Many of my students came into 6th grade confident that “two negatives make a positive”. Some thought this applied to everything from -4 + -5 to -3 - (-4) to multiplication and division. They had learned tricks but they were combining the fact that -x * -y = x*y and x - (-y) = x + y and lumping it all into “two negatives make a positive” for any and all operations. Obviously, I wanted them to understand the why behind these tricks so they could apply them appropriately. But figuring out the best models to develop that understanding was trickier than expected.
The first year I taught 6th grade, I taught adding and subtracting integers with counters. For subtracting, we added however many zero pairs were needed in order to “take away” the subtrahend. Most students got it but the next day and days after that they were getting mixed up with all the different rules to remember with seemingly little understanding.
The next year after reading this paper1, I decided not to use counters. Instead I used number lines for adding and subtracting to more clearly make the connection between adding and subtracting as inverse operations. It was still a lot for my students to remember what to do and when. Also, number lines were a pretty abstract thing for them to understand out of context.
The year after that, I found an analogy to go with the number lines that made everything click for my students. It’s silly but it works wonders.
The Analogy that Works
First, draw a vertical number line that goes from 10 (happy) to -10 (sad).
For Addition
Make up an imaginary person and their life on this happy-sad scale is whatever the first addend is. Then the second addend is either good or bad things that are being added to this person’s life. Brainstorm some good and bad things that could be added to someone’s life and as ridiculous as it is, and agree to treat all of them as 1 unit. Ask students if adding these things will make their life better or worse and to draw it as an arrow starting at their current state on the number line.
Here’s some examples:
On a scale of happy to sad this person’s life right now is at a 3. It’s just meh. Then you add 2 positive things to their life. Would their life get better or worse?
Now let’s say on the same scale, someone’s life is at a -5. They’re pretty depressed. But then you add 4 positive things to their life. Does it get better or worse?
For Subtraction
Make up an imaginary person and their life on a happy-sad scale is whatever the minuend is. Then the subtrahend is either good or bad things that are being taken away from this person’s life. Ask students if taking these things away will make their life better or worse and to draw it on the number line.
What if someone’s life is at a 4 and then you subtract 5 positive things from their life. Would it get better or worse?
What if someone’s life is at a 2 and you subtract 3 negative things from their life. Would it get better or worse?
What if someone’s life is at a -5 and you subtract 2 negative things from their life. Would it get better or worse?
The great thing about this is, it works perfectly for any addition or subtraction problem. It helps students understand WHY subtracting a positive has the same effect as adding a negative and subtracting a negative has the same effect as adding a positive. And students really only have to remember this one analogy instead of different rules for each scenario. I still have to remind them about it throughout the year, but for every student I’ve had who was confused about subtracting integers, this helps them get it immediately.
1 - I can’t find a link directly to the paper but here’s the citation:
Ulrich, C. (2012). The addition and subtraction of signed quantities. Quantitative reasoning and mathematical modeling: A driver for STEM integrated education and teaching in context, 2, 75-92.