'Creating Measures' Square-ness Task - Example #1 (solutions) Malcolm Swan Mathematics Education University of Nottingham Malcolm.Swan@nottingham.ac.uk Jim Ridgway School of Education University of Durham Jim.Ridgway@durham.ac.uk This problem gives you the chance to: · criticise a given measure for the concept of "square-ness" · invent your own ways of measuring this concept · examine the advantages and disadvantages of different methods. ____________________________________________________ Warm-up Use visual judgements to answer the warm-up questions. Which rectangle looks the most square? Which rectangle looks least square? Without measuring anything, put the rectangles in order of "square-ness." Comment: This first question is simply intended to orientate the students to the task in hand. It may be used as a class discussion. 1. Someone has suggested that a good measure of "square-ness" is to calculate the difference: Longest side - shortest side for each rectangle. Use this definition to put the rectangles in order of "square-ness." Show all your work. Solution: Using the measure 'Longest side - shortest side', the "square-ness" of each rectangle is given in the table below (using centimeters as the unit). Rectangle A B C D E F G H I Dimensions (cm) 3 x 3 1 x 8 6 x 2 4 x 1 3 x 4 3 x 2 6 x 5 4 x 2 12 x 4 Square-ness (cm) 0 7 4 3 1 1 1 2 8 Using this measure, the rectangles in order from most to least square are: A, E and F and G (tie), H, D, C, B, I. 2. Using your results, give one good reason why Longest side - shortest side is not a suitable measure for "square-ness." Solution: The above measure is unsatisfactory because: · It gives no indication of the overall 'proportions'. (E, F and G under this definition have the same square-ness yet are clearly different in shape, while C and I are similar in shape but give different square-ness measures). · It is dependent on the units used. If we use inches instead of centimetres we get a different "square-ness" measure. 3. Invent a different way of measuring "square-ness." Describe your method carefully below: Solution: There are many other ways of measuring "square-ness." Students might, for example, propose using: a) The ratio longest side/shortest side; b) The largest angle between the diagonals of the rectangle; c) The ratio of perimeter/area. a) and b) seem equally sensible. c), however, suffers the same problem as before. As it is not dimensionless, an enlargement of a rectangle will result in a different value for its "square- ness." If, however, we use d) the ratio (perimeter)2/ area then we would have a suitable, dimensionless measure. 4. Place the rectangles in order of "square-ness" using your method. Show all your work. Solution: Whichever measure we now use (a), (b) or (d), we obtain the same order for the rectangles. In order of "square-ness" they are: A (most square), G, E, F, H, C and I (tie), D, B (least square). Rectangle A B C D E F G H I Dimensions (cm) 3 x 3 1 x 8 6 x 2 4 x 1 3 x 4 3 x 2 6 x 5 4 x 2 12 x 4 Ratio: Longest ¸ Shortest 1 8 3 4 1.3 1.5 1.2 2 3 Largest angle between diagonals 90û 166û 143û 152û 106û 113û 100û 127û 143û Ratio: Perimeter2 ¸ area 16 40.5 21.3 25 16.3 16.7 16.1 18 21.3 5. Do you think your measure is a good way of measuring "square-ness?" Explain your reasoning carefully. Solution: Here we would like students to review their results critically and decide whether the results from their measurements accord with their intuitions. 6. Find a different way of measuring "square-ness." Compare the two methods you invented. Which is best? Why? Solution: This question provides an opportunity for students to look for an alternative measure.