FLAG - Tools: Math 'Creating Measures' Awkward-ness Task, Example #5 (solution)


Math 'Creating Measures' Awkward-ness Task, Example #5 (solution)



Square-ness, Example #1 (solution) || Steep-ness, Example #2 (solution)
Compact-ness, Example #3 (solution) || Crowded-ness, Example #4 (solution)
Awkward-ness, Example #5 (solution) || Sharp-ness, 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 "awkward-ness"
  • use your measure to put situations in order of "awkward-ness"
  • generalize your measure to work in different situations.



Warm-up


Here are four movie theater situations:

Comment:
The most awkard situation possible is shown below:

In this situation, A sits first, then


  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.)







  1. 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:







  1. Show how you can use your measure to place the four situations in order of "awkward-ness." Show all your work.







  1. 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).







  1. 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:

=  







Square-ness, Example #1 (solution) || Steep-ness, Example #2 (solution)
Compact-ness, Example #3 (solution) || Crowded-ness, Example #4 (solution)
Awkward-ness, Example #5 (solution) || Sharp-ness, Example #6 (solution)