This task authentically prompted many things:
1) As Dan suggested, kids get to think about what needs to be defined to make this question answerable, which gets them to “what is the perfect temp?”

2) Students quickly decided it was exponential decay, but when writing the function for no-Joulies, we have a serious excess data problem: we have the temp at every single minute! Exactly how many data points should we work with if we want to know the parameters in y=ab^x + c? Two? Three? Four?

3) Tutorial of Wolfram Alpha: Let’s go with 3 data points. We can plug each one into y=ab^x + c to give us three equations. Solve that system please, wolfram! And now we start thinking back to our big question: how do a, b, and c help us find our “cool-down” rate so we can triple it!

4) But also halve it at some point so the tea doesn’t get cold? Wait, what? And looking at the real Joulies graph, this isn’t a clean exponential anymore! Piece-wise functions, anyone?

5) Tutorial of Desmos: Think you found the function? Type it in and compare your graph to the real thing. Extensions: How would this be different if it were in a freezer? A sauna? Let’s make some parameters DYNAMIC.

So much fun!

My mistake on the graphs, I found the file that has the plain and joulies graph drawn, so ignore that bit of commentary.

As far as the domain goes, I thought it was important to see the tea level out to the ambient temp. There were many students (weak 11th and 12th graders) who made the mistake of not reading the scale and initially drew their graphs where the tea was at 0 F after a couple of hours. Seems like an important point.

I also agree that it is highly desirable for the students to come up with their own scale when they make their graphs in Act II, but this can be time consuming and/or a bit of a momentum killer depending on your class.

I see Dan’s lesson as a highly structured lesson with predictable responses that could be done in a class period. If you take away very much of this structure, I think it would be tough to get this done in a class period.

The six hour domain seems very un-Dan like — neglecting the central question in order to force in an unrelated topic.

To me the “best” solution would be to let students decide what their axes limits should be, then see the graph populated. A tablet-PC environment could make this happen. A static video takes this decision out of the hands of students because you’re forced to select this in advance, or to set up a limited number of options. The same is true for the vertical axis — my first reaction to the presentation was “Why does the vertical start at zero degrees Fahrenheit??” And what led to the maximum being 160 degrees?

I’d want students making those choices as well, ideally in an environment where a quick change doesn’t cost them anything. Even when students are asked in advance to create the initial graph, leave the axes totally unlabeled and let them make all the decisions.

I also think this flexibility would lead to students coming to different conclusions about the effectiveness of the Joulies. A one-hour or thirty-minute graph makes it look like the Joulies are doing a pretty good job, while the six-hour graph makes it look like they do nothing most of the time. It could even lead to a cool “how to lie with data” conversation, or at least an important conversation about the nonlinearity of the graph (to meet 8.F.5). Often students think all graphs and functions are linear. The short-term graph of the “no Joulies” seems linear enough… then boom it ain’t!

I’m also a little confused by your student work example — the graphs show that the Joulies version stays in the “perfect” zone for more than twice as long (75 minutes versus 30). So I would not agree with the student’s assessment that they “stay perfect for almost exactly the same amount of time”. The six-hour versus one-hour makes a big difference here, I suppose.

Last, two nitpicks: the video talks of coffee but then presents tea (no big deal but why use tea and not coffee?). And please show me an actual eighth grader with the quality handwriting exhibited in the “student work” ;)

Fair enough. I agree with you that the physics of the joulies (with the storage of energy in the solid-liquid transition inside) is probably beyond the scope of a high-school physics class.

If this were to be a consumer math class, I’d probably want to compare three approaches:

1) using a plain ceramic travel mug with lid.
2) using a stainless steel vacuum thermos cup
3) using joulies in the ceramic travel mug

(maybe adding a 4th approach, of using the thermos and joulies).

The students would have to be encouraged to come up with a relevant metric (perhaps how long the liquid is between 100 and 110 degrees Fahrenheit?).

This combined with a shorter time scale would at least be a more “realisitc” graph as to the effectiveness of the Joulies in my opinion.

The longer time with a full glass would make a nice control and a good way to show that 65 degrees as Dan points out.

My two cents.

Also, what’s your take on shortening the domain to one hour? That 65° realization might not surface.