When things don’t go so well.

I think I bit off more than I could chew this morning . . . .

We started a new section in our geometry book today and we were looking at an interesting question – prove that the side opposite the obtuse angle in an obtuse triangle is the longest side. My son did a nice job working through the problem. Even looking at the proof in two slightly different ways which was great:


But then for some reason I wanted to take a deeper dive, and oh my goodness did this deep dive not go well.

My idea was to get him to think about why some lines he drew in the prior proof were either inside or outside of the triangle. I just did a terrible job of helping him see the ideas when he was stuck and the whole discussion wasn’t that productive, unfortunately

Compounding the problem was that we had to get out the door for an appointment and we needed to stop the film. Ugh. I need to study this a little bit and figure out how to talk through the ideas here a little lot better next time.


Interleaved Practiced and math contest problems

Last week I read this interesting article posted by Pam Wilson on Twitter:

I don’t claim to know the ins and outs of interleaved practice, but I do something that sounds similar with my kids. Today seemed like a good day to write about it.

Each morning I have my kids spend a little time on some old math contest problems. Currently my younger son works on old MOEMS tests and my older son works on old AMC 10 problems. The goal is not speed or getting good scores, rather the goal is just to get exposure to math topics that we don’t happen to be covering just now. When problems given them a little extra trouble I spend a little time in the evening talking through them.

Today my older son struggled with an absolute value problem and my younger son struggled with a problem about averages. Here are those talks.

The AMC 10 problem with my older son is here:

Problem 13 from the 2010 AMC 10b

This is a tough problem for kids and requires a pretty good understanding of the absolute value function. After a rocky start by me (!) my son is able to talk through the basic ideas that are required to solve the problem:


After we have the four equations set up, we work through the solution for each 4. I went through this part in detail to emphasize that going back to the beginning to check the answers is important. Sometimes in equation solving you introduce solutions that are not actually solutions. We have one here that my son hasn’t noticed yet, but he does notice it later:


For my younger son, the MOEMS problems are wonderful. The two books we have with problems from this contest can be purchased here (oooh, and I didn’t even know until getting the link that there’s now a 3rd book out!!):

The MOEMS books at Art of Problem Solving

The problem that gave my son trouble today is about averages. We work through it two different ways and it seemed that the solutions made sense to him.


So, I like the concept of interleaved practice, and, I think, have been doing it unintentionally. I like using the old math contest problems for a couple of reasons. First, good problems for kids are hard to write and the problem from these contests are an easy source of well-written problems. Second, I like the variety – it has been quite a while since my older son and I talked about absolute value problems. This problem today gave me a nice opportunity to review that concept with him. With my younger son, I’m not sure that we’ve ever talked about averages formally. This problem served as a nice, informal way to talk through this concept.

I think there are lots of fun and important math ideas that kids can take away from these old contest problems.