More ways to multiply

If we have 2 groups and in each group I have 4, so that's one group of 4, and then here is my second group of 4, we already know that we could write this as 2 times 4, which is the same thing as 4 plus 4. Notice I have two 4's here. I have 4 plus another 4. Well, if I have 4 plus 4, or if I have 2 groups of 4, either way, I'm going to have a total of 8 things. And you see that right over here. We have 1, 2, 3, 4, 5, 6, 7, 8 things. What I want you to do is pause the video now and try to group these same 8 things, but to group it in other ways so that we can represent 8 as the product of whole numbers. Here I've represented 8 as the product of 2 and 4. 2 times 4 is 8. See if you can represent 8 as the product of other whole numbers, or as whole numbers in different ways, grouping it in different ways. So I assume you've paused the video. So let's try it out ourselves. So one thing we could do, we could view this instead of as 2 groups of 4, we can view 8 as 4 groups of 2. So that's 1 group of 2, 2 groups of 2, 3 groups of 2, 4 groups of 2. So we could write that 4 times 2 is equal to 8. And we could view this as the same thing as, literally, 4 2's. We have one 1, 2, 3, 4 2's. Each of these have 2 in them. So we're going to say 1, 2, 3, 4 2's. 2 plus 2, plus 2, plus 2 is equal to 8. These are both equivalent. 4 times 2, literally 4 groups of 2. That's the same thing as taking 4 2's and adding them together. Notice, we have 2 2's right over here. We added them together, 1, 2. Here, we have 4 2's and we're adding them together, 1, 2, 3, 4. We take our 4 2's and we add them together. How else could we represent 8? Well, we literally could view it as 8 groups of 1. So let's do that. So 8 groups of 1 would look like this. That's 1 group of 1, 2, 3, 4, 5, 6, 7, 8. So we could write this down as 8 times 1. 8 times 1 is, once again, equal to 8. And if we wanted to write this down as repeated addition, well, this is literally 8 1's. So 1 plus 1, plus 1, plus 1, plus 1, plus 1. Let's see. That's 1, 2, 3, 4, 5, 6, 7, 8. 1, 2, 3, 4, 5, 6, 7, 8. 1 plus 1, plus 1, plus 1, plus 1, plus 1, plus 1, plus 1 is equal to 8. Now, you might be a little stumped. Well, what's another way of getting to 8? Well, you could literally view it as 1 group of 8. So let me view it that way. So this is just 1 entire group of 8, the whole thing. The whole thing is a group of 8. So let me scroll over to the right a little bit. We could write this down as 1 times 8. And 1 times 8 is equal to 8. And how would we view that? Well, we only have one 8 now. We don't have to add that one 8 to anything else. So if we wanted to do it the way we've done the last few, we could literally write it down as we just have one 8. Well, one 8 is clearly going to just be equal to 8. So now let me ask you another question. So far we've been focused on each of these groups, but what if we actually view this as 4 groups of 8. Then how many things are we actually going to have? So let me make this very clear. So we have 1 group of eight 8, 2 groups of 8, 3 groups of 8, and 4 groups of 8. So we would view this as 4 times 8, or which is going to be the same thing as 8 plus 8, plus 8, plus 8. 4 8's. What is this going to be equal to? A And I encourage you to pause the video and figure it out right now. Well, there's a couple of ways that you could have thought about this. You could have literally just counted these. Or you could say, well, let's see, you can skip count by 8. 8, 16, 24, 32. Or you could have said, 8 plus 8 is 16, plus 8 is 24, plus 8 is 32. Or you could have literally just counted the triangles here.