**What's up with Hannah?**

Hannah is a Year 12 who I privately tutor. She is bright, articulate and hard-working. I was her maths teacher in Years 10 and 11 (she has now moved to another school). Last week I asked Hannah to factorise a quadratic. This is what she did:

x

^{2}+ 13x + 36
= x

^{2}+ 4x + 9x + 36
= x(x + 4) + 9(x + 4)

= (x + 9)(x + 4)

What's this got to do with trigonometry? Well in a subsequent conversation with Hannah she asked me about the upcoming changes to GCSEs, which will affect her younger sister. I told her that her sister will need to know how to find exact values of trig ratios such as sin30 and cos45. Hannah didn't understand what I meant by 'trig ratio'. She saw no link between what she saw as two distinct topics - ratio and trigonometry. She had absolutely no understanding of what sin30 is, though she is very competent at solving GCSE trigonometry problems. Again, this set off alarm bells in my head. I need to teach trigonometry differently.

**Does it matter?**

My primary role is to ensure that my students possess a toolkit of mathematical methods with which to solve problems accurately and efficiently, and an understanding of the mathematical concepts that underlie those methods.

Hannah got a good A* in her GCSE - this suggests I succeeded in equipping her with the required skills and knowledge for that qualification. But from my recent conversations with her, I realise that I had mixed success with the underlying concepts. I'm not going to beat myself up about it, but it gives me a focus - I need to think more about how I teach and assess for conceptual understanding.

When I introduce trigonometry I usually get my students to measure triangles and look for patterns (like this activity from Teachit Maths). Next time I introduce trigonometry, I'm going to try something a bit different. I'll show students these three triangles and ask what they have in common:

I hope they'll spot that they are similar triangles and that the ratio of the height to the diagonal is 1:2 (this would be a good time to introduce the terminology opposite:hypotenuse).

Then I'll show them the triangle below and ask for the length of the hypotenuse. I want them to realise that because it is similar (ie equiangular) to the three triangles above, we know that the ratio opposite:hypotenuse is 1:2. So the length of the hypotenuse must be 70.

We could say 'the ratio of opposite to hypotenuse in any right-angled with an angle of 30 degrees is 1:2'. This is a bit of a mouthful so instead mathematicians say 'sin30 = ½' (there's an interesting article here about the origins of terms sine, cosine and tangent).

I'll write sin30 = ½ on the board, then repeat this process for triangles with different angles, starting with 50

I'll ask my students to find the length of the opposite side in the third triangle and I hope they'll work it out based on their realisation that when the angle is 50

I'll continue with more sets of examples and end up with a list on the board that looks a bit like this:

sin10 = 0.174 sin30 = 0.5 sin50 = 0.776 sin 60 = 0.866 sin80 = 0.985 etc

Then I'll bring a 30

Hopefully someone will give the correct answer of 40cm - and then comes the important questioning. How did they know? Did they remember that the ratio is 1:2 when the angle is 30

Here I have a choice of where to go next with this lesson: either make a mini-project out of this (where my class make their own book of trigonometric ratios) or just show them a set of trigonometric tables ("here's one I made earlier!").

I think students will gain a better understanding of trigonometric ratios if they use tables, instead of calculators, to solve trigonometric problems for a few lessons (click here to see an example of a student's workings using this method). There's an online version of the tables here. Once students really understand what the ratios are and how to use them, reveal that all the numbers from these tables are saved in the memory of a scientific calculator.

I haven't tried this approach yet but I know other teachers do something similar. For example, the writer of this blog has his class create their own trigonometric tables. He says, 'I have found that by using a trig table my students concentrate on the concepts being studied rather than the calculator'.

The key point is that students need to understand that trigonometric ratios represent the ratios of the sides in right-angled triangles.

I'm interested to hear how others introduce trigonometry so please comment below or tweet me.

Now let's look at some good resources for teaching trigonometry. In my resource library I've made some recommendations so if you're planning any lessons on trigonometry, do have a look there. I've also found a few extras for you today. Did I mention that I love resources?!

Dan Walker has produced a brilliant PowerPoint on right-angled triangle trigonometry which starts off by introducing the ratios as I've described above. The whole PowerPoint is excellent quality and well worth a look.

Resources guru Don Steward gives us a range of fantastic activities. To practise calculating sides and angles in right-angled triangles, I like resources like these:

My students have a melt-down when I give them a problem that involves trigonometry and bearings. I suspect this is because my school doesn't teach bearings very well at Key Stage 3 (up until recently it wasn't even on our scheme of work), so they never know which angle they're meant to be calculating. In the question below they need to use alternate angles and trigonometry to calculate the bearing of town A from town B:

And in this more challenging question, they're asked to calculate the bearing of C from A. It can be done using right-angled triangles but it's much quicker to use the Sine and Cosine Rules.

MathsPad has a good range of trigonometry resources. I don't normally recommend resources that aren't free but I'm a big fan of MathsPad - a subscription costs £3 a month (please don't pay out of your own pocket - ask your boss). I particularly like the Sine Rule Codebreaker - it contains lots of practice questions in a more engaging format than standard worksheets.

If you haven't seen this resource before, written by Frank Tapson for TROL (teacher resources online), do have a look through it. It contains lots of helpful practice questions, including some on 3D trigonometry.

Finally, I'm looking forward to the next time I teach trigonometric graphs after watching Chris Smith's brilliant video in which he makes a spaghetti sine graph. I can't wait to try this.

Well that's it from me - I hope that's given you some ideas for teaching trigonometry. Please do let me know if you have any great ideas to share.

**Introducing trigonometry with similar triangles**When I introduce trigonometry I usually get my students to measure triangles and look for patterns (like this activity from Teachit Maths). Next time I introduce trigonometry, I'm going to try something a bit different. I'll show students these three triangles and ask what they have in common:

I hope they'll spot that they are similar triangles and that the ratio of the height to the diagonal is 1:2 (this would be a good time to introduce the terminology opposite:hypotenuse).

Then I'll show them the triangle below and ask for the length of the hypotenuse. I want them to realise that because it is similar (ie equiangular) to the three triangles above, we know that the ratio opposite:hypotenuse is 1:2. So the length of the hypotenuse must be 70.

We could say 'the ratio of opposite to hypotenuse in any right-angled with an angle of 30 degrees is 1:2'. This is a bit of a mouthful so instead mathematicians say 'sin30 = ½' (there's an interesting article here about the origins of terms sine, cosine and tangent).

I'll write sin30 = ½ on the board, then repeat this process for triangles with different angles, starting with 50

^{o}:^{o}, the ratio opposite:hypotenuse is 0.766:1.I'll continue with more sets of examples and end up with a list on the board that looks a bit like this:

sin10 = 0.174 sin30 = 0.5 sin50 = 0.776 sin 60 = 0.866 sin80 = 0.985 etc

Then I'll bring a 30

^{o}triangle back into play. I'll ask them to work out the height of this triangleSource: openlibrary.org |

^{o}? Or did they check the list on the board? Would they be able to do it from memory if the angle is 50^{o}? That ratio is much harder to remember. What if the angle was 52^{o}? We haven't worked that one out yet. Well, what we need is a reference list of**all**the ratios, like a bigger version of our list on the board. Then we'll be able to work out lengths in any right-angled triangle.Here I have a choice of where to go next with this lesson: either make a mini-project out of this (where my class make their own book of trigonometric ratios) or just show them a set of trigonometric tables ("here's one I made earlier!").

I think students will gain a better understanding of trigonometric ratios if they use tables, instead of calculators, to solve trigonometric problems for a few lessons (click here to see an example of a student's workings using this method). There's an online version of the tables here. Once students really understand what the ratios are and how to use them, reveal that all the numbers from these tables are saved in the memory of a scientific calculator.

I haven't tried this approach yet but I know other teachers do something similar. For example, the writer of this blog has his class create their own trigonometric tables. He says, 'I have found that by using a trig table my students concentrate on the concepts being studied rather than the calculator'.

The key point is that students need to understand that trigonometric ratios represent the ratios of the sides in right-angled triangles.

I'm interested to hear how others introduce trigonometry so please comment below or tweet me.

**Resources resources resources**Now let's look at some good resources for teaching trigonometry. In my resource library I've made some recommendations so if you're planning any lessons on trigonometry, do have a look there. I've also found a few extras for you today. Did I mention that I love resources?!

Dan Walker has produced a brilliant PowerPoint on right-angled triangle trigonometry which starts off by introducing the ratios as I've described above. The whole PowerPoint is excellent quality and well worth a look.

Resources guru Don Steward gives us a range of fantastic activities. To practise calculating sides and angles in right-angled triangles, I like resources like these:

**Bearings**My students have a melt-down when I give them a problem that involves trigonometry and bearings. I suspect this is because my school doesn't teach bearings very well at Key Stage 3 (up until recently it wasn't even on our scheme of work), so they never know which angle they're meant to be calculating. In the question below they need to use alternate angles and trigonometry to calculate the bearing of town A from town B:

And in this more challenging question, they're asked to calculate the bearing of C from A. It can be done using right-angled triangles but it's much quicker to use the Sine and Cosine Rules.

**The Sine Rule**MathsPad has a good range of trigonometry resources. I don't normally recommend resources that aren't free but I'm a big fan of MathsPad - a subscription costs £3 a month (please don't pay out of your own pocket - ask your boss). I particularly like the Sine Rule Codebreaker - it contains lots of practice questions in a more engaging format than standard worksheets.

**TROL**If you haven't seen this resource before, written by Frank Tapson for TROL (teacher resources online), do have a look through it. It contains lots of helpful practice questions, including some on 3D trigonometry.

**Spaghetti Graphs**Finally, I'm looking forward to the next time I teach trigonometric graphs after watching Chris Smith's brilliant video in which he makes a spaghetti sine graph. I can't wait to try this.

Well that's it from me - I hope that's given you some ideas for teaching trigonometry. Please do let me know if you have any great ideas to share.

I like that similar triangle idea.

ReplyDeleteAmong other things I use this http://burymathstutor.co.uk/UnitCircle.html

By the way, I wouldn't tell them:

"...that all the numbers from these tables are saved in the memory of a scientific calculator..."

I've read that calculators use Taylor series or the CORDIC algorithm.

Brilliant, thank you.

ReplyDeleteI just assumed that the tables were stored in the memory, didn't realise the calculator was computing them. I've learnt something new! Thanks.

Thanks to @letsgetmathing for sharing this excellent blog post which contains similar ideas, though a slightly different approach:

ReplyDeletehttp://danpearcymaths.wordpress.com/2013/04/17/introducing-trigonometry-taking-a-digital-leaf-out-of-fawn-nyugens-book/

I've tried the triangle-measuring approach before but I find that as soon as a calculator method is introduced, the concepts are quickly forgotten. Which is why my new approach attempts to embed the idea of ratio to a greater extent before moving onto more efficient methods.

Very useful resources. Thank you.

ReplyDeleteI also have a useful book related to this topic.

Trigonometry For Dummies (2014).pdf - 7.2 MB

http://www.anafile.com/hxsuffw53wx2.html

Great post! I think that Hannah's and my brain might have been made from the same mold: I followed all the rules and got high marks in trig but it took revisiting trig as an adult for the light to go on that sine referred to a fraction, which, or course, is a ratio. The fact that, in high school, all my "answers "were in the form of decimals made the whole ratio thing so opaque.In fact back then I don't think that it ever clicked that numbers written as decimals (and, in fact, all numbers) could be thought of in terms of a ratio. I had a nice little formula, I followed it, and everyone was happy and clueless. It all just seemed so random and arbitrary, like values that someone came up in some mysterious way.I was happy to just accept these values because that's the way it was. The trig values seem like logarithms in this way: I couldn't possibly compute their values, I just had to follow to rules to get the answer. The fact that there's a pattern to be found between the angles and ratios was lost in the lingo. One thing I would love to see in the teaching of trig is lots more emphasis on the circle that the triangle sits inside of, and also more emphasis on just sine and cosine, because once these concepts are fully grasped then the rest is a piece of cake. Trying to simultaneously make sense out of sohcahtoa, cosecant and all the strange looking fractions around the circle which don't intuitively feel that have any relationship to real numbers, well, teenagers think that they have more urgent issues to sort out and make sense out of. Oh, also, I wish that I had know back then that sine and cosine are used to describe cycles...which kind of makes the setting of the triangles inside of circles a bit more relevant.

ReplyDeleteThanks for your comment Paula, lots of very good points made and things for me to look into.

DeleteI introduced trig in this way to my top set year 10s today. They've all met it in year 9 and are familiar with SOHCAHTOA, but they engaged really well with developing the concepts from ratios. Students felt they had a better understanding of why we use the ratios at the end of the lesson and what the random decimals the calculators produce meant. They were fascinated by the tables and the fact that I had used them at school! I showed them that calculators don't just have all these values saved in them, but use power series to calculator and that the bigger the calculator/ computer memory the more accuracy they get. A lot of them do computing, so we're really interested in this. I also made the link to the unit circle which they seemed to understand quite instinctively. As a plenary I got them to see if they could spot pairs of angles from the display led trig tables that had the same ratio - ie sin 30 = cos 60. They started spotting lots of patterns and even 'guessing' at some that they couldn't see on the section of the table was displaying. I really enjoyed the lesson and it will be interesting to see if it makes a difference in the long run to their conceptual understanding.

ReplyDeleteAs an aside, the work was still on the board for my next lesson with year 12 further maths and they were asking lots of questions prompted from it, so some added interest there too!

Thanks for the ideas...Lisa

Thank you for this comment. This sounds like an amazing lesson - I'm so pleased to hear that my post gave you some ideas.

DeleteI agree that it is our job to help students understand mathematics and teaching just a method is wrong. I'm not sure that I agree that solving a quadratic by the method Hannah does not encourage understanding. It shows how the quadratic is split and can then be used in a grid method or other method, it is particularly useful when factorising quadratic where a > 1 ( whilst it is clunky it also shows how common factors work ). I think it could be used as a visiting method particularly when students don't know factors. And that leads to a question why don't students know factors or work with them well ?? I'm not saying your students cant but I know lots of mine struggle

ReplyDeleteThank you for this excellent breakdown, and lovely wealth of resources.

ReplyDeleteI have just been looking through the TROL booklet, which is great, however I am unable to find a link to where I might find the answers. I could of course sit down and work them all out, but it would save me a lot of time if anyone is able to let me know where I need to look. There is a description as to where answers can be found, but the links they advise checking do not actually exist.

Thanks in advance!

Hi. Thanks for the comment. I'm afraid I can't help with this - I'd love the answers too. I'd suggest sending a quick email to manager@cleavebooks.co.uk to request the answers, and if you hear nothing back try the affiliated feedback@cimt.org.uk. No harm in trying! Do let me know if you have any luck, and sorry again that I don't have the answers!

DeleteThis comment has been removed by the author.

ReplyDeleteLooking at the example student working using the similar triangles method, I was initially impressed with her apparent understanding of trig functions as ratios - this is not always evident to students when the ratios are presented as decimals, rather than fractions. But looking at her final answer made me wonder if she really understood what she was doing - her final answer should be a length! It could just be a careless mistake - it's always hard to tell, but I wonder if operating with trig functions in this way loses the functional aspect of the trig functions, namely that they are functions that convert angles to dimensionless numbers. The tables get produced by measuring lengths of triangles, and although the importance of angle is demonstrated initially the fact that the trig function is operating on an angle may get lost to the students.

ReplyDeleteSorry, I've not been clear. It's not a student's work, it's *my* example of what a student's work would look like if they used this approach. And the degrees symbol is a mistake - a 'typo' by me! I have corrected now (kind of, without rewriting it all), thank you for pointing it out!

DeleteWhen I wrote this post a couple of years ago I was thinking of new approaches but hadn't tried it with students yet.

Hi Jo

ReplyDeleteI have taught quite a low attaining set of y11 pupils trigonometry through ratio. It followed on nicely from our previous unit of work in which they asked "what's the point of writing a ratio in the form m:1?" What I am worried about now is that we didn't quite make the leap to using the formula algebraically. They currently feel confident in trigonometry which is a really big achievement for many of these pupils who came into my classroom with an inherent fear of maths. They can use ratio to find missing angles and missing side lengths. I feel that if I teach them the formulae this will undermine their new-found confidence, but worry that in an exam context they will be confused by the wording of some questions. Any advice is much appreciated.

It sounds like you’ve done a great job! What wording do you think will confuse them?

DeleteHey Jo, great post!

ReplyDeleteIt's helping with my planning for my top set year 9 who will be meeting Trigonometry for the very first time and I would love for them to appreciate the patterns and the mathematics behind it all rather than the good old 'here's SOHCAHTOA, now label the sides and find x' approach. I wondered where your triangle example images are from? You said you used examples for angles of 10, 60 and 80 degrees - did you just draw these freehand?

Thank you! :)

Hi. I can’t remember because this post is very old but I probably used CIMT images and edited them in Paint!

DeleteHi Jo,

ReplyDeleteAs always, a really useful post. Do you know when trig tables stopped being used in schools and calculators took over?

Thanks you

Hi. Sorry for the slow reply. Did you get the answer from Twitter in the end?

DeleteI am so excited to try this new angle for teaching trig (geddit). My brighter students definitely demanded a fuller understanding rather than just a method, so very excited to try out the similar triangle approach. Thank you so much!!!

ReplyDeleteHave you considered using a physics approach? In physics--I'm thinking of vectors primarily, but the word vector doesn't have to be used--a vector quantity is split into two orthogonal components because the two components are completely independent of each other. So the quantity is a %age of one and %age of the other. Whatever amount a vector is in one direction is shown by the projection of the vector onto either the x or y axis. This sounds complicated, but can easily be shown with a flashlight and shadows. Many real-world examples of this can also be introduced, with which many students will relate.

ReplyDeleteAlso, as an aside, I've struggled with the same thing you have explained here--the conceptual understanding of trig functions--but focusing just on triangles leaves out a lot of the story. Trig functions are really the link between circular or periodic motion in two dimensions and a one-dimensional representation of the motion. Without understanding this, how does the link to a sine or cosine wave function make sense?

Have you considered using a physics approach? In physics--I'm thinking of vectors primarily, but the word vector doesn't have to be used--a vector quantity is split into two orthogonal components because the two components are completely independent of each other. So the quantity is a %age of one and %age of the other. Whatever amount a vector is in one direction is shown by the projection of the vector onto either the x or y axis. This sounds complicated, but can easily be shown with a flashlight and shadows. Many real-world examples of this can also be introduced, with which many students will relate.

ReplyDeleteAlso, as an aside, I've struggled with the same thing you have explained here--the conceptual understanding of trig functions--but focusing just on triangles leaves out a lot of the story. Trig functions are really the link between circular or periodic motion in two dimensions and a one-dimensional representation of the motion. Without understanding this, how does the link to a sine or cosine wave function make sense?

Hi Jo. Just to let you know that the link to the online trig ratio tables is no longer live!

ReplyDeleteAnd this is a brilliant blog post, thank you!

ReplyDelete