This is an approach to introducing trigonometry I took from my previous department at Highgate. It focuses on just the tangent scale factor in this initial intro – we would then return to introduce sine and cosine a month or so later, following a spiral curriculum.
I would start with an introduction on the aim of the lesson: we want to find a link between the sides of a triangle and the angles within it. This could make use of the 3-act math format alongside a real life problem such as finding the height of a mountain: personally I often do this with one of my favourite tracks playing the background: Everest by Public Service Broadcasting.
The worksheet guides pupils to discover that if you divide opposite by adjacent for similar right angled triangles, you always get the same number. This gives us the link we are looking for.
Then for homework, pupils can be each given their own angle to work out this special number for. At the start of next lesson, I would tabulate the pupil’s results and encourage the class to look for patterns: any incorrect/wildly inaccurate values for tan will stand out and can be quickly corrected.
I’d then demonstrate how we can use this table (for now avoiding the calculator tan button) to estimate sides / angles. Using the pupils’ own numbers gives them a sense of ownership over the method, which adds a lot of value to the approach.
Suggestions for alternative approaches welcome as ever.