### 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 is well suited for
students who are working on the difference of two squares.

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

This task might ideally
be completed in groups of three or four.

Hand out a set of
building block cards (

Word,

PDF) to each group. (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.

### Key questions

What important
mathematical insights does my building block give me?

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

### Possible extension

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

### 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.

The final challenge could
also be done as a proof sorter activity using this set of cards (

Word,

PDF).