I can’t believe I didn’t realize you could think of subtracting negatives like this

A week ago, I thought I knew all the ways that kids could and would think about subtracting negatives. Then this happened and my world is a brighter place.

Today in class I asked kids to solve these in their heads, and to share how they thought about them:

The big question in my mind was, do kids make sense of (-5) – (-1) as taking one negative away from five negatives? In short, the answer was “yes.”

As I often do when I want to talk about strategies, I asked kids to shout out the numerical answer they got to (-5) – (-1) right away so it would be out of the way. Then, I asked for strategies.

The first strategy shared was the “two negatives make a positive” and variations on that. Then I called on J:

If 5 – 1 = 4, then you just flip the signs to get (-5) – (-1) = -4.

OK, great! Why? J struggled to articulate the reasoning. N stepped up to the plate, and put it plainly:

It’s like taking 1 negative away from 5 negatives.

L asked for clarification. N offered it. L, who hadn’t had a way to make sense of this, had a moment. I asked her to revise her quiz later in class, and she had no issue applying this reasoning to the relevant subtraction problems on the quiz.

Great!

Every new strategy delimits a problem type, because every strategy works for some problems and not others. The type of arithmetic problems that the above strategy (let’s call it “Taking Like from Like” or “Whole Number Analogy”) helps with is fairly specific:

• It works for (-5) – (-1) = (-4), but it doesn’t quite work well for (-1) – (-5) = (-4). You can’t really think of that as taking 5 negatives away from 1 negative without running into some issues.
• Of course, the model can be stretched to encompass (-1) – (-5), just as all models can. You just say you get -4 negatives. I wonder if my kids will find that useful? I don’t expect that it will, but I would love to be surprised!
• So the problem type this “Taking Like From Like” works for is subtracting a negative from a negative where the first term is larger than the second. It’s not really a useful interpretation for 5 – (-1) or (-1) – 5.

If we’re wondering what contexts could help develop this “Taking Like from Like” reasoning, they would have to be contexts in which this strategy is easier to come up with. Then the plan would be to draw on experiences with these contextualized problems in making sense of the formal arithmetic.

So, what sorts of contexts could support this? Elevation seems useless here, since the elevations aren’t really objects in the sense needed to “take away a negative from a negative.” For the same reason, temperature seems useless here. Money might be helpful here, since if you’re adding debt with debt (and savings with savings) you’re essentially adding/subtracting like from like. Maybe a problem such as “You owe 5 dollars and then someone takes $2 of IOUs away. How much do you owe?” would do the trick? Particles/charge problems, I think, are the most promising here. Both because the context unambiguously involves negatives and also because it’s easy to represent taking those charges away.

In sum: “Taking Like From Like” is a powerful strategy; it helps for (-a) – (-b) where |a| > |b|; particle/charge problems (and maybe debt/savings problems) can help support the development of this strategy by furnishing students with contexts they can later turn into metaphors and interpretations.