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:

______________________________________________

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

------------------------------------------------------------

And there you have it. Two negatives make a positive.

He now believes it and is properly applying it.

That's a nice proof. My kids have had no problem with the concept thanks to *Magic: The Gathering* card game. If your creature has a -2 counter on it and you take that counter off, obviously it's now 2 points more powerful than it was before. :D

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