Only J was in my lesson today as K was away. J asked to do some further revision on trig and log problems (following on from yesterday), as these are areas in which he is less confident.
Following yesterday's post, I was again interested in recognising-things-as. There were examples of recognising, and not recognising, as is to be expected given that we have not worked on these topics for a few months.
J started solving this problem by transforming tan^2(x) into sin^2(x) / cos^2(x) - see top right - but then realised this might not be useful. He then wrote what is in the red box (the blue happened later...). I was pleased that he drew the graph and found both solutions for x = tan^-1(√3) - an example of recognising that trig functions have multiple solutions, and using a visual image to find them.
In yesterday's post I described how J does not generally consider the negative root when square rooting. When he got to this point, he looked like he had finished. I asked him if he had finished. He then said: "I wonder if there is something I've missed with the squared?", and then started writing what is in the blue box. He had recognised that the action of square rooting results in a positive and negative solution.
This is an example of J becoming more aware of multiple 'hidden' solutions.
In yesterday's post I talked about the usefulness of having a visual image, as opposed to working purely algebraically. I was pleased to see that J had drawn the triangle (top right) when solving this problem. This is an example of recognising tan(x) = 4/3 as providing information about a right-angled triangle, from which the other trig ratios can be derived.
J sketched the first graph in an ingenious way. He made x the subject of the equation, and then drew the resulting log graph on y-x axes, then reflected this curve in the line y=x! He did the same for the second one.
I then explained that it would have been easier if he had recognised the question as really being about function transformations. In my experience, learners find it difficult to recognise where consideration of a function transformation might make a problem much simpler.
Function transformations appear throughout Higher and A-level maths (exponential, log, trig and polynomial functions), but their application is often implicit. I made a mental note to do some work on recognising function transformations in the future.