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:

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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:

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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:

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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

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Screen Shot 2014-05-08 at 9.43.53 PM

Graph 2

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Graph 3

Screen Shot 2014-05-08 at 9.50.18 PM

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Graph 4

Screen Shot 2014-05-08 at 9.49.30 PM

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Assessment Workflow Breakthrough

I think I had a breakthrough today.

I’ve been meaning to blog about the “assessment workflow” in my classroom, but I’ve been putting it off because (a) time is limited, especially at the end of the school year, and (b) I wanted to be mostly satisfied with my workflow before I shared anything (and I’m not there yet).

I’ll write up the full details of how assessment happens in my classroom (it’s been a major work-in-progress this year), but for now I want to share a tiny bit of background and then cut to today’s breakthrough.

The Background

Last Sunday I aired some of my thoughts and questions on this topic to @Mythagon. A few other thoughtful folks dropped by to share their own ideas and pose a few new questions for me to chew on. It left me with a clear sense (as have other conversations) that my assessment routine fails students in the category of self-feedback. I’ve been trying to foster more (and better) student reflection in our assessment routine for several months now, and those efforts are the reason I’ve pasted this quick reflection form…

quick reflection

…at the bottom of every new assessment I write. However, I was looking for a way to incorporate something that would require students to be more thoughtful (just shading in a couple of boxes doesn’t necessarily demand any careful consideration) and at the same time foster a growth mindset among my students.

The Breakthrough

At the end of today’s assessment (after grading them; more on that in the next post), just before collecting everything, I gave students the following directions:

Assessment Reflection.001

Two minutes later, I collected the papers and we moved on to something else. Later in the day I went through the papers to confirm the results and scores, to get a sense of common mistakes (again, more on this workflow later), and (this part was new today!) to read the SP and STI comments.

Growth Mindset, Plans for the Future

It’s early, but I’m sensing that this could be one of the most important features of my classroom in terms of developing a growth mindset among my students. I love the blend of looking back to celebrate something and looking forward at something (and how) to improve.

I’m wondering now about the best way to incorporate this SP/STI reflection into the “aftermath” of all my assessments. The comments (see below for some samples) were physically all over the place, with some easier to read than others. It might be worth the time (and “lost” space on the page) to add a little box near the top of the assessment with room carved out for the SP and STI comments. I’ll tinker with the layout and post an update if I come up with anything promising.

Student Comments, Round 1

Here are the SP/STI reflections from the first eight papers in the stack today. Some comments are decidedly un-profound, but others are exactly what I was hoping for right out of the gate. I’m hopeful that my classroom will become a more thoughtful and reflective place through this routine. We’ll see how it goes next time.

Update from the Interwebz

Pizza, Cookies, Ice Cream, and Measuring Tables in Caleb-Hands

My wife and I are blessed. Beyond measure. We get to hang out with four amazing kiddos every day:

Easter 2014

Tuesday evening this week my parents took three of the kids to their place so my wife and I could take our oldest (Caleb) out for a special Caleb-only dinner date. Let’s be clear from the start: It was, from the beginning, all about the dessert. Dinner was just a necessary formality. Fortunately, dinner included bacon, so it was a rather delicious formality… But I digress.

A TMWYK Measurement Opportunity

Shortly after we finished our burgers, pizza, and the like—and with a few minutes to spare before the real deal would arrive—I asked Caleb a few measurement questions about the width of the table. We regularly count things around the house and while we’re out running errands, but I haven’t asked a great many measurement questions. Inspired by some of the conversations I’ve heard and/or read from others in the #TMWYK corner of the #MTBoS, I figured “Why not start now?”

So I asked a few questions. And we had a blast. Here’s a partially-remembered record of what we talked about.

Ready, Set, Measure!

Michael: “Hey Caleb, I wonder how long this table is. How many of my hands do you think it would take to go from “here” to “there”? (I motion across the table.)

Caleb: “Um, I don’t know. 10?”

M: “Okay. Let’s check it out. Count with me… (Sliding my hands across the table…) One. Two.”

C: “Three. Four. Five.”

(At this point, I pause for a second to let him consider whether his guess was a good one. We’re a little past half way, so I think we’ll come in under 10. Then I resume…)

M: “Six.”

C: “Seven. Eight. Eight!”

M: “Wow, so you guessed 10, and it was eight. Very cool. What about Caleb-hands? How many Caleb-hands across?”

C: “Um, 11?”

(Here I missed a great opportunity to ask why he thought that. “You know, Caleb-hands are smaller than Daddy-hands, so we’ll need more of them,” or something like that. But he was a little distracted. We were at BJs, and there are like 42 televisions scattered across that place… So instead, I took a page out of Dan Meyer’s three act playbook and asked…)

M: “Caleb, what do you think would be a number that’s too high? Too many hands?”

C: “28.”

M: “Okay. And what about a number that would be too low?”

C: “Um, 6.”

(By the way, that would be his “too high” and “too low” guess for all of the remaining questions. He would alter the “just right” guess, but not the upper and lower bounds.)

M: “Okay, here we go. (I lead his hands, and we start counting…) One. Two.”

C: (He continues moving his hands and counting…) “Three. Four. Five. Six. Seven. Eight. Nine. Ten.”

(I pause him, and ask…)

M: “Okay, we’re at 10. What was your guess?”

C: “11.”

M: “Do you think it will only take one more hand?” (In my estimation we’re about 75% of the way there.)

C: “Uh.” (Squirm, squirm, squirm. Glance at a television. Then another. One more squirm for good measure. So I decided to abandon this tangential interrogation and get the party started again…)

M: “Okay, let’s finish. (I jumpstart his hands again…) Eleven. Twelve.”

C: “Thirteen! Yeah!”

M: “Very cool. Hey, what else could we use to measure?”

Rounds 3-7

We then played the same game with leapfrogging crayons, kids cups, napkins, half-napkins (folded over), and crayons again (the skinny way). All in all, we had a blast, and some nice conversations. And then dessert arrived. And it was fantastic. :)


Thanks for the inspiration, everyone in the #TMWYK steam (and especially Christopher, for sparking my thinking about much of this on your own blog and more recently over at If you haven’t been to that website, and you interact with little ones (your own, or someone else’s) on a regular basis, check it out. Great, great stuff.

P.S. There’s a video of one of our last exchanges, but it’s out of focus and Caleb is losing interest (“Squirrel!”), so I’m not sure it’s worth sharing.

10 Free Tools…

10 free tools

This summer I’m offering a two-day math and technology workshop series for JH and HS teachers. The purpose of the workshop? Equip teachers with digital tools and skills that will help them engage students, promote the CCSSM mathematical practices, and “multiply our hours” as teachers.

The workshop runs twice this summer: June 5-6 and June 23-24.

If you’re going to be in the area (Fresno, CA) this summer, or know someone else who will be, check out the the flyer for more details.