This post contains six accounts from Higher maths lessons with J and K this week.
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.
#3 Exploring (fail?)
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: