Editing assistance from Tom Davis, Katy Early, and Jim Tanton Graphics by Tom Davis
The three problems presented 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 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 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 triangular number. 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, one goal 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. 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 andstrengthen their ability to generalize and recognize underlying themes/parallels within various contexts.
Problem I: The Handshake Problem
Learning Objectives: By the end of this lesson students will be able to • • • Organize data obtained by thinking through a tangible experience Abstract the essentials of a mathematical analysis from a real-life situation See patterns in the numbersgenerated by shaking hands with an increasing number of persons
A Trio of Friendly Problems (Mary Fay-Zenk)
Materials Required: No specific materials are needed to present this problem other than paper and pencil for small groups of students to record their work and a whiteboard or overhead projector for the teacher to record and systematize the thinking of the class. Instructional Plan: Thislesson is primarily an interactive, teacher-directed activity from which students will generate number patterns to analyze.
1. Present students with this simple scenario.
Suppose you walk down to the corner of your street some afternoon and there are six of your friends standing around. How many handshakes would there be if everyone shook hands with each other person exactly once? (Shakinghands with yourself is not allowed!) Make certain students understand the situation clearly and assign them to groups of three or four to begin to act out/talk out the scenario and collect data. Encourage students to make a table recording the information they agree upon so they can share with the class after a few minutes. Although students may be tempted to join immediately with another group inorder to have a group of seven to act out the situation, persuade them to start with their smaller group of three or four students and try to find a way to find a pattern that would lead them to a solution they can defend. If students are insistent about tackling the larger problem, you might promise that they can try to act out the group of seven-at-the-corner after some initial pattern...