Total 58 Posts

## [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.

## Gas Station Ripoff

Here are three gas station pumps. Which ones are trying to rip you off? Can you tell just by looking?

After your students have that debate and share their reasons (expected: “the third is a ripoff because it’s moving faster”) invite your students to collect data for each pump and enter it at Desmos. Here we’re establishing a need for a graphical representation. It may reveal patterns that our eyes can’t detect.

The third act helps clarify the underlying trends. The third pump is spinning faster, but the price and the gas still exist in a proportional relationship. The first pump, meanwhile, pumps less gas per dollar the longer it runs.

I am indebted to William G. McGowan and Sean Berg, whose NCTM 2016 session description included the words “gas pumps have been hacked,” and there went my weekend.

Their description reminded me how important it is to expose students to counter-examples of the relationships they’re studying, protecting against over-generalization. (ie. “Everything is proportional. That’s the chapter we’re in!”) I’m becoming fascinated, in general, by problems that ask students to prove that a mathematical model is broken rather than just apply a model that works.

Featured Comment

I’ve written before about expanding teaching to the “neighborhood” of the special case. If we always show the highlight reel, students never get appreciation for how special and how powerful the ideas are. So I like that this lesson is about finding the non-proportional “ripoff” as it stands out in contrast to the “normal/expected” proportional relationships. (Ironically I would have said that proportionality is the special case, and nonproportionality is the ‘normal’– before thinking about what we expect as consumers. ‘normal’ is all subjective!)

## [3ACTS] Nissan Girl Scout Cookies

Treatment #1

A small rectangular prism measures 7 inches x 2.3 inches x 4.6 inches. How many times could it fit in a larger rectangular prism with a volume of 39.3 cubic feet?

Treatment #2

Nissan is going to stuff the trunk of a Nissan Rogue full of boxes of Girl Scout cookies. Nissan lists the Rogue’s trunk space as 39.3 cubic feet. A box of cookies measures 7 inches x 2.3 inches x 4.6 inches. How many boxes will they fit in the trunk?

Treatment #3

Show this video.

5. Ask for estimates of the capacity of the trunk and the dimensions of the box of cookies.

Hypothesis

Treatment #1 and Treatment #2 are as different from each other as Treatment #2 is from Treatment #3.

A layperson might claim that Treatment #2 has made Treatment #1 real world and relevant to student interests. But the real prize is Treatment #3, which doesn’t just add the world, but changes the work students do in that world, emphasizing formal and informal mathematisation.

“Real world” guarantees us very little if the work isn’t real also.

Design Notes

You can check out the original Act One and Act Three from Nissan.

I deleted this screen from Act One because I wanted students to think about the information that might be useful and to estimate that information. I can always add this information, but I can’t subtract it.

I added a ticker to the end of the video because that’s my house style.

I deleted a bunch of marketing copy because it was kind of corny and because it broke the flow of their awesome stop motion video.

I left the fine-print advisory that you should “never block your view while driving” because the youth are impressionable.

The Goods

[via whoever runs the Bismarck Schools’ Twitter account]

The speedometer in this video is broken.

Can you (or your students!?) fix it? Be careful: there are a couple of interesting ways to get this one wrong.

Also: what would the graph of speed v. time and position v. time look like here?

## The Math Problem That 1,000 Math Teachers Couldn’t Solve

I spent a year working on Dandy Candies with around 1,000 educators.

In my workshops, once I stop learning through a particular problem, either learning about mathematics or mathematics education, I move on to a new problem. In my year with Dandy Candies, there was one question that none of us solved, even in a crowd that included mathematics professors and Presidential teaching awardees. So now I’ll put that question to you.

Here is the setup.

At the start of the task, I ask teachers to eyeball the following four packages. I ask them to decide which package uses the least packaging.

With the problem in this state, without dimensions or other information, lots of questions are available to us that numbers and dimensions will eventually foreclose. Teachers estimate and predict. They wonder how many unit cubes are contained in the packages. They wonder about descriptors like “least” and “packaging.”

After those questions have their moment, I tell the teachers there are 24 unit cubes inside each package. Eventually, teachers calculate that package B has the least surface area, with dimensions 6 x 2 x 2.

We then extend the problem. Is there an even better way to package 24 unit cubes in a rectangular solid than the four I have selected? It turns out there is: 4 x 3 x 2. When pressed to justify why this package is best and none better will be found, many math teachers claim that this package is the “closest to a cube” we can form given integer factors of 24.

The Problem We Never Solved

We then generalize the problem further to any number of candies. I tell them that as the CEO of Dandy Candies (DANdy Candies … get it?!) I want to take any number of candies – 15, 19, 100, 120, 1,000,000, whatever – and use an easy, efficient algorithm to determine the package that uses the least materials.

Two solutions we reject fairly quickly:

2. Write down all the sets of dimensions that multiply to that number.
3. Calculate the packaging for that set of dimensions.
4. Write down the set that uses the least packaging.

And:

2. Have a computer do the previous work.

I need a rule of thumb. A series of steps that are intuitive and quick and that reveal the best package. And we never found one.

Here was an early suggestion.

2. Write down all of its prime factors from least to greatest.
3. If there are three or fewer prime factors, your dimensions are pretty easy to figure out.
4. If there are four or more factors, replace the two smallest factors with their product.
5. Repeat step four until you have just three factors.

This returned 4 x 3 x 2 for 24 unit candies, which is correct. It returned 4 x 5 x 5 for 100 unit candies, which is also correct. For 1,000 unit candies, though, 10 x 10 x 10 is clearly the most cube-like, but this algorithm returned 5 x 8 x 25.

One might think this was pretty dispiriting for workshop attendees. In point of fact it connected all of these attendees to each other across time and location and it connected them to the mathematical practice of “constructing viable arguments” (as the CCSS calls it) and “hypothesis wrecking” (as David Cox calls it).

These teachers would test their algorithms using known information (like 24, 100, and 1,000 above) and once they felt confident, they’d submit their algorithm to the group for critique. The group would critique the algorithm, as would I, and invariably, one algorithm would resist all of our criticism.

That group would name their algorithm (eg. “The Snowball Method” above, soon replaced by “The Rainbow Method”) and I’d take down the email address of one of the group’s members. Then I’d ask the attendees in every other workshop to critique that algorithm.

Once someone successfully critiqued the algorithm – and every single algorithm has been successfully critiqued – we emailed the author and alerted her. Subject line: RIP Your Algorithm.

So now I invite the readers of this blog to do what I and all the teachers I met last year couldn’t do. Write an algorithm and show us how it would work on 24 or another number. Then check out other people’s algorithms and try to wreck them.

Featured Comment

Big ups to Addison for proposing an algorithm and then, several comments later, wrecking it.

2015 Sep 25

We’re all witnessing incredible invention in this thread. To help you test the algorithm you’re about to propose, let me summarize the different counterexamples to different rules found so far.

20 should return 5 x 2 x 2
26 should return 13 x 2 x 1
28 should return 2 x 2 x 7
68 should return 2 x 2 x 17
222 should return 37 x 3 x 2
544 should return 4 x 8 x 17
720 should return 8 x 9 x 10
747 should return 3 x 3 x 83
16,807 should give 49 x 49 x 7
54,432 should return 36 x 36 x 42
74,634 should give 6 x 7 x 1777