Why do this problem?

This problem requires some simple knowledge of fractions and multiples but demands some clear strategic thinking. It may offer a good opportunity to compare methods between students - there isn't just one route to the solution. Once the problem has been solved, there is good opportunity to discuss the methods used and to decide which students prefer. Note that there is no need to use algebra in this problem.

Possible approach

This problem may be solved using a variety of approaches with different levels of stategic planning. For example, students could use trial and error from the starting number of counters, they could work backwards from the fact that they end up with the same amount or they could use the fact that each child passes a certain fraction to their neighbour. However they choose to work, it is important that students clearly understand the information given to them in the problem and then try to work out a solution strategy which is not totally ad-hoc.

A possible lesson breakdown might be:

[10 minutes] Show the students the problem and as a group discuss the important aspects ofthe problem and the possible initial suggestions of how to solve the problem.

[30 minutes] Ask students to try and find the answers in pairs

[10 minutes] As a group discuss the methods used. What worked? What didn't work? If faced with a similar problem in future, which methods would the class use?

Key questions

What number of starting counters would we need to check for Emma?
Could Ben possibly have started with 10 counters?
What possible numbers of counters could each child end up with?
Will there be one solution? No solution? Many solutions?

Possible extension

How many solutions would there be if the children had up to 100 counters between them?

Possible support

Group students in 3s and provide sets of 40 counters. Ask them to play this game as a group and challenge them to be the first group to find the answer. Use these rules

Decide who will be Emma, Ben and Jack.
Emmachooses any number of counters from the 40 counters
Jack then chooses any number of counters from those left over
Ben then chooses any number of counters from those left over
Emma, Jack and Ben
the second to choose as many as they like from those left over and the last student to choose as many as they like from those left over. Then ask them to pass a fifth, quarter or third of their counters respectively.

by choosing from the

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