Multiplying 2 fractions: number line

In a previous video, we saw that we could view 2/3 times 6 as whatever number is 2/3 of the way to 6 on the number line, which we saw is 4. Or another way to think about it is that 4 is 2/3 of 6. 2/3 times 6 can be viewed as-- well, how many do I have if I take 2/3 of 6? Now, what we want to do now is apply that same idea, but to multiply not a fraction times a whole number, but a fraction times a fraction. So let's say that we wanted to take 3/4 and multiply it by 1/2. And we know, of course, the order that we multiply doesn't matter. This is the exact same thing as 1/2 times 3/4. So to imagine where this gets us, let's draw ourselves a number line. And I'll do it pretty large so that we have some space to work in. So that's 0. And then that is 1. And of course, our line could keep on going. And let's first imagine 3/4 times 1/2 as 3/4 of the way to 1/2. So first let's plot 1/2 on our number line. Well, 1/2 is literally halfway between 0 and 1. So that's 1/2 right over there. And how do we think about 3/4 of the way to 1/2? Well, what we could do is think about well, what's 1/4 of 1/2? Well, we could divide this part of the number line into 4 equal sections. So that's 2 equal sections. Now that's 4 equal sections. And while we're at it, let's divide all of the halves into 4 equal sections. So let's divide all of the halves into 4 equal sections. So that's 4 sections. And now let's do this one. I'm trying my best to draw them equal sections. So I've taken each of the halves and I've made them into 4 equal sections. So this point right over here is 1/4 of 1/2. But that's not what we care about. We want to get to 3/4 of 1/2. So we want to get to 1, 2, 3/4 of 1/2. So this point right over here, this is literally 3/4 times 1/2. And this is, of course, 1/2 here. But what number is this? And let me do this in a new color. We can now visualize it on the number line. But what number is this actually? Well, a big clue is that, well, before we had the section between 0 and 1 divided into 2 equal sections when we only had to plot 1/2. But then we took each of those 2 equal sections and then split them into 4 more sections. By doing that, we now essentially have divided the section between 0 and 1 into 8 equal sections. So each of these is actually 1/8. So this point right over here is 1/8. This is 2/8. And then this is 3/8. And that's in line with what we've seen about multiplying fractions before. This should be equal to 3 times 1 over 4 times 2, which is equal to 3/8. And everything that we're talking about, so we don't get confused, this is all referring to this point right over here on the number line. But what if we thought about it the other way around? What if we thought about it as 1/2 of the way to 3/4? So we could divide the space between 0 and 1 into fourths. So let's do that. So that is 1/4, 2/4, 3/4. So this right over here is the number 3/4. And we want to go half of the way to 3/4. Well, what is half of the way to 3/4? Well, we split this section into 2 equal sections. So we could split right over there. And we want to go exactly one of those sections. 1/2 of 3/4 gets us, once again, right over here to this point-- 3/8. So either way you imagine it, whether you're essentially taking 3/4 of 1/2, or saying I'm going to go 3/4 of the way to 1/2, or you say I'm going to go 1/2 of the way to 3/4, either way, hopefully it now makes conceptual sense. You can visualize it, and it makes numeric sense that this is going to be equal to 3/8.