In response to connections, fluency, coherence

With the last post in place I started the term with my year 10s in mind. With trigonometry in mind. With understanding in mind. I felt a panic/rush/excitement/worry about this and about problems that I may have caused with previous lessons.

We started the lesson with a discussion of what trigonometry evoked in the students: I asked them to think and discuss for a few minutes. Students suggested triangles, angles, labelling sides, lengths, inverses, right angles... they also suggested SOH CAH TOA... "But what does that mean" a lot of them asked. A good question. What does that mean?

The students were then asked to think of questions in which they would use trigonometry (they have had some experience of it before). Lots of triangles were drawn (right-angle on the right). "OK... what can you do now? Try something!" Most students' examples gave the lengths of 2 sides (e.g. opposite and hypotenuse) and an angle to which they were trying to apply sine. "But what do I have to work out?" This gave a good discussion point...why use sine here when you have the angle, the opposite and the hypotenuse? What do you want to work out? Why would you want to include an unknown? How would that help?

Rearrangement of the quotients led to discussion. "Why have you done it like that?" "I looked at what I did before and I swapped them" It seemed that thinking of sin 46 as a number proved challenging. Thinking of sin as a function was also challenging: "Can you divide sin x by sin to get x?"

There was a lot of positive discussion and a lot of challenge. The class were at such different points in this topic that using this method of teaching gave the class an opportunity to work on aspects which they themselves found challenging. Whilst I took this opportunity to clarify some methods and why we did them, some students plotted sine as a graph and experienced it as a function.

The richness of looking at a small group of functions for an extended period of time excites me and helps me to realise connections and coherence. I also see points at which other 'topics' can be assessed as they come into use throughout. I can see myself doing Trigonometry for a while!

That Which is Always there... (part 2)

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 21º 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.