'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.