Summary: A puzzle, Traffic Jam, is used to motivate pattern recognition and generalization. The original lesson is on the web here at the Math Forum.

Students learn the rules of the game by watching the teacher demonstrate it on an overhead  projector. The students work on solving the puzzle using people pieces on a textbook page. This is followed by a computer simulation of the same activity. Once they have solved the puzzle, they are asked to predict the outcome of a more difficult version of the puzzle. Students then, working in groups, collect data in the activity and make conjectures about what would happen for the more difficult version of the puzzle without having to actually do it. They are asked to come up with an algebraic rule that helps them make predictions for any number of people.
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Here is the puzzle challenge:

There are seven stepping stones and six people. On the three lefthand stones, facing the center, stand three of the people. The other three people stand on the three righthand stones, also facing the center. The center stone is not occupied. The challenge is exchanging places. Everyone must move so that the people originally standing on the righthand stepping stones are on the lefthand stones, and those originally standing on the lefthand stepping stones are on the righthand stones, with the center stone again unoccupied.

The rules:

1. After each move, each person must be standing on a stepping stone.
2. If you start on the left, you may only move to the right. If you start on the right, you may only move to the left.
3. You may jump another person if there is an empty stone on the other side. You may not jump more than one person.
4. Only one person may move at a time.

The goal is to have the students come up with a pattern and see that the number of moves can be predicted based on the number of pairs there are on each side. After solving the puzzle for one, two and three pairs, they are to come up with a way to generalize these patterns using the language of algebra.

Suzanne was kind enough to send me a videotape of these activities. The images on this page are taken from the video tape. In the future we hope to have video clips available right here on the web so you can watch the action yourself.