The past couple of days I have been thinking a lot about trigonometry. This has been spurred in part by the fact that I will be doing some lessons on it soon and in part by Anne Watson's excellent opening address at the recent ATM conference. The address forced me to think about how joined-up maths really is, but also about how disconnected it is often taught. Starting with this post I am going to work on ways of encouraging 'Relational Understanding' in my students, an ability to see these connections for themselves.This post looks at the links between linear sequences, gradients, algebra and trigonometry.
I was struck yesterday by the image of someone using a right angle triangle to calculate the gradient of a line. "Hold on" I thought, "are they not calculating the Tangent of some angle? And more to the point isn't that the angle that defines the steepness of the line? And more to the point do we not call this the gradient?" This was quite striking to me. Stop me if this is painstakingly obvious, but here are two processes completed to varying degrees of success by thousands of students around Britain that I have not once seen taught as interconnected.
Now let me re-assure those of you who are reading this, mouth agape, ready to fire off an email telling me that you've been teaching this for years, I am a mere NQT my experience is sorely lacking. However this was an important realisation for me. The fact that I had students using the same skill in several areas but who were unaware of it was an exciting prospect!
My mind then moved from gradients and tangents to linear sequences. How could this be connected to both trigonometry and gradients? Think about the connection between similar triangles and trigonometry... any thoughts? Leave a comment.
Now when introducing tangent to my class I presented a set of right-angled triangles all with a 21ยบ angle and let them work out opposite over adjacent...surprise!!! They all give the same thing! I'm starting to wonder now whether surprise was the best emotion to connect with this fact however. Based upon the experience that they already have (linear sequences, similar triangles) could they not have been scaffolded towards an understanding that things cannot be otherwise? That the tangent of an angle is necessarily unique? I suppose that this is something to think about.
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Some Year 11s were working on a past GCSE paper recently and part of the problem was deciding what 'topic' some of the questions were 'on'. They had two similar triangles and were given two sides of the first and one side of the second and were asked for a missing side on the second triangle. They were trying to use trig but the triangles weren't right-angled. I found myself saying, "Ah, it's not a trigonometry question; it's a similar triangles question." What a daft thing to say - isn't trigonometry all about similar triangles? Should these 'separate' topics feel as separate as they do to me?
I find myself grappling with these emotions all of the time. This is shape and this is algebra...*this* is crazy! I'm currently fighting against lots of subconscious pigeon-holing. It's a battle but it's one that I am learning a lot from.
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