## Circle Theorems: Should we bother?

On the most recent edition of “Wrong, but useful” cohost Colin Beveridge (@icecolbeveridge) had a bit of a rant about circle theorems. He feels they are pretty pointless, and he is in a fairly good position to discuss this, as he spent a decade researching the topology of the sun, basically circular in nature, and never used any of them. He says that he has only ever used one once, and that was to find the centre of a circle (this use is the most practical use if a circle theorem I can think of).

The discussion came about because the other cohost, Dave Gale (@reflectivemaths) was talking about when he trained and the things he hadn’t encountered before. His experience reminded me very much of my own. While I was training to teach I was also working to ensure my subject knowledge was entirely up to scratch, and that I was familiar with the syllabus. There I discovered Circle Theorems, and they were pretty new to me. I don’t know if I’d ever been taught them, I did know that diameters make right angles at the circumference, and that chords make the same angle in the same segment, so I suppose I may have learned then forgotten them. The one in particular that I was certain I’d never met was “Alternate Segment Theorem”, infact it was something that at first confused me and I spent a long time investigating it during my PGCE year before I was completely satisfied that I understood it fully and could teach it.

These Circle Theorems seem to stand alone in the syllabus, they seemingly have no direct link to any other area of the maths GCSE, and they certainly seem to have no real practical use at all and if there is an answer (other then “never”) to the question, “when will we ever use Alternative Segment Theorem in ‘real life’?” I’d love to here it!

**So, should we be bothering with them?**

The usual pro-circle theorem argument goes along these lines: “It’s a nice introduction to mathematical proof which gives students a good grounding for future proofs.” This doesn’t really wash with me. The questions we ask are not very stringent as far as proving goes and more often than not the teaching is focused around “this is how you get the marks on this question,” than any actual proof.

However, I do feel that they have a place on the GCSE syllabus. I used to sit in the anti-circle theorem camp, but my views on this have changed. The more I get into circle theorems the more I love them. And the fact that they don’t have a point just adds to it. The circle theorems are beautiful. They show geometry at its finest, and they have been derived purely because someone wondered about the properties of a circle, not because there was a problem that needed fixing.

Students should be exposed to this kind of maths. They should be allowed to investigate these theorems, and allowed to conjecture about them, before trying to prove them. I think they very definitely should stay in our syllabus, but that we need to address the way they are taught, and assessed. Take the mechanical nature away from the topic and allow the beauty of the maths to prevail.

Reblogged this on The Echo Chamber.

I contend that there is only one true circle theorem from which all the others are derived. I teach it from that angle.

Ptolemys theorem?

No, angle at centre is twice the angle at the circumference. Quite like the blog, by the way.

Thanks! And yes, I can see that one could derive the others. I’ve never thought of it like that, how comes you came to that conclusion? is there a historic reason for it?

“when will we ever use Alternative Segment Theorem in ‘real life’?”

You can ask that about huge swathes of the maths curriculum – and the French curriculum, the biology curriculum, the history curriculum and all the others. Unless you’re planning on making a living off Pointless, there’s an awful lot that you learn at school that you will never use again. That doesn’t mean you shouldn’t learn it, because

(i) you don’t know at the time what will be useful and what won’t

(ii) sometimes the curriculum content is incidental in enabling kids to learn skills and techniques.

Circle theorems are fantastic – they are relatively easy to derive, they are beautiful and understandable, pure mathematics at their best. It’s a topic that can really lend itself to collaborative work and discussion, because it is one where kids can discover the rules through experimentation and then prove them from first principles.

Aye, I certainly agree! Students tend only to ask about it though. The reason I mentioned it in this context was that Colin’s point was that he researched an area of maths based on.circles for ten years and never used them. I don’t think that matters much, and love them as they are!

Nothing historic to be honest, just thinking about the Einstein idea of make everything as simple as possible but no simpler. Plus you can get them to use it to derive the other ones, which means you can et them to think about how to apply it, which leads to better long term learning.

Aye, I’ve not tried that approach but think I may next time.

I have come across recent edexcel (modular higher?) GSCE paper that asked the student to prove angle at centre twice angle at circumference – and the next paper took this as a given ie you could offer it as a reason for a step in a proof in a different question. Is this not unusual?

Students are allowed to recall and use the circle theorems. They are also expected to be able to prove them, they wouldn’t ask both in the same question.