Lesson 11The Distributive Property (Part 3)

Learning Goal

Let’s practice writing equivalent expressions by using the distributive property.

Learning Targets

I can use the distributive property to write equivalent expressions with variables.

Lesson Terms

equivalent expressions

Warm Up: The Shaded Region

A rectangle with dimensions 6 cm and $w$ cm is partitioned into two smaller rectangles.

Explain why each of these expressions represents the area, in cm $^{2}$ , of the shaded portion.

A rectangle with a shaded blue part. The width is w and the white segment is 4. The height is 6.

$6 w - 24$
$6 (w - 4)$

Activity 1: Matching to Practice Distributive Property

Match each expression in Column 1 to an equivalent expression in Column 2. If you get stuck, consider drawing a diagram.

Column 1

$a (1 + 2 + 3)$
$2 (12 - 4)$
$12 a + 3 b$
$\frac{2}{3} (15 a - 18)$
$6 a + 10 b$
$0.4 (5 - 2.5 a)$
$2 a + 3 a$

Column 2

$3 (4 a + b)$
$12 \cdot 2 - 4 \cdot 2$
$2 (3 a + 5 b)$
$(2 + 3) a$
$a + 2 a + 3 a$
$10 a - 12$
$2 - a$

Activity 2: Writing Equivalent Expressions Using the Distributive Property

The distributive property can be used to write equivalent expressions. In each row, use the distributive property to write an equivalent expression. If you get stuck, consider drawing a diagram.

product	sum or difference
$3 (3 + x)$
	$4 x - 20$
$(9 - 5) x$
	$4 x + 7 x$
$3 (2 x + 1)$
	$10 x - 5$
	$x + 2 x + 3 x$
$\frac{1}{2} (x - 6)$
$y (3 x + 4 z)$
	$2 x y z - 3 y z + 4 x z$

Are you ready for more?

This rectangle has been cut up into squares of varying sizes. Both small squares have side length 1 unit. The square in the middle has side length $x$ units.

A rectangle divided into 9 squares of varying sizes.

Suppose that $x$ is 3. Find the area of each square in the diagram. Then find the area of the large rectangle.
Find the side lengths of the large rectangle assuming that $x$ is 3. Find the area of the large rectangle by multiplying the length times the width. Check that this is the same area you found before.
Now suppose that we do not know the value of $x$ . Write an expression for the side lengths of the large rectangle that involves $x$ .

Lesson Summary

The distributive property can be used to write a sum as a product, or write a product as a sum. You can always draw a partitioned rectangle to help reason about it, but with enough practice, you should be able to apply the distributive property without making a drawing.

Here are some examples of expressions that are equivalent due to the distributive property.

$\begin{aligned} 9 + 18 & = 9 (1 + 2) \\ 2 (3 x + 4) & = 6 x + 8 \\ 2 n + 3 n + n & = n (2 + 3 + 1) \\ 11 b - 99 a & = 11 (b - 9 a) \\ k (c + d - e) & = k c + k d - k e \end{aligned}$

Lesson 11The Distributive Property (Part 3)

Learning Goal

Learning Targets

Lesson Terms

Warm Up: The Shaded Region

Problem 1

Activity 1: Matching to Practice Distributive Property

Problem 1

Activity 2: Writing Equivalent Expressions Using the Distributive Property

Problem 1

Are you ready for more?

Problem 1

Lesson Summary