I'm teaching exponential functions and logarithms in pre-calc right now. That means it's time to pull out my murder mystery, in which they will use logarithms to solve an important problem - which of their classmates killed John Doe? Since the murder mystery uses temperature to find the killer, I want to lead in with some thinking about how temperature changes over time.

On Wednesday and Thursday, I told my classes a story, and asked them to draw a graph. I said I was mixing some cake batter up to make a Halloween cake. I asked what temperature it should cook at. We decided to set our oven at 350 degrees. (In one class, I talked about how silly the Fahrenheit temperature scale is, but how, even with Centigrade, zero is just attached to water freezing. It's not the same as zero length, volume, or weight. Temperature is different...)

I also asked what temperature the batter was now. They told me room temperature, and we decided that was about 70 degrees. Then I drew axes on the board, labelled them, and asked the students to graph the temperature of the batter over time. Only one person (out of over 50 in the two classes) came close to the right shape. No one seems to have much intuition about how temperature changes. I did this once before, with the cooling coffee we always think about, and got slightly better results.

Here are my approximations of what students thought:

The green one may have been influenced by our attention in the past week to exponential growth, while the purple one seems to have taken the exponential growth we were studying and limited it by the temperature of the oven. I have often seen students give a linear graph like the blue one, and a logistic-like graph like the orange one. No one wants stuff to heat up fast at first, and then slower.

What makes their intuition bad here? Is there a physical experiment / demonstration we could do to improve their intuition? What would make exponential decay feel like the natural choice to them? Maybe cake is the wrong object to be heating?

Please help me think about this.

## Sunday, November 3, 2013

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Sue,

ReplyDeleteAs a physics teacher, I think the phenomena of heat and temperature qualify as things that students are deeply familiar with but don't understand. They confuse the two quantities, and we have to work really hard to establish ideas like the idea that heat transfer requires a temperature difference between two objects, or that the rate of heat transfer depends on the temperature difference between the two objects. Pretty much the only thing that convinces them is actual experimentation. We do things like heat up water with a hot plate or immersion heater and see how the rate of temperature increase gradually decreases as the water warms up.

Might it be worth it to actually do some of these experiments in your class? If your science department has any sort of data logging equipment (vernier labquests or pasco temperature probes) it should be possible to set up a quick and simple experiment to help students see which of those potential curves is correct.

I ask for volunteers to measure the temperature of water cooling off after it's been boiled. No one volunteered, but I have data from a previous semester. I'll give them the data. (I think they might be more likely to intuit a good graph for cooling than for heating.)

ReplyDeleteI think that with food, the "amount of cookedness" maybe actually does go more like the logistic curve. That is, at first it is cold so it cooks slowly, then it gets hot so it cooks quickly, and then it's done so it's not cooking any more? At least that would explain the first part well, cookedness being like the integral of temperature perhaps, though maybe at the end it is still getting more cooked. But it does seem to take a long time to get that last center bit to finish, so if I ignore the way the outside is getting overcooked, maybe the logistic curve is just about perfect.

ReplyDeleteI know that my naive assumption was that heat would transfer through the object more quickly as the heat "penetrated" the cake. Even though I have a sense from math that the rate of transfer depends on the heat difference, I had some thought that we were talking about the center of the cake and that at first it would be the same temperature, then once the molecules around it heated up it would heat very quickly til it leveled off and was the same temperature as its buddies.

ReplyDeleteI wonder if part of the trickiness is thinking about the wrong parts of the cake? Like the center of the cake will start to cook later because it's not very different of a temperature from the cake around it. The outer cake will cook quickly and then slow as it becomes the same temperature as the oven air -- meanwhile passing its heat to its cooler inner friends.

When we cook cake, we're very focused on the center, in large part because we stick the tester in there to make sure it's not gloopy. I wonder if (thin-crust!) pizza, which is almost all in contact with heat at all times, will elicit different models?

Thank you both! Josh, I told X that you liked her model - and called it cookedness. Max, I'll try pizza next time, to see if that helps.

ReplyDeleteHmm, I thought I had written a comment, but I don't remember doing the captcha, so it's possible I clicked to submit and neglected to verify my humanness...

ReplyDeleteI think one example of exponential decay could be water draining out of a fixed hole in the bottom of a container. IIRC, flow ∝ pressure ∝ height ∝ remaining volume (assuming constant horizontal cross-section), and since dV/dt ∝ V, the solution should be exponential in time.