Drawing an area model for the product of two unit fractions

0:00- [Instructor] All right, so let me draw two identical squares. So this is meant to be a square. I'm hand drawing it with this art tool. It's not gonna be perfect, but we're just going to assume that that's a square. Pretty, pretty good there. And I said two identical squares, so let me just actually copy and paste this one.

0:20So copy and then paste. There you go. I have two squares. And if I wanna make it extra clean, I can cut off this little thing here that I didn't need to draw. Let me cut that off too. I don't want this to be an art video. We have to get into the math soon, but then let me delete this one too.

0:41All right, so let's just assume these are two identical squares. And now this one on the left, I'm going to start with one-half of that square. So to look at one-half, I'm just gonna divide it straight down the middle. I could go a little bit further to the right, straight down the middle.

0:58And so what I am shading in right over there, that is one-half of the square. I essentially just took half of this side up here and I just divided the square right down there. Now this one on the right, I'm gonna think about it in terms of fourths. So let me do that.

1:17So let me first divide it into a half. I could go a little bit. No, it's hard to eyeball. Okay, that's half. And then let's see, if I divide each of those sides, that looks about right. All right, so now here, I'm going to shade in a fourth of the square. You'll see where this is going in a second.

1:42Now on this left one, what I wanna do is take one-fourth of that one-half. How do I do that? Well, what I could do is I can divide the square this way, first in half, and then into fourths. That is looking pretty good if I say so myself, not perfect. And so what you see is I've divided into fourths.

2:06So if I were to take one-fourth of that one-half, that one-half here is in orange, it's into four equal sections. So one-fourth of that one-half I could say is this right over here. So that's one-fourth of that one-half. And you can see what's going on here. This doubly shaded area, that is equivalent to one-fourth of that one-half.

2:29And we can see what that is in terms of the whole square. The whole square is now divided into 1, 2, 3, 4, 5, 6, 7, 8 equal rectangles or areas I guess you could say. And this is one of those. So this is one-eighth of the entire square. Now let's go the other way around.

2:50Here, we started with a fourth. And what I wanna do is I wanna take half of that. So let me now divide it this way into two. And so half of that fourth is where this overlaps. If I were to take half of the whole square, I would shade all of that, but I wanna take half of this fourth.

3:08So one way you could think about it, I took this fourth, I divided it into two, and I'm just gonna take half of it. And so that is that over there. Now this doubly shaded area is one-half times one-fourth. You could view that as, you could view that as half of one-fourth.

3:25There's other ways you can interpret it. Now, why is this interesting? Well, we can once again see when we look at the entire square, it's in 1, 2, 3, 4, 5, 6, 7, 8 equal rectangles or eight equal areas or eight equal sections. And this is one of them. So this is also going to be equal to one-eighth.

3:46So hopefully what you just saw there is there's different ways of visualizing unit fractions, that's what these are it, with area models. But we also saw that you're going to get the same thing if you multiply one-half or one-fourth times one-half as you're going to get from one-half times one-fourth.

4:04And you've probably seen this already in multiplication. It doesn't matter what order you multiply things in. If you take one times two, that's the same thing as two times one. If you take 587 times 326, that is the same thing as 326 times 587. And you see that that's also true with these unit fractions.

4:25I've just given you examples, but you could keep doing it with others and you'll keep getting that same result.