Subtracting integers should not be difficult for most if they make sense to them. In first grade, pupils learn that 100 – 92 means take away 92 from 100. The minus sign (-) means take away or subtract.

After two or three birthdays, pupils learn that 100 – 92 means the difference between 100 and 92. The minus sign (-) means *difference*. The lucky ones will have a teacher that would line up numbers on a number line to show that the difference is the distance between the two numbers.

After a couple of birthdays more, pupils learn that you can actually take away a bigger number from a smaller number. The result of these is a new set of numbers called negative numbers. That is,

*small number* – *big number* = *negative number*

The negative numbers are the *opposites* of the counting numbers they already know which turn out to have a second name, *positive*. The positive and the negative numbers can even be arranged neatly on a line with 0, which is neither a positive nor a negative number, between them. The farther left a negative number is from zero the smaller the number. Of course, the pupils already know that the farther right a positive number is from zero the bigger it is. It goes without saying that negative numbers are always lesser than positive numbers in value. This is easier said than understood. When I tried this out, it was not obvious for many of the learners I have to give examples of each by comparing the numbers and defining that as the number gets further to the left the lesser in value.

Now, what is 92 – 100 equal to? The difference between 92 and 100 is 8. But because we are taking away a bigger number from a smaller number, the result must be a negative number. That is **92 – 100 = -8**. Notice that the meaning of the sign, -, before 8 is different from that between 92 and 100.

What about -100 – 92? Because -100 is 100 units away from the left of 0 and 92 is 92 units away from the right of 0, the total distance or difference between them is 192. But because we are taking away a bigger number, 92, from a smaller number, -100, the answer must be negative (-). That is, **-100 – 92 = -192**.

And -100 – -92? Easy. Both are on the left of 0. The difference or distance between them is 2 but because -92 is bigger than -100, the answer should be a negative number. That is,** -100 – -92 = -8**.

We shouldn’t have a problem with 100 – -92. These numbers are 192 units apart and because we are taking away a small number from a bigger number, the answer must be positive. That had always been the case since first grade.

Who says we need rules for subtracting integers?

Click the links for other ideas for teaching integers with conceptual understanding

- A problem solving approach for introducing positive and negative numbers
- Assessing operations involving integers
- Algebraic thinking and subtracting integers – Part 1
- Algebraic thinking and subtracting integers – Part 2
- Subtracting integers using numberline – why it doesn’t help the learning

An interesting way to look at it. These are still rules, though, even if you can boil all of them down to “smaller number – bigger number = negative”. The only problem I have with this as a middle school math teacher is that I think it disregards the mathematical ideas going on behind why, for example, -100 – 92 = -192. Our goal should be for students to have a mental picture of -100 being way to the left of 0 on the number line, and then understand that by subtracting 92 more we are getting 92 units smaller, which would send us even further to the left. Subtracting negatives is always the trickiest of all, but the idea we teach with algebra tiles, that subtracting negatives = adding positives because you sometimes must add zero pairs to actually come up with your negatives to subtract, is an important concept that reinforces the idea that positives and negatives are opposites and cancel each other out. Our goal should be to build these understandings before teaching any rules, then allow students to use rules as a shortcut.

Thanks for sharing Samantha. I agree with you re about the zero pairs. They are indeed important ideas in mathematics. Smaller number – bigger numbers = negative is true all throughout whether it involves both negative numbers or not. It is not a rule but one of the ways that shows the existence of another set of numbers – the negatives. The only way learners will have difficulty accepting it is when we have been telling them that you can’t subtract a bigger number from a smaller number. They might think that there is no number that can express that difference.

I learned of a way of teaching subtraction by reading the subtraction sign as “in relation to”. So, 5 – (-8) would be read “5 in relation to -8”, which would give the answer “13 away”. I have never taught this way and I suppose there might be a signs issue that comes up: (the difference between (-13 – (-8) vs. -6 – (-8)), for example). Overall it seems like a good model.

Check out my video Subtracting integers – using three models. I include the idea of thinking about difference. But I feel the “number line jumps” OR the shortcuts are in many cases easier to work with. For example, the way I solve -100 – -92 is:

-100 – -92 = -100 + 92 because double minus reverts to plus

-100 + 92 is like a number line jump where you’re at -100 and jump 92 towards the zero. You won’t quite reach zero, therefore the answer is negative 8.

I did interactive number lines to show my students and they thrived with something they really did not think they would understand. The internet and sites like this give teachers solid ideas.

Cool! There’s one less class of kids I’ll have to re-teach negatives to in the 11th grade! Go YOU!!

Because my students have such a hard time with negative numbers (ie: solve for y in y + 25x = 3x + 7), I started thinking about what the problem was. I would get answers like “y = -28x + 7” or “y = 22x + 7” so it was obvious there was a lack of understanding of negatives.

For my thesis, I began looking into when negative numbers are taught- 7th grade! What?? That’s too late in my opinion. Then I began to look into HOW they are taught- with a number line. But at the very beginning of the first lesson in 7th grade, there is a picture of a boy with a caption above his head reading “I owe my dad $4. I have -$4”

So this idea of owing is tied directly into negatives. So I thought about owing someone some money, paying some back, and figuring out how much more I owed.

If I borrowed $12 and paid you back $7, the problem would look like “-12 + 7” but I would solve the problem, in my head, by counting from 7 to 12. This is not the way we are taught in school. The way we are taught in school is to “find -12 on the number line, count 7 to the right, see what number you land on.” But this isn’t what we do in real life!

Absolute value is the answer. Although “take the difference between the absolute values of the two numbers” is a bit of a mouthful, it is the way to go. This way both numbers, -12 and 7, are treated as real numbers instead of -12 being treated as a number and 7 being treated as a movement. I really think that if we teach kids this way they will begin to see the relationship between positives and negatives and no longer make mistakes when they get to me!

I just wanted to add a comment here to mention thanks for you very nice post.

Your 4th paragraph, 1st line, states that the difference between 92 and 100 is 2… I think you meant 8.

http://mathmaine.wordpress.com

I did. Thanks.

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