# The Return of the Venn Diagram

MrBartonMaths
11th September 2015 at 16:01

November 1st, 2013. It’s one of those “do you remember where you were?” moments. This time, however, it is not Super Saturday at the 2012 Olympics or Nasty Nick being evicted from the first ever series of Big Brother. No, it is the day it was confirmed that Venn Diagrams were making a return to the Maths GCSE.

I love a Venn Diagram. In fact, they are probably my second favourite type of diagram, right behind the all-time classic Tree Diagram. There’s something about their symmetry, their logic, their power to sort and display, that means nothing else quite compares. Their inventor, John Venn, was seemingly so proud of his creation that he took the rather immodest decision to name it after himself. I don’t blame him.

Whilst Venn Diagrams have not been on the GCSE syllabus for some time, I still sneak them into my teaching each year. Why wouldn’t you? The students have experienced them at primary school, those going on to do Maths A Level will meet them in the Statistics module, so it seems only fair to keep them fresh in students’ minds during those intervening years.

But their most-welcome reintroduction got me thinking: are there more ways to use this lovely diagram? Is there more potential in those lovely interlocking circles than I at first thought?

It turns out, the answer is a very definite yes.

Venn Diagrams are used to sort objects into categories, depending on their key characteristics, and Maths is full of things that can be put into categories. Of course, there are the obvious ones, like properties of numbers. Have a look (and a go!) at this simple double Venn Diagram below:

Inspired, as I often am by my Maths hero, Don Steward, who created this lovely Straight Lines Venn Diagram (http://donsteward.blogspot.co.uk/2012/01/linear-equations-and-mr-venn.html), I started to think a bit outside the box to try to come up with other topics that could be “Venn-ed”. It turns out, you can do nice things with fractions:

Or, how about one of my personal favourites, averages and range:

Finally, one of my Year 8s created an “Extreme Quadruple Venn Diagram. She was very proud, and so was I:

The more I have used Venn Diagrams over the last few months – and believe me, my students have seen more than their fair share recently – the more I started to realise just why they are such a valuable, rich, worthwhile activity.  Here are a few of the reasons I love Venn Diagrams so much:

1) Students can always make a start. If they can think of a number/expression/object or whatever it might be, it has to belong in one of the regions on the diagram, so they are up and running. ¼ belongs somewhere in the fractions Venn Diagram, just as the data set 1, 2, 3, 4, 5 belongs somewhere in the averages one.

2) The more regions students fill, the more challenging the task gets, which adds a nice element of differentiation. Moreover, asking questions like “which region contains the most possible objects”, or “convince me a region is impossible” prompt students to think even deeper.

3) Venn Diagrams are incredibly versatile, and can be used for almost all maths topics for all ages and abilities. At the time of writing, I have Venn Diagrams for 24 topics, and I am convinced that many more are possible. I have used them with primary school students, and my Year 13 Further Mathematicians. Everyone loves a Venn Diagram!

4) Venn Diagrams are very quick to create and require no special resources – just print the sheet out and you are good to go. Once you have the template, it is just a case of choosing the labels. And the beauty is, if you come up with a set of labels that leads to some impossible regions, all the better! This just means there is more to discuss!

5) They are easy to tweak by simply changing one of the circle labels if you find they are too difficult/easy. I like the say to my students things like: “what should I change this label to make this task harder?”

6) Students can create their own Venn Diagrams as a worthwhile, challenging extension task. My current favourite is to ask students to create a triple Venn Diagram with exactly one impossible region. This gets even the most able thinking.

Venn Diagrams are now and integral part of our new scheme of work that I wrote about in the previous post: https://www.tes.com/news/blog/what-makes-a-good-maths-scheme-work  For the reasons given above, I believe they are an ideal rich activity, and hence they make up several of the compulsory rich tasks across all year groups and ability ranges.

I have made all the Venn Diagrams I have created so far available, for free, on TES, and you can access them via this link: http://www.mrbartonmaths.com/venn.htm

If you use them, please let me know how you get on. And likewise, if you or your students create your own, then please share them. The more Venns, the better!