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Getting Started with PROBLEM SOLVING: A Trio of Friendly Problems By Mary Fay-Zenk
Editing assistance from Tom Davis, Katy Early, and Jim Tanton Graphics by Tom Davis

The three problemspresented for this extended lesson have both individual and cluster appeal. The Handshake Problem is a natural opener for the beginning of the school year, but could easily be presented to studentsat any time. It’s a situation-friendly and simple question: Suppose you walk down to the corner some afternoon and there are six of your friends standing around. How many handshakes would there be ifeach 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 ina 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 triangularnumber. 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. In addition, each of theseproblems can be solved using a recursive pattern, which means that students who are organized and persevering can complete a great deal of the task and feel very successful. Ultimately, of course, onegoal is that students would learn to recognize patterns that can be generalized. Even better than this, however, is to have students begin to explain why the patterns hold true and how the situationmight change if one or more of the parameters were to vary. Throughout the lessons, encourage students to move from “what is happening” questions to “why are they happening” and “what if” questions. Byoffering 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...
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