Pull the Goalie

It’s late in the game. We are Team A and we are one goal behind Team B. We are considering pulling our goalie. Is it to our advantage? If so, when should we do it? How long before the end of the game?

This is an interesting problem. It’s an authentic modeling exercise in that there’s lots of thinking and playing needed to come to understand what exactly our objective is, what variables and functions to use, and how to put everything in place. It’s also a model that most students will be able to relate to.

We use a dynamic programming approach, dividing time into 10-second intervals, starting at the end of the game, calculating the strategy for each successive interval in terms of the later ones that we have already done. We use the NHL data for our scoring probabilities.

The students need to know how to handle independent event (rolling dice) and the sum of a finite geometric series. The rest of it is really about figuring how to work with a complex structure. We feel that it belongs in grade 12.