r numbers, sums and sequences

This week with the class of 2021 - I can't claim a deep understanding of the 'r' numbers that are being thrown around the media at the moment, but I think the basic maths is a perfect application for talking about geometric sequences. We have just been looking at  SL Geometric Sequences  in class. I used the classic  In the money activity which is a favourite of mine and we had some really good interaction about the key differences and the behaviour of arithmetic and geometric sequences that the activity invites. We looked at some abstract questions about geometric sequences, and established the formulae*. We have been coming back to these  Statistics telling Stories graphs that are tracking the pandemic for lots of reasons (log scales, modelling, questions we should ask etc...) and now because I wanted to have a look at exponential growth in this context. I am always wary of handling such contexts in the knowledge that they have been tragic for many, but it is all pretty important just now right? Anyway, we moved on to draw the relevant parallels between the 'r' common ration in the geometric sequence and the 'r number' about which we hear so much at the moment. The table function on the GDC gives us an excellent way to explore what happens to sequences with different r-values. We started with yesterdays' reported figure where we live of 26000 new confirmed infections and then mapped out would happen to that number under the influence of different values of r ranging from 1.01 to 1.1. This was a spontaneous choice because I wanted students to see the difference a seemingly small change in r would give rise to over time. This involved setting up some functions to put in the table. We looked at this in different ways...

1. Set up two (or more) functions with different r values and have the number of days as a the variable. The table function then allows for side by side comparisons of how the model grows (clearly this can be done ona spreadsheet too, but it is a good opportunity to practice the GDC and the formulation of functions)

2. Set up two (or more) functions with the r number as the variable and a fixed number of days ( a different function for a different number of days) so you can look for direct comparisons after a certain period of time for different r numbers.

3. Introduce a sum element to both of the above.

This all seems to me like an excellent way to put the key conceptual ideas of the course at the front and centre of our teaching and make the applications philosophy obvious.

Deriving formulae

This week also brought up a regular thing that happens for me. With activities like  Visual Sequences and  In the money it is a clear aim to derive the formulae involved. The nth term of both arithmetic an geometric sequences comes naturally, intuitively out of the activities. With the cubes we can run through a lovely derivation and justification of the sum of an arithmetic sequence. I know too that there are equally satisfying proofs for the sum of a geometric sequence. What is difficult on this course, particularly at SL, is making choices about when it is OK just to give a formulae. This has always been my strategy with the sum of a geometric sequence. I help students to recognise that they have derived the other formulae, and that it will be possible to do the same for the geometric sum, but that it is not always necessary. It is necessary to know that it can be done so that these formulae don't take on the appearance of magic tricks, but not always necessary to do so. what do you think?

For me, I recognise that students will not be asked to reproduce proofs, but if they can remember that there was a time that we did, then I think this provides a key root back to mathematical soundness.. This blog post is just a diary post and so these are genuine reflections rather than conclusions. Just sharing, that's all,

Have a good week,

Jim