Adding and Subtracting Positive and Negative Numbers

I asked Mary Cleare to take a look at a draft of the article and she wrote back reminding me of a simple and elegant approach we once discussed. She is happy to recommend this alternative approach - it's what she has used ever since:

"I believe that adding and subtracting with negative numbers makes sense.

I have a big number line ($^-10$ to $10$, say) above or along the top of my whiteboard. With the students, we brainstorm on things that are POSITIVE and things that are NEGATIVE. We talk about how you feel if someone gives you a positive thing, or if someone takes one away. We talk about how you feel if someone gives you a negative thing, or if someone takes one away.

I feel OK today, maybe I score $2$ (pointing to number line) on this happiness scale.
How would I feel if someone gave me $4$ chocolates (a generic positive!)? Yes, I move up $4$ to $6$.
Now how would it be if someone gave me a detention (negative)? Yes, down $1$, to $5$.
What if you took away $7$ of my chocolates? How would I feel? Sadder? Yes, I need to go down $7$, to $^-2$.
What if you gave me $3$ detentions? Etc.

At some point, I usually get all the students pointing the direction I should be moving along the scale, so it's easy to see who hasn't got the idea yet. Once the class are getting confident, I usually start recording some of the calculations on the board, or getting a student to do it for me! I usually let them suggest moves that will take my happiness off the scale that I happen to have on my number line.

As a finale before I ask them to do lots of standard + and - questions, we make up a problem like:
$6 - 7 + (^-2) -1 - (^-4) + 9 + (-3) -1 - (^-7) - 4 - (^-8) - 1 + (^-2) + 8 =$ ? to do together.

I believe that when multiplying and dividing with negative numbers some of the calculations don't make normal sense, they only make mathematical sense -ie they're about making sure patterns continue, about preserving the distributive law."