More Numbers in the Ring
Problem
More Numbers in the Ring printable sheet
Change the ring so that there are only three squares.
Can you place three different numbers in them so that their differences are odd?
Can you make the differences even?
What do you notice about the sum of each pair in each case?
Try with different numbers of squares around the ring.
What happens with 5 squares? 6 squares?
What do you notice?
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 these sheets of blank rings to help you try out some different numbers:
Sheet with 3-number rings
Sheet with 4-number rings
Sheet with 5-number rings
Sheet with 6-number rings
Student Solutions
Vidhya from Kensri School in India sent in a very well reasoned solution:
When we subtract an even number from an odd number, or vice versa, the difference is always odd. So if we fill up odd numbers and even numbers alternately, if there are an even number of squares, the differences will all be odd. But there is no solution (in other words the differences cannot all be odd) if there is an odd number of squares.Teachers' Resources
More Numbers in the Ring
More Numbers in the Ring printable sheet
Change the ring so that there are only three squares.
Can you place three different numbers in them so that their differences are odd?
Can you make the differences even?
What do you notice about the sum of each pair in each case?
Try with different numbers of squares around the ring.
What happens with 5 squares? 6 squares?
What do you notice?
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 builds on Ring a Ring of Numbers. It encourages children to start from different examples and then begin to draw some more general conclusions based on their understanding of odd and even numbers.
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 some time to try to make all odd differences or all even differences. 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? Is it possible to make an arrangement with three numbers with all odd differences? You can work through the rest of the problem in a similar way, drawing the whole class together as appropriate.
Key questions
Possible extension
The problem Number Differences makes a good follow-up challenge.
Possible support
Some children will benefit from spending more time on the Ring a Ring of Numbers problem. Having digit cards to move around on a large piece of paper will also help and pupils might benefit from having sheets of blank rings so that they can try different combinations of numbers: