Building Bridges
Like a bridge over troubled water, this article explores the transition from parrotrecall and repetition of known facts within geometry, to the complexities of demonstration and mathematical proof – a journey that leads to deeper conceptual understanding and, one hopes, engenders a wanderlust for further travel.
Ask Year 7 pupils “How many degrees do the angles in a triangle add up to?” and most will happily regurgitate “180º”.
Ask the question “Why?” and you get a different response ... usually a blank look up at the ceiling … nothing registers ... tumbleweeds roll ...
Ask “how many degrees do the angles on a straight line add up to?” and the answer will undoubtedly be “180º”. Your next question would then be “Why?” … and you can guess what happens next! Quite often the response is “because Miss told me”, or “because it just is”. Now ask “what is a degree?” and that is a real conundrum!
Think of what similar questions could be: “How many degrees do the angles around a point add up to?” “How many degrees do the angles in a quadrilateral add up to?” Knowledge of the angle properties of straight and parallel lines, knowing that angles around a point add up to 360º (which is four right angles), recalling accurately and reliably angle facts about triangles and polygons: all these underpin the higher level skills needed to understand and apply complex ideas such as the “angles in circles” theorems. The earlier we spark pupils’ curiosity, the more interest and enthusiasm they develop  and the more accessible the later, harder reasoning becomes.
Remember, recall and regurgitate: these are the thinking skills that relate to the lower end of Bloom’s Taxonomy. Pupils can remember and recall what they have been told is true, but usually few ask “why?” by themselves: precise questioning by teachers is key to building bridges.
Take a look at the Mathematical Association Postcard Sets for higher and foundation level; they are especially useful at this time of year for revision and memory jogging.
The Demonstration
So how do we help pupils develop their deeper understanding? The use of computer demonstrations is an obvious next step, and the excellent range of animated gifs available to support understanding is well worth exploring: suggestions are at the end of this article. Physical demonstrations are helpful too. You could link this to triangle constructions first to start to build your bridges: check out Mr Collins Comic Book Constructions.
If you tear off the corners of your triangles and place them together, they seem to form a straight line: ta da...
But do they really? And if they do, why do the angles on a straight line add up to 180º? And so the sequence of questioning continues – a good cue to remind ourselves of Malcolm Swan’s excellent eight effective principles of maths teaching of which “use of questioning” is but one.
The Proof
We need to use the angle properties of parallel lines. “But why are these true?” – well, that’s your cue to talk about Euclid’s axioms and the parallel postulate (the 5th axiom), and all the unsuccessful attempts of mathematicians to prove it from the first four axioms. And then, let’s consider:
By definition angle A + angle B + angle C = the sum of the angles in the triangle. Angles A are equal and angles B are equal as they are alternate angles defined by a pair of parallel lines. Therefore the sum of the angles within a triangle will equal the sum of the angles on a straight line; but we still need to explain why both equal 180º!
So start your lesson with that simpleseeming question “Why do the angles in a triangle add up to 180º?” and see where it takes you … hopefully, like a bridge over troubled waters (apologies Simon and Garfunkel) the ensuing reasoning will ease (and enlighten) your pupils’ minds.
Helpful Resources
 Angles In 2D Shapes from Great Maths Teaching Ideas
 Interior Angle Sums of Triangles, using Geogebra
 NRICH triangle resources
 Spot the Angle! A Mathspad activity showing that with limited information you can still deduce a lot
 Mr Reddy’s Geometry Toolbox, a simple interactive tool for constructions.
Image credit
Page header by Alan Levine (adapted), some rights reserved
