'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)	37˚	34˚	27˚	45˚	34˚	21˚	

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.