Multiplication on the number line

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 4 times 2 means, we've already seen in other videos you could view this as four groups of two. So we could have four groups.

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 2 there, you have 2 there, you have 2 there, and you have 2 there.

You could also view that as 4 twos, or 4 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 2 is equal to 8.

We can also think about that on a number line. So I'm going to make a little bit of a number line here. We can imagine 4 times 2 being, all right, 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 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 going to go there, and then I'm going to go there. I'm taking equal jumps every time.

Then I'm going to go there, then I'm going to go there, and then I'm going to go over there. So what would that represent if we use the same type of ideas that we just thought about?

Well, I went from 0 to 4, 8, 12, 16, 20. I'm skip counting by 4. So you can 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 twenty. 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, 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 7 times 3.

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 the skip counting: you went from zero to three, six, nine, twelve, fifteen, eighteen, and twenty-one.

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. We get to seven. Then if we take another hop of seven, we get to fourteen.

And then if we take another hop of seven, we get to twenty-one. Interesting! At least for this situation, whether we took 7 hops of 3 or 3 hops of 7, 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.