Visual Patterns… Now What?!

If the response to my post from Tuesday is any indication, people on the Internet Machine love Fawn Nguyen’s Visual Patterns.


Let’s say you’re one of these folks, and your students are now rocking this sweet set of challenges. Now what?!


Well, for one thing, don’t stop! These are rich enough problems to keep bringing them before your students. (In fact, the real fun begins when we break out quadratics, including my personal favorite: patterns involving triangular and other figurate numbers.)

A Means To Another End

But I would offer that Visual Patterns are not only an end in themselves, but also a means to another end.

  • “An end in themselves” because, let’s face it, they’re awesome all on their own.
  • “A means to another end” because they provide students with experience in approaching mathematics through multiple representations. And this visual-verbal-numerical-graphical-algebraic tag team effort translates to new scenarios far more powerfully than a single representation would.

This last point was on full display this morning in Math B (eighth grade) as my students worked on Dan Meyer’s High School Graduation task.

Here’s a sample of how things went down:

now-what.002 now-what.003

Look familiar? I sure hope so.

After several rounds of Visual Patterns, students have developed a framework for translating a text-dense, potentially-intimidating task into something they can explore, something they can understand. In fact, once students had the table of values (which was admittedly a team effort), they were off to the races.

While students in past years were able to answer some of the numerical questions (when did the name-reading begin/end), they typically struggled to do anything more than that, and were at a loss when it came to writing an equation to model the scenario.

A Well Worn Path

So why were my students this year able to hack it? Because we’ve worn that visual-verbal-numerical-graphical-algebraic path so well in just a couple of weeks that moving from one representation to the next—and turning back to make connections among various forms—is becoming second nature.

And while there’s more than one way to foster this kind of connected thinking, I’ve found Visual Patterns to be among the most engaging, powerful, and effective.


As you can tell, I’ve had fun with Visual Patterns this week and last. I have one more post in me on this topic, then I promise I’ll shift my rambling to something else. :)

10-Second Pause

We’ve been shoring up our differentiation and integration skills in AP Calculus. During the last two class sessions, I’ve intentionally avoided whole-class review. I wanted students to wrestle individually and in small groups.

Today, however, I shifted back to a handful of “let’s walk through these together” exercises for the first 15 minutes or so of class. But I added a twist…

2015-02-25 15.43.02

At the beginning of each exercise, I asked students to rate their understanding of the problem we were about to attack:

  • Beginning (B)
  • Developing (D)
  • Proficient (P)

These are the same descriptions I use on the proficiency scale for my SBG assessments, so students are familiar with them.

2015-02-25 15.48.03

Then, after walking through the problem as a class, I asked them to rate their understanding again.

2015-02-25 13.20.46

My goal? To push my students a little further down the road of reflecting on their understanding. In particular, I wanted them to have a sense of what they need to work on prior to our assessments at the end of the week.

I’m hopeful that the 10-second pause on each problem gives them some valuable insight, and possibly some more informed motivation for what comes next. Better yet, maybe this is something they’ll apply on their own initiative in another situation, whether in my classroom or another one.

Visual Patterns + Desmos = Amazing!

I’ve been a fan of Fawn Nguyen’s for several years. I use resources from the website on a regular basis in my own classroom and in teacher training. The conversations are always excellent, and the emphasis on multiple-representations is a huge benefit to students wrestling with ideas in an all-too-often isolated context. (Plus, creating your own patterns is a blast!)


I took the reins for a middle school math class a few weeks ago. Our emphasis for the past couple of weeks has been CCSS.8.F, and linear-based visual patterns have been a key part of our exploration.

I’ve abandoned Fawn’s original handout, and even the modified version I created a couple years ago, and instead launch each visual pattern by having students fold a blank sheet of paper into quarters.

The other element I’ve incorporated into my visual patterns routine this year: Desmos.

The New Play-by-Play

Here’s how Visual Patterns plays out in my classroom these days:

1. Setup

Distribute a clean 8.5 by 11 inch sheet of printer paper to each student. Students fold the paper in quarters, then unfold.


The beauty here is that my preparation for visual patterns no longer involves a trip to the copier. Instead, I grab a ream of paper, three-hole the whole stack, and we’re ready to rock for quite some time.

2. Draw What You See

Next, I display—one at a time—the images for Stages 1-3. Students are required to draw each stage in one quarter of their paper.


My goal for these three rounds of “draw what you see” is  to force students to attend the the structural details of the pattern before they begin extending the pattern visually or describing the structure verbally.

3. Predict and Describe What’s Next

Next, I display the following…


…and ask students to sketch and describe Stage 4. Their recent investment in observing the structure of Stages 1-3 usually pays dividends in Stage 4, both in making the predictive sketch and in describing their rationale.


After a moment or two, I collect a few responses, recording them in a Keynote slide. (Note: I only do this for some of the challenges.)

4. Fast Forward to Stage 10

This is where the rubber meets the road. Can students extend the pattern beyond simply “the next one”?


We flip the paper over and use the top left quarter as work space for figuring out how many items are in Stage 10. Some students sketch the image. Others wrestle numerically. Others skip this quarter for a time until they’ve done more work elsewhere.

5. Represent!

As I mentioned above, one of my favorite things about Visual Patterns is the way these mini-tasks lend themselves to multiple representations. Here’s what we do with the remaining quarters on the back:

Make a table (and find the rate of change, for linear patterns):


Sketch the graph:


Write an equation:


6. Desmos!

At some point, students fire up Desmos on a phone, tablet, or laptop to confirm their results.

Screen Shot 2015-02-24 at 4.06.12 PM

Aside from general Desmos-awesomeness, there are a few specific benefits here:

  • Students confirm the numerical work they’ve summarized in the table. Errors in a sequence are often easier to spot in graphical form than numerical form. Adding a table to the expression list while keeping an eye on the coordinate plane helps students identify potential errors in pattern-extending and/or arithmetic.
  • Students confirm the equation they’ve found actually fits the numerical data. I derive more than a little satisfaction from watching a line or curve pass through a set of ordered pairs? Based on my students’ reactions, I am not the only one.
  • Students tweak the window in order to confirm and/or help create their on-paper graphical representation. I’ve encouraged students to apply the “fill the frame” advice heard in photography circles as they make their window adjustments.

The End Result

I use Scannable (a free iOS app from Evernote that makes scanning and saving dead-simple) to capture 2-3 samples of student work. Here’s one in its entirety:

visual-patterns-student-work.006 visual-patterns-student-work.007

Task Delivery: Less is More

I can’t tell you how many times I’ve taken a solid task and whittled it away to almost nothing.


It’s easy to fail with a terrible task. But over the past few years, I’ve also found a number of ways to flame out with lessons that were packed with potential.

The Main Culprit?

My inability to “let go” during the launch has derailed more than a few lessons. Picture me as the 4×100 relay member who won’t let go of the baton, causing the team’s chances to crash in a heap of flailing limbs.

My Prescription

I’ve been stretching myself in recent months by giving as brief an introduction as possible before getting out of the way. It doesn’t always work out, especially if the task itself is unclear. However, sometimes the results are fantastic, as was the case last week with my Math B class (mostly 8th graders).

I distributed student handouts for Battery Charging (an Illustrative Mathematics task), asked them to read the directions to themselves, then directed them to work in their table groups. I announced: “You’ll be on your own for the first 10 minutes.” I then stepped out of the way and watched as they struggled, some frustratingly, and others very productively. Some even finished the task with half of this “introductory time” remaining.

Stepping Back In

At the end of the 10 minutes, I re-engaged, offering guiding questions to struggling groups and pushing those who had already finished to solve it using another approach. Eventually, we drifted toward Desmos as a way to summarize our findings in different representations. This stage of the lesson—synthesizing, connecting, closing—is another element with room for improvement, but I find I do less damage here than in the launch (thanks in part to the 5 Practices).


What About You?

Is lesson launch a place where you struggle? If so, try launching your next task with as few words as possible. (Better yet, try launching something without saying anything more than, “Go!”)

Are you adept at setting rich math tasks in motion? Drop a line in the comments to share your wisdom.

Do you struggle with the all-important elements at the end of a lesson? Or have your abilities here grown in recent years? Either way, I’d love to hear what you’re doing well and what you’re looking to improve.

“Everybody stand up…”

Sometimes the saying “better to be lucky than good” applies to teaching as well. Today I stumbled across a new routine by little more than blind luck.


Inspired by the sleepy looks on several faces, I interrupted my middle school class with a shout: “Everybody stand up! Head to the back of the room. Make a circle around those two tables.”

At this point, I had no idea what we were going to do. But it was going to be on our feet and it was going to involve everyone.

On the way to the back of the room, I snagged an empty water bottle. And then…

Round 1

Holding the plastic bottle in my hands, I announced: “2, 4, 6.” Then I passed the bottle to the student on my right, and gave her no directions.

Her response was beautiful: “2, 4, 6, 8?”

“Nice. But leave off the 2, 4, 6. Just say 8.” We started over. “2, 4, 6.” Then, “8.”

“Alright! Pass it along.”

The next student: “10.”

And with that, the rhythm was established. We went all the way around the circle. And guess what?! Eighth graders can count by twos!

Round 2

With the bottle back in my hands, I started a new routine: “5, 10, 15.” But then I passed it off to the left. And they rocked this direct variation sequence just as easily as the first round.

Round 3 (and a surprise!)

“Okay, let’s ramp up the difficulty just a bit. Ready? Here goes: 1, 4, 7.”

I passed the bottle along (back to the right now), and with no hesitation: “10.”

Then followed 13, 16, and 19 without any trouble. And to be honest, much of the progress was smooth, as you’d hope for a group of middle schoolers.

But once every third or fourth student, there was a pause. Not a long one. Not necessarily awkward. Just a pause. And that up-and-to-the-left-as-if-the-answer-is-on-the-ceiling look that means someone is lying (or telling the truth; I can never remember). There was a fair bit of whispering, followed by a shout: “20… 21… 22!” And even some twitching fingers as students accessed old-school strategies for continuing the pattern.

This was magic for me. I’ve only been teaching this group for about three weeks. (It’s a long story.) As such, I don’t know their strengths and weaknesses quite as well as if I’d been their teacher all year. But this simple activity gave me instant insight into the basic number sense skills my students possess.

There was another bonus at the end of this round. We briefly discussed the “starting number” and the “change” (1 and 3, respectively). Since we’ve been rocking linear visual patterns recently, we turned this into the equation y = 1 + 3x rather quickly and moved on. (Assuming that we’re beginning with the zeroth term here.)

Round 4 (another surprise!)

We had time for one more: “5, 9, 13.” I passed the bottle left, and we were off. “17,” “21,” and so forth. But then we hit a snag. Someone forgot the previous numbers. So we invented a new rule: If someone gets stuck, they can ask the previous three people to repeat their numbers. No other hints are allowed.

On track. Off track. Hint. Back on track. And so on until we make it back to the beginning.

Looking Ahead

I’m excited to try this again next week. I’ve already started thinking about ways to adjust and/or extend:

  • Introduce a higher starting number, and/or negative change. (A student actually suggested 10, 8, 6, etc. as we wandered back to our seats.)
  • Introduce sequences involving fractions or decimals.
  • Invite students to generate the pattern by kicking a round off with their own sequence of three numbers.
  • Mix things up—and simultaneously encourage more students to focus on each response—so that if a student needs help, I call on a random student to repeat the last 2-3 numbers.

One More Thing…

I can’t help but think I may be subconsciously ripping off Sadie Estrella’s counting circles here. Whatever the case, I’m excited to see where this routine leads us in the weeks ahead.

If you do something similar with your students, or if you decide to give this a try with your own class, drop a line in the comments so we can benefit from your experience.

And if your name is Sadie and you hail from the lovely state of Hawaii, there’s a special spot in the comments reserved just for you. Let me know what you think!

Blog Recap • January 2015 Edition

January was a busy month on the blog. Here’s a recap of the lessons, posts, and other math-related things that filled up those 31 days.



  1. Match My Line. The start of this recent “Match My Graph” madness.
  2. Match My Parabola. If the reaction on Twitter is any indication, hands down the most popular lesson I’ve posted.

Top Posts

  1. 5 Things Every Teacher Should Know About Twitter. A quick introduction to using Twitter. By no means comprehensive, but a nice start for those interested in expanding their online PLC.
  2. Two Wrongs and a Right. An error-analysis routine inspired by Michael Pershan’s work at
  3. Age-Appropriate? Not Exactly. Valuable? Absolutely. A linear graphing challenge experiment gone… What’s the opposite of “awry”?
  4. Twitter Chats vs Family Dinners: Do We Really Have to Choose? I love Twitter. And Twitter chats. But family life means I usually choose not to participate. Enter #slowmathchat, a weekly chat not tied to a particular time of day.
  5. Four Points, One Line. A work-in-progress linear graphing challenge for students, with a goal of eliciting a variety of equation forms.

Other Posts

The rest of the posts from January are available here.


Community High School District 117
January 16, 2015 • Chicago, IL

San Dieguito Union High School District
January 27, 2015 • Escondido, CA

5 Things Every Teacher Should Know About Twitter

Over the past few weeks, several people have asked me how to take the next step with Twitter, whether that meant joining for the first time, or taking their tweeting to the next level.

Assuming you already have an account (if not, go here), here are five things every teacher should know about Twitter.

1. Don’t skimp on your profile setup

There’s a lot to do when you first join Twitter. It’s easy to stop halfway and think, “I’ll go back and add that later.” Before you dive any deeper into the world of Twitter, make sure you’ve at least taken care of the following:

  • Add a profile photo. I recommend adding a photo of yourself (ideally, of your face). You don’t need a professional headshot, just something that adds a human element to your online interactions.
  • Add a profile description. A few words about your passions and the various hats you wear (professionally and/or personally) will help the Twitter world know who you are and what you’re about. If you’re looking to expand your online learning community, a description-less profile will hold you back.


2. Learn the different message types

Not all messages on Twitter behave in the same way. Here’s what you need to know:

  • Regular tweets. These will be seen by everyone who follows you.
  • Mentions. You can “mention” someone by including their name somewhere in the middle of your tweet. All of your followers will see these messages, and a special “ping” will go out to the person you mention (even if they don’t follow you).
  • Replies. When someone mentions you in a tweet, you have the ability to reply. Only those who follow both you and the person you’re replying to will see your message, unless…
  • Dot replies. If you add a character before the @ symbol in your reply (the most common approach is .@, or the “dot reply”), then all of your followers will see your message. (Warning! Use dot replies sparingly, as many of your followers will only see half the conversation—and that can be awkward, confusing, and annoying. The best use of a dot reply is to share a comment—or better yet, a resource—that would be valuable to many of your followers.)
  • Direct messages. If you follow someone on Twitter, and they follow you, you can send private tweets (called direct messages, or DMs). While some things are best shared privately (maybe an email address, or a link to a private document), unless there’s a specific reason for talking on the “down low,” I prefer mentions and replies rather than direct messages.



3. Hashtags are a simple way to achieve Twitter Ninja status

Ah, the hashtag. A source of much eye-rolling for those older than 17 and not named Jimmy Fallon. Believe it or not, hashtags have the potential to change your life as a teacher. Here are two reasons why:

  • Communities gather around hashtags. Have you ever seen #MTBoS at the end of a tweet? It stands for mathtwitterblogosphere, and it’s a way that a rather large and decidedly amazing group of math educators share ideas, questions, lessons, activities, and feedback with one another. Have a question about standards based grading? Include #sbgchat in your tweet. Need some advice or feedback from (fill in the blank), chances are there’s a hashtag that will allow you to speak into an ongoing conversation within a particular community.
  • Hashtags are used for weekly chats. Did you know that every Sunday night at 8 pm (PST), hundreds of passionate educators gather for an hour to discuss the latest developments in the California education landscape? It’s one of hundreds of education-themed chats taking place during each week, and all you need to get in is the “hashtag key” (in this case, #caedchat).
  • Bonus. For math teachers who can’t commit to a chat at a particular time of the day/week, check out #slowmathchat (details here)


UPDATE: Shortly after sharing this post online, I participated in a discussion of Twitter etiquette. We briefly discussed dot mentions, hashtags, and retweets. Kate Nowak captured the gist of our conversation with this comment about intentionality:

4. Follow to your heart’s content

I used to watch my “following” count, thinking that if it got too high I wouldn’t be able to keep track of everything. Recently, I let go of that fear and started following whoever I wanted to, whenever I wanted to. Here are two thoughts that may encourage you to do the same.

  • Skipping tweets is not a sin. I’ve tried two approaches: (1) Limit who I follow and read every tweet, and (2) Follow whoever I want and read as much—or as little—as I want. I’ve found this second approach to be particularly helpful, especially since you can…
  • Add your favorite Twitter folks to a list. Let’s say you’re following several amazing people (just for illustration, we’ll say @jstevens009, @mrvaudrey, @mr_stadel, @fawnpnguyen, @robertkaplinsky, @ddmeyer) and you don’t want to miss a single thing they say. Add them to a list, and add that list as a column in Tweetdeck. (Again, more on that in a moment.) Voilà! Now you have a slow-and-steady stream of goodness from your favorite folks on Twitter, without restricting the overall number of people you follow.

Here’s how to add people to lists in Tweetdeck:


5. Get better software

Twitter works in a standard web browser. It has its own free app for smartphones. It even works on flip phones (via text messaging)! But you’re shortchanging yourself (in my opinion, anyway) if you don’t download and use these tools:

  • Tweetdeck. An application for Macs and PCs, this is essential if you want to track multiple hashtags/chats/lists with absolute ease. Oh, and it’s free. You can learn more and download it here.
  • Tweetbot 3 for iOS. Are you using an iPhone? The official Twitter app has its pluses and minuses. But in my opinion, nothing on the iPhone comes close to Tweetbot. While it’s not free, when you’re talking bang-for-your-buck, this $4.99 is well worth it. You can learn more about the app right here.
  • Android users… I don’t have an Android phone (at least not after that incident with the washing machine), but word on the street is that Talon and Tweetcaster are worth a look.

Parting Shots

If there’s only one thing you pick up from the list above, make it Tweetdeck. You’ll probably be able to get up and running on your own. If not, come back to the blog in a week or so (or subscribe via email—there’s a form on the top right of the blog) for a walkthrough of how to set up Tweetdeck from scratch.

Update: One Two More Things…

Embracing Mistakes with a New Weekly Routine

Over the last 11 years in the classroom, I’ve had the same teaching schedule exactly zero times. Every time August rolls around, it’s a different ballgame. Some years the shifts are subtle. In other years they’re quite dramatic.

Within each year, I’ve never taught more than two sections of the same course. In fact, one year I taught seven different courses. (Assuming, of course, that you’ll allow me to count Algebra 2 Honors as distinct from Algebra 2.)

With all of this variety from year to year (and class period to class period), I’ve come to rely on classroom routines not only to preserve my sanity, but also to challenge myself to pursue growth.

The latest initiative I’m considering? It’s right in line with my recent bent on extracting every ounce of growth-mindset goodness from the abundance of mistakes we (teachers and students) all make in the classroom.

With that background in mind, here’s what I intend to put in front of my students (and myself) during the last few minutes of class each Friday for the rest of the semester.


There’s no guarantee I’ll stick with this routine, or that we’ll benefit from it. But I am holding out hope—on both counts, in fact.

As for my actual goals with this prompt, I’m looking to further develop my students’ growth mindset. In particular, I want to foster a sense—across the entire classroom—that mistakes are actually opportunities, and that they can help launch us into the next level of awesome-ness.

As for sharing my thoughts here… There’s a sense of commitment that comes with writing about something.

I’m counting on you to hold me to it in the comments.

Twitter Chats vs Family Dinners: Do We Really Have to Choose?

I love Twitter. I think I’m on record in several places with this. (Actually, it’s probably more along the lines of “I love the Twitter,” just because it’s more fun to say it that way. Go ahead, give it a try.)

I love Twitter chats. This is probably less well known. And the less-well-known-ness is likely a result of my general lack of participation in Twitter chats.

Here’s a run through of the scheduled chats I’ve (in)frequented over the past year or so:

  • #alg1chat (Sunday, 6 pm PST)
  • #caedchat (Sunday, 8 pm PST)
  • #msmathchat (Monday, 6 pm PST)
  • #connectEDtl (Tuesday, 7 pm PST)
  • #calcchat (Thursday, 6 pm PST)
  • #paedchat (Thursday, 6 pm PST)

Now consider the fact that I love eating dinner and that I have four small children and you might begin to see where I’m headed with this.

As much as I enjoy these conversations with digital colleagues, it’s a huge sacrifice to drop what I’m doing to engage online during these key family hours. Between dinner time and bath time and story time and bed time there are enough “times” in each evening without adding a “chat time” to the list as well.

So here’s my solution:

Launch a new chat, one that borrows the structure and intentionality of a scheduled chat (e.g., #caedchat), but marries that with the freedom and flexibility of an ongoing conversation (e.g., #MTBoS).

Intrigued? Click the #slowmathchat tab in the menu for more details. Read up, get psyched, and add a new column to your Tweetdeck. I hope you’ll consider joining in the fun this Monday.

Post Script

Before you go… Consider this a reverse-tl;dr:


Four Points, One Line

As my students have worked through a series of linear graphing challenges this month, I’ve been looking for a way to challenge them to synthesize (and hopefully even extend) what they’ve noticed over the past few weeks.

I think I’ve found my culminating challenge.

Four Points, One Line.001 Four Points, One Line.002

My Goal

My goal is to elicit a variety of equation styles (point-slope, slope-intercept, etc), and my hope is that the restriction (which numbers they may use in the equations) is not only clear enough, but also provides the right dose of structure to encourage students to think more deeply about the relationships between the rate of change, intercepts, non-intercept points, and the parameters in each equation.

To give it a test run before sharing it in my own class, I hereby offer you this:

Your Challenge

How many different equations can you write using only the numbers included in the ordered pairs? Can you get to three? How about five? Maybe even 10? Or more?!

Do the work in Desmos, and drop a line in the comments!

As always, feedback—on the challenge in general, or the restriction in particular—is 100% welcome.


I struggled with the wording in the original challenge. As I shared above, my goal is to draw out from students a variety of equation forms, each one utilizing information revealed by a particular point or pair of points. After some back and forth on Twitter, I settled on this reframing of the task:

Four Points, One Line.003

I’d love to know whether you think that drives more quickly and clearly to the heart of what I want students to focus on (while leaving it open enough that students will feel freedom to tinker and explore).