Lesson 9The Distributive Property (Part 1)

Learning Goal

Let’s use the distributive property to make calculating easier.

Learning Targets

I can use a diagram of a rectangle split into two smaller rectangles to write different expressions representing its area.
I can use the distributive property to help do computations in my head.

Lesson Terms

equivalent expressions
term

Warm Up: Number Talk: Ways to Multiply

Find each product mentally.

$5 \cdot 102$
$5 \cdot 98$
$5 \cdot 999$

Activity 1: Ways to Represent Area of a Rectangle

Select all the expressions that represent the area of the large, outer rectangle in figure A. Explain your reasoning.
1. $6 + 3 + 2$
2. $6 \cdot 3 + 6 \cdot 2$
3. $6 \cdot 3 + 2$
4. $6 \cdot 5$
5. $6 (3 + 2)$
6. $6 \cdot 3 \cdot 2$
Select all the expressions that represent the area of the shaded rectangle on the left side of figure B. Explain your reasoning.
1. $4 \cdot 7 + 4 \cdot 2$
2. $4 \cdot 7 \cdot 2$
3. $4 \cdot 5$
4. $4 \cdot 7 - 4 \cdot 2$
5. $4 (7 - 2)$
6. $4 (7 + 2)$
7. $4 \cdot 2 - 4 \cdot 7$

Activity 2: Distributive Practice

Complete the table. If you get stuck, skip an entry and come back to it, or consider drawing a diagram of two rectangles that share a side.

column 1	column 2	column 3	column 4	value
$5 \cdot 98$	$5 (100 - 2)$	$5 \cdot 100 - 5 \cdot 2$	$500 - 10$	$490$
$33 \cdot 12$	$33 (10 + 2)$
		$3 \cdot 10 - 3 \cdot 4$	$30 - 12$
	$100 (0.04 + 0.06)$
		$8 \cdot \frac{1}{2} + 8 \cdot \frac{1}{4}$
			$9 + 12$
			$24 - 16$

Are you ready for more?

Problem 1

Use the distributive property to write two expressions that equal 360. (There are many correct ways to do this.)

Problem 2

Is it possible to write an expression like $a (b + c)$ that equals 360 where $a$ is a fraction? Either write such an expression, or explain why it is impossible.

Problem 3

Is it possible to write an expression like $a (b - c)$ that equals 360? Either write such an expression, or explain why it is impossible.

Problem 4

How many ways do you think there are to make 360 using the distributive property?

Lesson Summary

A term is a single number or variable, or variables and numbers multiplied together. Some examples of terms are 10, $8 x$ , $a b$ , and $7 y z$ .

When we need to do mental calculations, we often come up with ways to make the calculation easier to do mentally.

Suppose we are grocery shopping and need to know how much it will cost to buy 5 cans of beans at 79 cents a can. We may calculate mentally in this way: $5 \cdot 79$ $5 \cdot 70 + 5 \cdot 9$ $350 + 45$ $395$

In general, when we multiply two terms (or factors), we can break up one of the factors into parts, multiply each part by the other factor, and then add the products. The result will be the same as the product of the two original factors. When we break up one of the factors and multiply the parts we are using the distributive property.

The distributive property also works with subtraction. Here is another way to find $5 \cdot 79$ : $5 \cdot 79$ $5 \cdot (80 - 1)$ $400 - 5$ $395$