Teaching Math: The Distributive Property

So many concepts are crammed into a middle-school math curriculum. The students-and the teacher-can easily become overwhelmed. Incorporating student involvement and memory tricks into any lesson is helpful. The distributive property is one math concept that can throw middle-school students for a loop. It is usually taught on the heels of the commutative and associative properties (click here for tips on teaching those properties). Middle-school teachers lay the foundation for the students’ smooth transition into algebra. These tips for teaching the distributive property may help your middle-school students relax and enjoy learning this property.

Mental Math Introduction

I begin the lesson by asking the students to use mental math to come up with some answers very quickly to problems such as the following:
8 x 24
3 x 56
9 x 53
There are generally a couple students who can spout off the answers in a flash. They explain their thinking to the class. Whether they know the term or not, as a general rule, they’re actually using the “distributive property”.

We discuss how the word “distribute” means to spread something out. When the distributive property is used, numbers are “spread out”, making them easier to work with.

In the first problem 8 x 24: Distribute the 8 over the other two numbers.
First think: 8 x 24 = 8 x (20 + 4)
8 x 20 = 160; 8 x 4 = 32
Add 160 and 32, and the answer is 192.

3 x 56: Distribute the 3.
3 x 50 (150); 3 x 6 (18)
150 + 18 = 168.

9 x 53: Distribute the 9.
9 x 50 (450); 9 x 3 (27)
450 + 27 = 477

Once students figure out the distributing “trick”, they will be anxious to raise their hands to answer similar problems.

The Distributive Property of Multiplication
Note: Remind students that if there is no operation sign, it signifies multiplication.

For real numbers a, b, and c:
a(b + c) = (ab) + (ac)
Example:
2(3 + 4) = (2 x 3) + (2 x 4)
2(7) = 6 + 8
14 = 14

For real numbers a, b, and c:
a(b – c) = (ab) – (ac)
Example:
5(8 – 6) = (5 x 8) – (5 x 6)
5(2) = 40 – 30
10 = 10

Danica McKellar’s Idea

In her 2008 book, “Kiss my Math”, Danica (yep, that’s Winnie on the old Wonder Years TV show) offers this explanation on how to distribute numbers in a problem:

She tells students to imagine they’re going to a costume party at a friend’s house. The hostess is dressed as a bride, so call her b. There’s only one other person at the party so far, and he’s in a cat mask, so call him c.
The house looks like this so far: (b + c), where b stands for bride and c stands for cat.
You dressed up like Ariel from The Little Mermaid, so what will you be called? You guessed it…a.
When you knock on the door just outside the house, it looks like this: a(b +c)
When you get inside, you hug your friends…first hug b, and then c:
ab + ac
If there had been another friend dressed as a dragon (d), you’d hug all three of them:
a(b + c + d) = ab + ac + ad. This is the distributive property.

I like the idea of having masks made ahead of time (Ariel, the bride, the cat, and the dragon) and choosing students to act out this scene in front of the class.

Short Video and Quiz

Use Glencoe’s Brain Pop video on the distributive property with the entire class. The accompanying quiz can be used as a class activity as well.

Kids love any escape from the mundane. Give these ideas a try…your middle-school math students just might relate.