A cliché of modern life — but yesterday I managed to find a use for it.
One of the questions the linear algebra students had to solve was to prove (from the axioms) that, if cv = 0, where c is a scalar and v a vector, then either c = 0 or v = 0. Some of them managed the algebraic manipulation correctly but were stuck on the logic. I improvised the following explanation, which I am quite pleased with.
You know that cv = 0, and you are given a form to fill in, with two boxes labelled c = 0 and v = 0; you must tick one of these boxes. So look at the first box. If c = 0, you can tick it, and you are finished with the form, you can hand it in. If not, then the manipulations you did show that in fact you can tick the other box v = 0.