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The proof is in the pudding?

My grandmother was a great one for mixing up sayings and would often say that the proof is in the pudding!

The first piece of content on the new A-level specification is “Proof”

The question is when should this be covered? Do you start with ‘proof’ in week one? If you did, did you have any students left in week 2? Should proof be a session at the end of the course when all of the content has been covered in order to draw on a variety of topics as context for the proof session? Should you not have a section of the scheme of work that is labelled as ‘proof’ at all? Instead treat proof as one of the overarching themes and introduce proof as it naturally occurs throughout the course?

All of these are tricky to answer. I prefer to develop the idea of proof as the course progresses weaving it into topics wherever I can. One of my favourite activities was developed from a series of books entitled “proof without words”. I give the students these two diagrams and ask them, in groups, to write down as much as they can. I explain that the two diagrams show the same rectangle split in two different ways. Maybe you would like to pause and give it a go.

Some groups concentrate upon finding angles in terms of θ. Some find lengths of sides using trigonometrical ratios. In some cases groups need a hint that maybe they could also find some areas.

Discussion results in the observation that the area of the right angle triangle, half the rectangle can be found in two different ways so the two expressions must be equal. Students have therefore found, and proved, that sin(2θ) ≡ 2sinθcosθ. Experience tells me that the way to totally destroy lessons like this is to start off by saying “use these diagrams to prove…” – much better to allow students to explore. Hence I will leave you with this.

Write down as much as you can from this diagram. What you find?

Find out more about the books Proof without words

And more resources to help teach proof on the STEM website

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Subject(s) Mathematics 14-16, 16-19 1