Jyoti Thyagarajan
All teachers of trigonometry start with right-angled triangles, and they draw a little square where the 90º angle is. Then they put θ at the other angle, which is the base angle. Then they ask, “Which is the hypotenuse?” A chime of young voices comes back with, “The longest side!” and occasionally, “The side opposite the 90º.”
I know this, because this is what I did for the first five years I taught math.
The title of this article comes from the large number of times that I have heard this question from students and the many times that I have smiled in response. In my mind, trigonometry was beautifully patterned. I did not ever ask the question, “How is it possible that there is always a right angle?” Neither did my students. They were delighted that there was a pattern and that they could extrapolate these patterns to “fake” an answer from it. (Sometimes, they even coloured the sides of the triangle different colours to make it look pretty!)
Then, I travelled through Italy, and saw the Leaning Tower of Pisa and thought, “Aha! I can be adventurous with right-angled triangles now!” and brought it into my classroom. I took a printout of the picture below and handed it to every student and we constructed right-angled triangles around it.
Today, this is not a “rocket-science type lesson plan”, but here is what happened on that day in class. Some started at the right end of the tower and drew a line parallel to the edge of the picture. That ensured a right angle, within reasonable doubt. Some started at the left end of the tower and did not quite see where the line would land if it was perpendicular to the ground.
I heard a couple of them saying, in answer to this possible inaccuracy about the perpendicular,
“Could we run parallel to the wall on the building on the left? That should make it perpendicular, right?”
And someone else said,
“How do you know it is also not leaning? It looks leaning.”
And someone came back with,
“Have you ever heard of the Leaning Big Building of Pisa?” (A little laugh runs around the class.)
“That is the Pisa cathedral,” said the class pedant.
And with each of these comments, I learned a little more about how students learned. And suddenly, I realized that I was a PHYSICS teacher! And we were in the Physics Lab, so I dived for twine and weight bobs, tied some string to each bob and handed them out.
“Does this help to find the perpendicular?”
And they looked puzzled, till one of them put the sheet up on the board and held the twine and bob on the picture. He turned around, his eyes almost luminous, and said, “Good heavens! I now know why we look for right-angled triangles.” AND IN THAT INSTANT, SO DID I! All this time, it was just a pattern to me.
So if we fast-forward to 3D geometry in high school, it became completely exciting, chaotic and adventurous. I would teach it like this, maybe: “Let us look at the International Space Station. How many countries are involved? And if they can work in outer space, completely trusting each other, it might be a good lesson for us to take away from this exercise.”
A downloaded image of the ISS can be the source file for this class. (https://commons.wikimedia.org/w/index.php?curid=61056806)
We establish that this station did not go up there, constructed. An adventurous question to ask now would be, “Why would you not build it, put it in a box and get four rockets to payload it into orbit?” A hint might be: The Earth is moving at 450 km PER SECOND in its rotation. It drags its atmosphere with it at roughly the same speed. The box would surely be battered by the atmosphere and if it survived all that, what would you do with the box after you unpacked the ISS?
So we know that it went up there in bits and it was put together. Each piece left the earth from a different point. The Japanese probably left from Tanegashima Space Center in southern Japan and the US Lab left from Cape Canaveral. I would allot countries to students and they come up with acceptable trajectories. They decide that if the Earth is spinning in a certain direction, they can use that direction to catapult the part into space and then course-correct. And they do not have to be accurate or even right. They must just be persuaded to think at the leading edge of their knowledge.
What is important to this discussion is the ubiquitous right-angled triangle and its accompanying trigonometry. And the title of this article is not a question that classes ask me too often after the first few lessons. If they do, then there is always Roberto Carlos’ banana-goal-kick against France. There are also the three-pointer shots for the basket from Steve Curry of the Golden State Warriors. Neither would have worked without trigonometry. And some of our famous cricket catches on the rope, with the catcher throwing back the ball to his backup fielder, who is exactly on the perfect spot to catch a desperate throw back from the first catcher just before he steps over the ropes! Slick moves in sports always sweep up the students and bring them back on point.
I once had to teach the whole of speed, velocity and acceleration to a class using skateboarding because one of my students wanted to be a professional skate-boarder and he was good at it. He turned out to be the best resource of examples and proofs. The basketball court became our lab with six skateboards and many kids with scraped knees and knuckles.
I am no longer in classrooms and I miss that more than I would like to admit. If I ever return to the classroom, I think it will be for that wonderstruck shine in the eyes of a student saying, “Ooooh. THAT is why we use right-angled triangles.”
And only a math teacher will understand why.
The author is a teacher who has been in classrooms for over 44 years. She has taught physics and maths in Africa and in India. She can be reached at jyoti@meghshala.org.