Why is subtracting a negative number the same thing as adding it?

I've always been a fan of the number line. So easy, right? To add, you go to the right. To subtract, you go to the left. 

But then we got this problem in our problem set:

 Explain, using both a chip model and a number line model, how it is possible to have a positive difference when subtracting two negative numbers—e.g., -2 – (-5).

  

If I solve -2 - (-5) using the number line rule above--i.e., subtraction moves to the left--I get the wrong answer:

Oh, dear.

It's only because I secretly know that subtracting a negative is the same as adding a positive that I was able to get this problem right: -2 - (-5) is the same as -2 + 5. And then the number line works:

That's better.

But why? Why doesn't it work to subtract -5 just by moving to the left on the number line? I guess the rules are different for negative numbers? 

As usual, I started with the good people at the Khan Academy. There's no obvious section for "why can't I just move to the left when I subtract -7?", so I tooled around a little until I found this in the Arithmetic Properties section: Apparently there's something called the Inverse Property of Addition? Sounds promising.

Hmmmmm. . . . 

So the inverse property of addition just says if you add the negative version of a number to itself, you get 0:

5 + (-5) = 0

17,462 + (-17,642) = 0

And as the video above shows, to get to 0 from 5 I have to move 5 spaces to the left:

Got it.

Plus I already know from my number line rules that to subtract I go to the left and to add I go to the right. I moved 5 spaces to the left to get to zero--which means I subtracted 5. 

So if

5 + (-5) = 0

and

5 - 5 = 0,

then

5 + (-5) = 0 is the same as 5 - 5 = 0. And that means adding a negative number is the same thing as subtracting it!

So then. . . . Might the inverse also be true? That is, is subtracting a negative number the same thing as adding it? 

"It's a long shot, but it just might work."

 Let's see how this might play out:

I secretly already know that -2 - (-5) = 3, and that on the number line it looks like this:

Oh, right, right.

 And what did I do to get to 3 from -2? Look at that--I moved 5 spaces to the right . . . which means (say it with me) I added 5!

So if

-2 - (-5) = 3

and

-2 + 5 = 3,

then

-2 - (-5) = 3 is the same as -2 + 5 = 3. And that means subtracting a negative number is the same thing as adding it!

BAM!

So it appears that the sacred rules of the number line,

add to the right

subtract to the left,

Still work with negative numbers, as long as we remember the following:

Subtracting a negative number is the same as adding it.

Adding a negative number is the same as subtracting it.

As long as we keep those rules in mind, we're good! 

Still seems cumbersome to teach a second grader, though. But I'll work on it. . . .