Multiplication with arrays

So, I have several groups of these ball-looking things. And let's think about how many balls are in each group. We have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 12. And what I want to do is think about the different ways of dividing these 12 balls into different numbers of groups. So, for example, I could view these 12 balls as one, so that's 1 group of 3, 2 groups of 3, 3 groups of 3, 4 groups of 3. So I could view 12 as being 4 groups of 3. And the way that we would write that is that 12 is equal to 4 groups of 3. Or another way of reading this is that 12 is equal to 4 times 3. If I have 1, 2, 3, 4 groups, and in each of those groups I have 1, 2, 3, objects, I'm going to have a total of 12 objects. But that's not the only way we can get to 12. We could also view it as 3 groups of 4. So, let's look at that. So we could have it as 1 group of 4, 2 groups of 4, 3 groups of 4. So now we could view 12 as being 3 groups of 4. Or we could say-- let me get the right tool out-- that 3 times 4 is equal to 12. So, whether we're doing 4 times 3 or 3 times 4, they're both going to be equal to 12. 4 groups of 3 is 12, 3 groups of 4. But we don't have to stop there. We could also view 12 as, well, we could view it as 2 groups of 6. Let's look at that. So this is 1 group of 6 right over here. So, that's one 1 of 6. That's another group of 6. So, once again, we could view this as 2 times 6. 2 times 6 will also get us a 12. Well, what about doing it as 6 groups of 2? Well, we can draw that out, too-- 6 groups of 2. So that's 1 group of 2. Let me do that in a different color. Let me do it in this purple color. We have 1 group of 2, 2 groups of 2, 3 groups of 2, 4 groups of 2, 5 groups of 2, and 6 groups of 2. So once again, this is all different ways of writing 12, something equivalent to 12. We could write 6 times 2-- 6 groups of 2-- 6 times 2 is also equal 12. But we don't have to stop there. We could also literally view 12 as 1 group of 12. So how would that look? So 1 group of 12. So this whole thing is just 1 group of 12 here. So we could literally say 1 times 12 is equal to 12. We have one entire group of 12. 1 times 12 is equal to 12. And we could think of it the other way around. We could view this as 12 groups of 1. Let me draw that. So 12 groups of 1. This is 1 group of 1, 2 groups of 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12-- 12 groups of 1. So we could also write 12. 12 groups, and in each one, I have 1. Well, that's still going to get me to 12.