
Tools  Math 'Creating Measures' Steepness Task, Example #2 (solution)
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 "steepness"
 invent your own ways of measuring this concept
 examine the advantages and disadvantages of different methods.

Warmup
Without measuring anything, put the above staircases in order of "steepness."
Comment:
This first question is simply intended to orientate the students to the task. It may be used as a class introduction.
 Someone has suggested that a good measure of "steepness" is to calculate the difference:
Height of step  length of step
for each staircase. Use this definition to put the staircases in order of "steepness."
Show all your work.
Solution:
Using the measure 'height of each step  length of each step', the 'steepness' 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 
Heightlength (cm) 
0.5 
0.5 
0.5 
0 
1 
2.08 
Using this measure, the staircases in order from most to least steep are:
D, A and B and C (tie), E, F.
 Using your results, give reasons why Height of step  length of step is not a suitable measure for "steepness."
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 "steepness" measure.
 It is usually negative, which is inelegant and awkward to use.
 Invent a better way of measuring "steepness." Describe your method carefully below:
Solution:
There are many other ways of measuring "steepness." Students might, for example, propose using:
 The angle of inclination;
 The ratio of 'step height'/'step length' (technically: riser/run);
 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.
 Place the staircases in order of "steepness" 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) 
37^{o} 
34^{o} 
27^{o} 
45^{o} 
34^{o} 
21^{o} 
This gives the order of steepness (from most to least steep) as:
D, A, B and E (tie), C and F.
 Do you think your measure is a good way of measuring "steepness?" 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.
 Describe a different way of measuring "steepness."
Compare the two methods you invented. Which is best? Why?
Solution:
This question provides an opportunity for students to look for an alternative measure.
Squareness, Example #1 (solution)  Steepness, Example #2 (solution)
Compactness, Example #3 (solution)  Crowdedness, Example #4 (solution)
Awkwardness, Example #5 (solution)  Sharpness, Example #6 (solution)
