Academic journal article Australian Mathematics Teacher

The 20 Matchstick Triangle Challenge; an Activity to Foster Reasoning and Problem Solving

Academic journal article Australian Mathematics Teacher

The 20 Matchstick Triangle Challenge; an Activity to Foster Reasoning and Problem Solving

Article excerpt

Introduction

In this article we look at a simple geometry problem that also involves some reasoning about number combinations, and show how it was used in a Year 7 classroom. The problem is accessible to students with a wide range of abilities, and provides scope for stimulating extensive discussion and reasoning in the classroom, as well as an opportunity for students to think about how to work systematically. Pat, the first author and a classroom teacher, used the problem with her students and we will present some of the strategies, solutions, and issues that they encountered and discussed. Helen, the second author who works with pre-service and in-service teachers, has used this problem with teachers and likes thinking about tasks that are good for fostering reasoning and problem solving.

Pat first encountered this problem during Helen's presentation at a local mathematics teachers' conference (and, unfortunately, Helen cannot remember where she first came across it). The wording of the original problem was to find as many triangles as possible with a perimeter of 20 cm, where the side lengths have to be whole numbers. When Pat decided to use the activity with her Year 7 class she had the students working in pairs and adapted the task to make it more hands-on. She called the task the "Triangle Challenge" to appeal to the students' competitive spirit and restated the problem in terms of building triangles out of matchsticks:

   Make all of the possible triangles that can be made from 20
   matchsticks. You must use all 20 matchsticks for each triangle. You
   must record which triangles you have made in some way. How will you
   know when you have all the possible triangles?

Pat's students took to the task with gusto. It was not long before students were asking, "Can we break the matchsticks?" She gave a follow-up instruction that no matchsticks could be broken and there were to be no gaps between the matchsticks. Some students had difficulties with the construction of the triangles, especially with ensuring the matchsticks were touching and that the sides were straight. Some students used rulers to help with straightening the sides, so this technique was shared with the whole class. Helen wonders if this awkwardness with the materials may actually help students bridge the concrete and abstract characteristics of the shapes, since the students have to start thinking about whether or not the edges will really join up even though it is not entirely clear that they will because of the practical limitations of the matchsticks.

Observed student approaches

When Helen had first posed the problem, she had only seen her own solution (although she was confident the problem would be a good one for students), and so was curious as to what strategies that students might use when tackling the problem. Pat gave her students no hints at all as to how the triangles should be recorded, and told them to choose any method that suited them. A summary of how students tackled the challenge and recorded their triangles is presented in Table 1. There were a total of 24 students, working in pairs, on this particular day.

After about 40 minutes, Pat stopped the triangle construction and recording stage and started a class discussion. The discussion also continued into the following day's maths lesson.

Helen and Pat both believe that class discussion provides an opportunity to make explicit and public the reasoning with which students have engaged in their problem solving process, and to stimulate further reasoning, conjecturing, hypothesising, and refutation. Pat started by asking questions about the actual construction of the triangles. Most students had enjoyed the construction process, but said it was difficult at times to keep the sides straight, while ensuring the matchsticks were touching and that the whole triangle was neat and complete.

Pat then turned to specific questions about how students recorded their triangles. …

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