Subtracting Negative Numbers

Dd10 and dd7 are starting to learn about negative numbers. Not because we've reached them in their math books. (Although, actually we have. Many of the "temperature" problems they do in their math worktexts require them to find differences between positive and negative numbers. It's sneaky, BJU Press, but I love it!) They overhear me talking about negative numbers during tutoring sessions with the teenage boy I tutor.

On Friday, I spent nearly an hour trying to help him see and internalize why subtracting a negative number is the same thing as adding the absolute value of that number. No matter how I approached it, he seemed to view it as some kind of mathematical black magic and not based on reason or reality. There are many ways of explaining why 3-(-2)=5 (this blog post has a few good ones), but nothing seemed to convince him. This is a big problem because he is currently working on line equations at school and has to be able to calculate the slope of a line when given two points on the line. It's difficult to correctly calculate "rise over run" if you can't properly find the differences between x- and y-coordinates that aren't all positive.

The last explanation I tried seemed to work. He is comfortable with the definition of zero and with the algebraic rule "If a = b, then a + c = b + c." So I showed him a brief version of this proof:
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x - x = 0                    (0 is always what we get if
                                 we subtract a number from itself)

(-x) - (-x) = 0            (ditto above)

                       Now add x to both sides of the second
                       equation, which we can do because of
                       the rule "if a = b, then a + c = b + c."

(-x) - (-x) + x = 0 + x

Which, because of the commutative property of addition (order of addition doesn't matter), is the same as...

x + (-x) - (-x) = 0 + x

Which simplifies to...

x - x - (-x) = 0 + x

Which simplifies to...

0 - (-x) = 0 + x

Which is the same as...

- (-x) = x
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And there you have it. Two negatives make a positive.

He now believes it and is properly applying it.