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Where Can We Visit?

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?


Can you find the values at the vertices when you know the values on the edges?

Babylon Numbers

Can you make a hypothesis to explain these ancient numbers?

Four Coloured Lights

Age 11 to 14
Challenge Level

Why do this problem?

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.

Possible approach

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.

Key questions

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?

Possible extension

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.

Possible support

Hand out four coloured pieces of paper or post-it notes for learners to record the information gathered about each light.