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Subtracting with integer chips

Let's model subtraction expressions using positive and negative integer chips. To subtract, we'll take away the appropriate number of chips. What if we don't have enough chips to take away? We can add as many zero pairs as we need, because they don't change the value of the expression. Created by Sal Khan.

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Video transcript

- [Instructor] Let's say that we wanna figure out what negative eight minus negative two is. Now there's a lot of ways to approach this, but what we're going to focus on in this video is to really build the intuition, and we're going to do that with something called number chips which you might have seen before. So the first thing is, I'm going to represent negative eight with number chips. Well number chips, I can either have a negative number chip like that or I could have a positive number chip like that, and if you have one negative and one positive, if you were to combine these two, they could essentially cancel out to just zero because they're opposites of each other, and we've looked at that in other videos, but let's go to actually representing this up here. So negative eight could be represented as eight of these negative number chips. So let me do that over here. So that's one, two, three, four, five, six, seven, and eight. Now if we're subtracting a negative number, this negative number right over here, you could view this as two negative number chips, but we're going to be taking those away because we have a subtraction right over here. So if we start with negative eight, if we wanna subtract negative two, we can just take away two of these negative number chips, so let's just do that. And then how many negative number chips are we left with? We have one, two, three, four, five, six. So this is going to be, is that going to be six or negative six? Well these are negative number chips, so this is going to be equal to negative six. Now let's up it a little bit and let's start with something that might seem a little bit intuitive, and once again, there's many ways to approach this. Maybe the number chip technique will resonate with you for building a little bit of an intuition. Let's say we wanna figure out four minus seven. What is that going to be equal to? Well four is a positive number, so I think I'm gonna use some positive number chips to represent four. I could represent that as one, two, three, and four positive number chips. Now, if I'm taking away seven, I could say hey, I'm gonna, and I'm taking away a positive seven here, so that would be equivalent to taking away seven positive number chips, but I don't have seven positive number chips here, so what could I do? Well, I talked before, if I have one positive and one negative, they cancel out, so I can add these pairs of positive and negative number chips until I get to seven so that I can then take them away. So let me do that. So I'm gonna add a positive and I'm going to add a negative. Let me make sure I'm doing that same color. So this, once again, this is still four, because these two right over here, these cancel out. These are just zero, so you're just left with positive four. So I can do that, frankly, as many times as I want, and I still have four. So let me do that again. This is still equal to four, because once again, each of these pairs cancel out to just a zero. And let me do that again. And once again, my goal is to have seven positive number chips 'cause I need to take away a positive seven, so let me put the negative right over here. So this is all, right over here, equivalent to four. It's equivalent to what we had before, but by adding these pairs, I now have seven positive number chips. Now, if I'm taking away a positive seven, I can take away seven, let me do this in another color, I can take away seven positive number chips, and I have seven now, so I'm gonna take away one, two, three, four, five, six, seven. Now, what am I left with? Well all the positive number chips are gone, and all I have left are three negative number chips, so this is going to be negative three. Three negative number chips is the same thing as negative three. Let's do another example. Let's say that I wanna do positive five minus negative one. Why don't you pause this video and see if you can do this with number chips. All right? Well let's first represent that positive five with five positive number chips, so that's one, two, three, four, and then five. Now, we're not taking away a positive number. If we were taking away a positive number, we could just take away some of these positive number chips. We're taking away a negative number, so we need to have negative number chips in order to take 'em away. So, and we're only taking away one negative one, or we're taking away one negative number chip, I should say, so let's add one of those pairs. Remember, we can keep adding pairs 'cause it doesn't change the value because they cancel out. So let's add a pair. This is one positive number chip, and then let's put a negative number chip right over there. This quantity that's represented with number chips is still five because these last two cancel out, and you're just left with one, two, three, four, five. The whole reason why I added this pair here is now I can take away a negative one. That negative one is represented by this negative number chip right over here, so if I take that away, this taking away negative one, that's taking away a negative number chip. And now what am I left with? Well I'm now left with one, two, three, four, five, six positive number chips, or that's just a positive six.