The (New) Running Game


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 ( 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:

The Running Game.001Q

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…

The Running Game.002Q

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:

The Running Game.002A

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:

The Running Game.005Q

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?

The Running Game.005A

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 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:

The Running Game.020Q


Post^2. Script.

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

running game work

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 and we played the free daily puzzle. Find the six sets in the set of 12 cards as fast as you can.

setgame website

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.


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.


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:

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

set app

  1. 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.)
  2. Next, I announce that “you may now collaborate.” I wrap up attendance and get ready to call on groups.
  3. I call on one group at a time, asking each group to announce the numbers of a single set.
  4. I record the results on the board (regardless of any mistakes students might make), and proceed to the next group.
  5. Once I get to the last group, I then open it up to “anyone in the class.”
  6. Students continue sharing.
  7. 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).
  8. 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.”)
  9. 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


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.


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:



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:

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.

Desmos: Dot Capture Game

I created a silly little game for my Algebra 1 students several weeks ago. The motivation? Five-fold!

  • We’re a little weak with graphing lines. Some open-ended, Desmos-driven, instant-feedback style practice may help.
  • Domain and range? Yeah, not so much.
  • Inequalities? Haven’t done them justice. Yet. (Growth mindset, baby!)
  • Vertex form for quadratics? Still struggling.
  • We tried Des-man a few days before this game and found nothing but pain and frustration. Some could be attributed to me (in particular, a bungled launch of the activity), some to students’ lingering struggles (noted above), and most of the rest to the declining state of our netbook cart. (But they seemed so cool in 2009!)

At any rate, to get that bad taste out of my mouth and set the stage for greater success on the next Des-man go around, I created the Dot Capture Game. Here’s what you need:

  • Students (working in pairs)
  • Devices (we actually used 50% smartphones, 25% tablets, 25% laptops)
  • The world’s greatest, most beautiful graphing calculator

And of course, the handout:

Dot Capture Game


Getting Started

Give a brief intro—or none at all—and turn ‘em loose. If your experience is anything like mine, you’ll find yourself the weaving in and out of some great (albeit trivially-inspired) conversations about slope, intercepts, point-slope form, domain, range, inequalities and shading, vertices, direction of opening, etc.

This is definitely not high-quality modeling stuff (it’s not even low-quality modeling stuff), but it proved a great way to engage students with meaningful (read: productive) practice on a variety of topics related to graphing.

Oh, and the winner in my class? Here you go:



Final Thoughts

After trying this out in Algebra 1, I thought I’d throw it at my Algebra 2 and Precalculus students to see what they would do with it. It turned out to be good practice in those settings as well. Before sharing with these followup classes, a quick tweak to the handout was in order. In my first class, several students lost their graphs and expressions after hitting a deadly combination of keys on their device, and only one or two had been keeping a shiny written record. So to protect against future heartache, I added a second page to the handout. Here’s what one of them looked like at the end of class:

2014-04-04 15.05.13



Here’s a sweet suggestion from Desmos:

Billion Dollar Bracket

This gets me every time. I suppose it goes without saying that I loved this, too.

I don’t know why, but there’s something about that noise (in a math problem, no less) that simultaneous makes me giggle and fires up the I-need-to-know-what-was-said corner of my brain.

So I made this:

Since I made the video1 a few things have happened in the world of college basketball.

  • UCLA won
  • UCLA won
  • UCLA lost
  • I’m sorry, did they keep playing the rest of the games after UCLA lost?
  • Nobody won. The bracket challenge, I mean. I think someone won the tournament. I’m not really sure.
  • Warren Buffett and Quicken picked up about a billion new home loan leads.

At any rate, I’m not entirely satisfied with the result. I mean, I was really hopeful UCLA could make it to the Elite Eight I could turn this into an engaging lesson hook, but the first group I tried this on kind of just stared at the screen after the Act 1 video ended.

So I’d love some feedback, either in general, or in response to some of these:

  1. Is there any potential here? (I think there is…)
  2. What bits have I got right, if any?
  3. How would you hook students with the “what are the chances…” question here?
  4. Am I missing the point? Is there a better question to ask besides “what are the chances…”?

Thanks in advance for your thoughts!

P.S. If you’re interested in a link to Act 1 and Act 3, you’re welcome. And here’s a notebook full of some links and images I gathered but never used.

  1. For the record, that was two months ago. This post has been sitting in draft purgatory for long enough. So it’s time to drop this in the urgent bin and get it out the door. 

All Mistakes Are Created Equal?


I’m a big fan of Michael Pershan’s project Math Mistakes. If you’ve never checked it out, it’s worth exploring. And while I’m meddling with your life, here’s a tip for your entire department: Start each meeting by spending five minutes exploring one of the mistakes posted on Michael’s site. On a rotating basis, have one member of the department share a “provocative” math mistake from the blog (or maybe even one from his or her own classroom). And once duly provoked… Cue the discussion!


I included the following uninspiring question on a recent assessment:

Screen Shot 2014-05-15 at 7.54.06 AM

The first two assessments I graded included the following responses:

Student 1


Student 21


Comment Fodder

So here’s my question (er, set of questions) for you:

  1. Are these mistakes equally egregious?
  2. What misconceptions are contained in the first mistake? How would you address them?
  3. What misconception are contained in the second mistake? How would you address them?
  4. Would you grant any credit for either response, and why?

  1. My apologies for the retype. I added some feedback before snapping a photo. 

Counting Within Twelve

This is Caleb, back in September 2009:


And here again a few months ago, with a haircut (or lackthereof) that might even make Matt Vaudrey proud:

2013-10-13 17.33.59

I think Caleb’s swell. Of course, being Caleb’s dad I’m more than a little bias. But I have it on good authority from many people who aren’t Caleb’s dad that my analysis is spot on.

Anyway, one of the swell things Caleb has been doing lately is counting. Everything. Cheerios, ice cubes, grapes, cookies, and all manner of things found at the kitchen table; white tiles, grey tiles, ceiling panels, and all manner of things found in the bathroom hallway at preschool; Legos, piles of Legos, boxes of Legos, and all manner of things found on the family room table while he waits for his younger brother to fall asleep. Asked a moment ago about his favorite thing to count, he responded with a list of several things, and then: “I like to count pretty much everything. Everything in the world.” Excuse me while I go get a tissue.

Lately Caleb has been getting some joy-filled counting workouts while we play a modified version of Monopoly that he and my wife invented a few weeks ago. He’s not quite ready for the paper-money, numbers-in-the-hundreds, mortgage/unmortgage, house/hotel dynamics. In fact, he’s even having trouble with the name (he calls it “Buh-noc-oly”). But he’s totally into rolling the dice, stomping around the board, and carrying out the “everything-costs-one-Chuck-E-Cheese-token” result of wherever he lands. While watching him play—and thinking back to my own childhood, which was probably filled with about 10,000 games of Monopoly—I’ve developed a few wannabe insights about what’s going on.

  • This sort of practice is super valuable because it’s fun, it has nothing to do with flash cards, there’s a point to it (in Caleb’s mind) that is much bigger than the counting itself, and there’s heaps and heaps of it built into even a relatively short game.
  • The board is basically a making-fives, making-tens, pattern-finding paradise. I mean, just look at it. You’re on Virginia Avenue. You roll a 6. Jackpot! Free Parking here we come! Or you’re on Marvin Gardens. You’ve got greedy eyes for Park Place. What are you hoping to roll? One step lands on “Go To Jail” and after that it’s 5 more to the railroad and 2 more to Park Place. That’s 1 and 5 and 2… 8! Better yet, you’re on Reading Railroad, you roll a 6 + 2, so that makes 8. You break 8 down into 5 + 3, jump 5 to “Just Visiting” and then 3 more to States Avenue. Granted, Caleb hasn’t tapped into anything quite as nifty as that last example, but that’s the kind of potential packed into the board, and it will be fun to see what kinds of patterns he discovers as he continues playing.


  • Subitizing and basic addition automaticity are undeniably cool, and counting pips on a pair of dice has ‘em both in spades. I’ve seen Caleb go from counting all the pips individually on each die, to recognizing 5 as 5 and 3 as 3 (and the sum as 8) without counting them one at a time, in just a few games. I love when students develop their own shortcuts—whether in Algebra 1 or in Calculus—and it’s been fun to watch my son develop some of his own speedy strategies without any promptings from other kids or adults.

2014-05-10 19.33.24

One day we’ll learn how to actually play the game. And one day he’ll probably tire of it (or of playing with me). But along the way, I plan to enjoy watching little a-ha moments flash across his face as he steers that big ol’ boat around the board.

2014-05-10 19.31.47

P.S. The other thing that’s great about Monopoly? Swindling Negotiation. As in, four player game, tough luck at the start, three properties to your name when the wheeling and dealing begins. And somehow—Somehow!—you weasel negotiate your way to three complete monopolies and total domination. And some upset family members who refuse to play with you in the future. But sometimes that can’t be helped.

Desmos Hack: One-Variable Inequalities

I was typing out solutions to an Algebra 2 assessment the other day. Question 3 on the assessment asks students to solve an equation involving absolute value. I began my solution with this…

Screen Shot 2014-05-08 at 9.53.34 PM

…and then launched into an algebraic confirmation of that solution.

Now on the one hand, throwing a Desmos-generated graph into a “detailed solutions” handout is a great move because, well, just look at it. It’s beautiful. And hey! Multiple representations! Plus it took about 30 seconds from start to finish. No brainer, right?

Well, on the other hand, including something like that is dangerous, because when you find yourself writing the solutions to questions 6 and 7 (as I did just a few moments later), and these questions ask for a graphical display of the solution to a one-variable linear inequality… Well now you’ve tasted greatness, and you won’t settle for anything else.

There’s just one problem: Desmos doesn’t do linear inequalities in one variable.

Okay, that last sentence is actually not true. Desmos will graph linear inequalities in one variable. You just have to ask nicely. Check it out:

Screen Shot 2014-05-08 at 10.02.00 PM

Screen Shot 2014-05-08 at 10.02.07 PM

I imagine I’m not the only one to do this (and it would still be pretty cool if Desmos would add one-variable number line graphing functionality… Pretty please?), but I thought I’d share how to do it anyway, just in case anyone is curious (and wants to give one-variable graphing a little Desmos-love).

Here’s How

The best way to explain is to throw a few images in here and let them do the talking. Drop me a line on Twitter (@mjfenton) or in the comments if you have any questions (or tips for how to make this even easier or more awesome). Or if your name is Eli and you have a new feature to announce.

Graph 1

Screen Shot 2014-05-08 at 9.44.18 PM

Screen Shot 2014-05-08 at 9.43.53 PM

Graph 2

Screen Shot 2014-05-08 at 10.09.59 PM

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Graph 3

Screen Shot 2014-05-08 at 9.50.18 PM

Screen Shot 2014-05-08 at 9.47.19 PM

Graph 4

Screen Shot 2014-05-08 at 9.49.30 PM

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