Sal uses the distributive property to break up 4x7 into smaller numbers.
- [Instructor] What we're going to do in this video is dig a little bit deeper into our understanding of multiplication. And just as an example, we're going to use four times seven. And some of you might know what four times seven is, but even in this case, I think you might get something from this video because we're gonna think about how you can break down a multiplication question into simpler parts, and that's going to be useful well beyond four times seven. It's going to be useful in your future when you're tackling more and more complicated things. Now there's a couple of ways that we can visualize four times seven. My favorite way is to visualize it with angry cats, so let's bring on the angry cats. (imitates cats meowing) Yep, they're still angry. And we can see that this is a representation of four times seven. We have four rows right over here. Four rows, and each of those rows have seven cats. And so you can see that right over here, each of those rows has seven cats. Some people would call this a four by seven grid or four by seven array, however you want to view it, but if someone were to ask you what's the total number of cats, it would be four rows times seven columns, four times seven. Now another way to represent four times seven is also with a tape diagram. You might see something like this, where here, we're visualizing it as seven fours, or you could view it as four plus four plus four plus four plus four plus four plus four. Now that's all well and good and you can add that up, if you like, but what I promised you is that we would figure out ways to break down things that might simplify things in the future. Well what if you didn't know what four times seven is, but you knew what four times five is and you knew what four times two is? Well what's interesting is that seven is five plus two, so what if we tried to first figure out this many cats, so four rows and five columns right over there, and then we tried to figure out this many cats? Four rows and two columns. And you can see that it's the exact same number of cats. So one way to think about it is four times seven is the exact same thing as four times, and I'm gonna use parentheses, and that just means to do that part first, is equal to four times, instead of seven, I could write that as five plus two, 'cause that's what seven is, so all this is saying is four time seven is the same thing as four times five plus two, where you do the five plus two first, 'cause we have those parentheses around it, and five plus two is, indeed, equal to seven, and we can see that that is equivalent to the total number of cats that we have here, which we could view as what we just circled off in this orangeish pink color which would be four rows of five, so that would be equal to four times five. Four times five, and then to that, we can add this second group of cat heads or angry cat heads, and that is four rows of two. So that's four times two, and we could put parentheses if we want, just to make it a little bit more readable. Now why did I do that? Well, some folks might find four times five a little bit more straightforward. I could skip count by five. I can go five, 10, 15, 20. Also four times two might be a little bit more straightforward, and so it could be easier to say hey, this is just going to be four times five, which is 20, plus four times two, which is equal to eight, and so that is just going to be equal to 28. And you could have thought about it the same way down here with what is sometimes called a tape diagram. We could say, all right, if I have five fours, that's this mount right over here. That is the four times the five, and then I could add that to the two fours, the four times the two right over here, and that's another way to get to four times seven. So the big picture here is even if you're not dealing with four times seven, if you're not dealing with angry cats, and in most of our lives, we actually try to avoid angry cats, there might be a way to break down the numbers that you're multiplying into ones that you might be more familiar with. I'll give you one more example. Let's say someone were to ask you, well, what is six times nine? Pause this video and see if you can break this down in some useful way. Well, maybe you know what six times 10 is, and you also know what six times one is, so you could rerwrite nine as 10 minus one. Well then this would mean that six times nine is the same thing as six times 10 minus one. Based on exactly what we just did up here, that says that this whole thing is going to be the same thing as six times 10. Six times 10 minus six times one. One way to think about it is, I just distributed the six. That's the distributive property right over there, and then six times 10 is equal to 60, and then six times one is equal to six. And it might be easier for me to say, hey, 16 minus six in my head, that's equal to 54. So I know what some of you are thinking. Six times nine seems so clean, and now I've involved all of this other symbolism, symbols, and I've written down more numbers, but at the end of the day, I'm trying to give you skills for breaking down problems and including ways that you might want to do it in your head. If you're like, hey, I'm kind of foggy on what six times nine is, but six times 10, hey, I know that's 60, and six times one, of course, that's six, well what if I view this as six times 10 minus one? And then I could tackle it and get 54. And once again, you might know six times nine. You might know four times seven, but in the future, it might be useful for bigger and bigger numbers to think about how could I break this down?