Equivalent fractions with models

So what we're going to do in this video is think about equivalent fractions.

Let's say we have the fraction three-fourths, and I want to think about what is an equivalent number of eighths. So three-fourths is equal to how many eighths? To represent that, how many? I could put a question mark there, but instead of a question mark, I'll just put a letter. So what should y be? Three-fourths is equal to y-eighths. What does y need to be to make this true?

Before I tell you to go pause this video and try to work on it on your own, which I will do in a little bit, I'll give you a little bit of a hint. So let's try to represent three-fourths—or I'll represent it for you. So I will do it with this rectangle. I'm going to divide it into four equal sections.

So let's see, that would be dividing it roughly in half. I'm hand-drawing it, so it's not perfect, but these should be equal sections. The areas of each of these sections should be equal. So there you go, this is my hand-drawn version of that. And so three of those four—and I will do that in purple—three of those four, it could be one, two, and then just for kicks, I will do this one out here. So that is three-fourths right over there.

Now, if I were to think about this in terms of eights, I'm going to draw another whole, but this time instead of just splitting them into fourths, I'm going to split it into eighths. So let's do the fourths first just because it's easy to look at the one above that. So that's my fourths, and then I'll divide each of the fourths into two. So that gives me eighths.

All right, almost there. The drawing is really the hardest part here. And so each of these is an eighth. It's hand-drawn, but imagine if there were eight equal sections. So how many eighths is equal to three-fourths? Pause the video and try to work it out on your own.

All right, well, we can just look at this visually. So this first fourth, we could say that's equivalent to filling out this eighth and this eighth. So that first fourth is equal to two-eighths. This second fourth is equal to another two of these eighths, and then this third fourth is equal to another two of the eighths.

So how many eighths have I shaded in? Well, I have one, two, three, four, five, six. So I have six over eight. So three-fourths is equivalent to six-eighths. So in this scenario, y is equal to six-eighths, or we could say six-eighths is equal to y.

Now, let's do another example. So we could see here in this top circle, we've divided it into six equal sections. So each of these are one of the six equal sections or one-sixth. We can see that one, two, three, four of them are shaded in. So we have represented in that top circle that is four out of six.

So what I want to think about is how many thirds are equivalent to four-sixths? Pause this video and think about it. So once again, how many thirds are equivalent to four-sixths? Instead of just putting a question mark there, I'll put the letter x. So what should x be for these two things to be equivalent?

Another way to think about it is four over six is equal to x over three. Four-sixths is equal to how many thirds? All right, now let's do this together. And so, one way you could think about it—let's see, for one-third first—to make this equal to one-third, it looks like that is equivalent to what I am circling in the orange up here.

And that also makes sense if I were to divide a third into two. So now I would have, this would be a sixth, and that would be a sixth. So you need two-sixths to make up a third, or each third is equivalent to two-sixths. So this is one-third right over here, and that is equivalent to two of these sixths.

But we are not completely done yet; we have another two-sixths. So we could say those two-sixths are equivalent to another third. It’s a little tricky because they didn't put this sixth next to that sixth, but you could imagine if we were to move—let's say we were to move this sixth.

So I'm gonna color this one in white. So if I were to move that sixth to right over here, so we're shading this one in instead, so then you can see that these two-sixths right over here, these two-sixths are equivalent to this third right over there.

So what you can see is that our four-sixths—and remember I moved this one over, so I'm not shading this one in anymore—but you can see the four-sixths that I've shaded in is equivalent to two-thirds. Or another way to say it is x would be equal to two.

x would be equal to two. Four-sixths is equal to x thirds, or four-sixths is equal to two-thirds.