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. At one level it provides self-checking practice of the
algorithms for column addition and short division. At its heart it
is about challenging and developing students' understanding of
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
) 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
Alternatively, the whole group could work together on all the
building blocks, ensuring that the group doesn't move on until
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
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
What important mathematical insights does my building block
How can these insights help the group tackle the final
Of course, students could be offered the Final Challenge
without seeing any of the building blocks.
After establishing a general proof, students might like to try
some similar problems:
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.
For students who have a weak grasp of place value, Diagonal Sums
gives addition practice in a context where the place value
structure is quite explicit.