Patricia's Teaching Blog: Prove something in the subway!

Today was a very fun class. We were able to finally dig into mathematical proofs.

Examples such as "if n is an odd number then the square is an odd number" were discussed.

We discussed the proofs together and one of the class activities was using a direct proof to show that sum of two odd integers is even.

I made sure to stress that most of the propositions we have to prove would be of the form:

∀x (P(x) →Q(x))

What do you understand by hypothesis? For which values of P and Q is P→ Q true?

I believe this activity opened an interesting discussion. A couple of students wanted to work with particular numbers, but I told them we needed something more general. I asked them to pick their two favorite letters as the odd numbers to start using the hypothesis. They realized that as soon as they started writing the hypothesis, they know that the following step is writing how the expression of the sum would look like.

We then recalled that proving the above conditional is equivalent to proving its contrapositive. We discussed a specific example where we got stuck on doing a direct proof and realized that an indirect proof using a contrapositive was more viable.

The class ended by writing on the board a statement I wanted them to prove by contrapositive.

I suggested getting a small notebook where they can write proof in the subway. I joked and said: "and you can send me a picture of you with your proof in the subway". A student asked if this was serious...now I believe this would be a good idea. Why not? Would not this be way more fun than looking out at your smartphone during a subway ride? It would be so stimulating to read and write proofs in the subway!!!

One caveat:: a student drives to school. Another student said: "do it in red lights!!!"