The Difference Between Math and Modeling with Math in Five Seconds

Jim Pardun sent me a video of a dog named Twinkie popping balloons in the pursuit of a world record. How you train a dog to do this, I don’t know. How there is a world record for this, I don’t know either.

What I know is that this video clearly illustrates the difference between math and modeling with math.

You can’t break math. Some people think they broke math but all they did was break ground on new disciplines in math where, for example, triangles can have more than 180° and parallel lines can meet.

Our mathematical models, by contrast, arrive broken. “All models are wrong,” said George Box, “but some are useful.” And we see that in this video.

Twinkie pops 25 balloons in 5 seconds. How long will it take her to pop all 100 balloons? A purely mathematical answer is 20 seconds. That’s straightforward proportional reasoning.

But mathematical modeling is less than straightforward. It requires the re-interpretation of that answer through the world’s imperfections. The student who can quickly and confidently calculate 20 seconds may even be worse off here than the student who patiently thinks about how the supply of balloons is dwindling, adds time, and arrives at the actual answer of 37 seconds.

Feel free to show your classes that question video, discuss, and then show them the answer video. Or if your class has access to devices, you can assign this Desmos activity, where we’ll invite them to sketch what they think happens over time as well.

The difference between the students who answer “20 seconds” and “37 seconds” is the same difference between the students who draw Sketch 1 and Sketch 2.

You might think you know how your students will sort into those two groups, but I hope you’ll be surprised.

That difference is the patience that modeling with math requires.

BTW. I’m very interested in situations like these where the world subverts what seems like a straightforward application of a mathematical model.

One more example is the story of St. Matthew Island, which dumps the expectations of pure mathematics on its head at least twice.

Do you have any to trade?

2017 May 19. Steve Rein asked for the data set. Right here.

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 wouldn’t file it under what most would consider traditional math modeling, but Peter Donnelly’s TED Talk “How Stats Fool Juries” contains a probability nugget with an accessible premise but with a surprise conclusion. The problem concerns coin-tossing: Which will more likely occur first in a string of coin tosses: HTH or HTT? Intuition says that each of these strings has a 1/8 probability of occurring in a string of 3 tosses, so how could our expectation of them be anything but equal. But giving kids some coins and having them record when these string occur lends some insight into their difference. Find the TED talk and forward to the 4 minute mark for the problem – or start from the beginning for some quite bad British humor (or humour…whatever…)

  2. Thanks for a great post. I loved the comic, although my kiddos might be a bit young for that. I don’t know.

    BTW, the Act 1 video link goes to Act 3.

  3. I think this video can easily lead to a great way to talk about theoretical vs experimental as well as how, in real life, constants of proportionality involving humans (or any other living creatures) are generally theoretical. If Javier can paint 1/2 of a room in 3 hours and Cici can paint 1/5 of a room in 2 hours, we really don’t know how long it would take them together, do we? What if the next room has lots of built-in shelving or what if Javier refuses to work with Cici because she is so slow and Cici ends up throwing her paint can at Javier, requiring a trip to the ER and then end or their painting relationship? And even with even machines, constancy is hard to find. Machines can break down, fail due to power outages, have to be moved to a different room so Javier can paint, etc. So, I wonder, is a truly linear function ever really possible?

    • Chester Draws

      March 10, 2017 - 12:08 pm -

      What if the next room has lots of built-in shelving or what if Javier refuses to work with Cici because she is so slow and Cici ends up throwing her paint can at Javier, requiring a trip to the ER and then end or their painting relationship?

      That’s a very negative way to look at modelling, and the power of Maths in general.

      If Javier can paint 1/2 a room in 3 hours and Cici can paint 1/5 of a room in 2 hours, then we would predict naively that they together would paint a room in 3.75 hours.


      We put them together and time them. If they finish in under 3.75 hours then we know that there is some synergistic effect, and they work better together than separately. If they take longer than 3.75 hours then we know that combining them is wasteful. That seems to me to be extremely relevant information if you want to be as efficient as possible.

      A simple linear model is required to make sense of the combined time taken. That the actual result is never going to match that model isn’t a problem at all — the only time we haven’t learned anything interesting is when our naive model is correct.

      Isn’t this what biologists and chemists actually do when measuring plant growth and reaction rates to determine whether combining nutrients or catalysts is having an effect?

    • That’s a very negative way to look at modelling, and the power of Maths in general.

      Chester, please try not to take such an embattled perspective around here. It undercuts the value of the rest of your comment, in particular the bit I’ve highlighted.

  4. Thanks for doing another 3-Act. In trade, an example from one I created last fall and a relative to your shower vs. bath problem.

    awesome modeling idea #1

    First, Halloween Scare was a 3-Act I put together showing our candy bowl along with a counter measuring the number of pieces remaining next to a clock/timeline showing the start and finish of trick-or-treating. The rate of “decay” dropped off dramatically (luckily) as my students predicted I would surely need to pilfer the fruit bowl.

    awesome modeling idea #2

    Second, and often while showering, I’ve noticed that there doesn’t seem to be an even or gradual increase in the water temperature in relation to the degree of turn on the dial. In fact, I’d predict that the perfect temperature is a one micro-adjustment short of way too hot. Maybe a personal preference variable in there, but having students figure out the water temperature vs. dial angle might be my next 3-Act venture. Thanks again, Dan.
  5. Thanks for the post. I’m struggling to follow your train of thought a bit where you contrast the “purely mathematical” answer with the modeling answer. It sounds like the “purest” (for want of a better word) answer is “not enough information is given.” It is not fair from a mathematical standpoint or a modeling standpoint to assume that we can find the answer by using a direct proportion. It seems your point is that students are actually misinterpreting the task. They implicitly assume it’s to use a familiar technique (linear modeling), but in reality the task is to determine a more realistic method (taking other factors into account) and then use it.


    If I’m understanding that correctly, then it seems like the the trouble arises not with the students’ ability to reason using any one method, nor their ability to determining which method is most applicable, but rather with their ability to interpret the task (namely, determining that their task *is* the latter) from an apparently straightforward question or problem statement. Am I understanding what you’re saying?
    • I think that’s right. Thanks for the restatement. I’m also linking that inability to interpret with a particular impatience, impatience that often derives from a strict focus on answer-getting in pure mathematics.

    • Yes. I’ve shared the dropbox folder link. Technically pretty basic, but have at it. Shower temps is in development. I’d love some tips on the apps and methods you use.

    • Ah, that makes sense. (Sorry if I was being redundant.) What are your thoughts on how much of the impatience lies with the how we phrase the question? Can we reduce the impatience by presenting the problem in a way that more explicitly highlights what assumptions the students should make?

  6. Thanks for another great post! In my college algebra class, I use a crime scene investigation activity to help me teach mathematical modeling. I borrow the leg of a skeleton from the biology department and bring it to class. I present the students with the ages and heights of four missing persons. Then, I ask my students how we can determine whether or not the leg belongs to any of the missing persons on the list. Through guided conversation and exploration, the students typically decide to create a scatter plot and draw a regression line. They use this line to help them determine who the owner of the leg is.

    that’s patience!

    Every time I do this activity, at least one student raises the point that, although there is a positive linear correlation between the length of the tibia and a person’s height, it is not a perfect linear correlation. You can have short people with long legs and tall people with short legs. It’s always exciting to see them discover the power and limits of mathematical modeling.

    I’m looking forward to using this new task in future classes!

  7. Erik Von Burg

    March 10, 2017 - 12:11 pm -

    Miles, here is where I see the problem. Pure mathematics, in my mind, is defined by unambiguous rules and permutations and declarations can be made because the rules are absolute. (I must postscript this by saying this is my interpretation of pure mathematics, and I welcome any clarification.) Modeling with mathematics at its core is the process of quantitatively defining a given situation. These messy situations have multiple factors at play, and these factors have varying levels of significance. And, to further complicate matters, the effect of certain factors run counter to one’s intuition. If a student applies a linear model to the balloon-popping dog, he/she most likely did not recognize the impact of certain factors (e.g. proximity to next available balloon, fatigue, attention, etc.) on the time it takes to complete the task. The maths doesn’t fail. But. the student does not recognize the limitation of a single pure mathematical understanding to fully describe the situation.


    This is one of the reasons that I think that computer programming is so incredibly important. My students develop games in the Scratch programming language, and it is really just one huge modeling problem. They need to mathematically model everything in the game from making a character realistically jump to altering the level of difficulty. It is painfully obvious when their models fail to achieve their expected result.
  8. good catch!

    This is the clearest illustration of exponential decay I’ve ever seen (if you try counting balloons remaining, instead of balloons popped). It’s so clear why the rate of popping is proportional the number of balloons remaining (with twice as many balloons, it’s twice as easy to find another balloon to pop).

    Relates so well to other exponential decays, e.g., the kidneys removing molecules of medicine from our blood, where the kidneys are analogous to the dog, and the medicine molecules are analogous to the balloons. But you can see it happening live in this video, and even a child could understand the reasons. Great. And yes, also great from the modeling end.

    • Oo. Good call. So I thought I’d find something logistic in there, then didn’t, and then chalked it up to a failed linear function, not a well-defined function of another kind.

      Is there some way I should fix up the media above (the video? the activity?) to better draw that model out?

      I could ask students to sketch a model of balloons remaining v. time, for instance.

  9. Yes, if you can edit the video to count balloons remaining rather than ones popped, and adjust the Desmos slide to ask for the sketch of balloons remaining vs. time, I think it’d be easier to connect this to content I’m required to teach. If in fact an exponential equation fits the data reasonably well, I’d like to have an extra slide that lets us drag points to try to fit an exponential curve to the data. Kind of like your “Will it Hit the Hoop?”.

    This could lead into some contrasting cases. E.g., a sequel with another video, this one being linear and asking students to make a prediction, watch the answer video, and explain which family of functions it was and why. Then another one, this time with another exponential. That would be a lot of extra work for you. For what it’s worth, I usually use this SportCenter commercial as one of those contrasting cases: . I think I stole that idea from you, but may have been another mtbos peep.

    And folks who just want modeling practice, rather than modeling + exponential functions, might be mad. They’d probably want you to keep the existing version of this activity available, too.

  10. And folks who just want modeling practice, rather than modeling + exponential functions, might be mad. They’d probably want you to keep the existing version of this activity available, too.

    I don’t actually think a lot of those people exist. I’ll fix this up.

    • This guy first tipped me to the video.

      Thanks for the incredible collaboration on this project. I too agree that an exponential decay model could make for a very effective lesson. I would be interested in using a re-edited version if it’s not too hard to create at this point in time. Thanks.
    • @Jim, check out Screen 3. Click “Exponential.” Pretend you only had data for the first 25 balloons. Model those data with the exponential function. Look how far you’d be off. I’m happy to take this further but I can’t figure out what to do with that discrepancy.

      @Kevin is on the case also.

  11. Harry O'Malley

    March 14, 2017 - 7:53 am -

    Here’s a function that starts off in a linear style and then transitions to an exponential style similar to this dog video. I just realized it while eating a bowl of cereal this morning.

    Function input: number of spoonfuls of cereal eaten

    Function output: amount of cereal pieces left in the bowl

    At the beginning, when there is an ample amount of cereal in the bowl, each spoonful basically removes the same amount of cereal (a full spoonful: maybe 30 cereal pieces or so). At some point, though, dipping your spoon in and pulling it out removes less than a full spoonful’s worth of cereal pieces because the cereal density has thinned out. After that, each spoonful eaten causes this density to lessen even more, which, in turn, causes the next spoonful to remove even less, etc. This is the annoying part of eating the cereal where you’ve got to push the cereal pieces to the side of the bowl and then use your fingers a little to push some of them onto your spoon. Here’s a picture of the bowl during its exponential phase:

    Exponential Cereal Pic

    And here’s a sketch model of the function in GeoGebra, where each piece of cereal is individually part of the simulation:

    Exponential Cereal Model

    Hopefully this continues to flesh out the ideas brought out by Kevin Hall. Thanks, Kevin.

  12. This could be a good exercise for a discussion of curve-fitting by linearization then regression. (At least that is how I would discuss this at the collegiate level in one of my classes.) Are the data available for the public to use for educational purposes?

    • Good catch. It popped at the same time as the 55th. I used that spreadsheet to drive the counter on the video. Now updated so it’s more accurate for other kinds of analysis.

    • In casual exploration the standard statistical approach (taking a log of each x = balloon number and y = time) gives too much weight to early datapoints and so the least-squares fit tends to poorly predict what happens at the end of the video. I’m needing to think about the best way to fit this trend. Maybe least squares without transforming the variables … but that is a less straightforward analysis unless one is, for example, a junior statistics major in college.

      If you want I can e-mail you code in R (free software that statisticians tend to use) that would help with analyzing these data.