The three problems presented for this extended lesson have both individual and cluster appeal. Since each of these problems can be visualized or acted out quite readily, the problems can be accessible at some level for virtually every middle school student. By offering all three of the problems to your students over a period of time, there is the opportunity to draw them into the problem-solving experience at a deeper level and allow students to practice and strengthen their ability to generalize and recognize underlying themes/parallels within various contexts.
The Handshake Problem is a natural opener for the beginning of the school year: Suppose you walk down to the corner some afternoon and there are six of your friends standing around. How many handshakes would there be if each person shakes each and every other person’s hand once? The second problem, All Possible Diagonals, asks students to draw all possible diagonals in eight regular polygons, record their findings in a table, and use this information to generalize the number of diagonals in an n-gon. The third challenge, entitled Triangular Numbers, asks students to generate a rule for finding the nth triangular number.