Multiplying with area model: 16 x 27
Sal uses an area model to multiply 16x27. Created by Sal Khan.
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- Is it permissible to use this area concept by subtracting a 'negative' area?
For example, using our problem of 16 x 27, we could actually make an area of 20 x 27. We can then subtract an area of 4 x 27 to yield our answer.
We can easily calculate 20 x 27 = 540, and then subtract a similarly easy calculation of 4x27=108 → 540-108=432.(179 votes)
- Good point, I think you are absolutely right. We can even set up a diagram like Sal has done, using negative numbers in one or both sides of the multiplication.
Eg. Using (30 - 3) for 27, and (20 - 4) for 16 in the diagram below
We get 600 - 120 - 60 + 12 = 432
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20 | +600 | -60 |
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-4 | -120 | +12 |
- Can you explain me how to do the area model negative numbers if it is possible?
Also what do you mean when you say subtracting a 'negative' area?(14 votes)
- I don't know what it means by subtracting a negative area, but the question basically says to multiply 16 by 27 which is 432. This probably doesn't help but I hope it does.(12 votes)
- 30 -3
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20 | +600 | -60 |
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-4 | -120 | +12 |
|______|_____| this is my area model khan sir(11 votes)
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So I'm going to multiply 16 times 27. I'm going to do it using something called an area model. And the whole point of an area model is to really understand what's going on in the multiplication process. So 16, you can represent literally as 10 plus 6. This 1 is in the tens place. It represents one 10. So we can represent that as 10 plus 6. Let me do the 6 in that same green color. 10 plus 6. And let me mark off 10 and 6 here. So this is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. So this part in blue, that's the 10. So I'm representing-- I've gone 10 slashes, or I'm representing 10 boxes right over there. And then the 6, which I want to do in that green color, the sixth, let's mark off 6 boxes. 1, 2, 3, 4, 5, 6. So 16 is this whole length. The blue part is the 10. The green part is the 6. The 10 comes from the one in the tens. This is literally 1 10 and this is literally 6 1s. Now let's think about 27. Well, we already know that the 2 in the 10s place is representing 20. So let's count off 20. So it's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 15, 16 17, 18, 19. 20. So up until that point right over here, this line right over here has a length of 20. But we're not just talking about 20. We're talking about 27. So it's 20 plus 7. So let's count off 7 here, 1, 2, 3, 4, 5, 6, 7. Now if we look at the total number. So if we have 16 times 27, the product 16 times 27, what gives you the area of a 16 by 27 rectangle. So let me draw that. So it's 16-- this is a width of 16. Let's bring a little line down right over here-- and then the height of my rectangle, the way I've drawn it is 27. This is l, 2, 3, 4, 5, 6, 7. So the area of this rectangle is going to be 16 times 27, the area of this entire rectangle. Now, we could do is break it up by parts, because it'll be easier to compute, and we can see what part of the area those different products represent. So, for example, we can think about what 10-- so let me separate out the sections. So let me draw it. So this section is 10 wide. And then let me draw a section like this. So we could figure out the areas of each of these sections, and then the area of the entire rectangle, which is going to be this product, is going to be the area of all of these rectangles combined. So we could first think about it-- well, let's think about what 20 times 10 is. Well 20 times 10 is a fairly straightforward thing. It's going to be 200. You could think of as 2 times 1, and you have two 0's there. Or you probably know in your head 20 times 10 is just going to be-- you're going to add a 0 here. So it's going to be 200. So 20 times 10 is 200. And let me highlight that in. So you have-- that's not the color I wanted to use-- let me use this blue right here. So that's the blue from the 10. Let me put some orange in there from the 20 to make it clear that this is the product of both of those numbers. So 20 times 10 is 200. Now what's 20 times 6? Well 2 times 6 is 12, plus you have this 0 here. So this right over here is going to be 120. And it has the orange in it. So this is a 20. And then let me put some green in for the 6. 20 times the 6. Now what's the area of this section right over here? Well, it's 7 high and it is 10 wide. So it's going to be 7 times 10 or 70. I'll have an area of 70 square units. So let me do it in this purple, and I'll thrown some blue in there, too. It's kind of a fun art project. I'll throw some blue in there too. And then finally, what's the area of this little section right over here? It's 7 high and it's 6 wide. So it's going to be 7 times 6 or 42 square units of area. Let me color it in. So I got some magenta in there, and then I got some green in there right over there. So what's the area of this entire thing going to be? Well, it's going to be the 200 plus the 120 plus-- let me do it this way-- it's going to be 200 plus 120 plus 70 plus 42. When you add that up, you get, let's see, in the ones place, you get a 2. Then you get 2 tens plus 7 tens is 9 tens, plus 4 tens, is 13 tens, which is the same thing as 3 tens and 1 hundred. And then this is a 4. Did I add that up right? Let's see, this will be 11. Yup, that looks right. So this is 432. So this is going to be equal to 432. And you might be saying, hey, Sal, why did we go through all of this business? I've seen before that if I take 16-- I could take 16 times 27 like this. You probably learned this type of a process. 16 times 27. Then you say, OK, let's start with the 7 in the ones place and you do 7 times 6 is 42. And you carry the 4. You're really just putting the 4 in the tens place because it's a 40. But right when you did that 7 times 6, we essentially calculated this right over there. And then when you multiply the 7 times the 1, you really multiplying 7 times 10, and then you're adding the 4 from the 42. So when you do the 7 times this 1. This is you're actually calculating this area. And then when you carry it when you add this carried 4, and put it right down here. You're essentially taking the sum of both of these things, because you're multiplying 7 times the 16 to get this area. So let's just do it. 7 times 1 is 7 plus 4 is 11. So 112, what you just figured out right over here, is this area Right over here. So 7 times the 10 or the 70, plus 7 times 6, the 42. Now when you go into the tens place you've probably always said, oh, you know, I just throw a 0 down there. But why do you throw a 0 down there? Because this 2 is representing 10. So if you multiply 2 times the 6, and you put a 12 and put a 2 down and carry the 1. That wouldn't be right. This is a 20. So that's why you put the 0 there. So let's scratch this out so we don't get confused. 2 times 6 is 12. So we're used to carrying the 2 down here and carrying the 1. But notice we're really thinking about 20 times 60 is 120. Just doing that we just calculated that right over there. And then we do 20 times 10 is 200. But then we had that one from the 120. So this is going to be 300. So what we just did when we multiplied the 2 times the 16, we just calculated this total area. We did the 20 times the 6 to get that. And then we had the 20 times a 10 to get that. When we did the 20 times the 6, we carried this 1 into the hundreds place. And so we added it all together and got 320. And then this step right over here, you're literally just finding the combined area-- the area of this plus the area of that the area. So that's going to get us to--- we deserve a drum roll now. 2 plus 0 is 2. 1 plus 2 is 3. 1 plus 3 is 4.