If everyone in the class shook hands with everyone else, how many handshakes would there be? As you solve this problem, be sure you explain clearly your reasoning. Use some kind of representation to help explain your reasoning. Make comments about any patterns you see.
This task can be given to students with some sense of number and patterns. Students with knowledge of the area of rectangles and triangles may make some strong connections that allow them to come to a generalization of the solution. This problem makes a powerful statement to students that the use of diagrams and charts can help make connections to patterns and generalizations. It also shows students that acting out a problem often helps formulate a strategy that can solve the problem. As students solve this problem, they may very well act out the problem. Others will break the problem down to simpler cases and look for a pattern. Students will often begin by making some kind of drawing, chart or diagram to help them sort out the problem. Often times there is a discussion of what constitutes a handshake. Students need to come to a resolution as to whether when two people shake hands it counts for one or two handshakes. 60 minutes This task can be used to demonstrate a number of different lessons. In science or health it can show how disease spreads across a population. It could be useful to show how AIDS has become so widespread among some populations. It might also be used in social studies in a discussion of how people greet one another in different societies or simply to demonstrate the "social niceties". If the question of what a handshake is comes up, allow students to give their opinions and try to convince others of their point of view. Students can be given a great deal of latitude in solving this problem - there are a number of strategies that are appropriate, from acting out the problem to using more refined approaches.
- Graph
- Lined and blank paper
- Pencil
- *Compass
*Some students see people standing in a circle and each person is a dot on the circle and gets connected to every other dot. Students who use a chart of simpler cases should be encouraged to look for patterns. | Number of People | Number of Handshakes | | 1 | 0 | | 2 | 1 | | 3 | 3 | | 4 | 6 | | 5 | 10 | | 6 | 15 | | . | .. | | . | .. | | . | .. |
Students who use the rectangular representation (see Expert Benchmark) should be encouraged to connect their understanding of area to simpler cases to come up with a generalization. (number of people) x (number of people - 1)) / 2 = handshakes
This student used an inappropriate approach. Finding the number of handshakes at each group of desks and adding the results together will not lead to a correct solution. The representation of the desks and students at the desks does not help solve the problem.
This student had a strategy that was partially useful. They left out the last handshake. There is some evidence of mathematical reasoning. The solution lacks a complete explanation.
The solution and comment on finding a pattern show that the students have a broad understanding of the problem. They use a strategy that leads to a solution using effective mathematical reasoning. There is a clear explanation and appropriate use of a chart.
The solution and generalization show a deep understanding of the problem. Using a formula to solve the problem is a sophisticated and efficient strategy that leads directly to the solution. The solution shows refined and complex reasoning. There is a clear and effective explanation of the solution. The mathematical representation is actively used as a means of communicating ideas related to the solution of the problem.
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