More on conceptualizing negative numbers

In my last post I explored how to add and subtract negative numbers on the number line. I got to a place where I think I can teach that to my students, but it still feels cumbersome--maybe because I can see on a number line that subtracting a negative is the same as adding it, but I still don't understand why.

As I was scrolling through the lessons with my good friends over at the Khan Academy, I found the Negative Numbers and Coordinate Plane section. Sounds promising. . . .

Oh my gosh they have a whole practice section on section on negative numbers on the number line! Let me click through that right quick . . .

Question 1

Question 2

Etc etc etc. . . . These are not what I need, although I do like how they do the number lines both horizontally and vertically which may give students the chance to conceptualize numerical order slightly differently.

Let's see what else they have. . . . A-HA! Here's an introductory video on negative numbers. This might be just my speed:

The most helpful explanation of the concept of negative numbers starts for me around 2:10: the higher the negative number, the smaller it is. He says it this way:

"Negative 100 is a lot less than negative 1."

He adds that we can think about this idea as a negative being a lack of something. So if I lose $100 from my bank account, I have way less money than if I lose $1 from my bank account. I might say it this way:

"How not-much of something do I have? If I have 100 not-muches of something, and you have 10 not-muches of something, who has less?" (Now that I see that all typed out, it could be even more confusing. But who knows? Maybe it unlocks something for someone. . . .)

I can also connect this concept to my last post about the number line: When I subtract, I go to the left--and when I go to the left, the numbers get smaller. So anytime a student is confused by this concept, she can simply draw a number line--all she has to do is memorize what the number line looks like, i.e., that the numbers immediately to the left of the 0 start at -1. So which number is smaller, -10 or -30? And she will see that the answer is -30, because she has to go to the left of -10 to get to it:

Gotcha.

I'm really starting to see how grasping the concept of negative numbers--especially adding and subtracting them!--can be very confusing for students. After all, I'm a student, and I'm confused myself!

Next time let's look in more depth at adding and subtracting negative numbers. Thanks for stopping by!

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. . . .