'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).