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.