
Tools  Math 'Creating Measures' Squareness Task, Example #1
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."
 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.
 Using your results, give one good reason why Longest side  shortest side is not a suitable measure for "squareness."
 Invent a different way of measuring "squareness." Describe your method carefully below:
 Place the rectangles in order of "squareness" using your method. Show all your work.
 Do you think your measure is a good way of measuring "squareness?" Explain your reasoning carefully.
 Find a different way of measuring "squareness."
Compare the two methods you invented. Which is best? Why?
Squareness, Example #1 (solution)  Steepness, Example #2 (solution)
Compactness, Example #3 (solution)  Crowdedness, Example #4 (solution)
Awkwardness, Example #5 (solution)  Sharpness, Example #6 (solution)
