# Venn diagrams

Take a look at the interactivity below. Before you try anything, have a think about these questions:

What do you see?

What do you wonder?

We call this way of sorting information a Venn diagram (named after the mathematician John Venn).

Can you drag the numbers from 1 to 30 into their correct places in the Venn diagram?

How do you know where to put each number?

Here is another one for you to try.

How do you know where to put each number this time?

If you would prefer to work away from a screen, you could print off these sheets, which have a copy of each Venn diagram on them.

Here is a screenshot (a picture) of the interactivity, but the labels of the Venn diagram are missing.

What could the labels be?

Here is another version of the interactivity for you to try.

What do you notice this time?

Can you explain your 'noticings'?

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

Where would you put a number which is a multiple of 5 but not even?

Where would an even multiple of 5 go?

Where would a number which is odd but not a multiple of 5 go?

Where would you put a number which is not a multiple of 5 and not even?

Thank you to everybody who sent in their answers to these questions.

Tetsu from St Michael's International School in Japan sent us some ideas about how a Venn diagram might work. Tetsu noticed that there are two circles that overlap, and the middle part where they overlap is where we write numbers that are both even and are multiples of 5. Well done for writing down what you could see and what you wondered when you looked at the diagram!

Harry from Copthorne Prep School in the UK noticed that only the numbers that met the description of the circle were allowed in. He worked out that if they met the descriptions of both circles then they went into the crossover part, and if they didn't meet the description of either circle then they had to stay outside. He made this video explaining how to put the numbers into the Venn diagrams:

Well done, Harry! We've also uploaded an image of Harry's solutions.

Lyra, Giselle and Taylor from Westridge in the USA explained their ideas for the first three Venn diagrams:

You would put the not even numbers and multiples of 5 in the right circle (multiples of 5), put the both even and multiples of 5 numbers in the middle (where the circles combine), put the even numbers that are not multiples of 5 in the left circle, and you would put the numbers that are neither even or multiples of 5 outside of the circle.

In the second diagram, put the numbers that are less than 20 that are not odd in the right circle, the numbers that are odd and not less than 20 you put in the left circle, put the numbers that are odd and less than 20 in the middle (where the circles combine), and put the numbers that are neither odd or less than 20 outside of the circles.

In the third diagram, the labels should be: the left one should be labeled "odd", and the right one should be labeled "higher than 20".

Thank you all for explaining this so clearly.

Abbie, also from Westridge, noticed something interesting about the last diagram:

Also, for the diagram that says "even numbers" and "odd numbers", there can't be an even AND odd number unless you are going to count zero. That would be a trick question to those who don't know their numbers well enough yet.

That's a really good idea, Abbie - I wonder if we would count zero as even number, an odd number, both or neither!

Haleema from Pierrepont Gamston Primary School also had an idea about the last diagram:

The thing I noticed on the last one was that if the labels are even and odd numbers there can't be anything in the middle. I don't think a Venn diagram should have been used for that. Maybe you could draw circles and label them but there just shouldn't be a middle section.

This is a good idea, Haleema - if we drew a diagram with two circles that didn't overlap, would that still be a Venn diagram or would it be a different type of diagram? I wonder if there are any other types of diagram we could use for two types of numbers where there isn't any overlap.

Finally, Sarah from the ABQ Seeb International School in Oman sent us this video explaining how to put the numbers into the first Venn diagram and into her own Venn diagram:

Thank you for that very clear explanation, Sarah.

**Why do this problem?**

This problem provides an opportunity for children to become familiar with Venn diagrams, whilst reinforcing knowledge of number properties. Placing numbers in a Venn diagram requires children to consider more than one property of a number at the same time. This problem is also a good context in which to encourage learners to articulate their reasoning.

### Possible approach

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

You could introduce the task by projecting the first version of the interactivity using an interactive whiteboard. Without doing anything at all, invite learners to consider what they see and what they wonder.

After giving some time for children to talk in pairs, bring them together to share their thoughts and any questions they have. Through this whole group discussion, you can draw out the features of Venn diagrams and you may like to invite some children to come and drag a number to the correct place, 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.

You can set the children off on the task, either using the interactivity if you have access to tablets/computers, or on paper. This sheet contains copies of the first two diagrams in the problem. If children work in pairs it will encourage them to construct mathematical arguments to convince each other where on the diagram each number belongs. Explaining out loud in this way often helps to clarify thinking and will give a purpose for accurate use of mathematical vocabulary.

You could listen out for misconceptions or disagreements to share in a plenary so that all children become involved in the reasoning. You may also like to look at the image of the diagram with missing labels in the plenary. Give learners time to talk in their pairs before inviting a few to share their solution. You will find that different pairs have 'homed in' on different numbers to help them solve this part of the task, therefore it is worth hearing from a few so that more than one strategy is shared.

You can end the lesson with the final interactivity. What do they notice? You may need to encourage the class to start dragging numbers into the correct places before they realise that there won't be any numbers in the intersecting region in the middle. Why not?

Clicking on the purple cog of the interactivity allows you to change the settings so you can tailor the labels and sets of numbers to suit your learners.

Key questions

### Possible extension

Children can create their own Venn diagrams for others to complete. Alternatively, you could challenge them to create Venn diagrams which have certain criteria e.g. an empty intersection (like the final example in the problem); all items in the intersection; three overlapping sets etc.

Possible support

Some learners might find it easier to collect numbers with a certain property, for example, even numbers or numbers less than 10, in single circles. Then they can look at those that should go in both circles.