Properties of multiplication

So if you look at each of these 4 by 6 grids, it's pretty clear that there's 24 of these green circle things in each of them. But what I want to show you is that you can get 24 as the product of three numbers in multiple different ways. And it actually doesn't matter which products you take first or what order you actually do them in. So let's think about this first. So the way that I've colored it in, I have these three groups of 4. If you look at the blue highlighting, this is one group of 4, two groups of 4, three groups of 4. Actually, let me make it a little bit clearer. One group of 4, two groups of 4, and three groups of 4. So these three columns you could view as 3 times 4. Now, we have another 3 times 4 right over here. This is also 3 times 4. We have one group of 4, two groups of 4, and three groups of 4. So you could view these combined as 2 times 3 times 4. We have one 3 times 4. And then we have another 3 times 4. So the whole thing we could view as-- let me give myself some more space-- as 2 times-- let me do that in blue-- 2 times 3 times 4. That's the total number of balls here. And you could see it based on how it was colored. And of course, if you did 3 times 4 first, you get 12. And then you multiply that times 2, you get 24, which is the total number of these green circle things. And I encourage you now to look at these other two. Pause the video and think about what these would be the product of, first looking at the blue grouping, then looking at the purple grouping in the same way that we did right over here, and verify that the product still equals 24. Well, I assume that you've paused the video. So you see here in this first, I guess you could call it a zone, we have two groups of 4. So this is 2 times 4 right over here. We have one group of 4, another group of 4. That's 2 times 4. We have one group of 4, another group of 4. So this is also 2 times 4 if we look in this purple zone. One group of 4, another group of 4. So this is also 2 times 4. So we have three 2 times 4's. So if we look at each of these, or all together, this is 3 times 2 times 4, so 3 times 2 times 4. Notice I did a different order. And here I did 3 times 4 first. Here I'm doing 2 times 4 first. But just like before, 2 times 4 is 8. 8 times 3 is still equal to 24, as it needs to, because we have exactly 24 of these green circle things. Once again, pause the video and try to do the same here. Look at the groupings in blue, then look at the groupings in purple, and try to express these 24 as some kind of product of 2, 3, and 4. Well, you see first we have these groupings of 3. So we have one grouping of 3 in this purple zone, two groupings of 3 in this purple zone. So you could do that as 2 times 3. And we have one 3 and another 3. So in this purple zone, this is another 2 times 3. We have another 2 times 3. Whoops. I wrote 2 times 2. 2 times 3. We have another 2 times 3. And then finally, we have a fourth 2 times 3. So how many 2 times 3's do we have here? Well, we have one, two, three, four 2 times 3's. So this whole thing could be written as 4 times 2 times 3. Now, what's this going to be equal to? Well, it needs to be equal to 24. And we can verify 2 times 3 is 6 times 4 is, indeed, 24. So the whole idea of what I'm trying to show here is that the order in which you multiply does not matter. Let me make this very clear. Let me pick a different example, a completely new example. So let's say that I have 4 times 5 times 6. You can do this multiplication in multiple ways. You could do 4 times 5 first. Or you could do 4 times 5 times 6 first. And you can verify that. I encourage you to pause the video and verify that these two things are equivalent. And this is actually called the associative property. It doesn't matter how you associate these things, which of these that you do first. Also, order does not matter. And we've seen that multiple times. Whether you do this or you do 5 times 4 times 6-- notice I swapped the 5 and 4-- this doesn't matter. Or whether you do this or 6 times 5 times 4, it doesn't matter. Here I swapped the 6 and the 5 times 4. All of these are going to get the exact same value. And I encourage you to pause the video. So when we're talking about which one we do first, whether we do the 4 times 5 first or the 5 times 6, that's called the associative property. It's kind of fancy word for a reasonably simple thing. And when we're saying that order doesn't matter, when it doesn't matter whether we do 4 times 5 or 5 times 4, that's called the commutative property. And once again, fancy word for a very simple thing. It's just saying it doesn't matter what order I do it in.