"Man, I didn't do nothing," students often said to me when I spoke with them about their behavior. "My point exactly," I replied. "If you didn't do nothing, you must have done something."

It was after one of those exchanges when it occurred to me that the English language might provide a better way of helping kids add and subtract integers (really all real numbers) than the many approaches--from rules to manipulatives to real-life illustrations such as temperature and money--I had tried up to that point.

I started by asking students, "If Fluffy is not not a dog, what is she?" And after a few minutes of kids guessing every animal but dog, a boy named Derrick said, "Wait, she's a dog." Other students gave him a funny look and said, "Can't you read? It says she's NOT a dog." But Derrick barked right back: "Can't you read? It says 'Fluffy is NOT not a dog,' which means she is a dog."

This exercise not only helped students finally get why subtracting a negative is the same as adding a positive, but also inspired an approach to adding and subtracting positive and negative numbers that's more effective than anything else I've tried or seen. What distinguishes this approach--which my students and I called "The System"--is that it involves combining numbers rather than adding or subtracting them, as illustrated here:

If you've read my prior posts where I excused Aunt Sally and crumpled FOIL, you know I'm no fan of gimmicks that undermine mathematics and can cause misconceptions. But that's not the case with "The System," which improves students' computation and comprehension. It's also a good lead-in to combining like terms in Algebra.

So if your students have trouble adding and subtracting positive and negative numbers, you definitely won't want to not teach them "The System."

Image provided by GECC, LLC with permission

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