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## [Updated] Will It Hit The Hoop?

Six years ago, I released a lesson called Will It Hit The Hoop? that broke the math education Internet. (Not a big brag. It was a much smaller Internet back then.)

I think the core concept still works. First, students predict whether or not a shot goes in the hoop based on an image and intuition alone. Then they analyze the shot using quadratic modeling and update their prediction. Then they see the answer. For most students, quadratic modeling beats their intuition.

The technology was a chore, though. Teachers had to juggle two dozen different files and distribute some of them to students. I remember loading seven Geogebra files onto student laptops using a thumb drive. That was 2010, a more innocent time.

So here’s a version I made for the Desmos Activity Builder which you’re welcome to use. It preserves the core concept and streamlines the technology. All students need is a browser and a class code.

Six year older and maybe a couple of years wiser, I decided to add a new element. I wanted students to understand that linears are a powerful model but that power has limits. I wanted students to understand that the context dictates the model.

So I now ask students to model this data with a linear equation.

Then I show students where the data came from and ask them to describe the implications of their linear model. (A: Their linear ball goes onwards and upwards forever.)

And then we introduce parabolas.

## [WCYDWT] Will It Hit The Hoop?

Is he going to make it? Can you draw me the path of a shot that will make it? That will miss it?

How about now? Can you draw me the path of a shot that will make it? That will miss it?

How about now? Can you draw me the path of a shot that will make it? That will miss it?

A little more obvious, isn’t it? And like that, we’ve derived illustrated the fact that, while one point is enough to define a point, and while two points are enough to define a line, you need three points to define a parabola.

Here are seven versions of the same problem. Each one contains:

1. the half video, for asking the question,
2. the half photo, for giving the students something to work with,
3. the geogebra file, one use for the half photo, featuring a dynamic parabola in vertex form.
4. the full video, for showing the answer,

Attachments

## #BottleFlipping & the Lessons You Throw Back

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.

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 x2 + 6x + 8 bottles in x + 4 minutes. 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

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.

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.

## Starter Pack

Hi! This blog is over ten years old and has hosted 1,500 posts during its first ten years. Let me pull out some highlights.

I taught high school math for six years and shared much of my curriculum online. My most popular lessons included Graphing Stories, Stacking Cups, Will It Hit The Hoop?, and The Feltron Project. I created a style of real-world lesson plan called Three-Act Math, which a bunch of educators enjoy and a bunch of educators find totally weird. (The former group has a website clubhouse with 30,000 members.) I was very skeptical about the value of homework and I assessed students using a process that is known as Standards-Based Grading.

During my last year in the classroom, two events transformed my career. First, I was laid off from classroom teaching. Second, I gave a talk about math education that went online and has been viewed a couple of million times. (You can watch other talks I’ve given also.) Those events put me on a path to a) speak publicly about instructional change in math education in forty-nine states and four continents, and b) earn a doctorate studying with some of the best math education researchers around.

I now work for a math edtech startup called Desmos where we try to help students learn math and love learning math. I have opinions about math education technology, most of which are pessimistic, though occasionally I’m exuberant about its possibilities.

Miscellaneous:

## Other People’s Problems

Draw a shape on squared paper and plot a point to show its perimeter and area. Which points on the grid represent squares, rectangles, etc. Draw a shape that may be represented by the point (4, 12) or (12, 4). Find all the “impossible” points.

I like this problem a lot (I’ll spoil some of the fun in the comments) even though it’s fundamentally dissimilar to most of the problems I write about here.

One of the best parts about my working life right now, including grad school and my work with publishers, is my daily exposure to the vast set of answers to the question, “What makes for good math education?” That exposure has helped me find the edges of the usefulness of my own answer to that question, one which I’ve been developing for years, and for that I’m grateful. There’s nothing more pathetic than walking around convinced you’ve found the answer, forcing yourself to perceive other answers as either inferior to or derivative of your own, missing out on the bigness of the work of math education, on its richness and difficulty. Realizing the smallness of my own work relative to the whole has made me a much happier worker. That’s the odd thing.

So I’m grateful to instructors like Labaree and Stevens who urged us all to quit trying to solve the problem and focus first on describing the domain of the problem and its range of solutions. With that focus, I started to see fundamental similarities between this problem above and other problems I like.

1. They all reveal their constraints quickly and clearly. They’re brief. This one gives you a compelling task in a handful of words. You don’t have to meander through four or five steps to understand its point. See also: “How many pennies is that?”; “Will it hit the hoop?”; “How long will it take him to go up the down escalator?” Also notice how none of the questions Bowen Kerins poses in this comment would exceed the 140-character limit on a tweet.
2. They all use images to express their tasks concisely and to perplex the learner. That image alone would be enough to provoke some productive mathematical questions, even if none of them would necessarily be as productive initially as “Where are all the impossible points?” Perplexing images have that power. Also, imagine expressing that task using words alone. How much longer would that postpone the student’s encounter with the point of the task?
3. They all feature low threat levels and low barriers to entry. The task above allows the student to come up with a hypothesis and test it with new data instantly, from scratch, using nothing more than her mind, a piece of paper, and a pencil. The task allows a teacher to encounter a struggling student and say to her, “Would you just draw a rectangle for me? Any rectangle.” and start there. Once the student has graphed that rectangle on the plane the teacher can say, “Would you do that with five more rectangles and let me know what you notice, if anything?” The student can basically generate her own second act, which is better than most of the problems I design, which often require the advance knowledge of a certain mathematical model, without which you’re basically screwed.
4. They all ask you to understand the math forwards and backwards, inside and out. First it asks, “Given a shape, where’s the point?” Later it asks, “Given this point, what’s the shape?” This reversal of the question and the answer encourages students to understand their own thinking comprehensively.

Let me close with a tweet from David Cox, a math teacher who also gives a damn about design.

Know what tasks you like. Know why you like them. Know the similarities between tasks you like. And, special notes to myself:

1. Know the research that describes those tasks.
2. Keep a loose grip on your own sack of solutions.

BTW: Here’s an e-mail Alan Schoenfeld sent our problem-solving class describing the aesthetic of problems he likes.

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