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

Comments 7

  1. Great post again – thanks! You’ve reminded me of the importance of getting students to actually draw out the first few patterns. I really like the simple idea of folder a piece of paper into 4 and will be using it today!

    There are some videos on the nrich site here
    which I think are nice because they show students that there is more than one way of skinning a cat (sorry – English expression, hope that translates OK!)

  2. I have just discovered your blog and posts. I am a former high school teacher turned and now I work with lots of middle and high school teachers. You provide a rich source of reflective food for thought. I am guessing you may have seen Learning and Teaching Linear Functions (older source on DVD) by Branca, Seago and Mumme (sp?) which has a similar approach to using visual sequences to develop students abilities to understand linear functions and graphs.

    I wonder if you had some students make some visual connections. In your post above, I can see a way that students can visually see when the function is y = 3x + 2. One might ask, “Where is each part of that equation in the picture? Where is the 2 for example?” They can find where there are 2 circles, and notice that there is a way to see the 2 circles in each succeeding stage, that they are constantly there (pun intended), and they can also see that there is a 3 by x rectangular array of circles, and that the width of that array is increasing (hence that dimension changes and varies).

    Thank you for sharing your thoughts about having students draw the sequence for themselves, and then how you have them draw the next stage, and then leap to 10, make a table and a graph. Desmos looks like a great tool–I’ll check that out.

    Do you do this for quadratic functions too?

  3. Post

    @Deb, thanks for stopping by the blog. And double thanks for the comment!

    I’m not familiar with the resource you mentioned. It sounds like a great resource, though I like the price tag of Fawn’s website better than this:

    My introduction to these sorts of patterns (and extending them, then developing rules for them) began with a series of activities from the AIMS Education Foundation while studying the Grad Math program at Fresno Pacific University. I fell in love with figurate numbers there…

    We definitely move into quadratics. I suppose you could explore additional function types, though I feel the geometric/area connections that can be made with linear and quadratic visual patterns are the most enjoyable. In fact, we just had a conversation about this on Twitter last night! Here’s a link: (The comments I’m referring to arose during a chat with @reilly1041.)

    We usually discuss the geometric connection/interpretation in class, but it’s something I decided not to highlight in this post.

    Here’s a small set of images suggesting how such a quadratic discussion might play out in class or in a workshop:

    Thanks again for stopping by! Looking forward to more comments in the future. 🙂

  4. Wow-I had no idea about the price on the Learning and Teaching Linear Functions (LTLF). I better put my copy in a safe place, as apparently it is a collectors’ item! Last time I ordered the participant’s version, it was $15, and now it has gone to about $30, which is much better than the facilitator’s version you pulled up. Geez. The main difference between the two is that the facilitator’s version describes how to conduct a series of professional learning activities for teachers. On both there are videos of teachers implementing with students which are interesting, and materials for a whole unit (which are very similar to what you and Fawn already have). Here is the url (forgive me for not being as adept at making this a live link.),+Participants

    Nice graphic on the quadratic example. Thanks for sharing. I have used these successfully this way too.

    You probably have already done what I will share next, something I picked up from LTLF. One of the things that Branca and Mumme and Seago suggested, was to make the table have 3 columns, one in the center for the “Indicated Arithmetic”. I sometimes change that language in the heading for kids (“show you work”, or “computation”), but the idea is to have them write an arithmetic expression for how they can count the number of circles in the figure (without just counting each one by one), and then for them to use a common method of determining the number of circles in each figure resulting in similar expressions.

    It is a bit hard to explain here. I’ll refer you to a document
    that was created for teachers who, after doing some face-to-face pd, were going to team teach a lesson for students. It may make it clearer. If you have done something similar I would love to hear about it. I find this extra column useful for supporting students in moving to the algebraic, or to support those recursive thinkers (seeing the adding on pattern) to moving to an explicit function (seeing that multiplication can be used as well.) As I read a bit of Fawn’s website, I believe she noticed that delaying when they write the differences on the tables helps with that too.

    Thanks for the conversation!

  5. Post

    @Deb, everything you share reminds me more and more of how things played out in one of my classes at Fresno Pacific. We often added an “arithmetic column” to help generalize, and regularly used a “helping column” to “check the neighbors” (numbers above and/or below) and perform various other strategies aimed at finding the generalization numerically.

    There are so many valuable things to do with Visual Patterns. I really appreciate your comments, as they’re getting my wheels to turn about what else I can bring into my classroom.

    I hope this is the first of many conversations! Thanks again for sharing your thoughts. 🙂

    Take care,

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