Why do this 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.
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
There are at least three levels of approach to this
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
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
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
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
For what sorts of crazy curves might these analogies not
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
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.
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
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