ZPC Podcast Appearance

Earlier this week I gave a talk at the AIMS Center on using Desmos to facilitate classroom discussion around digital activities. While on campus, I stopped by their podcast booth for a 20-minute conversation with Chris Brownell.

Dr. Brownell is the Director of Mathematics and STEM Education Programs at Fresno Pacific University—home of the AIMS Center—and also the host of the podcast Zone of Potential Construction. (He’s also a former instructor of mine.)

If you’re interested, you can listen to the podcast here.

You can also subscribe to the podcast. Just search for “zone of potential construction” in your favorite podcast app.

WNBA Scoring Averages – Activity Analysis

Here’s an image from an activity (WNBA Scoring Averages) that I wrote earlier this year.


When we dropped the activity into the search pool at teacher.desmos.com, I didn’t think it was a world-changer, but I was pretty sure it wasn’t untouchably awful.

Hang on a sec while I fire up our database to check how many times it’s been run since it arrived on the scene in May 2016 .

Seven, eight, carry the one… Round up for good measure… Almost there… Aaaaaand…

Zero. Zero sessions.

Needless to say, that’s not a lot of sessions.

So what went wrong? I’ll offer a few theories, and along the way highlight a couple of things that might be worth preserving. I’d also love to hear your thoughts on how to make this—and activities like it—better.

Screens 1-3

Bummer. 27% of the activity just to set the context. That’s killing my discussion-to-screen-count ratio.

screenshot-2016-09-21-21-49-51 screenshot-2016-09-21-21-49-53 screenshot-2016-09-21-21-49-55

Screen 3 (above)

Before asking for a calculation, ask for a prediction. Nice!

Screen 4

An opportunity for students to wrestle with the meaning of a scatterplot in a setting they haven’t seen a thousand times. Also, a chance for teachers to highlight interesting responses, starting with informal ones and progressing to formal ones. That being said, I can’t shake the feeling that nobody cares about the patterns they see on this screen.


Screen 5

This screen’s a mess. “The red point means WHAT? My original prediction? I can’t remember that!” And a purple point that I might move? This is just awful. (P.S. What’s the input field for on this screen?)


Screen 6

I can just hear the conversation now… Me: “In row 1, write the equation…” Students: “Why?” Me: “Because I said so.” That line hasn’t been working with my three year old twins at bedtime, and I don’t think it’s going to play out well in the classroom. There’s got to be a better motivation for plotting this line now, something beyond the “trust me, just do it” rationale I offer here. Maybe the solution is to skip having students enter the line, and just jump from Screen 5 to 7?


Screen 7

Continuing that thought… I propose deleting Screen 6, and revising Screen 7 to read: “(1) How do the points compare to the line y = x (shown in black)? (2) What does this mean in context?” Even still, I’m not sure how strong a screen that leaves us with.


Screen 8

“Boom, a line of fit! Now use it! ” This screen reveals a couple of major open question I have about modeling activities in Desmos Activity Builder. (1) Should students generate the line or curve of fit? Always? Sometimes? Never? I think “sometimes” is the right answer here. But I’m not clear (yet) on when it’s the right move. Here I offer it to students “for free” so they can focus on using and interpreting the line. But I’m not sure that’s the right move. (2) Many of the modeling tasks I’ve build in Activity Builder feel a little too linear and granular (as in, one tiny step, then the next tiny step, and another, and another, with all students moving through in the more or less the same manner and sequence). Is this the best approach? I doubt it. But what’s the alternative? I haven’t figured that out yet.


Screen 9

Interpret the parameters in context. Yay!? But—students had no role in generating that value, so it feels uninteresting to ask about it. I’m wondering about an alternative screen—or sequence of screens—where students use sliders to create their own line of fit. Then answer a question or two about the meaning of their parameters in context, ideally in contrast to a classmates’ parameters. Hm. Still feels lackluster.


Screen 10

I’ve thrown this discussion-prompting screen into a handful of activities over the last several months. Do you think it’s helpful? My hunch is that some teacher tips go unread, but a student-facing screen encouraging discussion at this point is sure to be noticed.


Screen 11

As was shared recently over on the Desmos blog, we strive to activities that are easy to start and difficult to finish. Low floor, high ceiling, etc. Extension screens are one small part of that puzzle, and I’ve dropped one in here to wring a little more value out of the context, and to spark (possibly) a conversation about residuals. I like this screen. At least I think so. I’d love to know what you think.



I’ve rambled a bit about what I do like, and what I don’t like. And in the latter, hinted at ways that I might be able to improve this activity. Unfortunately, even with those improvements, I don’t think this is a particularly strong activity. It lacks a compelling problem. There’s very little cognitive conflict, even with the predictions at the beginning. (Maybe I could hold those up at some point early on: “Here’s your prediction. Here’s all your classmates’ predictions. Let’s see who’s closest!” Or maybe not.) Students don’t have very many opportunities (if any) to be right or wrong in interesting ways.

Over To You

So what do you think? Is this salvageable? What does it suffer from in its current form? What strengths does it have, and how could we build on those?

Thanks in advance for chiming in!

Home Run Kings – Activity Analysis

Earlier this year, I wrote a Desmos activity called Home Run Kings. Here’s the blurb from the activity page:

In this activity, students interpret quantitative data in order to predict whether Bryce Harper—a promising young professional baseball player—will break the all time record for most career home runs.

I like some things in the activity. I’m not so sure about others. I wonder if you’ll help me give it an upgrade?

I’ll start by adding some screen-by-screen commentary. After that, here’s how you can help. Let me know…

  • What you like
  • What you don’t
  • What you’d add/edit/remove

Bonus points if you try this out with students and share a summary of their feedback!

Screen 1

A few words (and an image) to set the context.


Screen 2

I offer students a bit more context (in this case, a graph showing Harper’s home run totals for the first few years of his career) and ask them what they notice. While I’m interested in the full range of responses, I’m expecting quite a few students to focus on the big jump from 21 to 22 years.


Screen 3

Next up, I use a sketch screen to capture informal student thinking about the relationship between home runs and player age. One of the things I love about sketch is that students don’t have to worry about function families, equations, formal domain restrictions, or anything. Just sketch the relationship. (Side note: Don’t conflate formality with richness, here or in other activities. There’s plenty of fodder for rich discussion, uncovering misconceptions, and developing ideas in informal student responses—sketches and otherwise. Of course, building toward formality is a noble goal, but informality is a great place to build from. This concludes my soap box tangent.)


Screen 4

We’ll circle back to Bryce Harper in a moment. But first, a screen to draw out student observations on a pair of graphs showing full (and home run-prolific) careers. The heart of this activity is all about interpreting graphs in context. My hope is that this screen helps move students along toward that objective.


Screen 5

Here I bring Harper back, with five other players’ career totals shown. I’m concerned that there’s too much going on visually on this screen. Would you second that thought? Or push back against it?

Concerns aside… This screen asks students to use the graphs to pick a side and defend their answer. Actually, it asks them to play their own devil’s advocate and construct an argument on both sides. Too much for one screen? Again, I’d love your input here.


Screen 6

The reveal. Not as flashy as some other things I’ve seen online. All I could muster is a screenshot. Any thoughts on how to improve the reveal here? Or does this simple approach serve its purpose?


Screen 7

One challenge I’ve had in thinking through how Desmos activities might play out in other teachers’ classrooms is how best to communicate “Hey, a discussion would be really great right here!” We use teacher tips to that effect. (Successfully? I’m not sure.) But I’ve also tinkered with a discussion-prompting screen like this one a few times as well.


Screen 8

I’ve made a habit of including an extension or two at the end of Desmos activities to allow students who finish a bit earlier to occupy themselves with something related and worthwhile as classmates finish the core part of the activity. And to allow teachers to assign some followup thinking/exploring for home.

I’d love to hear your thoughts on that approach in general, as well as how it plays out in this particular activity. (Though, based on the screen title, it seems I had plans for a second part of the extension that I never got around to building.)


Your Turn

What did you like/dislike? What would you change? Thanks in advance for your comments below and on the Twitter.

Dusting Off the Blog

Hi there!

It’s been a little while since I posted anything here. That’s about to change. Here’s why, in bullet-list form:

  • When I started this blog, it was a place for me to share and reflect on what I was doing in my classroom. Successes. Failures. And everything in between.
  • When I joined Desmos last August, I was no longer sure what to write about here. (Maybe I was afraid to share my failures in a new role?)
  • I’m learning as much as ever from my colleagues—at Desmos, on Twitter, and at conferences.
  • I’m dusting off the blog so I can reclaim this space as an opportunity to share, to reflect, and to learn.

So stay tuned for a slight uptick in posts. I’m looking forward to the discussions that will follow.

Conference Time!

If you haven’t picked up on this yet, I love math education. And while I love connecting with folks on Twitter and through blogs, conferences are the absolutely best. Hands down, no contest.

I’m pretty pumped for this week. Here’s a few reasons why:

  1. NCTM in CA! I get to attend the biggest math conference in the land—in my home state!
  2. Desmos Happy Hour and Trivia Night. Sessions and talks are great. Conversation with colleagues is even better. Plus, trivia! Thursday, 6:30 pm, SoMa StrEat Food Park.
  3. Ignite talks. 10 presenters. Five minutes each. 20 slides that auto-advance every 15 seconds. I gave an Ignite talk at CMC North in 2014, and it was both terrifying and exhilarating. Hoping for a repeat (well, at least the exhilarating part). Also hoping to see you there. Friday, 9:30 am, 134 (Moscone).
  4. My Journey… The #MTBoS has had a massive influence on me over the last three years. Here I’ll share some insights from the journey. Friday, 12:30 pm, 3003 (Moscone).
  5. The Desmos booth! In addition to the usual Desmos booth goodness (free swag! new features!), we’re hosting Activity Builder office hours. Stop by Friday between 1 and 5 pm to hang with members of the Desmos Teaching Faculty. Bring an in-progress activity, an idea, a question, or all of the above!
  6. ShadowCon. I missed this at NCTM Boston, and am so excited to hear six inspiring speakers share their passion and issue a call to action. Friday, 5 pm, Marriott Yerba Buena 7.

Looking forward to seeing you there!

And if you can’t follow along in person, consider keeping an eye on #NCTMannual.

Coin Combinations



What do you notice?

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

What do you wonder?

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

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

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

Sequel #1

Then I noticed something else on the machine:


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

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

Sequel #2

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

Post Script

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

Tesla Model 3

I’m fascinated by the pre-order hype surrounding Tesla’s latest car, the Model 3.

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


Throw that image on the screen and ask students:

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

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



When they’re ready for the reveal…

Sequel #1

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

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

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

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


Sequel #2

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


Based on the information in the comment above:

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


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

Cue Rockstar Math Edition

Last February I attended and presented at the CUE Rockstar event in Petaluma, CA. It was a blast.

The only way it could have been any better was if it was 100% focused on math. (No offense other disciplines. Math is just my first love when it comes to teaching.)

Well lo and behold, guess what’s coming up on May 14-15?! An all-math CUE Rockstar!

And check out the crew:

Screen Shot 2016-03-17 at 9.26.31 AM

Not too shabby, eh?

If you’re going to be anywhere near Los Gatos, CA in the middle of May (or if you’re willing to make the trek so that you are close to Los Gatos for that weekend), head over to cue.org/rsmath for info (schedule, speaker bios, registration details, etc).

Hope to see you in May!

Points, Lines, and (Maybe Not) Inequalities

One of the things I love about Twitter is the opportunity it provides for feedback on creative work. And thinking about teaching and learning mathematics is some of the most creative work I know. I’m still amazed when folks I’ve never met in real life take time out of their day to help me improve.

Here’s a case in that beautiful point.

Points, Lines, and Inequalities

Earlier this week I created an activity designed to help students make connections between individual points and graphs of linear equations and inequalities.

Here’s the first screen:

screen 1

On its own, pretty simple. But imagine a class full of students doing this. And then imagine directing their attention to a graphical overlay of every student response.


Now we’re talking. Literally. At this point, I’m asking students:

  1. What do you notice?
  2. What do you wonder?
  3. What do these points have in common?
  4. What do they not have in common?
  5. Is there a way we could summarize all of these points algebraically?

And then… Turn them loose on this:

screen 3

My hope is that the previous discussion laid the groundwork for the idea that these are all points where x equals 3, so x = 3 is an (obvious?) way to summarize them algebraically. That’s how it plays out in my head, anyway. More on that in a moment.

Students then move on to the next (rather similar) prompt:

screen 4

(And the activity continues in a similar fashion for a total of six scenarios—four lines, two inequalities. For a closer look, check out the links at the end of the post.)

Room for Improvement

Let’s circle back to Twitter. Yesterday I shared a link to the activity, invited folks to give it a test run, and hoped a few might offer some feedback.

The response proved incredibly helpful. In particular, Bowen Kerins offered not one or two, but nearly a dozen comments about what works, what doesn’t, and how I could make this activity even better.

I have a few takeaways from that conversation about how to make this activity better. I’ll share them here in order to clarify (for myself) and share (with others) some design principles that I think may prove helpful the next time we fire up the old Activity Builder.

I’ll share the first takeaway here, and a couple more in the next few days.

Takeaway #1

The cycle in screens 1-3 (place a point, imagine all the points, summarize algebraically) is missing a crucial step. It begins simply and somewhat informally, which is helpful. And then it invites students to imagine/predict, which I also think is helpful. But then it jumps straight to the summary, without pausing long enough on the “discussion” that I described above.

My current fix. Insert this between screens 1 and 2:

new screen

By asking students to name these points (note: not just their own, but also several others), I think they’ll be more likely to see (or “hear”) the repetition that leads to our algebraic summary later on:

  • “x is 3, y is mumble-mumble…”
  • “x is 3, y is something-something…”
  • “x is 3, y is whatever…”
  • “Hey, x is always 3! So then, x = 3!”

My future fix. This activity uses an existing (and lovely) feature called copy previous, where whatever a student did on one graph screen can be pushed ahead to (or “served up” on) a later one. You’ll see this in action between the screens 1 and 2 in my original activity.

What I would love here is another feature (even more lovely?) where the graphical responses from every student are served up to kiddos on a later screen. In other words, bring the graphical overlay feature that we have in the teacher dashboard into Activity Builder itself. Apply that to my original screen 3, and I think we’re in much better shape.

And the good news? Something like this is already in development. (Side note: I continue to be amazed at the engineering skill—wizardry?—of my colleagues.)

Closing Thoughts (For Now)

I think these proposed changes will prove helpful. But I also think they stretch the activity out a bit. So maybe it’s not realistic (or even helpful) to work through vertical lines, horizontal lines, linear functions, and linear inequalities all in the same activity. Maybe this is two separate activities. Or even three.

I take some comfort knowing I’m not the only one who thinks a narrower focus here could be helpful.

With that in mind, here is my new-and-hopefully-improved activity. Give it a whirl, and let me know what you think in the comments.

Next Time

I’ll share some thoughts on making the middle section (linear functions) stronger.


I almost forgot! You can get the original activity here. Or try it out as a student here.

Sliders with a Purpose

Back in August 2015, Desmos released its Activity Builder, a tool by which teachers (and Desmos folks like myself) can build custom Desmos activities. Over the past few months, I’ve seen a number of Activity Builder screens like this:


In fact, I’ve created a number of Activity Builder screens like that.

However, I’ve also seen quite a few screens like this:


Notice the difference? It’s subtle, but powerful.

The ingredients are largely the same, but the vague objective is replaced by a clear target.

In each case, my followup screen probably looks something like this:


In other words, my ultimate goal here is to invite students to observe and then describe a parameter’s impact on a given graph. But my path to that end has shifted from “poke around and see what you see” to “complete this specific task, now reflect on how you made it happen.”

Here are a couple more scenarios where I’ve found this approach to be helpful:

image4 image5 image6 image7


So, what would you do with this?


P.S. For a closer look at the screens above, go here.