(p/q)2 = 2
p2 = 2 q2
so p2 has 2 as a factor, which means that p must have 2 has a factor. So p=2r for some r.
We now have
(2r)2 = 2 q2
4r2 = 2 q2
2r2 = q2
and therefore q has 2 as a factor.
So both p and q have 2 as a factor, but we assumed that they were relatively prime, which is a contradiction.
The kids didn't really think much of this one. They seem take it for granted that √2 is irrational, so why do they need a proof. Or maybe they don't like proof by contradiction. (Maybe they are constructionist mathematicians...)