ôô

# What is the Distributive Property? Learn what the distributive property is, how to picture it, and what it means to say that multiplication is distributive over addition.

By
Jason Marshall, PhD
Episode #043 A while back, we talked about the commutative property of addition and we used this property to figure out how to add quickly. But the commutative property isn’t the only math property, which might lead you to wonder what great tricks the others have to offer. Well, your wait is over. Because today we’re talking about another one of those properties: the distributive property.

The podcast edition of this article was sponsored by the AARP Auto Insurance Program from The Hartford, which can save drivers, over the age of 50, \$384 on auto insurance. Get an 8-minute quote at aarp.thehartford.com/podcast.

## Review: What is Area?

Do you remember when we talked about area? Well, that topic is going to make another appearance in this article, so let’s take a minute to make sure we’re all on the same page. The idea of area is used all the time in everyday life. For example, let’s say you need to figure out how much money to save for new carpeting in your bedroom. When you go to the store, you see that carpet is sold by the square foot. So, to estimate the total cost for your room, you need to multiply the price per square foot by the size of your bedroom—and by size, I mean the area in square feet. So how do you calculate the area? Well, you just measure the length and the width of the room in feet, and then multiply these two numbers together. That means that for a rectangle, the area is just the length of the rectangle times its width.

## How to Picture the Distributive Property

Okay, now that we’ve got a handle on the idea of area, I’m going to describe a drawing that I want you to picture in your mind—or follow along with a pencil and paper and make the sketch as I describe it. First, draw a large rectangle. It doesn’t matter if it’s wider than it is tall, or taller than it is wide—picture it however you want. Now, draw a vertical line inside this large rectangle that extends all the way from its top edge to its bottom edge. Again, it doesn’t matter exactly where you position it from left to right—that’s entirely up to you. So what do you have? Well, you have a picture of a large rectangle that contains 2 smaller rectangles inside of it Now, let’s name some of the features in our drawing. Why? Well, just as with people, it will allow us to talk about them without having to say awkward things like: “look at the tall guy with the short red hair and glasses.” We can just say: “look at Bob.” But in our rectangles case, let’s not use names like “Bob.” Let’s use letters of the alphabet—just because they’re a lot shorter to write.

Okay, let’s call the height of the rectangles “a” (they all have the same height, right?), and the width of the 2 smaller rectangles “b” and “c” (from left to right). Got all that? All right, now let’s see what we can do with these names.

## The “Left Side” of the Distributive Property

Here’s what I have in mind. Let’s first calculate the area of the large rectangle. As we talked about earlier, the area of a rectangle is just its height times its width. Remember, we named the height of the large rectangle “a,” and the width is…well, we actually didn’t name the total width, did we? But we did name the widths of the 2 smaller rectangles: “b” and “c.” So that means that the total width of the big rectangle is just the sum of these 2 small widths: “b” + “c”. Make sense? And with that, we can write the area of the big rectangle as height times width—that is:

“a” x ( “b” + “c” )

Okay, so that’s the area of the large rectangle. For future reference, let’s go ahead and call this “the ‘left side’ of the distributive property.” I know that might seem like a mysterious name right now, but hold on a minute and it’ll make sense.

## The “Right Side” of the Distributive Property

Now, think about this: Couldn’t we also have written the total area of the large rectangle as the sum of the areas of each of the small rectangles? If you think about it, you’ll realize that you absolutely can—the total area has to equal the sum of the individual areas. So what’s the area of the 2 smaller rectangles? Well, for each of them, their area is just their height—which is “a” just like for the large rectangle—times their width. The width of the rectangle on the left is “b,” so its area is “a” x “b”. Similarly, for the rectangle on the right, its area is “a” x “c”. So if we add these 2 areas together, the total area of the large rectangle must be

“a” x “b” + “a” x “c”

As before, let’s enigmatically call this “the ‘right side’ of the distributive property” for future reference. But wait—hold on a minute! This expression is different than what we got before for the total area! Before we said it was “a” x (“b” + “c”). And now we’ve said that it’s “a” x “b” + “a” x “c”. But we said they had to be the same since they both represent the total area. So which is right? Well, actually, they’re both correct and they’re not really different—in fact, they’re exactly the same.

## What is the Distributive Property?

And the reason they’re the same is called the distributive property. We can combine the “left” and “right” sides of the distributive property that we calculated from the area of the large rectangle and the sum of the areas of the small rectangles, and we can write the distributive property like this This says that multiplication is distributive over addition. And that means that if we take the sum of some numbers—in our case “b” + “c” (although it doesn’t have to be 2 numbers, there could be as many as we want), and multiply this sum by some other number—in our case “a,” the result is the same as if we first individually multiplied each number in the sum by “a” and then added these all up.

## How Does the Distributive Property Work?

[[AdMiddle]Just to make sure this really works, let’s try it with a few numbers. How about 2 x (3 + 5). Well, we have to do whatever is in the parenthesis first, so start by adding the 3 and 5 to get 3+5=8, and then multiply this by 2 to get 2x8=16. Now let’s do it the other way. It’s just 2 x 3 + 2 x 5, which simplifies to 6+10=16. So, yes, we get the exact same answer either way—the distributive property works. Of course, we already knew that since our picture with rectangles showed us that it had to be true!

## Wrap Up

Okay, that’s all the math we have time for today. But that’s not all we have to say about the distributive property. So be sure to check in on future episodes to hear about a real life interpretation of the distributive property, and a way that you can use it to perform lightning-fast multiplication in your head. 