Division as equal groupings

So it looks like we have some angry cats on our hands. Yeah, yeah, they seem angry. What we want to do is think about how can we separate these angry cats. Because the only thing worse than an angry cat is 12 of them coordinating potentially to take over the world.

And so what we're going to think about is: can we divide these 12 angry cats into equal groups? Just for the sake of this video, let's say we want to divide them into three equal groups. How could we do that? Well, let me see if I can do it.

Let's see, maybe this is one group right over here; maybe this is another group right over there, and then this is another group. Are those three equal groups? Well, see, this top group has four, this group has three, and this one has five. So no, these aren't three equal groups, so that won't work. It has to be not just three groups but three equal groups.

So let me see if I can do something different. I could have these four in a group, and then I could have these four in a group. I think this is going to work out this time. And then I could have these four in a group. As I just said out loud, all of these groups have exactly four angry cats in them, which is a little bit more manageable.

So this works out. I have just divided the 12 into three equal groups. How many cats are in each of those equal groups? Well, we can see very clearly that there are four cats in each of those equal groups. So the way to think about division in this example is: you started with 12 and you divided into three equal groups, and we ended up with four cats, four cats per group.

Now, what if we wanted to interpret it slightly differently? So let's bring in 12 more angry cats, which is always a dangerous thing to do, but we're doing it for the sake of learning, so it's worth it. Meow! Yep, more angry cats!

Now let's imagine 12 divided by 3. 12 divided by 3, but this time we're not going to think about the 3 as 3 equal groups. We're going to think about it as 12 divided into groups of three. So groups, groups of three.

So each of the equal groups will have three in them, and so then that's going to tell us how many equal groups we were able to have. So let's do that; let's create groups of three. So let's see, that's a group of three angry cats right over there, this is another group of three angry cats, this is a group of three angry cats, and then this is a group of three angry cats.

So how many equal groups of three angry cats do I have? Well, I have one, two, three, and four equal groups. So if I want to take 12 angry cats and divide them into groups of three, I end up with four equal groups.

So there you have it! There's two different ways that we can imagine division and the same exact division expression: 12 divided by 3. You could view it as 12 being divided into three equal groups, and then the answer would be how many things per group; or you could say, "Hey, 12 divided into groups of three," and then that would end up with four equal groups.