As I drag the corner of the right-angled triangle on my whiteboard, creating an infinity of similar triangles, I draw the students’ attention to the calculation of the ratio of the lengths of two of the sides. Yes, through all this change, it remains constant. How do I envisage my students reacting? I want them to say what they think, not what they think I want to hear. What is a reasonable reaction to my use of a dynamic geometry software package? Well, ‘so what?’ springs to mind.

Mathematically, the proportionality of similar shapes is interesting. My lesson, or my demonstration invites a termination to the discourse. I ask the question, how do I *teach* the students trigonometry? Steer, de Vila & Eaton (Mathematics Teaching 214) describe a method referred to as ‘MNO’ (Burke and Olley, 2008). I will describe and exemplify this method for teaching Trigonometry at GCSE, and then ask whether this still invites a shrug of the shoulders.

The M in MNO stands for *map*, and what I want to do here is lay down the maths that is Trig at GCSE level. I have clustered these around nodes, the clusters representing maths that I think can be taught concurrently, and by implication, the separation indicating the maths that I think can’t. My map is not set in stone, is negotiable and evolving.

I move onto the N, the narrative. I think that most maths could be introduced from a historical perspective, in this case, the trig ratios from Antiquity, and the relatively modern view of Trigonometry as a Function. I have decided to take a problem solving approach this time, therefore, the History is bypassed. My narrative looks like this:

The O is for Orientation, I want the students to engage with the maths I want them to learn by doing something. This is where the plethora of activities for learning maths that has been built up over the years comes in. One of my first stops will be www.resourceaholic.com . The activities I get my students to do will include the use of problems to be solved (first stop nrich.maths.org), as well as class textbooks. All this will be clarified by my strong explanatons, and students will have plenty of time to reflect and ask questions. I will write about the activities I use another time.

So can the students still reasonably respond with ‘so what?’ Of course they can, they can say* and think what they want. Given the thought I have put into my plan for teaching the topic, I am confident that I can contextualise their concerns and reorient them towards mastering trigonometry.

*Clearly there are some things one can’t say in a classroom, a reorientation towards the maths seems to me to be the best response.

Note spelling of http://www.resourceaholic.com/ 🙂

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Thanks. Sorted now

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I’m going reread your post, and maybe even comment again after visiting your links, but now I’d like to say what was so surprising to me about trig when I decided to look at it again after being away from math for decades. Once I realized a few things, thing that will seem so elementary to you, it was like I was struck by lightning. First, and this will be no surprise to you, I got hung up, really stopped in my tracks, on trying to keep the terms straight: sine, co secant, tangent AHHH! All 6 of the basic trig terms were give equal weight in my high school trig class, but now I’m wishing that there was just a focus on sine and cosine. Once those two most used concepts are embedded, then mention the others. If sine and cosine are thoroughly grasped, the rest is just an extension of the concept, right? Then, the circle disappeared from the conversation way too soon. Now, I find it useful to see every triangle firmly locked into a circle that is fixed on to the origin of the coordinate plane. This makes so much sense as I finally got the memo that the line extending from the origin is the most phenomenal equivalent fraction wizard in the universe. What? no more having to find common denominators to find equivalent fractions? Just identify points on the line and your get equivalent fractions? Yes! If that doesn’t get a student’s attention nothing will. And what are those fractions? They are the x.y points, in other words, the length of the triangle’s legs. And here’s where the lightning made the most direct hit, when I realized that all those trig terms were just fractions. Numbers less than one. Less than one because in relationship thinking ther only space that exists happens between nothing and everything, in other words, between nothing and the whole things, oh, between zero and one. I do know why I missed that bit about sine and cosine being fractions, it was because (and this will date me) I learned trig before everyone had calculators and we looked up trig functions in tables of decimals, so I was never aware of any patterns, just right and wrong answers,

I am really sympathetic to the position of trig teachers. You are competing with everything else a teenager is thinking about.

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Thanks for the reply; this has given me a lot to think about

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I like the approach you have taken to in making connections. I have used a similar approach to map and locate trigonometry. I recommend that you look at the follow (see p. 203 and on) which has a very comprehensive of research in to teaching and learning trig.

https://global.oup.com/academic/product/key-ideas-in-teaching-mathematics-9780199665518?cc=gb&lang=en&#

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Thank you

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FWIW I have used the following successfully (I think) in China: use the London Eye & distance from the center to the side (r cos x) and distance up (r sin x). and then define sin and cos using a unit circle with r = 1. Maybe can have printed picture of the London Eye and the students make measurements for various angles. Perhaps add a picture of another Ferris Wheel with different radius and have students calculate the ratios equalling sinx and cox for both, hence deducing the ratios are similar for the same angle but different radii. I like your map because it shows other related ideas. Maybe you should add trig identities to your map.

Nice blog.

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Thanks Dave!

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