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
If Anna leaves the group, what happens to the average?
If Cayley leaves the group instead, what happens to the
If Erin leaves the group instead, what happens to the
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
This question may need fleshing out -
What are the possibilities?
What are the variables that can be
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
In what ways can the means change?
The open nature of this problem offers opportunities for students
to think of other effects and test whether they are possible. For
Is there a limit to the number of Charlie's friends who can leave
Emerald and have the same effect?
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
You may choose to offer the following specific example.
Group 1: ages 2, 8, 10, 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.