Order when multiplying commutative property of multiplication

In this video, we're going to talk about one of the most important ideas in mathematics, and that's whether order matters when you multiply two numbers. So, for example, is 3 times 4 the same thing as 4 times 3? Are these two things equal to each other? And regardless of whether these two are equal to each other, is it always the case that if I have some number times some other number, if I swap the order, am I going to get an equivalent number?

Well, pause this video and see if you can work through that. Try to think about that a little bit before we think through it together.

Well, let's think through this particular example, and we're going to do so with the help of some angry cats. So, we clearly see some angry cats here. Yes, they are angry! We could view this as three groups of four. So this is one group right over here of angry cats: four angry cats. This is two groups of four angry cats, and this is three groups of four angry cats.

If we view the first number here as groups of the second number, but we could also view it as four groups of three. How would we do that? Well, we could have one group of three angry cats, we can have two groups of three angry cats, we can have three groups of three angry cats, and we can, of course, have four groups of three angry cats.

So based on that, if you think of it—the first number as groups of the second number—well, it seems like the order doesn't matter. Another way you could think about it is here you have four rows of three angry cats. You have one, two, three, four rows of one, two, three angry cats.

And so to figure out how many total cats you have, you multiply four times three. But you could view this same group of angry cats but just view it with a slightly different perspective.

So here we have our angry cats, and then let me rotate our angry cats, probably risking making them a little bit angrier. Let me move them back in, and now we could view this as three rows—one, two, three—of four cats in each row. So let me put all of these upright.

So we have one, two, three rows of one, two, three, four cats. Don't want to look at those because it might make us a little bit confused. And we're dealing with the exact same number of cats, and so I'm only dealing with three and four here.

But what you will see is order doesn't matter when you are multiplying two numbers. And we could also see that on a number line. We could do that with multiple examples. I'll keep a couple of angry cats here looking at us just to keep us in check.

If we want to think about 3 times 4, we could view it as 4 threes: 3, 6, 9, 12. Or we could view it as three fours: 4, 8, and 12. And I focused a lot on three and four, but we could do it with any two numbers that we're trying to multiply together.

So, let's say we wanted to do that with, I don't know, let’s see, six. Whether six times four, whether it's the same thing as four times six. Well, you could view it as six fours: 4, 8, 12, 16, 20, and then 24. Or you could view it as four sixes: one six, 12, 18, and 24.

So, big takeaway: order doesn't matter when you are multiplying numbers like this. And this is sometimes referred to as the commutative property. It's a fancy word, but it's really just saying that whether you're doing 6 times 4 or 4 times 6, the commutative property of multiplication says, "Hey, those two things are going to be equivalent."

Meow! Yes, they are angry for being rotated.