A New Way to Introduce Newton's Law of Cooling

by Joel S. Foisy, SUNY Potsdam

For a couple of semesters, I had been introducing calculus students to Newton's Law of Cooling through Smith and Moore's Project CALC cooling body scenario (in Calculus: Modeling and Application (1996) Houghton Mifflin). In their book, Smith and Moore describe a murder mystery, and the only way to narrow down the list of suspects is to use Newton's Law of Cooling. This approach, along with several months worth of bombardment by the Simpson case, inspired me to have my students act out their own murder scenario, instead of reading a hypothetical one. I call this the "Trial of the Semester". I ask one student to volunteer to be the defendant. This student is accused of the murder of his or her mathematics teacher from the previous semester. I then divide the other students up into teams: the prosecution team, the defense team, the media team, and the jury. For all teams, except the jury, membership is limited to no more than 3 members, and the members of these teams are all required to speak in court. This works best in a small class, where most students have a chance to address their classmates. I presented the following scenario to my differential equations class, which met from 8 to 8:50 AM on MWF.
The dead body of Steve's Calc II teacher was found by Crumb Library at 9:10 AM on Monday, October 7, 1996. The cause of the death was determined to be lethal injection with a substance that causes immediate death upon entering the bloodstream. The syringe was found at the scene with Steve's fingerprints on it. Needless to say, Steve is a prime suspect. In fact, two reliable eyewitnesses saw Steve near the library at 9AM that same day. Steve's alibi of being in Differential Equations class can be verified, but only from 8:00 to 8:50. The purpose of this trial is to determine Steve's innocence or guilt, using Newton's Law of Cooling. In other words, you will attempt to determine whether or not the crime was committed after 8:50 AM.
Additional Facts:

Steve works in a pharmaceutical lab and handles a lot of syringes, so it is not unlikely for a syringe to have his fingerprints on it.

All DNA evidence in this case has been thrown out due to multiple lab errors.

The body temperature of the professor was 96 degrees at 9:10 AM. The body temperature was recorded again at 9:16 AM and found to be 95 degrees. (The teacher may modify time and temperature readings.)

The official temperature on Monday morning was recorded to be 30 degrees at 8AM, and 35 degrees at 9AM.

Three days before the murder, the professor visited a doctor, at which time the professor's body temperature was recorded at exactly 101 degrees.

On the morning of the murder, before coming to work, the professor wrote the following journal entry: "I have felt terrible ever since I had that horrible Calc II student, what was his name, was it Scott? And for the past few days, I had that terrible fever. I finally feel better today, not 100 percent, but nearly back to normal."
I then give each team a short explanation of what will be expected of them. For example, I tell the prosecution team the following: "You are going to be the first to officially present in the trial. You want to establish Steve's guilt in the jurors' minds. You will use Newton's Law of Cooling to do this. You will also want to anticipate the arguments of the Defense Team in your opening arguments. For the written portion of the project, write up the calculations, using Newton's Law of Cooling, which you used to estimate the time of death. In addition, include any other arguments you are going to use to prosecute Steve (try to stick, as much as possible, to mathematical arguments)."
I also tell each team the following order of presentation:

Media Commentators read the facts, and make a presentation explaining Newton's Law of Cooling.

Prosecution.

Media Commentators.

Defense.

Media Commentators.

Prosecutionclosing arguments.

Defenseclosing arguments.

Media Commentators, and jury decision.
The media team is responsible for introducing Newton's Law of Cooling in a way that the public can understand. The prosecution may then try to show that if the assumptions of the model are correct, and assuming the professor's body temperature at the time of death was 98.6, then Steve appears to be guilty. The defense team may then attack the assumption that the professor's body temperature was 98.6 degrees at the time of death, or the assumption of a constant outside temperature. It is really up to the students involved to decide exactly what details to attack.
I have used this trial twice, giving the students about a week to prepare, with some inclass time spent preparing. In both cases the defendant was found not guilty. I do not tell students what to decide; they must base their judgment entirely on their peers' arguments. The first time I used the trial in a 30student first semester calculus class, and for the most part the students did an excellent job. The defense lawyer was so charismatic and eloquent that I was not sure if he would ever give up the floor. The students really demonstrated an understanding of Newton's Law of Cooling. The second time I used the trial in a small Introduction to Differential Equations class. This time, the presentations were not quite as polished. The prosecution team produced temperatures that did not steadily decreaseobviously they had made some sort of calculation error. This made the presenters feel awkward temporarily. In addition, the defense lawyer had trouble distinguishing between the temperatures given by the model and the actual temperature of the body. In both cases, the experience of doing the trial helped the students better understand the implications of making simplifying assumptions when formulating a mathematical model. Both the level of preparedness and the public speaking abilities of the students influenced how smoothly the court cases went. The first trial lasted about 30 minutes, whereas the second trial lasted an entire 50minute class period. In each case, students learned how to use mathematics as a tool in a teamoriented setting and demonstrated their understanding (or lack of it) in front of their peers.
Joel S. Foisy
Department of Mathematics State University of New York, College at Potsdam
Potsdam, NY 136763176
foisyjs@potsdam.edu
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