We went straight into this task at the start of the lesson. It was designed to provide some consolidation of the procedure for transforming functions of the form acos(x) + bsin(x) into either k.cos(x+c) or k.sin(x+c), as well as introducing a means of checking (by expanding the solution). It was also designed bringing together various 'uses' of this transformation.
I expected the students to find the first few parts task relatively routine, following our work yesterday (and some additional homework). However, the students found it harder than I had expected. Whilst we hope students have internalised what has been encountered, there is a danger in expecting students to be able to do something. With expectation can come disappointment, which can lead to negative interactions.
There were a number of instances of non-routine mathematics, which are hard to document. There was a 'mixing-up' of various parts of this procedure, unsuccessful short-cuts being taken, things were written that did not follow, mathematically.
They could be viewed as a number of (disconnected) errors, or a lack of 'rigour', but I suspect that many 'errors' made are - in this case, and often more generally - indicative of the complexity of the problem as a whole. I have found that students often do surprising things (both negative and positive) when working on complex problems.
I was aware that one of the students was feeling less than confident as the lesson went on. I felt his mood deteriorate. He usually presents as a confident person, but here was an emotional sensitivity I had not experienced before. I suspected, but could not be certain, that his mood was a direct result of - and contributing to - the difficulties he was finding with the mathematics.
Following this task, both students worked on this exam problem:
Both students performed the transformation in part (a) with little difficulty, and made excellent attempts at sketching the graph in part (b). They have improved considerably at making (reasonably accurate) sketches over the couple of months we have been working together. Among other things, I suspect this is due to the extensive use of interactive geogebra applets; I feel as though the canonical images are becoming ingrained.
When they had finished I asked them to look at what they had done. It is a joy to watch their mathematical development, and I wanted them to share in that joy. By their smiles I suspect they felt it too, although it might have just been relief.