I’ve been a fan of Fawn Nguyen’s visualpatterns.org 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:
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.
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:
At some point, students fire up Desmos on a phone, tablet, or laptop to confirm their results.
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.