Why do this problem?
This problem may support a deeper understanding of averages. The unexpected result may lead students to ask what else might be possible. Encouraging students to ask questions and organise their work in a systematic way in order to draw conclusions are all key mathematical skills that can be encouraged.
Before introducing the problem, revisit finding the mean through this simpler question:
Draw up the following table on the board:
The table shows the ages of five children. What is the average (mean) age?
If Anna leaves the group, what happens to the average?
If Cayley leaves the group instead, what happens to the average?
If Erin leaves the group instead, what happens to the average?
Ensure that students notice that the average can stay the same, go up or go down depending on whether the age of the child leaving is the same as, less than or greater than the average.This is the focus of the main problem.
Introduce the main problem.The first part should now be straightforward, so students can devote their thinking to the follow up question:
What other effects can moves between countries have on average incomes?
This question may need fleshing out -
What are the possibilities?
What are the variables that can be altered?
Students (perhaps working in pairs) could be asked to present their findings. This may offer an opportunity to reflect on the value of approaching the work in a systematic way.
What could cause the mean to increase?
What could cause the mean to decrease?
Is it possible to double one country's average income whilst halving the other?
Interested students may also wish to consider whether there are contexts where this statistical manipulation may be used to advantage.
You may choose to offer the following specific example.
Group 1: ages 2, 7, 11, 14 and 16 (average 10)
Group 2: ages 5, 11, 12, 13 and 19 (average 12).
Investigate what happens to the averages when different members move from one group to the other.