'Creating Measures' Square-ness 
Task - Example #1 (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:
· 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
Use visual judgements to answer the warm-up questions.
Which rectangle looks the most square?
Which rectangle looks least square?

Without measuring anything, put the rectangles in order of "square-ness."

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:
 A, E and F and G (tie), H, D, C, B, I.
 
 
 
 
 
2. 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:
· It gives no indication of the overall 'proportions'. (E, F and G under this definition have 
the same square-ness yet are clearly different in shape, while C and I are similar in shape 
but give different square-ness measures).
 
· It is dependent on the units used. If we use inches instead of centimetres we get a different 
"square-ness" measure.
 
 
 
 
 
3. 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:
 a) The ratio longest side/shortest side;
 b) The largest angle between the diagonals of the rectangle;
 c) 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 
 d) the ratio (perimeter)2/ area
 
 then we would have a suitable, dimensionless measure.
 
 
 
 
 
4. 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
90û
166û
143û
152û
106û
113û
100û
127û
143û

Ratio: Perimeter2 
¸ area
16
40.5
21.3
25
16.3
16.7
16.1
18
21.3


 
 
 
 
5. 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.
 
 
 
 
 
6. 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.