The 12ths* had done a whole lot of differentiation the previous year, and so I decided to begin my teaching with Integration. After a few revision classes on differentiation, I used the ideas in the TAM’s Integral resources on introducing integration as the reverse of differentiation.
The lesson idea begins with a bit of fun, sending two students outside while the class comes up with a gradient function of a given curve (Picture 1)
The students were called back in after the curve was erased and they had the task of recreating it using only the gradient function and the co-ordinates of one point on the curve. On our board was “dy/dx = 2x – 6 and (2,1) lies on the curve”.
Starting from a graph brought about a rich discussion. The two students discussed what dy/dx meant at particular points, x = 5 and x = 2 (4 and -2 respectively), in terms of the steepness of the graph. A bit of prompting led them to think about dy/dx = 0 and the stationary point. I then asked students to come up with the curve with the gradient function 2x+3 (a type of ‘surprising’ or ‘confounding’ question as described by Watson and Mason in Questions and Prompts for Mathematical Thinking). Students pointed out that without the information of a point on the curve there was not a single answer, and this then led to thinking about families of curves.
The text books prescribed for the ISC have few graphs. Many students enjoyed making the links between the function and gradient function as graphs and I realised the need to do this often, given that the texts lacked this fuller picture. They proceeded to use the skill of reversing differentiation to find simple integrals.
After a good half hour of practice on mini slates (the environmentally friendly version of MWBs which are used a lot in primary school across India and thus easily available) and tables in their books, (picture 2) we moved to looking at the link between integration
and area (Picture 3)
I had to do this by frantically drawing on the blackboard as the school has no projectors in the classrooms. Although this is not the most efficient way, there is some benefit I feel (apart
from improving my drawing skills!) for students to watch curves being sketched in real time.
This paved the way to introduce the Fundamental Theorem of Calculus in the next class, following the structure on the Integral website.
The next few classes were spent on activities. Many of these I adapted from the integral website. Students in the class had only ever worked from text books or worksheets. They took to activities like Tarsia and Venn Diagrams with relish. The discussions amongst students was vibrant. I had to adapt the resources substantially to include the many more techniques and methods included in the ISC that are not part of the A Level, but the originals gave me excellent ideas to work from.
Especially useful was the activity to sort integrals according to the technique needed to solve them. (This can be accessed from the MEI SoW Resources page http://mei.org.uk/files/sow/30-integration-res.pdf).
The discussions were useful for students who were struggling with all the myriad techniques – they could glimpse into the minds and thinking of those who had got a good handle on it, seeing what these students noticed or looked for when deciding on a particular technique. The activity also generated good debates for integrals which could be approached in more than one way.
Integration is a huge chunk of the curriculum, not every class can be brimming with varied activities. I do feel however that a good tone has been set for the rest of the year.
* a few differences between Indian and UK acronyms
-ISC ≈ (is equivalent to) AS and A-level
-11th ≈ year 12 and 12th ≈ year 13