Heat conduction as a linear regression exercise

The pre-assessment at term start suggested that the class was particularly weak with respect to mathematical graphing skills. As a result I have sought to move more linear regressions and plotting exercises into the course. I decided to revamp laboratory six and shift from the previous structure of having groups share data and invent charts to depict the shared data. I brought along timers and had the students record the time versus the rise in temperature.



I knew that the data would rise at a declining rate to a fixed maximum value and then undergo cooling curve decay back to room temperature. To try to eke out a somewhat linear relationship, I had the students only plot the initial rise in temperature over the first six to nine minutes. This still had a logistic look to it for some students.


Allston and Berry


Lucas and Dwayne


Brinando and Megan



I noted on the board the behavior that I expected to see. The red box indicates the domain and range in which I expected somewhat linear behavior.



As data came in, the data appeared to follow a logistic model. Initially flat as heat had not yet transferred along the rod connecting the cups. Then a rapid rise which eventually decays to what appears to be a constant value. Only after a time does that value decay - the hot water cup keeps providing heat energy that tends to hold the cool side bath at a constant temperature.


Actual data from a student is seen as blue boxes above. A linear regression does not fit quite as well as a logistic in terms of the behavior of the data.



Afternoon board.


Note the estimated slopes for five materials - each pair of students worked with one conductor of heat.

The laboratory went well. Although the lab was less oriented towards collaborative generation of a way to report data (the old structure to four), the gain was that another laboratory focused on the potential power of linear regressions to model nature, even if only over a limited time and temperature range. I am not certain that the old structure produced any particular learning, certainly nothing that would be easy to measure. There is value in the old structure - students engaged in discussing how to present results to the rest of the class. Teams of scientists forging a common understanding of their data. Most groups, however, settled on column charts of start and end temperatures. Few groups ever stumbled on the more informative "delta temperature" column chart. And those that attempted to use an xy scattergraph of start versus maximum temperature wound up with a somewhat difficult to read chart where a 45 degree slope equals one line is a perfect non-conductor. The above brings another linear regression to the course. Coupled with the planned linear for lab seven, this really puts basic linear modelling into almost every week.

One side note. I fiddled and tweaked my way to the initial logistic expression by using a computer in the possession of Brinando and Megan. They watched as I fumbled and fought my forward to a design that would behave in a way similar to their data. I explained what I was doing and why, including referring to the (h - k) portion of the graph. I gather one or both had MS 101 Alg and Trig, so they were not necessarily completely lost in my ramblings. I thought that interchange was valuable, that exercise was valuable. They were able to see their instructor struggle with a mathematical model, not have it quite right at first, and then iterate his way to a reasonable solution.

I have always disliked simply showing students pat solutions, I think they learn more from watching one struggle with an equation, trundle up and down some dead ends, and then eventually find a way through the mathematics to a solution. Unfortunately only two could really watch what I was doing. This was a small group exercise. Yes, I know it could theoretically be done on a projected laptop or SMART board, but I know that would not be the same.

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