The idea of division

We've got 24 triangle things right over here. And what I want to do in this video is to divide it into different numbers of groups. So the first thing I want to do is I want to divide this 24 triangle things into 3 groups and think about how many do I end up per group. So let's try that out. So I'm going to divide it into 3 equal groups. So that is one equal group right over there. Then I have another equal group right over here. And then I have a third equal group right over here. So if I divide 24 into 3 equal groups-- 1, 2, 3-- how many are going to be in each group? Well, we can count that. We have 1, 2, 3, 4, 5, 6, 7, 8 in each group. So we could say that 24 divided by 3 is equal to 8. Now, you might say, hey, this is very similar to what we saw in multiplication. In multiplication, we said if we have 3 groups of 8, we could view that as 3 times 8 and get 24. And you are exactly right. Let me do those same colors-- we could also write that 3 times 8-- so if I have 3 groups of 8, that that is going to be equal to 24. So when we started in this video, we had 24 things. We want to divide it into 3 equal groups. We got 8 in each group, or you could say 3 equal groups of 8 is equal to 24. But there's even other ways of thinking about this. So let me clear this up a little bit. So let me clear that. So in the first example, I divided 24 into 3 equal groups. But you could also view 24 divided by 3 as dividing 24 into groups of 3. So let's think about what that looks like. So if we divide it into groups of 3, then, for example, this is a group of 3. That is a group of 3. This is a group of 3. You might see where this is going. That's a group of 3. That is another group of 3. And we're going to think about how many groups of 3 we're actually going to get. So this is another group of 3. And that's another group of 3. So how many groups of 3 did we get? Let's see, we have 1, 2, 3, 4, 5, 6, 7, 8 groups of 3. So another way of viewing 24 divided by 3 is divide 24 into groups of 3. And then you will have 8 groups of 3. And one way of thinking about this-- if you want to express the same thing in terms of multiplication-- is if you have 8 groups of 3, that is also going to be equal to 24. Whether you have 3 groups of 8 or 8 groups of 3, either way, you're going to have 24. Now, let's make things more interesting. What I want you to think about is, based on what we just saw, what is 24 divided by 12? And I encourage you to pause the video, draw out 24 triangles like this, and try to figure out what 24 divided by 12 is. Well, I assume you've paused the video. And there's two ways to think about 24 divided by 12. You could say, well, let's divide 24 into groups of 12 and think about how many groups we have. So we could do that. So let's see. This is one group of 12 right over here. That's one group of 12, and then here is another group of 12. So how many groups of 12 do we have? Well, we have 2 groups of 12. So we could say 24 divided by 12 is 2. But another just as reasonable way of doing this is you could have said, well, let me divide 24 into 12 groups instead of groups of 12. So if I want to divide it into 12 groups, 12 equal groups-- well, let's see. This is 1 equal group, 2 equal groups-- actually let me do it this way. Well, let me do this-- 2 equal groups, 3 4, 5, 6, 7, 8, 9, 10, 11, 12. So once again, if you say, oh, I'm going to divide 24 into 12 equal groups, how many do you have in each group? Well, you have 2. So once again, 24 could be viewed as 24 divided into 12 equal groups. And how many do you have in each group? Or 24 divided into groups of 12, and how many groups would you have? And that's what we saw in the last example. So now, let's make things even more interesting. What I want you to think about-- a couple of things. I want you think about what 24 divided by 6 is. And I also want you to figure out what 24 divided by-- let me use that same color-- 4 is. And once again, I encourage you to pause the video, draw these triangles, and figure it out. What is 24 divided by 6 and 24 divided by 4? So let's tackle 24 divided by 6 first. And let's try to divide 24 into 6 equal groups. So let's see. This could be 1 equal group, 2 equal groups-- in fact, each group here is a group of 4. And we have 6 rows. So 3 equal groups, 4, 5, and 6. And so if you divide 24 into 6 equal groups, how many do you have in each group? Well, it's pretty clear you have 4. You have 4 In each group. Another way we could have thought about that is we could have said, let me divide 24 into groups of 6. So if you divided 24 into groups of 6, you could have viewed it like this. So that's 1 group of 6 right there. That's another group of 6 right over here. That's another group of 6. And I think you see how many groups of 6 we have. How many groups of 6 do we have? We have 4. We have 4 groups of 6. Well, now let's think about what 24 divided by 4 is. Well, if I view 24 divided by 4 as taking 24 and dividing it into 4 equal groups, I've just drawn that. I have 4 equal groups, and in each group I have 6. So notice 24 divided by 6 is 4. 24 divided by 4 is 6. And that's because I could view this as 4 groups of 6 or say that 4 times 6 is equal to 24. Or you could just as equivalently say that 6 times 4 is 24. You could equivalently say that 6 times 4 is equal to 24.