Using the distributive property when multiplying

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. 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 going to 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 are 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. 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. So you can see that right over here, each of those rows have seven cats. Some people would call this a four by seven grid or four by seven array, however you'd 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: 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 going to 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, because that's what seven is.

So all this is saying is four times seven is the same thing as four times five plus two, where you do the five plus two first because we have those parentheses around it. And five plus two is indeed equal to seven. 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 orangish-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, ten, fifteen, twenty. 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 twenty plus four times two, which is equal to eight. And so that is just going to be equal to twenty-eight.

You could have thought about it the same way down here with what is sometimes called a tape diagram. We could say, "Alright, if I have five fours, that this amount 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, even 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 ten is, and you also know what six times one is. So you could rewrite nine as ten minus one. Well then, this would mean that six times nine is the same thing as six times ten 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 ten 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 ten is equal to sixty, and then six times one is equal to six.

It might be easier for me to say, "Hey, sixty minus six," in my head that's equal to fifty-four. 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, including ways that you might want to do in your head.

If you're like, "Hey, I'm kind of foggy on what six times nine is," but six times ten, hey, I know that that's sixty, and six times one, of course, that's six. Well, what if I view this as six times ten minus one, and then I could tackle it and get fifty-four. 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.