**2016 Oct 7**. I was wrong about everything below. After admitting defeat to #bottleflipping, my commenters rescued the lesson.

I’m sorry. I went looking for a lesson and couldn’t find it.

Relevant background information:

Last spring, 18-year-old Mike Senatore, in a display of infinite swagger, flipped a bottle and landed it perfectly on its end. In front of his whole school. In one try.

That thirty-second video has six million views at the time of this writing. Bottle flipping now has the sort of cultural ubiquity that can drive even the most stoic teacher a little bit insane.

my brother's teacher banned bottle flipping what

— maya !! ebg (@booseungkwon) October 5, 2016

Some of my favorite math educators suggested that we turn those water bottles into a math lesson instead of confiscating them.

I was game. Coming up with a math task about bottle flipping should be easy, right? Watch:

Marta flipped

xbottles in^{2}+ 6x + 8x + 4minutes. At what rate is she flipping bottles?

Obviously unsatisfactory, right? But what *would* satisfy you. Try to define it. Denis Sheeran sees *relevance* in the bottle flipping but “relevance” is a term that’s really hard to define and even harder to design lessons around. If you turn your back on relevance for a second, it’ll turn into pseudocontext.

For me, at the end of this hypothetical lesson, I want students to feel more *powerful*, able to complete some task more efficiently or more accurately.

Ideally, that task would be bottle flipping. Ideally, students who had studied the math of bottle flipping would dazzle their friends who hadn’t. I don’t think that’s going to happen here.

But what if the task wasn’t *bottle flipping* (where math won’t help) rather *predicting the outcome of bottle flipping* (where math might). You can see this same approach in Will It Hit the Hoop?

The quadratic formula grants you no extra power when you’re in mid-air with the basketball. But when you’re trying to predict whether or not a ball will go in, that’s where math gives you power.

**Act One**

So in the same vein as that basketball task, here are four bottle flips from yours truly. At least one lands. At least one doesn’t. Each flip cuts off early and invites students to predict *how will it land?*

**Act Two**

How can I shoot a slow-mo video of a projectile and have it appear over a coordinate plane in the background? @ddmeyer #bottleflips

— Mike Linskey (@MikeLinsk) October 3, 2016

Okay. Here’s a coordinate plane on top of each flip.

If you’ve been around this blog for even a day, you *know* what’s coming up: we’re going to show which flips landed and which flips didn’t. Ideally, the math students learn in the second act will enable them to make more confident and more accurate predictions than they made in the first act.

*But what is that math?*

I asked that question of Jason Merrill, one of the many smart people I work with at Desmos. I won’t quote his full response, but I’ll say that it included phrases like “cycloid type thing” and “contact angle parameter space,” none of which fit neatly in any K-12 scope and sequence that I know. He was nice enough to create this simulator, which has been well-received online, though even the simulator had to be simplified. It illustrates *baton* flipping, not *bottle* flipping

**Act Three**

Here is the result of those bottle flips. For good measure, here’s a bottle flip from the perspective of *the bottle*.

I’m obviously lost.

Here’s a link to the entire multimedia package. Have at it. If you have a great idea for how we can resurrect this, let me know. I’m game to do some video editing on your behalf.

But when it comes to bottle flipping, if “math” is the answer, I’m not sure what the question is. Please help me out. What is the lesson plan? How will students experience math as power, rather than punishment.

Sure, it’s probably a bad idea to destroy the bottles. But it’s possible we shouldn’t turn them into a math lesson either. Maybe bottle flipping is the kind of silly fun that should stay silly.

**2016 Oct 7**. Okay: I was wrong about #bottleflipping. A bunch of commenters came up with a great idea.

**Featured Comments**

I see a couple students playing the game during some down time and my immediate reaction is, “There’s gotta be some great math in there!” One of the boys who was playing sees my eyes light up. He looks at me in fear and says, “Mrs. Raskin. Please. I know what you’re thinking. Please don’t mathify our game. Let us just have this one thing we don’t have to math.”

Mr K:

I suspect I should put as much effort into making this teachable as I would for dabbing.

Meaghan found a nice angle in on bottle flipping, along with several other commenters:

It would be neat if you could spend a, for example, physics class period talking about experimental design (for fill ratio questions or probability questions) and collecting the data, and then troop right over to math class with your data to figure out how to interpret it.

Paul Jorgens has the data:

It started with an argument in class last week with 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. Thought about problems like skill of tosser, variation in bottles, etc. We started with 32 bottles filled to varying levels. During class over 20 minutes 32 students flipped bottles 4,220 times.

## 62 Comments

## Kyle Pearce

October 5, 2016 - 6:50 pm -That is probably the most well put together, unfinished 3 act task I’ve ever seen!

I will say that I have put some thought into how math fits and I went straight for probability distributions. I’m thinking if we have enough trials, we could head down the binomial or even extend to binomial approximation to the normal distribution? It’s been a while since I played in that space, but I’m feeling it could be a fun one.

## Dan Meyer

October 6, 2016 - 9:08 am -Lol. Thanks, Kyle. I put some time in on it, which hurts to throw away. But I learned some math and got to stress test my framework for curriculum design, so it’s hard for me to count it all as a loss.

I’m coming around to your idea of probability distributions. It’s one of the more promising angles I’ve heard.

## JS

October 5, 2016 - 7:26 pm -I’m not a math educator; I teach high school physics. But here’s an idea (couched in high school terms I can muster): if you can chart the trajectory of the spin centre or get the vertical velocity, that lets you calculate the time of flight. And if you can work out the rate of spin, that lets you work out whether the bottle will land upright or at an angle.

That’s two things to extract. The first could be done with Tracker or Desmos but the second could be a bit tricky …

## Dan Meyer

October 6, 2016 - 9:10 am -Right. Nice. Then my question is: is the math we learn worth the hassle it took to us to learn it? (eg. Desmos, Tracker, lots of terminology that’s extraneous to math.) No easy thumbs up / thumbs down answer here, I suppose. Just a bunch of competing interests.

## Jimmy

November 11, 2016 - 11:07 am -Mr. Meyer,

Perhaps we could justify working out the mathematics if we were attempting to build a machine that could “flip” a bottle repeatedly with landing success. Understanding those variables would then be key! Let’s dy/do it!

## Joshua

October 5, 2016 - 7:35 pm -I don’t have the answer to the main question (how do you make this into a curriculum-aligned lesson), so I’ll do some notice and wonder based on the act 1 video and the act 3 gif.

Notice:

– The bottle isn’t full (looks about 1/3 full, though the gif seems more than half full)

– The throw involves spinning the bottle “backward”

– Each try seems to show about the same amount of time in flight

– In the first three tosses, the bottle cap has rotated less by the end of the clip. The fourth clip seems similar rotation to the first.

– the third toss is lower than the others

– The bottle seems to move about the same distance to the right in each clip

– I can see the blue bottle top tracing out a curve

– (a meta notice): like shooting a basketball, this is a skill that humans can improve with practice, though we aren’t consciously/explicitly solving the equations when we do so.

– The bottle slides along the table after it lands

Wonder

– can you successfully stick the throw with a forward rotation? How much harder is that?

– am I really correct about my time observation (thus can I extract information about relative amounts of force used in the throws)?

– Is there an amount of water in the bottle that is optimally stable?

– what is the curve made by the bottle cap?

– how precisely do I need to throw the bottle (what is my margin of error)? What is more important, being precise about rotation, how hard to throw forward, how hard to throw up, relative height of target and release, horizontal distance between target and release?

– do the answers to the previous question depend on friction of the bottle bottom and table?

– What is the success rate of an “untrained” bottle thrower? How quickly does that skill improve? What is a reasonable success rate for a trained thrower?

– What about other throwing games: throw a bottle through a hole with diameter close to that of the bottle (vertically or horizontally oriented), throw a partially full (thus moderately heavy) bottle into a light wastebasket without tipping over the waste basket, knife throwing, screwdriver throwing, etc…

– is there a simple heuristic I can use as a thrower to increase my chances of success? To be clear, I’m thinking about the constant angle heuristic that works for catching a fly ball in baseball.

## Dan Meyer

October 6, 2016 - 9:12 am -Thanks for your comment,

Joshua. This raises to mind for me a Venn diagram about questions. It has three circles:Ideally, there are a few questions in the overlap there.

## Mr K

October 5, 2016 - 8:01 pm -Oh good.

I’ve been dealing with trying to math this craze as well, and I’m drawing blanks. With my genius kids, I was thinking at least I could do a lesson on moment of inertia of asymmetric objects, but even that misses the fluid dynamics aspect.

I suspect I should put as much effort into making this teachable as I would for dabbing.

## Dan Meyer

October 6, 2016 - 9:13 am -Comment featured for dabbing reference.

## Heather Kohn

October 6, 2016 - 9:56 am -We measured the angle of our dabs in my precalc class and it was pretty fun. Check out what we did here: https://growingexponentially.wordpress.com/2016/09/18/the-golden-dab/

## Robert Kaplinsky

October 6, 2016 - 10:17 am -I’m scared to ask, but what the heck is dabbing?

## Dan Meyer

October 6, 2016 - 11:17 am -*looks nervously at Heather & Mr. K*

## Heather Kohn

October 6, 2016 - 12:17 pm -@Robert It’s the dance move made popular by Cam Newton after he scored a touchdown. But it looks like you’re sneezing into your arm.

## Elizabeth Raskin

October 5, 2016 - 8:21 pm -I teach 7th graders. Last year they were obsessed with slitheri.o (a video game where you compete with other players to create the biggest snake on a screen by eating colored pellets and not running into another snake.) I see a couple students playing the game during some down time and my immediate reaction is, “There’s gotta be some great math in there!” One of the boys who was playing sees my eyes light up. He looks at me in fear and says, “Mrs. Raskin. Please. I know what you’re thinking. Please don’t mathify our game. Let us just have this one thing we don’t have to math.” Note that these were my advanced math kids, so I didn’t let them get off that easily. I asked what math they saw in the game. They gave some suggestions about percent increase and geometric possibilities and yet none of the ideas we came up with sparked enough interest in them that they wanted to pursue. I let them keep their game this time.

I definitely think relevancy is important to our kids, but I don’t want to ruin their fun, either. We bottle flip like mad during break time and lunch, but for now…I just ask that it not happen in my class. They seem cool with it.

## Dan Meyer

October 6, 2016 - 9:14 am -Ha! Comment featured for that awesome student-teacher exchange. Especially for the fact that your student

knewwhat you were up to without you having to say anything.## Robert Kaplinsky

October 6, 2016 - 10:18 am -This is hilarious. Elizabeth.

## Sarah Caban

October 6, 2016 - 3:02 am -I am currently co-teaching a 6th grade math class and we are in the middle of a unit on ratio and proportions. I have been wondering if the ratio of water to air makes a difference in how the bottle lands. I am not sure how we would prove this, but I think it would be relevant if we could get to a place that we had a better understanding of equivalent proportions . If the bottle is too “watery”, will it mess up my flip? If I find a good fit ratio of water to air, can I extrapolate the ratio to other bottles? I don’t know if any of this is applicable because I haven’t had time to flip enough bottles myself. I am hoping to explore this soon because I am curious. If anyones has explored these questions, I would be interested in hearing what you discovered about whether the ratio of water to air matters.

## Dan Meyer

October 6, 2016 - 9:19 am -Thanks for the comment, Sarah. Your approach intrigues me here. “How do you know the perfect fill height for any bottle?”

Vox suggests that 1/3 is the right proportion of water in the bottle. Maybe you could give students lots of different bottles and ask them to locate the right fill height.

## Rob Eby

October 6, 2016 - 10:15 am -Oh, there we go. Pick several different fill levels. For each one, do several flips, and count the proportion that land. Now, chart the data (or maybe even plot the data) what patterns emerge? If we did say 0, 1/3, 2/3, 3/3, would going to 1/4’s help, or maybe we start with 0, 1/2, 1, then the thirds, then fourths, and maybe finally the fifths.

So depending on grade level, we have data analysis, probability, including binomials, maybe even curve fitting with parabolas.

Like you noted above Dan, I am not sure the work is worth the mathematics, but maybe.

## Glen Lewis

October 6, 2016 - 10:19 am -Rob – This is similar to what I commented below. I don’t see the work being too extensive. There is some really good math/science involved (importance of precision) and I picture the engagement being rather high.

## Glen Lewis

October 6, 2016 - 3:51 am -I picture a room of kids, each with a full bottle. Flip it five/ten times each, how often did the bottle land. Take out 1/5, 1/4, or 1/3 of the water, and repeat. Keep taking out water in the same ratio, until the bottle is empty, flipping and finding how often it lands.

Not sure the exact place for a lesson like that, but students experience ratios, probability and measurement, at the least. Some great places for discussion, and a lot of places to apply different math that students may or may not already know.

## Galen

October 6, 2016 - 10:09 am -Yes, ratios and percentages was my thought too. Adding the water level/bottle size variables is a great idea. Science class could probably pitch in with the center of gravity and rotational stuff if needed. There is a decent amount of skill among the students at this point so no need for practice or demonstration, just a big tarp for ruptures.

## Cathy Yenca

October 6, 2016 - 3:51 am -Hi Sarah, here’s a link to the Boston Globe article originally shared on Twitter. They seem to have an ideal ratio for you, but in traditional textbook fashion, they’ve spilled the beans and revealed it before giving folks the chance to test it. So… happy flipping… if you dare!

https://www.bostonglobe.com/metro/2016/09/29/bottle-flipping-becames-rage-with-middle-schoolers/1INRaYEj9dQILseR5cgVUK/story.html

## Howard Phillips

October 6, 2016 - 4:43 am -“Breaking down the bottle toss

The act of flipping a water bottle through the air to achieve a perfect, upright landing on the table in front of you is part artform, part science.”

As the bottle falls and the liquid drops to the bottle bottom the angular momentum, of the bottle stays more ore less constant. The liquid moves up and down. Not much rotation at all. The dropped liquid helps the bottle stay upright at the end as the impulsive force of the water is dissipated.

## Dave Johnston

October 6, 2016 - 4:30 am -Stuff I’m thinking about…

– Does Dan have an empty model home he uses to film his stuff?

– Not all water bottles are created equal. I’m guessing this would be really difficult with those short little 8 oz bottles.

– What if I was trying to design the perfect “flipping bottle?” Would I increase the size of its base? Make it taller?

– Is there an optimal amount of water in the bottle?

Paths I might pursue in a lesson:

– Predictions…

o How does our bottle flipping improve over time? Flip 10 bottles each day for a couple of weeks. Based on our performance the first 10 days, how many will land successfully on day 14?

o Our research team flipped 10 empty bottles, 10 bottle 10% full, 10 bottle 20% full… look at the data figure out the optimal water ratio.

– Graphing…

o Am I noticing in Jason’s simulation that the center of gravity travels through a parabola? There’s some exploration there.

– Bait-and-Switch options. Use the bottle flip to generate questions other than “will it land upright?”

o Imagine a video outdoors showing bottles being flipped to different heights and falling to the ground: Flipped 3 ft in the air, 6 ft in the air, 10 feet in the air. Then flip one 15 feet up and freeze the video at its apex. Can we use the rest of the video to determine the fall time?

o Geometric probability… if I flip several bottles onto a square table, how many will land in the inscribed circle?

o What happens after the flip? My teacher’s trash can fits [x] discarded bottles, how many are in the overflowing recycle bin outside her door?

o Did you know they changed the design on many water bottles within the last few year? the screw on portion is shorter than it used to be. (And I had to build an adapter for the school’s water rocket launcher.) Why? How much plastic does that save?

o And if we’re going really far afield, price or weigh a case of 8 oz bottles vs a case of 16 oz or 24 oz…

## Mr K

October 6, 2016 - 4:32 am -Additional thought:

I’m expanding into science this year, after 12 years of only teaching math. I’ve realized that what is taught in school as science are the hard sciences – those distinguished by experiments with fairly high correlation factors. Those whose research involves much lower correlations – psychology, sociology, economics, are not mentioned as science at all until maybe you get to some AP classes. But that’s where the really meaty questions about this lie: What is it about our brains that makes tossing a bottle of water so compelling?

## Dan Meyer

October 6, 2016 - 11:04 am -That may be more interesting to me than the question, “How do you best flip the bottle?”

## W. Ethan Duckworth

October 6, 2016 - 5:36 am -I don’t know about making a good lesson plan out of it, but as far as the math goes here’s my take. The center of mass will follow a parabola. A computer graphic could be overlaid to show where the center of mass is (this would probably be faked a little bit). That parabola could be modeled in the same way as in the “will it hit the hoop” demo.

While the center of mass moves along the parabola, the bottle rotates at a fairly constant angular velocity. To figure out that velocity, count the number of rotations, and divide by the total time of the bottle flight.

Now, for the bottle to land, we basically need two things to come together. Measure the hight from the center of mass to the table top. When the center of mass is at the right hight* from the table top, the bottle needs to be vertical. So, find the time taken for the center of mass to reach the correct height, and see if this also equals the amount of time for the bottle to rotate into a vertical position.

* right hight = ? Just like with the basketball hoop, there is a small range of values that will succeed. I don’t know what that range is, but it might be something like 0.25–0.5 bottle heights.

## Howard Phillips

October 6, 2016 - 5:45 am -A bit of geometry will give an approximation to the angle where the centre of mass is over one edge of the bottle so the range of values for uprightness can be ascertained. This will not just be “upright”.

## Dan Meyer

October 6, 2016 - 11:05 am -Some trouble we had in our conversation at Desmos was how best to mark the center of mass of a water bottle where the water sloshes around. For that reasons, we ran with a baton instead of a bottle. Ideas?

## GenFunctions

October 6, 2016 - 5:51 am -This feels like it should fit with parametric equations. Probably because I talk about cycloids when we do parametric equations.

The obvious question is, “When does the bottle land?” Breaking that down, we need to know the starting position, angle, force, etc. If we can assume that the bottle does indeed follow a cycloid, we could find parametric equations for them. Then figure out what the starting position on the cycloid needs to be – including how far above the table – in order for the bottle to land.

But that’s a big if – does it actually follow a cycloid? I suspect it does – or we could make some assumptions, like a full water bottle, that would simplify to a cycloid. But it’s not going to be lined up nicely along a horizontal line, because of gravity. So then we’re talking about what the relative angle between the table and the cycloid is. We probably need to treat the table as a tangent line, and ask where the tangent line needs to be along the cycloid.

So now we’re talking about Calc BC. I have one student in Calc BC, and she is both highly motivated and highly curious. If we have a spare day, I might explore this; she’ll probably find it interesting and she could use a good, hard problem to chew on. But I think she’d probably be interested in learning about parametric equations and cycloids just because those things are beautiful in themselves and useful for more straightforward reasons.

## Dan Meyer

October 6, 2016 - 11:06 am -Interesting conclusion. It’s possible the water bottle context is clouding math that (for some students) is super interesting on its own. If you run through any of this with your Calc BC students, I hope you’ll come on back and let us know how it goes.

## J. Palmquist

October 6, 2016 - 6:32 am -I am thinking statistics lesson. What questions can we ask? What do we control for? How do we hold constant any variables? How many trials? Do some data gathering (now)? and bring the data back when we have learned applicable Hypothesis Testing. My mind is spinning like a bottle…. Thx Dan

## Dan Meyer

October 6, 2016 - 11:07 am -Yeah, I think this is the big lesson idea that I totally missed when I was thinking this through. I was pursuing modeling, but hypothesis testing in statistics seems like a much more natural fit. Thanks for your thoughts here, J.

## Robert Kaplinsky

October 6, 2016 - 10:21 am -I admire the amount of effort you put into this just to show how it is hard to even know what the question is. I have had plenty of contexts that seem intriguing, but I’ve come to realize how important it is to figure out what question I want to address early on. Otherwise I find that I’ve invested a lot of time building a problem that I can’t use.

Also, your video from the bottle’s perspective is all kinds of nauseating awesomeness.

## Scott

October 6, 2016 - 12:44 pm -Now I admit I haven’t read all the comments above me so I don’t know if this idea has already been said:

The math in bottle flipping isn’t the math we are used to… but the students have the opportunity to INVENT “the math of bottle flipping”.

What I mean is, they can develop formality by building from their informal knowledge and experience. Isn’t that “math” ? We try to do this all the time! Its just we adults and teachers get confused about what students have informal knowledge and experience with. Take the course of Geometry– it tries to assume we all know about shapes and stuff and we’re gonna now codify it into a system. However, many students don’t know about shapes’n’stuff so we get confused and we start teaching properties of triangles and area formulas as if THAT’S the important thing about geometry (it’s not).

Throw out all the parabolas and probabilities (parabobabilities?). Throw out the physics and the fractions. We can come back to that later.

Bottle Flipping. What is it? What are the rules? What are the kinds of flips? What is a legal bottle? What’s a good outcome? Your students have answers for these questions– but they aren’t likely to be the SAME answers. Well, lets have a discussion. What SHOULD these definitions be? What things must we assume?

We could try to start axiomatically defining these things and we maybe get some sort of logic leadin– “What are the axioms? What are the theorems? What are the undefined terms?” But more generally, I see this as a math allegory. We will negotiate terms, we will draw out properties and abstract them, we will look for underlying structure.

We may find that students start bringing up the “useful” math concepts themselves. All the ideas I see mentioned above the students can likely feel some need for. Then the student start looking themselves at the connected concepts. If they do, let them bring it in! But if they don’t, it would be disingenuous of us to graft unrequested structure onto the students’ fledgling systems.

tl;dr have the students build their own concepts of what bottle flipping is, thereby engaging with formalization and abstraction.

## Meaghan

October 6, 2016 - 1:37 pm -It would be neat if you could spend a, for example, physics class period talking about experimental design (for fill ratio questions or probability questions) and collecting the data, and then troop right over to math class with your data to figure out how to interpret it.

(Great preparation for real-life science!)

## Dan Meyer

October 7, 2016 - 2:40 pm -Love it. Check out Paul Jorgens below who appears to have gathered data with his class. What do you make of it?

## Paul

October 14, 2016 - 8:26 am -I think that is a seriously awesome idea! I remember having a teacher in high school telling us about a class she used to teach that was combined with Math and Physics. It was actually a double block, and the first block was more of learning the math and the second block was more of an application/lab using the math. Now that I’m studying education and am seeking to be a math teacher, this thought gets me really excited! I would honestly love teaching in a class where this sort of stuff happens because I feel like it would just be an amazing opportunity for students to engage with what they’re learning. Have you ever been in or worked with a school that has something similar to this? I’m wondering if it is implemented frequently anywhere.

## Ken

October 6, 2016 - 4:22 pm -I’m a left-leaning, wet blanket English teacher. I might have my students research the number of plastic water bottles purchased every day in the U.S. and where those bottles end up, and then write an essay on the immorality of purchasing bottled water and leaving just enough water in the bottle to make it more flippable.

## Dan Meyer

October 7, 2016 - 2:38 pm -Welcome to our math party, Ken! Thanks for sharing what the humanities might do with the bottle flipping phenomenon.

## Paul Jorgens

October 6, 2016 - 7:17 pm -It started with an argument in class last week with 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 (due to a staff development day we had an extra 3rd period) to try to answer the question. We met and designed an experiment. Thought about problems like skill of tosser, variation in bottles, etc. We started with 32 bottles filled to varying levels. During class over 20 minutes 32 students flipped bottles 4220 times. We the all filled in our data on a google sheet. I enjoyed watching the cells fill on the screen as 32 students entered data. It is here http://bit.ly/bottletoss

We spent time some “noticing” with the data. We looked at the data in desmos. Percent filled vs. Success rate. It looks bell curvish I guess. https://www.desmos.com/calculator/bh6hzkw1fv

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? I don’t know.

## Dan Meyer

October 7, 2016 - 2:47 pm -Oh awesome. Love the hustle. Your students’ data collection gives some empirical support to Vox’s intuitive claim.

## Denese

October 10, 2016 - 8:08 am -This looks like great fun and I am planning to do this in the next week with my classes. I have many senior “bottle flippers” and as long as they stop when the bell rings I only give a warning that if the bottle breaks, they are cleaning up the water! We just started a one-to-one initiative and all of our juniors and seniors have their own Chromebooks so this will be a great way to incorporate them. I do have a couple of questions…what is the data in columns labeled “rate” and the very last column. I am sorry if it is obvious and I am missing it! Thanks for the idea!

## Tiffany

January 18, 2017 - 10:06 am -Now I guess the follow up to this is “why?” Why would about a third of the way filled result in the greatest bottle-flipping success rate?

## Chris Heddles

October 7, 2016 - 2:43 am -As someone who spans maths and science (physics) teaching (with a background as an engineer) my first instinct is to use this as a great example of a problem that needs simplifying before it can be explored meaningfully. Even if the problem itself yields no useful specific content, this problem-solving approach is worth having in the arsenal.

To a point, I think that Dan’s initial direction is leaning towards pseudocontext because it isn’t helping kids answer the question that they actually care about. They (mostly) don’t care about predicting whether a bottle in flight will land on its end. The truly powerful exploration is helping them work out how to make a successful throw more often. That is more challenging, for sure but is the question that all the bottle-tossers actually care about.

As others have already identified, the biggest complexity here is the sloshing water. It’s relatively unpredictable and nearly impossible to measure what’s going on with simple video analysis. This makes it an ideal candidate for simplification. Tossing an empty bottle is not an option as it behaves too differently but maybe the water can be “fixed” using jelly (jello) or agar.

With this physical simplification, the bottle will act more like the baton model discussed by Dan and his colleague but still feel like the original problem to the students. They can then play with it to see whether it is easier to toss successfully (probably) and then the video analysis becomes much simpler for a rigid object in flight. Different throwing methods should yield more-repeatable behaviour, allowing students to see and test patterns.

This should then become a problem that straddles the divide of “feels interesting and useful to explore” and “easy enough to make progress”. If time and interest permits, students could then see which of the patterns they find are applicable to the water-filled version.

That’s about as far as my musings go – hopefully it triggers some better ideas in others.

## Dan Meyer

October 7, 2016 - 2:48 pm -We talked about your idea around the lunch table at Desmos today, Chris. Everyone thought it was a useful simplification of a model that definitely needs simplifying.

## Paul Jorgens

October 11, 2016 - 5:46 am -Denese – That label wasn’t helpful. It should have said fill ratio or some label to indicate how much of the bottle was filled with water. The next column is the rate of successful bottle flips. It wasn’t clear at all. Thanks for the help.

## Ryan Allen

October 7, 2016 - 4:41 am -I am thinking that you could run an experiment about how much water in the bottle makes for optimum chances of landing a flip. Students could have a water bottle and a pitcher of water and they get 10-20 attempts at each volume. They could graph the results (dot plot perhaps) and my guess is that it would be a normal distribution around the “optimal” volume left in the bottle.

I hope this works, I only had the tab open on my browser that said bottle flipp… and within two minutes of students seeing only that, I had everyone’s attention. I hadn’t even mentioned anything about bottle flipping, I was just watching your 3 acts on a different tab before they came into class! I feel like there is something there, even if it is only for the satisfying of curiosities.

## Xavier

October 7, 2016 - 6:45 am -Perhaps we could simplify the problem (as Polya suggested). What about vertical thrown? What are the conditions for having a “upside-down” or “upside-up” sit of bottle for *vertical* hit?

## Jon Orr

October 7, 2016 - 8:30 am -I’m wondering if fitting a model is where we want to go. Maybe we can use bottle flipping to engage students in some math. Today our class made predictions on what height of water will be ideal. They drew lines on the bottles, found the volume of the water. Measured poured that water in there. They started practicing for the contest next class. Where we’all have time intervals to see who can get the most number of “lands” per minute(rates). Highest rate wins. Then let’s create equations to see how many we should land after ____ minutes. Let’s test that out. What happens if we race and one of us gets a head start? What’s the new equation? Where will we tie? Let’s test this out.

## Dan Meyer

October 7, 2016 - 2:48 pm -What’s the objective you’re pursuing here, Jon? Seems to be experimental design at first. Rates later. Then systems of equations?

## Dan Meyer

October 7, 2016 - 2:58 pm -This is Episode #11,030 of the TV show called “Dan Gets Blown Away by the Commenters.” I’m enormously impressed by your angles on experimental design and hypothesis testing, both of which eluded me when I was first thinking about content objectives here. (Check out Paul Jorgens, whose students flipped 4,000 bottles in a period.)

Thanks, as always, for bringing the heat.

## Jim Foley

October 8, 2016 - 8:39 am -Here’s a different approach: connect students with an Engineering class/teacher, and preferably in a Makerspace.

Can they construct a device that will perfectly land a bottle flip every time?

Difference between applied math and math theory, it seems to me. There will by necessity have to be math of various sorts performed along the way, and of course there’s so many other interesting problems to solve and questions to pose and answer, too: what kind of bottle? How full? What constitutes a “device?” And so on…

## Cathy

October 8, 2016 - 8:44 am -Jim – now THAT would be amazing! Can you imagine a school robotics team/class designing a water-bottle-flipping robot? Our high school students have a robot that shoots hoops… wonder if they’d be up to this!

## Dan Meyer

October 9, 2016 - 2:39 pm -Big fan. I’m very disappointed with my search for “bottle flipping robot.” Someone should make this happen.

## Mary Auger

October 9, 2016 - 12:15 pm -Hello. I teach grade 6 math. Couldn’t we just do trials and have fun while writing our results as fractions, convert them to decimals and finally to percents? We could discuss many of the added skills/concepts above, too. I think we could do this and have a water bottle flipping champ. Relevance makes math fun!

## Mary

October 9, 2016 - 3:21 pm -This isn’t a 3-acts though .. oops. My apologies!

## Dan Meyer

October 9, 2016 - 3:25 pm -No need to apologize, Mary. I was looking for

anykind of mathematical lesson in the bottle flipping, not necessarily a three-act task. I like where you’re going with the bottle flipping contest.## Denise Wright

October 11, 2016 - 9:17 am -My daughter came home with a science lab lesson on bottle flipping they used different amount of water in the bottles as a variable in the experiment. None of her group could complete the flips successfully. In her lab conclusion she wrote about how this was a game of skill and it depended on too many variables to be a reliable scientific experiment. Bah !

## Emily

February 19, 2017 - 3:44 pm -I like where Mary was going with this. You could incorporate the relevance of stats in athletics too. Instead of Field Goal Attempts (FGA) you could have Bottle Flip Attempts (BFA) and instead of Effective Field Goal Percentages (EFG%) we could do an Effective Bottle Flip Percentages (EBF%). This could cover ratios, part to whole relationships, and converting fractions into percentages. You could also have each person find their BFA and EFG% on different water levels which would give them more practice with these skills. Therefore, even though this may be a game of skill, you should still have some good data to conclude one water level being more successful.