Pythagoras and the four colours

Laao ar NAA AL, LA PYTHAGORAS AND FOUR COLOURS Hasan Unal describes a spatial activity based on partitioning a square into four figures and constructing two new squares from them. One wav to teach Pythagoras’ Theoremís through use of puzzles. Marshall 2004: 1) points out that, ‘in creating their individual solutions to puzzles, students may reveal mathematical thinking on which approaches to the standard curriculum could be based." This article describes a puzzle-like spatial structuring activity related to Pythagoras’ Theorem, The tasks were tried with pre-service middle and secondary mathematics teachers in the USA and We gave the students a 5-by-5 square and asked them to construct twe squares From thegiven square by dividingit into four regions so that these four regions can be rearranged into two squares (4by-4 and 3-by-3}.! This activity can be done with manipulatives or just plain paper. There are multiple answers to the problem and students’ thinking skills get challenged. It also helps to improve spatial ability, since students need to rotate Turkey. There are two cases: Case 1: IF] give you two squares, you need to give me one square with an area equal te sum of those twe squares (sce Figure 1). to Figure 1 D The Task = the images in their minds when they construct new squares. The task is fun and intrinsically motivating, and it offers just the right amount of challenge. Students’ solutions fall into two categories (A and B) and colour is used below to illustrate the wav in which the 5-by-5 square has been dissected. Table 1: Summary of Type A solutions 5° Case 2: IF] give vou one square, you need to give me two squares whose areas sum to the area of the original square (see Figure 2). 4° OO 2 pumber of solutions four colours three colours one € our 9 tour colours one colour three colours I Table 2: Summary of Type B solutions number of solutions Figure 2 fourr colours We have explored the second case in this article, but our contention is not that one way is superior to the other. The two cases provide different perspectives that might be useful for teaching Pythagoras’ Theorem and developing students’ visualisation skills. Furthermore it also encourages divergent thinking. MATHEMATICS TEACHING INCORPORATING MICROMATH 206 / JANUARY 2008 two colours two colours

PYTHAGORAS AND FOUR COLOURS Hasan Unal describes a spatial activity based on partitioning a square into four figures and constructing two new squares from them. One way to teach Pythagoras’ Theorem is through The Task use of puzzles, Marshall (2004:1) points out that, ‘in creating their individual solutions to puzzles, We gave the students a 5-by-5 square and asked students may reveal mathematical thinking on them to construct two squares from the given which approaches to the standard curriculum could square by dividing it into four regions so that these be based.’ This article describes a puzzle-like spatial four regions can be rearranged into two squares (4- structuring activity related to Pythagoras’ Theorem. by-4 and 3-by-3).! This activity can be done with The tasks were tried with pre-service middle and manipulatives or just plain paper. There are secondary mathematics teachers in the USA and multiple answers to the problem and students’ Turkey. thinking skills get challenged. It also helps to improve spatial ability, since students need to rotate There are two cases: Case 1: IfI give you two squares, you need to give me one square with an area equal to sum of those the images in their minds when they construct new squares. The task is fun and intrinsically motivating, and it offers just the right amount of challenge. two squares (see Figure 1). Students’ solutions fall into two categories (A and B) and colour is used below to illustrate the Figure I way in which the 5-by-5 square has been dissected. a |+ b’ = Table 1: Summary of Type A solutions © 2 Case 2: If I give you one square, you need to give me two squares whose areas sum to the area of the original square (see Figure 2). 2 32 number of solutions four colours three colours one colour 9 four colours one colour o three colours 1 Table 2: Summary of Type B solutions Figure 2 52 42 32 four colours two colours two colours We have explored the second case in this article, but our contention is not that one way is superior to the other. The two cases provide different perspectives that might be useful for teaching Pythagoras’ Theorem and developing students’ visualisation skills. Furthermore it also encourages divergent thinking, number of solutions

Type A solutions The characteristic of this type of solution is that four colours from the main square (5-by-5) are distributed as three colours in the first square (4by-4) and one colour in the second square (3-by-3) or vice versa — see figures 3-12. The partitioning in the main square in Figure 6 was carried out using only rectangles and a square. The solution in Figure 7 was the first solution students came up with, and we find that the majority of students arrive at this solution at their initial attempt. The solutions in Figures 9, 10 and // took more time to discover than did the previous ones. In Figure 12, it is the 4-by-4 square that is all one colour rather than the 3-by-3 one. Figure 8 m ur Figure 3 Figure 9 Figure 4 Figure 10 Figure 5 Figure 11 5°=25 4 =16 3°=9 Figure 6 Figure 12

Type B solutions both of the two smaller squares contain exactly two colours each. Figure 16 The solutions in Figures 13 to 15 are different from the ones above. In the previous solutions, one of the two smaller squares (either the 3-by-3 or the 4-by-4) was all one colour and the other square contained three colours. In these type-B solutions, Figure 17 Conclusion We believe that this approach to Pythagoras’ Theorem makes the lesson more active and encourages students to consider why it works. Prese! nting Figure 14 geometry in ways that stimulate curiosity and encourage exploration can enhance students’ learning and improve their attitudes towards mathematics, Hasan Unal is an assistant professor in the mathematics department at Yildiz Technical University, Istanbul, Turkey. Figure 15 Note 1 The problem was originally investigated by Bolt (2001). For a summary of the solutions and an activity sheet to use An interesting feature of some of the students’ with students, go to www.atm.org.uk/mt206 solutions (both A- and B-type) were ones where pieces from the 5-by-5 square needed to be turned References over — as opposed to merely rotated: see figures 16 Bolt, B. (2001) Two squares from one, Mathematics in School, (type A) and 17 (type B), for instance. In other November, 30-31 Marshall, A. J. (2004) Construction of meaning: urban words, both sides of the paper would need to be elementary coloured, These solutions were not valued by the Journal of Mathematical Behavior, 23 (2), 169-182 students; they felt that allowing these made the task insufficiently challenging. It will be up to the teacher to decide at the beginning of the activity if flipping the figures is going to be allowed or not. students’ interpretation of geometric puzzles, The

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