'Creating Measures' Awkward-ness Task - Example #5 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: · invent your own measure for the concept of "awkward-ness" · use your measure to put situations in order of "awkward-ness" · generalize your measure to work in different situations. ____________________________________________________ · Have you ever arrived at a packed theater after the show has started? · You have to make everyone stand while you squeeze past to take your seat. · Imagine that five people A, B, C, D and E each arrive to take their seat in a theater. · They are not allowed to take different seats to the one they have been allocated. This diagram shows the order in which they arrive and their seating positions: · So, D arrives first and sits in the second seat from the right hand end of the row. · Then E arrives. D has to stand up while E squeezes into the last seat in the row. · Then A arrives. She sits on the first seat of the row. · Now B arrives and makes A stand, while he takes the second seat in. · Finally C arrives and makes both A and B stand up while she takes her seat. Warm-up Try out this situation from different starting points using scraps of paper labeled A, B C, D and E until you can see what is happening. What is the most awkward situation you can devise? Draw it below: Here are four movie theater situations: 1. Place the four situations in order of "awkward-ness." · Which is the easiest situation for people? · Which is the most awkward? · Explain how you decided. 2. Invent a way of measuring "awkward-ness." This should give a number to each situation. Explain carefully how your method works. 3. Show how you can use your measure to place the four situations in order of "awkward-ness." Show all your work. 4. Adapt your measure so that the minimum value it can take is 0 (where no-one is made to stand up) and the maximum it can take is 1 (the most awkard situation possible). 5. Show how your measure in part 4 may be generalised for any number of people entering a row. ( That is when n people enter a row with n available seats).