[pingucrimson]

We used Z a lot when dealing with modular arithmetic and complex roots of unity, mostly just to quantify our variables. I can't recall ever using N or Q in high school, though.

Also, you don't need to mention Dedekind cuts at all when dealing with R - it can be defined by the fact that it's the smallest extension of Q that's closed under limit-taking (and I think most high school math students do understand that).