The last time I effectively taught adding & subtracting with negative numbers I was on to a good idea, but there were a couple of faults – only the kids actually walking the number line were engaged, and they lost their positional cues as soon as they went back to their desk.
So this time, every kid gets a go:
Get a bucket of Lakeshore kid counters, and print out a bunch of number lines1. Have a talk about the difference between positive/negative and adding/subtracting. (i.e one thing is a property of a number, the other is something you do between two numbers). Once they kind of get it (it doesn’t need to be perfect, they just kind of need to know that there is a difference), go through a couple of example expressions just having them identify whether the expression involves addition or subtraction, and whether each term is positive or negative. Once they get pretty good, have them them fill in the hint blanks with walking forward/backward2 for positive/negative, and facing right/left for adding/subtracting.
Now comes the easy part. They find the first number on the number line, decide which way their kid should face, and whether he should go forwards or backwards. By carefully choosing mirroring problems, they can see both the difference and the similarity between swapping both operation and sign. Eventually sneak in problems that drift off of the number line, and let them start proposing rules for how to solve those problems.
2 A couple of kids in my third period were fixated on forward=right, backward=left, regardless of which way their kid was facing. With them I had better luck using walk/moonwalk. Feel free to demonstrate as your ability allows.