This problem allows students to explore the process of making
mathematical conjectures based on visual intuition. The conjectures
can then be proved/disproved following a straightforward exercise
in finding turning points (whilst taking care over distinguishing
the various cases). This is good training and practice of key
mathematical skills.

#### Possible approach

Have a class discussion about the positions of X that might
lead to a minimum of the function before anyone actually writes
anything down. Consider and discuss what happens to the function as
X moves along the line segment from P to Q. What conjectures do the
class come up with? Is there a general agreement or are multiple
conjectures stated?

After the students have found the turning points have the
class reflect on whether their findings agree with the original
conjecture(s) and if not why not?

#### Key questions

- What is a conjecture?
- What features of the geometric situation do you notice?
- How do you expect the function to change as X moves from P to
Q?
- Would you expect the position of X giving a minumum value of
the function to be dependent or independent of scaling, that is
would this position of X be the same relative to the endpoints for
a short line segments as when the line segment was stretched to
make it longer

and, after doing the calculus:

- What does the graph of the function look like as x varies?
- Does this graph look the same whatever the length of the line
segment?
- How many turning points are there?
- Does the nature of the turning points, and/or the position of X
where the turning points occur, depend on the value of the constant
a?
- Did we miss any of the possibilities when we made our initial
conjecture and if so why?

Possible extension

Consider generalisations of this type of problem.

Possible support

Students might struggle with the concept of a conjecture as a
precise mathematical statement which can be proved or disproved
following further investigation. You might like to suggest these
possibilities and to vote on them

- I think that the location of the minimum does not depend on the
value of a
- I think that the location of the minimum does depend of the
value of a
- I think that the minimum values are always located in the
centre of the line.