Lesson 17 CCLS Equivalent Expressions 6.EE...23 Apply the distributive property to write an...

©Curriculum Associates, LLC Copying is not permitted.L17: Equivalent Expressions166

Part 1: IntroductionLesson 17Equivalent Expressions

In Lesson 17 you learned to read, write, and evaluate expressions with variables. Now, take a look at this problem.

Write an expression that is equivalent to (3 14) 27.

Explore It

Use the math you know to answer the question.

Which two terms in (3 14) 27 could you add to get a multiple of ten?

Rewrite the expression so that these two terms are next to each other.

Did rewriting the expression change its value? Explain.

Rewrite the expression with parentheses to show that the two terms with the sum that is a multiple of 10 should be added first.


What is the common factor of the numbers in parentheses?

Rewrite the expression so that it is a number plus the product of a number and a sum.

Does rewriting the expression this way change its value? Which property of operations supports your answer?

Explain why you might want to write an equivalent expression.

CCLS6.EE.36.EE.4

©Curriculum Associates, LLC Copying is not permitted.167L17: Equivalent Expressions

Lesson 17Part 1: Introduction

Find Out More

In the problem on the previous page, you applied properties of operations to an expression with all constant terms to create equivalent expressions.

(3 14) 27 5 (14 3) 27 Commutative property of addition Reordering the terms does not change the

value of the expression.

(14 3) 27 5 14 (3 27) Associative property of addition Regrouping the terms does not change the


14 (3 27) 5 14 3(1 9) Distributive property Distributing the common factor does not

change the value of the expression.

The same is true for expressions with variable terms. You can apply properties of operations to a variable expression to create equivalent variable expressions.

(3x 14) 27 5 (14 3x) 27 Commutative property of addition

(14 3x) 27 5 14 (3x 27) Associative property of addition

14 (3x 27) 5 14 3(x 9) Distributive property

Reflect

1 Henry says that you can apply the commutative and associative properties to 5x 10 and get 10x 5. Is Henry correct? Explain.

Lesson 17

©Curriculum Associates, LLC Copying is not permitted.

L17: Equivalent Expressions168

Part 2: Modeled Instruction

Read the problem below. Then explore how to use properties of operations to write equivalent expressions with variables.

Jamie has 4 bags of apples. Ashley has 3 bags of apples. Each bag has the same

number of apples in it.

Write an expression for the total number of apples. Then simplify it to create an

equivalent expression.

Picture It

Draw a picture of the bags of apples.

Jamie Ashley

xapples

xapples

xapples

xapples

xapples

xapples

xapples

Model It

Model the apples with math tiles.

Jamie x x x x

Ashley x x x

Lesson 17


Part 2: Guided Instruction

Connect It

Now solve the problem.

2 In both the Picture It and the Model It on the previous page, what does x represent?

3 Write an expression for Jamie’s total number of apples. Write an expression

for Ashley’s total number of apples. Write an expression for the combined

total of Jamie’s and Ashley’s apples.

When two or more terms in a variable expression have the same variable factors, they are called like terms. You can use the distributive property to simplify an expression with like terms.

4 What is the common factor for each term in your expression from problem 3?

5 Distribute the common factor and simplify the expression.

6 What does the simplified expression mean?

7 Explain how to simplify an expression with like terms, such as 6g 5g.

Try It

Use what you just learned about writing equivalent expressions to solve these problems. Show your work on a separate sheet of paper.

8 A school cafeteria has 30 boxes of Wheaty Squares cereal and 20 boxes of Mighty O’s cereal. Each box has the same number of ounces of cereal. Write an expression to represent the total ounces of cereal. Then simplify it to create an equivalent expression.

9 The inspector at a bottling plant checks 25 bottles. Two bottles do not pass inspection. All the bottles hold the same number of milliliters of sports drink. Write an expression for the milliliters of sports drink that pass inspection. Then simplify it to create an equivalent expression.

Lesson 17




Read the problem below. Then continue exploring how to use properties of operations to write equivalent expressions with variables.

Javier creates a rectangular painting. The painting is 3 feet long and more than

2 feet wide. The expression 3(2 x) represents the area of the painting.

Write an expression equivalent to 3(2 x).

Model It

The multiplication expression 3(2 1 x) means three groups of 2 1 x. Use math tiles to model three groups of 2 1 x.

2 1x 2 1x 2 1x

Reorder and regroup the tiles.

2 1 2 1 2 x1x1x

Compare 3(2 x) and (2 2 2) (x x x).

Picture It

Draw and label a picture of Javier’s painting. Imagine dividing the painting into two smaller rectangles.

2 ft x ft

3 ft

Imagine dividing the rectangle into two smaller rectangles.

The area of the whole painting is 3(2 x).

Lesson 17



Connect It


10 The expression 3(2 x) is a product of the factors 3 and .

11 In the Model It, what expression is equivalent to 3(2 x)? Explain.

12 Look at the Picture It. Explain why the area of the whole picture is 3(2 x).

13 In the Picture It, the area of the left side of the rectangle is . The area

of the right side is . Write an expression for the area of the whole

painting: .

14 Compare 3(2 x) to the equivalent expression in 13. What property did you apply?

15 Simplify the expression from problem 14.

16 Is 3(2 x) equivalent to your simplified expression? Explain.

Try It

Use what you just learned about using the distributive property to write an equivalent expression to solve these problems. Show your work on a separate sheet of paper.

17 Use the distributive property to write an expression that is equivalent to 5(2x ] 1).

18 Use the distributive property to write an expression that is equivalent to 18 24x.

Lesson 17




Read the problem below. Then explore how to determine if expressions are equivalent.

Is 5h 2h2 equivalent to 7h? Explain.

Picture It

Imagine line segments that are h, 5h, and 7h units long.

h

5h7h

Imagine rectangles that are h2 and 2h2 square units in area.

h

h

h2

h

h h

2h2

Model It

Use math tiles to model 5h, 2h2, and 7h.

hh h

5h

h h h

7h

h h h h h hh2

2h2

h2

Lesson 17



Connect It


19 Look at Picture It. If you combine the line that is 5h units long and the rectangle that is 2h2 units in area, do you get a figure that looks like the line that is 7h units long?

20 Look at Model It. If you put the tiles representing 5h together with the tiles representing 2h2, do you get a set of tiles that represents 7h?

21 Richard says that 5h and 2h2 are like terms because they both have the variable h. Is Richard correct? Explain.

22 Is 5h 2h2 5 7h a true statement? Substitute a non-zero value for h and evaluate 5h 2h2 and 7h to support your answer.

23 Apply the distributive property to write an expression that is equivalent to 5h 2h2. Show your work.

Try It

Use what you just learned to solve these problems. Show your work on a separate sheet of paper.

24 Are 3x 6 x and 2(2x 3) equivalent expressions? Use substitution to check your answer.

25 Are 8(w 6) and 5 8w 1 equivalent expressions? Use substitution to check your answer.

Part 5: Guided Practice Lesson 17



Study the student model below. Then solve problems 26–28.

Are y 2y 2 3 and 3(y 2 1) equivalent expressions? Explain.

Look at how you can apply the properties of operations to show your answer.

Solution:

26 Simplify 6x3 6x2 6x to get an equivalent expression. Label any properties of operations that you use.

Show your work.

Solution:

Which property of operations allows you to combine like terms without changing the value of an expression?

Pair/Share

Are 6x3 and 6x2 like terms? Why or why not?

Pair/Share

y 1 2y 2 3 5 y(1 1 2) 2 3

5 3y 2 3

5 3(y 2 1)

Yes; I can apply the properties of operations to

y 1 2y 2 3 without changing its value and get 3(y 2 1). If I let

y 5 2, y 1 2y 2 3 5 2 1 2(2) 2 3 5 2 1 4 2 3 5 6 2 3 5 3

and 3(y 2 1) 5 3(2 2 1) 5 3(1) 5 3. Since 3 5 3, I know

y 1 2y 2 3 5 3(y 2 1).

If the terms of an expression have a common factor, you can apply the distributive property to simplify the expression.

Two expressions are equivalent if you can simplify one expression and get the other expression.

Part 5: Guided Practice Lesson 17


27 Dalia’s living room is 10 feet long and 12 feet wide. Her dining room is also 10 feet long. Write two equivalent expressions that each represent the combined area of the two rooms.

Show your work.12 ft

LivingRoom

DiningRoom

10 ft

x ft

Solution:

28 Which expression is equivalent to 2 3n 2 9n?

A 16n

B 3n 8

C 4(3n 1)

D 4(3n 4)

Anya chose D as the correct answer. How did she get that answer?

How could Anya check her answer?

Pair/Share

How could you show that the two expressions are equivalent?

Pair/Share

Combine like terms and apply properties of operations to simplify an expression.

Draw and label a picture to help you organize the given information.

Part 6: Common Core Practice Lesson 17


L17: Ratios176

Solve the problems. Mark your answers to problems 1–4 on the Answer Form to the right. Be sure to show your work.

1 Which expression is equivalent to 3x 5 2x x2 1?

A 12x

B 5x 5

C 6x 6

D x2 5x 6

2 Which lists all the like terms in the expression 4 x4 4x 4x2 2x?

A 2x, 4x

B x4, 4x, 4x2, 2x

C 4, x4, 4x, 4x2

D 4, 4x, 2x

3 The expression 0.25(2d 1) represents the fines per day, d, for overdue books. Which expression is equivalent to 0.25(2d 1)?

A 0.252d 1

B 0.50d 0.25

C 2d 0.25

D 0.50d 1

1 B C D

2 B C D

3 B C D

4 B C D

Answer Form

Number Correct 4

Part 6: Common Core Practice Lesson 17

©Curriculum Associates, LLC Copying is not permitted.177L17: Ratios

Self Check Go back and see what you can check off on the Self Check on page 143.

4 A game company makes a board game that comes with 2 dice and a card game that comes with 3 dice. Which expression shows the total number of dice in b boxes of the board game and b boxes of the card game?

A 5b

B 5(2b)

C 5 b

D 2b 3

5 Taylor writes an expression with 5 terms. All 5 terms are like terms. How many terms are in the equivalent expression with the least number of terms? Explain.

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6 Kari uses substitution to decide whether x2 x is equivalent to x(2x 1). She says the expressions are equivalent because they have the same value when x 5 0. Is Kari correct? Explain.

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L17: Equivalent Expressions 169©Curriculum Associates, LLC Copying is not permitted.

Lesson 17

equivalent expressions(Student Book pages 166–177)

Lesson objectives

• Understand that the properties used with numbers also apply to expressions with variables.

• Recognize and generate equivalent expressions.

• Substitute values into expressions to prove equivalency.

PReRequisite skiLLs

• Recognize that variables stand for numbers.

• Understand properties of operations and apply each of them in numeric representations.

• Substitute values into expressions.

vocabuLaRy

commutative property of addition: reordering the terms does not change the value of the expression. E.g., a 1 b 5 b 1 a

associative property of addition: regrouping the terms does not change the value of the expression. E.g., (a 1 b) 1 c 5 a 1 (b 1 c)

distributive property: distributing the common factor does not change the value of the expression

like terms: two or more terms in a variable expression that have the same variable factors

the LeaRning PRogRession

In Grade 5, students wrote, interpreted, and evaluated numerical expressions. Students in Grade 6 will apply the properties of operations to generate equivalent expressions. In Grade 7, students will continue using the properties of operations to generate equivalent expressions.

This lesson focuses on understanding that the properties used with numbers also apply to expressions with variables. Students will show properties of operations. They will recognize equivalent expressions via modeling with manipulatives, diagrams, or story contexts. Students will also apply the properties of operations with expressions involving variables to generate equivalent expressions and substitute values into expressions to prove equivalency.

Toolbox Teacher-Toolbox.com

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✓ ✓

Prerequisite Skills

6.EE.3 6.EE.4

Ready Lessons

Tools for Instruction

Interactive Tutorials ✓

ccLs Focus

6.EE.3 Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2 1 x) to produce the equivalent expression 6 1 3x; apply the distributive property to the expression 24x 1 18y to produce the equivalent expression 6 (4x 1 3y); apply properties of operations to y 1 y 1 y to produce the equivalent expression 3y.

6.EE.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y 1 y 1 y and 3y are equivalent because they name the same number regardless of which number y stands for. (see page A9 for full text)

aDDitionaL stanDaRDs: 6.EE.6 (see page A32 for full text)

stanDaRDs FoR MatheMaticaL PRactice: SMP 2, 3 (see page A9 for full text)

L17: Equivalent Expressions170©Curriculum Associates, LLC Copying is not permitted.

Lesson 17Part 1: introduction

at a gLance

Students view an expression and are guided to write equivalent expressions using mathematical properties.

steP by steP

• Tell students that this page guides them in writing an equivalent expression using mathematical properties.

• Have students read the problem at the top of the page. Explain that by using properties of numbers we can write equivalent expressions.

• Work through Explore It as a class.

• Ask, Which property allows values in an addition problem to be added in any order? [commutative property]

• Ask, Which property allows values in an addition problem to be regrouped? [associative property]

• Ask students how they might prove that the expressions have the same value. [possible answer: using manipulatives to show the combinations]

• Ask student pairs or groups to explain their answers for writing equivalent expressions.

Connect the words commutative to commuting (changing positions), associative to associating with people, and distributive to distribute (sending to others). Relate these base words and real-world examples to the mathematical definitions.

eLL support• Of the expressions you wrote on this page, which 

expression is easier for you to evaluate? Explain.

Listen for student use of number sense and the composition of numbers.

• Explain why the commutative and associative properties do not apply to subtraction and division. How can you support your reasoning?

Look for students to justify their reasoning with models or by disproving a property with an example.

Mathematical Discourse

©Curriculum Associates, LLC Copying is not permitted.L17: Equivalent Expressions166

Part 1: introductionLesson 17

equivalent expressions

in Lesson 17 you learned to read, write, and evaluate expressions with variables. now, take a look at this problem.

Write an expression that is equivalent to (3 1 14) 1 27.

explore it

use the math you know to answer the question.

Which two terms in (3 1 14) 1 27 could you add to get a multiple of ten?

Rewrite the expression so that these two terms are next to each other.


Rewrite the expression with parentheses to show that the two terms with the sum that is a multiple of 10 should be added first.


What is the common factor of the numbers in parentheses?

Rewrite the expression so that it is a number plus the product of a number and a sum.

Does rewriting the expression this way change its value? Which property of operations supports your answer?

Explain why you might want to write an equivalent expression.

ccLs6.ee.3

6.ee.4

3 and 27

3

(14 1 3) 1 27

14 1 3(1 1 9)

14 1 (3 1 27)

no; (3 1 14) 1 27 5 (14 1 3) 1 27 because the order of the numbers doesn’t

matter when adding.

no; (14 1 3) 1 27 5 14 1 (3 1 27) because you can add any two terms first;

the sum stays the same.

no; 14 1 (3 1 27) 5 14 1 3(1 1 9) by the distributive property.

the expression could be easier to simplify or combine with other expressions.



at a gLance

Students will explore the usage of the commutative, associative, and distributive properties in writing equivalent expressions. They will understand that the properties used with numbers also apply to expressions with variables.

steP by steP

• Read Find Out More as a class.

• Ask students to explain how the equivalent expressions illustrate the property. Review that a term is a part of an expression separated by addition or subtraction.

• Point out the similarities and differences between the comparable expressions. Guide students to realize that the operations on expressions with numbers can also be performed on expressions with variables.

• Ask groups or pairs of students to share answers and to explain their reasoning for Reflect by using their mathematical definitions.

sMP tip: Reading arguments of others and deciding if they make sense allows students to construct their own viable arguments (SMP 3) by using mathematical definitions. Occasionally ask students to determine the validity of the reasoning of their peers: Ask students, Does the reasoning make sense? Does that always work?

Ask students to provide real-world situations which can be represented with numerical or algebraic expressions that can be rewritten using one of the mathematical properties.

Example: 5 families ordered the same meal of 6 chicken sandwiches, 3 desserts, and 2 drinks. What expressions would let you find the total number of food items the family bought? 5(6) 1 5(3) 1 5(2) 5 5(6 1 3 1 2)

Real-World connection



Find out More

In the problem on the previous page, you applied properties of operations to an expression with all constant terms to create equivalent expressions.

(3 1 14) 1 27 5 (14 1 3) 1 27 commutative property of addition Reordering the terms does not change the


(14 1 3) 1 27 5 14 1 (3 1 27) associative property of addition Regrouping the terms does not change the


14 1 (3 1 27) 5 14 1 3(1 1 9) Distributive property Distributing the common factor does not

change the value of the expression.

The same is true for expressions with variable terms. You can apply properties of operations to a variable expression to create equivalent variable expressions.

(3x 1 14) 1 27 5 (14 1 3x) 1 27 Commutative property of addition

(14 1 3x) 1 27 5 14 1 (3x 1 27) Associative property of addition

14 1 (3x 1 27) 5 14 1 3(x 1 9) Distributive property

Reflect

1 Henry says that you can apply the commutative and associative properties to 5x 1 10 and get 10x 1 5. Is Henry correct? Explain.

no; 5x is a term in the expression 5x 1 10. the commutative and associative

properties say you can reorder and regroup the terms of an expression

without changing the value of the expression. but if you reorder or regroup

the factors of the terms of an expression, such as changing 5x to 10x, the

value of the expression may change.

172 L17: Equivalent Expressions


Lesson 17Part 2: Modeled instruction

at a gLance

Students will use properties of operations to write equivalent expressions with variables by modeling the distributive property with pictures and tiles.

steP by steP

• Read the problem at the top of the page as a class.

• Review Picture It as a class. Ask, Why is an x used on each bag? [The number of apples in each bag is unknown.]

• Lead students to identify that the bags of apples are the same and contain the same unknown number of apples.

• Direct students to Model It. Emphasize that the tiles used for Jamie and Ashley are the same shape. Ask, Why are the tiles all the same shape? [The tiles represent the bags, which contain the same unknown number of apples.]

Model the distribution of factors.

Materials: mathematical situations; colored candies or counters, including:

three groups of 2 yellow and 1 red

two groups of 1 brown and 5 blue

three groups of 3 orange and 2 green

two groups of 5 red and 3 blue

twelve groups of 4 green and 8 yellow

• Ask students to work in pairs to use colored candies to model the mathematical situations.

• Ask pairs of students to write two equivalent expressions. The second expression should demonstrate the application of the distributive property on the first expression.

• Have volunteers share their models and expressions with the class.

visual Model

Lesson 17



Part 2: Modeled instruction

Read the problem below. then explore how to use properties of operations to write equivalent expressions with variables.

Jamie has 4 bags of apples. Ashley has 3 bags of apples. Each bag has the same

number of apples in it.

Write an expression for the total number of apples. Then simplify it to create an

equivalent expression.

Picture it

Draw a picture of the bags of apples.

Jamie Ashley

xapples

xapples

xapples

xapples

xapples

xapples

xapples

Model it

Model the apples with math tiles.

Jamie x x x x

Ashley x x x

• What can you say about the diagrams shown in Picture It and Model It? 

Listen for student discussion supporting ideas such as: the bags contain an equal number of apples; the diagrams in Picture It and Model It are equivalent; and the value of x must be a positive number.

• How would the diagrams be different if Jamie’s bags held a different number of apples than Ashley’s bags? How would the expression be different?

Listen for responses that show that students understand how to use variables to represent a real-world problem situation.



Lesson 17Part 2: guided instruction

at a gLance

Students revisit the problem on page 168 to write equivalent expressions by identifying like terms and applying the distributive property to expressions with variables.

steP by steP

• Read Connect It as a class. Be sure to point out that the questions refer to the problem on page 168.

• Identify the terms in the expression. [4x and 3x]

• Point out to students the new vocabulary like term as connected to 4x and 3x and that both terms have a common factor of x which makes them “like.”

• Ask, What does the common factor of x represent in the real world? [the number of apples in each bag]

• Read Try It as a class. Prompt students to use pictures or tiles to model the expressions, if necessary.

tRy it soLutions

8 Solution: 30c 1 20c 5 c(30 1 20) 5 50c; Students may combine like terms without directly using the distributive property. They might also create a model to support their work.

9 Solution: 25m 2 2m 5 m(25 2 2) 5 23m; Students may combine like terms without directly using the distributive property. They might also create a model to support their work.

Review the meanings of term, like terms, common factor, factor, and distribute.

eLL support

sMP tip: Students reason quantitatively when they use properties of operations to recognize or write equivalent expressions (SMP 2). Regularly ask students to look for ways they can compose or decompose numbers or use properties of operations to simplify calculations.

ERROR ALERT: Students who wrote 10c(3 1 2) need to be reminded that the second factor should represent the number of boxes while the common factor is the value both terms have in common, c.

Lesson 17


Part 2: guided instruction

connect it

now solve the problem.

2 In both the Picture It and the Model It on the previous page, what does x represent?

3 Write an expression for Jamie’s total number of apples. Write an expression

for Ashley’s total number of apples. Write an expression for the combined

total of Jamie’s and Ashley’s apples.

When two or more terms in a variable expression have the same variable factors, they are called like terms. You can use the distributive property to simplify an expression with like terms.

4 What is the common factor for each term in your expression from problem 3?

5 Distribute the common factor and simplify the expression.

6 What does the simplifi ed expression mean?

7 Explain how to simplify an expression with like terms, such as 6g 1 5g.

try it

use what you just learned about writing equivalent expressions to solve these problems. show your work on a separate sheet of paper.

8 A school cafeteria has 30 boxes of Wheaty Squares cereal and 20 boxes of Mighty O’s cereal. Each box has the same number of ounces of cereal. Write an expression to represent the total ounces of cereal. Then simplify it to create an equivalent expression.

9 The inspector at a bottling plant checks 25 bottles. Two bottles do not pass inspection. All the bottles hold the same number of milliliters of sports drink. Write an expression for the milliliters of sports drink that pass inspection. Then simplify it to create an equivalent expression.

the unknown number of apples in each bag.

(4x) 1 (3x) 5 x(4 1 3) 5 7x

there are 7 total bags of apples.

Factor out the common factor, g, and simplify. 6g 1 5g 5 g(6 1 5) 5 11g

30c 1 20c 5 c(30 1 20) 5 50c

25m 2 2m 5 m(25 2 2) 5 23m

4x

x

3x

4x 1 3x




at a gLance

Students will write equivalent expressions from tiles and area models while applying the distributive property to expressions with variables.

steP by steP


• Guide students to connect “more than 2 feet wide” with “2 1 x.”

• Direct students to the tiles in Model It. Remind students that one small square represents 1 and the rectangular tile represents x.

• Ask students which expressions they could write to represent the tile model before rearranging the tiles. [possible answer: 2 1 x 1 2 1 x 1 2 1 x] Help students to see that the reorganization of the tiles did not change the value.

• Ask pairs or groups of students to explain connections see between the two expressions.

• Review Picture It as a class.

• Guide students to determine the area of each smaller rectangle within the picture. [3 ? 2 5 6 and 3 ? x 5 3x]

• Ask students to compare the expressions developed in Model It and Picture It. What do they notice? [There are several ways to write the same expression.]

calculate area using the distributive property.

Materials: graph paper, scissors

• Draw a rectangle on graph paper with a length of 12 units and a width of 9 units. What expression represents the area of the rectangle?

• Draw a second rectangle with a length of 12 units and a width of 15 units. What expression represents the area of this rectangle?

• Cut out the rectangles and place them next to each other to create a rectangle which is still 12 units long. What expression represents the area of this rectangle?

• What equivalent expressions can you create from this area model?

visual Model

Lesson 17




Read the problem below. then continue exploring how to use properties of operations to write equivalent expressions with variables.

Javier creates a rectangular painting. The painting is 3 feet long and more than

2 feet wide. The expression 3(2 1 x) represents the area of the painting.

Write an expression equivalent to 3(2 1 x).

Model it

the multiplication expression 3(2 1 x) means three groups of 2 1 x. use math tiles to model three groups of 2 1 x.

2 1x 2 1x 2 1x

Reorder and regroup the tiles.

2 1 2 1 2 x1x1x

Compare 3(2 1 x) and (2 1 2 1 2) 1 (x 1 x 1 x).

Picture it

Draw and label a picture of javier’s painting. imagine dividing the painting into two smaller rectangles.

2 ft x ft

3 ft

Imagine dividing the rectangle into two smaller rectangles.

The area of the whole painting is 3(2 1 x).

• Describe in your own words how the picture in Picture It shows 3(2 1 x). 

Listen for responses that students see (2 1 x) as a single entity that represents the length of the painting, and that it can be multiplied by 3, the height of the painting.

• How do the different ways to show the problem lead to different equivalent expressions?

Listen for responses that indicate that the model helps you see equivalent expressions for 3(2 1 x) that use either addition (because you can add the tiles) or multiplication (because you can think of 3 times 2 and 3 times x).




at a gLance

Students revisit the problem on page 170 using the distributive property to compare and write equivalent expressions involving variables.

steP by steP

• Read Connect It as a class. Be sure to point out that the questions refer to the problem on page 170.

• Refer students to the various expressions and models used in Model It and Picture It.

• Emphasize that applying the properties of operations to expressions does not change the value of the expression. This can be proven in many ways as shown in Model It and Picture It.

tRy it soLutions

17 Solution: 10x 2 5; Students may evaluate the expression by 5(2x) 2 5(1) or use a model.

18 Solution: 6(3 1 4x); Students may also write 6(4x 1 3) or use a model.

use the distributive property to write expressions.

Materials: colored counters

• Sort the counters by color. Count the number of counters of each color.

• Ask, If you were given two identical bags of counters, how could you use the distributive property to write an expression to represent the total number of counters? For example, 2(10r 1 15g 1 12y).

• Ask students to share other expressions to represent the total number of counters in the two bags. For example, 20r 1 30g 1 24y.

• Use the distributive property to write expressions showing how many counters of each color would be in 5 bags, 10 bags, and so on.

hands-on activity

ERROR ALERT: Students who wrote 3(6 1 8x) did not choose the greatest factor of the two terms.

Lesson 17



connect it


10 The expression 3(2 1 x) is a product of the factors 3 and .

11 In the Model It, what expression is equivalent to 3(2 1 x)? Explain.

12 Look at the Picture It. Explain why the area of the whole picture is 3(2 1 x).

13 In the Picture It, the area of the left side of the rectangle is . The area

of the right side is . Write an expression for the area of the whole

painting: .

14 Compare 3(2 1 x) to the equivalent expression in 13. What property did you apply?

15 Simplify the expression from problem 14.

16 Is 3(2 1 x) equivalent to your simplifi ed expression? Explain.

try it

use what you just learned about using the distributive property to write an equivalent expression to solve these problems. show your work on a separate sheet of paper.

17 Use the distributive property to write an expression that is equivalent to 5(2x ] 1).

18 Use the distributive property to write an expression that is equivalent to 18 1 24x.

2 1 x

(2 1 2 1 2) 1 (x 1 x 1 x); adding three 2s is the same as 3 times 2. adding

x 1 x 1 x is the same as multiplying 3 by x.

the width is 3 ft, the length is 2 1 x feet so the area is 3(2 1 x) square feet.

the distributive property, 3(2 1 x) 5 (3 ? 2) 1 (3 ? x), distributing 3 to the 2 and x.

5(2x 2 1) 5 (5 ? 2x) 2 (5 ? 1) 5 10x 2 5

18 1 24x 5 (6 ? 3) 1 (6 ? 4x) 5 6(3 1 4x)

yes; applying the properties of operations to an expression does not change

the value of the expression.

3 ? 2

3 ? x

(3 ? 2) 1 (3 ? x)

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




at a gLance

Students will prove the equivalency of expressions using models and substitution.

steP by steP


• Review the line segments in Picture It. Have a volunteer confirm the accuracy of the lengths.

• Remind students that 5h and 2h2 are being combined. Ask a volunteer to show this combined model.

• Direct students to the area models with a particular focus on h2. Discuss the area of h2 as the length multiplied by the width. Show students the algebra tile corresponding to h2.

• Direct students to the tiles in Model It. Emphasize the use of various shapes to represent each term.

• Clarify that the terms are not being combined but instead being compared; 5h and 2h2 are in one grouping, and 7h is in another grouping.

use the distributive property to prove the equivalency of expressions.

Materials: tiles, graph paper, expressions

• Put students into pairs or groups.

• Display the following expressions:

4x 1 4, 3(2x 1 5), 2(8 1 x), 3(2x 1 0),

0.5(x 1 1), 2x(8x 1 x), 4(x 1 1), 6x 1 15,

16 1 2x, 6x, 0.5x 1 0.5, 16x2 1 x2,

17x2, 7x2 2 3x2, 4x2, x2(7 2 3)

• Assign each pair or group of students one of the expressions you displayed.

• Ask each pair or group of students to create a model which represents the expression they have been assigned.

• Ask each pair or group of students to find another expression which is equivalent to their own.

• Students should share equivalent expressions and models with the class.

visual Model

Lesson 17




Read the problem below. then explore how to determine if expressions are equivalent.

Is 5h 1 2h2 equivalent to 7h? Explain.

Picture it

imagine line segments that are h, 5h, and 7h units long.

h

5h7h

imagine rectangles that are h2 and 2h2 square units in area.

h

h

h2

h

h h

2h2

Model it

use math tiles to model 5h, 2h2, and 7h.

hh h

5h

h h h

7h

h h h h h hh2

2h2

h2

• When using the tiles to model 5h 1 2h2, one student models h2 with two h tiles. What is wrong with this representation?

When using two h tiles, the measure of the square is 2 units by h units, which is not the same as h2.

• Why can’t you use the line segments to represent h2? Can you show any multiplication with line segments?

Students’ responses should include that h2 means h times h. To represent multiplication with the line segments, you have to know how many segments of length h to show. You can use a line to represent a number times a variable, but not a variable times a variable.




at a gLance

Students revisit the problem on page 172 to determine if expressions are equivalent using mathematical properties, modeling, and substitution.

steP by steP

• Read page 173 as a class. Be sure to point out that Connect It refers to the problem on page 172.

• Guide students to use Picture It to prove the expressions are not equivalent.

• Guide students to use Model It to prove the expressions are not equivalent.

• Review the definition of like terms. Students should share their reasoning as to why the expressions are not equivalent, using the phrase “like terms” in their reasoning.

• Problem 22 introduces substitution as a way of testing for equivalency. Point out that equivalent expressions will have equal values. Review the concept of a variable having any value. Students will replace (substitute) the variable with any non-zero value to test for equivalency. Ask, Why do you think the value of the variable should not be 0? [Unlike a non-zero value, 0 would make the statement true.]

tRy it soLutions

24 Solution: Yes; Students should substitute a non-zero value for x.

25 Solution: No; Students should substitute a non-zero value for x.

Work with students to connect the word substitute to “replace.” Offer real-world examples, such as a substitute teacher who replaces a teacher.

eLL support

apply substitution and the distributive property to determine temperatures.

Materials: formula card: °C 5 5 ··

 9 (°F 2 32)

• Use your knowledge of the distributive property and substitution, along with the formula card, to convert the following temperatures to degrees Celsius:

9°F, 32°F, 23°F, 24°F, 245°F

• Show at least two different ways to evaluate each expression.

• As a class, discuss the method(s) which are most efficient in evaluating each expression.

hands-on activity

ERROR ALERT: Students who wrote the correct answer may have substituted zero for the variable, coming to the correct answer using the wrong reasoning.

Lesson 17



connect it


19 Look at Picture It. If you combine the line that is 5h units long and the rectangle that is 2h2 units in area, do you get a fi gure that looks like the line that is 7h units long?

20 Look at Model It. If you put the tiles representing 5h together with the tiles representing 2h2, do you get a set of tiles that represents 7h?

21 Richard says that 5h and 2h2 are like terms because they both have the variable h. Is Richard correct? Explain.

22 Is 5h 1 2h2 5 7h a true statement? Substitute a non-zero value for h and evaluate 5h 1 2h2 and 7h to support your answer.

23 Apply the distributive property to write an expression that is equivalent to 5h 1 2h2. Show your work.

try it

use what you just learned to solve these problems. show your work on a separate sheet of paper.

24 Are 3x 1 6 1 x and 2(2x 1 3) equivalent expressions? Use substitution to check your answer.

25 Are 8(w 1 6) and 5 1 8w 1 1 equivalent expressions? Use substitution to check your answer.

no

no

no. For terms to be like terms the variables must be exactly the same. the

variable h in the term 2h2 is raised to the second power. the variable h in the

term 5h is not.

no. For example, let h 5 3, 5h 1 2h2 5 5(3) 1 2(3)2 5 15 1 2(9) 5 15 1 18 5

33, but 7h 5 7(3) 5 21. because 33 Þ 21, 5h 1 2h2 Þ 7h

5h 1 2h2 5 (5 ? h) 1 (2 ? h ? h)

5h 1 2h2 5 h(5 1 2h)

yes. student should substitute a non-zero value for x.

no. student should substitute a non-zero value for w.



Lesson 17Part 5: guided Practice

at a gLance

Students will apply properties of operations to show equivalency in expressions.

steP by steP

• Ask students to solve the problems individually to apply the properties of operations to show equivalency in expressions.

• When students have completed each problem, have them Pair/Share to discuss their solutions with a partner or in a group.

soLutions

Ex A problem that applies the properties of operations is shown.

26 Solution: 6x(x2 1 x 1 1); distributive property; Students could solve the problem by rewriting 6x3 1 6x2 1 6x as (6 ? x ? x ? x) 1 (6 ? x ? x) 1 (6 ? x) 5 6x(x2 1 x 1 1).

27 Solution: The combined area of the two rooms is 10(12 1 x) or 120 1 10x; Students could solve the problem by showing that combined area is length times combined width. Combined Area is Area of Living Room 1 Area of Dining Room: A 5 (10 ? 12) 1 (10 ? x) 5 120 1 10x.

28 Solution: C; Anya did not factor 4 out of 4 when she applied the distributive property.

Explain to students why the other two answer choices are not correct:

A is not correct because 2, 3n, 2, and 9n were combined, but they are not all like terms.

B is not correct because the factor 4 was not multiplied by 3n and then was just added to 4.

Part 5: guided Practice Lesson 17


27 Dalia’s living room is 10 feet long and 12 feet wide. Her dining room is also 10 feet long. Write two equivalent expressions that each represent the combined area of the two rooms.

Show your work.

12 ft

LivingRoom

DiningRoom

10 ft

x ft

Solution:

28 Which expression is equivalent to 2 1 3n 1 2 1 9n?

a 16n

b 3n 1 8

c 4(3n 1 1)

D 4(3n 1 4)

Anya chose D as the correct answer. How did she get that answer?

How could Anya check her answer?

Pair/share

How could you show that the two expressions are equivalent?

Pair/share

Combine like terms and apply properties of operations to simplify an expression.

Draw and label a picture to help you organize the given information.

combined area is length times combined width: A 5 lw 5 10(12 1 x)combined area is area of Living Room 1 area of Dining Room: A 5 lw 1 lw 5 (10 3 12) 1 (10 3 x) 5 120 1 10x

10(12 1 x) or 120 1 10x

Possible answer: she did not factor 4 out of 4 when she

applied the distributive property.

Part 5: guided Practice Lesson 17



study the student model below. then solve problems 26–28.

Are y 1 2y 2 3 and 3(y 2 1) equivalent expressions? Explain.

Look at how you can apply the properties of operations to show

your answer.

Solution:

26 Simplify 6x3 1 6x2 1 6x to get an equivalent expression. Label any properties of operations that you use.

Show your work.

Solution:

Which property of operations allows you to combine like terms without changing the value of an expression?

Pair/share

Are 6x3 and 6x2 like terms? Why or why not?

Pair/share

y 1 2y 2 3 5 y(1 1 2) 2 3

5 3y 2 3

5 3(y 2 1)

yes; i can apply the properties of operations to

y 1 2y 2 3 without changing its value and get 3(y 2 1). if i let

y 5 2, y 1 2y 2 3 5 2 1 2(2) 2 3 5 2 1 4 2 3 5 6 2 3 5 3

and 3(y 2 1) 5 3(2 2 1) 5 3(1) 5 3. since 3 5 3, i know

y 1 2y 2 3 5 3(y 2 1).

If the terms of an expression have a common factor, you can apply the distributive property to simplify the expression.

Two expressions are equivalent if you can simplify one expression and get the other expression.

6x3 1 6x2 1 6x 5 (6 ? x ? x ? x) 1 (6 ? x ? x) 1 (6 x)

5 6x(x2 1 x 1 1) distrib. prop.

6x(x2 1 x 1 1)


Lesson 17Part 6: common core Practice

at a gLance

Students solve problems concerning equivalent expressions that might appear on a mathematics test.

steP by steP

• First, tell students that they will write equivalent expressions. Then have students read the directions and answer the questions independently. Remind students to fill in the correct answer choices on the Answer Form.

• After students have completed the Common Core Practice problems, review and discuss correct answers. Have students record the number of correct answers in the box provided.

soLutions

1 Solution: D; combine like terms and use the commutative property: x2 1 (3x 1 2x) 1 (5 1 1) 5 x2 1 5x 1 6

2 Solution: A; 2x and 4x have the same variable factor.

3 Solution: B; 0.25(2d) 1 0.25(1) 5 0.5d 1 0.25.

4 Solution: A; b(2) 1 b(3) 5 5b.

5 Solution: Possible solution: The equivalent expression with the least number of terms will have 1 term. Terms will be all constants or all variable terms. For example, 1 1 2 1 3 1 4 1 5 has 5 constant terms, and can be simplified to a single term, 15. For example, x2 1 2x2 1 3x2 1 4x2 1 5x2 has 5 variable terms, and can be simplified to a single term, 15x2.

6 Solution: No; The expressions have the same value when x 5 0, but Kari should substitute a non-zero number for x. For example, let x 5 4. Then, x2 1 x 5 (4)2 1 4 5 16 1 4 5 20. But, x(2x 1 1) 5 4(2[4] 1 1) 5 4(8 1 1) 5 4(9) 5 36. Since 20 Þ 36, x2 1 x cannot be equivalent to x(2x 1 1).

Part 6: common core Practice Lesson 17

©Curriculum Associates, LLC Copying is not permitted.177L17: Ratios

self check Go back and see what you can check off on the Self Check on page 143.

4 A game company makes a board game that comes with 2 dice and a card game that comes with 3 dice. Which expression shows the total number of dice in b boxes of the board game and b boxes of the card game?

A 5b

B 5(2b)

C 5 1 b

D 2b 1 3

5 Taylor writes an expression with 5 terms. All 5 terms are like terms. How many terms are in the equivalent expression with the least number of terms? Explain.

6 Kari uses substitution to decide whether x2 1 x is equivalent to x(2x 1 1). She says the expressions are equivalent because they have the same value when x 5 0. Is Kari correct? Explain.

Possible answer: the equivalent expression with the least number of

terms will have 1 term. the terms will be all constants or all variable

terms. For example, 1 1 2 1 3 1 4 1 5 has 5 constant terms, and can be

simplified to a single term, 15. or, x2 1 2x2 1 3x2 1 4x2 1 5x2 has

5 variable terms, and can be simplified to a single term, 15x2.

Possible answer: no; the expressions do have the same value when

x 5 0, but kari should substitute a non-zero number for x. For example, let

x 5 4. then, x2 1 x 5 (4)2 1 4 5 16 1 4 5 20. but, x(2x 1 1) 5 4(2[4] 1 1) 5

4(8 1 1) 5 4(9) 5 36. since 20 Þ 36, x2 1 x cannot be equivalent to x(2x 1 1).

Part 6: common core Practice Lesson 17


L17: Ratios176

Solve the problems. Mark your answers to problems 1–4 on the Answer Form to the right. Be sure to show your work.

1 Which expression is equivalent to 3x 1 5 1 2x 1 x2 1 1?

A 12x

B 5x 1 5

C 6x 1 6

D x2 1 5x 1 6

2 Which lists all the like terms in the expression 4 1 x4 1 4x 1 4x2 1 2x?

A 2x, 4x

B x4, 4x, 4x2, 2x

C 4, x4, 4x, 4x2

D 4, 4x, 2x

3 The expression 0.25(2d 1 1) represents the fi nes per day, d, for overdue books. Which expression is equivalent to 0.25(2d 1 1)?

A 0.252d 1 1

B 0.50d 1 0.25

C 2d 1 0.25

D 0.50d 1 1

1 A B C D

2 A B C D

3 A B C D

4 A B C D

answer Form

numbercorrect 4

Differentiated instruction

L17: Equivalent Expressions180©Curriculum Associates, LLC Copying is not permitted.

assessment and Remediation

hands-on activity challenge activityWrite a problem to match the expression.

• Display for students the following expression:

3(y 1 1 ··

 2 y2)

• Ask students to create a word problem which models and represents the given expression.

• Help students to think of real-world situations (for example, ways to repackage sets of items) which might help them think of suitable problems.

• Encourage students to be creative with their word problems.

create a mini-poster.

Materials: algebra tiles, graph paper, colored paper, chart paper for posters, colored pencils, recycled and/or crafting objects

• Display for students the following expression:

16 1 12y 1 18y 1 20 1 4y2

• Ask students to create a mini-poster which models and represents equivalent expressions for the given expression in as many ways as they can.

• Suggest that students choose a different object for each of the terms in the expression.

• Help students to think of real-world situations (for example, shopping for groceries or gifts) to represent the expressions, which might help them picture a model.

• Encourage students to be creative with their models.

• Ask students to determine if the given expressions are equivalent and to provide their reasoning: 2(2x2 2 2) and 2x2 2 2x 2 4 1 2x2 1 2x. [Yes; 2x2 2 2x 2 4 1 2x2 1 2x 5 4x2 2 4 5 2(2x2 2 2)]

• For students who are struggling, use the chart below to guide remediation.

• After providing remediation, check students’ understanding. Ask students to determine if the given

expressions are equivalent and to provide their reasoning: 4(5x – 6) and 1 ··

 2 (30x – 12) 1 5x.

[No. Substituting x 5 1, 4(5x – 6) 5 24 and 1 ··

 2 (30x – 12) 1 5x 5 14. Since 24 Þ 14, the expressions are not

equivalent.]

if the error is . . . students may . . . to remediate . . .

yes, because both expressions equal 24.

have tested for equivalency with zero.

Model terms such as 2x and x2 which are not equivalent. Substitute zero for x showing that zero is not a valid value for substitution.

no, because the first expression does not have any x terms while the second expression does.

not recognize the result of opposites values when combining like terms.

Model expressions with algebra tiles with a focus on subtracting and negative values. Show students that 22x and 2x are opposites resulting in zero x terms. Practice similar expressions which require subtraction of terms.

Lesson 17 CCLS Equivalent Expressions 6.EE...23 Apply the distributive property to write an...

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