Supplemental Materials: Grade 6 Module 4 Lesson 12

Supplemental Materials

Adapt Optimizing Instruction, 6–9

A Story of Ratios® A Story of Functions®

Supplemental Materials

Adapt: Optimizing Instruction, 6–9

Contents Analyze Student Work .................................................................................................................................. 3 Grade 6 Module 4 Topic C Overview ........................................................................................................ 3 Grade 6 Module 4 Lesson 12 Overview .................................................................................................... 5 Achievement Descriptor – 6.Mod4.AD6 ................................................................................................... 6 Achievement Descriptor – 6.Mod4.AD7 ................................................................................................... 7 Achievement Descriptor – 6.Mod4.AD8 ................................................................................................... 8 Grade 6 Module 4 Lesson 12 ........................................................................................................................ 9 Credits ......................................................................................................................................................... 30 Works Cited ................................................................................................................................................. 30

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Topic C Equivalent Expressions Using the Properties of Operations In topics A and B, students learn the order of operations and write algebraic expressions from descriptions and real-world situations. In topic C, students build upon these skills when they use the order of operations to evaluate algebraic expressions and when they use properties to write and identify equivalent expressions.

As topic C starts, students write, interpret, and manipulate algebraic expressions that involve multiplication and division. Students develop an understanding of the importance of efficiency, realizing that expressions with fewer factors and symbols are easier to both

_​​ 1​​ · 12w and 3x · 3y are more efficiently expressed as 3w and 9xy, respectively. Students write interpret and evaluate. For example, students understand that the expressions

4w square w units units w + 7 units

4

and identify equivalent expressions, and they learn that equivalent expressions evaluate to the same number for all possible values of the variables.

Next, students build upon their work with the distributive property from grades 3 and 4 when they use the distributive property to write algebraic products as sums or differences.

By examining a rectangle composed of two smaller rectangles, students explain how different expressions, one written as a product and one written as a sum or difference, can represent the same area. Students use the distributive property to justify that expressions are equivalent. They develop fluency with using the distributive property to write equivalent expressions.

28 square 7 units units 4 units

In lesson 14, students use the distributive property to factor expressions. By using pictorial models and real-world situations, students develop a foundational understanding of how and why we write sums of whole numbers that have a common factor as products. They compare and interpret the real-world meanings of equivalent expressions such as 12 + 30 and 6(2 + 5). After writing numerical expressions as products of factors, students transition to writing algebraic sums and differences as products. Students develop fluency with factoring algebraic expressions such as 7x + 14 and 42m − 18.

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In the final two lessons, students learn how to combine like terms. Through a digital interactive, students use area models and angle relationships to learn how to combine like terms. Students develop the understanding that combining like terms is an application of the distributive property. After developing fluency with combining like terms that have multiple variables, students combine like terms in expressions that require distributing first, such as 5(2x + 3) + 2(4x + 1).

Expression Written Expression Written as a Sum as a Product

12p + 30c

3(4p + 10c)

In topics D and E, students apply their understanding of equivalent expressions when they write and solve one-variable equations and write two-variable equations.

Progression of Lessons Lesson 12 Applying Properties to Multiplication and Division Expressions Lesson 13 The Distributive Property Lesson 14 Using the Distributive Property to Factor Expressions Lesson 15 Combining Like Terms by Using the Distributive Property Lesson 16 Equivalent Algebraic Expressions

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LESSON 12

Applying Properties to Multiplication and Division Expressions Write and identify equivalent algebraic expressions involving multiplication and division by using the properties of operations.

Write algebraic expressions that represent real-world situations.

EUREKA MATH2

6 ▸ M4 ▸ TC ▸ Lesson 12

EXIT TICKET Name

Date

12

1. For parts (a) and (b), write three expressions that are each equivalent to the expression. a. 14b Sample:

___ 2 · 7b, 7 · 2b, 28b

1 · 36x b. _ 3

2

Lesson at a Glance This lesson begins with students working with a partner to write and evaluate multiplication and division expressions that represent real-world situations. Students write terms efficiently by using as few factors as possible and learn to write terms as products of their factors. In a class discussion, students explain that terms written with as few factors as possible allow for efficient evaluation. In a game of bingo, students identify equivalent algebraic expressions.

Key Questions

Sample:

36x , 12x, 36 ___ __ x 3 3

• How can we write a term so that we can evaluate it as efficiently as possible?

2. For parts (a) and (b), write an equivalent expression with as few factors as possible. a. 5g · 7h · 2

70gh

• What strategies can we use to determine whether expressions are equivalent?

3m __

Achievement Descriptors

3 m · 1_ b. _ 4 5 20

6.Mod4.AD6 Evaluate expressions, including those within formulas,

at specific values of their variables. (6.EE.A.2.c) 6.Mod4.AD7 Generate equivalent expressions by using the properties

of operations. (6.EE.A.3) 6.Mod4.AD8 Identify equivalent expressions. (6.EE.A.4)

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6.Mod4.AD5 Identify parts of an expression by using mathematical terms.

RELATED CCSSM

6.EE.A.2.b Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.

Partially Proficient

Proficient

Highly Proficient

Identify parts of an expression by using mathematical terms.

Consider the description of an expression.

The product of 5 and the sum of 4x and 8.

What are the factors of the expression?

Choose all that apply.

A. 5 B. 4x C. 8 D. 5 + 4x E. 4x + 8

6.Mod4.AD6 Evaluate expressions, including those within formulas, at specific values of their variables.

RELATED CCSSM

6.EE.A.2.c Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).

1 For example, use the formulas V = s 3 and A = 6s 2 to find the volume and surface area of a cube with sides of length s = ​​ ​​ .

__ 2

Partially Proficient

Proficient

Highly Proficient

Evaluate expressions, including those within formulas, at specific values of their variables.

The expression 25.50 + 7n represents the monthly cost of a water bill in dollars when n water bottles are used in one month. What is the cost of the water bill when 14 water bottles are used in one month?

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6.Mod4.AD7 Generate equivalent expressions by using the properties of operations.

RELATED CCSSM

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

Partially Proficient

Proficient Generate equivalent expressions by using the properties of operations.

Write three expressions that are each equivalent to x + y + 3(x + y).

Highly Proficient Identify and correct mistakes made when generating equivalent expressions by using the properties of operations.

Ryan writes an equivalent expression for the given expression.

4(x + 3) − 2x + 1 He uses the following steps.

Step 1

4x + 3 − 2x + 1

Step 2

4x − 2x + 3 + 1

Step 3

2x + 4

Part A In which step does Ryan make a mistake?

A. Step 1 B. Step 2 C. Step 3 Part B Explain Ryan’s mistake.

Part C Write a correct equivalent expression.

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6.Mod4.AD8 Identify equivalent expressions.

RELATED CCSSM

6.EE.A.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 + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.

Partially Proficient

Proficient

Highly Proficient

Identify equivalent expressions.

Which expressions are equivalent to 4(x + 2) − x?

Choose all that apply.

A. 4x + 2 − x B. 4x + 8 − x C. 3x + 8 D. 3x + 2 E. 4x + 8

6.Mod4.AD9 Determine whether a given value is a solution to an equation.

RELATED CCSSM

6.EE.B.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

Partially Proficient

Proficient

Highly Proficient

Determine whether a given value is a solution to an equation.

Which value of g makes 5 + 2(g + 4) = 13 a true number sentence?

A. 13 B. 11 C. 2 D. 0

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LESSON 12

Applying Properties to Multiplication and Division Expressions Write and identify equivalent algebraic expressions involving multiplication and division by using the properties of operations.

Write algebraic expressions that represent real-world situations.

EUREKA MATH2

6 ▸ M4 ▸ TC ▸ Lesson 12

EXIT TICKET Name

Date

12

1. For parts (a) and (b), write three expressions that are each equivalent to the expression. a. 14b Sample:

___ 2 · 7b, 7 · 2b, 28b

1 · 36x b. _ 3

2

Lesson at a Glance This lesson begins with students working with a partner to write and evaluate multiplication and division expressions that represent real-world situations. Students write terms efficiently by using as few factors as possible and learn to write terms as products of their factors. In a class discussion, students explain that terms written with as few factors as possible allow for efficient evaluation. In a game of bingo, students identify equivalent algebraic expressions.

Key Questions

Sample:

36x , 12x, 36 ___ __ x 3 3

• How can we write a term so that we can evaluate it as efficiently as possible?

2. For parts (a) and (b), write an equivalent expression with as few factors as possible. a. 5g · 7h · 2

70gh

• What strategies can we use to determine whether expressions are equivalent?

3m __

Achievement Descriptors

3 m · 1_ b. _ 4 5 20

6.Mod4.AD6 Evaluate expressions, including those within formulas,

at specific values of their variables. (6.EE.A.2.c) 6.Mod4.AD7 Generate equivalent expressions by using the properties

of operations. (6.EE.A.3) 6.Mod4.AD8 Identify equivalent expressions. (6.EE.A.4)

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Agenda

Materials

Fluency

Teacher

Launch Learn

5 min

30 min

• None

Students

• Real-World Multiplication and Division Expressions

• Bingo board (1 per student)

• Equivalent Expressions

• Copy and cut out bingo boards (in the teacher edition). Distribute equal numbers of the four types of bingo boards among the class.

• Expressions Bingo

Land

10 min

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Lesson Preparation

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Fluency Multiply Rational Numbers Students multiply rational numbers to prepare for writing products and quotients of factors.

Directions: Evaluate. Write your answer as a whole number as needed.

1.

2.

1 ​​ ​​ · 2 · 6 2

_

2 ​​ ​​ · 3 · 5 3

_

6

10

3.

1 1 ​​ ​​ · ​​ ​​ · 12 2 3

2

4.

1 3 ​​ ​​ · ​​ ​​ · 20 2 5

6

5.

4 · _8​​ ​​ · _6​​ ​​ 7 1

7 __ ​​ ​​

6.

1 3 2 ​​ ​​ · ​​ ​​ · ​​ ​​ 5 4 7

3 __ ​​ ​​

_ _ _ _

_ _ _

12

70

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Launch

6 ▸ M4 ▸ TC ▸ Lesson 12

5

Students write an algebraic expression to represent a real-world situation.

Display problem 1 and read the prompt aloud to students. Then allow students a couple of minutes to complete the problem with a partner.

1. Kelly’s city wants to increase the amount of fresh fruits and vegetables that people eat.

Kelly works with the city to estimate the number of pounds of fresh fruits and vegetables ​​ 1 ​​ of the foods that people plan to eat are that people eat each day. Kelly learns that about _ 3 fresh fruits and vegetables.

Differentiation: Support For students who need additional support starting Launch, consider providing sample values for p and f. Then connect how students reason with the concrete values to how they can reason with the variables.

Let p represent the number of people in Kelly’s city. Let f represent the number of pounds of food one person plans to eat in one day.

Which expressions can Kelly use to determine the number of pounds of fresh fruits and vegetables that people eat every day in her city? Choose all that apply.

_1 A. ​​ ​​ · f · p 3

_1 B. ​​ ​​( fp) 3

_1 C. ​​ ​​ f · p 3

1 _____ D. ​​ ​​ 3·f·p

_f E. ​​ ​ · p 3

_1 F. ​​ ​​ ( f + p) 3

G. 3 · f · p

H.

3 ·p f

_

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_3 I. p​​ · f

J. f · p ÷ 3

K. 3( f + p) After a minute or two, bring the class together to discuss which expressions are correct and how students know they are correct. Display the correct equivalent expressions. Discuss which expressions are the simplest to understand and why. Ask students whether they can determine an even more efficient way to write the expression. Share the expressions _ ​​ 1 ​​ fp 3

and __ ​​ ​​ with students. fp 3

Why might the expressions _ ​​ 1 ​​ fp and __ ​​ ​​ be considered the simplest to understand?

3

fp 3

The only symbol other than the number and the variables is the fraction bar. These expressions don’t have extra symbols or parentheses.

Today, we will learn different ways to write expressions involving multiplication and division.

Learn Real-World Multiplication and Division Expressions Students write and evaluate algebraic expressions by using multiplication and division.

The expressions _​​ · p and _ ​​ 1 ​​ fp are equivalent. The two expressions are equivalent f 3

3

because for any values that f and p represent, the two expressions evaluate to the

same number. Let’s look at an example.

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Teacher Note To prove that two expressions are equivalent, we would need to evaluate each expression for every possible value of the variable, which is not possible. However, although substitution does not prove equivalency, substitution can be a useful strategy to check whether two expressions are not equivalent. To ensure that students use this strategy appropriately, encourage students to substitute at least two values for each variable in the expressions to check whether the expressions are equivalent. If students evaluate two expressions for the same value of the variable and get different numbers, then they know that the two expressions are not equivalent.

In a later activity, students look at an example of misused substitution to draw an inaccurate conclusion about the equivalency of two expressions.

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Substitute 4 for f, the typical number of pounds of food that one person plans to eat in one day, in both expressions. Substitute 12,000 for p, the number of people in Kelly’s city, in both expressions. What is the value of each expression?

Both expressions evaluate to 16,000.

Display the work for substituting 4 for f and 12,000 for p in each expression.

_f ​​ · p = ​​ 4_ ​​​(​12,000​) 3

3

48,000 ​= ​_____ 3

= 16,000 ​​ _3 ​​ fp = ​​ _3 ​​(4)​(​ ​12,000​) 1

1

= ​​ 3_ ​​​(​48,000​) 1

= 16,000 When we substitute 4 for f and 12,000 for p, both expressions evaluate to 16,000. Did we prove that these expressions are equivalent? Explain.

No. To use substitution to prove that these expressions are equivalent, we would have to evaluate both expressions at every possible value of each variable instead of evaluating at just one value.

Choosing values for the variables and substituting them into the two expressions to see whether the expressions evaluate to the same number does not actually prove that the two expressions are equivalent. However, substitution is a useful strategy to check equivalency. If the expressions evaluate to different numbers when we substitute the same values for each variable in both expressions, then we know that the expressions are not equivalent.

What is the meaning of 16,000 in this situation?

On a typical day, the people in Kelly’s city eat 16,000 pounds of fresh fruits and vegetables.

Direct students to complete problem 2 with a partner.

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2. A tennis club buys boxes of tennis balls. Each box has C containers that hold 12 tennis balls each. The club divides the tennis balls equally among its 4 locations. Write an expression to represent the number of tennis balls that each location receives. Let B represent the number of boxes of tennis balls the club buys.

12 · B · C ​​ 4

_______

When students are finished, bring the class together. Invite students to share their expressions. Display the expressions for the class. Anticipate the following expressions:

12 · B · C ÷ 4 12BC ____ ​​ ​​ 4

3BC

1 ​​ ​​ · 12BC 4

_

Teacher Note Students may use lowercase or uppercase letters as variables, but whichever they use must match the variable defined. Sometimes, choosing between uppercase and lowercase helps avoid writing letters that look like numbers. For example, the letter b can easily be confused with the number 6. While variables can be uppercase, as in problem 2, they are most commonly lowercase. Consider pointing this out to students as they complete problem 2.

How many terms are in each expression? Explain.

Each expression has one term because each expression can be written as a product of numbers and variables.

We can write our expression efficiently as 3BC, which only has a single number multiplied by the variables. We write the coefficient before the variables. So we wouldn’t want to write this term as BC3 or B3C.

Display the expressions _​​ 1 ·​ 12 · B · C​ and 3BC. 4

Ask students to think–pair–share about the following question.

3BC is simpler to interpret because it has fewer factors than _​​ 1 ​​ ·​ 12 · B · C​and 3BC doesn’t

Teacher Note In grade 6, students are not required to write multiple variables in a term in alphabetical order, but it is a useful habit to develop for the future.

Which expression is simpler to interpret? Explain.

use the multiplication dot.

4

What is the real-world meaning of 3BC? The number of tennis balls that each location receives is 3 times the number of boxes the club buys multiplied by the number of containers in each box.

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3BC is more efficient for evaluating than _​​ 1 ​· 12 · B · C​because 3BC requires fewer Which expression is more efficient for evaluating? Explain.

4

calculations to determine the number of tennis balls that each location receives.

Allow students several minutes to complete problems 3 and 4 with a partner. Circulate as students work, asking the following questions: • How do you know what is being multiplied and what is being divided?

• How can you write the expression without multiplication or division symbols?

3. Jada has 24 boxes of toys with t toys in each box. She gives one-third of the boxes to a charity.

Write an expression to represent the total number of toys that Jada gives to the charity.

24t 1 ​​ 3 ​​, 24 · t · ​​ 3 ​​, 8t​​

___

_

If each box has 9 toys, how many toys does Jada give to the charity?

____

24​(​9)​ ​ ​ = 72​ 3

Language Support Charities, or charitable organizations, are organizations that strive to improve the wellbeing of people, animals, or the environment.

Charities raise money for the causes they support. Ask students whether they can identify any charities in their community.

Jada gives 72 toys to the charity.

4. Noah earns $14 per hour. He deposits _​​ 3 ​​ of his earnings into a savings account.​​ 4

Write an expression to represent the amount of money Noah deposits into his savings account when he works for h hours.

3 3 ​​4​​ · 14h, 1​ 4 · h ·​ ​​ 4​​

_

_

Suppose Noah works for 30 hours. How much money does Noah deposit into his savings account?

_​​3​​ · 14(30) = 315 4

Noah deposits $315 into his savings account.

When most students are finished, bring the class together. Invite students to share their expressions for part (a) of each problem. Display the expressions for the class to compare.

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Highlight which terms are simplest to interpret and work with when evaluating. For example, note that the term 8t is simpler to interpret and evaluate than the term 2 ​ 4 · t · _1 ​.​ 3

Equivalent Expressions Students write and identify equivalent algebraic expressions.

Ask students to turn and talk about what a factor is. Then ask the following questions to discuss equivalent expressions.

What are the factors of 12?

1, 2, 3, 4, 6, and 12 2·6

What different ways can we write 12 as a product of factors?

3·4

12 · 1

2·2·3

Now consider the expression 12x. What are the factors of 12x?

1, 2, 3, 4, 6, 12, and x The factors of 12x include the factors of 12. But the factors of 12x also include a factor of x. How can we write 12x as a product of factors?

12 · x

Sample:

4 · 3x 2x · 6

2·2·3·x

How can we write the expression __ ​​ ​​ as a product of factors?​​ 3 ​​ · f 2

_

3f 2

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3 · ​​ _2​​ f

3 · ​​ _2​​ · f 1

Notice that we can write __ ​​ ​​ as a product of factors in more than one way, just as we 3f 2

can write 12x as a product of factors in many ways.

Display the expressions __ ​​ ​​, _ ​​ 3 ​​ · f, 3 · _​​ ​​, and 3 · _​​ 1 ​​ · f . Ask students to turn and talk about which 3f 2 2

f 2

2

We can write 25 as 5 · 5, and we can write 81 as 9 · 9. How can we write x 2 as a product of factors?

expression they think is the most efficient to use for evaluation and why.

x·x

Are 2x and x2 equivalent expressions? Explain.

No. 2x represents 2 times x, and x 2 represents x times x.

What expression can we write that is equivalent to x · x · x ?

x3

When we combine all numerical factors into a single coefficient and write any repeated multiplication of a variable as a single base with an exponent, such as x 3, we are writing an equivalent expression with as few factors as possible. How would you write 5 · 5 · x · x · x with as few factors as possible?

UDL: Representation Before discussing how to write variables with exponents, consider activating prior knowledge by reviewing exponents.

Consider reviewing the terms power, base, and exponent as well as the fact that the exponents represent repeated multiplication.

Use the following examples.

• 3 · 3 · 3 · 3 = 34 = 81

• 32 = 25 = 2 · 2 · 2 · 2 · 2

25 x 3

Allow students to complete problems 5–15 with a partner. Circulate as students work, asking the following questions: • How can you use the commutative property of multiplication to write an equivalent expression?

• How do you determine the coefficient of the term?

• What factors can you use to write

?

• How can you use substitution to check your answer?

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For problems 5–8, write an equivalent expression with as few factors as possible.

_3 5. ​​ ​ · 8 · y 4

_3​​ ​ · 8 · y​ = 6​ ·​ y

6. 5 · 8 · r · r

5 · 8 · r · r = 40 · r · r

4

= 6y

_9 7. ​​ ​ · d · 10 5

_​​9​ · d · 10​ = _9​​ ​ · 10 · d 5

__3 8. 4 ​ · v · ​ ​ · 2 11

= 40 · r 2

= 40r 2

3

24 __ ​= ​ ​ · v 11

3

__ = ​ ​ ​v​ 11 24

For problems 9 and 10, fill in the blank to make the equation true.

7

If students are not yet ready to answer problems 9–15 independently, choose a few problems and work through them as a class.

4​ · v · ​ __ ​ · 2 = 4 · 2 · ​ __ ​·v 11 11

5

= 18d

9. 56w = 8w ·

Teacher Note

10. 45y = 3 ·

UDL: Representation As needed, consider reframing problems 9 and 10 in the following way.

• 8w times what number equals 56w?

15y

• 3 sets of what factor make a total of ​45y? ​

Sample: 5 · 7x, 7 · 5x, 5 · 7 · x

11. Write three expressions that are each equivalent to 35x.

Differentiation: Challenge

12. What is the value of 35x when x = 5?

35(5) = 175

13. Are 3x · 3y and 3xy equivalent expressions? How do you know?

No. The expressions are not equivalent because 3x · 3y = 3 · 3 · x · y = 9xy.

For students who complete problems 5–13 quickly, consider asking the following question.

• Are the expressions ​​ y · y · y ​​ and y2

_____ y

sometimes, always, or never equivalent?

How do you know?

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14. Noah says that 4x and x4 are equivalent because when x = 0, the expressions evaluate to the same number. Is Noah correct? Explain.

Noah is not correct. The expressions evaluate to the same number when x = 0 but not for every value of x. For example, when x = 1, 4x evaluates to 4 and x4 evaluates to 1.

15. Which expressions are equivalent to 63gh? Choose all that apply.

A. 7g · 9h

B. 3 · 21 · g · h C. 7h · 9g

D. 63 + g + h E. 63g + h

F. 7gh · 9gh

G. 63h · g

When most students have finished, share the answers for problems 5–10. While discussing the problems, consider asking the following questions: • What strategies can you use to write an equivalent expression with as few factors as possible?

• Is there more than one possible solution for problems 9 and 10? Explain.

Ask a couple of students to share their responses to problem 11. Display a list of correct products as students share. Consider discussing the fact that an infinite number of multiplication expressions are equivalent to 35x. Add on to the students’ list, including answers such as 35 · x and _ ​​ 1​​ · 70x.

2

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Invite students to share their strategies for determining that the two expressions in problem 13 are not equivalent. Anticipate that some students will use substitution to show that the expressions are not equivalent while other students will use the commutative property of multiplication.

Display problem 15.

Invite students to share their thoughts about which expressions are equivalent to 63gh.

Encourage students to debate and to add on to one another’s responses. As the class discusses the answer choices, consider asking the following questions: • How do you know the expression is equivalent to 63gh?

Language Support As students discuss problem 15, encourage them to use the Agree or Disagree section of the Talking Tool to support the discussion.

Promoting the Standards for Mathematical Practice

• How do you know the expression is not equivalent to 63gh?

• What properties can you use to help you determine whether the expression is equivalent to 63gh?

Sana substitutes 1 for g and 1 for h. She determines that the expression 7gh · 9gh is equivalent to 63gh because both expressions evaluate to 63. Is she correct? Explain.

Ask students to think–pair–share about the following question.

No. If Sana substitutes other values for g and h, such as 1 for g and 2 for h, she will see that the expressions are not equivalent because they do not evaluate to the same number. 63gh evaluates to 126 and 7gh · 9gh evaluates to 252.

When using substitution to determine whether expressions are not equivalent, test at least two different values for the variables. At this point in our mathematics education, we can evaluate expressions by substituting a few different values for the variables and then make a reasonable assumption about what’s true. Substitution does not prove that two expressions are equivalent, but it lets us know that they’re not equivalent.

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When students listen to and analyze their peers’ ideas about which expressions are equivalent to 63gh, they are constructing viable arguments and critiquing the reasoning of others (MP3).

Ask the following questions to promote MP3: • Is that selection a guess or do you know for sure that the expression is equivalent to 63gh? How do you know for sure?

• Why does your strategy work? Convince your partner.

• What questions can you ask your partner about why they think the two expressions are equivalent?

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EUREKA MATH2

6 ▸ M4 ▸ TC ▸ Lesson 12

Expressions Bingo Students fluently match equivalent algebraic expressions.

Randomly distribute the four different versions of bingo boards to students. Direct students to insert the bingo boards into their personal whiteboards. Note that on each bingo board, some of the squares are already filled in with answers. Have students use dry-erase markers to randomly fill in the remaining eight answers on their bingo board.

Display the remaining eight answers.

m·p÷2

​m​

2p ​​ ​​ 2

4mp

8mp

​6mp​

__

2mp

3 2 ​2 ​​m · ​3 ​​p

_

_

Teacher Note Because students insert the bingo boards into personal whiteboards and use dry-erase markers to fill in the remaining bingo answers, the bingo boards can be reused in multiple classes if needed. Collect the bingo boards at the end of each class. If there is not enough time for the bingo game, these questions can be answered on personal whiteboards.

On your bingo board, eight of the answers are already filled in. Write the remaining eight answers randomly on your board. Fill in one answer per square.

If there is not enough time for the bingo game, these questions can be answered on personal whiteboards.

I will read a clue. If you see an expression that is equivalent to the given clue, mark that square on your bingo board. The first person with five marked expressions in a line wins.

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EUREKA MATH2

6 ▸ M4 ▸ TC ▸ Lesson 12

Read aloud the clues for the game in the left column. The answers on the boards are in the right column. Circulate while reading clues, correcting students who mark incorrect answers.

Clue 12m

8 · 4m

​​ _4​​ · 24m

Answer ​2 · 6m​

​​ _2​​ · 16m 1

4m · ​​ _2​​ 1

m·m

4·m·0

6m

Answer m·p÷2

​​ _2 ​​mp 1

m·1

32m

1

​m​

Clue

​m​

__

2p ​​ 2 ​​

​p​

​​ _6​​ · 6m

4m · ​​ _2 ​​p

​2mp​

2m

2 · 2 · 2mp

8mp

1

8m

m2 0

1

2m · 2p

​4mp​

​​ _2 ​​m · 4p 3

6mp

2 3 ​m · ​ 3​p 2

_

​mp​

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_

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EUREKA MATH2

6 ▸ M4 ▸ TC ▸ Lesson 12

Land Debrief

5 min

Objectives: Write and identify equivalent algebraic expressions involving multiplication and division by using the properties of operations.

Write algebraic expressions that represent real-world situations.

Use the following prompts to facilitate a class discussion about equivalent expressions.

How can we write a term so that we can evaluate it as efficiently as possible?

We can eliminate multiplication symbols and use a fraction to represent division. We can multiply all numbers to determine the coefficient.

What strategies can you use to determine whether expressions are equivalent?

I can use the commutative property of multiplication to rewrite expressions. If I evaluate two expressions by substituting the same value for each variable in both expressions and those expressions evaluate to different numbers, then I know that the two expressions are not equivalent. If I choose different values to substitute for the variables and the expressions evaluate to the same number for every different value of each variable, then I can be fairly confident that the two expressions are equivalent.

Exit Ticket

5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

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Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

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EUREKA MATH2

6 ▸ M4 ▸ TC ▸ Lesson 12

Recap

EUREKA MATH2

RECAP Name

Date

12

56r 3 = 8r ·

used properties of operations to write and identify equivalent algebraic expressions. wrote algebraic expressions that represent real-world situations.

a. Write an expression to represent the number of pencils each student receives. Let b represent the number of boxes of pencils Miss Song gets. · 12 · b _______ Sample: 10 24

Examples For problems 1 and 2, write an equivalent expression with as few factors as possible. Show how you know.

1. 30 · 1_ g 5

30 · 1_ g = 30 · 1_ · g 5

5

= 6g

b. Suppose Miss Song gets 5 boxes of pencils. How many pencils does each student receive? Show your work. · 12 · 5 10 · 12 · b = 10 _______ _______ 24 24

2. 6x · 7y · 1_ 2

6 · x · 7 · y · 1_ = 1_ · 6 · 7 · x · y 2

30 · _ g = 30 · _ (10) 1 5

1 5

Write an equivalent expression with as few factors as possible.

Multiply and divide to create a term with a single coefficient and a variable.

10 · 12 · b _____ 120 · b _______ = 24

___ = 600

24

= 5b

24

2

=3·7·x·y

= 25 Each student receives 25 pencils.

= 21xy Two expressions are equivalent if they evaluate to the same number for every possible value of the variable. It is impossible to substitute every value for g. However, it is reasonable to substitute a few values to check whether the expressions are not equivalent. For instance, substitute 10 for g in both expressions.

7r 2

4. Miss Song gets boxes of pencils for her class. Each box has 10 packages of pencils. Each package has 12 pencils. Miss Song splits the pencils equally among 24 students.

In this lesson, we •

A factor pair of 56 is 8 · 7. We can write r3 as r · r · r and r · r · r = r · r2.

3. Fill in the blank to make the equation true.

Applying Properties to Multiplication and Division Expressions •

EUREKA MATH2

6 ▸ M4 ▸ TC ▸ Lesson 12

6 ▸ M4 ▸ TC ▸ Lesson 12

Consider also using the equivalent expression 5b. Substitute 5 for b to determine that each student receives 25 pencils.

5b = 5(5) Use the commutative property of multiplication to rearrange the expressions. Then multiply the numerical factors and write the expression as a single coefficient multiplied by one or more variables.

= 25

= 6(10) = 60

6g = 6(10) = 60

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R E CA P

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EUREKA MATH2

6 ▸ M4 ▸ TC ▸ Lesson 12

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

PRACTICE Name

Date

EUREKA MATH2

6 ▸ M4 ▸ TC ▸ Lesson 12

6 ▸ M4 ▸ TC ▸ Lesson 12

12

For problems 3–8, write an equivalent expression with as few factors as possible. Show how you know.

3. 2 · 5 · 10n

4. 1_ g ⋅ 16 4

2 · 5 · 10n = 10 · 10 · n

1. Eddie earns $15 per week for w weeks. He saves half of what he earns. Which expressions represent the amount of money that Eddie saves? Choose all that apply.

15w A. ___ 2

= 100 · n

_1 g ⋅ 16 = 1_ ⋅ 16 ⋅ g 4 4 = 4g

= 100n

B. 2 · 15 · w

5. 8x ⋅ 4y ⋅ 1_ 2

C. _1 (15w) 2

6. p ⋅ 7_ ⋅ 12 6

8 ⋅ x ⋅ 4 ⋅ y ⋅ 1_ = 1_ ⋅ 8 ⋅ 4 ⋅ x ⋅ y

15 · w D. __ 2

2

E. _1 w 2

=4·4·x·y 2

p ⋅ 7_ ⋅ 12 = p ⋅ 14 6

= 14p

= 16xy

F. _1 · 15 · w 2

3 ⋅ 5_ ⋅ 8 f 7. _ 4 6

G. 15w ÷ 2 15 H. ___ 2w

8. 6 ⋅ x ⋅ 3_ ⋅ 5_ 4 6

6 ⋅ x ⋅ 3_ ⋅ 5_ = 6 ⋅ 3_ ⋅ 5_ ⋅ x

15 _3 ⋅ 5_ ⋅ 8 f = __ ⋅8⋅f 24 4 6

4 6

= 5_ ⋅ 8 ⋅ f

4

8

2. An owner of burger restaurants receives boxes of burger buns. Each box has 12 packages of burger buns. Each package has 8 burger buns. The owner splits the burger buns among 3 restaurants.

4 6

18 5_ = __ ⋅ ⋅x 15 = __ x

= 5f

6

4

9. Are _1 x ⋅ 4 ⋅ 6 and 12x equivalent expressions? Explain.

a. Write an expression to represent the number of burger buns each restaurant gets. Let B represent the number of boxes of burger buns the owner receives.

12 ⋅ 8 ⋅ B Sample: _______

2

of multiplication to show that _1 x ⋅ 4 ⋅ 6 = x ⋅ _1 ⋅ 4 ⋅ 6 = x ⋅ 12 = 12x. So for any possible value of x, Sample: Yes, they are equivalent expressions. I can use the commutative property

3

2

b. Suppose the owner receives 10 boxes of burger buns. How many burger buns does each restaurant get? Show your work.

12 ⋅ 8 ⋅ B = _______ 12 ⋅ 8 ⋅ 10 _______ 3

960 = ___ 3

2

both expressions evaluate to the same number.

10. Are 9x · 8 and 81x equivalent expressions? Explain.

3

Sample: No, they are not equivalent expressions. I can substitute 10 for x in each expression.

The expressions do not evaluate to the same number because 9(10) · 8 = 720 and 81(10) = 810.

So the expressions are not equivalent.

= 320 Each restaurant gets 320 burger buns.

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156

P R ACT I C E

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EUREKA MATH2

6 ▸ M4 ▸ TC ▸ Lesson 12

EUREKA MATH2

6 ▸ M4 ▸ TC ▸ Lesson 12

For problems 11 and 12, fill in the blank to make the equation true.

11. 28t = 4t ⋅

12. 48r 3 = 8r ⋅

7

6r 2

Remember For problems 13–16, multiply.

1 × 12 13. _ 4

1 × 25 14. _ 5

1 × 80 15. _ 8

1 × 81 16. _ 9

3

5

10

9

17. A fruit punch recipe calls for 7 ounces of apple juice for every 2 ounces of grape juice. Write an algebraic expression that represents the number of ounces of apple juice for g ounces of grape juice.

3.5g 18. Tyler uses 3 cups of milk to make 4 batches of pancakes. How many cups of milk does Tyler need to make 1 batch of pancakes?

Tyler needs 3_ cups of milk to make 1 batch of pancakes.

4

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P R ACT I C E

157

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EUREKA MATH2

1 ​​ ​​ · 6m 6

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6m

2m

2 · 6m

m2

32m

2m

8m

32m

m2

2 · 6m

Bingo Board 2

Bingo Board 1

_

0

1 ​​ ​​ · 6m 6

_

0

8m

6m

6 ▸ M4 ▸ TC ▸ Lesson 12 ▸ Bingo Boards

28

This page may be reproduced for classroom use only.

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2 · 6m

32m 6m

2m

0

_

1 ​​ ​​ · 6m 6

_

6m

8m

1 ​​ ​​ · 6m 6

Bingo Board 4

8m

32m

0

2m 2 · 6m

m2 Bingo Board 3

m2

EUREKA MATH2

6 ▸ M4 ▸ TC ▸ Lesson 12 ▸ Bingo Boards

This page may be reproduced for classroom use only.

EM2_0604TE_C_L12_bingo_boards.indd 234

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Supplemental Materials

Adapt: Optimizing Instruction, 6–9

Credits Great Minds® has made every effort to obtain permission for the reprinting of all copyrighted material.

If any owner of copyrighted material is not acknowledged herein, please contact Great Minds for proper acknowledgement in all future editions and reprints of this handout.

Works Cited Great Minds. Eureka Math2TM. Washington, DC: Great Minds, 2021. https://greatminds.org/math.

© 2022 Great Minds PBC

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