## Tools - 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

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:

D, A and B and C (tie), E, F.

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:

• 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.

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:

D, A, B and E (tie), C and F.

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)

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