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Calculus Analogies

Age 16 to 18 Challenge Level:

Why do this problem?

This fun problem will hopefully prove incredibly useful to all students: having a sound geometrical visualisation for concepts in calculus is essential in any application beyond the simplest algebraic examples and also proves very useful in checking that calculations make sense. It will also be very useful for uncovering misconceptions about calculus.

Possible approach

You might want to hand out these cards in Word 2003 or pdf format so that students can more easily consider the statements under discussion. 
This need not be a long activity and can be used at any point in the curriculum where the concepts in any of the 5 analogies have been encountered. You can focus on a couple of the most relevant analogies if desired.
You could simply set up the situation and let the students enter into discussion. Students can think about the ideas in small groups and sketch 'road maps' on which to test their ideas.
Alternatively, you can sketch a curve with, say, 4 turning points on the board and ask for a volunteer to model the motion of the imaginary steering wheel as you trace your finger along the curve. Another volunteer can record the motion of the steering wheel, paying particular attention to the direction or speed of turn. You could then sketch a more 'demanding' road and repeat the exercise.
There are at least three levels of approach to this problem:
1) Once students are intuitively clear as to which analogies are largely reliable the lesson can move on and the analogies can be referred to as a guide throughout subsequent study of calculus.
2) Students can try to construct convincing justification that the analogies are sound, including some thought on when the analogies break down (i.e. what sorts of roads do the analogies work for, and what sorts of 'pathological' roads do the examples not work for?)
3) Students might try to come up with some analogies of their own which others might test out. For example, other analogies for the sign of the gradient might involve mountains, valleys or hills.
Note that various misconceptions might be unearthed during this task, and many more advanced concepts in mathematics might be raised by students. See the possible support below for some of these.

Key questions

Who can drive a car? Who can describe the motion of a wheel through a journey? 
Can you imagine driving along the road indicated on this map?
For what sorts of crazy curves might these analogies not work?
Can you give a clear justification for you results (using words or algebra)?
What can we say about a car which is moving due north at some point?

Possible extension

The key advanced extension is to try to create analogies for other concepts in calculus. This is very open ended, but will really get students thinking about calculus as the mathematics of rates of change. 
All students might naturally move on to the problem Patterns of Inflection after trying this problem.

Possible support

Some students, who equate mathematics with algebra, might struggle to see this as 'mathematics'. Reassure them that the visualisation practiced and the explanations constructed are a key part of advanced mathematical thinking.
Some students, even the most traditionally 'able', might find the visualisation aspect of this problem extremely difficult. Such students need to be encouraged not simply to give up and to exercise this part of their mathematical brain. Perhaps others in the group might try to explain the concepts to them?
Misconceptions or errors to look out for are:
1. The steeper the gradient the more the wheel needs to be turned
2. A function can be used to describe, say, a circle (No: A function is single valued)
3. A point of inflection must also be a stationary point (No: That is a stationary point of inflection)
Advanced concepts in mathematics which might be raised in some form are:
1. What is a function as opposed to a curve?
2. What is a continuous / differentiable function?
3. What is curvature?
4. Are there functions which are only twice differentiable?