This is an account of a 'higher' lesson with two students, J and K. The lesson was an introduction to basic differentiation, and an exploration of (the gradients of) polynomials of varying degrees.
I asked them to read the sections in a textbook on differentiation in preparation for the lesson, but both students sent me an email saying they could not understand much of it. As a result of this, I prepared these notes, which we read and discussed at the start of the lesson, and which they said were much clearer.
Then I offered this set of tasks, a guided exploration of gradients of polynomials from quadratics up to quintics, designed to be worked through using Geogebra. Notice how there is no mention of differentiation from first principles, and only a brief mention of rates of change and tangents. The aim is to first develop a rationale for differentiation (to find the gradient function), and become fluent in differentiating.
At some point I intervened. We discussed the graph, and this lead to the realisation that the gradient function was the same for all values of k. This led to a discussion of how to differentiate various expressions, which formed the examples from which they differentiated everything that followed. Although it didn't turn out how I expected, the task was a valuable one: it provided the basis for discussion of various examples, after which I invited J and K to make their own notes. Here are K's:
I felt I had to intervene to provide further/more representative examples, or model the use of geogebra for exploring the gradient at all points on the curve. The way I did this was to focus attention on a statement J had made: "The gradient of all cubics is always positive." As soon as I re-stated it, he modified his conjecture, noticing that he had differentiated -7x^3 incorrectly (see above left).
At this point, I drew attention away from substituting values, towards considering how gradient was changing along their graphs. I selected K's example (above right) as it was a cubic with two turning points. I generally seek to use a learner generated example that is representative: this always seems to create more interest than using an example I have created. You can see J's notes on what we discussed at the top of his example (above left); K started making notes in his book:
It is noticeable how their explorations are becoming more directed towards expressing generality.
For homework, they are going to practice differentiating a wide range of expressions with negative and fractional powers, and algebraic fractions.