Why do this problem?
provides an experimental context to introduce students to one method of estimating populations, where direct counting is not feasible. The method relies on proportional reasoning, and emphasises the value of using an average of several samples to provide a more reliable estimate.
Provide each group of students with a bag (not transparent) containing a reasonably large number of coloured counters (or small multi-link cubes or similar). The bag should contain a number of red (or some other particular colour) counters or cubes - the colour of the rest doesn't matter.
The bag represents the area being sampled.
The red counters/cubes represent the badgers which have been tagged.
The rest of the counters/cubes represent the rest of the population in the given area.
Students start by counting the number of red counters/cubes.
They then put all the counters/cubes back in the bag, and shake them up.
They remove a handful and count how many red ones there are in their sample.
They then count how many counters/cubes there are altogether in their sample.
From the three recorded figures - the number of tagged badgers, the number of tagged badgers in the sample, and the total number of badgers in the sample, they should be able to estimate how many badgers there are altogether in the area:
We assume that the proportion of tagged badgers in the sample is equal to the proportion of tagged badgers in the area.
If you know the proportion of tagged badgers in the sample, how does that help you to estimate how many badgers there are altogether in the given area?
Why might one sample only not provide a good enough estimate?
If you take several samples, how can you combine the results to give a better estimate?
Students should criticise the model and its assumptions.
Proportional reasoning can be conceptually difficult for students.
Rather than using a formula, which simply obscures what is going on, help students who find this difficult with a series of questions:
If half the badgers in the sample are tagged, what proportion of the whole population of badgers would we expect to be tagged?
How many badgers were tagged?
So how big would the population be?
Then vary the proportion, perhaps to a third, or a quarter, to establish the principle that whatever proportion are tagged in the sample, we would expect the same proportion to be tagged in the population. We know the proportion in the sample, so we use that to estimate how many badgers there are in the population.