## Saturday, May 14, 2016

### An Alternative to "Add the Opposite"

I've always been a little bothered by how textbooks (and presumably, teachers) explain subtracting integers on a number line. Here's an excerpt from a recent Pearson textbook which has been aligned to the Common Core:

From this, we see that 9 - 5 = 9 + (-5), and from that we conclude that we can always subtract numbers by adding the additive inverse. This makes sense, but what about subtracting a negative? We're just supposed to accept that it is the same as adding a positive? Or what if we are subtracting negatives from a positive? How do you take something away when it's not even there? (I know...zero pairs.)

So how do you explain this without simply telling students to "add the opposite"? Wouldn't it be better if students were comfortable with subtracting negatives?

I teach adding and subtracting integers by having students locate the first number on the number line. You then have two options...you're either going left or right. To do this, they look at the operation. If they see +, they think that they need more of something. If they see -, they think that they need less of something. If we see plus a positive, we need to go in a more positive direction (right). If we see plus a negative, we need to go in a more negative direction (left). For minus a positive, we go less positive (left). For minus a negative, we go less negative (right). And that's it. It makes sense to them and we don't have to be afraid of the subtraction sign.

From here, students use number lines to solve addition and subtraction problems, and eventually, they start to make their own connections. They see that subtracting a negative has the same effect as adding a positive. They see that subtracting a positive has the same effect as adding a negative. As we work with larger numbers, students become less reliant on the number line and use their intuition.

One of the best things about teaching this way is that some of my struggling students can always fall back on the number line. Don't get me wrong, it can be painful to watch a student solve -27-1 by extending a number line far out to the left. I let them do it and then ask them to try a similar problem without writing anything down. Over time, they learn to trust themselves and do it mentally.

Another nice thing about teaching this way is that you can easily extend these ideas to multiplying integers. Positive times negative means more negative. Negative times negative means less negative. You can show how this works with repeated addition/subtraction: -3(-4) = -(-4) - (-4) - (-4).

I hope this provides you with a better alternative than the standard textbook explanation. If you try this, please leave a comment below on any insights that you have.

1. This reminds me of an activity where I want to go outside and spray paint some number lines and have students act it out. Let's take 2 examples, 2 - 3, 2 + (-3), and 2-(-3) since they all start at 22.

For the first one start at 2. If your subtracting face the negative numbers. Move 3 steps in that direction. You're at -1.

For 2+(-3). Start at 2. Face the positive direction since your adding. Take 3 steps backwards.

For 2-(-3). Start at 2. Face the negative direction. Walk 3 steps backwards. You are at 5.

I don't know if that is just rephrasing what you just said, or making it more clear, or muddling it, but that's how I look at it.

Although, in the long run, I think the plus minus tiles make more sense.

2. There's also the option to have students think of subtraction as "the distance between" using a number line. Using all positive numbers, they might see it as a familiar counting up strategy: 9-5 is the distance from 5 to 9, which is 4. Throw negative numbers in the mix, like 9-(-5), and now you have to find the distance from -5 to 9, and hopefully students use zero usefully in this strategy. As usual, though, some students might struggle when the subtrahend is greater than the minuend, such as 5 - 9. It's still the distance from the second number to the first, but the direction matters, and so there's still a need to remember a negative when moving right to left.

3. This post has really got me thinking... and that's good. You're using simple and helpful language. Integers can be messy and I appreciate your approach.
As I read this, I was thinking how integers are not necessarily a common means of communication. How often do we order something and say, "I'll have a cheeseburger and take away two negative cheeseburgers for my friends."?
NEVER!
Every teacher I talk to or blog post I read about integers, it seems that everyone is trying to connect the language, symbols, concept, and representations used with integers. And that's important. However, it's really easy to not connect all four! One thing I wonder is how much of this disconnect between the language, the symbols, the concept, and the representation are screwing kids (and us) up.
In the classroom, I often found myself telling kids how I think of it, and now think that was dumb of me. Furthermore, I made the mistake of just starting with expressions that were already made, like -5-3= . I realize now that it would have been better to ask students how they think of it. Let me give you an example.
I wonder what language/thinking students would come up with if you placed them on a number line and asked them to get to another place on the number line. For example, place the student on -5 and ask them to get to -8. Listen to what they come up with and ask them to write a math sentence.
Would some say,
"I took 3 away from -5"
"-8 is 3 less than -5"
etc.
Okay, now write a math sentence that makes sense to you. Maybe we'd see:
-5-3=-8
-8=-5-3
-5+ (-3)=-8
Or place them on -4 and ask them to get to 0. Might a student actually say, "I took away 4 negatives."?
That would be awesome and I'd buy that kid a cheeseburger.
It took me way too long to realize that students can often come up with more logical thinking/language than me because their first instinct is to keep it informal or conversational. My role was better suited toward formalizing their thinking along their journey. Or in this case, once they share their thinking, you step in with your simple and awesome language for those that need "my teacher says that you..."
What do you think?

4. I really like the idea Andrew has about accessing the language that the students would use to explain it to themselves. More and more, I think my students already have understanding of most of what I'm teaching, and my job as a teacher is to figure out how to access that understanding and help them clean it up (and address the misunderstandings).

In the end, though, under the skin much of what is proposed is in fact "adding the opposite" in some form or another. It may be nice to tell them that after they work it out in their own language, for a couple of reasons. One, that is the mathematical definition of subtraction. While I often avoid introducing new topics in terms of formal and precise definitions, I think it's important that they know that those definitions exist, and they exist for good reason. Also, it can be helpful to practice translating between our internal explanations and the technical and precise language of a subject. Finally, it's extremely likely that they're going to see that precise definition again (because it's the way every textbook says it), so it's good to have laid the seed of that formal definition somewhere within the comfort of their own explanation.

Tangentially related, I have wondered about how we can make better connections between the two notions of subtraction: one, the movement along a number line, and two, the distance between two numbers on a number line. Because thinking about subtraction as the distance between two numbers makes much of the confusion about subtracting negatives go away. But that causes new confusions, for example distinguishing between 8–(-2) and -2–8. They're 10 units apart on the number line, but it requires addressing that subtraction is always a *signed* distance.

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