TITLE: Significant Proofs Understandable to Middle School Students with Pre-Algebra Background
QUESTION [15 upvotes]: I'm interested in introducing my middle school children to proofs using significant examples (i.e., not just basic geometry proofs in an intro to trig textbook). They have a pre-algebra background (so, polynomials, powers, greatest common factor, etc.). I can think of several proofs off the top of my head:
Proof of the irrationality of $\sqrt{2}$
Cantor's diagonalization proof
Pythagorean's rearrangement proof (and other proofs of the Pythagorean theorem).
What other significant proofs are there that would be explainable to someone with a pre-algebra background?

REPLY [1 votes]: In Ontario I have met many high-school graduates who are vague or confused about $\Bbb R,$ especially on the issue of infinitesimals, that is, whether a number can be positive but less than every member of $\Bbb Q+.$ I think this is because they were not taught properly. I'm sure they never heard of a Dedekind cut. I suggest you give an axiomatic def'n of $\Bbb R,$ and show from the def'n that (i) the Archimedean property, (ii) the non-existence of infinitesimals in $\Bbb R$, (iii) the order-density (in $\Bbb R$) of $\Bbb Q$ & of its complement, (iv) every non-empty subset of $\Bbb R$ with an upper (lower) bound has a lub (glb) and from this deduce that $\sqrt 2$ exists in $\Bbb R.$
I think it would be helpful to emphasize that these follow from the def'n of a certain structure ($\Bbb R$) but that we can also define larger structures (e.g. the hyper-reals), so we cannot derive properties of the "real" numbers without a definition. Without a def'n it's like trying to prove that widgets are green.
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