On Memorising

I remember taking over a Higher Level IB maths class where the previous teacher had told them that for the trig questions on the non calculator paper they really had to memorise the unit circle with the values of the special angles.

I personally, despite having a pretty comprehensive understanding that the unit circle is the basis of the trig ratios, have never been able to remember which numbers go where. I remember there are some √2 and √3s in there somewhere, and some symmetry, but actually which numbers go with which angles? Nope!

Some of the students in the class had repeatedly tried to memorise these special values but found that they were prone to mistakes. So I showed them my trick. 

I told them that I have two favourite triangles; one is half a square and one is half an equilateral triangle. You can label them as you wish (as the fractions cancel down at the end) but I like to label the square with side length 1 and the triangle with side length 2. 

Then with a little Pythagoras' Theorem, the fact that angles in a triangle add up to 180° and the definitions of sine, cosine and tangent you can find out all you need to know about these triangles and deduce the required ratios. 

In order to figure out the signs for angles over 90 degrees I do have to think of the graphs too but not in that much detail really. I usually just visualise the following in my head:

As I went through this explanation many of the students were pleasantly surprised at the neatness and even those who had managed to learn the original circle were intrigued by this other approach.

I know that some people are good at memorising lists, values, formulae, times tables etc. We should be encouraging them to do this. And, for those people, I imagine that once they have memorised these things, they pop out at exactly the right moment when solving a problem.

However we have other students, maybe like me, who find memorising words, numbers, symbols and formulas incredibly difficult and perhaps prefer a diagram, some understanding of the underlying structure or a way of figuring something out from first principles. We should be giving these students opportunities to feel successful too.