I like this problem

We were driving back from Boston this morning, so we didn’t have our normal morning math class. Instead I spent about an hour talking through some of the problems from the 2010 AMC 10a. Sort of a different approach to math contest problems – my son would read the problem out loud and we’d talk about each one for a few minutes.

This one caught my attention:

Problem #13 from the 2010 AMC 10 A

Here’s the problem in full:

“Angelina drove at an average rate of 80 kph and then stopped 20 minutes for gas. After the stop, she drove at an average rate of 100 kph. Altogether she drove 250 km in a total trip time of 3 hours including the stop. Which equation could be used to solve for the time t in hours that she drove before her stop?”

\mathrm{(A)} \ 80t + 100(\frac{8}{3} -t) = 250 \qquad \mathrm{(B)} \ 80t = 250 \qquad \mathrm{(C)} \ 100t = 250

\mathrm{(D)} \ 90t = 250 \qquad \mathrm{(E)} \ 80(\frac{8}{3} -t) + 100t = 250

I realize the difficulties that can arise from the phrasing “could be used” but putting that aside for now, I really liked the discussion that arose from this particular problem.

Two things that I enjoyed especially were:

(1) For lack of a better word, I’ll call it math fluency. The conversation about the five answer choices helped me understand that my son does not see equations like these in the same way that I do. Perhaps not the exact right comparison, but I thought it was similar to watching someone who is just learning a language reading along with a native speaker.

I steered the conversation to talking about the 3 simple equations first. I made this decision to avoid having the conversation stop once he recognized (a) was the right answer, but it proved useful from the fluency side, too. Working through the three easier equations helped me get a better feel for what he thought the equations were saying. This part of the conversation was actually made a little easier since the solutions to the three straightforward equations are large (compared to 3 hours) and that makes it not so hard to see that these times cannot be correct answer to the question.

(2) The two remaining complicated equations were really confusing to my son, and finding a way to talk through them without just telling him what every term meant was a challenge for me. I decided to ask him where he thought the 1/3’s came from in those two equations. In retrospect I’m not sure how I arrived at this question for him, but thinking about those fractions did seem trigger some new ideas for him. Eventually he was able to see that (a) did indeed represent the correct answer.

This conversation today was unusual since we didn’t have any way to write down what we were talking about – we really just talked through the problems. Talking through the problems gave me a different view of his understanding of math – more from a reading comprehension side than a math comprehension side in this setting. The lesson for me from today is to look more carefully at the fluency side and pay more attention to each of the kid’s first impressions on encountering a problem.