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. It can be followed up (or proceeded) with a note on why you would expect triangles with the same internal angles to have the same scale factor.
Then we can split up the job of working out these scale factors: give each pupil an angle. I then tabulate the pupil’s results and encourage the class to look for the pattern: 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. I have heard that at Highgate, from where I got this approach in the first place, they now also graph the tan values at this stage, to make the pattern even clearer and then use the graph to find the required tan values, but I haven’t tried that myself.
Personally, I then apply the same approach to introduce the sine and cosine scale factors, only returning to show how the calculator knows all the values later in the year.
Suggestions for alternative approaches welcome as ever.