Grade 6 Module 4 Lesson 13 by General

LESSON 13

The Distributive Property Use the distributive property to write the product of two factors as a sum or difference.

Lesson at a Glance EXIT TICKET Name

Date

1. The rectangle is made up of two smaller rectangles. Write two expressions that each represent the total area of the rectangle in square units.

m units

2 units

1.5 units 1.5(m + 2) 1.5m + 3 2. Use the distributive property to write an equivalent expression.

4(2b − 5) 4(2b − 5) = 4(2b) − 4(5) = 8b − 20

In this lesson, students consider rectangles made up of two smaller rectangles. At first, both side lengths of the rectangle are known; later, one side length is unknown. Students write two expressions that each represent the area of the new rectangle formed: one that is a product of two factors and one that is a sum of two products. They write two expressions that each represent the area of one of the smaller rectangles: one that is a product of two factors and one that is a difference of two products. Students use the distributive property to show why the expressions are equivalent. In pairs, students complete several sequences of problems to practice using the distributive property to write equivalent expressions.

Key Question • How can we use the distributive property to write a product as an equivalent expression?

Achievement Descriptors 6.Mod4.AD2 Express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor by using the distributive property. (6.NS.B.4) 6.Mod4.AD7 Generate equivalent expressions by using the properties

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Agenda

Materials

Fluency

Teacher

Launch Learn

5 min

30 min

• None

Students

• Areas of Rectangles

• None

• Area of the Shaded Rectangle

Lesson Preparation

Land

• None

10 min

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Fluency Evaluate Numerical Expressions Students evaluate numerical expressions by using the order of operations to prepare for applying the distributive property to algebraic expressions. Directions: Evaluate. 1.

2(4 + 5)

2(4) + 2(5)

(3 + 7)3

(3 + 7 + 5)3

5(2 + 1) + 5(6 + 3)

3(2 + 3) + 9(2 + 3)

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Launch

Students determine the areas of rectangles. Direct students to problem 1. Have them complete the problem individually or in pairs. Circulate as students work. For part (c), identify students who find the area of the rectangle by using addition and those who use multiplication. 1. Find the area of each rectangle. a.

7 units

8 units

7 units

13 units

7 units 8 units

13 units

7(8) = 56 56 square units

7(13) = 91 91 square units

56 + 91 = 147 7(8 + 13) = 7(21) = 147 147 square units

Invite the identified students to share their thinking about part (c). Purposefully choose work that allows for rich discussion about connections between strategies. Why does the expression 7(8 + 13) represent the area of the rectangle in part (c)?

7(8 + 13) represents the area of the rectangle in part (c) because it is the product of the rectangle’s length, which is 7, and its width, which is the sum of 8 and 13.

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Why does the expression 7(8) + 7(13) represent the area of the rectangle in part (c)?

7(8) + 7(13) represents the area of the rectangle in part (c) because it is the sum of the areas of the two smaller rectangles. Because the numerical expressions 7(8 + 13) and 7(8) + 7(13) both represent the rectangle’s area, what can we conclude about the two numerical expressions? The expressions must be equivalent. Ask students whether they remember the name of the property that allows us to write the product of a number and a sum, such as 7(8 + 13), as a sum of two products, such as 7(8) + 7(13). Confirm that this is an example of using the distributive property to write an equivalent expression. Today, we will write equivalent algebraic expressions by using the distributive property.

Learn Areas of Rectangles Students write expressions to represent the area of a rectangle composed of smaller rectangles. Display the picture of the two rectangles from problems 2(a) and 2(b). Have students record answers in their books throughout the discussion of problem 2. What is the area of each rectangle? How do you know? The area of the yellow rectangle is 28 square units because 4(7) = 28. The area of the blue rectangle is 4w square units because 4(w) = 4w.

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Teacher Note

7 units

w units

4 units 4 units

In earlier grades, students write the product of a number and a numerical sum as the sum of two products. They also write the product of a number and a numerical difference as the difference of two products. Encourage students to use the same understanding when they write equivalent algebraic expressions by using the distributive property.

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Display the picture of the rectangles joined together from problem 2(c). UDL: Representation

The two rectangles are put together to form a new rectangle. The length of the new rectangle is 4 units. What is the width of the new rectangle?

w units

The width of the new rectangle is w + 7 units.

4w square units and 28 square units inside

If students make the mistake of saying the width of the new rectangle is 7w units rather than w + 7 units, have them look back at problem 1(c). Tell them to think about how they found the width of that rectangle. If necessary, have them substitute a value for w in 7w and check whether that makes sense as the width of the new rectangle. Have students think–pair–share about the following prompts.

Draw students’ attention to the idea of adding the areas of the two smaller rectangles to find the area of the largest rectangle by writing the smaller rectangles as shown. Draw grid lines and draw braces showing the widths of

7 units 4 units

Can you think of two algebraic expressions that each represent the area of the new rectangle in square units? If so, what are the expressions?

w units, 7 units, and w + 7 units.

4w square w units units w + 7 units

Yes. Both 4(w + 7) and 4w + 28 represent the area of the new rectangle. Display the expressions 4(w + 7) and 4w + 28. Why does the expression 4(w + 7) represent the area of the new rectangle?

4(w + 7) represents the area of the new rectangle because it is the product of the new rectangle’s length, which is 4 units, and its width, which is the sum of w units and 7 units. Does the expression (w + 7)4 also represent the area of the new rectangle? Why?

28 square 7 units units 4 units

Yes. (w + 7)4 also represents the area of the new rectangle. This expression is equivalent to 4(w + 7) because multiplication is commutative. Why does the expression 4w + 28 represent the area of the new rectangle?

4w + 28 represents the area of the new rectangle because it is the sum of the areas of the two smaller rectangles. Because 4(w + 7) and 4w + 28 represent the same area for any value of w, what can we conclude about these two algebraic expressions? The expressions must be equivalent.

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What property allows us to write the product of a number and a sum, such as 4(w + 7), as a sum of two products, such as 4w + 4(7)?

Language Support

The distributive property. Consider the rectangle in problem 3. Does it matter whether the rectangle is horizontal, as in problem 3, or vertical, as in problem 2? Explain. No. We still find the area by multiplying the rectangle’s length by its width. Have students complete problems 3 and 4 in pairs. Circulate as students work. Ask the following questions to advance student thinking:

Consider creating an anchor chart with students to help them understand how using the distributive property with variables is like using the distributive property with numbers. Later, add examples showing the distributive property with subtraction.

• How do you find the area of a rectangle?

The Distributive Property

• What are the length and width of the rectangle?

5(8 + 3) = 5(8) + 5(3) = 55

• What are the areas of the smaller rectangles?

5(x + 3) = 5(x) + 5(3) = 5x + 15

• How do you use the distributive property?

a(b + c) = a(b) + a(c) = ab + ac 5(8 − 3) = 5(8) − 5(3) = 25 5(x − 3) = 5(x) − 5(3) = 5x − 15

2. Find the area of each rectangle. a.

a(b − c) = a(b) − a(c) = ab − ac

7 units

w units

4 units 4 units 7 units 4 units 4(7) = 28 28 square units

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4(w) = 4w

4(w + 7) or 4w + 28 square units

4w square units

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3. The rectangle is made up of two smaller rectangles. Both 3(6 + 2x) and 18 + 6x represent the total area of the rectangle in square units.

6 units

Differentiation: Challenge

2x units Consider having students work the following problem to challenge their thinking.

3 units

The rectangle is made up of three smaller rectangles.

a. Which of the two expressions represent the total area of the rectangle as the product of the rectangle’s length and width?

a units

b units

c units

3(6 + 2x)

3.5 units

b. Which of the two expressions represent the total area of the rectangle as the sum of the areas of the smaller rectangles?

18 + 6x c. Use the distributive property to show that the expressions are equivalent. Fill in the blanks.

3(6 + 2x) =

(6) + 3(

)

= 18 + 6x

• Write two expressions that each represent the total area of the rectangle in square units.

3.5a + 3.5b + 3.5c 3.5(a + b + c)

4. The rectangle is made up of two smaller rectangles.

2.5 units

• Use the distributive property to show that the expressions are equivalent.

3.5(a + b + c) = 3.5(a) + 3.5(b) + 3.5(c) = 3.5a + 3.5b + 3.5c

h units

• Choose values greater than 0 for a, b, and c. Substitute the values into both expressions and evaluate. Sample: Let a = 1, let b = 2, let c = 3.

4 units

3.5(1) + 3.5(2) + 3.5(3) = 3.5 + 7 + 10.5 = 10.5 + 10.5 = 21 3.5((1) + (2) + (3)) = 3.5(3 + 3)

a. Write an expression to represent the total area of the rectangle in square units as the product of the rectangle’s length and width.

= 3.5(6) = 21

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b. Write an expression to represent the total area of the rectangle in square units as the sum of the areas of the two smaller rectangles.

2.5h + 10 c. Use the distributive property to show that the expressions you wrote in parts (a) and (b) are equivalent.

2.5(h + 4) = 2.5(h) + 2.5(4) = 2.5h + 10 When most students have finished, invite students to share their answers. Ask the following questions to debrief the problems. In problem 3, what is the area of the rectangle with side lengths 3 units and 2x units? How do you know? The area of the rectangle is 6x square units because it is the product of the side lengths. When we multiply a number by a term with a coefficient and a variable, we multiply the number by the coefficient of the term. In problem 4(a), what are the factors of 2.5(h + 4)?

2.5 and h + 4 Although h + 4 is an algebraic expression, the sum h + 4 is being multiplied by 2.5, so we can think of h + 4 as a single factor. Could we have written (h + 4)2.5 to represent the total area of the rectangle? If so, why? Yes. We can multiply the factors h + 4 and 2.5 in either order because multiplication is commutative.

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Area of the Shaded Rectangle Students write expressions to represent the area of a shaded rectangle within a larger rectangle. Direct students to problem 5. Have them complete the problem individually or in pairs. Circulate as students work. Listen for different strategies students use to find the area of the shaded rectangle.

2 units

5. Find the area of the shaded rectangle.

2(15 − 7) = 16 2(15) − 2(7) = 16

7 units

16 square units

15 units

Identify a few students to share their thinking, such as students who used 2(15 − 7) and students who used 2(15) − 2(7). Purposefully choose work that allows for rich discussion about connections between strategies. Why does the expression 2(15 − 7) represent the area of the shaded rectangle?

2(15 − 7) represents the area of the shaded rectangle because it is the product of the shaded rectangle’s length and width. Why does the expression 2(15) − 2(7) represent the area of the shaded rectangle?

2(15) − 2(7) represents the area of the shaded rectangle because it is the difference between the areas of the largest rectangle and the unshaded rectangle.

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Because the numerical expressions 2(15 − 7) and 2(15) − 2(7) both represent the shaded rectangle’s area, what can we conclude about the two numerical expressions? The expressions must be equivalent. Display the rectangle from problem 6. Have students record answers in their books throughout the discussion of problem 6. Have students think–pair–share about the following prompt.

2 units w units

The length of the shaded rectangle is 3 units. What is the width of the shaded rectangle? How do you know? The width of the shaded rectangle is w − 2 units. We can subtract the width of the unshaded rectangle, 2 units, from the width of the largest rectangle, w units, to find the width of the shaded rectangle.

3 units

Label the width of the shaded rectangle as w − 2 units. The shaded rectangle’s length is 3 units. We found that the shaded rectangle’s width is w − 2 units. Use the length and width to write an expression on your whiteboard that represents the area of the shaded rectangle in square units. Have students show you their whiteboards. Check that they have written 3(w − 2). What is the area of the largest rectangle? How do you know? The area of the largest rectangle is 3w square units. We can multiply the length of the largest rectangle, 3 units, by its width, w units, to find its area. What is the area of the unshaded rectangle? How do you know? The area of the unshaded rectangle is 6 square units. We can multiply the length of the unshaded rectangle, 3 units, by its width, 2 units, to find its area.

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Have students think–pair–share about the following prompt. How can we use the expressions that represent the largest rectangle’s area and the unshaded rectangle’s area to write another expression that represents the shaded rectangle’s area? We can subtract the expression that represents the unshaded rectangle’s area, 6, from the expression that represents the largest rectangle’s area, 3w, to find an expression that represents the shaded rectangle’s area, 3w − 6. Because the expressions 3(w − 2) and 3w − 6 both represent the shaded rectangle’s area for any value of w, what can we conclude about the two expressions?

Promoting the Standards for Mathematical Practice

When students apply their understanding of using the distributive property with numerical expressions and the areas of rectangles to write equivalent algebraic expressions by using the distributive property, they are looking for and making use of structure (MP7).

The expressions must be equivalent.

Ask the following questions to promote MP7:

How can we show that the expressions 3(w − 2) and 3w − 6 are equivalent by using the distributive property?

• How can what you know about the distributive property help you write equivalent expressions?

We can multiply 3 by w and multiply 3 by 2 and subtract the products. The verb distribute describes using the distributive property to write the product 3(w − 2) as the equivalent expression 3w − 6.

• What’s another way you can think about the area of the rectangle that will help you write equivalent expressions?

Have students remain in pairs to complete problems 7 and 8. Circulate as students work and offer support as needed. Verify that students use proper notation when they use the distributive property to find equivalent expressions in problem 8. As needed, prompt students to draw and label rectangle diagrams in problem 8 to help determine the product. 6. The rectangle is made up of two smaller rectangles. a. Write two expressions that each represent the area of the shaded rectangle in square units.

2 units

3(w − 2)

w units

3w − 6 b. Use the distributive property to show that the expressions you wrote in part (a) are equivalent.

3(w − 2) = 3(w) − 3(2) = 3w − 6

3 units

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7. Which expressions represent the area of the shaded rectangle in square units? Choose all that apply.

A. 5(3x − 11)

Differentiation: Support

3x units

B. 55 + 15x

5 units

C. (3x − 11)5

11 units

D. 15x − 55

To support students, have them choose at least two reasonable values of the variable in problem 7. Have them evaluate each expression to determine which expressions are equivalent.

E. 5(11) − 5(3x) F. 5(3x) − 5(11) 8. Use the distributive property to write an equivalent expression as a sum or difference. 3 a. _ (x − 12) 4

3 3 3 (x − 12) = (x) − (12) 4 4 4

= _x − 9 3 4

b. (2 − 3n)7

(2 − 3n)7 = 2(7) − 3n(7) = 14 − 21n c. 18(0.5a + 0.5b)

18(0.5a + 0.5b) = 18(0.5a) + 18(0.5b) 1 d. 8(_x + 2y + 1) 4

= 9a + 9b

8(_ x + 2y + 1) = 8(_ x) + 8(2y) + 8(1) 4 4 1

= 2x + 16y + 8

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When most students have finished, choose students to share answers. Consider asking students to share the reasons each expression does or does not represent the area of the shaded rectangle in problem 7. Use any of the following questions to debrief the problems. • In problem 7, what is the area of the largest rectangle? How do you know? How does that relate to the previous lesson? • What is different about problem 8(d)? How did you apply the distributive property? How do you know that the distributive property lets us multiply the number outside of the parentheses by the three terms inside the parentheses?

Land Debrief

5 min

Objective: Use the distributive property to write the product of two factors as a sum or difference. Initiate a class discussion by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Display the picture of the two side-by-side rectangles. How can we write two different expressions that each represent the area of the largest rectangle in square units? What are the two expressions?

b units

c units

a units

We can multiply the largest rectangle’s length by its width, which results in a(b + c). We can add the areas of the two smaller rectangles, which results in ab + ac.

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Display the picture of the two side-by-side rectangles with one that is shaded. How can we write two different expressions that each represent the area of the shaded rectangle in square units? What are the two expressions?

c units a units

b units We can multiply the shaded rectangle’s length by its width, which results in a(b − c). We can subtract the area of the unshaded rectangle from the area of the largest rectangle, which results in ab − ac. How can we use the distributive property to write a product, such as 4(w + 7), as an equivalent expression? To write an equivalent expression, we can use the distributive property to multiply the factor 4 by each term of w + 7. Then we can add the two products to get 4w + 28.

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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Recap

RECAP Name

Date

2. Consider the shaded rectangle.

t units

The Distributive Property 3 units

In this lesson, we •

wrote a product as an equivalent sum or difference by using the distributive property.

2 units

Examples a. Write two expressions that each represent the area of the shaded rectangle in square units.

1. The rectangle is made up of two smaller rectangles.

x units

8 units

4 units

a. Write an expression to represent the total area of the rectangle in square units as the product of the rectangle’s length and width.

4(x + 8) b. Write an expression to represent the total area of the rectangle in square units as the sum of the areas of the two smaller rectangles.

4x + 4(8)

rectangle as the product of the shaded rectangle’s length, 3 units, and its width,

3t − 6

t − 2 units.

b. Use the distributive property to show that the expressions you wrote in part (a) are equivalent.

4(x + 8) represents the total area of the

3(t − 2) = 3(t) − 3(2)

rectangle because the total area is the product of the rectangle’s length, which is 4 units, and its width, which is the sum of x units and 8 units.

Use the distributive property to multiply the factor in front of the parentheses, 3, by each term inside parentheses. 3(t − 2)

4 x + 4(8) represents the total area of

and 3t − 6 are equivalent expressions by the distributive property.

the rectangle because the total area is the sum of the areas of the two smaller rectangles.

For problems 3 and 4, use the distributive property to write an equivalent expression as a sum or difference. 1 4. 10(6x + _ y) 5

8(3f − 5) = 8(3f ) − 8(5) = 24f − 40

3t − 6 represents the area of the shaded rectangle as the difference between the area of the largest rectangle, 3t square units, and the area of the unshaded rectangle, 6 square units.

= 3t − 6

3. 8(3f − 5)

3(t − 2) represents the area of the shaded

3(t − 2)

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R E CA P

10(6x + 1_ y) = 10(6x) + 10(1_ y) 5

= 60x + 2y

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

PRACTICE Name

Date

3. The rectangle is made up of two smaller rectangles. Which expressions represent the total area of the rectangle in square units? Choose all that apply. A. 6x

1. The rectangle is made up of two smaller rectangles.

p units

2 units

B. 3(x + 2)

9 units

C. 3(2x)

5 units

x units

D. 3(6x) E. 3x

a. Write an expression to represent the total area of the rectangle in square units as the product of the rectangle’s length and width.

F. 3(x) + 3(2)

3 units

G. 3x + 6

5(p + 9) b. Write an expression to represent the total area of the rectangle in square units as the sum of the areas of the two smaller rectangles.

4. The rectangle is made up of two smaller rectangles.

5p + 45

m units

2. Which expressions represent the area of the shaded rectangle in square units? Choose all that apply.

4 units

A. 3(12) − 3(2)

5 units

B. 3(12) − 3(10)

a. Write two expressions that each represent the area of the shaded rectangle in square units.

C. 3(10) + 3(2)

4(m − 5)

D. 3(10)

4m − 20

12 units

E. 3(12 − 2)

b. Use the distributive property to show that the expressions you wrote in part (a) are equivalent.

F. 30 G. 36 − 6 H. 30 + 6

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4(m − 5) = 4m − 4(5)

2 units

= 4m − 20 3 units

169

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P R ACT I C E

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15. A bottled water company charges $9 per month for a water cooler. The company also charges $5.95 per bottle of water used in one month.

For problems 5–10, use the distributive property to write an equivalent expression as a sum or difference. 1 (p − 12) 5. _ 2

2 (15 + n) 6. _ 3

_1 (p − 12) = 1_ (p) − 1_ (12) 2 2 2

Complete the table for the given numbers of bottles of water.

_2 (15 + n) = 2_ (15) + 2_ (n) 3 3 3

= 1_ p − 6

= 10 + 2_ n

7. 6(2f − 4)

6(2f − 4) = 6(2f ) − 6(4)

3 8. _ (20 − 4g) 4

_3 (20 − 4g) = 3_ (20) − 3_ (4g) 4 4 4

= 12f − 24

1 9. 9(2x + _ y) 3

= 15 − 3g 10. 20(0.1r + 2.5t)

Number of Bottles of Water Used

Total Cost (dollars)

14.95

38.75

68.50

20(0.1r + 2.5t) = 20(0.1r) + 20(2.5t) = 2r + 50t

9(2x + _1 y) = 9(2x) + 9(1_ y) 3 3 = 18x + 3y

16. Write the ordered pair of the point that results from each reflection. a. Reflect the point (5, 4) across the x-axis.

(5, −4)

Remember

b. Reflect the point (−3, −6) across the y-axis.

For problems 11–14, multiply. 1×1 11. _ 4

1×3 12. _ 5

1×7 13. _ 8

1 × 10 14. _ 9

_1 4

_3 5

_7 8

10 __ 9

(3, −6) c. Reflect the point (7, −2) across the x-axis.

(7, 2) d. Reflect the point (−8, 1) across the y-axis.

(8, 1)

P R ACT I C E

171

172

P R ACT I C E

253