Maths Teaching Idea #6
I have started doing some revision with a GCSE class and decided to pose a standard problem without much of the usual padding to see what students do with it.
The diagram above was drawn by hand (ie no scale) with the words:
x²+y² = 5 and y = 2x + c are drawn above.
Give a condition for c so that the two graphs have no points of intersection.
I had anticipated a fairly quick substitution and a reduction to a quadratic with too many unknowns for the class to deal with. This was not what happened.
Learners were generating coordinates, drawing different lines with gradient 2, assuming that c ‘must be root 5’ and relating it to a vertical line problem we had done recently. This was an interesting situation and reminded me of a chapter I had just been reading in Lerman and Davis (2009) where Tim Rowland talks about his overreliance on algebraic methods to solve geometric problems (pp 49–61).
One of the students suggested drawing a triangle which was very exciting! “Which triangle?”, “what do we know about the triangle?”, “how does this help us with our problem?”
Eventually we were dealing with similar triangles, scale factors, surds, area, circle theorems, Pythagoras’ Theorem.
The problem resolves when the original ‘algebraic’ triangle with known height is compared with our simplified similar triangle, and the inequalities follow easily. This genuinely felt different from my own default approach and seeing the connections and thinking that was going on during and after the problem was solved has reinforced my desire to continue looking for ways to stay within the context of the problem when trying to solve it.
Reference: Lerman, S. and Davis, B. (eds) Mathematical Action and Structures of Noticing: Studies on John Mason’s Contribution to Mathematics Education (2009)
