'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.

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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	90˚	166˚	143˚	152˚	106˚	113˚	100˚	127˚	143˚	
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.