Ring a Ring of Numbers
Here is a picture of four numbers placed in squares on a circle so that each number is joined to two others:
What do you see?
What do you notice?
Choose four numbers from this list: 1, 2, 3, 4, 5, 6, 7, 8, 9 to put in the squares so that the difference between joined squares is odd.
Only one number is allowed in each square. You must use four different numbers.
What can you say about the sum of each pair of joined squares?
What must you do to make the difference even?
What do you notice about the sum of the pairs now?
Here are some sheets
for recording your solutions.
This problem is based on an idea taken from "Apex Maths Pupils' Book 2" by Ann Montague-Smith and Paul Harrison, published in 2003 by Cambridge University Press.
Why do this problem?
provides a context in which children can recognise odd and even numbers, and begin to think about their properties. It also offers practice in addition and subtraction.
It would be good to show the image in the problem to the class and ask what they notice, and whether they have any questions. Give them time to consider on their own, then to talk to a partner. Invite learners to offer their noticings and questions but try not to say anything more than "thank you" as they share their thoughts with everyone. Rather than answering any questions yourself,
encourage other members of the group to respond.
Use the ideas that have been offered to build up to introducing the task as stated and give pairs of children chance to find at least one way of making odd differences. They could be using this sheet of blank circles and/or digit cards. You could invite pairs to record arrangements that work on the board as they find them and invite everyone to check that they are indeed solutions.
Once you have several ways on the board, invite learners to comment on what they notice. What do all the arrangements have in common? You can work through the rest of the problem in a similar way, drawing the whole class together as appropriate.
It is important to encourage the children to explain why the arrangements of odd/even numbers produce these results. You could make drawings like these using paired joined squares to help deepen their understanding.
What do you notice about the numbers in the ring when the difference between joined pairs is odd?
What do you notice about the numbers in the ring when the difference between joined pairs is even?
Can you explain why?
Some learners might benefit from having counters or other objects to help with their addition and subtraction.