Last night I tutored my geometry student, and so I told him about the Putnam problem that I mentioned here on this blog yesterday. He figured out that the answer was "N must not contain any zeros" right away, but of course, many of this proof techniques -- proof by induction, proof by contradiction (indirect proof), and even proof of the converse (since the statement was given as a biconditional) -- were new to him.
I gave a very common analogy for proof by induction -- the initial or base case of an induction is like knocking down the first domino, and the inductive case is domino n set up to knock down domino n+1. For indirect proof, I told him how to prove, indirectly, that he is dreaming -- assume that what you see is real, until you reach a contradiction such as seeing pigs fly in the distance, then conclude that you are dreaming all along.
I believe that my explanations would be easier for my student to understand if I wasn't giving so many different proof techniques in the same day. And so I focused on the one most relevant to his current geometry class -- biconditionals, or "if and only if" statements. I gave him my worksheet from Section 2-5 of the U of Chicago text, which I've posted right here on this blog. He appeared to figure out most of the problems on that worksheet.
Returning to Chapter 7 of the U of Chicago text, we see that there are two sections left. Of these, Section 7-7 is far more important, as we use the theorems SSS, SAS, and ASA to prove that a figure is a parallelogram.
But what is in Section 7-8? This section covers the SAS Inequality -- often known as the Hinge Theorem, according to the Exploration Question at the end of the section. Dr. Franklin Mason also calls this theorem the Hinge Theorem.
I'm of two minds as to whether to include Lesson 7-8 on this blog. Notice that Dr. M includes the Hinge Theorem, Triangle Inequality, and all other inequalities related to triangles in his Chapter 5. But we've been waiting because many of these theorems depend on indirect proof, so we were waiting for Chapter 13. Yet on this blog, I posted the Circumcenter Concurrency Theorem even though it also requires an indirect proof (but, as Dr. Wu pointed out, the part that requires indirect proof can be handwaved over).
Once again, this is why I discuss what I post here with my geometry student. How important is it for me to avoid or delay indirect proofs in order to avoid confusing him? Last night, it appeared that he sort of understood indirect proof, but combining it with a lesson on induction and biconditionals can only confuse him more.
In the end, I post Section 7-8 today. If one wants, one can just skip to the review questions. By getting 7-8 out of the way, I can spend the rest of the week on the more important Section 7-7.