Reason and Wonder

Where’s the emphasis? Right where it belongs.

One of the things I love about Desmos is that it allows students and teachers to keep the focus where it should be. Working on a linear approximation problem in Calculus? It’s easy to get caught up in algebraic and numerical details and lose sight of the big (somewhat amazing) picture: We can approximate a crazy curve with a simple line! (Provided we stay in the neighborhood, of course.)

And if you make a mistake, it can be an absolute bear to track it down. Is your issue differentiation? Evaluating a function? Or is your weak spot related to what’s happening visually in linear approximation?

Here’s a problem from last year’s AP Calculus review workbook:

There’s a lot of great work on the page.

Take the derivative? Check.
Find the slope of the tangent line? Check?
Find the equation of the tangent line? Check.

But that’s where things fall apart.

Now, imagine you’re a calculus student. You’ve been hammering away at this thing for several minutes. Maybe you don’t even remember what the problem’s asking for in the first place. You get to this point, you scan the original problem, see an input value of 4.2, and presto-whammo, plug it in and out comes 0.4. “Great news, everyone! That’s on the list! Well done, folks. On to the next problem!”

Only, that answer is entirely wrong. So how do you debrief this problem with a student, small group, or class, so they can see the source of the error? For problems with even a sliver of something graphical, Desmos has become my go-to tool for helping students find and fix their errors.

Here’s what I built with a pair of students last year (with a link to the live graph here):

Students can’t use Desmos on the AP exam (for now, anyway), so I’m not trying to permanently sidestep what they ultimately must be able to do sans technology (or with a device from that “other” graphing calculator company). But what we can do in class with Desmos is build a better visual/conceptual sense of what’s happening in this problem so they’ll be more prepared for something similar in the future.

Here’s a short list of what this Desmos graph did for us in this scenario:

We offloaded the algebraic and numerical work of finding the derivative (and evaluating it for a particular x-value) and built the tangent line in a matter of seconds (rather than minutes). Mentally, we’re still fresh, and ready to focus on what the problem is really asking us to do: compare the function and the tangent line.
We gave things specific names so we could call on them in our time of need. Okay, that may sound a little dramatic. But think about why we even bother with function notation. Why give a function a name? Well, why did your parents give you a name? So they could call on you! (“Alfred! Get down here and pick up your comic books!”) So then, why do we give functions names? Because function notation is on the Chapter 8 test in Algebra 2? No! We give functions names so we can call on them. So they can do our bidding. If you don’t name it, it’s difficult to put a function to work for you. Give it a name? Now our wish is its command. (It sounds a little bit like we’re going to take over the world, with math as our trusty sidekick.)
We gave things specific names so we could keep clear in our own minds the various moving pieces in the problem. There’s a function. We called that f. There’s a tangent line. We called that l (or t, or whatever). As we approach the end of the problem, and we start looking for the error, it’s easier to avoid simply evaluating f(4.2) or l(4.2), because we know we’re dealing with both f and and l. (How could we forget?! We named them! We practically gave birth to them.)
We visualized the error with a beautiful little orange bar, and in doing so imprinted on our minds (for future problems) what linear approximation error looks like.
We dropped a slider in so we could answer a hundred related problems in a matter of seconds, further clarifying for the confused student (or teacher; I was terrified of linear approximation my first two years of teaching Calculus) what the problem is really about, and all with a nifty, dynamic burst of compare-and-contrast.

Do you need Desmos to teach this stuff? Maybe not. But given the option, I’ll use it every time. Last year in my classroom, we got a lot more out of a five minute Desmos-supported conversation than we did in a whole day of business-as-usual notes and examples.

So this year? We went straight to Desmos and put the emphasis right where it belongs.

P.S. I love GIFs.

A Good Day in Precalculus

After struggling a bit earlier in the week, it was nice to end with a positive experience in Precalculus on Friday. There’s nothing profound in the sentences below, but I still think it’s worth writing down, if for no other reason than increasing the likelihood that I recall some of this as I plan future lessons.

Context (for the Year)

We made the move to block schedule this year. After years of being opposed to the idea, I recently softened my stance and even began to think a block schedule approach might be helpful. I’m still learning the ropes, but the “early returns” in my own classroom (from me as well as my students) are positive.

Context (for the week)

We’ve been working on trigonometric properties and identities. I had a rather disappointing experience in Precalculus on Wednesday, and though I can’t recall the details of last Monday, I’m pretty sure it was far from amazing.

The Daily Plan

As students shuffle into class each day, I throw “The Daily Plan” on the projector. Friday’s looked like this:

Mini Exploration

Some of what I disliked about my lessons from earlier in the week related to the all-eyes-on-me approach I took. So I decided to start Friday with a miniature exploration. Again, nothing profound, but something that would:

Provide students with a brief review of the properties they would need to have at-the-ready for today
Give each group (or pair) of students an opportunity to proceed as quickly/slowly as they needed to
Challenge students to express their thinking in writing and in discussion
Allow me to wander through the room, checking progress, lingering with students who needed extra support (which I tried to supply via questions, not statements)
Require students to do some individual and small group thinking and wrestling before our whole class discussion/recap

Here’s what the handout looked like:

And here’s a link to the two-page PDF (in all of its non-glory).

Reordering Task

After debriefing the mini exploration (which included a whole-class conversation and a “puppet volunteers” work-through of Problem 6), we moved on to a reordering task. It’s an idea I had been thinking about for a few months (years?), but one I had never put into action. After using this on Friday, I think I’m sold on its quality—at least for some problem types.

Here’s a look at the student page:

And a link to the handout, for closer inspection.

Properties Quiz

I should have done this each day for the last three days, but my moments of genius are few and far between. On several occasions in class we’ve described the properties as “puzzle pieces” and proving identities as “solving a puzzle.” If you don’t have the basic properties memorized, it’s like trying to do a puzzle with pieces missing. Apparently that metaphor was insufficiently inspiring, as many of my students spent little to no additional time at home committing the reciprocal, quotient, and Pythagorean properties to memory. And this lack of recall presents a problem for the work at hand.

An older version of me would have tried to remedy that problem by lecturing the class about the importance of blah blah blah. On Friday, I decided to skip over that part and instead gave students a few minutes of class time to work on committing these bad boys to memory. “5/5/5” on the daily agenda meant five minutes of silent and individual study, five minutes of partners quizzing one another verbally and/or in writing, and five minutes of do-the-best-you-can-on-your-own quizzing. I may have only given them 3 or 4 minutes for each stage, but the results were great. Most students now have the “puzzle pieces” in hand, and those that do not know exactly where their weaknesses lie.

Visual Patterns

In a block schedule setting, I’m finding that focusing on trig properties and identities for the entire class period is just too much of the same thing. To mix things up—and to plant some seeds for an upcoming functions- and graphing-heavy chapter—we worked through a Visual Pattern (our first one in months). As an aside, I wish I did a better job of sticking with my start-of-the-year resolutions (e.g., “Visual Patterns will be a regular feature in such-and-such class this year.”) I suppose it’s not too late to bring it back into the mix…

Personal Takeaway

If my second sentence in this post is going to be true, I need to nail down why I think Friday was better than the other days in Precalculus last week. I think it boils down to two things:

In designing the lesson, I endeavored to make my students the key do-ers throughout the lesson (whether via thinking, writing, arguing, sorting, explaining, or defending)
In an effort to maintain student focus and fight off the feeling of the class “dragging on and on,” I provided students with several distinct (though still related) tasks

As I create my next set of daily plans, I’ll try to keep these little victories from Friday in mind.

Teaching vs Student-ing

(In all likelihood, this will be the shortest blog post of my blog-posting career.)

I like being a teacher of high school mathematics more than I enjoyed being a student of high school mathematics. What does that say about the nature of the classroom as I’ve experienced it on either side of the desk? And what does it say about the nature of the classroom I should endeavor to create?

Try, Try Again

Monday wasn’t the greatest of teaching days. But in teaching we have a thousand opportunities to try again. If at first you don’t succeed… Write a long blog post reflecting on what went wrong, and then go back to the drawing board.

Tomorrow’s Game Plan

When I taught this lesson three years ago, it lasted 40:01. I’m not kidding. (By the way, that link is a whole other coming-over-from-the-dark-side post that I don’t have time for right now…) Two years ago and last year? Basically the exact same lesson. Six problems. Work through ’em as a class. Everyone stick together. Eyes on me. And so on. Bell rings. Head home. Practice. Try not to cry.

A quick preview:

And if you’re curious, a closer look, compliments of your friend the PDF.

If you read last night’s post, possibly the only thing that I made clear was my desire to shift away from “all eyes on me” instruction where it makes sense. (Which, as it happens, is in quite a few places.) With that in mind, here’s my plan for tomorrow:

Use Desmos, Keynote, and MathType to build a set of cards containing: graphs (12), integral expressions (12), and directions (1)
Print, slice, and drop in Ziploc sandwich bags. (Hooray for Costco!)
Distribute one bag to each group (~4 students).
Give them about 5 minutes to sort. Then 3 minutes to describe in writing their thinking. (“How’d you match each graph with its integral expression?”) Then 2 minutes to debrief as a class.

Now, what to do with that extra half hour…? I’m still working on that bit. In the meantime, I’ll share some of the cards. Enjoy!

Wish Me Luck!

I’ll check back in tomorrow with a brief report on whether tomorrow is a success, or another borderline failure.

The Goods

All of the resources I created are available here.

The script is dead. Long live the prompt!

Period 2

Well that wasn’t any good.

Here’s a picture of the main part of my Calculus lesson from today:

One example. That’s it. The central element of my entire lesson was… one example. And not even a task. Just an example. A watch-me-as-I-carefully-walk-through-every-step-of-this-sucker and make-sure-you’re-on-guard-in-case-I-ask-you-any-leading-questions example. Oh my.

Granted, there was more after that example, just not a great deal more. And none of it great. We actually wrapped up the example, started the “next thing,” and quickly abandoned ship after some “Show me on your fingers how you’re doing” feedback from students revealed that all was not well (not by a long shot).

This changing of gears led to a somewhat-useful last 10 minutes of class (thanks in part to Desmos), which in turn led me to wonder: What did the end of class have that the start of class was missing? For one thing, after seeing the first part flop I had to clarify in my mind the bottom line goal for the lesson. I settled on this: If students left my room with the ability to translate verbal and algebraic problem descriptions into graphs, and those graphs into integral expressions, we’d be golden.

Period 3

Well that wasn’t any good, either.

I went overboard Sunday evening creating a slide deck that (I thought) would help me lead students through a carefully crafted conversation on the topic of trigonometric properties and identities. The slide deck was slick as all get out. But the lesson was boring. You could see it on their little compliant faces. They didn’t even complain. They just sat there. Copying a property or two here, sketching a graph or two there, dutifully jotting down an observation or two when I asked, and so on, for the better part of half an hour. Argh!

Inspiration ≠ Incorporation

At this point I have no idea if I’ve painted a clear picture of what took place today in my classroom. Even less so what’s going on in my head right now. If you’re feeling uncomfortable, abandon ship now, ’cause this is about to get even less coherent.

You see, I’ve been struggling with a number of thoughts over the past few months. To name a few:

Sam Shah is awesome. As some teachers call out worksheets on Twitter, Sam is busy packing one aha-moment after another into carefully crafted mathematical adventures for his students, all on that oldie-but-goodie 8.5 by 11 format. I want to bring these kinds of things into my own classroom!
Dan Meyer is awesome. As some teachers give the all-call for real world applications over all else, Dan is blasting through pseudocontext and drawing attention to the heart of the matter (engagement), regardless of whether a task is labeled “real world” or “fake world.” I want to make genuine student engagement a central (and regular) feature of my teaching practice!
Karim Ani and Team Mathalicious are awesome. As some teachers are firing up Khan Academy accounts and printing off hundred-of-the-same worksheets from Google search results, Karim and the crew are laying siege to the notion that math is uninteresting, unengaging, unimportant, or unworthy of our attention. I want to bring Mathalicious-style conversations into my own lessons!
Jonathan Claydon is awesome. Seriously, have you seen the pictures of his class over at Infinite Sums? His students do stuff. All the time. They’re active, they’re involved. I don’t care what the lesson is, they’re at the heart of it. I want this to be true of my own students!

And I haven’t even mentioned Stadel, Nguyen, Kaplinsky, Vaudrey, Stevens… The list goes on. And while my inspiration grows, my frustration does too, because I can’t find a way to incorporate all of this awesome into a coherent whole in my own teaching world.

That’s really the issue. And I’m just using a frustrating Monday morning to process what I’ve been struggling with for months in the hope that I can make some sense of it all.

The Challenge

(Still with me? Awesome. Hang in there, we’re almost done.)

So let me try to name my struggle, clearly and succinctly, so I can go about the task of moving beyond it. Here goes:

For the past 500 days I’ve been inspired daily (literally, every single day) by what I see in the MTBoS. At the same time, I have yet to find a way to weave that inspiration into my own practice in a coherent, compatible way.

The Way Forward

I don’t know the entire solution, but I know it starts with this: I’m done designing scripted lessons, those awful handouts with eleven-teen examples that we’ll walk through. Together. All of us. At the same pace. (I’ve created enough of those to last a lifetime, and they don’t develop in students any of what I’m after.) I’m done drawing up anything where I can predict with 99%+ accuracy what the students will be thinking at any given point. I’m done throwing together slide decks that demand students focus on the same thing at the same time. I’m done throttling their insights, their noticings, and their wonderings by squeezing out of them a certain style of efficiency that is anything but effective.

Instead, I’ll be spending my time infusing worksheets with aha-moments and did-you-just-see-that mathematical surprises. I’ll be on the lookout for visuals that mess with students minds and spark dozens of questions they actually want to answer. And I’ll expand my teaching skillset so that I can navigate the waters of a class full of students exploring different problems inspired by the same visual. I’ll take risks, push the boundaries of what I’m currently capable of, and through it all develop my ability to orchestrate rich mathematical discussions, whether they’re centered around a thought-journey disguised as a worksheet, a rich and who-cares-if-it-has-no-context problem, an engaging and demanding task, or an honest-to-goodness real-world scenario. And whatever I do, I’ll make sure my students are at the center of it.

In short, I’m done with trying to script their thinking. I’m going all in with prompting them to think. “The script is dead. Long live the prompt!”

CMC South 2014 Recap: The Sessions

Last weekend turned out to be one of my favorite weekends of my teaching career. I haven’t exactly been at this for decades, but 11 years is no short span either. The weekend was that enjoyable, at least for me. If I’m going to do any justice to the recap, I’ll need to split it into two parts. So much of what was wonderful about last weekend was the sessions, and so much of what was wonderful about last weekend had nothing to do with the sessions. I suppose that means I have the breakdown for my recap:

Part 1: The Sessions

Part 2: Everything Else

Onward to the sessions-focused recap!

Session 1

La Cucina Matematica: Free Tools for your Math Kitchen
(Matt Vaudrey, John Stevens • Friday, 8:30 am)

After cramming down some continental breakfast (thanks, Comfort Inn!) and making the trek from our hotel to the Hard Rock Hotel’s basement, I sat down for a mathematical treat: John Stevens and Matt Vaudrey’s 90-minute version of La Cucina Matematica. They battled through some technical difficulties at the start and soon found their rhythm, taking turns running the show, moving the participants through a series of mathematical appetizers, entrees, sides, and desserts. If you don’t follow their blogs (here and here) you’re missing out.

My highlight from John’s segments was seeing how he facilitated a couple of Would You Rather Math discussions. I’ve been a fan of these since last winter (even contributing a prompt or two myself), but it was a delight to see John in action, leading teachers through a couple of mini-discussions, regularly zooming out to discuss the moves he was making and the opportunities inherent in the format so we might take these tools back into our own classrooms.

My highlight from Matt’s segments? Easy: Mullets. But maybe not in the sense that you’re thinking. I was already familiar with his mullet ratio work (as well as its popularity with teachers and students), so the highlight for me didn’t occur until about 20 minutes into the mullet conversation. It was then, near the end of this segment, that I realized how much work Matt has put into weaving this ridiculously-engaging context into a rich sequence of mathematical topics. I used to think of the mullet ratio lesson as a great one or two day task. Now it looks more like a swiss army knife scenario, useful in developing maybe a dozen key ideas in middle school mathematics. Well done.

Session 2

Reasoning, Discovering, and Critiquing with Networked Tasks
(Eli Luberoff • Friday, 10:00 am)

Eli Luberoff is a legend. (He’s like the Madison Bumgarner of ed tech startups. Minus the snot rockets. I think.) And Desmos is simply a math teacher’s dream. 90 minutes sitting in a room listening to Eli talk about Desmos turns out to be pretty fantastic as well. Highlights from the session? Three come to mind:

Regressions. Yep, it’s live. In fact, Eli and Team Desmos launched it around 4 pm the day before the conference began. Someone give that man a raise!
Desmos Activities. I’ve tinkered with all of these in the past, including briefly highlighting them in various workshops and conference sessions, but I’ve only tried one with my own students. Seeing a whole “class” in action at once was magical, and inspired me to bring more of these into my classroom in the near future.
Because I had been tinkering with the regressions preview throughout the week as I prepped for my own CMC South sessions (one of which featured Desmos), Eli dropped my name during his session. Something along the lines of, “He’s on there practically 24 hours a day!” All I could think was, “He said my name! He said my name! Okay, breathe… Calm down… But he said my name!”

Sessions 3-5

Desmos: Infinite Graphing Power on Every Device
(Me • Friday, 1:30 pm)

Turning Students Into Posers + Solvers
(Me • Friday, 3:30 pm)

Desmos: Infinite Graphing Power on Every Device
(Me • Saturday, 8:30 am)

I was pretty excited for these sessions, my first conference presentations since joining the MTBoS. I plan on writing more detailed recaps in the near future. For now I’ll just say that I had a blast, and feedback was super positive.

Session 6

Offering a Thought-Provoking Experience Through Math
(Edward Burger • Saturday, 10:30 am)

I packed up from my last session, still riding high after showing off Desmos to a small-but-packed-room of teachers. I wasn’t quite sure what to attend next (I confess to not doing much homework in regards to which sessions to attend, as most of my time was spent in last-minute slide deck prep). After narrowing it down to two options, I saw Fawn Nguyen on her way to one of my two choices: Edward Burger’s session on “Offering a Thought-Provoking Experience Through Math.” I may have been one of only a few people in the room who hadn’t heard of Dr. Burger before, but after about 15 minutes I realized I was in for a treat. I gave the whole live-tweeting a session thing a try, so rather than recount the highlights anew, I’ll just drop a few of my favorite quotations:

"Try to remove the word *problem* from your lexicon. Replace with: Challenge, puzzle, riddle, conundrum, exercise.” – Burger #CMCS14

— Michael Fenton (@mjfenton) October 25, 2014

"Understanding is not binary. It runs along a spectrum. Where are you today? How can you go deeper tomorrow?” via @ebb663 #CMCS14

— Michael Fenton (@mjfenton) October 25, 2014

“Failure is a mistake that will lead to a new insight” via @ebb663 #CMCS14

— Michael Fenton (@mjfenton) October 25, 2014

Idea: Ss solve exercise, make one intentional mistake, trade papers, find peer’s mistake. (They usually find many!) via @ebb663 #CMCS14

— Michael Fenton (@mjfenton) October 25, 2014

From @ebb663: Engineering curiosity? Never ask: “Do you have any questions?” Presupposes there might not be any. #CMCS14 (1/2)

— Michael Fenton (@mjfenton) October 25, 2014

One from Matt Vaudrey:

When @ebb663 got me at #CMCS14: In 20 years, Ss will forget formulas. What will they take from your classroom to make them their best self?

— Matt Vaudrey (@MrVaudrey) October 25, 2014

Paraphrased Q from @ebb663 at #CMCS14: “In my own classroom, does the math serve our thinking, or does our thinking serve the math?"

— Michael Fenton (@mjfenton) October 25, 2014

And the clear fan favorite, based on the fact that this received more retweets and favorites than anything I’ve ever posted before:

Idea: Student stumped, frozen? Ask for an answer “you know is wrong.” Unfrozen. Then ask about why it’s wrong. – via @ebb663 #CMCS14

— Michael Fenton (@mjfenton) October 25, 2014

Session 7

Transformulas: Simplifying Relationships with Hi & Lo Tech
(Jedediah Butler • Saturday, 1:15 pm)

I met Jedediah Butler for the first time in Vaudrey and Stevens’ La Cucina session the day before. He was tinkering with something in Desmos, and I was blown away. (And a little nervous, since I was supposed to be wowing people with my Desmos chops later that day.) When I heard that Jed was giving a session on Geogebra—one of my confessed math/tech weak spots, despite my interest and affection—I knew I couldn’t skip it.

Jed didn’t disappoint. In fact, I was even more impressed with the sheer quantity (and the consistently excellent quality) of the Geogebra applets he’s created. It’s really a remarkable collection. Go check it out. Plus, it was loads of fun to sit in on a session of a MTBoS colleague who made the jump from “semi-make-believe Internet friend” to “hey, we’ve actually met in the flesh friend” just 24 hours earlier.

Session 8

Teaching Math Using Real-World Topics
(Karim Ani • Saturday, 3:15 pm)

What’s the best way to wrap up an amazing conference? With an amazing last session. And Karim Ani (founder of Mathalicious) certainly didn’t disappoint (even with all the hype being pumped out by the Mathalicious Twitter account). I was blown away by the presentation as a whole, and in particular:

Outstanding story telling, compelling visuals, and (most importantly) genuine thoughtfulness in @karimkai's #CMCS14 session.

— Michael Fenton (@mjfenton) October 25, 2014

The thought I couldn’t shake toward the end of the session, and one that captures what I believe to be the strongest feature of not only Karim’s session but also the lessons Mathalicious keeps churning out, is that a day (or better yet, a week) as a fly on the wall in the Mathalicious offices could literally change a math teacher’s life. If there’s ever a summer internship program, I’ll be the first one to camp out on the office doorstep in the hopes that they’ll let one more aspiring mathematical conversation-starter in the room. After listening to Karim for 90 minutes, I was hungry for more. And not only for well-designed lessons with slick visuals and applets and thoughtfully-crafted teacher and student resources… It’s the whole Mathalicious way of thinking through lesson design and student engagement that has me most excited. I hope to stoke that excitement spark into a full-fledged flame in the near future.

The (New) Running Game

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Background

A little over a year ago I created a set of Nike+ Running-themed proportional reasoning challenges for my middle school students. The idea originated as a failed attempt at a Three Act task, and eventually turned in to a somewhat-useful website (proportionplay.com). You can read more about the project here.

Even after using these problems occasionally throughout last year, and also in various teacher workshop settings, I was never satisfied with the results. Things usually drifted into uncomfortable arithmetic, and my focus from the word “Go!” was always on the proportional reasoning element.

So after mulling things over during last year’s spring semester—and trying out some of those ideas at the start of this year—I’ve settled on a new approach to the same set of problems. I’m optimistic that this new twist will make the problems much more engaging to my own students (and maybe even to some of yours as well).

The (Old) Running Game

The old me would start the conversation with an image like this:

Then I’d ask students what they noticed about the images. There would often be a wide range of answers (“I see orange!” “Hey, the phone battery is draining!”), but eventually we’d have covered the basics: “After 10 minutes, some mystery man (or woman) had run 1.88 km. This person apparently kept running for another 10 minutes, and then for another 5. We don’t know the total distance traveled at either of those latter points.”

Next up: Find what’s missing!

This is where we’d wander off track from my primary goal (develop proportional reasoning!) and end up stuck in some tedious arithmetic (with the tediousness increasing slightly each day in the series of 20 challenges). There’s nothing wrong with having (or building) proficiency with decimal arithmetic. However, my goal here was to emphasize and develop proportional reasoning. The doubling, the halving, the combining of these pieces to make five halves of the original… Students would stare into the blinding light of ninety-four hundredths and somehow lose track of what they were after in the first place.

Good conversations often ensued, but I was after great conversations. Something had to change.

The (New) Running Game

Here’s how we roll now: I still begin with an image…

And we still do the whole noticing thing (at least on the first couple of challenges). But then I invite (encourage? demand?) students to cut some corners.

21 minutes and 5 seconds? Nah! That’s basically just 21 minutes.
Double that? 42 minutes. Boom! We have our first answer: “This runner probably took about (hugely important word for the rest of our running game conversations) 42 minutes to run 8 km.”
Back to 4 km in 21 minutes. Half of 21? Let’s call it 10. So 2 km in about 10 minutes (maybe a little more).
So 42 plus 10… That’s 52 minutes.
But maybe our runner is no Madison Bumgarner. Maybe fatigue actually affects our faithful jogging friend. Let’s hedge our bet. Maybe… 54 minutes?

And then, the reveal:

And it turns out we’re wrong. Totally and completely wrong. But now the fun begins, because we can ask:

What happened?!
Why were we so off?
Did our arithmetic betray us? (No. We took some liberties with rounding so the arithmetic would be easy-peasy.)
So what gives?
Did this guy (or gal) speed up? Or did he/she slow down? (To me, this conversation alone is worth all the screenshots I took while running with a phone in my hand.)

Maybe we made some assumptions about our runner. Maybe we need to blow up some of those assumptions so our future “off-ness” isn’t so pronounced. (And isn’t that one of the best parts of modeling? Figuring out the subtleties of the situation… Testing and either retaining or tossing out our assumptions…)

So we play again:

And this time we come to the table with some additional information. This dude slows down (and seriously) as the seconds tick by. Let’s keep that in mind as we think through the next round:

1.89 miles… That’s pretty close to 2 miles in 16 minutes
That’s 1 mile every 8 minutes
24 minutes… Let’s chalk that up as 3 miles
32 minutes… That would be 4 miles
What about 36 minutes? That’s another 4 minutes (half of 8!) so I’m thinking an extra 0.5 mile. All told: 4.5 miles.

Of course, that would be if our runner followed a predictable pattern. Now the fun comes in. This is where every single student in the class could end up with a unique and not-unjustifiable answer. How do you want to hedge your bet in the end? Will the runner slow down? Have we already accounted for that with our rounding up to 2 miles? (Or have we gone the wrong way?!) And so the customization of the answers begins (ideally with much arguing and hoping and anticipating).

Me? I’m thinking we should tone down the distances… 2.7 miles in 24 minutes, 4 miles in 36 minutes. So, how did I do?

Now when you throw that next image on the board, and one kid in the back slams his fist on the table in defeat and the other pumps her fist in the air in triumph… You know you’ve got them.

The Road Ahead

The challenges are all still available at proportionplay.com. And I’ve replaced the original (and terrible) handout with this shiny new one. I think the new handout does a reasonably good job supporting my updated approach to these tasks by emphasizing the reasoning involved in the entire process. That—combined with my wide open invitation to estimate since the runner is almost certainly not a same-pace-all-day robot—should shift the focus from arithmetic to proportional reasoning.

Post. Script.

The last two challenges are both borderline terrible. Or wonderful. Or both. Here’s a preview of Day 20:

Post^2. Script.

Another favorite aspect of these problems? Whether you round or not, there are several ways to tackle each challenge:

Disclaimer: The whiteboard image is from one of the courses I teach in the grad math/science department at Fresno Pacific University.

Our New Set Game Routine

For years I’ve used the Set Game to grab students’ attention at the start of class—in a playful, mind-sharpening way. (As an aside, I’ve often wondered whether I should have some sort of bell-work-problem-set-something-or-other, but I’ve never moved even an inch on that wondering… at least not thus far.)

A while back I wrote about how this lovely game ventured from my childhood and into my classroom. My approach has shifted quite a bit (and for the better, I think) so I thought I’d share how we do things in class now.

Super Previously

When I first introduced this to my students, iOS didn’t exist. So each day we played, I simply pointed my web-browser (pre-Google Chrome, if you’ll believe it) to http://www.setgame.com and we played the free daily puzzle. Find the six sets in the set of 12 cards as fast as you can.

Students would raise their hands, I’d call on them as fast as my mouth could manage, they would shout three numbers quick as can be, and I’d click the corresponding cards on the screen. If all was well, we repeat that process five more times and cross our fingers in the hopes that we had the fastest time of the day/week/month/year/ever.

Previously

Several years later I got an iPad. And an Apple TV. And I forked over the 5 bucks for the iOS version of the Set Game. Instead of working through the free daily puzzle, I would offer my students a 60-second challenge. The beauty of this approach is that I could control the amount of time we spent playing Set on a given day (super previously, we would sometimes play for several minutes if the students had trouble finding the last couple of sets). Now it was 60 seconds, period. No more, no less. Aside from that, our style remained the same: Find a set, raise your hand, shout numbers. Repeat.

Now

One of the drawbacks of these original approaches is that they played to the strengths of the fastest students and ignored those who need more time to process. The start of class was a flurry of pattern-finding and number-shouting (quite impressive to watch as a visitor unfamiliar with the game), but it really depended on a select few; the majority of my students barely participated at all.

With that in mind, here’s how we play now:

I display a random, annotated image (hooray for screenshots on the iPad) from a ready-and-waiting Keynote file:

I give students “30 silent seconds to search” during which (as the description implies) they are to search, um, silently, for as many sets as they can find. (By the way, I’m taking attendance while they do this.)
Next, I announce that “you may now collaborate.” I wrap up attendance and get ready to call on groups.
I call on one group at a time, asking each group to announce the numbers of a single set.
I record the results on the board (regardless of any mistakes students might make), and proceed to the next group.
Once I get to the last group, I then open it up to “anyone in the class.”
Students continue sharing.
Once no more students are willing to share (and by this time we usually have 6-9 sets), I ask “Does anyone see any imposters?” (our name for mis-identified sets).
Imposters are identified, with brief explanations as to why the proposed trio of numbers is not actually a set. (Ex: “1, 2, 11, because two of them are singles.”)
Then we move on to whatever is next.

Advantages to this approach? I see a few:

All students are given an opportunity to play
Students are also given an opportunity to collaborate
No one student or group is allowed to dominate when sharing proposed answers
Students have a regular, bite-sized, built-in opportunity to “critique the reasoning of others” (SMP 3) when they identify imposters
One last thing… Students love those black numbers on yellow squares, as it shifts the entire focus to visual pattern recognition (though I’m a bit sad that they benefit so greatly from this little trick, as I would prefer that they recognize the structure build into the 3×4 array)

In the Future

Recently, I’ve wondered about challenging students with the task of creating a set of 12 cards with exactly six sets. I anticipate this would be quite challenging, and there might be some interesting patterns and discussions that would arise along the way. I’ll keep you posted on this front. Or, if you beat me to the punch, let me know how you and your students fare.

Beyond Set

For me, this post is about much more than detailing the evolution of our Set Game exploits. The most important thread of this development is one that I hope will run through most of the lessons in my classroom. The challenge: While creating opportunities for engaging experiences, carve out space where every student has an opportunity to participate in a meaningful way. As I turn my attention from the Set Game to student activities, rich math tasks, and class discussions, I’ll pay closer to attention to whether my teaching decisions help or harm students in this regard. I challenge you to do the same.

Noticing and Wondering with Trigonometric Identities

Friday

I had a sub Friday in Precalculus. My students were set to take a group of mini-assessments for Chapter 3. I expected they would have some additional time at the end of class, so (out of love for the substitute, and a desire not to waste class time) I wanted to launch Chapter 4 with an exploratory activity. I think it has some potential for improvement, but this is what I scraped together while getting my sub plans ready.

Monday

Today my goal was to move toward proving trigonometric identities (see below for a handout), but first I wanted to debrief from their experience on Friday. I gave students three minutes to share their observations and questions in their groups, then I asked each group to share one noticing and one wondering. We ended up with this:

Onward

My favorite part of all of this? Built on their questions, we’re now headed into Chapter 4 with a bit more motivation than we might have otherwise had. Also, there were a few wonderings that surprised me (at least in subtle ways), and these will almost certainly enrich our conversations over the coming days.

P.S. Up next, this.

P.P.S. That last handout is far from my best work, but it helped us transition from the noticing and wondering into some basic proofs.

Desmos and the Diagonal

This morning I ran across an entertaining tweet from someone with a sweet first name:

Took an old rectangle/diagonal problem and made an interactive on Desmos https://t.co/P0ZdpSAvSI #mathchat

— Mike (@mikekaechele) October 8, 2014

I was reminded of Dan Meyer’s treatment of this problem, and began wondering about adding shading to Mike’s work in Desmos. In between meetings today at school I tinkered a bit, and discovered a totally-inefficient, but still-effective way of getting it done.

Here’s a glimpse of what I ended up with…

…and a link to my graph in Desmos.

Show Off

Of course, @Desmos went about it in a much more intelligent way… But I’d venture that I derived more joy from finishing my long way round.

P.S. This is my first blog post since June 5. I hope to sit down soon to write about the reasons for my time away. Needless to say, the gap between posts has been far too long. But life and work have their demands, and sleep is nice at times, too.

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