Subtracting Integers Using Algebra Tiles

So, in this video we are going to some samples of some questions we can ask students to do with subtracting integers. And we'll talk a little bit about how that helps them develop the rule on their own. So, let's get started with some examples.

So, we're going to start of with a very simple subtraction problem. We're gonna start with positive eight minus positive three. And we know students already know the answer to this, but what we're going to do with this example is to show what the idea of subtraction actually means, and how to use the materials with this. So, when we have positive eight minus positive three, we always start with the first number, representing that with the algebra tiles. So here we have positive eight. Now, instead of representing positive three, what we're going to do is we're going to take that many away from the group. So, we're gonna take away three positives and just move them to the side for now. Our final answer is going to be whatever we have remaining here. So, in this case it's going to be positive five. Just what we would expect.

Here's another example that we can do. Negative nine minus negative three. Now, what's interesting with this example is that usually we don't subtract negatives until much later on. But with the materials, this is going to make a lot of sense. So, just like when we had a positive minus a positive, we're going to start with our first number. In this case, it's negative nine. And now we're going to take away, or remove, three negatives, or negative three. So, we put those to the side and what do we have left? Negative six.
Here we have positive six minus negative three. So, we'll start off in the same way we did for the other subtraction ones. We start off by making our positive six. And now we have to subtract away negative three. In other words, take away three negatives. But we don't have any negatives here to take away. But that isn't a problem if students remember the very first lesson we did with integers. We were able to represent integers in many different ways. So, this is one way to represent positive six. But if we remember that we can add a zero pair and not change the amount then this is still positive six. And indeed, we do have a negative here. 

So, I could take that one negative away if I wanted to. But that's not gonna do us enough good. We need to take away three negatives. So, if we make three zero pair, this number that we see is still positive six. And the students should be used to seeing this and reading this as positive six. Now, we can take away those three negatives and we can see what we have left. Positive nine. So, positive six, take away three negatives, or negative three equals positive nine.

Here's another example. Negative three minus negative five. And, so we'll start with the negative three, and now to take away five of the negatives, students will do this in one of two ways usually. And both ways are very good. The first way that they?ll do it is they'll say well I need to take away five negatives and I only have three, and so what they'll do is put on two more zero pairs. They'll then take those five negatives and they'll see the answer is positive two. So that's the first way they might do this. The second way is once they have their negative three here, they may just look at that negative five and say oh I need to put five pairs of zero out here. So, there's my one zero pair, two, three, four, and five. And then they'll just take away these five negatives and then they'll see what they have left, and then they?ll make the zero pairs, they have three zero pairs here to take away. And They'll left with positive two once again.

The first way is obviously much quicker, but the second way actually gets to the heart of what they'll be doing when we're talking about adding the opposite. And we'll see that in a little bit.
As students work through more and more examples on their own, they're going to see that the same thing is happening over and over again. That what they're doing is they're adding on the number of zero pairs to whatever is what they want to subtract. Once they subtract away what they want, they end up combining together their original amount plus the opposite of whatever they took away. So, they're going to see that every time they're doing a subtraction problem, they're really adding the opposite. And as the students work more and more, they're gonna be getting faster and faster with that. And some of them may come up with that rule on their own, and others when we see them working the right way, we can help them along.
So, let's look at an example of how a student might make this transition from doing it the long way to shortcutting to this rule that they have discovered. When students first do positive seven minus negative two, they?ll probably put the positive seven out and then they?ll try to make the zero pairs. 

They know they need two negatives, so they'll put out two pairs. And they give up all the positive and then they?ll subtract the two negatives giving them positive nine. But as students get faster and faster with the work, they're going to see what's happening over and over again. Instead of putting out the whole zero pairs, instead of putting out the entire zero pairs, they're just gonna take these negative anyway. So rather that put them out and take them back, they?ll just put the two positives out right away. And that's when we can make the connection that taking away these two negatives is really the same as adding two positives. Which is what we saw in the addition work actually. That taking away is the same thing as adding the opposite.

So, what we can do is we can revisit those original questions again, but with the idea of subtracting as adding the opposite in mind. For those students who have kind of discovered that on their own, this is a good way to reinforce that. For student who are still working a lot and kind of struggling to see this idea, then we can revisit this lesson again but ask the question in a new way.
So, when we revisit positive eight minus positive three, we can put out those positive eight like we did before and say last time we actually had to physically remove these three. But is there another way to make those three go away? And the answer is yes we could add three negatives. And by adding three negatives that's gonna have the same effect because those are gonna cancel with three of the positives making those zero pairs and what we have left is our positive five. So, when we add these negative threes, that has the same effect as subtracting away. And we can see that very clear with the materials.

So, if we were to revisit negative nine minus negative three, what we did before was we put out nine negatives and then we took three of the negatives away. But is there another way to get rid of these three negatives? Yes, we can add three positives on. And by making those zero pairs, that in essence takes away those negatives and the positives go away anyway with the zero pair. So, again, I can get the same result instead of actually removing these three negatives by adding these three positives.

So, throughout all the examples students have been doing, they?re seeing this link between subtracting and adding the opposite. And that's very powerful. We don't have to actually give them that rule. They'll come up with it on their own through the work. And if students are coming up with it on their own, then we can give them more examples, ask them questions to help them get there on their own. And that's very powerful for the students to be able to do that. If they ever forget the rule, they can draw up pictures.

If they're in the classroom, they can go grab the materials and recreate that experience for themselves once again. And every time they do that, it will come back faster and faster until the students truly have mastered subtracting integers.
So, that's it for this lesson. I hope that you enjoyed it. And next time we could do some multiplication with integers. Thank you very much.