1 Going with
In today's lesson with J and K, we started by going over a couple of questions that J and K had found difficult from some exam questions, and then we (me and them) created a couple more similar examples to 'consolidate' the ideas, including some examples that drew attention to the structure of the questions (see K's notes on this in the image below).
J and K said they found it useful, but I was concerned that it required the accommodation of a large number of ideas. What would they take away from this? Whilst going with what they wanted to work on felt useful and 'natural', I was concerned that there was a lack of the coherence that was a prominent feature of yesterday's lesson (in which we worked on one exercise). There was a sense in which it was a drop in an ocean.
I often start a lesson by reviewing homework, but I wonder if it can detract from the coherence of the lesson. The question, I suspect, is whether I am choosing to review homework, and whether it is useful for the learners, rather than doing it as a matter of course.
Working on problems involving differentiation is an exercise in learning what to do, when to do it, and why. The following actions may be useful at various times, depending on the problem:
- Solving f(x)=0, to find roots.
- Solving f'(x)=0, to find the x-coordinate of stationary points.
- Solving f(x)=k, to find the x coordinate of a point given the y coordinate.
- Solving f'(x)=k, to find the x coordinate of a point with a given gradient.
- Substituting a value for x into f(x) to find the y-coordinate of a point given an x coordinate, often a stationary point.
- Substituting a value for x into f'(x) to find the gradient at a point.
- Substituting x = 0 into f(x) to find the y-intercept.
Often problems involve a number of these actions. It is unsurprising that learners find it difficult to know what to do and when. A question that J and K keep asking themselves is: "What does this [action] tell us?" A deep understanding of functions/graphs and gradient functions is required, and a range of visual images.
3 Complicating matters
J and K both tried to solve the equation 3x - x^3 = 0 by factorising into two brackets, like this: (-x^2 ...) (x ...) = 0. I have noticed that they often do not spot 'simpler' approaches to solving (quadratic and cubic) equations where possible, such as this - where a factor of x can be taken on the left hand side.
We have talked about looking for simple approaches a few times, but it is very difficult to break routine and spot a simpler approach in the moment. I presented the problem x^2 - x = 0. Slightly surprisingly, J immediately transformed the equation into x(x-1)=0, perhaps recalling a conversation from the past, and then could immediately see how to solve the original equation. He sighed and said he was "complicating matters".
4 Working on, not through
The equation above came from this problem, which I gave to build on work on stationary points, and specifically points of inflexion, that we had first encountered yesterday:
Both students found the addition to the problem (in red) challenging. It was designed to make them consider other forms this type of question might take, as well as offering an opportunity to be creative.
Here is K's first attempt, setting f'(x) = x + 3, which I particularly liked because it revealed the min/max structure of quadratic stationary points in quadratics. After he had drawn the graph, I asked him why this function could not have this graph, and he circled the power 2 in f(x), as can be seen:
J's first attempt at the problem is shown below. Reasonably early on he conjectured that f'(x) could be of the form (x-1)^2 (see top right of the image below) and went from there. It was an interesting moment when he found the quadratic 1/3x^2-x+1 had no solutions, meaning the cubic only had one root, which he later reflected must be the case, due to its having a point of inflexion!