yes, they both simplify to 20 1 4x; another equivalent expression is (4 • 5) 1 (4 • x).

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Solving

Unit 3 Exp

ressions and Eq

uations Unit 3



Solve.

5 Look at the expressions s2 1 2s2 and 3s2

a. Draw math tiles to model s2 1 2s2 and 3s2 Does your model show that they are equivalent expressions? Explain

b. Use substitution to check your answer in part (a)

6 Use the terms 4, 12a, 6, 2a, and 24 to make equivalent expressions Use each term only once Use substitution to prove that the expressions are equivalent

Show your work.

Solution:

7 Bethany says that 3x 1 6 1 x and 3(x 1 2) are equivalent expressions She used substitution to support her answer Explain what Bethany did wrong

Let x 5 2

3x 1 6 1 x 5 3(2) 1 6 5 6 1 6 5 12

3(x 1 2) 5 3(2 1 2) 5 3(4) 5 12

12 5 12

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Yes, the expressions are equivalent; both models represent 3s2.

Possible answer: Let s 5 3

s2 1 2s2 5 32 1 2 • 32 5 9 1 2 • 9 5 9 1 18 5 27

3s2 5 3 • 32 5 3 • 9 5 27

27 5 27, so the answer is correct.

6(2a 1 4) 5 12a 1 24

Let a 5 2: 6(2a 1 4) 5 6(2 • 2 1 4) 5 6(4 1 4) 5 6(8) 5 48

12a 1 24 5 12 • 2 1 24 5 24 1 24 5 48; 48 5 48

She did not substitute 2 for the third term, x, in 3x 1 6 1 x.

The expressions 6(2a 1 4) and 12a 1 24 are equivalent.

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2s2

s2 s2

3s2

s2 s2 s2

s2

s2


Name: Lesson 17

Determine Whether Expressions Are Equivalent

Study the example problem showing how to determine whether expressions are equivalent. Then solve problems 1–7.

1 Are the terms 2s and 3s2 like terms? Explain

2 Explain how the tiles show that 2s 1 3s2 is not equivalent to 5s

3 Use substitution to prove that 2s 1 3s2 is not equivalent to 5s

4 Use the distributive property to write an expression that is equivalent to 2s 1 3s2

Example

Is 2s 1 3s2 equivalent to 5s?

Use math tiles to model 2s 1 3s2 and 5s

s s

2s 5s3s2

s s s s ss2 s2 s2

The expression 2s 1 3s2 is not equivalent to 5s

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No; The variables in like terms must be exactly the same. The variable s in the term 3s2

is squared. The variable s in the term 2s is not.

Possible explanation: There are two s-tiles in both 2s 1 3s2 and 5s, but when you take

those away you are left with 3s2 and 3s, which are not equivalent.

Possible answer: Let s 5 4; 2s 1 3s2 5 2(4) 1 3(42) 5 8 1 3(16) 5 8 1 48 5 56 and

5s 5 5(4) 5 20. Because 56 20, the expressions are not equivalent.

2s 1 3s2 5 (2 • s) 1 (3 • s • s) 5 s(2 1 3s)

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Solving

Unit 3 Exp

ressions and Eq

uations

Unit 3



5 Which expression is equivalent to 6 1 7n 1 4 1 8n? Select all that apply

A 13n 1 12n

B 5(3n 1 2)

C 5(3n 1 10)

D 15n 1 10

Solve.

3 Look at the expression 1 ·· 2 (c 1 8) Tell whether each statement about the expression is True or False.

a. 1 ·· 2 (c 1 8) and c 1 8 ····· 2 are

equivalent expressions u True u False

b. 1 ·· 2 (c 1 8) and 1 ·· 2 c 1 4 are

equivalent expressions u True u False

c. The only terms in 1 ·· 2 (c 1 8)

are c and 8 u True u False

d. You can multiply c and 8

by 1 ·· 2 in 1 ·· 2 (c 1 8) to find

an equivalent expression u True u False

4 The expressions a(8x 1 7) and 4x 1 3 5 are equivalent What is the value of a?

Show your work.

Solution:

Look at the expressions. Do you have to distribute a to find its value?

How can you use the distributive property to find equivalent expressions?

How do you know that expressions are equivalent?

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3

3

3

3

a(8x 1 7) 5 4x 1 3.5

8ax 1 7a 5 4x 1 3.5

For this to be true, 8ax must equal 4x and 7a 5 3.5, so a 5 0.5.

a 5 0.5

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Name: Lesson 17

Equivalent Expressions

Solve the problems.

2 The picture shows the dimensions of a vegetable garden and a fl ower garden

6 ft

7 ftVegetable garden

Flower garden x ft

Which expression represents the combined area of the gardens in square feet? Select all that apply

A 42 1 6x

B (6 • 7) 1 (6 • x)

C 13 1 6 1 x

D 6(7 1 x)

William chose C as a correct answer How did he get that answer?

1 Are 5n 1 9 1 n and 3(2n 1 9) equivalent expressions? Use substitution to check your answer

Show your work.

Solution:

What value will you substitute for to check your answer?

How do you find the area of a rectangle?

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No; Let n 5 4.

5n 1 9 1 n 5 5(4) 1 9 1 4 5 20 1 9 1 4 5 33

3(2n 1 9) 5 3(2 • 4 1 9) 5 3(8 1 9) 5 3(17) 5 51

33 Þ 51

The expressions 5n 1 9 1 n and 3(2n 1 9) are not equivalent.

He added the lengths when he should have multiplied each length

by the width of 6.

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lesson 17 - amazon s3 · practice lesson 17 equivalent expressions 188 lesson 17 equivalent...

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