Fractions as division
When we were first exposed to multiplication and division, we saw that they had an inverse relationship. Or another way of thinking about it is that they can undo each other. So for example, if I had 2 times 4, one interpretation of this is I could have four groups of 2. So that is one group of 2, two groups of 2, three groups of 2, and four groups of 2. And we learned many, many videos ago that this, of course, is going to be equal to 8. Well, we could express a very similar idea with division. We could start with 8 things. So let's start with one, two, three, four, five, six, seven, eight things. So now we're going to start with the 8. And we could say, well, let's try to divide that into four groups, four equal groups. Well, that's one equal group, two equal groups, three equal groups, and four equal groups. And we see when we start with 8 divide it into four equal groups, each group is going to have 2 objects in it. So you probably see the relationship. 2 times 4 is 8. 8 divided by 4 is 2. And actually, if we did 8 divided by 2, we would get 4. And this is generally true. If I have something times something else is equal to whatever their product is, if you take the product and divide by one of those two numbers, you'll get the other one. And that idea applies to fractions. It actually makes a lot of sense with fractions. So for example, let's say that we started off with 1/3 and we wanted to multiply that times 3. Well, there's a couple of ways we could visualize it. Actually, let me just draw a diagram here. So let's say that this block represents a whole, and let me shade in a 1/3 of it. So that's 1/3. We're going to multiply by 3. So we're going to have 3 of these 1/3's. Or another way of thinking about it, it's going to be 1/3 plus another 1/3 plus another 1/3. That's our first 1/3, our second 1/3, and our third 1/3. And we get the whole. This is 3/3, or 1. So this is going to be equal to 1. So you use the exact same idea. If 1/3 times 3 is equal to 1, then that means that 1 divided by 3 must be equal to 1/3. And this comes straight out of how we first even thought about fractions. The first way that we ever thought about fractions was, well, let's start with a whole. And that whole would be our 1. And let's divide it into 3 equal sections, the same way that we divided this 8 into 4 equal groups. So if you divide this into 3 equal sections, the size of each of those sections is going to be exactly 1/3. Now, this leads to an interesting question that might be popping in your brain. Notice, we have 1 is the numerator, 3 is the denominator, and we just said that this is equal to the numerator divided by the denominator. 1 over 3 is the same thing as 1 divided by 3. Is this always true for a fraction? Well, let's just do the same thought experiment, but let's do it with a different fraction. Let's take 3/4 and multiply it by 4. So multiply it by 4. So once again, let's see if I could draw 1/4 here. Let me do this in a new color. So let's say that this block right over here is a whole. We'll divide it into four equal sections. So now I've divided it into fourths. And let me copy and paste it so I can use it multiple times. So copy. All right. Now, 3/4, that's going to be-- we can assume-- I didn't draw it perfectly. Actually, I could draw it a little bit better than that just to make the four equal sections actually look equal. So that looks like a little bit better of a job. I'm trying to make them four equal sections. Let me copy that one. So let me use it for later. Now, 3/4. This is four equal sections, and 3/4 represents three of them-- one, two, three. But now we're going to multiply it by 4. So we're going to have 3/4 four times. So we're going to need some more wholes here. So let's throw in another whole. So this is one 3/4. Now let me do the next 3/4 in another color. So that's a 1/4, that's a second 1/4, that's a third 1/4. That's another 3/4. And now let's do-- so we've done two 3/4 just now. Let me make it clear. This is the first 3/4, and then this plus this is the second 3/4. Now let's do a third 3/4. And we're going to have to use another whole right over here. And I will do that in this color. So my third 3/4, so here's a 1/4, here's my second 1/4, here's a third 1/4. So in green, I have another 3/4. And now we need four 3/4. So let's do that in a color I have not used yet, maybe white. So that's a 1/4, that's two 1/4, and that is three 1/4. So notice, now I have now I have one 3/4, two 3/4, three 3/4, and four 3/4. And what did I do when I got those four 3/4? Well, it's pretty clear. This is turned into 3 wholes. So this is equal to 3 wholes. Well, if 3/4 times 4 is equal to 3, that means that 3 divided by 4 is equal to 3/4. So the same idea again. 3 over 4 is the same thing as 3 divided by 4. And in general, this is true. The fraction symbol here can be interpreted as division. And looking at this diagram right here, it made complete sense. If you started with 3 wholes, and you want to divide it into 4 equal groups, one group, two groups, three groups, four groups, each group is going to have 3/4 in it.