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Sal subtracts numbers using models and number lines.   Created by Sal Khan.
Video transcript
Let's review a little bit about what we know so far about subtraction. So if I say 5 minus 3, what does that mean? Well, there's a couple of ways to think about it. I could have 5-- let's say I had to 5 berries. So 1, 2, 3, 4, 5. So I could have 5 berries, and when I say minus 3 you're subtracting 3 from it. I can view that as saying I'm going to take away 3 of these berries. So if I take away that berry, that berry, and that berry. So I took away 1, 2, 3 berries. How many berries do I have left? Well the only berries I have left are right here-- 1, 2. So I have 2 berries left just like that. Now the other way, the other way that I could visualize or think about 5 minus 3, I'll do it over here. 5 minus 3-- is to think about what the difference between 5 and 3 is. So let me draw this. So let's say I have 5 berries. 1, 2, 3, 4, 5. And let's say that you have 3 berries. Here's a slightly different color. You have 3 berries. So another way to think about 5 minus 3 is how many more berries do I have than you have? And if you look right here, well, you see, this berry is another-- you have also one berry there. We both have one berry there, we both have one berry there. But I've got 1, 2 berries that you don't have. So once again, I have 2 more berries than you have. Now we can also think of this from the number line point of view. So let me draw a number line just like that. It's my number line. We've learned on the addition videos we can keep going off forever. And actually, we could even go to the left of 0 and go into negative numbers, which we'll see in future videos. But I'll start at 0. 0, 1, 2, 3, 4, 5-- I'll just go up to 7. So if we do 5 minus 3, if we view 3 as being taken away from 5, 5 minus 3 means start at 5. If I did 5 plus 3 I would jump 3 spots to the right because that's increasing the number of things I have. But since I'm subtracting 3, I want to decrease by 3. So I decrease by 1, 2, 3 and I get to 2 just like that. Now if we visualize it from this point of view, let me draw another number line. I want to show you. I mean this is, I'm taking away 3 and here I'm saying, how many more is 5 than 3? Even though they're the exact same answer, but there are two different ways to think about it. Let me draw a number line here again. Let me draw the same number line. I have 0, 1, 2, 3, 4, 5, 6, 7. So if I were to plot where 5 is on this number line, so this is the 5 right there. I'll put a little pink square around it. 5 is right there. Now 3, let me do 3 in this yellow color. 3 is right here on the number line. So in this way of thinking about 5 minus 3 you're saying, what is the difference-- let me write that down. Here we're saying, what is the difference between 5 and 3? And to figure out the difference you actually have to say, how much do you have to add to 3 to get to 5? So the difference here, how different is 5 than 3? Well you have to go up 1 and then up 2 to get to 5. So the difference between 5, which is all the way over here, and 3, which is just that far, is 2, just like that. That right there is 2. Let me draw that in another box. So that's 2 right here. I want to make this difference between subtraction and difference-- I want to make it at least reasonably clear to you because these are two different ways of viewing subtraction, but it ends up being the exact same operation. You're going to get the same answer regardless of which way you think about it. Now, I could view-- let me do different numbers now. Let me do 7 minus 4. So I could view this as, maybe I have a 7 foot long piece of wood. It's 7 feet long. If I put a ruler up against it I would have 0, 1, 2, 3, 4, 5, 6, 7. So I have a 7 foot long piece of wood. And then I could saw off 4 of those feet. So if I were to saw off 4 of these feet-- so I saw off 1, 2, 3, 4. How much wood do I have left? So all of this stuff right here, I'm eliminating. I'm sawing it off. I'm sawing it off of the wood. Maybe I should do that in a darker color to show that I'm sawing it off. So all of this stuff is going to disappear. I'm grinding it away. I'm sawing that off. So I'm just left with-- after I saw the 4 inches or feet or whatever of the wood, I'm left with 1, 2, 3 inches of wood. So this is 3. So 7 minus 4 is equal to 3. This is viewing subtraction as literally taking away. I sawed off the wood, so I took away wood. Now I could think of it in a slightly different way of thinking about it, but give you the exact same answer. We could say 7 minus 4. So once again, I could have the 7 inch long piece of wood like that. So if I put a ruler here that's 1, 2, 3, 4, 5, 6, 7. So once again, a 7 inch long piece of wood. And now instead of taking 4 away of it I'm comparing it-- so that's a 7-- I'm comparing it to a 4 inch long piece of wood. So I have another 4 inch long piece of wood right there. That's my 4 inch long piece of wood. that's 7, this is 4. You could view 7 minus 4 as taking 4 inches away from the long piece of wood. Or you could view seven minus 4 as the difference between the 4 inch piece of wood and the 7 inch piece of wood. So in this case, what's the difference? To go from the 4 inch piece of wood to the 7 inch piece of wood I would have to grow by 3 inches, or I would have to add a 3 inch piece of wood somehow. Or the wood would somehow have to grow by 3 inches in order to become 7 inches. So these are 2 completely equivalent ways to view subtraction. That's all a little bit of a review from the last video. Now what I also want to do in this video is start tackling slightly larger problems. But you'll see that really, the number line applies just equally as well as to kind of the simpler problems that we've done before. Let's do 17 minus 9. So just like everything else, there's two ways we could've done it. You know, the more slow way is you could draw 17 objects. Let's say I have 17 chips. 1, 2, 3, 4, 5, 6, 7, 8, 9 10, 11, 12, 13, 14, 15, 16, 17. And I'm going to take away 9 of them. So I'm going to take away 1, 2, 3, 4, 5, 6, 7, 8, 9. How many am I left with? I'm left with 1, 2, 3, 4 5, 6, 7, 8. So 17 minus 9 is equal to 8. But that took a long time and you could imagine, if this number was a lot bigger it would've taken me forever to draw all of these circles and then scratch out things. And it would've wasted paper and time. And we have other things to do. So another way you could do it, and maybe this would be easier for you to visualize, is to draw the number line. You always don't have to start at 0. So if we draw the number line, if we say that's 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7-- you could imagine, I could keep going to the left all the way to 0. But I start at 17. I could start at 17 and take away 9 from it. So I go 1, 2, 3, 4, 5, 6, 7, 8, 9. And once again, we are left at 8. Now this was, at least in my head, a little bit cleaner and faster than this one. But in either case, you don't want to do this every time you have to subtract 9 from 17 or want to find the difference between 17 and 9. And then to realize that's 8. So this is something that eventually you want to internalize. You'll want to know by heart that, oh, 17 minus 9? I know that is 8. And by the way, 17 minus 8? What's 17 minus 8? Well, that is 9. And now why does all of this make sense? Because 8 plus 9 is equal to 17. So 17 minus 9 is 8. Or 17 minus 8 is 9. When I say 17 minus 8, I'm essentially saying that is equal to some number that if I were to add to 8 will equal 17. Well, that's 9. When I say 17 minus 9 that's saying, there's some number, that if I were to add it to 9, I'll get 17. And that's 8. So all of these, all of these statements, are kind of saying the same thing. That 8 plus 9 are 17. Or the difference between 17 and 9 is 8. Or the difference between 17 and 8 is 9. Hopefully I'm not confusing you. So for most of these subtraction problems where the answer is a one-digit answer, you should eventually have them memorized, but in your head it's good to be imagining this number line. Let's do a couple more of these. And then, once we have these memorized or at least be able to do a number line if we forget, I'll show you have to do any subtraction problem, arbitrarily for super large numbers. So let's say we're going to do 13 minus 5. So once again, I'm not going to do the whole circles or the berries this time. I'm just going to draw the number line. Just draw the number line like that. Let's start at 14, 13, 12, 11, 10, 9, 8, 7, 6, 5-- and you just can keep going lower and lower. You can go to 0 or you can even go past 0. We'll talk about that in the future. But we start at 13. We're starting at 13. And we're going to take 5 away from it. So this is the subtraction view of subtraction; we're taking away. 1, 2, 3, 4, 5 and we land at 8. So 13 minus 5-- let me do this in a new color. 13 minus 5 is equal to 8. Now another way we could have thought about that, I plotted where 13 is. I can plot where 5 is. I could say look, this is 5. 5 is right here on my number line. What do I have to add to 5 to get to 13? So let's see. I would have to go 1, 2, 3, 4, 5, 6, 7, 8. I have to add 8 to 5 to get to 13. 5 plus 8 is equal to 13. So that tells me that 13 minus 5 is equal to 8. This also tells me that 13 minus 8 is equal to 5. All of these, are on some level, telling me the exact same thing. But the difference between 13 and 5 is 8. The difference between 13 and 8 is 5. 5 plus 8 is 13. So hopefully you have the hang of that and if you haven't done so already, it'll be good to practice all of these. Taking a teen number and then subtracting any of the one-digit numbers from those teen numbers. That's in general, very, very good practice for you.