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.
Over the past few months we’ve talked about how to think about three of the big four arithmetic operations: addition, subtraction, and multiplication. Which, perhaps not surprisingly, means that a conversation about division is on tap for today.
A lot of people despise division because they think it’s hard. This perception isn’t wrong—it usually is harder to do than the other arithmetic operations. And the truth is that once you know how to do long division by hand, you may as well use a calculator for the rest of your life (except in those increasingly rare occasions in which you don’t have one in your pocket).
If you, like so many other people, don’t have warm-and-fuzzy super-fond memories of division, rest assured that today we’re not talking about the mechanics of doing long division. Instead, we have the more noble goal of attaining a new vision for division. By which I mean we’re going to talk about the underlying meaning of division.
So, how should you think about division? Let’s think about it!
How to Think About Division with Numbers
To begin with, let’s think about what we’re really doing when we divide one number by another. And, as we’ve done with the other arithmetic processes, let’s think about all of this in terms of manipulating stacks of blocks.
First of all, we know that we can think of a problem like 3 • 5 = 15 as being represented by a grid of 15 blocks arranged into an array that’s 3 blocks high by 5 blocks wide. We’ve talked about how we can think about this as stretching the row of 5 blocks until it's 3 times its original height. Thus, we can view multiplication as a process that scales or stretches a number. So why are we reviewing multiplication in a discussion about division? Because the two processes are intimately related. In fact, they’re what we call inverse processes.
And that means that in the same way that subtraction “undoes” addition, division “undoes” multiplication. In other words, we can think of division as a process that scales down—or compresses—a number until it is some other number of times smaller.
In this case, we can think of 15 / 3 visually as a process that takes a grid of 3 rows of 5 blocks and compresses it until it’s squished down to 1 row of 5 blocks. Which means that 15 / 3 = 5. No surprise, but it is a great way to think about the meaning of division.