Multi-digit multiplication: place value and area models
- [Voiceover] What I hope to do in this video is get a little more practice and intuition when we're multiplying multi-digit numbers. So let's say that we wanted to calculate what 7,000 times six is. 7,000 times six. Now, for some of you, it might just jump out at you. That, hey look, if I have seven of anything, and here I have seven thousands, and I multiply that by six, I'm now going to have seven times six of that thing, or 42 of that thing, and in this case we have 42 thousands. So you might just be able to cut to the chase and say, hey look, six times seven thousands is going to be 42 thousands. And that's great if you can just cut to the chase like that, and another way to think about it is like, look, six times seven is 42, and then since we're talking about thousands, we're not just talking about seven, we're talking about seven thousands. I have three zeros here, so I'm going to have 42 thousands, three zeros there. But I want to make sure that we really understand what is going on here. This will also help us with a little bit of practice of our multiplication properties. So 7,000 is the same thing as 1,000 times seven, or seven times 1,000. It's seven thousands. Or you could view it as a thousand sevens, either way. So this is the same thing as 1,000 times seven times six. And so you could view it as, you could do 1,000 times seven first, which would be 7,000, and then times six. Or you could do the seven times six first, and this is, this right over here is the associative property of multiplication. It sounds very fancy, but it just says that, hey look, we can multiply the seven times six first, before we multiply by the thousand. So we could rewrite this as 1,000 times, and if we're going to do the seven times six first we can put the parentheses around that, times seven, times six. Seven times six. Notice, it's 1,000 times seven times six. I could do 1,000 times seven first to get 7,000, or I could do the seven times six first to get, and you know where there is going, so if you multiply the seven times six first, you're going to get 42, and you're gong to have 1,000 times 42. 1,000 times 42. So you can view this as a thousand 42's or maybe a little bit more intuitively you could view this as 42 thousands. So, once again, we get to 42,000. And so the whole reason, some of y'all might have just been able to do this immediately in your head and that's all good, but it's good to understand what's actually going on here. And the reason why I also broke it up that way, this way, is that the exercises on Khan Academy make you do this to make sure that you really are understanding how to break up these numbers and how you could re-associate when you multiply. Let's do another one. Let's say that we wanted to figure out, let's say that we wanted to figure out, let me give ourselves some space, let's say that we wanted to figure out what 56 times eight is. And there is a bunch of different ways that you could do it. You could say that, look, 56, this is the same thing as 50, five tens, that's 50 plus six ones, so 50 plus six, and all of that times eight. And then you could distribute the eight and you could say, look, this is going to be 50 times eight, So it's going to be 50 times eight, plus six times eight. Plus six times eight. And 50 times eight? Well, five times eight is 40, but we're not just saying five, we're saying five tens, so five tens times eight is going to be 40 tens, or it's going to be 400. Another way to think about it, five times eight is 40, but we're not talking about five, we're talking about five tens, so it's going to be 40 tens. So 50 times eight is 400, and then six times eight is, of course, equal to 48. So this is going to be equal to 448. This is actually how I do things in my head. When I do it in my head I obviously am not writing things down like this but I think, okay, 56 times eight, I could break that up into 50 and six, and eight times 50, well that's 400, or 40 tens you could say. Eight times five tens is going to be 40 tens, or 400. And then eight times six is going to be 48, so it's going to be 400 plus 48. Once you get some practice you're going to be able to do things like this in your head. And if it helps, we can also visualize this looking at an area of a rectangle. So imagine this rectangle right over here, and let's say that this dimension right over here is eight, it is eight units tall. So that's the eight. And this entire dimension, this entire length here is 56. So the area of this rectangle is going to be 56 times eight, which is what we set to figure out, and to do that, well we could break it up into 50 and six, so this first section right over here, this has length, we could say this has length 50, that has length 50, and then this second section, this has length six. This has length six. And the reason why we broke it up this way is cause we can, maybe in our heads, or without too much work, figure out what eight times 50 is and then separately figure out what eight times six is. So separately figure out the areas of these two pieces of the big rectangle and then add them together. So what's eight times five tens? Well it's going to be 40 tens, or 400. This is going to be 400 square units, is going to be the area of this yellow part. And then what's eight times six? Well we know that's going to be 48 square units. So the entire rectangle is going to be the eight times the 50, or the 50 times the eight, the 400, this area, the yellow area, plus the magenta area, plus the eight times the six, the 48, which is 448. 448 is going to be the area of the whole thing. Eight times 56.