If you had to pick the most frustrating part of your Secondary School Maths education, what would it be? For me, it is hands down the trigonometric ratios. You can probably guess how it was taught. The teacher simple wrote SOH CAH TOA on the board, gave some examples and then set us off with work from the textbook.
I remember exactly how I felt at the time. Hold the phone, what are these things sin, cos and tan all about?
I didn’t often question what we were told in Maths class because it simply wasn’t the done thing. This is partly because we had to move quickly through the syllabus to take our GCSE’s a year earlier than normal (not a good thing). Most of the time I didn’t worry too much about not knowing where something came from but it seemed a bit too far out using buttons on a calculator when I had no idea what they were. What was the secret?
It took quite a while before I figured it out. In fact, I think it was when a friend asked me at University what they were. At the time I embarrassingly had no simple answer based on his reference frame (he had no idea what I meant by a point moving round a unit circle). I spent the evening trying to connect the dots with right-angled triangles and it suddenly came to me as soon as I stopped thinking about it (as things usually do). Hence, it wasn’t until I was 20 years old that I understood why the trig ratios worked with right angled triangles!
It gets worse…
When I started teaching, I taught it in exactly the same way that I’d hated as a student, with absolutely no understanding! Why would I do that? My feeble excuse is that in my first two years of teaching, I taught in a school that only managed to get 15-20% of its students a grade C in GCSE Maths – let me say that again – the top 15 students in each year group would get a grade C. Hence, I taught the “difficult” concepts in a way that I felt would least confuse the students and prepare them best for the GCSE exams in which they didn’t need to know what sin, cos and tan were to answer the questions. The assessment items, coupled with the pressure of achieving results in a school league table system did have a negative effect on my teaching (not to mention the behaviour problems which I struggled to control).
Anyhow, no matter how the students are assessed now, I aim to promote understanding and since I changed my teaching strategy for this topic, I thought I’d share. One other thing I should note is that there are a few things that I wish I’d done slightly differently so I’ll actually be describing how I’m going to do it next year.
Introducing Trig – 2 lessons
To get started, I needed a context. I took a picture of a crane which you can see outside of our classroom window.
How tall do you think the crane is?
We went through the normal first act routine of guessing etc. and then I drew this picture on the board to move the converstation forward.
Me: Can we use Pythagoras to find this missing length?
Of course not, so we need some new Mathematics then. Let’s get to it.
Firstly, one thing I did differently this year was to get students measuring the lengths of triangles. Last year I got them straight onto investigating in Geogebra. That was fine but I think there’s something more concrete about physically measuring lengths as opposed to moving a slider and seeing the lengths change on a screen. I think you’re more engaged with the whole process of measuring and recording in this case and have more time to conjecture. Once the conjectures have been thrown out there, use technology to justify.
For the triangles, I used sheets provided by nzmaths.co.nz which has 8 triangles with a 30 degree angle, 8 triangles with a 40 degree angle, 8 triangles with a 45 degree angle, etc. Each pair were given one of the sheets. They had to meausure the lengths and fill out the recording sheet below. (This is a slightly edited version to the one provided by nzmaths)
Of course, once they’d filled out the sheets using ruler and pencil, they started to conjecture about what was going on: “It seems like the ratios are all pretty similar but they’re not exactly the same. Are they supposed to be the same?” At this point, get them on Geogebra to find out what the actual ratio should be for the angle they are working with(Click the picture to go to the file).
This confirmed that no matter how large or small the right angled triangle is, if it has a 45 degree angle in it, the ratios will always be exactly the same. It was obvious for most students to understand this due to the similiarity of the triangles.
At this point we’re half way there. Fortunately, as you’d expect, someone hits the nail on the head and asks how this seemingly useless fact will help us find the height of the crane. I draw the following triangle on the board.
Me: Which group has the recording sheet for the 30 degree triangle? Right, can you give me the most accurate answers you have (i.e. from Geogebra) for the triangle ratios.
Opp/Hyp = 0.5; Adj/Hyp = 0.866; Opp/Adj = 0.577
I write this on the board and invite them to figure out h. Within 30 seconds the first pair had an answer and in the next 2 minutes, every pair but one had the correct answer. I gave the first pair a chance to describe their reasoning.
Student: Since h is the opposite and 20 m is the adjacent then h/20 = 0.577 so h = 11.54 m.
I’m over the moon! We’ve just used trigonometry to find a missing length without even mentioning the words sin, cos and tan! Up comes the next integral question.
Student: How would we do this in an exam without memorizing the ratios for a 30 degree triangle?
Here comes the piece de la resistance:
Me: “Can you copy this down please?” I need you to memorize all of this.
For a 1 degree triangle: Opp/Hyp = 0.017; Adj/Hyp = 1; Opp/Adj = 0.017
For a 2 degree triangle: Opp/Hyp = 0.035; Adj/Hyp = 0.999… I’m finally interrupted.
Student: NO WAY SIR! Don’t you dare tell me we have to memorize every number all the way to 90 degrees????!!!????
Me: Don’t worry, it’s no where near that bad, only to 89 degrees.
ALL students: NO WAY, that’s impossible. Right, I might as well pack up now because I’m failing. There must be a better way. (refer to Fawn Nguyen’s post at this point – Thanks for the idea Fawn)
Me: Okay guys, do you think that you’re finally ready to see how this is done without having to memorize hundreds of numbers?
I get the TI-Nspire calculator software up on the board and type in tan(30), the answer is 0.577.
Me: Tell me what I just did.
IMMEDIATE ANSWER: “The 30 stands for the angle in the triangle and because Opp/Adj = 0.577 for a 30 degree triangle, tan on the calculator must stand for Opp/Adj.”
Let me share some student comments with you at this exact moment.
That is so cool!
The students have complete clarity of understanding!
YES, YES, YES! The students went on to figure out with considerable ease which triangle ratio is sin and which is cos. I couldn’t have asked for anything more in this intro to trigonometry! Next lesson is going to be about answering the initial question which will require making some simple theodolite’s, going outside and measuring the angle of elevation up to the tip of the crane. To make the theodolite, you need string, bluetack (which acts as the weight at the bottom of the string), a template protractor and cardboard(to stick the protractor on). I’ll stick a picture up of a student’s thedolite once they’re made.