One thing I struggle with is getting my pupils to make the leap from creating examples of a proposition to giving a proof of it. I found the ‘Make 37‘ activity was great for getting them to start to explain ‘why it will never work’, instead of producing endless examples of it not working.
However, few of them were able to move from ‘an even number of odds can’t make an even’ to an algebraic proof (some version of 10(2n+1) = 37 -> 20n=27 -> no integer solution or “10(2n+1) divides by 2, so it is even, so it can’t be 37” would have sufficed).
In addition to appreciating any advice or anecdotes on how to take the step to formal proofs, I would also like to know people’s opinions on ‘picture proof’ (e.g. proving that angles in a triangle must have a sum of 180 by taking the exterior angles and ‘shrinking’ them inwards to show they make a full turn – there is an animated example of this on mymaths). Mathsisfun do what is, to my mind, a more rigorous approach, but it still relies on ‘looking right on this occasion’.
Extension task:) I puzzled over this with a colleague from another school – we could only show it with a picture. Is there a better way? Can it be done other than by pointing at an illustration? To our embarrassment, it’s from the Junior section of the UKMT mentoring resources. I’m avoiding the advanced section for now.
Answers to this questions would be appreciated, as the solution won’t be released for a few more weeks!