I Was Wrong About #BottleFlipping

I didn’t think there was a useful K12 math objective in bottle flipping. My commenters served their usual function of setting me straight.

I was asking the question, “Can you predict whether or not a bottle will land?” A modeling problem.

Commenters like Meaghan asked the question, “What conditions will set yourself up for success in bottle flipping?”

How much water in the bottle? What kind of angle on the toss? Clockwise or counterclockwise? These are statistical questions.

Paul Jorgens followed that angle with his class:

It started with an argument in class last week about the optimal amount of water in the bottle. Should it be 1/4 filled? 1/3? Just below 1/2? I told the group that we could use our extra period to try to answer the question. We met and designed an experiment. We thought about problems like skill of tosser, variation in bottles, etc. We started with 32 bottles filled to varying levels. During 20 minutes of class, 32 students flipped bottles 4,220 times. We the all filled in our data on a Google Sheet.


I meet again next week with the small group that had the idea. I think they want to produce something for school news. Did we answer the question about how much water to put in the bottle?

Check out the graph of their data.


The Paper Helicopter is a similar exercise in experimental design. These activities come from the same template. If we understand that template, we can swap lots of different questions into the experiment, including those that seem most interesting to students in this moment.

We can teach students how to use mathematical tools to answer questions that interest them. We can also assign detentions. If there’s any middle ground, I’m not seeing it on Twitter right now.

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BTW. Here’s a Desmos activity you can use to facilitate data collection in your class. Your students add their data on the first screen. Then they see the sum of their class’s data on the second screen.

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


  1. Chester Draws

    October 8, 2016 - 1:56 pm -

    Clockwise or counterclockwise? These are statistical questions.

    I would say engineering myself. I have watched a LOT of bottles thrown. I find it really tedious. This is how I think it works, and how I might model it.

    Whether it will land or not depends on whether the motion of the base of the bottle is more or less vertical at the moment of landing.

    Try to land a bottle by throwing it without flipping. You soon see that the problem is that a parabola has constant horizontal motion, so the bottle will always slide on landing.

    The flip is essential so that the base of the bottle at the moment of landing is just before vertical, the mass of the water then forces is down — there is little or no horizontal movement at the moment of impact so the bottle doesn’t slide, and the rotational movement is still backwards if just before vertical so the bottle doesn’t tip over.

    If the spin it too fast you get too small a margin for error at the moment of landing, so a slow spin is needed to maximise the sweet spot. One complete spin is easiest to control — the bottle starts vertical at the base of the throw, and ends vertical again.

    Parametrically we have a classic parabola : x = t, y = 0.5 t^2 combined with a classic circular motion of a point rotating about the centre of x = sin t, y = cos t.

    We can play around with variables modelling the spin speed inside the trig function, height of parabola etc, so a=Curve[t+sin(3 t),-0.5 (t-3)^2+4+cos(3 t) 0.5,t,0,6] in Geogebra shows the retrograde motion of the base relative to the centre of mass and how spinning too fast leaves little margin for error.

    A solution around a=Curve[t+sin(0.6 t),-0.5 (t-3)^2+4+cos(0.6 t) 0.5,t,0,6] seems viable as a practical path, although in general the kids throw a lower loop.

    Note if you spin the bottle the “other” way, replacing sin with -sin in the equation you get a curve that cannot cancel the motion without doing more than one gentle spin, which becomes impossible in practice.

    I have taken no account of sloshing because 1) we are modelling at school level, not NASA level, and 2) because any throw with more than a little slosh tends to fail anyway, as the continued motion of the water in the bottle tends to make it topple even if you land it right.

    I also haven’t taken into account the distance of the base of the bottle relative to centre of mass. Again, describing circular motion on top of parabolic motion is the aim: that is what little Maths there is in the this thing — the rest is engineering.

    The amount of water is important because without enough water in it any plastic bottle will bounce and we need the mass of water pushing into the landing to keep it stead. But with too much the centre of mass is moved further from the base of the bottle, which makes the sweet spot for landing much smaller (as the mass of the water upon landing is much harder to cancel). But that’s physics, not Maths.

    Feel free to delete if this comment is off topic for the thread Dan, but you asked how I might deal with it parametrically.

  2. Cristina Antoniolli

    October 12, 2016 - 6:10 am -

    Just did this with my students. You definitely need to use Ice Mountain water bottles. Great data analysis followed with students!

  3. Reply

    This is so great to implement such a figure of pop culture into a math lesson! Being a student in my school’s math education program, these kinds of ideas always piques my interest becasue I want to know what the next big thing is in culture. If there is anyway to implement pop culture into math class, I feel like my job as a math teacher to engage students in their learning is done. I love how Paul Jorgens took the extra step to answer questions about the volume of the water and how it could affect the success rate of flipping the water bottle. Something that started as a lesson turned into a full-blown experiment that students can enjoy. I’m curious, though. As you mentioned at the end about assigning detentions, how can you as a teacher make sure that students aren’t just flipping water bottles around school or in the classroom? I feel like they would just make excuses and say that we did it in class, so it’s okay.

  4. Reply

    I teach 8th grade supporting math classes. Currently I have a group of students with more skill, leadership and curiosity than most. Obviously, I will not be talking parabolas. They have been very in to bottle flipping. to the point that their general ed math teacher banned water bottles from the class. I thought I should capitalize on their interest and do some math! I like Meghan’s question: What conditions will help one be the most successful when bottle flipping? We would then set up some experiments and collect data, eventually using the data to draw conclusions. Any thoughts/advice?

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