Jo Tomalin describes both the rewards and the frustrations that can be generated by a tweak... or two
(ProQuest: ... denotes formulae omitted.)
John Mason and Anne Watson presented us with a problem which they had met somewhere, and had tweaked themselves, to bring out issues they thought were worth following. I found that as I worked on the problem, first in a group and then on my own, sometimes my attention was following a direction which had been set by John and Anne, and at other times the questions they offered gave me opportunities to tweak the task for myself, and to explore possibilities that intrigued me.
We were told that duckweed covered 1 % of the surface of a pond, and that its area increases by 50% every 1 .5 days. We were asked what questions might be asked, and some questions were suggested. John told us that the problem was crafted to avoid some issues with the number 2, namely that 2 + 2 = 2x2 = 22, which might muddy the pond waters.
We spent some time looking at the problem together, using John's applets (main article MT231), then we worked in groups with some written questions as a stimulus. My group found that we moved between asking our own questions and using some of John's questions as a springboard for our thinking.
My first response to this problem was that many of the questions could be seen as standard A-level questions that would be solved using logarithms. However, John and Anne had specifically said that the task had arisen in the context of exploring the thinking of primary school children about exponential growth, and I was working in a group with primary colleagues, so I decided that for me one of the challenges was to think about how far I could approach this task without thinking about logarithms, and how this might deepen my understanding of exponential growth.
The first hurdle thrown up by John's statement was that we were told what happened every 1 .5 days, rather than every day. At this point I conjectured that I would need to use logarithms to solve how much the weed grew every day, and that the numbers would be messy. Our group decided to avoid the issue, at least at first, by making a table to explore what happened for each period of 1 .5 days, rather than thinking about single days. Generally, unless I have noted otherwise, my diagrams and tables are based on periods of 1 .5 days, rather than on single days.
This was fairly easy to calculate, as for each period we added on half the previous total. This allowed us to answer, to the nearest 1 .5 days, questions such as how long it would take to cover the whole pond. It also helped to draw a diagram, showing the section added each day. Our original diagrams were hand-drawn, but I have used Excel here to reproduce them more neatly, finding that the colours help to illustrate some of the relationships we explored.
In a short plenary we were then shown a visual representation by another group, who had physically produced the bars from paper, using folding to increase the length of each bar by half the length of the previous bar each time. In the same plenary we discussed how far we were seeing the relationship as additive, add on 50%, or multiplicative, multiply by 1 .5. This then raised the question: if you want to halve the time taken to reach a particular value, should you double the additive part, 50%, or the multiplicative factor, 1.5? Our group puzzled on this for some time, feeling that it 'ought to be the multiplicative factor, until Tandi Clausen-May suggested looking at the extreme example of duckweed that increased by 0% each day, equivalent to multiplying by 1 . This would take an infinite time to cover the pond, whereas doubling the multiplicative factor to 2 would take a finite time, which is nowhere near half of infinity.
We then went back to our diagram, and while playing with it realised that if things were to happen twice as fast, what happened in 3 days had to happen in 1 . …