We had some more fun tonight working through three old AMC 8 and 10 problems that gave my kids a little difficulty.

First up, my younger son working on problem #4 from the 1989 AJHME (the predecessor to the AMC 8). The problem asks you to estimate the value of 401 / .205. This is a really nice estimation example.

His solution is interesting because he decides to simply the problem by multiplying the denominator by 1,000. That leads to some difficulty later on when we have to go from that number back to the solution to the original problem. Definitely some instructive difficulty, though.

I helpfully show him a different way to solve the problem that involves multiplying by 10 incorrectly . . . .

Next up was problem #6 from the 1989 AJHME. This is a problem about a number line that you can find here:

This is a classic “fence post” problem. I’m not sure what gave him trouble when he was working through it the first time around, but it sounds like he may have miscounted the fence posts. He doesn’t have much trouble this time around.

Last up was my younger son working on problem #18 from the 2007 AMC 10 B. The problem is a nice geometry problem. You can find it here:

The arithmetic and algebra on this problem gave him some difficulty while he was working on the problem during the day. He manages to work through that algebra here, though, and presents a nice solution. We also get to talk about a nice alternate solution that he thought about early in the video before going in a different direction.

From my perspective, I found it very interesting to hear about the various choices that he thought he had along the way. For example, at one point he’s trying to decide whether or not to use the quadratic formula or taking the square root of both sides of an equation.