Why do this problem?
draws learners into ideas concerning curve fitting and
the way in which curve fitting problems can be greatly simplified
by using 'sensible' coordinate systems and the way in which
length-preserving transformations can be used to create these.
An important part of this problem is students coming to the
realization that the problem is greatly simplified by choosing the
right sorts of coordinates. Students might benefit from doing this
question individually and it might make a good starter activity to
return to once students have had a chance too think about it.
How can the transformations be used to create a 'simple' cubic
The extension referred to in the equation is very interesting.
You might like to inform students that there is in fact an
algebraic solution to any cubic equation, which they might use in
One way into this problem might be to draw a few simple cubic
equations, find their turning points and then work out the squared
distance between them. Alternatively, first try the easier problem
which will start to get students to think about
cubic equations and turning points.