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This problem involves a
significant 'final challenge' which can either be tackled on its
own or after working on a set of related 'building blocks' designed
to lead students to helpful insights. It requires a lot of mental
calculations involving money, and could provide good practice of
these skills while students also work on problem solving
strategies.

Initially working on the
building blocks then gives students the opportunity to work on
harder mathematical challenges than they might otherwise
attempt.

The problem is structured
in a way that makes it ideal for students to work on in small
groups.

Hand out a set of
building block cards (Word, PDF) to each group of
three or four students. (The final challenge will need to be
removed to be handed out later.) Within groups, there are several
ways of structuring the task, depending on how experienced the
students are at working together.

Each student, or pair of
students, could be given their own building block to work on. After
they have had an opportunity to make progress on their question,
encourage them to share their findings with each other and work
together on each other's tasks.

Alternatively, the whole
group could work together on all the building blocks, ensuring that
the group doesn't move on until everyone understands.

When everyone in the
group is satisfied that they have explored in detail the challenges
in the building blocks, hand out the final challenge.

The teacher's role is to
challenge groups to explain and justify their mathematical
thinking, so that all members of the group are in a position to
contribute to the solution of the challenge.

It is important to set aside some time at the end for students to share and compare their findings and explanations, whether through discussion or by providing a written record of what they did.

A teacher comments:

Working on this problem reminded students that maths is sometimes best done by guessing and trying to improve on that guess. Could be used for sequences as well as Trial and Improvement.

What important
mathematical insights does my building block give me?

How can these insights
help the group tackle the final challenge?

How many £10 shall I
include?

If I swap a 50p for a
10p, what does that do to the total amount of money?

Of course, students could
be offered the Final Challenge without seeing any of the building
blocks.

What happens when the
prices change to:

£10 for adults

£1 for
pensioners

50p for children

How many solutions are
there this time?

Encourage groups not to
move on until everyone in the group understands. The building
blocks could be distributed within groups in a way that plays to
the strengths of particular students.