# The AMC 8

Had a good morning with the boys today.   One thing that is really fun for me teaching them is that I never really know where the conversations are going to go.  Sometimes, and probably quite often, I mistakenly think that a concept has sunk in when in reality it needs quite a bit more review.

Today with my younger son the topic was divisors.  He has a hard time getting the words right – 2 is a divisor of 8 gets translated into 8 divides into 2.  He seems to understand the basic idea of divisors.  Questions such as “find the divisors of 20” are not that difficult for him, but something like “how many integers n are there so that 20 / n is an integer” are still difficult.

The bulk of my time with him this morning was spent on the following questions – If n is a divisor of 20, is n also a divisor of 60?  Similarly, if m is a divisor of 60, is m also a divisor of 20? We ended up listing out all of the divisors of each number and them comparing the lists.  It was interesting to see him wrap his mind around the 2nd question.  After that we made this movie about the divisors of perfect squares:

The morning with my older son was spent reviewing the quadratic formula.   We derived the formula yesterday and I wanted him to give a “lecture” about it today.  It is interesting to watch higher level ideas come together in his mind.  The derivation isn’t especially difficult, but there are a couple of ideas that you wouldn’t likely just stumble on all by yourself!

By coincidence there was a little bit of discussion on twitter this week about completing the square.  Though we spent all of last week on that topic, it isn’t a topic that I’d thought was all that interesting.  In a FB conversation, though, my friend Julie Rehmeyer pointed out that was the most interesting part of the quadratic formula for her.  That comment made me rethink what I wanted my son to get out of these two weeks, so I put more emphasis on completing the square at the end.

Julie thought that the final derivation of the formula was more about manipulating symbols than it was about an interesting mathematical fact.  I don’t feel as strongly about that point, but I don’t think she’s wrong.  I actually began the discussion of completing the square last week with this fun “paradox” to emphasize what can go wrong when you just blindly manipulate symbols:

The conversation with Julie made me think back to what I remember from learning about the quadratic formula as a kid.  What came to mind was the relatively simple sum and product of roots formulas (and their generalizations to equations with degree greater than 2).   For some reason I always found it amazing that you could pick out these facts about an equation just from the coefficients.  As I mentioned to Julie in our conversation, these simple facts show up again when you learn about Galois theory and help explain why you can’t write down the roots (in general) for polynomials of degree greater than 5.    I plan to talk about some of these fun details on Friday.