It's revision time and Year 11s haven't grasped trigonometry yet. Do you ignore it, leaving them to think that yet another aspect of maths is a random set of meaningless symbols and formulae? Or grasp it as an opportunity for the integration (sorry) of practical maths, historical anecdotes, algebra, ICT, and shape and space? There is some sound software available for individual practice of techniques, but most students enjoy experimenting, measuring and using their calculators, so...
* Why did carpenters, builders and sailors, as early as 3000bce (so the pyramids tell us), use ropes with evenly spaced knots? Get students to draw a triangle with sides a variety of Pythagorean triples (good practice), then tell them they've all been rather particular - you want one that doesn't have a right angle: a prize to the student who can produce one. Why not? And where do the knotted ropes come in? Ask them about the angles in the triangles drawn - you should have sets for each "family" of triples. Do they need to measure the third angle in the triangle? Why (not)? Revise similar triangles and ratios while you're at it.
* Start with tangents: these relate directly to gradients of hills ("Low gear: 14 per cent"). All that measuring is very time-consuming, and probably not too accurate with most Year 11s at the helm, so how can they use a calculator to work out missing sides? Angles? Try it for the triangles they already have. Then extend to work out the longest rod that would fit in the classroom, or (on a sunny day when it's far too good to be inside) a tree, or the school.
* Having concluded that all right-angled triangles with one angle of, say, 30x are similar, so each is just a scale version of any other, corresponding (matching) sides must be in a fixed ratio.
Draw some triangles with 30x, work out some ratios: start with sine because the numbers are easy to work with, then go on to cosine because that illustrates the nature of most trig ratios. You'll need to develop some vocabulary to refer to the sides (and if the word hippopotamus comes to mind, at least you'll have a laugh).
Somehow they're going to need to remember which trig ratio is which: I survived for many years on "Some old hags can always hide their old age", but now that's getting a bit too pertinent I encourage my students to be more inventive: you could always run a competition with a geometry set as a prize.
Jennie Golding is an AST, and head of maths at the Woodroffe School, Lyme Regis