The starting point is a class discussion on making a simple net for an open rectangular box beginning with a 20cm by 30cm rectangle and cutting square pieces from the corners. There's always a valuable exploration about "what shape?" to cut away and how to get the dimensions of the box from the size of the cut-out - which at some point leads to a formula for the volume.
There's a magical moment when pupils fully realise that the single starting rectangle can produce boxes with different volumes. This is when we head for the computer room to use the Grasshopper Spreadsheet (which they learned about in IT in Year 8) in order to work on a decimal search and a graph - choices arrived at by judicious questioning. Tables of values are produced and, by the end of the second lesson, we have usually amended the number of figures being used and got print-outs of tables and graph.
Discussion of the limitations of the spreadsheet leads us back to calculators for the decimal search, and their varying number of displayed figures into a discussion of whether they will ever get to "the answer" for the size of cut-out which gives the maximum volume. Someone always produces a graph where the volume formula has produced negative answers . . .
Then it was another set's turn in the computer room so we were due back in the classroom when the inspection began - and that's when I had the idea of using our new graphic calculators for the "extensions" of the problem. This set had explored linear relations on them previously. I had expected the inspector to go to see the set in the computer room, but no - my set had print-outs, evidence of their competence on the spreadsheets, and he wanted to see how would extend the problem!
These pupils are used to thinking what "given" factors might be changed to create a further problem, so after only a brief discussion exploration began. Some kept the area of the rectangle on 600 square cm and so explored the shape of y = x(15-2X)(40-2X); some were enlarging the rectangle and began with 7 = x(40-2x)(60-2x); others only stretched the rectangle. All the pupils studied the effect on the maximum point of their graph. Katie got others interested in what made y = x(20-20x)(30-2x) go below the x-axis and where the x-intercepts would be on similar graphs.
It was the speed with which the graphs could be produced and analysed using "zoom" and "trace" that led to a high level of engagement and momentum in their progress with the problem. The volume formula could be adjusted and the graph redrawn in a matter of minutes, and they also had the new starting point for their decimal search. The explorations and write-ups were done in a couple of lessons.
There was satisfaction as they realised how powerful the calculators were for what otherwise would have been repetitive and time-consuming work.
Next time there is at least one change that I will make - I left sharing the extension ideas until I had graded the work. By then they were occupied with other ideas and couldn't recapture their excitement in their general results. But two pupils had found out that calculus was helpful in this problem - one had written "I can't wait to learn about that!"
Margaret Poston is head of maths at St Michael's Catholic grammar school in north-west London