Multiplication on the number line

- [Instructor] What we're going to do in this video is think about different ways to represent multiplication. And especially connect it to the notions of skip counting and the number line. So if we were to think about what four times two means, we've already seen in other videos, you could view this as four groups of two. So we could have four groups, so one group, two groups, three groups, and four groups, and each of them have two of something. I'll just put two little circles here. So you have two there, you have two there, you have two there, and you have two there, and you could also view that as four twos, or four twos added together. So we could view it as two plus two plus two plus two. And this, of course, is going to be two plus two is four, four plus two is six, six plus two is eight. We see that over here, we could even skip count. Two, four, six, eight. Four times two is equal to eight. We can also think about that on a number line. So I'm gonna make a little bit of a number line here. And so we could imagine four times two being, all right. So this is one times two, two times two, three times two, and four times two. So we started at zero, and we took four hops of two along the number line to end up at eight. We went from zero to two, four, six, eight. We just skip counted our way to eight. So if I were to ask a similar question, actually, let me draw a little series of hops, and I want you to think about it the other way. What multiplication does that represent? So let's say I start here and then I'm going to hop like this. So I'm gonna go there and then I'm gonna go there. I'm taking equal jumps every time. Then I'm gonna go there. Then I'm gonna go there. Then I'm going to go over there. So what would that represent, if we used the same type of ideas that we just thought about? Well I went from zero to four, eight, 12, 16, 20. I'm skip counting by four. So you could imagine this is probably something times four. Now how many hops did I take? I took one, two, three, four, five hops of four. So this is five times four. And we can see that we ended up at 20. We could also view this as being the same thing as five fours or four plus four plus four plus four plus four, and you see that over here. You have, we're starting at zero, we're adding four, then another four, then another four, then another four, and another four. We have five fours here. Let's do one more. So I'm gonna have a number line here and think about what it would mean to say, do something like seven times three, times three. Well, we could view that as seven hops of three starting at zero, seven equal hops. So one, two, three, four, five, six, and seven. We end up at 21, so this is equal to 21. You could also view this as we took seven threes and added them together. And you could also view it as skip counting. You went from zero to three, six, nine, 12, 15, 18, and 21. Now just out of interest, what if we went the other way around? What if we were to take three hops of seven? What would that be? Well, we would start here and so we would take our first hop of seven right over there. Get to seven, then if we take another hop of seven, we get to 14, and then if we take another hop of seven, we get to 21, interesting. At least for this situation, whether we took seven hops of three or three hops of seven, we got to the exact same value. I encourage you to think about whether that's always going to be the case. I'll see you in a future video.