FLAG - Tools: Math 'Creating Measures' Compact-ness Task, Example #3


Math 'Creating Measures' Compact-ness Task, Example #3



Square-ness, Example #1 (solution) || Steep-ness, Example #2 (solution)
Compact-ness, Example #3 (solution) || Crowded-ness, Example #4 (solution)
Awkward-ness, Example #5 (solution) || Sharp-ness, 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 "compact-ness"
  • invent your own way of measuring this concept
  • refine your scale so that it measures from 0 to 1.

Over recent years, a number of geographers have tried to find ways of defining the shape of an area. In particular, they have tried to devise a measure of 'compactness'. You probably have some intuitive idea of what "compact" means already. Below are two islands. Island B is more compact than island A. "Compact-ness" has nothing to do with the size of the island. You can have small, compact islands and large compact islands.


Warm-up

 


  1. Calculate the "compactness" of each of the following 'islands' using the above definition.





  2. Use your results to explain why Area ÷ Perimeter is not a suitable definition for "compactness."





  3. Invent your own measure of "compactness".
    Put the shapes A to F in order of "compact-ness" using your measure. Discuss whether or not your measure is better than 'Area ÷ Perimeter.'





  4. Adapt your measure so that it ranges from 0 to 1.
    A perfectly compact shape should have a measure of '1,' while a long, thin, shape should have should have a measure near to 0.





Square-ness, Example #1 (solution) || Steep-ness, Example #2 (solution)
Compact-ness, Example #3 (solution) || Crowded-ness, Example #4 (solution)
Awkward-ness, Example #5 (solution) || Sharp-ness, Example #6 (solution)