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

Warmup
Use visual judgements to answer the warmup questions.
Which rectangle looks the most square?
Which rectangle looks least square?
Without measuring anything, put the rectangles in order of "squareness."
Comment:
This first question is simply intended to orientate the students to the task in hand. It may be used as a class discussion.
 Someone has suggested that a good measure of "squareness" is to calculate the difference:
Longest side  shortest side
for each rectangle. Use this definition to put the rectangles in order of "squareness."
Show all your work.
Solution:
Using the measure 'Longest side  shortest side', the "squareness" 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 
Squareness (cm) 
0 
7 
4 
3 
1 
1 
1 
2 
8 
Using this measure, the rectangles in order from most to least square are:
A, E and F and G (tie), H, D, C, B, I.
 Using your results, give one good reason why Longest side  shortest side is not a suitable measure for "squareness."
Solution:
The above measure is unsatisfactory because:
 It gives no indication of the overall 'proportions'. (E, F and G under this definition have the same squareness yet are clearly different in shape, while C and I are similar in shape but give different squareness measures).
 It is dependent on the units used. If we use inches instead of centimetres we get a different "squareness" measure.
 Invent a different way of measuring "squareness." Describe your method carefully below:
Solution:
There are many other ways of measuring "squareness." Students might, for example, propose using:
 The ratio longest side/shortest side;
 The largest angle between the diagonals of the rectangle;
 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 "squareness."
If, however, we use
 the ratio (perimeter)^{2} / area
then we would have a suitable, dimensionless measure.
 Place the rectangles in order of "squareness" 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 "squareness" 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 
90^{o} 
166^{o} 
143^{o} 
152^{o} 
106^{o} 
113^{o} 
100^{o} 
127^{o} 
143^{o} 
Ratio:
Perimeter^{2} ÷ area 
16 
40.5 
21.3 
25 
16.3 
16.7 
16.1 
18 
21.3 
 Do you think your measure is a good way of measuring "squareness?" 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.
 Find a different way of measuring "squareness."
Compare the two methods you invented. Which is best? Why?
Solution:
This question provides an opportunity for students to look for an alternative measure.
Squareness, Example #1 (solution)  Steepness, Example #2 (solution)
Compactness, Example #3 (solution)  Crowdedness, Example #4 (solution)
Awkwardness, Example #5 (solution)  Sharpness, Example #6 (solution)
