[OJFord]

Hm, that's a good one, I'm struggling to explain it to myself now!

My first thought is area, if you imagine four quadrants, draw a rectangle with side lengths positive from the origin, it's top right and multiplying the lengths gives you the area. If you instead take both sides negative then it's bottom left, but the area is the same.

However.. it's also the same if only one side extends negatively, so this is not at all satisfying.

(If that's made it even more confusing, the misleading error there is that side lengths are multiplied to give area, not positions on an axis relative to origin - the rectangle centred at the origin also has the same area.)

[IntrepidWorm]

It sometimes gets easier if you first define a negative number as an "opposite positive." You can then engage in a discussion of what the opposite of an opposite is. Ultimately though, finding multiple approaches for the teaching the same concept is going to serve you better than locking in one specific analogy.

There's no one right way to teach math (or anything, really) because teaching is a dynamic relationship between teacher and student that is highly contextual and subjective. That, I think, is one of the underlying issues with standardization in schools. Something that works for 80% of kids will unfortunately fail that remaining 20%.