### Why do this problem :

Beyond the reasoning challenge to sort out how the fraction
re-caught suggests an estimate for the size of the whole
population, this problem introduces thoughtful students to the
concept of 'confidence' in hypothesis testing ' consideration of
how likely an estimate is to be wrong by some specified
amount.

### Possible approach :

Present the group with this problem on paper and ask them to read
it and discuss in pairs what the situation is and what is asked
for. This may lead some pairs to successfully solve the problem, in
which case the main activity now becomes the task of explaining not
just the calculation, but also the justification, to the other
students in the group. If however this is a problem that isn't
quickly solved a simulation with counters or coloured cubes is an
excellent aid to visualisation.

### Key questions :

- Describe the procedure used.What is
this procedure trying to do?

- Is it the actual population or an
estimate?

- How close do you think it is?

### Possible extension :

Conduct the same simulation as below for 'Possible Support' but
draw attention to the variation that occurs as the simulation is
repeated, and invite students to investigate how much their
calculated estimates vary and in general use of the term how
confident they might think it safe to be with their estimate. For
example what 'plus or minus' amount might they attach to their
answer. This situation is then gradually generalised to different
size populations and different relative size of sample.

### Possible support :

Simulation with counters or coloured cubes is the most useful
aid to visualisation. For example put 20 counters into a bag and
explain that the bag is the pond and the whole fish population in
this instance is 20. Remove 5 counters and replace them with
counters of a different colour, explaining that these five are the
first sample, and the different colour allows the counters 'caught'
for a second time to be identified. Now make the second sample of
five.

This establishes the context or procedure being discussed so that
attention can now rest on solving the problem.

A population of 100, with a sample size of 20, might give
estimates closer to the actual population, and this may perhaps
help students to see how to use the fraction re-caught.