'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.
____________________________________________________
· 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:
=