# Some fun geometry and a challenging number theory topic

Some days teaching the boys seem to go super well – could be some interesting ideas from them, or just general enthusiasm, but days like this make me really happy that I have the opportunity to teach them math.

Yesterday my son struggled with this AMC 10 problem:

Problem #14 from the 2012 AMC 10 B

It is a pretty challenging problem. We were all pulled in slightly different directions today and didn’t get a chance to do any regular school work until I got home from work tonight. I thought it might be interesting to revisit a different version of this problem and hear him talk through it.

Here’s the problem I came up with and his initial thoughts:

I was interested to see that his initial approach was to try to compute, and the desire to compute was driven by remembering the formula for the area of a rhombus. Drawing the long diagonal of the rhombus gives him a different geometric idea, though. I been trying to emphasize geometric ideas over computation, so I was happy to see the change in approach.

The new approach led us down the path of congruent triangles and then principles of counting. Fun!

So, I stopped the last movie after about 5 minutes. The final part of his solution is here. At the end I ask him to re-explain the geometric idea he’s using ( over counting, I guess), and then showed him an alternate geometric solution.

So that was fun. I love how the geometric ideas pulled him away from the straight computation.

Next up was a new section in our Number Theory book – linear congruences. This is (obviously) a pretty advanced topic, and I actually skipped it with my older son when we went through the same book a few years ago. But, my younger son has found this introduction to number theory to be really interesting, so what the heck.

I just love his enthusiasm when he sees that 2x = 3 (mod 4) has no solutions.

Happy to head into the blizzard on this math high note tonight 🙂