# Question of the Week 5: Circle Theorems

The following question is taken from my website Diagnostic Questions. Here you will find 1000s of high quality maths multiple choice diagnostic / hinge questions, ideal for assessment for learning, which have been created and shared by maths teachers all over the world.

Circle Theorems are a notoriously troublesome topic for many students. Personally I love the challenge of trying to spot which theorems have been used, and trying to fill in the missing angles one step at a time. But that could well be due to the fact that I am a massive maths geek who loves any kind of puzzle, and I have come to realise throughout my teaching career that this is not a character trait shared by the majority of adolescents.

I wrote this question specifically to use with my lovely Year 11 class in the build up to their recent GCSE. We had covered all the theorems a few times, but many of them were still have difficulty spotting which ones to use. Perhaps unsurprisingly, the Alternate Segment Theorem was proving the least popular of the theorems amongst the class, and as I looked through their work in their books I was noticing that they were misapplying the rule left, right and centre. Specifically, they had real trouble identifying which two angles were equal.

So, I wrote this question, and I was pleased to see that it split the class.

Here is what I learnt from their responses and the class discussion that followed:

After students had voted, I then did the usual and asked the students who had voted for a to explain their reasons, then b, etc. The discussion was one of the most interesting (and heated!) that we had had. And when we came to the re-vote afterwards, not only did everyone agree that d was correct, we had also discussed most of the other major circle theorems, and what they would tell us in the diagram, in arriving at our answer.

So, a key message here is never to be afraid to have "not enough information" or "none of the above" as the correct answer in a question. It can really challenge the students as they have to rule out each of the other answers to get to it.

Whilst we are talking circle theorems, I might as well plug (it is free, so I hope you will forgive me!) my series of interactive Autograph activities that look at each circle theorem dynamically. You don't need Autograph installed to use them, and they can be used in the classroom to demonstrate and extend knowledge, or given to the students to play around with at home.

You can access them here

And if you have used the question, or have any thoughts or comments about it – perhaps you would include a different incorrect answer – then please share your comments below. And please consider creating a question yourself on the website!

Circle Theorems are a notoriously troublesome topic for many students. Personally I love the challenge of trying to spot which theorems have been used, and trying to fill in the missing angles one step at a time. But that could well be due to the fact that I am a massive maths geek who loves any kind of puzzle, and I have come to realise throughout my teaching career that this is not a character trait shared by the majority of adolescents.

I wrote this question specifically to use with my lovely Year 11 class in the build up to their recent GCSE. We had covered all the theorems a few times, but many of them were still have difficulty spotting which ones to use. Perhaps unsurprisingly, the Alternate Segment Theorem was proving the least popular of the theorems amongst the class, and as I looked through their work in their books I was noticing that they were misapplying the rule left, right and centre. Specifically, they had real trouble identifying which two angles were equal.

So, I wrote this question, and I was pleased to see that it split the class.

Here is what I learnt from their responses and the class discussion that followed:

**Answer a)**students seemed to be muddling up the tangent/radius theorem that states the tangent meets a radius at 90 degrees, and thus there were 20 degrees left over.**Answer b)**the most common reason for this were a misapplication of the alternate segment theorem, or thinking that the lines were parallel and hence alternate angles were equal.**Answer c)**this revealed some muddled reasoning about opposite angles in a cyclic quadrilateral**Answer d)**although it goes against natural human instincts that seem to demand that an answer much be found, this is indeed to correct answerAfter students had voted, I then did the usual and asked the students who had voted for a to explain their reasons, then b, etc. The discussion was one of the most interesting (and heated!) that we had had. And when we came to the re-vote afterwards, not only did everyone agree that d was correct, we had also discussed most of the other major circle theorems, and what they would tell us in the diagram, in arriving at our answer.

So, a key message here is never to be afraid to have "not enough information" or "none of the above" as the correct answer in a question. It can really challenge the students as they have to rule out each of the other answers to get to it.

Whilst we are talking circle theorems, I might as well plug (it is free, so I hope you will forgive me!) my series of interactive Autograph activities that look at each circle theorem dynamically. You don't need Autograph installed to use them, and they can be used in the classroom to demonstrate and extend knowledge, or given to the students to play around with at home.

You can access them here

And if you have used the question, or have any thoughts or comments about it – perhaps you would include a different incorrect answer – then please share your comments below. And please consider creating a question yourself on the website!