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This problem is well
suited for those who are about to begin to learn the
concepts of calculus. It is easy to access, yet offers many
insights into the relationships between funtions and their
derivatives. The language of calculus - change, derivative, turning
points, maximum, minimum, curve, functions, equations, axes, zeros,
continuity etc. - should naturally arise in the exploration
of this task and it should provide a natural framework onto which
to build the formality of calculus at a later date.

As with most NRICH tasks,
this problem is low threshold-high ceiling, so it also will prove
an interesting exploration for the more sophisticated
thinker.

You can use this task before the words
'turning points' and 'derivative' have been introduced to the
students, although he concept of 'continuous' might need explaning.
This condition is quite restrictive but does allow for curves which
are not differentiable, such as the modulus function.
Throughout the task, encourage discussion and explanation of
results.

This problem works well
as a group task.

Start by suggesting that
students draw a pair of coordinate axes and roughly sketch a curve
with one point of zero gradient. Ask them to locate the places on
the x-axis where either the sign of their curve or the sign of
its gradient changes. Collect responses concerning the
various numbers of zeros in the class, and sketch on the board
those with exactly two zeros. Discuss whether there are any other
possibilities, with students sketching any other possibilities
found on the board.

Once the group is
convinced that all possibilities have been found, move on to the
next question where students are asked to sketch a curve with
exactly two points of zero gradient. Ask them to try to make the
most interesting curve they can with exactly two points of zero
gradient.

Discuss the interesting
curves arising and sketch on the board those with exactly 2
zeros. Discuss whether there are any other possibilities, with
students sketching any other possibilities found on the
board.

Once this part is
complete, allow students to continue with the full task
individually or in small groups.

It is expected that questions
concerning the notion of differentiability, turning points,
points of inflection and asymptotes will emerge. These are the key
geometric concepts of A-level calculus. Higher levels of continuity
will also emerge if students create curves with vertical asymptots,
which are not continuous at the asymptotes.

Can you describe to a
friend what a continuous curve is?

What can you say about a
curve which alternates in sign?

What happens to your
results if you shift the axes up or down?

What is the possible
behaviour of a curve as it tends to infinity and minus
infinity?

Particularly interested
students can be encouraged to construct
particularly interesting examples: pushing the boundaries of
the problem is a good thing and they should be encouraged to create
particularly clear explanations that their catalogues are
complete.

You can also extend the
task to involve asymptotes and ask, "Explain, with justification,
for which whole number values of $A, B, C$ can you create a
continuous (except at the asymptotes) curve with $A$ zeros, $B$
turning points and $C$ vertical asymptotes

If people are struggling
to start, you can sketch a few arbitrary curves on the board and
ask students to locate the points of zero gradient and the zeros.
Then suggest that they work in pairs and do the same to get a feel
for the problem.

Some precise and neat
students might be resistant to making rough sketches. Encourage
them and make it clear that it is allowed and expected in this
task! Conversely, some rough and messy students might be resistant
to neatly sketching the results of their experiments. Encourage
them to understand that writing up neatly is an essential part of
this task.

Students who are
algebra-focussed might not perceive some of this task to be
'maths'. Reassure them that mathematical reasoning is taking place
and that constructing clear, precise examples and explanations is
highly mathematical and will provide a very solid foundation for
the algebraic calculus which will follow.