Multiplying with area model: 78 x 65

I'm going to multiply 78 times 65 in a little less than standard way, but hopefully it'll make some sense, and you realize that there's multiple ways that you can multiply. And this is actually the way that I multiply numbers in my head. So 78 times 65. And then we're actually going to think about what the different parts of this process represent on this area model. So 78 times 65. So I'm going to start just the way we normally start when we multiply. I'm going to start with this 5 in this ones place, and I'm going to say 5 times 8 is 40. And instead of just writing a 0 and carrying a 4 right over here, I'm just going to write the number 40. So this was the 8 times the 8. Now, I'm going to multiply the 5 times the 7. And we have to be a little bit careful here because 5 times-- this isn't just any 7, this is a 70. So what is 5 times 70? Well, 5 times 7 would be 35, so five times 70 is 350. So I'll write that down, 350. So just as a review, 5 times 8 is 40, 5 times 7 is 350. If you add these two together, this is going to be 5 times 78. Now let's go over to the 6. So let's multiply the 6 times the 8. Now we have to be careful again. This 6 is not just a regular 6, it's in the tens place. This is a 60. 60 times 8. Well, 6 times 8 is 40, so 60 times 8 is going to be 480. So it's going to be 480. And then 6 times 7. Well, that would be 42, but we have to be careful. This is 60 times 70, so we're going to have two zeroes at the end. This is 4,200, not just 42. So 6 times 7 is 4,200. And now we can add everything together. And this is a very similar process to what we do when we do the traditional method of multiplying. I just made it a little bit more explicit what parts are from multiplying which digits. But we can add everything together. In the ones place, we have a 0. In the tens place, we have 4 plus 5 is 9. 9 plus 8 is 17. And now we can carry a 1. 1 plus 3 is 4. 4 plus 4 is 8. 8 plus 2 is 10. Carry a 1, regroup of 1 even, and then you have a 5 right over there. So you get 5,070. Now, I want to think about-- I want to visualize what was going on here using this area model. So once again, we had 78. So I'm going to make this vertical length represent 78. So this distance right over here represents the 70. That's the 70, and then we'll make this distance right over here represent the 8. Let me make that a little bit cleaner so you see what I'm talking about. So this distance right over here represents the 8 and then we're going to multiply that times 65. So this distance right over here-- it's not drawn perfectly to scale, but it gives the idea-- this is 60. And then this distance right over here is the 5. So this whole distance is 65. This entire distance is 78. So when you multiply-- if you had a rectangle that's 65 units wide and you multiplied it, and it had a height of 78 units, then its area is going to be 78 times 65. It's area is going to be 5,070. Now, each of these parts we can map to one part of this area model right over here. When we multiplied the 5 times the 8 and got the 40, that was this section right over here, 5 times 8 is equal to 40. When we multiplied 5 times 70, and got 350, that's this right over here. 5 times 70, and we got 350. When we multiplied 60-- the 6 in the tens place-- 60 times 8 and got 480-- that's this right over here-- this is 60 times 8 is equal to 480. And then finally when we multiplied 60 times 70 and got 4,200-- that's this area right over here-- this area is 60 wide, 70 tall. So this is 60 times 70, which is equal to 4,200. And then when we added everything up to get 5,070, we were essentially just adding up the areas of each of these tiles. This big one is 4,200, that makes up most of the area, and we get 480 from this magenta one, then 350 from this greenish-yellow one, and then 40 from this greenish-blue one to get 5,070. So the area of this entire thing is 5,070 square units.