Watch this one minute clip and if you find yourself wondering, “Do Joulies really work?” ask yourself, “What would that kind of temperature graph look like?” and then click on through to the third act of the lesson plan to find out.

A few release notes:

**The Goods**. My favorite part of the task is how much work the students have to do to translate the inventor’s claims to mathematics. He says, “When coffee is poured into a travel mug with Joulies inside, the coffee cools down to a perfect temperature three times faster than normal. Then, when the coffee would normally cool off, the heat that was captured is released actually keeping your coffee pleasantly warm for twice as long.” ¶ So students have to make an assumption about “perfect temperature.” Is that a range? Is it a single temperature? Then they have to make an assumption about the initial temperature of the liquid and its final temperature. Then they have to create their initial graph. ¶ They get to decide and own*all*of that. We don’t care. We care about*their transformation of their initial graph*and whether or not it fits the inventor’s claims.**Formative Assessment**. Why did I close the first act video with this frame and not this frame?**The Competition**. Sometimes I wonder why we should bother with that level of precision, why we should analyze these videos on a frame-by-frame basis when our competition in the video-based math curriculum space is basically drooling all over itself.**Citation**. Marco Arment performed a similar experiment with Joulies. I got the idea to use rocks in the sequel from Jeff Ammons.**Feedback From Pearson**. They told me I should consider changing the domain of the temperature graph from six hours to one, because it’s rare to drink the same cup of coffee for six hours, and to be a little kinder to ELL students by using “joulies / no joulies” rather than “joulies / plain.” Other than that, they get what I’m trying to do here and they support it.

**Featured Comment**

Criticism from Bowen Kerins is one of the big reasons why I bother posting this stuff. Here’s his entire comment:

I think the one-hour timeframe is better than six hours. More importantly, though, you’re running across limitations of video technology by having to make this decision at all.

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” ;)

## 27 Comments

## tony

February 16, 2012 - 1:11 pm -My question to you, and your readers: what would be different about this lesson if it were a science class instead of a math class?

## Michael

February 16, 2012 - 1:42 pm -For our middle school science class, we might change the amount of Joulies and compare trying to find the optimal number of Joulies for a certain amount of coffee. Or we could use one Joulie and find out what happens to different amounts of coffee with just one Joulie.

## mr bombastic

February 16, 2012 - 2:05 pm -The ending frame is not a minor detail – everyone in the room is thinking it so give them an opportunity to say it and be a part of the discussion.

I agree with pearson on the domain being too large. The two curves together are fairly confusing to think about. The more the graph corresponds to the actual act of drinking a cup of tea, the easier it is to interpret. The utility of the joulies depends on how much more quickly it cools to drinking temperature & how much longer it stays at a nice drinking temperature. It is too tedious to estimate these times with the 6 hour domain.

Less importantly, the large domain gives away the answer to a nice little extension question: what would happen if we extended the graph out from a 30 minute domain to a 6 hour domain.

I would like the Joulies line to be blue on all graphs. It is gray on the graph with the rocks.

I like this a lot. Much more sophisticated and interesting than some dumb graph about a bike ride or a floating balloon.

## Mark Watkins

February 16, 2012 - 3:48 pm -Act one is a good set up but there’s so much more that could be done and still be math relevant. And every idea I’d have for this involves actually taking the temperature measurements in class scientific method style.

Act two almost tricks the hypothesis/null-hypothesis out of the students without them knowing it; I like it.

Act three, however, feels like a student-less science lab that could have easily tested convection, conduction and radiation losses next to the product if desired. I’m dying to see this part in the hands of the students rather than passively watching it on the screen. I know it’s not feasible to do a six hour lab but I also can’t imagine you took six hours of manual data in order to do this experiment (well, you’re pretty driven, so I guess maybe it’s possible).

I guess I feel like the 3 Acts template is a little constraining this time around.

## Dan Meyer

February 16, 2012 - 4:05 pm -I took six hours of measurements (one second intervals) and six hours of photos (five second intervals) for each condition – with Joulies, without Joulies, and with rocks. A set of five Joulies costs $50. If you’d like to do this one in class, be my guest. The obstacles seem fairly impressive to me.

## louise

February 16, 2012 - 4:19 pm -Applied math/science (physics) go together. I keep trying to beg the administration – run them back to back. We get 2 hours to run experiments and do the math, write up the work, create a tech demonstration… kids get 2 or 3 separate credits in the system, experiences are rich and real.

Visit a research university, you see 15 “different” disciplines all getting PhDs from the same work. Division of subjects is arbitrary and capricious.

## James C.

February 16, 2012 - 4:46 pm -$50 for the Joulies?!

I bet when the inventor came out with this silly product he didn’t expect his primary market to be math and physics teachers trying to prove just how silly it is. ;)

## mr bombastic

February 16, 2012 - 5:43 pm -I can’t see doing this as an experiment. Seems like an awful lot of busy work to have the students design and carry out an experiment where they look at a watch & thermometer and write down the numbers that appear on the watch & the thermometer & then plot the points. It is an interesting graph, but a very boring experiment.

So many administrators seem to have this knee jerk reaction that it is better if the students do it. Often it is. But often the students are actually half socializing while half engaged in a low level activity. Meanwhile the administrator oohs & ahhs as they babble about Piaget and all the higher order thinking going on.

## IanR

February 16, 2012 - 7:39 pm -Chemist here; your chemistry department might have a datalogger that would make this easier. This is similar to heat capacity labs in chemistry, and “dropping metals into boiling water” is often a type of problem in calorimetry sections. What’s also interesting is that the volume of water matters, and it’s most certainly different in each case. Usually there is a curve in these types of measurements and you have to extrapolate back to the mixing time. The hypothesis here–that less water with a metal in it will cool down faster–is rather obvious, since metals are a poor insulator; I’m not sure that this would be considered a inquiry lab or not in the sciences. But it might depend on the level of the students, and their misconceptions. Many students think that metals conduct heat poorly, when actually they are great conductors.

## gasstationwithoutpumps

February 16, 2012 - 8:56 pm -Also, from the standpoint of a tea drinker, the tea needs to steep at high temperature for 5 minutes before adding the joulies. Rapid cool-down of the tea water before steeping is a bad thing.

I agree with IanR that the experiment needs to be tried with identical water amounts.

If I were to get joulies (which I thought of doing at one point), it would be to put them in a thermos which I would pour already steeped tea into. The thermos would hold the temperature fairly constant, and the joulies would get the initial temperature about right. The combination would hold the tea at the right temperature for a very long time. Currently, I don’t mind drinking cold tea, and setting up and cleaning out a thermos is too much trouble, except when I’m going to be out judging a science fair or otherwise away from a source of tea for more than a couple of hours.

## k morrow-leong

February 17, 2012 - 6:44 am -“Formative Assessment. Why did I close the first act video with this frame and not this frame?”

This is the important piece for mathematics classes. Mathematizing a context can be done many ways, but only if you think about it first!

## Mark Watkins

February 17, 2012 - 6:56 am -But the data process was automated, right? That’s all I was saying. You didn’t read it/record it by hand over a 6 hour time frame; something electronic did the grunt work there. The only reason I say I’d want to do this in my class is that I have actually done something like this in my (physics) class. LoggerPro and some temp sensors and you’re good to go. My set up was about what causes the most heat loss: convection, conduction, or radiation. Isolate your variables, exstablish a control, blah blah blah… and you’ve got yourself a thermodynamics lab. Removing the scientific from the project for the sake of the math feel artificial to me… but then again my degree is in physics, not math; I just play a math teacher on TV.

## Mark Watkins

February 17, 2012 - 8:08 am -@Mr Bombastic

I agree, you can’t expect students to sit and watch the data collection for this experiment as is. But I’ve never met a researcher assistant that sat and watched the data collection happen, either. They calibrated the tool, set up the data collector, and then found something more productive to do for the next however-many hours. My personal experience with this lab is that it’s right at the sweet spot for teaching the scientific method, data analysis, and research methods all at once without being too abstract or too complicated for arriving at an effective conclusion within the limits of a classroom setting. This works in the 3Acts framework (honestly), but I want cross-curricular home runs here. Not ground rule math doubles.

## mr bombastic

February 17, 2012 - 10:26 am -@Mark, We are probably just looking at this from different view points.

In a math class, I would view this as a one day standalone lesson about how well the julies work — not a unit on thermodynamics. I want my class to come away thinking that they did a fairly quick analysis of a new situation, looked at some real data, and were able to use mathematical evidence to decide whether they agreed with an advertisement.

I don’t think a unit on thermodynamics in a math class is a good way to convince students that math is useful. I also think it would very likely end up being a unit on thermodynamics that happened to use a little bit of math as opposed to a math unit centered around thermodynamics. In a science class, data collection and so forth would make much more sense to me.

## Mark Watkins

February 17, 2012 - 10:43 am -That’s a fair point. Maybe what I’m really seeing is a part that fits in a math class that can be book-ended by a science class. Much of math can probably be put into this format I imagine.

## gasstationwithoutpumps

February 17, 2012 - 11:10 am -I have a little problem with mr bombastic’s insistence on a firewall between math and science classes. I would much rather see one lesson that teaches both good math concepts and good physics concepts, than one that is artificially forced to cover only math or only physics.

Physical phenomena are a rich source for real-world mathematical modeling, and Dan has done a good job of setting students up to ask questions that lead to models that they can handle with the math they have. The cooling using joulies is a bit messier than some of the other problems he has put forward, but some playing around with log scales could lead them to a simple formula for the control condition and the tail of the graph with the joulies. Even if the students are not advanced enough for log scales on plots, there is some ability to compare the time courses and see the value of plotting data.

The splitting of math from science has done damage to both fields. Teaching calculus without physics and physics without calculus results in a lot of wasted effort in both classes. I’d like to see more integration of science and math content, not less.

## Dan Anderson

February 17, 2012 - 11:49 am -I tried this activity out with two groups of Consumer Math today. It was a nice activity, and the kids were interested across the board. I agree with mr. bombastic from comment 2 about having a consistent graph line color in each video. It’d also be nice to have a big image of each final graph (instead of just the control). I found myself taking screenshots of the finished graphs so we could talk about the what happened.

Otherwise, big success, some great discussion between the students on why their drawn lines shouldn’t dip below 65 or so, and why the steepness of the joulies’ line is different. We also visited kickstarter.com to find out how much $ he raised and talked about how his company made money on an ineffective product.

Thanks Dan.

## mr bombastic

February 17, 2012 - 1:15 pm -I hope I didn’t come across as insisting on a firewall between math and science. If you want to build an algebra 2 exponential model unit around Newton’s law of cooling I am with you (but not if they are going to use log plots to develop the formula). I just don’t see the julie lesson fitting into that unit. We have no way to model the graph for the julie. It is true that the tail of the julie case matches the normal case, but the tail has nothing to do with the main question about whether the julies work or not.

## Dan Meyer

February 17, 2012 - 7:11 pm -Thanks for the feedback,

Dan. I haven’t begun the Pearson revisions yet and I’d like to pack in your suggestions at the same time. Are you saying you’d like a screenshot of the last frame of each graph video? What do you need that pausing the video couldn’t accomplish?Also, what’s your take on shortening the domain to one hour? That 65Â° realization might not surface.

## DMT

February 18, 2012 - 12:39 am -The first question that comes to my mind is whether or not removing liquid peridically (mimicing actual drinking) would have an effect. If we assume that some heat is stored in the rocks/Joulies and that heat is later transfered back to the liquid, then it seems reasonable there would be more of an effect with less liquid over time.

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.

## gasstationwithoutpumps

February 18, 2012 - 10:16 am -@mr bombastic.

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?).

## Bowen Kerins

February 18, 2012 - 10:49 am -I think the one-hour timeframe is better than six hours. More importantly, though, you’re running across limitations of video technology by having to make this decision at all.

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” ;)

## mr bombastic

February 18, 2012 - 2:17 pm -One advantage of video is that it gives instant credibility to the situation being modeled. On the other hand, comparing the same curve with different domains is a very, very appealing idea to me for a class that can handle it.

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.

## Dan Anderson

February 19, 2012 - 8:06 am -@Dan Meyer

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.

## Dan Meyer

February 19, 2012 - 8:54 am -@

Bowen, thanks for your thoughts on the piece. I especially like the picture you paint of dynamic changes and re-adjustments of the graph. Hopefully, our best and brightest are hard at work on that kind of platform. Fingers crossed.## Bowen Kerins

February 19, 2012 - 10:47 am -It also fits in well with the “make a guess” entry point (and general perserverance). Interactive tech is great for this — I think a big reason is its personal nature. Only Angry Birds will know you sucked at that level, and only the grapher will know your initial choice of axes was terrible. No matter how much I try in classrooms, kids remain afraid to jump in with initial guesses, fearful that it’s being graded or something.

## Zack Miller

December 5, 2012 - 11:41 pm -I put this one to work in a two-hour Alg 2 block yesterday. The hook makes this one of my favorite tasks. Kids plead to know, “Does this really work??” After we chuck around some predictions, we got down to business. We need to create a function that models the regular cup of tea, and then modify the parameters to create a new function modeling a cup of tea with Joulies if Joulies worked perfectly as advertised (of course so we can compare to the real “tea w/Joulies” data).

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!