Spaghetti Trigonometry - instrumental vs relational understanding

I've been really excited about all my y10 lessons this week. I love planning for them and teaching them because they're great students. My lessons with them are never perfect by any means, but although they're a pretty diverse bunch (32 students, FFTs from E to A) they've got a great attitude to learning and they're just lovely to work with. I also generally plan y10 lessons with a colleague of mine who teaches the parallel middle set group and I really enjoy collaborating with her. 

This week and early next week we're doing Trigonometry and we've had 4 lessons to play with. The students had never heard of it before, so we needed to think carefully about what we could actually fit into the 4 lessons without over-doing it and making everyone stressed. I also wanted to think about how we could get students to have a genuine understanding of trig, rather than just viewing it as a topic where they "follow the steps" and get the answer.


I took to the web and looked around for good trig lesson plans. I found this video (https://www.teachingchannel.org/videos/introduction-to-trigonometry) where a very confident American maths teacher gets out a set of bongo drums and chants Soh Cah Toa, Soh Cah Toa, Soh Cah Toa! with her class. It was nice, but this approach only generates instrumental understanding, rather than relational understanding. (If you watch the video, those of you who know me may be thinking that using bongos sounds like just the sort of thing I might do and well ...... yes... I was quite taken by the bongos part, so I may or may not have had a go on my desk......)

Then I found this video and got very excited http://www.youtube.com/watch?v=AUOgD_Hq70A . Spaghetti trig! I recommend watching the video if you want to do it and need a more detailed explanation, but basically it works like this:

You draw out a unit circle and mark off 15 degree intervals on the circumference. The circle I used had a radius of 12cm, so we took that to be 1 unit. You snap off a piece of spaghetti to be the size of the radius and move it around the circle. You then use other pieces of spaghetti to construct right angled triangles like the one in the diagram.

Since sin(x) = opp/hyp, if hyp = 1 then sin(x)= opp, so the length of the piece of spagetti is the value of sin(x). You can then stick each piece of spaghetti that you use onto the appropriate place on the axis (as you can see in the diagram).

I was really excited about using this idea, but I wasn't sure how it would work out. Usually I don't teach trig graphs to middle ability groups until near the end of their course. It's classed as A/A* grade, so I leave it till the end "if we have time", but it got me thinking that, stuff the supposed "grade" of the topic, it's actually not that difficult, and it could really help students to understand what they were doing. 

So this is how we approached the lessons

Lesson 1: Discovering trig ratios by measuring triangles of 30, 45 and 60 degrees.
I also included a quick intro showing students real life situations where trig is used so that they have an idea of the big picture. I always like to give them an idea of where we're going at the start of each new topic.

Lesson 2: Find missing sides using the Soh Cah Toa formula triangles
They got the hang of the method pretty easily, but then, the method is pretty straightforward if you use the "cover up method" in the formula triangles. They could then answer questions, but genuine understanding was still not there. (Despite the use of desks as bongo drums)

Lesson 3: The graph of Sin(x)

Spaghetti trig! The one I was really looking forward to!

So, how did it go? Was it a useful approach?

I'm going to stick my neck out and say it definitely was a useful approach. Of course it took longer than the standard "use your calculator and plot points on the graph approach", but the pieces of spaghetti snap pretty easily so it didn't take too long. More importantly, it generated LOTS of really useful discussions. Students quickly realised that there were pairs and indeed quads (is that the right word?!) of lengths that came in 90 degree intervals. They had really useful discussions with each other about what might happen at 90, 180 and 270 degrees (is it infinity? is it 0? is it1?) and they appreciated that it would turn into a wave.

We ran out of time at the end of the lesson to go any deeper into the discussion, but I intend to use the graphs at the start of next lesson and stick at least one on the wall. When I teach it again, I reckon I can speed it all up by being more organised with the resources.  Questions I'm thinking about for next lesson are: how could we do cos(x)? What happens when we go over 360? What is the amplitude of the graph? If you know the value of sin(10) what other angles can you tell me?

Hmmm. I'm getting carried away! This is getting into A level terriorty rather than GCSE, and I'm still not convinced that they all understood the point of what we were doing. I'm going to have a good long think about next lesson and figure out a way to draw everything together. Can we find missing lengths without using calculators, just using the graphs? Maybe if we used non scientific calculators then they would have to refer back to the values on the graphs. We have got sets of non scientific calcs in the department........... (voice trails off as Miss King stares into middle distance and gets lost in mathsland.........)