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Sal uses pictures and practice problems to see commutativity and associativity in multiplication.   Created by Sal Khan.
Video transcript
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 or the product of 3 numbers in multiple different ways and it doesn't matter which products you take first, or what order you actually do them in. Let's think about this first. So the way that I've colored them in I have these 3 groups of 4, if you look at the blue highlighting. This is 1 group of 4, 2 groups of 4, 3 groups of 4. Actually let me make that a little bit clear: 1 group of 4, 2 groups of 4, and 3 groups of 4. So these 3 columns you can view as 3 times 4. Now we have a another 3 times for over here. This is also 3 times 4. We have 1 group of 4, 2 groups of 4, and 3 groups of 4. So you could view these two 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 get my... let me get 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 can see it based on it was colored. And of course if you do 3 times 4 first you get 12, and if you multiply that by 2 you get 24 which is the total number of these 3 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 at the purple grouping in the same way that we did right over here, and verify that the product still equals 24. Well I'll assume that you have paused the video so you see here in this first, I think you should call it a zone we have 2 groups of 4 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 that is also 2 times 4 if you 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. 2 times 4. Notice, I did a different order, and here I did 3 times for first, here I did 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, as 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 had these groupings of 3. So we had one grouping of 3 in this purple zone. two groupings of 3 in this purple zone. So you could view that as 2 times 3. And we have one 3 and another 3 so in this purple zone it's another 2 times 3. We have another 2 times 3, oops, 2 times 2. We have another 2 times 3. 2 times 3, and finally we have a four 2 times 3. So how many 2 times 3's do we have here? Well we have one, two, three, four 2 times 3. So this whole thing could be written as 4 times 2 times 3. 2 times 3. Now what's this going to be equal to? Well it needs to be equal to 24, and we can verify it: 2 times 3 is 6 times 4 is indeed 24. So what I The whole idea of what I am trying to show here is that the order in which you multiply does not matter. How you associate it, this is ehm, let me make it this very clear. So whether you do, let me pick a different example a completely new example. 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, we have seen that multiple, 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, 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 are 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 a fancy word for a reasonably simple thing. And when we're saying that order doesn't matter, that it doesn't matter whether we do 4 times 5, or 5 times 4, that's called the commutative property. And once again, a fancy word for a very simple thing, just saying it doesn't matter what order I do it in.