As an introduction to Higher integration, I presented this task to J and K (click to view gallery), with very little introduction:
Each subset is followed by a blank space, inviting the learner to create and solve their own example that is representative of the ones above. A close look reveals some interesting examples. There are also some 'surprises', some functions that are impossible to integrate. These features are designed to stimulate the use of initiative.
Whilst I employ variation quite often, it is unusual for me to create such a large set of 'basic' problems like this. However, I felt that it would be useful for the learners in this case, and this is all that matters. In his farewell address to the ATM conference in 1988 (which you can read in the free ATM publication Working with Rods and Why), Caleb Gattegno said:
People often say: ‘I teach them but they don’t learn’. Well, if you know that, stop teaching. Not resign from your job: stop teaching in the way that doesn’t reach people...
This is not to say that I would always teach every topic in this way, but that my experience tells me that the process of integration can be learned efficiently through this kind of activity. After doing this work, J and K were able to complete a number of exam problems with no further teaching, and with little difficulty: