Sorting the numbers
Can you put the numbers in the correct place in this Carroll diagram?
Age
7 to 11
Challenge level
Problem
Can you put the numbers in the correct place in this Carroll diagram?
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You could print off this sheet to complete.
Now have a look at this Carroll diagram. This time, the idea is to fill in the labels for the rows and columns.
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You could print off this sheet to complete.
How are you deciding where to place the numbers?
Getting Started
What do you know about the number $1$, for example? So where would you put it in the diagram?
You could look at each number in turn in this way.
In the second Carroll diagram, what do all the numbers in the top left box have in common?
How are they different to the numbers in the bottom left box, for example?
Student Solutions
Rowena from Christ Church Primary Schoolexplained how she filled in the first diagram:
I first looked at the two columns. Between $5$ and $15$ you don't use $5$ and $15$. So I started at $6$ for the right hand column, it went in the top row, $7$ in the bottom etc. When you count up it goes from bottom to top and back again until you reach a multiple of $10$, then it stays in the box you just put a number in. After I did this for $6$ up to $14$, I moved to the left hand column and repeated the same process for $1-5$ and $15-30$.Well explained, Rowena, thank you. Here is the picture Rowena sent of her completed diagram:
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Thank you also to Cong from St Peter's Roman Catholic Primary School and Callum and Katie from Eynesbury CE(C) School for their solutions. Callum and Katie, you were so close in the second Carroll diagram, there was just one heading which you had slipped up on. Perhaps you can see why Cong's works? Here is Cong's completed diagram for the second challenge:
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Teachers' Resources
Why do this problem?
This problem gives children a way of sorting numbers according to different properties and also forces them consider more than one aspect at once. It also provides opportunities for children to explain their placing of the numbers, using appropriate language.
Possible approach
There are two aspects to this problem: Firstly, it focuses on sorting numbers according to certain properties and secondly, it requires a knowledge of how a Carroll diagram works. If your class has not had much experience of Carroll diagrams, it might be useful to look at the problem Carroll Diagrams which also has suggestions in the notes of how you could go about
introducing the relevant ideas.
This problem offers more of a challenge in that children are asked to identify the criteria by which the numbers have been sorted. Encourage them to talk to each other about how they might work out the labels for each row and column. There are many different approaches and sharing some of their ideas with the whole group would be beneficial. Try to focus on the clarity of their arguments,
thereby encouraging well-reasoned solutions.
Alan Parr, who has contributed many great ideas to NRICH, has sent in this Word document of further Carroll diagram sorting activities which you may like to use as follow-up to this problem.
Key questions
Do you think that the number "5" is between 5 and 15 or not?
What can you say about the numbers is this box that is different from the numbers in that box?