I'm doing a mini-quiz each day right as students come in, to get them to come on time. It's also helping me see what needs more work. I make up the problems pretty much on the spot. I gave the same type of problems today that I gave yesterday - 4 basic log problems:
- log24 (Oops, way too easy, I meant this to be log42!)
Then we started in on the murder mystery. [Read my previous post for details.] We had discussed it a bit before, and they were supposed to have completed assignment 1:
We want to think about how a hot cup of coffee cools off.Many of them had answers that were way off, so we discussed en masse. We talked about the physics involved (though we have not yet mentioned Newton's Law of Cooling), and I got estimates from them for the first 3 questions and built the beginnings of a graph, with (0,160) and (60,80) plotted, and a dotted line at y=80. [We were saying the coffee started out at 160 degrees, and after 60 minutes it had cooled to the air temperature, 80 degrees.]
1. What would be a reasonable starting temperature?
2. After about how long would it be cold?
3. About what temperature is it when it’s cold? (Why?)
4. Now let time be the x-axis (t-axis) and temperature (T) be the y-axis, and (on graph paper) graph temperature versus time for a cop of coffee, using what you know from common sense. Does a straight line graph make sense for this?
Then I had them get in their groups (front two people push their desks sideways, and all four push the desks closer, it's very quick), and draw a quick graph using that framework. Lots of hesitation, I had to prod them to just guess. I told them I had seen two types of graphs as I'd walked around the room:
As we talked about them, it turned out we needed two more graphs:
I labeled them A, B, C, and D, and asked the groups to discuss and each group would vote on the one they thought best represented the cooling coffee. (A is two straight line segments, a common hypothesis from students. B is exponential decay. C is a straight line segment, then a curve, then another straight line. D looks kind of logistic.) As I waited, I realized this was much like the Peer Instruction championed for physics courses by Eric Mazur. Interestingly, D got the most votes. (It was 0, 2, 2, and 5.) We talked some more about the physics of it, and decided to measure actual coffee the next day. (They had gotten their 30th donut point the day before for catching my 30th mistake, so we decided to have donuts and coffee on Thursday.)
I left this question open at the end of class. In the past, students have often looked up rate of cooling in the textbook or online, and have mentioned Newton's Law of Cooling. The initiative they take is great, but I'm sorry to see them just following a formula after that. This class came in Thursday not having done that. We had hot coffee and a thermometer from the chem lab. We got our data, and will look at it on Monday. This class has engaged more with the project than any other class in my memory. (Granted, I do have a bad memory.)
If you're interested in thinking about good questions for this vote-and-discuss type of interaction, join us at the free webinar I'll be hosting tomorrow. I'll be interviewing Maria Terrell, founder of the Good Questions Project, as part of the MathFuture webinar series. It starts at 11am Pacific time / 2pm Eastern time.