# Carroll Diagrams

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.

What do you know about the number 1? Where would it go?

How about the number 1? And 3?

You could look at all the odd numbers first, for example, and decide where they go in the diagram.

We had a lot of answers to this activity sent in, but only a few children explained their thinking - we would love to hear your ideas about the way you answered these questions!

Lyra from Westridge in the USA explained her method for putting numbers into these Carroll diagrams:

I know the number 1 is odd and less than ten, so I know that I should put it in the box that says "odd, less than 10". I continued with this strategy with the rest of the numbers. So: All the numbers that are odd and less than 10 go in the "odd, less than 10" box. All the numbers that are not odd and less than 10 go in the "not odd, less than 10" box. The numbers that are not less than ten and are odd go in the "odd, not less than 10" box. And, all the numbers that are not odd and not less than 10 go in the "not odd, not less than 10" box. The same thing happens with the next diagram.

Well done for explaining how to put numbers into a Carroll diagram, Lyra. I wonder if anybody else approached this in a different way?

We had a lot of solutions sent in from pupils at the ABQ Education Group in Oman - thank you all for your ideas. Taqiya made a video explaining how to put the numbers from 1-20 into the first Carroll diagram:

Well done, Taqiya! Taqiya has made the diagram simpler by changing the label 'Not odd' to 'Even'. Do all numbers have to be either odd or even? Can you think of a number that isn't odd or even?

**Why do this problem?**

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.

### Possible approach

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

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

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.

### Key questions

Why have you placed that number there?

### Possible extension

### Possible support

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.