I recently taught integration to my Y12 class. To make sure I hit all the misconceptions from the previous lesson, I crafted my board example from a function and differentiated it. The final integration problem had a function to integrate and a point it went through, enabling the constant of integration to be found.
The class managed really well with the problem, but I felt from their questions, that they weren’t ready for independent work. Off the cuff, I explained how I’d constructed my example. Then it struck me – get them to use the same process:
1. Think of a function f(x) – the difficulty level is up to you.
2. Pick an x value, then work out f(x), to give you a point (x,y).
3. Differentiate f(x)
4. Give your point (x,y) and f'(x) to a partner.
5. Your partner works through your problem trying to find f(x).
6. Check your partner’s method and solution.
7. If they didn’t get it right, go through their method and see if you can see if they went wrong.
8. If you can’t spot their mistake, did you go wrong?
I thought getting students to differentiate as part of an integration lesson would be a recipe for disaster, but it actually helped consolidate the links between these two processes. The functions that the students thought up were far worse than anything I’d used previously – they had brackets that needed expanding, fractional indices, negative indices, decimal constants etc. The conversations about the work and levels of engagement (competitiveness) between partners was brilliant.
I’d recommend trying this style of activity and I know I will be adapting it for other topics.