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How to Think About Division: Part 1

What does it really mean to divide two numbers? Or two variables? How can you visualize these processes? And how are they related to the process of multiplication? Keep on reading to find out.

By
Jason Marshall, PhD,
Episode #280

How to Think About Division with Variables

Let’s now extend this idea to variables. The good news is that nothing really changes. For example, let’s think about the problem (• y) / x:

Division with Variables

We can picture the numerator of this expression as a rectangle whose area is • y. That is, we’ve taken the length y and stretched it until it is x times taller. But now the question is what happens when we divide this by the variable x? Well, it’s basically the same as when we divided (3 • 5) by 3—we compress the rectangle by the amount we divide by.

So, in this case, we’re compressing (• y) by an amount given by the value of x—which means that we “undo” the stretching that the multiplication originally did. And the answer to the problem (• y) / x is therefore simply y. Again, I know this is a fairly obvious result, but the point is that we now have a nice way to visualize what we’re doing when we do division.

Division, Multiplication, and Numbers

With that basic idea now firmly tucked away in our mathematical tool belts, it’s time for me to let you in on a big “secret” about division: It’s pretty much the same as multiplication. What do I mean? Well, even though this might sound kind of counterintuitive, I mean that stretching and compressing (i.e., multiplication and division) are actually one in the same process—in the sense that each of these things can be turned into the other.

To see what I mean, let’s take a look again at the problem 15 / 3 = 5 that we talked about earlier. But instead, let’s think about the problem 15 • ⅓ = 5 … because these two problems are really one and the same. In our way of thinking about the world, the problem 15 / 3 is asking you to figure out how many blocks you'll have left after compressing a 3 by 5 array of blocks into a stack that's 3 times smaller than it was to begin with.

Division and Multiplication with Numbers

But what about 15 • ⅓—how is that the same? Well, as we've seen, when we multiply one number by some other number we are scaling that first number to be some number of times bigger. And when we multiply some number by a fractional number like ⅓, we are scaling the first number to be some number of times—⅓ in this case—smaller than its original size. So scaling by ⅓ has exactly the same effect as compressing the number until it's 3 times smaller. Obvious, yes—but powerful too.

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