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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 functions 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 an natural framework on 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. Throughout the task encourage discussion and explanation
of results either in pairs or to the class. An important aspect of
the task is that students will discover, and struggle with
describing, the concepts which will later become formalised with
calculus.

Start by suggesting that
students draw a pair of coordinate axes and roughly sketch a curve
which turns once (gradient changes sign). Ask them to locate the
places on the x-axis where either the sign of their curve or the
sign of its gradient changes.

Continue with a
class discussion of which combinations of signs are possible for
curves which turn once. Follow any interesting conjectures and
spend time to unpick any confusion in definitions.

Elementary calculus concerns curves
which are continuous, differentiable and have no asymptotes.
However, perfectly reasonable curves exist which violate any or all
of these properties (mod function, tan etc.). Furthermore, the
concept of a function far transcends the notion of a curve (some
functions don't have a curve) and there are nice curves (such as
circles) which are not functions. In trying to create their curves
in this problem students will most likely introduce examples which
are not functions, not continuous, not differentiable or have
asymptotes. If this happens, great! Discussion concerning the sorts
of curves which are appropriate for calculus will be a valuable
learning point.

Once the concept is
firmly understood, students can explore individually or in small
groups the possibilities for curves which turn twice and then
try systematically to find examples with 1 through to 6
possibilities for sign change.

Students might get
involved with this task and start to experiment with curves or
functions which break the conventional rules for functions. If this
occurs, great! (as part of the subtlety with calculus concerns
curves which are non-differentiable.)

How can you sketch a
curve where the sign of the gradient changes at some point?

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

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

Can you think of any
different 'types' of curves which have a particular combination of
sign changes?

Skilled students can be
encouraged to construct particularly interesting examples:
pushing the boundaries of the problem is a good thing. For example,
might there exist a curve with this combinations of signs?:

Sign of function | + | + | + | - | - | - | + | + | + |

Sign of function | + | - | + | - | + | - | + | - | + |

Why or why not?

If people are struggling
to start, sketch a curve with a few turns on the board and
partition into regions of sign change collectively. Then suggest
one person draws a curve and explains to the group the regions of
sign change.

Students who are
algebra-focussed might not perceive this 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.