
Tools  Math 'Creating Measures' Awkwardness Task, Example #5 (solution)
Squareness, Example #1 (solution)  Steepness, Example #2 (solution)
Compactness, Example #3 (solution)  Crowdedness, Example #4 (solution)
Awkwardness, Example #5 (solution)  Sharpness, Example #6 (solution)
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 task gives you the chance to
 invent your own measure for the concept of "awkwardness"
 use your measure to put situations in order of "awkwardness"
 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.
Warmup
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.
 Place the four situations in order of "awkwardness."
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.)
 Invent a way of measuring "awkwardness." This should give a number to each situation. Explain carefully how your method works.
Solution for Questions 2 and 3:
A suitable measure of "awkwardness" 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.
 Show how you can use your measure to place the four situations in order of "awkwardness." Show all your work.
 Adapt your measure so that the minimum value it can take is 0 (where noone 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 "awkwardness" score for five people = 10 (see Warmup).
 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 "awkwardness" = 0.
For 2 people, the maximum "awkwardness" = 1.
For 3 people, the maximum "awkwardness" = 3 (= 1 + 2).
For 4 people, the maximum "awkwardness" = 6 (= 1 + 2 + 3).
For 5 people, the maximum "awkwardness" = 10 (= 1 + 2 + 3 + 4).
...
For n people, the maximum "awkwardness" = (= 1 + 2 + 3 + ... n).
Thus, if s = The number of occasions on which people have to stand;
we can define our measure of "awkwardness" for a given situation to be:
=
Squareness, Example #1 (solution)  Steepness, Example #2 (solution)
Compactness, Example #3 (solution)  Crowdedness, Example #4 (solution)
Awkwardness, Example #5 (solution)  Sharpness, Example #6 (solution)
