Ring a ring of numbers
Problem
Ring a Ring of Numbers printable sheet
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 out of 1, 2, 3, 4, 5, 6, 7, 8 and 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.
Getting Started
Try putting one of the numbers in any square to start with. What numbers could go on each side of it?
When you add the numbers in two joined squares, what kind of number do you get?
You might like to print off this sheet of blank rings to help you try out some different numbers.
Student Solutions
Thank you to everybody who sent in their ideas about this task. Harper from Bedfordshire shared what they noticed:
I found that if you put odd next to odd it makes an even difference. If you put even next to even it makes an even difference too. But if you put even next to odd you get an odd difference. So if you want an odd difference you have to put even, odd, even, odd. If you want an even difference you have to put all odd or all even.
Rukmini from Hopscotch Nursery noticed that there is a pattern in the sums of the pairs of numbers:
When the differences are all odd, the sums are all odd.
Well spotted! I wonder why the sums are all odd when the numbers are even, odd, even and odd?
Teachers' Resources
Ring a Ring of Numbers
Ring a Ring of Numbers printable sheet
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 out of 1, 2, 3, 4, 5, 6, 7, 8 and 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?
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.
Possible approach
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.
Key questions
Possible extension
More Numbers in the Ring allows children to investigate different numbers of numbers in the ring.
Possible support
Some learners might benefit from having counters or other objects to help with their addition and subtraction.