From Maura:
We went over a few homework problems, mostly talking about issues with the different graphs and remembering how to select non-adjacent columns. We then reviewed – again – linear and exponential functions. We brainstormed properties of linear functions, then did the same for exponential functions. We did a few simple examples, looking at exponential functions and extracting initial value and relative change (we could have called it growth rate, but I’d rather use relative change). The students had a few questions about the doubling time and half-life examples from last time, so we spent a fair amount of time on that. We looked at a population that doubles every 6 months and we wrote out that information in a table. We figured out the basic function P = C 2^T where T is the 6-month time period. But what if 11 months pass? Or 15 months? We figured out that we could evaluate P=C 2^(11/6), or P = C 2^(15/6). The jump was to figure out that the function could be written as P = C 2^(t/6) where t is the actual time in months. Then we rearranged things using algebra to get P = C (2^(1/6))^t = C 1.122^t, so the monthly relative change is just over 12%. We did another example, just to see how it works, and then also did this using the Excel spreadsheet. Half-life works in a similar way.
Other comments: some students are annoyed with Excel. They just don’t like it and don’t like spending time on it. I try to be encouraging: Excel does a lot of the work for us, making it easier for us to think about what really happens; Excel lets us see patterns; Excel is a tool that you may use in your life. Nothing seemed to convince them. Ethan’s group seems much more comfortable with Excel.
I then started Ethan’s class. It’s always a challenge to teach the same subject twice in a row – no matter how hard I try, I usually go faster the second time. This time there were no homework questions, so we jumped right into the review of linear and exponential. But it was like pulling teeth to get them to tell me about rates of change. I had to walk around and bother people to pay attention. I don’t think it’s that they knew the answers – they just weren’t focusing. Not an easy day of teaching. We finally picked up the pace and were going strong when Ethan walked in.
From Ethan:
Maura taught the first half of my class. Her blog posting should precede this one, but I’m not sure she’ll get around to it since she’s away until next week.
When I arrived halfway through the class she’d been working with the linear vs exponential spreadsheet, so I spent the rest of the time playing with it. Several interesting things emerged.
One student asked about fitting a trendline to the linear function. So we did. I asked them first to predict what would happen – an exercise I wanted to carry out before they tried it in Excel, to model the principle that suggests that predicting the answer to a problem is a good way to start. With a little tooth pulling we decided that the slope and intercept should be ABSCHANGE and START from the spreadsheet (labelled cells!) and the R^2 should be 1. Wonder of wonders we were right. You couldn’t see the trendline because it was right on top of the data line. Then we moved one of the data points up a bit (artificially) and correctly predicted what would happen to the trendline and to R^2.
Discovered (on the mac) how to show all the formulas in a spreadsheet instead of the numbers. That was cool.
Experimenting with really long time periods for exponential growth (increasing STEP from 1 to 200 to simulate 3,000 years of accumulation) produced interesting results. The linear graph and all but the last entry in the exponential graph are invisible (they’re de facto on the x axis) because the last point is 10,000 times as big as the next to last. And Excel actually dips the interpolating curve down below the x axis in order to make it smooth enough. The analogy is a big truck swinging to the right before making a wide left turn.
We finished with experiments on doubling times, and discovered the rule of 70.
I did all of this with Excel play, none with formulas. That may put my students at a disadvantage on the common final, but give them a style and a skill that might serve them better in the long run.
I need to see how much of this material should be incorporated in the text.
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