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.

Here are their responses to the first task (click images, use left/right arrows to view):

The second task was intended to be an exploration of the gradient of f(x) + k. I had assumed/hoped that they would identify how the gradient for any x value was the same, regardless of k, by considering the graph. This didn't quite happen the way I expected. One of the reasons was that they were attending to the function rather than the graph, and did not know how to differentiate the constant term.

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:

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:

Here are their responses to the third task. This task is intended to be more open, to allow J and K to explore their own examples:

One of the drawbacks with J and K generating their own examples was that they only chose one or two examples from which to base their conclusions about the nature of cubics. The cubics they chose were not representative of all cubics. It may also be seen that they initially explored the gradient of these cubics by substituting a couple of values into the gradient function.

I felt I had to intervene to provide further/more representative examples, or model the use of geogebra for exploring the gradient at

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:

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:

The fourth task was an open exploration of quadratics. You can see that J (left) has started creating algebraic versions of the gradient function, whilst K worked with examples before making notes in his book:

Notice that K's (right) notes are not quite correct. I pointed this out, and he corrected his notes:

It is noticeable how their explorations are becoming more directed towards expressing generality.

The fifth task was designed to extend these ideas to quartics and quintics. We had not yet looked at polynomials of order >3 yet, but I felt this provided a good opportunity (quartics are part of the syllabus in Highers). Here are their notes:

We then had a discussion about what we had done today. I encouraged them to compare the similarities and differences between the features of the various curves. This led towards a brief discussion of the relationships between the degree of the polynomial and the numbers of factors and turning points. I asked them to summarise what they had learned:

Finally, I invited them to reflect on how they felt about working in this exploratory way:

Following this, we talked about the shape of the gradient function, and the connection between the roots of the gradient function and the turning points. This is something we will return to later when finding minima and maxima.

For homework, they are going to practice differentiating a wide range of expressions with negative and fractional powers, and algebraic fractions.

For homework, they are going to practice differentiating a wide range of expressions with negative and fractional powers, and algebraic fractions.