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!

That’s legit!

Boom!

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.

Love this! Thanks for sharing :). Looking forward to reading more of your blog this weekend!

Thanks Jim, also looking forward to reading some of your blog posts.

This is fantastic. I’ve done something similar in the past – having students construct the triangles, make measurements and calculate the ratios. What I haven’t done is then have them use that information to solve a problem before going to the calculator and the definitions of sine, cosine and tangent. I love how you set this up so nicely to hook them in!!

That’s exactly the way I’ve taught it in the last few years. Thanks for your comment – I need to read over your blog posts.

Pretty much identical to the way I teach trig other than that I get the classes to draw up trig tables (in 5 degree increments) through the measuring process and answer loads of questions using those.

That leads to students wanting a ‘more accurate’ table in 1 degree increments (which I project from the www) as some of the angles are not multiples of 5 etc.

A bit further and then the ‘ do we have to learn these’ comes and out comes the magic calculator.

An interesting point Steve – I’ll have to consider getting them to do more examples before the calculator comes out. Having said that, the students “seemed” ready for the calculator at the end of the second lesson.

How about this as a next measuring activity? I taped string from the top of the locker down to somewhere on the floor, forming a right triangle. Made enough of these triangles-all differently angled-so each pair had their own triangle. They had to measure angle and a length to calculator locker height. Of course, all groups should get same answer.

Love love the crane idea, something they see every day!

Nice idea Reilly! How great would it be if the students then went to find their own object to make a right triangle with, take a photo and find missing lengths/angles.

“For a 1 degree triangle…” Too cruel, Mr. Pearcy. :)) Absolutely love this, Dan. Thank you!

Ha, I’d say that part of the lesson was influenced by you – make them beg! 🙂

Laughed out loud when you started the 1 degree increments! Awesome lesson.

Ha, cruel but really fun. I just wish I could have recorded the reactions! Thanks for the comment Darren.

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Just tried this today with 3 other teachers as a lesson study. We have 50 minute periods and found that we only got to either calculating the mean, and checking the Geogebra values, A few classes got to trying to calculate the triangle, or it was assigned for homework. Here are some of our thoughts. 1. only give out the 30 and 45 degree triangles. Have each person in a group of 4 measure only 1 triangle and then only average 4 instead of 8. I know it leads to possibly less accurate answers, but it also allows time for discussion about “what if your averages are far from the accurate decimals.

Some teachers felt pressured to blaze through, and some students want to wait to be given answers, but those are growing pains that we address, and build more activities like this into our routines. Ample time and encouragement needed to be given for the kids to discuss and think about the comment section, to see similar triangles, different lengths, and similar ratios…

We are doing CRISS framework for staff development this year, so we added a prior knowledge piece as a warm up, solving some one step solving such as .876 = x/4, problems, and thinking about how the words hypotenuse, adjacent side and opposite side apply to an angle in a right triangle.

Tomorrow we plan to have them “beg” for the short cut and end with a reflection piece about what they learned.

Do you have another suggestions or have you thought about changes you would make?

THanks!

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execellent idea to introduce the three ratios

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Thanks. Excellent intro. for today’s lesson. It gives students some perspective into the “old” Trig tables of values.

Dan, this is absolute gold. I have been looking for an inquiry based lesson in trig for a long time. Even came up short with Dan Meyer! I I am going to try this with my students tomorrow. Thx!

Daniel, I am new to using Geogebra and to this approach of teaching math – I LOVE LOVE LOVE your lesson ideas! I thank you so much for sharing all of your wonderful ideas. I can see through your lessons how I can transform my teaching to be more powerful and to involve the students completely. Thank you!!

Hi Christine, thanks for your kind words – it’s really nice to get positive feedback. I think you’ve hit the nail on the head with student involvement – I made a video recently in which I talk about ‘dialogic teaching’ which is what I wholeheartedly believe in as a way to make students integral to their own learning and create a ‘conjecturing/thinking/ideas sharing” atmosphere in the classroom. https://danpearcymaths.wordpress.com/2015/10/23/geogebra-pedagogy-workshop/

Will this lesson work well in an interview lesson? an introduction to Trigonometry to top set year 9?

I am a little worried, students may only have enough time to complete the recording sheet, where can i show differentiation and progressions? any suggestions? my interview is tomorrow.

Thank you

Hamara

Hi Hamara, possibly a shortened version of it. You could maybe use parts of it but I highly doubt you could fit it all into one lesson. To do it properly, I think you need at least two full lessons.

Hope the interview goes okay!

Trigonometric ratios, fast app to find: sine, cosine, tangent, cotangent, secant and consecante of an angle

https://play.google.com/store/apps/details?id=com.anazco.juan.razonestrigonometricas&hl=en

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