FLAG - Tools: Math 'Creating Measures' Square-ness Task, Example #1 (solution)


Math 'Creating Measures' Square-ness Task, Example #1 (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 problem gives you the chance to:
  • criticise a given measure for the concept of "square-ness"
  • invent your own ways of measuring this concept
  • examine the advantages and disadvantages of different methods.



Warm-up

Comment:
This first question is simply intended to orientate the students to the task in hand. It may be used as a class discussion.


  1. Someone has suggested that a good measure of "square-ness" is to calculate the difference:

    Longest side - shortest side

    for each rectangle. Use this definition to put the rectangles in order of "square-ness." Show all your work.



Solution:
Using the measure 'Longest side - shortest side', the "square-ness" of each rectangle is given in the table below (using centimeters as the unit).

Rectangle
A
B
C
D
E
F
G
H
I
Dimensions (cm)
3 x 3
1 x 8
6 x 2
4 x 1
3 x 4
3 x 2
6 x 5
4 x 2
12 x 4
Square-ness (cm)
0
7
4
3
1
1
1
2
8

Using this measure, the rectangles in order from most to least square are:







  1. Using your results, give one good reason why Longest side - shortest side is not a suitable measure for "square-ness."



Solution:
The above measure is unsatisfactory because:







  1. Invent a different way of measuring "square-ness." Describe your method carefully below:



Solution:
There are many other ways of measuring "square-ness." Students might, for example, propose using:

  1. The ratio longest side/shortest side;

  2. The largest angle between the diagonals of the rectangle;

  3. The ratio of perimeter/area.
a) and b) seem equally sensible. c), however, suffers the same problem as before. As it is not dimensionless, an enlargement of a rectangle will result in a different value for its "square-ness."

If, however, we use

  1. the ratio (perimeter)2 / area

then we would have a suitable, dimensionless measure.







  1. Place the rectangles in order of "square-ness" using your method. Show all your work.



Solution:
Whichever measure we now use (a), (b) or (d), we obtain the same order for the rectangles. In order of "square-ness" they are:

    A (most square), G, E, F, H, C and I (tie), D, B (least square).
Rectangle
A
B
C
D
E
F
G
H
I
Dimensions (cm)
3 x 3
1 x 8
6 x 2
4 x 1
3 x 4
3 x 2
6 x 5
4 x 2
12 x 4
Ratio:
longest ÷ shortest
1
8
3
4
1.3
1.5
1.2
2
3
Largest angle between diagonals
90o
166o
143o
152o
106o
113o
100o
127o
143o
Ratio:
Perimeter2 ÷ area
16
40.5
21.3
25
16.3
16.7
16.1
18
21.3







  1. Do you think your measure is a good way of measuring "square-ness?" Explain your reasoning carefully.



Solution:
Here we would like students to review their results critically and decide whether the results from their measurements accord with their intuitions.







  1. Find a different way of measuring "square-ness."
    Compare the two methods you invented. Which is best? Why?



Solution:
This question provides an opportunity for students to look for an alternative measure.







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)