FLAG - Tools: Math 'Creating Measures' Steep-ness Task, Example #2 (solution)


Math 'Creating Measures' Steep-ness Task, Example #2 (solution)




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 "steep-ness"
  • invent your own ways of measuring this concept
  • examine the advantages and disadvantages of different methods.



Warm-up

Comment:
This first question is simply intended to orientate the students to the task. It may be used as a class introduction.


  1. Someone has suggested that a good measure of "steep-ness" is to calculate the difference:

    Height of step - length of step

    for each staircase. Use this definition to put the staircases in order of "steep-ness." Show all your work.



Solution:
Using the measure 'height of each step - length of each step', the 'steep-ness' of each staircase is given in the table below (using centimeters as the unit).

Staircase
A
B
C
D
E
F
Height (cm)
1.5
1
0.5
1
2
1.25
Length (cm)
2
1.5
1
1
3
3.33
Height-length (cm)
-0.5
-0.5
-0.5
0
-1
-2.08

Using this measure, the staircases in order from most to least steep are:







  1. Using your results, give reasons why Height of step - length of step is not a suitable measure for "steep-ness."



Solution:
The above measure is unsatisfactory because:







  1. Invent a better way of measuring "steep-ness." Describe your method carefully below:



Solution:
There are many other ways of measuring "steep-ness." Students might, for example, propose using:

  1. The angle of inclination;

  2. The ratio of 'step height'/'step length' (technically: riser/run);

  3. The ratio of 'height of whole staircase'/ 'length of whole staircase';
These are equally sensible, and equivalent, except is may be sometimes unclear what we measure as the 'length' of the staircase.







  1. Place the staircases in order of "steep-ness" using your method. Show all your work.



Solution:
Whichever measure we now use (a), (b) or (c), we obtain the same order for the staircases.

Staircase
A
B
C
D
E
F
Height (cm)
1.5
1
0.5
1
2
1.25
Length (cm)
2
1.5
1
1
3
3.33
Height ÷ length
(2 d.p.)
0.75
(3/4)
0.67
(2/3)
0.5
(1/2)
1
(1/1)
0.67
(2/3)
0.38
(3/8)
Angel of inclination (nearest degree)
37o
34o
27o
45o
34o
21o

This gives the order of steep-ness (from most to least steep) as:







  1. Do you think your measure is a good way of measuring "steep-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.







  1. Describe a different way of measuring "steep-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.







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)