For those of you that have been around a bit … (I mean teaching for a while and I’m not insinuating anything else 😉 !!) you will have taught these before. They’re a bit of a hokey-cokey topic – one minute they’re in the formal programme of study as dictated by the DFE then they’re out and for the minute they’re in! Under the “probability” section you will see:

If like me, you don’t have a clue what “enumerate” means – it basically means to list or count. Interestingly something called an “enumeration” is an actual thing … it’s a complete, ordered listing of all the items in a collection.

Whilst the above is new to BOTH tiers the bold type of the below (point 9) means that “Only the more highly attaining students will be assessed on the content identified by bold type. The highest attaining students will develop confidence and competence with the bold content” but like I said about the product rule that whilst it’s a Higher tier concept only I might not limit teaching conditional probabilities with Venns just to those doing the higher tier. I love a good Venn diagram and I think they’re a really useful tool to have in your mathematical armoury.

Whilst the birth of Venn diagrams as we know them today can be almost precisely dated, using diagrams to visualise relationships between groups of things, concepts and ideas has existed for a long, long time.  For example, the use of diagrams to demonstrate valid logical arguments appeared in the works of Gottfried Wilhelm Leibniz – in a 1686 fragment Leibniz illustrated all of Aristotle’s valid syllogisms (google it!) through circle drawings. Later, in 1761, Euler used almost identical diagrams to explain the same logical syllogisms but it was in his paper in 1880 that John Venn altered what he called “Euler circles” to become the diagrams that bear his name. 

Venn himself did not use the term “Venn diagram” in On the Diagrammatic and Mechanical Representation of Propositions and Reasonings in the Philosophical Magazine and Journal of Science and whilst similar concepts were used prior to this, he most certainly was the first person to formalise their use and generalise them. The first time the term “Venn diagram” was used was in a 1918 book, “A Survey of Symbolic Logic” by Clarence Irving Lewis.

Just to clarify, there is a distinct difference between a Venn diagram and a Euler diagram: A Venn is a mathematical illustration that shows all of the possible mathematical or logical relationships between sets and whilst a Euler diagram resembles a Venn diagram, it does not necessarily show all possible intersections of the sets and some would argue that a Euler diagram is more useful for showing real world data, because not all sets partially overlap with all other sets. This site explains the differences very simply.

Many secondary teachers will be aware that students are taught “Carroll diagrams” and I have even heard Venn diagrams referred to as Carroll diagrams by students …. They are not the same thing but that’d take up a whole other blog post!

I love teaching Venn diagrams. Students find the topic quite easy to access and it’s what I would call a low threshold topic .. that said you need to consider carefully the questions you use to both “model” with the class but also the ones that you ask them to complete. Often, it isn’t the Venn diagram that they will struggle with it will be remembering what a square or prime number is and this is a topic that suits itself really well to the use of “Boarding cards”.

I’ve done loads of these (one for each of the Super 60 topics and more!) and the one below is one that I did for some resources for the Pearson Getting Ready to teach GCSE Statistics events that they’ve been doing recently (so I hope they’re Ok with me sharing here!). The idea behind these is that I use them at the end of a lesson before launching into a new topic to enable me to understand what pre-requisite knowledge the students do or don’t have. They take 5 minutes for the students to complete and the information is invaluable in planning whether I need to do a quick reminder of key information / skills that they may have forgotten.

I’m gonna say it again: choose your questions carefully! Actually, no! …make sure the students have the knowledge needed to categorise the items correctly. The last thing you want is students not getting the right answers and therefore thinking they “can’t do Venn diagrams” because they have forgotten that 1 is NOT a prime number. The questions below are just to illustrate a possible route through the topic and are a combination of questions that I might use as part of whole-class explanations (Read that as teacher modelling with questioning!) or for them to have a go themselves, making little, but not uncomfortable cognitive jumps themselves.

I’d introduce them to the idea of categorising what I call a “Two’fer”. Two circles with a simple concept that they can understand the context surrounding it, such as the number of brothers and sisters a group of people might have. During this I will of course introduce the correct terminology and use of epsilon or the “curly E” which students are fascinated with … it seems to embed the term “universal set”. I used to be wary of terminology (irrational I know!) but am now of the mind that it is important not to be scared of introducing them to maths terms. I find it enriches the subject! NB: With each of the below questions I’m not sure of the sources but if anyone knows I’m happy to give credit where it’s due so do let me know (I just have them collated in my progressive examples document that I use for modelling my examples):

Before moving on to them having a go I might do a whole class exercise and see if we can produce the same information using the number of brothers and sisters they have. They’ll then have a go themselves using a similarly easy context and then they’ll have a go at reading information from a completed diagram. We’ll then look at an example that involves listing elements in the correct sections and I’d add on being able to draw conclusions about probability and this is where the intersection/union and even “given that” terms are used (even for Foundation students! Why not? At the stage this gets first taught, we probably don’t know what tier of entry these students will be entered for anyway and I don’t think we should be limiting their access  to the higher tier later in their schooling based on what they weren’t taught lower down the school)

After we’ve nailed the idea of drawing and extracting information I’m conscious that I want to get to this question …

I love this one!! It’s a real brain hurter!! It really emphasises the importance of reading the question carefully and concluding that if 80% pass the numeracy (and everyone passed at least one thing!) that must mean that 20% didn’t pass. It takes a while and if needs be I’ll get a group of students up at the front of the class to illustrate it physically too! After this there would be a suitable period of practice where the students are able to “show what you know” before we move on to “three-fers” … yep, you guessed it three circles! I might however, use something like the below which I’ve used several times that involves them devising and drawing their own diagrams and then using the information to answer statements.

All of this may have taken me two or three lessons to get to this point and I’ll want to check their understanding before looking at where we go next. This worksheet is ideal to assess their understanding and worked really well when I’ve used it ..

It dawns on me that some of you may think that I’m advocating there is just one way to teach a topic … I’m not! We all teach topics differently I’m just enjoying documenting my chain of thoughts! I’ve also pulled attached the questions from  Edexcels exam wizard together that I put together for GSCE Statistics events (but they’re still useful) -> Venn Diagrams – Exam Style Questions )  and put some solutions ( Venn Diagrams – Exam Qs 2 – SOLUTIONS  ) together that you may find useful but it’s also worth checking out the other questions from all the boards in the “Questions by topic blog post” (-> here ) to put your own route through the topic together. Note that some of these are from GCSE Statistics so may take it further than you’d choose to take the topic but that’s the beauty of teaching – you get to make the choice!

Enjoy