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### Number and algebra

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### Advanced mathematics

# Carroll Diagrams

## Carroll Diagrams

**Why do this problem?**

### Possible approach

*This activity featured in an NRICH Primary webinar in March 2021.*
### Key questions

### Possible extension

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Age 5 to 11

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

Take a look at the interactive below.

What do you notice?

What do you wonder?

This kind of table is called a Carroll diagram (named after the mathematician and author Lewis Carroll).

Can you drag the numbers into their correct places?

How do you know where to put them?

If you would prefer to work away from a computer, you could print off this sheet.

How about this Carroll diagram? The interactivity is below, or you could print off this sheet.

You may like to print off this sheet, which is a picture of a completed Carroll diagram, but the labels are missing. Can you give the rows and columns labels by choosing from the list at the side?

If you click on the purple cog of the interactivity, you can change the settings and create your own Carroll diagrams for someone else to complete.

This problem gives children a way of sorting numbers according to different properties and also provides a situation in which they need to consider more than one attribute at once. In addition, it gives children the chance to explain their placing of the numbers, using appropriate language.

There are two aspects to this problem: Firstly, sorting numbers according to certain properties and secondly, knowledge of how a Carroll diagram works.

Project the first interactivity on to the board, trying not to say anything other than:

- What do you notice?
- What do you wonder?

Give everyone a minute or so to think on their own, then encourage a pair of children to talk to each other. Bring everyone together after a suitable period of time and invite pairs to share their thinking and noticings, and any questions they may have. You could jot down all that is said on the board. Try to value everybody's contribution, even if it does not seem relevant, or even if it is
something you have not thought about yourself.

By drawing on the learners' comments and questions, you can introduce the idea of a Carroll diagram (or remind the class if they have met them before). You might invite a member of the class to come and drag a number to the correct cell, and ask someone else to explain why that is correct. Equally valid is to ask someone to drag a number to an incorrect cell, again asking for an
explanation from a different learner.

This could then lead on to pairs of children completing the Carroll diagram itself, either using the interactivity on a tablet/computer, or on paper. When you bring everyone back together again, you might like to ask which numbers were easier to place and why.

Depending on their experience, you can then offer the second interactivity and/or you could create your own Carroll diagram for completion using the Settings menu (purple cog). You can include some examples of Carroll diagrams which have lost their labels. Here is a printable example but you could also do this in the Settings by leaving the label text blank and just inputting the numbers. You can then drag the numbers to the correct positions yourself and invite the class to decide what the labels are. (You could present the class with all the numbers in the correct places, or you could add numbers one by one as they watch and see how long it takes them to work out the labels.) There are many different approaches to a 'no labels' version of the problem, 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.

What do you know about the number...? Where would it go? Why?

Is the number 10 less than ten?

Where will odd numbers go? Why?

Why have you placed that number there?

Why have you placed that number there?

Alan Parr, who has contributed many great ideas to NRICH, has sent us these further Carroll diagram sorting activities which you may like to use as follow-up to this problem.

You could suggest a particular way of starting, for example, looking at all the odd numbers first and deciding whether each is less than 10 or not. The interactivity allows users to change their mind about the positioning of a number, so learners who do not like committing ideas to paper might benefit.