This post contains six accounts from Higher maths lessons with J and K this week.

## #1 Coming-to-action

J and K are stuck on the problem

K says:

*'Prove that the line y = 4x - 2 is tangent to the circle with equation (x - 6)^2 + (y - 5)^2 = 17'.*I ask them what they are thinking about. This is a simple question, but I sometimes forget to ask it.K says:

*"I want to set them equal to each other, but I can't."*I have noticed before that this seemingly habitual phrase, "Set them equal to each other," is preventing them from solving a wider range of simultaneous equations. We have discussed and used the substitution method for solving such equations a few times, but it does not yet come-to-action for J and K.## #2 Note-making

This year, I am giving more emphasis and time to reflection. I am pleased that J and K now automatically make excellent notes, without prompting - and always containing pertinent examples - such as this:

## #3 Exploring (fail?)

I decide to use 'guided exploration' to introduce the chain rule, but it becomes an exercise in

*pattern spotting*. I hoped they would spot the pattern when differentiating expressions with brackets, but they didn't:What there have been any more benefit to them spotting the pattern than me telling them the chain rule directly? I'm not sure, and am concerned that the time could have been used more effectively.

However, there is value in J's conjecture - visible in the image - that (x^2+3x+4)^2 has derivative

**k**(x^2+3x+4). This idea forms the basis of subsequent work. J and K do become fluent in using the chain rule next lesson, perhaps as a result of this exploration. I remind myself not to make premature judgements about the usefulness of something.

## #4 Exploring (success?)

One of the most successful aspects of my teaching this year has been creating opportunities for J and K to use Geogebra to explore functions. I invite them to use it to explore the gradient function for functions of the form

*a.sin(k.x+c)*. This exploration is 'immediately successful', meaning that J and K make accurate conjectures about the derivates, and have reasons for doing so. I think this is because there is a (visual) basis for the pattern spotting which was not present in the previous exploration.

## #5 Not f'(x) = 0

J and K are solving this unusually worded problem:

They both find f'(x), and then try to solve f'(x) = 0, rather than f'(x) = 1.

## #6 It

I presented this problem

*before*discussing increasing/decreasing functions, as a means of introducing the concept:J and K try to determine various things, but without any clear rationale. As the level of uncertainty increases, they become confused as to whether they are trying to show whether g(x) or g'(x) is increasing. J finds the turning point of g'(x), and concludes:

*"It is not always increasing."*I ask him what he is doing. He says:*"I'm trying to show that it is always increasing."*I ask him what 'it' is.