Download a pdf for all the Activities and Solutions.
Aim: Explore the gradient of a chord or secant and generalise. Visually see the gradient of the tangent as the limiting value as the two points get closer together.
Apps: InteractiveDiffCalc, Main
InteractiveDiffCalc is an application that was added in version 3 of the Classpad operating system. The strength in the InteractiveDiffcalc app is being able to follow the first principles approach visually. By dynamically moving the points closer together students can see the secant getting closer to the tangent.
It also supports the development of the idea of the gradient function and taking time to understand the plotting of the gradient at specific points is a neat way of encouraging students to develop this concept.
I would use this to introduce differential calculus at Year 11 or as a refresher in Year 12.
It has been a busy time. There are areas where I have been able to use the activities very effectively and it has integrated nicely into the classroom sequence. It has worked best in my Methods units 3&4 class where I felt I have had more space and time to teach the way I want to teach. My students have gradually got used to me, and I to them. My adjustments are that I talk too much. I would prefer to be allowing the students more time to figure things out amongst themselves.
The last two lessons we were starting integration and we were working through “Are we there yet?” It is a delight to appreciate one’s work, the way it starts simple and then opens up so dramatically.
It begins by using the average speed in an interval to estimate the distance travelled, and doing by-hand calculations. It then develops a spreadsheet to automate the process and it is particularly cool as the function is defined in Main and it is possible to simply redefine the function for a new scenario. That is one of the nice features of the integrated machine. They were much better at copying cells in the spreadsheet by dragging, so they were teaching me something too.
The next question is to think about what happens when the number of intervals changes and leading up to the limit as the width approaches 0. While we are using the trapezoidal rule there is no mention of it as it is average speed that is being used. Of course this is tedious to carry out in the spreadsheet because we either need to be very clever in the spreadsheet and as you get more cells the handheld does slow down. So it is an ideal time to introduce a program. This is show off time.
In my class we have a Class OneNote (Office 365) where I can share the resource. The students copy the OneNote page into their own section and can then go about writing in their answers to the questions in the activity (they have a stylus and a tablet computer), as if it was paper based. In preparation I had the text of the program available on the OneNote for the students to copy, rather than having to enter the program step by step which invariably leads to errors and a considerable time debugging. This worked for all the students who were able to use the emulator on their laptops. By the way, to copy right-click on ClassPad Manager and then choose Paste Special. This also dealt with a little inconsistency in the capitalisation in the notes. Now there is a powerful tool for looking at the limit and off we go. It was pleasing to have someone ask “How does the program work?” “Here look in the Learning notes”. Natural curiosity piqued!
Hey look we can run the program from Main and set up some parameters, even change the function, so we now have a numerical integration program. Such a long way from the by-hand starting point.
This then has been two 60 minute lessons. They had previously been introduced to Integral Calculus in Year 11 last year and could no doubt have ploughed on with a more traditional approach. From my initial discussions with students I thought they would already have covered the ideas in the activity. If they had it wasn’t evident in their work. Instead we have discussed the meaning of the area of squares on the distance-time graph, explored the idea of area under the curve and that the area is a defined value that can be determined as a limit. I think my students have had the opportunity to conceptually explore Integral calculus while having some fun. While there won’t be an assessment question based on this, I would argue strongly it was two lessons well spent.