I Have The Coolest Hobbies

I threw twelve different blocks of cheese in the microwave last weekend. Here is video of five of them, 1-cm, 2-cm, 3-cm, 4-cm, and 5-cm cubes:

Cheese Cubes from Dan Meyer on Vimeo.

My goal is to use those data to predict how long it’ll take this guy to fully melt:

For fun, I also threw in two blocks with the same surface area and two blocks with the same volume, just to test out my two prevailing hypotheses, neither of which played out.

Here are the numbers in a Google spreadsheet. You could help me out by coming up with a model that fits the data well and (especially) explaining why your model makes sense in the world of microwaving cheese. I’ll post my own model as a spoiler in the comments. I have no idea why it works, though.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. I eyeballed the following models:

    melting time v. height of block (I knew this wouldn’t work)
    melting time v. surface area
    melting time v. volume
    melting time v. exposed surface area (ie. excluding the bottom face)
    melting time v. ratio of volume to surface area
    melting time v. ratio of volume to exposed surface area.

    That last one is a remarkably good fit for an exponential (R-squared is like 91% or something) and predicts the big block well also. I just don’t understand it in the context of a microwave. Why does it work?

  2. Hey Dan, not sure what I can do with this information, but I love that you did it, and it will make a cool math lesson.

    Meanwhile, it made me really want some cheese.

  3. How do you come up with these ideas?! Very neat stuff. We’re just finishing up exponential equations in my classes right now, I’d like to try to use this somehow. I’m curious though, any hypothesis as to why the smallest block of cheese melted 2nd?

  4. “I have no idea why it works, though.”

    If that’s not a hook into mathematical inquiry, then I don’t know what is. The fact that it’s genuine (I think) makes it even that much better, rendering useless even the must stubborn pseudocontext detectors.

  5. When I first saw the video via your tweet, I immediately thought “Exposed surface area to volume ratio!” And I was going to tweet that back.

    But then I said, “Nah. It’s a MICROWAVE, not an OVEN. Exposed surface area shouldn’t matter.” I guess I was wrong.

    And now I think I know what your next experiment should be….

  6. There’s a heat transfer problem going on here. You’re basically raising the energy of the cheese until it undergoes a phase change (not sure if it’s precisely a phase change because cheese isn’t a pure substance, but we’ll run with it). Energy is transferred to and from the cheese by three mechanisms:

    1: Energy from the microwave emission is absorbed uniformly throughout the volume of the cheese block.
    2: Heat is conducted between the bottom face of the cheese block and the plate. Note that the plate also absorbs some of the microwave energy and will likely warm up.
    3: Heat is also transferred by convection from the cheese block to the air inside the microwave. The air doesn’t absorb much of the microwave energy, so it stays relatively cool until the cheese and plate warm it up.

    If the cheese and plate are at roughly the same temperature, then there’s not much conductive heat transfer (#2). I’m guessing that the dominant effects are the energy absorbed by the volume of the cheese block (#1) and the energy lost by cheese block through convection to the air in the microwave (#3).

    As you move to bigger and bigger blocks of cheese, I think the air in the microwave has time to warm up, so it doesn’t affect things as much. I bet if you put an upside down tupperware over the cheese block so that it is in contact with a much smaller volume of air that it would melt faster.

  7. Mind-boggling. In the name of good inquiry I’m going to throw out the following hypothesis:

    Just because the microwave is heating with microwaves rather than heat and “could” heat the molecules on the inside of the block and the outside at the same time doesn’t mean they do. I’m guessing that the microwaves are being absorbed by the water molecules near the outside of the block of cheese before they ever reach the inside thus the exposed surface of the block of cheese would matter.


  8. @Steven: While the plate will warm up slightly, remember that microwaves work by exciting water molecules, of which I would guess that there are relatively few in ceramic as compared to cheese. I’m guessing that #2 is going to be a relatively large effect as well, and probably even larger than the convection in a relatively small microwave.

    There’s also going to be a radiative component, but that one I am okay ignoring.

  9. Microwaves don’t heat the cheese uniformly. The center of the block is not exposed to as much microwave radiation, since the cheese around it has absorbed most of it. Thus the radius of the block (distance from center to surface) is important. If I’m reading the numbers right, this seems to be the strongest effect.

    The total energy absorbed by the cheese is proportional to the time, so if the block were being heated uniformly, only the volume would matter.

    I suspect that determining when the cheese has melted is also subject to surface tension effects, with smaller blocks holding their shape better even when they are nearly liquid.
    Melting ice, rather than cheese, might make for more obvious determination of when everything is melted, since ice does not form a skin the way melted cheese sometimes does.

  10. This is one of my favorite videos you’ve done yet. The mathematician in me likes the naturalness of the question and the scaling, but the kid in me likes the cheese and the microwaving.

    Seriously, I often feel this relief as a pure mathematician that I don’t have to worry about the problems you present: they’re applied, and I work in a realm where beautiful simplifications make my life easier, and my work more aesthetically pleasing and natural. Still, actually figuring real world stuff out holds tremendous appeal for me.

    Incidentally, have you seen Conrad Wolram’s Computer Based Math stuff? He has a nice TED talk on the importance of doing what you’re doing, essentially.

  11. Nice!

    Here is what I know about microwaves. The more evenly distributed the molecules are, the more likely the food will heat from the outside in. If the food is drier on the outside then the food will heat from the inside out.

    My guess was that looking at the surface area would have made the most sense. It seems that your data supported that. Unless you made a mistake, time to surface area was identical to time to ration of exposed surface area to volume.

    I don’t understand how they would have had identical results but you are showing that they do.

    Here is what I wonder:

    How are the microwaves emitted. My guess is that the are emitted at and angle pointing down on both sides to cause bouncing of the waves until in contacts a water molecule. This would support top-down, lateral-in heating.

    Now that I am typing this I am questioning your results even more. Can you double check that the data for the two graphs are correct:

    The only thing that differs is the x-axis label.

    Thanks for the food for my thoughts!

  12. A quick bit of playing in Excel shows that a Logarithmic function appears to be a decent model for melting time vs. ratio of volume to surface area. Hmm… I wonder what the relationship is between that log function and the exponential function for melting time vs. ratio of surface area to volume…

    Thanks for another great lesson Dan!

  13. @Tom, good catch. I re-uploaded the graph.

    Frank: And now I think I know what your next experiment should be….

    Me too. Trying to get back in my wife’s good graces after smelling up our home with melted cheese.

    @Math Zombie and anybody else planning on using this in class, keep an eye on this spot. Two days and I’ll have all the videos up and running.

  14. I found a nice quadratic with comparing volume and time to melt. It had an r squared of 95%.

    It would be good for students who know many types of relationships to look for different ones with different comparisons.

  15. Please tell me you ate the cheese after, because if not, that is a waste of perfectly good cheese…even if it was for math.

  16. Cheese is conductive.

    So the microwaves do not penetrate all that far into the surface of the cheese, so the microwave energy absorbed is going to be largely a function of the exposed surface area of the cheese.

    The energy required to melt the cheese is going to scale linearly with the volume.

    So the time it takes to melt should be roughly V/ESA.

    As for the little guy, that’s interesting, but microwave wavelengths are about 12 cm, so from a node to a high point is about 6 cm. I imagine the little guy took longer because it happened to be near a node in the microwave.

    You actually should be able to verify this pretty easily. If you make the little 1cm^3 blocks, and distribute a handful of them on a nonrotating plate (I don’t know if you can turn off the rotation, or remove the rotating platter and place another one in there), you ought to notice that there will be a large variation in the melting times for the little guys.

    In fact, I’ve seen a demo where you take a plate full of minimarshmellows, and then by looking at the distance separating the ones that grow the largest, looking on the back of the microwave for the frequency, and knowing about the wave nature of light, you can estimate the speed of light to within 10%

    In short, the small one melts slowly because our assumption of uniform micowave absorption breaks down on small length scales

  17. Dan,
    I tweeted these, and then I realized I read it here! Duh.

    So I put the data in JMP, and then calculated volume and also surface area.


    I think the volume is far more indicative and predictive of time to melt than surface area! Both equations are natural logs, and the graph without the regression of the volume v. time is a much tighter and better graph than the surface area v. time.

  18. Hey Dan, maybe to get back in your wife’s good graces, you could sing her a song! Here is a good one:

  19. As The Virtuosi pointed out, the nodal distance of the microwaves is about 6cm so that distance is likely to be critical for any pattern or calculation, irrespective of the area/volume ratios. Placing several blocks around the microwave would be illustrative and would test this theory. Of course turning the rotating plate off is critical!

    As an enrichment activity with my Gr.8 Science students, we calculate c using a plate of chocolate chips and v=frequency*wavelength. Usually results are +/- 20% error. There’s a great math/science inquiry in here somewhere relating surface area to volume formulae and ratios with these science concepts. It’s easy to find this demo on YouTube.

    Is there a point where the ratio of mass to melting time jumps up as the probability that a randomly placed block contains more than one nodal point increases? How much of an effect does the rotating plate have? If the emitter is in the side of the microwave would a large sheet of cheese (high SA:V) laid flat melt more quickly than a vertically oriented sheet of the same dimensions. How would you correct for gravity? Does a microwave work on its side?? I’m going to be spending a lot of money on cheese this weekend ;)

  20. From what I know of microwaves interacting with metals (i.e. conductive things and electromagnetic radiation at GHz frequencies), “The Virtuosi” has a good starting hypothesis.

    Said another way, the electronic properties of the cheese will limit the heat flux into the bulk material. The thermal conductivity of the material limits the diffusion of heat into the center. We could probably rig a small 3D thermal conductivity model (assuming no melting) to examine how the heat flux and volume are related in a (highly) idealized way.

    The cavity in the oven can, indeed, have hot spots. Think of a string vibrating and you’ll have the mental picture of “nodes” within the microwave cavity.

    Nice experiment.

    If you do the many small cheese experiment, I’d like to see that data, too.

  21. Scott MacClintic

    February 23, 2011 - 5:56 am -

    Have you considered using spheres of cheese as opposed to cubes? The penetrance of the microwaves to middle of the cheese would be more uniform with sphere.

  22. I wonder if any students, once they figure out that melting time v. ratio of volume to exposed surface area is the best fit, will make the connection to shredded cheese….

  23. Very cool idea and seems to have generated lots of interest.

    I’ve been reading the blog off and on, and I would like to push the conversation a bit further by asking about a specific learning goal for a lesson that includes this video. I’m interested in how everyone would complete this sentence: As a result of this lesson, students should understand that…

  24. oh, sorry that wasn’t clear. I meant a lesson that included this video and work on a problem related to the video.

  25. The Virtuosi’s argument that melting time should be proportional to volume / exposed_surface_area rings true to me. The only difficulty is explaining why the relationship appears to be exponential, not linear!

    From a theoretical point of view, melting cheese is hard to model, because there’s so much going on. If you want a nice theoretical model, you might want to look at simpler things, like time-lapse videos of melting ice cubes.

    I would guess that if you have an ice block shaped like a box, the length, width, and height of the block will all decrease at the same constant rate as the block melts—until the corners of the block get rounded enough that it’s no longer even approximately box-shaped. I’ve never tried it out, though, and I was never good at thermodynamics…

  26. Is this real cheese, or processed cheese food?

    Also, how long was the microwave off/empty between trials? I know when I do popcorn in the microwave, I put a cup of water in the microwave for 1 minutes first, then take it out and put the popcorn in using the popcorn button on the microwave. Seems to give me better results than using a cold microwave.

  27. It was mentioned here that the magnitude of the electric field inside the oven may not be homogeneous, i. e. ‘hot spots’ occur where there are antinodes in the electric field.

    Furthermore, Rob suggested that the radiation would mainly absorbed close to the surface of the cheese.

    I agree with those theories, but that’s not what I see in the movie. For each cube, most of the melting process takes place at the bottom, most clearly seen at 00:33 (3 cm cube).

    So my model was plotting melting time vs. HEIGHT^2 x LENGTH x WIDTH which results in an affine function.

  28. I love this. :) I like the split-screen thing (David Cox also did something like this a while back, no?) but most of all I like how the explanation (whatever it may be) is based in some science, as opposed to some arbitrary factory setting (ie. for a toaster).

    Good work!

  29. Dan, this is the reverse of the “Baby in the Car” problem.
    Here the small cube has the ability to lose heat faster (and survive) because of its greater surface area to volume ratio.
    In a hot car a baby loses fluid much faster because of the same larger SA to Vol ratio. A baby in a hot car will expire sooner than an adult in the same car through dehydration.
    I’ve thought of time lapse photography for apples of different sizes in a warm oven but this is much better, even though it “appears” to work in reverse.

  30. Hi Dan,

    I fit your data by hand to an exponential with the following equation:

    melting time = 4.55 exp(3.93*volume/surface area) – 1.15

    As a metric, I minimized the sum of the squares of the differences between my model and the actual data for each point.

    The reason I think the exponential is a reasonable model is that it is a solution to the heat equation, which is more or less what we’re dealing with here. Temperature should rise as a function of volume (microwave heats polar molecules in the cheese), and temperature should decrease as a function of the difference in temperature between the cheese and the environment, subject to the amount of surface area available for radiation from cheese to environment. (You’ve got a partial differential equation in space and time.)

    As a sidebar, I teach biology in addition to math, and this problem is similar to the one cells in your body face. In the cell case, surface-to-volume ratio is the main factor which limits cell size. The idea is this:

    1) cells need to transport food and oxygen into the cell to make energy

    2) cells need to transport waste (such as heat and carbon dioxide) out of the cell to avoid poisoning themselves

    3) the rate of transport processes is proportional to the surface area of the cell

    4) the rate of waste generation and energy useage is proportional to the volume of the cell

    End result: if a cell gets too big, it generates waste and uses energy so fast that it can’t transport food and waste fast enough to survive. In practice, that means that most cells are less than 1mm in diameter. We as large animals require a circulatory system to get around this difficulty.

    If you’re ok with it, I would love to develop this idea into an inquiry-based lab for my students. Please let me know if you have suggestions.

  31. Dan,

    Great site. I’m looking forward to using some of the ideas in my lesson.
    What software did you use to combine all the videos together?



  32. Having thought hard about the all-important issue of cubeyness (link below), I am delighted to see it in application here. My students’ cubeyness measure (sum of the reciprocals of the dimensions) needs to be adapted for this problem. I will get to work on the exposed-cubeyness-measure and its correlation to cheese melting time.

    On why it works, it seems the exposed surface area is how the microwaves get into the cheese. I imagine heat loss to be a minimal concern over the time scales involved here, and in comparison to the bombardment of microwaves.

    And the exposed surface area idea is lovely. Kids often wonder about orientation of prisms. Abstractly, orientation doesn’t matter-a triangular prism is still a triangular prism no matter what face it is resting on (or whether it is resting on a face at all). But in the applied problem, this matters very much. Did you try different orientations of your cheese? The melting time of your cheese ought to vary greatly depending on how it stands. And what is your cheese budget, anyway? Can you afford several dozen more trials so we can examine variation in your data?

    More on cubeyness:

    More on middle schoolers and prisms:

  33. I’m interested in Belinda’s question about the mathematical learning goal(s) of the lesson in which the activity is embedded. Carmen has terrific ideas about science content knowledge and I can imagine a lesson on hypothesis testing. What’s the *math* goal you’re after?

  34. Simon: What software did you use to combine all the videos together?

    I used Adobe AfterEffects. There has to be a cheaper, simpler solution for the same effect, though.

    Christopher: Did you try different orientations of your cheese? The melting time of your cheese ought to vary greatly depending on how it stands. And what is your cheese budget, anyway? Can you afford several dozen more trials so we can examine variation in your data?

    Different orientations of the same block of cheese is inspired. My wife has put some serious constraints on my cheese budget, though, in light of my smelling up our place, so the reshoot of this activity, while inevitable, will have to wait.

    (PS. Do they only sell cheese in my part of the world? Doesn’t someone else have sufficient curiosity to test this out.)

    Karen: What’s the *math* goal you’re after?

    Seriously, I haven’t thought about it. I’m not sure where the idea comes from that this is in any way a lesson. I was curious. I followed up on my curiosity. I got confused. I came here to share my curiosity and confusion with you folks. That’s about all I have right now.

  35. The very simple model:

    13.071507 + 0.18126714*(volume)*(height)

    has R^2 of 0.977, not bad. Found using Eureqa.

  36. Dave, if you’re going to have a huge y-intercept, I’d like to see some data points closer to 0.

    Without a physical explanation and very few data points, fitting an arbitrary curve with high R^2 values is pretty meaningless.

    I can see volume being an important parameter, linearly related to time (total energy needed to melt that much cheese).

    I can also see “thickness” as a important parameter (as energy is preferentially absorbed nearer the surface).

    I have trouble with the y-intercept, though. Why should melting microscopic amounts of cheese take 13 seconds?

  37. gasstation writes: “Without a physical explanation and very few data points, fitting an arbitrary curve with high R^2 values is pretty meaningless.”
    I agree wholeheartedly.
    Sort of.
    I have raised this critique of the (Dan’s word) pseudocontexts in my college’s College Algebra text. I suppose your critique of Dave’s model, which I sort of share, is better described as one of “pseudomodels”. If we can’t say anything about why the function type we use should apply, then is it really a meaningful model?
    I shall have to think about this and write in more depth when a bit of time opens up for me.
    A few conversations in the last year have convinced me that I have in some ways been too harsh on pseudomodeling. My expectation matches gasstation’s; I expect that we should be able to explain why a particular function type is used in our model. But maybe this explanation doesn’t always need to be grounded in the context (e.g. a cubic because of volume or an exponential because of repeated multiplication). Maybe the explanation for a particular function type can be rooted in (for example) the calculus properties of the relationship. Maybe we start looking for polynomials when we notice that the rate of change is increasing (i.e. the derivative is non-constant).

  38. Yeah who knows. Maybe the y-intercept is capturing some sort of ramp up time of the microwave, or some average delay in recognizing something is melted and stopping the timer.

    What’s interesting to me is the (volume)*(height) term itself. Volume alone makes intuitive sense to me (the more cheese the more energy needed). But seeing that the height factor is predictive suggests some sort of asymmetry. Perhaps the microwave is effectively aiming down, or it’s just that the cheese is placed at the bottom?

    It’s fun to think about.

  39. Actually, I’m not opposed to models that have no physical explanation. I use them all the time in my research. The key point is that using max R^2 to pick a model out of many hypotheses makes it very likely that you will pick a bogus model, particularly if there are few data points. (With 5 data points having different x values, I can always find a 4th-degree polynomial that fits, but it may have no predictive value for new points.)

    When choosing models, one either needs to have a convincing mechanism OR one must test the model on data not used to choose the model. (Best is to have both.)

    In my field, cross-validation is the standard, since we often have lots of data from past experiments, but collecting new data is slow and expensive. In cross validation, you do all your model building and model selection from part of the data, then check to see how well the model fits on the remaining data.

    You have to be very careful when doing this, since you can only use the verification set once. If you go back and change your model, based on a failure on the verification set, then you’re using all the data for training, and have no independent verification.

    If the model-building and selection process is automatic, you can do random splits of the data, and see the distribution of how well the data fits. This can give you a good idea whether you have enough data to fit the sort of model you are using.