'Creating Measures' Awkward-ness 
Task - Example #5 (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:
•	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:

Comment:
The most awkard situation possible is shown below:

In this situation, A sits first, then
•	A stands while B takes her place
•	A and B stand while C takes his place
•	A, B and C stand while D takes his place
•	A, B, C and D stand while E takes her place.

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.
	
	Solution:
	The above measure is unsatisfactory because:
	The easiest situation is situation (3), because this results in only one person having to stand on one occasion (person D has to stand while E squeezes by). 
	
	The most awkward situation is probably (4) because people have to stand on five occasions. (A has to stand while B sits down, then A, B, C and D all have to stand while E sits down.)

	

2.	Invent a way of measuring "awkward-ness."  This should give a number to each situation. Explain carefully how your method works.
	
	Solution for Questions 2 and 3:
	A suitable measure of "awkward-ness" would be to count the number of times a person makes someone stand up to let them pass. This would give, for situations 1 to 4:
	
	Number of times person makes someone else stand	
Situation	A	B	C	D	E	Total	
1	0	1	2	0	1	4	
2	0	0	2	3	3	8	
3	0	0	0	0	1	1	
4	0	1	0	0	4	5	
	
	Using the totals, we have, from least to most awkward:
	Situations 3, 1, 4 then 2.

	

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).
	
	Solution:
	To make the measure range from 0 to 1, we could divide the totals above by the maximum possible "awkward-ness" score for five people = 10 (see Warm-up). 

	

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).
	
	Solution:
	If there was just one person, the maximum "awkward-ness" = 0.
	For 2 people, the maximum "awkward-ness" = 1.
	For 3 people, the maximum "awkward-ness" = 3 (= 1+2).
	For 4 people, the maximum "awkward-ness" = 6 (= 1+2+3).
	For 5 people, the maximum "awkward-ness" = 10 (= 1+2+3+4).
	...

	For n people, the maximum "awkward-ness" =  (= 1 + 2 + 3 + ... n).
	
	Thus, if s = The number of occasions on which people have to stand;
	we can define our measure of "awkward-ness" for a given situation to be:

	
	=