The Tower of Hanoi is a
well-known mathematical problem which yields some very interesting
number patterns. This version of the 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.

Initially working on the
building blocks gives students the opportunity to then 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.

Start by explaining how
the Tower of Hanoi game works, making clear the rules that only one
disc can be moved at a time, and that a disc can never be placed on
top of a smaller disc. This interactivity could be used
to show how the game works.

Hand out a set of
building block cards (Word, PDF) to groups 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.

What important
mathematical insights does my building block give me?

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

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.

Handouts for teachers are
available here (word document,
pdf
document), with the problem on one side and the notes on the
other.