by Mark Illingworth

I gave my students the Squirrel Problem years ago as an experiment. I wanted to be able to analyze an original problem for the end-of-the-year project, but I knew I should be doing things along the way to help them get to that point. I was starting to have them do Explorations, but I wondered what would happen if I gave them a very open-ended problem that they could work on in teams. (I should mention that this problem was given to an Honors Geometry class.)

You can download and use the entire problem here, but I’ll summarize for this discussion. Suppose we know that squirrels cross the road under cars on purpose as part of their initiation to become full members of the scurry. (a) If a squirrel crosses under a moving car, what is the probability that it will make it safely to the other side? (b) Your company is interested in selling life insurance policies prior to a hundred thousand squirrels prior to their crossing. The policies will each be worth $1000. How much will you charge in premiums?

I gave them time to work on the challenge once a week for six weeks, and they were supposed to get together outside of class to develop their analyses and solutions. I had no idea that they’d run a far as they did with this opportunity.

I think you can imagine how textbook publishers would treat this problem. First of all, they wouldn’t. The problem is too “out there” and there are too many steps. If they did, pose the problem, they’d provide all the data the students needed: the car’s velocity, the squirrel’s running speed, the distance between the tires, etc. In other words, most of the opportunities to think would be removed from the problem, and students would rely on the single skill on which that lesson was focused. What a squandered opportunity!

All five of the teams were able to decide what variables were important in order come up with a solution they were able to rationalize, but I want to focus on the groups that left my expectations in the dust. Three groups taught themselves the Python programming language so that they could create simulations in which they randomized variables. One group took it even farther. They realized that it didn’t make complete sense to make variables completely random. In other words, consider the angle at which the squirrel crosses relative to the direction of travel of the car. Let’s say that this angle would be (assuming you have a reasonably intelligent squirrel) between 85° and 95°. Rather than to make all possibilities in this range equally likely, they taught themselves how to randomize the variable within a normal distribution so that 90° was much more likely than 85°. I had never talked about normalized distributions!

There’s no turning back, right. Once I saw what students could do when I gave them the opportunity to analyze and evaluate creatively, I started creating more problems—both small and big—in which they needed to engage much more than the part of their brains that was good at following recipes. I'll include some problems of this type in the Problems section of the site, but you can easily turn many of the great problems you already use by removing much of the information that you provide. I recommend that you try some of the ones you provided to both to get the flavor of these problems and to see for yourself the great thinking that students will experience in order to reach solutions.

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This article describes The Squirrel Problem, which involves the probability of a squirrel crossing underneath a moving car safely.

This open-ended problem differs from textbook problems in that it does not provide all the data needed to solve the problem. The technique(s) needed to solve the problem are also not specified by the lesson from which it was taken.

The students ran with this problem in amazing ways, which included teaching themselves coding and normal distributions.

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