'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.
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· 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).