Why do this problem?
was designed for the 2018 National Young Mathematicians' Award
, so it might be a particularly useful activity for a small group of your highest-attaining pupils to work on. It encourages a systematic approach and provides an opportunity for pupils to explain how they know
they have all the solutions.
Begin by showing children a paper number strip with numbers from 0-12 equally spaced. Fold the strip in half so that the 0 is on top of the 12, and talk about the pairs of numbers that are together. (In this case, stack totals all equal 12 except for the 6 which is on its own.)
Next, unfold the strip paper and re-fold it at 8 and 4. Discuss the phrase 'stack totals' - we want to know the totals of the numbers which are in a 'stack'. Here, the stacks are: 0 and 8; 4 and 12; 1, 7 and 9; 2, 6 and 10; 3, 5 and 11. Ask pupils to work out what the stack totals would be (these are 8, 16, 17, 18 and 19). Is there a pattern here? Why? Encourage pupils to look at how the numbers
in each stack change as you move down the strip of folded paper.
Once pupils have understood how the folding and stack totals work, give them time to work in small groups on the three challenges.
Explain what you are doing to find the next solution.
Tell me about the stack totals you've got.
How are you trying to get a particular total?
Tell me about how you decide where to make the folds.
Pupils could explore what happens to stack totals if the length of each fold can vary. For example, in the case of a 12 number line, you could have folded at 2 and 8. In this case you have 2 and 8 each on their own, then: 1 and 3; 11 and 5; 10 and 6; 9 and 7; 0, 4 and 12. In that case, the stack totals would be 2, 4, 16, 16, 16, 16 and 8.
Some pupils may need help in producing the number strips so that they can start the task. Providing lined or squared paper will help the children space the numbers equally on the strips.
Sharing different ways of thinking about the problem and different ways of recording that thinking can be very useful in developing all learners' problem-solving skills.