Multi-digit multiplication: place value and area models
Let's multiply 6 times 7,981. And the way we're going to do it right now is just to represent or expand out 7,981 as 7,000 plus 900 plus 80 plus 1. And so multiplying 6 times 7,981 is the same thing as multiplying 6 times 7,000 plus 6 times 900 plus 6 times 80 plus 6 times 1. You'd essentially distribute the 6. And to help us keep track of things, let me draw a little grid right over here. So this is the 6, and we're going to have to think about what 6 times 7,000 is, 6 times 900, 6 times 80, and 6 times 1. So I'll make a little square for our rectangle for each of them. Let me do that. So here we go. And so we just need to think about, what is 6 times 7,000? Well 6 times 7 is 42. So 6 times 7,000 is 42,000. 6 times 900, well once again, 6 times 9 is 54. So 6 times 900 is 5,400. 6 times 80, well 80 is eight 10s. So 6 times 8 is 48, but since it's six times 80 or eight 10s, this is going to be 48 10s, or 480. And then finally, 6 times 1, of course, is equal to 6. So to find what this product is, we just have to take the sum of each of these numbers. What 6 times 7,000 is plus 6 times 900 plus 6 times 80 plus 6 times 1. So let's do that right over here. So it's going to be 42,000 plus 5,400 plus 480 plus 6. And we get, let's see, in the ones place, we just have a 6. In the tens place, we just have an 8. In the hundreds place, 4 plus 4 is 8. In the thousands place, 2 plus 5 is 7. And then finally, the ten thousandths place, we still have a 4. So we get 47,886. So this Is equal to 47,886. And what I encourage you to do is to think about how this is really underlining what we're doing here. It's not that different than what you might have done with the traditional multiplication techniques. And this is a useful way of thinking about it because now you really understand what's going on. And actually, when you start doing things like this in your head, at least for myself, this is actually how I try to tackle the multiplication problem. When someone says 6 times 7,981, if I was just looking at this and I didn't have any paper, I would say, OK, what's 6 times 7,000? I'd say, OK, that's 42,000. I'll try to remember that. What's 6 times 900? Oh that's 5,400. Well if I add that to the 42,000, I get 47,400. Then, what's 6 times 80? 480. Have to add that to the 47,400 to get to 47,880. And then, what's 6 times 1? Well that's 6. Well add that to the 47,880, which I've been keeping in my brain, and that's going to be 47,886. So this helps you understand what's really going on when you multiply multiple digits, and it's a useful technique for doing mental multiplication.