What do you notice?

My first observations included, “Hey! Sodas cost $1.50.”

What do you wonder?

The first thing I wondered was, “How many coin combos are possible?”

And immediately after that, “Are pennies allowed?” (Check the picture for the answer.)

And with my teacher hat on… “What strategies would other students and teachers use to answer the “how many combos” question?

Sequel #1

Then I noticed something else on the machine:

How many coin combinations are possible now? Do you think it will be…

• More?
• Double?
• Less?
• Half?
• The same?

Sequel #2

The “use exact change” light isn’t lit. How does that affect our original answer(s)?

Post Script

True, true, this is an age-old problem. Maybe even a tired problem. But I still love it, especially in this visual form. Also, I’ve never noticed the coin icons on the payment panel before. That’s pretty nifty if you ask me.

Model 3 design sketches pic.twitter.com/P5ucOBRUZ7

— Elon Musk (@elonmusk) April 3, 2016

With a little help from Skitch, let’s turn this scenario into a math problem.

Throw that image on the screen and ask students:

• How many orders in 24 hours?
• What info would help you figure that out?

Ideally, after making some predictions (and writing them down!) students make a request for average price per vehicle, and you deliver:

When they’re ready for the reveal…

Model 3 orders at 180,000 in 24 hours. Selling price w avg option mix prob $42k, so ~$7.5B in a day. Future of electric cars looking bright!

— Elon Musk (@elonmusk) April 1, 2016

Sequel #1

Let’s see what else we can do with this…

Definitely going to need to rethink production planning…

— Elon Musk (@elonmusk) April 1, 2016

Now 232k orders

— Elon Musk (@elonmusk) April 2, 2016

253k as of 7am this morning

— Elon Musk (@elonmusk) April 2, 2016

276k Model 3 orders by end of Sat

— Elon Musk (@elonmusk) April 3, 2016

Pre orders began on Thursday, March 31. Tesla promised a numbers update on Wednesday, April 6. How many pre orders do you think will have been placed by then?

• Make a prediction.
• Use math to find a more accurate answer.
• Explain your thinking.

I’ll drop an update here once we know the answer.

Update!

Over 325k cars or ~$14B in preorders in first week. Only 5% ordered max of two, suggesting low levels of speculation.

— Elon Musk (@elonmusk) April 7, 2016

Sequel #2

Tesla aims to sell 500,000 cars per year by 2020. Consider this comment from CEO Elon Musk:

Based on the information in the comment above:

• Do you think Tesla will meet its 2020 goal?
• What sort of year-over-year percentage growth will this require?
• If you think they’ll miss the mark… by how much?
• If you think they’ll surpass the target… by how much?

Invitation

Drop an answer to one of the questions above in the comments below. Or, share another idea or two for how this Model 3 craze could play out in a math classroom.

Let’s say you’re one of these folks, and your students are now rocking this sweet set of challenges. Now what?!

Well, for one thing, don’t stop! These are rich enough problems to keep bringing them before your students. (In fact, the real fun begins when we break out quadratics, including my personal favorite: patterns involving triangular and other figurate numbers.)

A Means To Another End

But I would offer that Visual Patterns are not only an end in themselves, but also a means to another end.

• “An end in themselves” because, let’s face it, they’re awesome all on their own.
• “A means to another end” because they provide students with experience in approaching mathematics through multiple representations. And this visual-verbal-numerical-graphical-algebraic tag team effort translates to new scenarios far more powerfully than a single representation would.

This last point was on full display this morning in Math B (eighth grade) as my students worked on Dan Meyer’s High School Graduation task.

Here’s a sample of how things went down:

Look familiar? I sure hope so.

After several rounds of Visual Patterns, students have developed a framework for translating a text-dense, potentially-intimidating task into something they can explore, something they can understand. In fact, once students had the table of values (which was admittedly a team effort), they were off to the races.

While students in past years were able to answer some of the numerical questions (when did the name-reading begin/end), they typically struggled to do anything more than that, and were at a loss when it came to writing an equation to model the scenario.

A Well Worn Path

So why were my students this year able to hack it? Because we’ve worn that visual-verbal-numerical-graphical-algebraic path so well in just a couple of weeks that moving from one representation to the next—and turning back to make connections among various forms—is becoming second nature.

And while there’s more than one way to foster this kind of connected thinking, I’ve found Visual Patterns to be among the most engaging, powerful, and effective.

Postscript

As you can tell, I’ve had fun with Visual Patterns this week and last. I have one more post in me on this topic, then I promise I’ll shift my rambling to something else.

Which is closer to 1? Does it depend on n? Explain. (n-1)/n or n/(n-1) Inspired by Q2 on this: http://t.co/zUp670WyqS

— Michael Fenton (@mjfenton) April 23, 2014

Just in case Twitter messes up the formatting on whatever device you’re reading on, here’s the question again, as well as my challenge to you and a solution of my own:

The Question

Which is closer to 1? Does it depend on n? Explain.

My Challenge

I’m curious to know how others reason through and/or visualize this problem. If you find yourself with 5-10 spare minutes this week, scribble out an approach/solution/representation (or two) on a napkin, take a photo, and drop a link to the image in the comments. I’ll update this little ditty on Friday by embedding images in the post.

Pictures, words, algebra, numbers, proof with words, proof without words… It’s all fair game here.

A Solution

Don’t peek until you’ve given it a go yourself (and hopefully shared your solution in the comments), but here’s my visual argument, sans words. (As in, “I’m not providing any words.” Let me know if you think it needs a comment to clarify.)

With that in mind, I’ve been frequenting Geoff Krall’s fantastic curriculum maps rather, um, frequently as of late. My latest joy-filled find: Don Steward’s Complete the Quadrilateral (via Fawn Nguyen).

Inspired by these posts—and a separate domino-style activity shared by Mike Chamberlain at a common core workshop last semester—I spent a good portion of my three-day weekend turning the complete-the-quadrilateral task into 21 dominos of self-checking goodness.

Set Up

Teachers, print this…

…and use these…

…to get a stack of these…

…which you’ll shuffle and put in one of these…

…and hand out to groups of these:

How to Play

Working in groups of 2-4, students will:

• Join dots to complete the indicated quadrilateral on each domino (Multiple possibilities? Maximize the area!)
• Determine the area and perimeter of each figure
• Connect the domino chain, from start to finish

Just to clarify the whole “domino chain” bit, each domino contains one answer on the left (to a previous card’s question) and one question on the right (with an answer to follow on the next card). I’m fairly certain that last sentence doesn’t make any sense, so… “Hey-look-a-picture!”

Solutions

If you need a hand with completing the quadrilaterals, check out the source, or even the original source. (Note that I removed four problems—Fawn’s #9-11 and 14, due to identical figures mucking up the uniqueness of my domino chain—and that my final problem matches Don’s, not Fawn’s.) For help with the area, perimeter, and domino sequence, note that my handouts contain the dominos in order (reading one column at a time, from top to bottom):

Variations

The teachers in my geometry class served as guinea pigs for this revamped version of the activity, and after a debriefing discussion, I think it would be wise for me to create a few variations to allow for more students to participate without becoming too overwhelmed/frustrated. (Translation: The task took a very long time, and with the four required steps—complete the quadrilateral, find the area, find the perimeter, match the domino—I’m afraid student interest will fizzle out as frustration grows.)

Here are several I’m wondering about (including one I’ve already added):

• Split the deck into two groups of 10 dominos (Set A, a bit easier; Set B, more challenging), allowing for three levels of difficulty to differentiate between or within classes (Set A only, Set B only, Sets A and B)
• Reduce the deck to 10 dominos, remove perimeter from consideration (if the self-checking domino action is to stay in tact, I think I should make sure I have 10 distinct areas)
• Same as #2, but remove area from consideration (not sure if I would then make the rules “maximize perimeter” or if that’s just too awkward)

Request for Feedback

There’s a lingering fear in my mind that I took a great activity and ruined it. If you think that’s the case (or not the case), I’d love to hear why in the comments. Also, if you have any other suggestions for tweaking the length of the activity without losing its original challenge or appeal, let me know.

I received a reply, but I’m more interested in a tangential question @NatBanting asked in the middle of the exchange:

@mjfenton Curious: what makes a task “rich” in your eyes? I’m trying to synthesize my view.

— Nat Banting (@NatBanting) August 16, 2013

I’ve been throwing that phrase (“rich tasks”) around a lot more lately, in tweets, blog posts, workshops, and conversations at school and at home (my wife is either amazingly longsuffering or genuinely interested—maybe both?—when it comes to talking about math education). So what exactly is a “rich task”? What do other people have in mind when they use the phrase? What do I mean when I use it?

I told Nat that I’m still working that out in my own mind, but I think it’s time to clarify what I mean, if for no other reason than to have a better sense of what I’m looking for when I try to find, adapt, and/or create rich tasks for my own students.

Here’s What I Do Mean

A rich mathematical task is one that…

1. Has a low floor and a high ceiling. The first of many ideas I’ve stolen from others, this one from so many sources I don’t even know who to credit anymore. Most recently, Dan Meyer has me thinking about this in his Makeover Monday series. The bottom line: everyone can start, no one can truly say they’ve exhausted the problem’s potential (at least not in a 50-minute period).
2. Has multiple entry points, invites use of multiple representations. Student A starts by exploring numerically, Student B begins by investigating graphically, Student C jumps in by reasoning algebraically, and I don’t have to tell two of them that their approach is a dead end because—even if they don’t always make it—there is fruitful territory a little further down the path in any of their approaches.
3. Has multiple solution paths, provides opportunity for rich discussion. If there’s only one way to solve the task students lose out on the rich discussion of making connections between various approaches and teachers lose the opportunity to build a mathematically coherent, concrete-to-abstract storyline as they orchestrate these discussions.
4. Integrates multiple topics. I owe a lot of what I’m thinking here to a single word Daniel Schneider used in a post about assessment. After my initial foray into standards based grading left me dissatisfied with an overly fractured curriculum, I’m now placing a high priority (philosophically, at least) on tasks and assessments that bring multiple topics together. A rich task, in my estimation, should demand that students wrestle with multiple topics from multiple domains (if I can use the term in the CCSSM sense).
5. Engages student interest, is mathematically/cognitively challenging. I’m a little mixed up here, because I believe engaging students’ interest is massively important, but I’m not ready to throw away tasks that fail to generate buzz among students if I know they nevertheless provide great opportunities for exploration and discussion.

Here’s What I Don’t Mean

To clarify what I think a rich task is, I’ll share a few thoughts on what I think a rich task is not:

1. A well-crafted, but constrained guided-discovery activity. I value this kind of activity, but when students are guided along a specific path to a specific goal, it pushes the lesson into a different category for me.
2. A thoughtfully constructed lecture or an engaging presentation. If the instructor is doing the heavy lifting during class, I would say the students are not engaged in a rich mathematical task. I’m not opposed to heavy lifting, especially in preparation outside of class, but students need to play an active, central role in exploring/solving/reporting if I’m going to use the “rich task” label for an activity.
3. A challenging problem for which students already have a tried-and-true method. I have a large stack of started-but-not-finished books, some related to math and education, others not. George Polya’s How To Solve It is on the list, though I’ve read enough of it to be provoked and inspired, particularly by the distinction Polya provides between a problem (solution/method not known) and an exercise (solution/method already known).

I’ll Close with a Link…

Incidentally, after typing out the post, I Googled “what is a rich mathematical task” and found this.

…And an Invitation

I would love to hear what you think are the key characteristics of a rich mathematical task. Drop a line in the comments or send me a line on Twitter.