Equivalent fractions and different wholes

- [Voiceover] If I were to give you the fraction 1/3, I could rewrite it. If I multiply both the numerator and the denominator by two, one times two is two, and three times two is six. So we know that 1/3 is the same thing as 2/6. This is true, 1/3 is just another way of saying 2/6. Now given that we know that, which of these, I guess you could say, picture equations actually show us that that is true? Show that to us visually? We could start with this first one. We have three equal sections, so each of these sections are 1/3. And we have one of those 1/3 shaded in. If we go on the right side, we have six equal sections, so it's divided into 1/6, and we have two of them shaded in. One, two. And the important thing to realize, is over here, we divided the same rectangle to the 1/3 as we divided into 1/6 over here. We are comparing fractions of the same whole. So you see that if you were to take these two magenta boxes and put them together, they actually do have the same area as this one larger rectangle. So this one looks like a good picture explanation or a visual explanation for why 1/3 is equal to 2/6. You take the same rectangle, divide it into 1/3 and take one of them, and then you take that same exact rectangle, divide it into 1/6 and pick two of them, you have shaded in the same amount. 1/3 over here is the same thing as 2/6 over here. The important thing to realize is that you're taking 1/3 and 2/6 of the exact same rectangle. When you look over at these figures, you see that this is 1/3 of this figure right over here. It's split into three equal sections, and this is one of them that's shaded in. And this is, right over here, 2/6. You have six equal sections and we've shaded in two of them. And we know that 1/3 is equal to 2/6, but this picture isn't true. This shaded area right over here is less than this shaded area over here. So you say, "Why doesn't it work? "Why isn't this 1/3 the same thing as 2/6?" Because you're taking 2/6 of a larger hexagon here. Here, you're taking 1/3 of a smaller figure. Here, you're taking 2/6 of a larger figure. You can't compare fractions of different wholes. 2/6 of a larger whole is going to be larger than 1/3 of a smaller whole. The only time that 1/3 and 2/6 are going to be equivalent is if they are of the same whole, like we saw up here. So this is not true. And for the same reason, this right over here is 1/3 and this over here is 2/6. This is 1/3 of a smaller circle than this 2/6 is. So this is not true. By that same logic, we have these other pictures over here. And over here, we're not talking about 1/3 and 2/6. Here, we're talking about, let's see, splitting something into one, two, three, four, five, six, seven, eight. So we're talking about 1/8, and in this case, it's one, two, three, four, five, six, 6/8, and we're comparing it, let's see over here, they're dividing into four and they've shaded in three. Now it is true that 6/8 is equal to 3/4. Six divided by two is three. Eight divided by two is four. So if you multiply or divide the numerator by the exact same thing, then you're going to get an equivalent fraction. Now 6/8 is equal to 3/4, but this picture isn't true because 6/8 would be equal to 3/4 if you're talking about taking 6/8 and 3/4 of the same size, in this case, diamond. These things are not the same size, so you can't make this picture statement. Same thing for this one over here. These circles are of different sizes. So taking the same fraction of different sizes, you can't say that they're going to be equal. This last one, you're taking the fractions of the same whole. It's kind of this weird left-pointing arrow-looking shape, but they are the same shape. And you see here, we've divided that shape into 1/8, and we have shaded in six of them. One, two, let me do that in a yellow so you can see it. One, two, three, four, five, six. Over here, we have divided it into 1/4, and we have shaded in three of them. So this statement is true. We already knew that 6/8 and 3/4 are the same thing, but this picture right over here visually shows us that, because we're taking 6/8 of the same whole as we're taking 3/4 of it, and you see that here. This area, let me shade it in a different color. So the area that I'm shading in, six of these triangles with equal area is the same thing as three of these parallelograms, or three of the four. These have the same area, you could see that. One of these, like that one right over there, could be equivalent to this over here, equivalent to that over there. Actually, let me do that for all of them. This one over here could be equivalent to, let's see, if you were to flip it over, it would be equivalent to that over there. And finally, I think you see where this is going, this one over here is equivalent to this one over here. So we have the same fraction shaded in. We just divided the one on the left into more equal pieces than we did on the right but these are equivalent fractions. And this picture shows us that 6/8 is, indeed, equal to 3/4. Once again, 6/8 of the same whole is equal to 3/4 of that same whole.