'Creating Measures' Steep-ness 
Task - Example #2 (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 "steep-ness"
· invent your own ways of measuring this concept
· examine the advantages and disadvantages of different methods.

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Warm-up

Without measuring anything, put the above staircases in order of "steep-ness."

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:<BR>
D, A and B and C (tie), E, F.
 
 
 
 
 
2. 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:
· It gives no real indication of the steepness. Using this measure, A and C are labeled 
as equally steep, which does not fit with intuition. 
 
· It is dependent on the units used. If we use inches instead of centimetres we get a 
different "steep-ness" measure.
 
· It is usually negative, which is inelegant and awkward to use.
 
 
 
 
 
3. 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:
a) The angle of inclination;
b) The ratio of 'step height'/'step length' (technically: riser/run);
c) 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. 





4. 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)

Angle of inclination 
(nearest degree)
37û
34û
27û
45û
34û
21û


This gives the order of steep-ness (from most to least steep) as:
D, A, B and E (tie), C and F.
 
 
 
 
 
5. 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.  
 
 
 
 
 
6. 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.