Step 1. Throw this in front of your students:

Step 2. Give students some think time. Then collect responses however you want. Here are two options:

• If you have enough whiteboard space, give the entire class 60 seconds to write one of their equations on the board. Set a timer. Or count down during the final 10 seconds. Or both. Whatever the case, this needs to be quick. If time allows and you’re interested, select 2-3 equations for the class to verify.
• Are your students arranged in groups? Ask each group to contribute their favorite equation. Then give the class a couple minutes to hunt for “imposter” (i.e., incorrect) equations.

Samples. Here are some responses you might see for today (May 13):

Can I use your classroom as a remote laboratory? If you’re game, I’d love for you to run this by your students sometime this semester. If you do, drop a few words about your experience in the comments below.

Cheers!

I had prepared two related, but non-identical handouts, each with ten problems related to CCSS.8.F.04. Prior to class, I decided I would use the first handout as source material for a few examples and the second handout as our pool of practice problems.

After the first example, I paused. Instead of moving immediately on to a second example, I told my students:

“Alright, kiddos. Look through the second handout and put a mark next to every problem you think you’re now equipped to tackle.”

No big deal, right? Well, I’m starting to think it might be. This simple request produced a not-so-subtle shift in their approach, one that I think may have had an important impact on their mindset.

Instead of moving through a full set of examples, and then turning our attention to a full set of practice problems, where comments like “I’m confused,” “I’m stuck,” “I don’t know how to do this,” and (especially) “You never showed us one like this!” might abound, my students were actively hunting for problems within their reach. And if my informal observations are on track, then in the context of that active hunting, my students extended their reach a bit farther than normal.

Is this a one-time fluke? Or is asking students to search for what they can do a subtle way of boosting what they’re capable of?

Call for Comments

If you have any thoughts on what I’ve described above, whether anecdotes from your own class or links to research, drop a line in the comments!

I’ve presented the lesson in a variety of workshop settings (each time it was well received), but this week marks the first time I used it with my own students. It was an absolute blast.

The full lesson is available here. I’d love to know what you think, as well as what you’d add or tweak (especially if you test it out with your own students).

Post Script

You can get more lessons on the Lessons Page.

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:

1. 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.
2. 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.

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:

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.

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!

A little background… My mom teachers middle school math, just a few miles from where I do. When I started out teaching, she was my go-to resource for teaching questions (especially classroom management). I’ve often been a resource for her regarding conceptual development or activity ideas for a current or upcoming topic.

Here’s what she sent me last night at the end of an email:

Lacking any particular inspiration, I directed her to Robert Kaplinsky’s PrBL Search Engine, and then hopped on Twitter:

Hey #MTBoS, what are your favorite lessons on area and circumference? #msmathchat

— Michael Fenton (@mjfenton) December 10, 2014

The response? Pam Wilson to the rescue!

In particular, this seemed like a really cool idea:

So what happened the next morning? This!

Now that’s a pretty cool Professional Learning Community/Family.

P.S. Here’s a glimpse at the handout my mom created.

P.P.S. And her thoughts on the lesson, including what she’ll tweak for next year:

A little over a year ago I created a set of Nike+ Running-themed proportional reasoning challenges for my middle school students. The idea originated as a failed attempt at a Three Act task, and eventually turned in to a somewhat-useful website (proportionplay.com). You can read more about the project here.

Even after using these problems occasionally throughout last year, and also in various teacher workshop settings, I was never satisfied with the results. Things usually drifted into uncomfortable arithmetic, and my focus from the word “Go!” was always on the proportional reasoning element.

So after mulling things over during last year’s spring semester—and trying out some of those ideas at the start of this year—I’ve settled on a new approach to the same set of problems. I’m optimistic that this new twist will make the problems much more engaging to my own students (and maybe even to some of yours as well).

The (Old) Running Game

The old me would start the conversation with an image like this:

Then I’d ask students what they noticed about the images. There would often be a wide range of answers (“I see orange!” “Hey, the phone battery is draining!”), but eventually we’d have covered the basics: “After 10 minutes, some mystery man (or woman) had run 1.88 km. This person apparently kept running for another 10 minutes, and then for another 5. We don’t know the total distance traveled at either of those latter points.”

Next up: Find what’s missing!

This is where we’d wander off track from my primary goal (develop proportional reasoning!) and end up stuck in some tedious arithmetic (with the tediousness increasing slightly each day in the series of 20 challenges). There’s nothing wrong with having (or building) proficiency with decimal arithmetic. However, my goal here was to emphasize and develop proportional reasoning. The doubling, the halving, the combining of these pieces to make five halves of the original… Students would stare into the blinding light of ninety-four hundredths and somehow lose track of what they were after in the first place.

Good conversations often ensued, but I was after great conversations. Something had to change.

The (New) Running Game

Here’s how we roll now: I still begin with an image…

And we still do the whole noticing thing (at least on the first couple of challenges). But then I invite (encourage? demand?) students to cut some corners.

• 21 minutes and 5 seconds? Nah! That’s basically just 21 minutes.
• Double that? 42 minutes. Boom! We have our first answer: “This runner probably took about (hugely important word for the rest of our running game conversations) 42 minutes to run 8 km.”
• Back to 4 km in 21 minutes. Half of 21? Let’s call it 10. So 2 km in about 10 minutes (maybe a little more).
• So 42 plus 10… That’s 52 minutes.
• But maybe our runner is no Madison Bumgarner. Maybe fatigue actually affects our faithful jogging friend. Let’s hedge our bet. Maybe… 54 minutes?

And then, the reveal:

And it turns out we’re wrong. Totally and completely wrong. But now the fun begins, because we can ask:

• What happened?!
• Why were we so off?
• Did our arithmetic betray us? (No. We took some liberties with rounding so the arithmetic would be easy-peasy.)
• So what gives?
• Did this guy (or gal) speed up? Or did he/she slow down? (To me, this conversation alone is worth all the screenshots I took while running with a phone in my hand.)

Maybe we made some assumptions about our runner. Maybe we need to blow up some of those assumptions so our future “off-ness” isn’t so pronounced. (And isn’t that one of the best parts of modeling? Figuring out the subtleties of the situation… Testing and either retaining or tossing out our assumptions…)

So we play again:

And this time we come to the table with some additional information. This dude slows down (and seriously) as the seconds tick by. Let’s keep that in mind as we think through the next round:

• 1.89 miles… That’s pretty close to 2 miles in 16 minutes
• That’s 1 mile every 8 minutes
• 24 minutes… Let’s chalk that up as 3 miles
• 32 minutes… That would be 4 miles
• What about 36 minutes? That’s another 4 minutes (half of 8!) so I’m thinking an extra 0.5 mile. All told: 4.5 miles.

Of course, that would be if our runner followed a predictable pattern. Now the fun comes in. This is where every single student in the class could end up with a unique and not-unjustifiable answer. How do you want to hedge your bet in the end? Will the runner slow down? Have we already accounted for that with our rounding up to 2 miles? (Or have we gone the wrong way?!) And so the customization of the answers begins (ideally with much arguing and hoping and anticipating).

Me? I’m thinking we should tone down the distances… 2.7 miles in 24 minutes, 4 miles in 36 minutes. So, how did I do?

Now when you throw that next image on the board, and one kid in the back slams his fist on the table in defeat and the other pumps her fist in the air in triumph… You know you’ve got them.

The Road Ahead

The challenges are all still available at proportionplay.com. And I’ve replaced the original (and terrible) handout with this shiny new one. I think the new handout does a reasonably good job supporting my updated approach to these tasks by emphasizing the reasoning involved in the entire process. That—combined with my wide open invitation to estimate since the runner is almost certainly not a same-pace-all-day robot—should shift the focus from arithmetic to proportional reasoning.

Post. Script.

The last two challenges are both borderline terrible. Or wonderful. Or both. Here’s a preview of Day 20:

Post^2. Script.

Another favorite aspect of these problems? Whether you round or not, there are several ways to tackle each challenge:

Disclaimer: The whiteboard image is from one of the courses I teach in the grad math/science department at Fresno Pacific University.

• We’re a little weak with graphing lines. Some open-ended, Desmos-driven, instant-feedback style practice may help.
• Domain and range? Yeah, not so much.
• Inequalities? Haven’t done them justice. Yet. (Growth mindset, baby!)
• Vertex form for quadratics? Still struggling.
• We tried Des-man a few days before this game and found nothing but pain and frustration. Some could be attributed to me (in particular, a bungled launch of the activity), some to students’ lingering struggles (noted above), and most of the rest to the declining state of our netbook cart. (But they seemed so cool in 2009!)

At any rate, to get that bad taste out of my mouth and set the stage for greater success on the next Des-man go around, I created the Dot Capture Game. Here’s what you need:

• Students (working in pairs)
• Devices (we actually used 50% smartphones, 25% tablets, 25% laptops)
• The world’s greatest, most beautiful graphing calculator

And of course, the handout:

Getting Started

Give a brief intro—or none at all—and turn ’em loose. If your experience is anything like mine, you’ll find yourself the weaving in and out of some great (albeit trivially-inspired) conversations about slope, intercepts, point-slope form, domain, range, inequalities and shading, vertices, direction of opening, etc.

This is definitely not high-quality modeling stuff (it’s not even low-quality modeling stuff), but it proved a great way to engage students with meaningful (read: productive) practice on a variety of topics related to graphing.

Oh, and the winner in my class? Here you go:

Final Thoughts

After trying this out in Algebra 1, I thought I’d throw it at my Algebra 2 and Precalculus students to see what they would do with it. It turned out to be good practice in those settings as well. Before sharing with these followup classes, a quick tweak to the handout was in order. In my first class, several students lost their graphs and expressions after hitting a deadly combination of keys on their device, and only one or two had been keeping a shiny written record. So to protect against future heartache, I added a second page to the handout. Here’s what one of them looked like at the end of class:

Update

Here’s a sweet suggestion from Desmos:

@mjfenton What if Ss rolled two dice to determine which curves they had to use and the numbers also represented the coordinate to capture?

— Desmos.com (@Desmos) May 22, 2014