How Many Solutions?

I’ve been thinking about systems of linear equations for the past couple of days. Most of my focus has been on systems of two equations, but this morning I wrote a question about a system of three linear equations that might spark some interesting and revealing classroom conversations:


And now I’m wondering…

  • How many solutions does that system have? Drop your thoughts in the comments, would you?
  • If you put this prompt in front of your students, I’d love to hear a recap of how that plays out.

Resource Countdown • December 24, 2015

For a tiny bit of background, check out last week’s inaugural “resource” post.

Now, on to the countdown!

3… Tweets

2… Posts

22? 30? 50? 100?, by Joe Schwartz . A wonderfully thoughtful post, followed by a flood of wonderfully thoughtful comments.

Good Mathematician vs. Great Mathematician, by Ben Orlin. In shifting from Reeder to Feedly and back to Reeder, I somehow (argh!) temporarily (phew!) lost my RSS feed to Ben Orlin’s delightful Math With Bad Drawings. Resubscribed. It’s good to be back.

1… Book

Several weeks ago, Andrew Stadel suggested I read Weekend Language. I procrastinated for a while, but finally picked it up. It was as good as advertised. I highly recommend it to anyone who wants to become a better presenter.


That’s all for now!

Resource Countdown • December 17, 2015

I’d like to try something new on the blog. Throughout the week, I typically stumble across a handful of tweets, blog posts, and activities that catch my eye.

I’ll gather than up in a countdown style (the Internet loves a good list, doesn’t it?) and share them here. Every Thursday? Some Thursdays? We’ll see, but I’m hoping for the former.

Countdown to what, you ask? The end of the post, I suppose. Or the weekend. Or whatever comes after you read the post. 🙂

Here goes…

3… Tweets

2… Posts

Project Pentagon, by Christopher Danielson. At NCTM Nashville I had the opportunity to tinker with Christopher’s tiling turtles and pentagons. They’re lovely. He writes more about his growing obsession with pentagons here.

Interesting Problem, by Jonathan Claydon. Jonathan raises an interesting question about the role of technology in the classroom, and its implications for the questions we ask students when technology is so readily available.

1… Activity

Steve Leinwand mentioned this MARS task (Calculating Volumes of Compound Objects) during his talk at CMC North. I’m intrigued. Would you use this as-is? Or would you make adjustments? Which ones and why?


That’s all for now!

Desmos Marbleslides

This week, Desmos released a new activity called Marbleslides. As you might have heard (or seen), it’s delightful:


Dan Meyer wrote about the activity on his blog, and our resident wordsmith dropped a marbleslides post on the official Desmos blog.

[Psst: If you haven’t played yet, and want to… Go here.]

In reflecting on what I love about marbleslides, my thoughts go back to the session Katie Reneau and I co-presented at this year’s CMC North and South conferences. We presented four principles that we’ve found effective in task selection and implementation with our own students.

  1. Use tasks that build capacity for SMP 3.
  2. Use tasks that are accessible and extendible.
  3. Use tasks that encourage iteration.
  4. Use tasks that have room for multiple approaches.

There’s not much in the way of SMP 3 (constructing viable arguments…), but marbleslides is an absolute gold mine when it comes to the other three principles.

Accessible and Extendible

The activity kicks off with a challenge that “even your baby cousin” could tackle. Provided he or she is willing to hit the “launch” button.


From there, the challenges grow. Marbleslides adds one layer at a time, providing students with digital sandboxes in which to explore constants and coefficients, translations and dilations, concavity and restrictions.

As students build their skills, Marbleslides pulls back the supports. Students are asked to find (and reflect on) solutions to increasingly complex challenges.


In our conference session, Katie and I used an Open Middle problem to illustrate the meaning (and value) of iteration in math tasks. We both love problems that offer students some early reward, and them immediately invite them to dig deeper, work smarter, etc.

For Marbleslides, imagine this sequence:

  • A student builds a solution, clicks launch. No stars. (Bummer.)
  • But she tries again. And gets a star. (:internal-fist-pump:)
  • She adjusts her graph (or graphs), and gets two more stars. (Nice!)
  • So close… Just one more… A few more tweaks (and attempts), and she’s like:


  • But then a classmate throws down a challenge: Five parabolas? Betchacant with just three.
  • And the tweaking, the improving—the iterating—continues.

Multiple Approaches

Something else I love about marbleslides? Students can legitimately take alternate paths to the solution. Years ago I began asking students (almost like a broken record), “Great! Can you solve it another way?”

For the record, this prompting works out much better when there’s more than one mathematically meaningful approach.

And on most marbleslides levels (especially later in the activities), I’m pretty sure there are. From different paths, to different functions, to minimizing equations, to minimizing the time it takes to get all four stars… There are a lot of options for pressing deeper on any part of these tasks.


If you haven’t tried out marbleslides yet, pick your flavor (lines, parabolas, exponentials, rationals, periodics), and let me know what you think.

If you’re already tried it, do me a favor: Save a GIF of your favorite solution (yours, a colleague’s, or a student’s) and drop it in the comments.

Instructional Shifts Challenge

I attended CMC North last weekend in Asilomar, CA. Not surprisingly, it was a fantastic conference. From Matt Larson’s opening keynote Friday evening, to Steve Leinwand’s talk on Sunday morning, to all the sessions/ignites/meals/conversations in between, it was an exciting and energizing weekend.

Also not surprisingly, I found Steve’s presentation to be a wonderful blend of encouragement, challenge, and inspiration.

A Challenge

Toward the end of his talk, Steve invited everyone in the room to identify their 2015-2016 instructional shifts to-do list.

I’m still wrestling with the contents of my own list (and am pretty sure that’ll be the next post). In the meantime, let me pass my own version of Steve’s challenge along to you:

Before the week is out, write down your own instructional shifts to-do list for Spring 2016.

Don’t try to change everything all at once; you’ll likely end up with no change at all if you bite off more than you can chew. But commit to changing at least one (or maybe even 2-3) of your instructional practices in the next semester. Write them down. Pursue them. Use them to make a difference for your students.

Go one step further and share them. In the comments. On your blog.* On the Twitter. Better yet, share them with your colleagues and develop your practice together!

*What’s that! You don’t have a blog? Sounds like you have a great idea for your first post… 🙂

Some Inspiration

Coming up short on ideas? Get started with something here:


Good luck with your goal-setting, and your goal-pursuing! I’ll be back soon with my own to-do list, tweaked for my own current role at Desmos.

Which reminds me… If you’re not in the classroom, but you still interact with teachers on a regular basis, you’re not off the hook! Consider how this challenge might translate into your current setting.


Gearing Up for CMC North 2015

One of my favorite conferences is the one hosted by the California Math Council in Asilomar, CA. This is the first math education conference I started attending regularly, and it’s in one of the most beautiful locations on the planet.

I’m presenting twice this year:


I’m also co-presenting with a former colleague (Katie Reneau) who did an amazing job presenting at CMC South in Palm Springs. Here are her session details:

  • Rich Discussions and Rich Tasks in the MS Math Classroom
  • Saturday, 11:00 am to 12:00 pm
  • PC Middle School, Room 29

The One Conference Non-Negotiable

What you attend throughout the day on Saturday is up to you (though of course I’d love to see you in one of the sessions I’m involved in). However, there is one non-negotiable here:

No matter what, make sure you attend the IGNITE talks.

Saturday evening, get yourself to Merrill Hall. I promise you won’t be disappointed. The “party” starts at 7:30 pm, but you might want to get there a bit early.

Still Game Planning?

If you’re attending and still figuring out what sessions to attend, or you’re just lurking from a distance and want to add a little fuel to your #CMCN15 jealousy fire, you can access the full conference program as a PDF right here.

P.S. Sometime before the conference is over,  I’ll post session slides and resources on my speaking page.

Time for a #slowmathchat Shift

I’ve shared here on the blog, and also in several recent conference sessions, that Twitter has changed my life. No exaggeration. Not even in the slightest. I’ve heard similar sentiments from quite a few others, including this from John Stevens:

I’ve learned a ton from John. I’m a huge fan of his work at and with La Cucina Matematica, and if memory serves, he’s the one who introduced me to the idea of a Twitter chat. John and some other outstanding folks run the #caedchat which drops every Sunday evening at 8 pm (PT). I joined in once, and it was great. (Side note: If Twitter is like drinking too much awesome from a firehose, then #caedchat is that to the extreme.)

I enjoyed the conversations and connections, but the whole synchronous chat thing didn’t work for me. Most chats are in the evening, and in my house that time is dinner time/bath time/story time/bedtime/put-the-house-back-together time. (My wife and I have four little ones, and the oldest just turned six. Yep. Intense is the way to describe it.)

#slowmathchat is Born

Anyway, a year ago I thought of a way that I could engage in Twitter chats without locking in to a specific time: #slowmathchat

I wrote about it here and with the help of Cole Gailus even archived some of the chats here.

#slowmathchat was a blast. I learned a lot in dreaming up topics and writing questions, and also through engaging in conversation. Sometimes I just learned by watching other folks’ responses. Either way, it was fantastic.

Time for a Shift

But it’s time for me to make a shift. I have some other projects I’d like to explore with the small pockets of discretionary time that I currently have. So I won’t be writing any more “This week’s #slowmathchat topic…” tweets, and I won’t be tweeting out #slowmathchat questions every Monday through Friday.

Several of you have shared that #slowmathchat helped you enter into this crazy online community, that the structure of these chats was helpful, even inviting. I still think the asynchronous chat format is really appealing, as is the focus on questions and discussions.

Call To Action

So if you have a math education question you’d love others to weigh in on, I’d love it if you kept using #slowmathchat. And if this thing keeps on ticking for a few more months, or even years, great! And if not, you know you can always find a great group of passionate math educators over on the #MTBoS hashtag.

So long, #slowmathchat! It’s been fun.

NCTM Nashville 2015

I spent Wednesday through Friday of last week in Nashville, TN for the 2015 NCTM Regional Conference and Exposition.

Instead of writing a full-blown recap (bedtime approaches) I’ll simply share a question that’s been on my mind since my return flight touched down in Fresno late Friday night. Okay, it’s not really a question, but a series of questions. Here goes:

  1. What makes a conference like NCTM Nashville (or conferences in general) special?
  2. What can you get at a conference that you can’t get on Twitter/blogs?
  3. What can you get on Twitter/blogs that you can’t get at a conference?
  4. Is there anything these two formats could learn from one another that would make each one even better?

I’m still formulating my own answers to these questions. I suspect I’ll write another post in a few days with my thoughts. In the meantime, I’d love to hear what you think. Drop a comment below, or give me a holler on Twitter.

P.S. I’ve really enjoyed Cathy Yenca’s and Tracy Zager’s recap posts.

P.P.S. The #MTBoS booth was so much fun!

True Confessions

True confession #1: Twitter changed my life.

That may sound strange, considering the following:

  • I generally hate social media.
  • I joined that Facebook thing all you kids talk about all the time, then decided it was a waste and deleted my account.
  • When I first heard about Twitter, I thought, “Who cares what Bieber had for breakfast?!”
  • I didn’t start texting until 2009. When I was 27. I know, I know! Crazy, right?
  • I like complete sentences, decent grammar, and punctuation.

But literally, Twitter changed my life.

Somewhere around the beginning of 2013, I started using Twitter to connect with other math teachers. I ended up in this crazy-wonderful community that calls itself the mathtwitterblogosphere (MTBoS, for short). I’ve always had amazing colleagues at my small school, but usually on the order or one or (if I was lucky) two other math teachers across the entire JH and HS staff.

When I hopped on Twitter, I suddenly found myself in a community of dozens (rather, hundreds) of folks who were as nerdy and weird and interested in getting better at this teaching thing as I was. I loved it. I still do.

In fact, I’m pretty well convinced that jumping into this community (whether by starting a blog, getting more active on Twitter, or both) is among the best ways to supercharge your teaching, whether we’re talking skills or passion or enjoyment.

Anyway, that’s my rambling attempt to convince you that you should consider jumping into this mix of mathy folks.

And thanks to some amazing MTBoS folks, there’s a pretty sweet guide to getting started right here:

So whether you’re brand new to the MTBoS, a dabbler who wants to dive a bit deeper, or a grizzled veteran looking to help others with their first steps, give that link a try. I promise you won’t be disappointed.

Hello Again (and Histograms)

Hi there!

So I haven’t exactly been pumping out record numbers of blog posts since I shifted from the classroom to my current gig at Desmos. This lack of posts is partially related to general busyness, but it’s also related to this question that I’ve been wrestling with for the past few months:

I’m not in the classroom, so… what would I even blog about?

Over the past few weeks, it’s gradually dawned on me that my next hundred posts can be about the same things as my first hundred or so: trying and tinkering and learning and struggling and failing and succeeding, and so forth.

So here goes…

Custom Polygraph

Back in December 2014, Desmos released Polygraph, a small collection of activities designed to develop students’ informal language into formal vocabulary. If you’ve ever played Guess Who, you’ll understand the rules right away: use yes/no questions to narrow down the field until you’ve singled out the winning face (or graph).

Near the start of the summer, Desmos released Custom Polygraph so teachers build their own polygraphs to address whatever vocabulary they want.

Polygraph: Histograms

Earlier today I built a Custom Polygraph focused on histograms. My hope is that it will get students talking about shape, center, spread, and related concepts/vocabulary. I’d love some feedback, especially if you use it with students (or if you play a practice round with colleagues).

You can take a look at it here:

Also, I used this graph to make the pictures:

In a future post I’ll share my thoughts on what makes an effective Custom Polygraph, though that’s a work in progress, so I welcome your own insight, either on Twitter or in the comments below.