Question of the Week 11: Area of a Circle

The following question is taken from my website Diagnostic Questions. Here you will find 1000s of free, 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.

I created the above question especially for my delightful Year 11 set. We were going through a practise GCSE exam paper, and we came to a question about calculating the area of a circle. Under normal circumstances, this would cause them no issues – they would reach for their trusty calculators and everyone would be happy. However, this particular question appeared on the non-calculator paper, and contained those 7 dreaded words: “give your answer in terms of pi”.

This was met with a combination of sighs, shrugs, a few tears, and crucially very little logical mathematics that would have gained them any marks.

It got me thinking that giving answers in terms of pi should make circle questions easier, as it avoids the potentially distracting never-ending stream of decimal places, but it can actually complicate things in the minds of students.

Having spent the remainder of the lesson ensuring students were happy with the idea that the pi symbol is just a number, I followed it up with the question above at the start of the next lesson.

Here is my logic behind each of the answers:

Answer A) here, students may simply square the diameter instead of finding the radius, hence revealing that they have a misconception with the area of a circle formula itself

Answer B) may imply that students are muddling up the formula for area of a circle with that for working out the circumference

Answer C) is the correct answer

Answer D) reveals an entirely different kind of misconception. As well as indicating they have forgotten the formula for area of a circle, students have also revealed a key misconception when it comes to squaring, possibly believing the 12 squared is 24, and not 144.

I got a complete mixed bag of responses, and the students’ reasoning was very revealing.

I find the last answer particularly interesting. Whereas answers A and B suggest more work is needed on working out the area of a circle, Answer D implies that students also need to do some work on squaring. This could be further assessed by using a question like the one below:

That is what I love about diagnostic questions. Not only do they reveal misconceptions students have with the current content they are studying, but they can also help identify fundamental underlying misconceptions with the kind of baseline knowledge that is crucial in order for students to understand more complex concepts. Yes, I love them. But, as we established last week, I am pretty biased.

If you get chance, please let me know how your students get on with this question, and the reasons they give for their answers.

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