Many standard questions give exactly the information required
to solve them, with one or two standard approaches signposted in
the question. This problem is different, in that learners are given
a large quantity of information to sort through and make sense of
for themselves in order to reach a solution. Along the way,
learners will have to make choices about how to proceed - the
opportunity to make such choices in problem solving is an important
part of every child's educational experience.

Begin the lesson by dividing the board into two columns, one
headed with a tick and the other headed with a cross.

Ask learners to suggest numbers, and write each suggestion in
the appropriate column according to a rule of your own choice. Make
it clear to the class that the activity is designed to model
scientific enquiry, so they can come up with a hypothesis for your
rule, but you will not confirm their hypothesis, you will only
place numbers in the appropriate column.

Here are some suggestions for rules which will not come up in
the main activity:

- Odd numbers
- Negative numbers
- Numbers which are not whole numbers
- Prime numbers
- Triangular numbers
- The sum of the digits is odd
- The numbers are always one more than multiples of 3

Once the class have tried the activity with a couple of rules
until all are reasonably convinced their hypothesis holds, move on
to the main task.

For the main activity, arrange the class into pairs or small
groups. Hand out a set of these cards (cut out in
advance) to each group, and introduce the task as it is described
in the problem.

Towards the end of the time working on the problem, leave some
time for the class to come together to discuss how they approached
the task, the decisions they made about how to organise themselves,
and the justifications for the conclusions they came to.

The second set of
cards could be used as a follow-up some time later, with
discussion afterwards focusing on whether they worked more
efficiently having attempted a similar problem before.

How will you sift through the data?

How will you record your current thoughts?

How will you check your hypotheses?

Are there any cards that don't fit in with your hypotheses?The problem Charlie's Delightful
Machine offers a similar task based on an interactive
environment.

There is a follow-up problem, A Little Light
Thinking which would be suitable for the higher attaining
students.