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.

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.

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.

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.

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.

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.

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?

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 !