### Why do this problem?

This problem starts with a simple situation which can be
analysed quickly using mental methods, but which provides a
starting point for tackling a more challenging problem. The
challenge can be tackled at many different levels, using trial and
improvement (perhaps using spreadsheets), looking for number
patterns, or with a more formal algebraic approach.

### Possible approach

Introduce the first problem. Solve it together with the class.
Can anyone explain why everything works out so neatly?

Now share the second problem. Students may wish to use trial
and improvement to solve it (using spreadsheets might make things
quicker). Alternatively, they may use insights from the first
problem to suggest starting points. For those who are
confident using algebra, the problem can be expressed and solved
using equations.

Share approaches and answers to the second problem and
encourage the students to use insights from this discussion to help
them solve the third problem.

After solving the third problem, students who are less confident
with algebra could be encouraged to look at the patterns in the
results so far and to suggest generalisations based on what they
have seen. Then they could test their conjectures with a variety of
examples of their choice.

### Key questions

How can we express each part of this problem algebraically?

### Possible Extension

Students who are more confident can be challenged to write their
generalisation algebraically, and manipulate the algebraic
expressions to produce a rigorous proof that the results will
always hold.

### Possible Support

Peaches Today,
Peaches Tomorrow.... might offer a suitable introduction to the
fraction skills required in this problem.