[jacobajit]

The usual problem with this pedagogy is that kids get confused when presented with improper fractions (eg. 11/8), and so on. I guess this is probably still a good intuition to start with, but how did you tackle those extensions later on?

[DoreenMichele]

I don't recall that coming up as an issue for him. I don't see why that would be a problem. You can just explain that 8/8 is one whole, leaving 3/8. I don't see why that would be hard to demonstrate.

"These are two pies. Let me cut them into eight pieces each. Let's count to eleven pieces. That's one whole pie plus 3 parts of the second pie, so it's the same as 1 3/8."

[OJFord]

"but we cut 16 slices this time so isn't it 11/16"

[dvfjsdhgfv]

Three additional pizza slices automatically appear in my mind when you mention this. I found it more difficult to explain why the multiplication of two negative numbers gives a positive one.

[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%.

[relaxing]

First, accept that a negative is the inverse, or opposite of something.

We can intuitively understand that multiplying slices by a negative inverts them - turns them from slices given into slices taken away. (Distribute the multiplication into addition if necessary to drive the point home.)

So it follows that multiplying negative slices by a negative inverts them again, turning them back into slices given.

It’s not a convenient visual, but if a negative slice represents the “taking away” of a slice, then a negative negative takes away the “taking away.”