Often I have worked in a maths and science department within schools and colleges. I have zero science knowledge and the thought of even being mistaken as a science teacher terrifies me. In the spirit of doing something different and stepping out of my comfort zone I observed quite a few science lessons when working in an academy. I saw speed, distance, time being taught. I felt comfortable with this as we also teach the topic, although I admit I teach it badly. The teacher gave each student a triangle of SDT, you know the ones.
She said cover over the one you are working out, if it's on top it's a divide, if its side by side it's a multiply. The speed (no pun intended) the students were able to answer questions was astounding. Part of the ‘bad’ part of the lessons when I teach speed, distance and time is getting students to set up the calculation correctly. Forever encountering divides when we were after multiplication! Yes the students in this science lesson had calculators but they could set the calculation up correctly, my mind was blown! After some pupil feedback it became clear that this is the way speed, distance and time (SDT) was taught throughout their school lives. It made me wonder, why do we try to do things differently in maths then? How many times had I previously taught SDT with algebraic manipulation under the algebra topic rather than a stand alone SDT lesson? Why had I never used these triangles before? Was I comfortable with students being able to solve SDT in this way without knowing how to rearrange equations? I think I was willing to give it a go as an alternative for those who struggled with algebraic manipulation. I wasn’t planning on lowering my aspirations but sometimes when students just can’t access it, they need to be shown another way. Could I start with triangles and then explain the algebraic manipulation afterwards? I think I had found a way.
It got me thinking, not just about SDT but what else we could use these magic triangles for. I stumbled upon someone's kind free resource on TES that introduced trigonometry in the same way. I will forever be grateful for the contributor to TES for these magic trigonometry triangles.
In the same way as I saw SDT taught. If you are calculating a missing hypotenuse you would divide the opposite by the Sine of the angle.
Bravely (the same day!) I set about teaching trigonometry to my bottom set year 10s (this was in the old spec days where pretty much everyone did higher tier in the schools I worked in. We can debate the morals of that widespread practice another time, I offer this information to give context to this example) my bottom set year 10s knew that I believed in them and they had developed a level of resilience where they would be willing to give it things a go. True to their reputation and our relationship they gave the triangles a go. The magic triangles were a hit! They could calculate with trigonometry! Bottom set year 10 could find missing sides of triangles!
Over the next few lessons we even got to inverse operations and calculating missing angles. Reflecting on the book work we still had a problem in that we couldn't label triangles correctly. Once labelled correctly we could answer missing angle, missing side almost all types of question, it was just how to label triangles that was letting us down now. I then began to think about how can I help learners accurately label triangles? I realised they could do the opposite with ease (the one opposite the angle with writing on!). Then they could label the hypotenuse (the one opposite the right angle!). Finally the missing side would be the adjacent. We had cracked it, they could label triangles accurately!
Reflecting on my success I happily sat down to mark and realised we were still making mistakes. Students were using the adjacent when it should be the opposite. They were using all 3 sides of the triangle regardless what the question was asking. I decided to have a chat with my students and see if we could work out why they were using all 3 sides. Ellie immediately became frustrated and said “just tell me what information do I actually need Miss?” this got me thinking, we don’t need three sides to solve the problem. We only need two sides. Once Ellie and I worked through this she flew! Immediately Ellie began eliminating a side, always correctly. I asked her how she knew which one to cross out and she said “Miss, there’s always one without any writing on!” She was right!
I had great success with trigonometry this year. Was I to ask a student a trigonometry question after the exam would they be able to answer it? Probably not, but if we have them a magic triangle could they? Probably yes. Reflecting on my practise though, did the students learn trigonometry or did they learn how to use a magic triangle? We all know what the answer to that is. This was quite a few years ago and keen to develop conceptual understanding I now I teach trigonometry quite early on. It wins students over in the early weeks. Trigonometry is often a struggle and with magic triangles students seem to be able to attempt it easier. I then move into algebraic manipulation from the magic triangles rather than the other way round as it was before I taught Ellie and bottom set year 10. Last year a resit student of mine in her late 50s said, “ I didn't know why things change from a multiply to a divide and that they undid each other, but it's just magic triangles isn't it?” I replied “yes it is!”
Students can often recall sohcahtoa with ease, with a slight change in the way you say it, making the o in soh and the a in cah with the o in toa slightly higher when you say them you can write them like this naturally making magic triangles. Meaning you can always grab a magic triangle to help you get started!
