'Creating Measures' Steep-ness Task - Example #2 (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 "steep-ness" · invent your own ways of measuring this concept · examine the advantages and disadvantages of different methods. ____________________________________________________ Warm-up Without measuring anything, put the above staircases in order of "steep-ness." Comment: This first question is simply intended to orientate the students to the task. It may be used as a class introduction. 1. Someone has suggested that a good measure of "steep-ness" is to calculate the difference: Height of step - length of step for each staircase. Use this definition to put the staircases in order of "steep-ness." Show all your work. Solution Using the measure 'height of each step - length of each step', the 'steep-ness' of each staircase is given in the table below (using centimeters as the unit). Staircase A B C D E F Height (cm) 1.5 1 0.5 1 2 1.25 Length (cm) 2 1.5 1 1 3 3.33 Height-Length (cm) -0.5 -0.5 -0.5 0 -1 -2.08 Using this measure, the staircases in order from most to least steep are:
D, A and B and C (tie), E, F. 2. Using your results, give reasons why Height of step - length of step is not a suitable measure for "steep-ness." Solution: The above measure is unsatisfactory because: · It gives no real indication of the steepness. Using this measure, A and C are labeled as equally steep, which does not fit with intuition. · It is dependent on the units used. If we use inches instead of centimetres we get a different "steep-ness" measure. · It is usually negative, which is inelegant and awkward to use. 3. Invent a better way of measuring "steep-ness." Describe your method carefully below: Solution: There are many other ways of measuring "steep-ness." Students might, for example, propose using: a) The angle of inclination; b) The ratio of 'step height'/'step length' (technically: riser/run); c) The ratio of 'height of whole staircase'/ 'length of whole staircase'; These are equally sensible, and equivalent, except is may be sometimes unclear what we measure as the 'length' of the staircase. 4. Place the staircases in order of "steep-ness" using your method. Show all your work. Solution: Whichever measure we now use (a), (b) or (c), we obtain the same order for the staircases. Staircase A B C D E F Height (cm) 1.5 1 0.5 1 2 1.25 Length (cm) 2 1.5 1 1 3 3.33 Height ¸ Length (2 d.p.) 0.75 (3/4) 0.67 (2/3) 0.5 (1/2) 1 (1/1) 0.67 (2/3) 0.38 (3/8) Angle of inclination (nearest degree) 37û 34û 27û 45û 34û 21û This gives the order of steep-ness (from most to least steep) as: D, A, B and E (tie), C and F. 5. Do you think your measure is a good way of measuring "steep-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. Describe a different way of measuring "steep-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.