Extending Problems

In my mathematics classes I try to carve out a day or half a day here and there to do off-topic problem solving.

The usual routine in a math class is: teacher teaches new topic, skill, fact, etc, and then the students solve a bunch of problems or answer a bunch of questions applying that skill or fact.

But there is something missing…  Students don’t have to select what skill or tool or fact to apply. They just pick the one they were just taught, or the one before that. In fact, year end or semester end exams are a little better, but still the students are picking from a group of related skills, and the questions posed usually make obvious which tool to use.

That’s where the off-topic problem solving comes in. What if I ask a group of you a question, unrelated to what we did yesterday, seemingly out of left field, that requires only math that you already know, but without any of the usual cues about what tool to use?  That’s what I do. Questions mix counting, arithmetic, organization, and visualization skills. They require reasoning, planning. In a better world, with a less dense curriculum, I would do a whole lot of this.

With freshman classes I have a few favorite problems. Each asks “how many” which is a question we don’t ask nearly often enough. Ghost the Bunny. The checkerboard. How many subsets. We solve maybe two of these in the fall. A few years ago I started asking kids to turn in a write up of the problem, including process and solution.

But for the last two years I have asked the students to go further… to devise their OWN problem, as an extension of something we have already done. This year, they extended Ghost the Bunny, or the Checkerboard.

Laura’s pet bunny, Ghost, hops up a flight of 12 stairs. Ghost hops up one step or two steps at a time, and never hops down. How many ways can Ghost reach the top step?

How many squares are on an 8 x 8 checkerboard?

What would your students come up with? (I’ll tell you mine in a follow-up post)