Struggling through an AMC 10 geometry problem

This week instead of his regular homework I’ve been asking my older son to work through problems 11 through 20 from old AMC 10’s. These problems are on the edge of what he’s able to do right now, but he’s learning to struggle through these problems.

Every night when I get home from work I ask him to pick one that he wasn’t able to solve and we talk through it. Today’s problem was #18 from the 2011 AMC 10b:

Problem 18 from the 2011 AMC 10b

Here’s the problem:

Rectangle ABCD has AB = 6 and BC = 3. Point M is chosen on side AB so that \angle AMD = \angle CMD. What is the degree measure of \angle AMD?

Part of the reason why this problem is challenging is that it requires you to pull together a couple of different ideas from geometry into the solution. It isn’t even really clear what you are supposed to do to start in on the solution. Here’s the first part of our conversation tonight – my son’s main strategy here is to look for similar triangles:

In the last video we looked at a few different idea, but unfortunately none of them worked. What can we do now? In this part of the talk we switch directions a little and try a little bit of angle chasing. As we chase a few angles, my son discovers that a few of the angles in the diagram are equal. So, with the angle chasing strategy we have learned a little bit!

One thing that was cool here is that the diagram we end up with reminds my son of the proof of why angles in a triangle add up to 180 degrees.

The last step to solving the problem eludes him for a bit, though:

New we have to try to figure out what else is going on – what haven’t we noticed? What else is there to see?

Then, about 1:00 in he sees the isosceles triangle! That’s it! The rest is just a little arithmetic (and remembering where the 30 degree angle is in a 30 – 60 – 90 triangle!)

It is nice to see the 15 minute struggle get rewarded.

Sometimes it is hard for me to sit on my hands during this type of project, but I enjoy watching this struggle. Hopefully it is productive and helps connect a few different ideas from geometry in my son’s mind.

Honestly, I’d be pretty happy spending all my time working with kids on problems like this – well, maybe coaching a little ultimate, too 🙂

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