Subtracting with integer chips | Integers: Addition and subtraction | 7th grade | Khan Academy

Let's say that we want to figure out what negative 8 minus negative 2 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 8 with number chips. Well, number chips can either be a negative number chip like that, or I could have a positive number chip like that. 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 are opposites of each other. We've looked at that in other videos.

But let's go to actually representing this up here. So, negative 8 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 could be viewed as 2 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 8, if we want to subtract negative 2, 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 6 or negative 6? Well, these are negative number chips, so this is going to be equal to negative 6.

Now, let's up it a little bit and let's start with something that might seem a little bit intuitive. 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 want to figure out 4 minus 7. What is that going to be equal to? Well, 4 is a positive number, so I think I'm going to use some positive number chips to represent four. I could represent that as 1, 2, 3, and 4 positive number chips.

Now, if I'm taking away 7, I could say, "Hey, I'm taking away a positive 7 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. I'm going to add a positive and I'm going to add a negative.

Let me make sure I'm doing that the same color. So this, once again, is still four because these two right over here are cancelling out—these are just zeros. So you're just left with positive four. 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 cancels out to just zero. And let me do that again. And once again, my goal is to have seven positive number chips because 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 7, I can take away seven. Let me just use another color; I can take away seven positive number chips. And I have seven now, so I'm going to 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 want to do positive 5 minus negative 1. 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 them away. 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 because it doesn't change the value since they cancel out. So let's add a pair here. 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.