Reason and Wonder

Visual Patterns + Error Analysis

Most of the work my students do with Visual Patterns occurs in small groups, where collaboration not only is encouraged, but often required. Today I steered things in a slightly different direction.

I wanted to see what students were capable of on their own. No peer discussion. No Desmos. I was particularly interested to see whether they could write an equation to describe the relationship they see unfolding visually, numerically, and graphically. Some could. Some could not.

As I walked throughout the room, I took note of the incorrect equations students were writing down. There were four in particular that caught my eye. Students were clearly doing some relevant work with calculating slope and (trying, anyway) to identify the y-intercept… However, the way they mashed it all together left something to be desired.

On the Fly…

So as my last few students were finishing their work, I threw together this slide, featuring the pattern they had been working with, four “out-in-the-wild” incorrect equations, and a not-so-accidental suggestion that one of the equations is correct (just to keep ’em on their toes):

We then took a blank sheet of printer paper, folded it in half, and unfolded it. Two workspaces on the front, and two on the back. We then filled each workspace with an equation as well as a table of the actual values…

…plus a show-your-thinking-on-the-page run through of each x-value, evaluated in the given equation.

In each case, the expression values didn’t match the actual values, and the equation proved to be an imposter. While this “let’s evaluate” approach was rather typical, and the discussion was somewhat predictable, the results were nevertheless quite powerful. The major issues we identified and addressed were:

Students lacked a go-to strategy for verifying whether their equation was valid. It’s almost as if they viewed the progression (from numerical to algebraic) as a one-way street. Our conversation helped them see how to go from numerical to algebraic, and then back again to confirm. I probably should have equipped them with this skill some time ago. Better late than never.
Students were working with some faulty y-intercept assumptions. The evidence suggests (and their comments confirm) that of those who got this wrong, most went hunting for the y-intercept where it’s often found: at the top of the table. “The y-intercept occurs when the x-value is zero…” is not something that has stuck with them, despite visual, verbal, and numerical attempts to drive that home. I’m cautiously optimistic that today’s conversation smashed that misconception for most students.

Side Benefit

While most students wrote an equation with a rate of change of 4 new circles per stage, I did have a few “slope stragglers” suggest equations like y = x + 4 and y = 10x + 4. As we evaluated multiple x-values in each of four equations, and left a record of our evaluating on the board, the similarities and differences rose to the surface. The connection between the coefficients of our faulty equations (1, 4, 4, and 10) and the common differences between expression results caused a few more light bulbs to turn on.

Future Use

We’ll continue to explore Visual Patterns in small groups most the time, but I think I’ll include a dose of individual formative assessment, followed by small-group or whole-class error analysis now and again. We unpacked a lot of misconceptions today, and made a number of valuable connections as well, all in a rather short period of time. Anything that draws out misconceptions so we can smash ’em to bits through class discussion is worth bringing back for an encore.

Ignite Talk (Video) • Technology and the Curious Mind

Last December I joined nine other educators in giving an Ignite talk at the CMC North 2014 conference in Asilomar, CA.

The event was organized by the wonderful folks at The Math Forum. They’ve begun posting video of the talks on their website. I’ve embedded mine below for your viewing and/or heckling pleasure.

Technology and the Curious Mind

Postscript

I wrote a few words about the experience (and posted the slides and manuscript) in a previous blog post.

Match My Line • Create Your Own

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Last month I shared Match My Line and Match My Parabola, a pair of Desmos-driven challenges (rather, sets of challenges).

They’ve quickly become two of my favorite ways to build or deepen graphical understanding, whether working with middle school students or older kids (who “should know this stuff” but commonly do not).

I brought a do-it-yourself extension of the “Match My Line” challenges to Math B a couple weeks ago.

It. Was. Awesome.

How To Play

Step 1: Everyone starts out flying solo. Fire up Desmos. Add a pair of points. For this first-ever-instance of MMLCYO, I required one of the points to be on the y-axis (but not at the origin).

Step 2: Record the ordered pairs in the first two rows of the table. (Want the handout? Click here.)

Step 3: Find an equation that passes through the points. Confirm in Desmos.

Step 4: Trade your points (but not equation) with a partner. Hunt for their equation, using a new Desmos graph to confirm. Record the points and equation in “Their Challenge #1.”

Step 5: Find a new partner to trade with. Repeat this until you’ve filled out “Their Challenge #1-4.”

Step 6: Create a new equation, and run through four rounds of trading/graphing again, this time recording the results on the back (“Their Challenge #5-8”).

Success!

Every year since I began teaching, I’ve tried to help students develop proficiency with finding an equation to model two or more collinear points. The results have always been hit and miss. Until this year. Granted, this “Create Your Own” activity was not their first introduction to rate of change, intercepts, and slope-intercept form, but my students absolutely rocked this activity.

After the wrap up (details below), I asked students to rate their “before” and “after” understanding on a 1-5 scale (5 = high). The results were encouraging, with the typical student expressing a shift from about 2-3 to about 4-5 (with most giving an “after” rating of 5). Woohoo!

Wrap Up

I think the combination of minimal teacher talk and active students (mentally and physically) made this a success. Plus, having students confirm their results in Desmos pushed students to work on the math, rather than settling for simply “completing the page.”

We wrapped things up with four rounds of whole-class “here are my two points, what do you have to say about that” gauntlet throwing. For the first two, I took the challenge, modeling aloud the thinking I had heard around their tables throughout the class period.

For the next two challenges, I asked two students to narrate their thinking as they found the equation. They rocked it.

Looking Ahead

I plan on bringing “Match My Line • Create Your Own” back in a few weeks, but we’ll shift our attention to two non-intercept points and point-slope form.

I think this “Create Your Own” approach is also packed with potential for quadratics and other Match My Function categories, and I can’t wait to weave it into my Precalculus course later this year.

Comments?

Ideas for how to extend or improve this with lines, parabolas, or something else? Drop a line in the comments!

Student Sample #1

I snapped some scans of student handouts at the end. Here’s one:

Student Sample #2

And another:

Blog Recap • February 2015

February was the biggest month on the blog, almost by a factor of two. Here’s a recap of the lessons, posts, and other math-related things that went down.

Lessons

Learn Desmos. A great place to start if you want to learn more about this free online graphing calculator.

Speaking

39th Annual Spring Math/Science Conference (CMSEMC) | Invited
▸ Turning Students Into Posers + Solvers
February 7, 2015 • Redwood City, CA

CUE Rock Star Teacher Camp Petaluma | Faculty
February 13-15, 2015 • Petaluma, CA

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:

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.

Postscript

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…

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.

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

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!

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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:

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.

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:

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.

Lessons

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

Speaking

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

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

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On the Fly…

Side Benefit

Future Use

Technology and the Curious Mind

Postscript

How To Play

Success!

Wrap Up

Looking Ahead

Comments?

Student Sample #1

Student Sample #2

Lessons

Top Posts

Other Posts

Speaking

A Means To Another End

A Well Worn Path

Postscript

The New Play-by-Play

1. Setup

2. Draw What You See

3. Predict and Describe What’s Next

4. Fast Forward to Stage 10

5. Represent!

6. Desmos!

The End Result

The Main Culprit?

My Prescription

Stepping Back In

What About You?

Round 1

Round 2

Round 3 (and a surprise!)

Round 4 (another surprise!)

Looking Ahead

One More Thing…

Lessons

Top Posts

Other Posts

Speaking