Why do this problem?
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.
Possible approach
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.
Key questions
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?
Possible extension
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?
Possible support
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.