Fraction multiplcation on the number line

So we're going to think about, in this video, is multiplying fractions. So let's say that we wanted to take two-thirds, and we want to multiply it by four. What is this going to be equal to? Pause this video and try to think about it on your own.

All right, now let's work through this together. And to help us, I will use a number line. Let's say that each of these hash marks represent a third. So this is 0, this is one-third, two-thirds, three-thirds, four-thirds, five-thirds, six-thirds, seven-thirds, eight-thirds, and nine-thirds.

So where is two-thirds times one? Well, two-thirds times one is just going to be two-thirds. We just take a jump of two-thirds, so that is times one. If we multiply by two, or if we take two-thirds times two, that'll be two jumps. So one, two-thirds, two, two-thirds, three, two-thirds, and then four, two-thirds.

So we just took four jumps of two-thirds each. You could view that as two-thirds plus two-thirds plus two-thirds plus two-thirds. And where does that get us to? It got us to eight-thirds. So notice two-thirds times four is equal to eight-thirds.

Now we could go the other way. We could look at a number line and think about what are ways to represent what the number line is showing us. On Khan Academy, we have some example problems that do it that way, so I thought it would be good to do an example like that.

And so let's label this number line a little bit different. Instead of each of these lines representing a third, let's say they represent a half. So zero, one-half, two halves, three halves, four halves, five halves. What did I write? Five, six? My brain is going ahead: five halves, six halves, seven halves, eight halves, and nine halves.

And let's say we were to see something like this. So if you were to just see this representation—so I'm going to try to draw it like this—if you were to just see this representation, what is that trying to represent? What type of multiplication is that trying to represent?

Well, you could view that as three halves plus another three halves plus another three halves because notice each of these jumps are three one-halves or three halves. So you could view this as three halves plus three halves plus three halves, or another way of thinking about it is this is three jumps of three halves.

So you could also view this as being the same thing as three times three halves. And what are these equal to? Well, three halves plus three halves plus three halves, or three times three halves, it gets you to nine halves.