Desmos Art 2.0

One of the hallmarks of the MTBoS is constant refinement and reflection – taking something of your own or someone else’s and making it better.

The conics unit has come and gone in my Algebra II classes, and like last year I want to do a performance assessment. Back in the day this assessment was Amy Gruen’s piecewise functions picture. With the advent of Desmos it’s now a digital version of the same project. (I wrote about last year’s here). Then in early summer I saw the tweet that let me know how much better my project could be for my students.

Dropping the image into Desmos first, then creating the equations to match the image? Brilliant! That led to a pretty productive online conversation, and to me making some slight changes to my plan for this year. My big takeaways from last year were:

  1. my students selected some very cool but also very challenging pictures to duplicate
  2. they needed massive amounts of support writing equations to match lines and curves
  3. probably not everybody did their own work

Providing massive amounts of support is what Desmos does best. That scaffolding probably means less frustration, and less cheating. At least that’s what I’m telling myself.

Fingers crossed
Via Tenor

Started before break with a functions review (Alg II (3) Functions one-pager), not only of conics but of all the functions we’ve learned this year. The day back from spring break we learned how to match equations with lines or shapes in a picture with this Desmos activity.

Then I introduced the project, and offered a carrot (it’s a quiz grade, you guys!). And away they went, seeking pictures.

 

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They found standard-issue high-school-kid stuff: lots of cartoon characters, superhero or sports team logos, palm trees and flowers. I had them make a (rough) sketch of the image on grid paper, then try to identify equations of four functions that would be included in the final product. I wanted them to get used to the idea of seeing small sections of the larger whole, and finding ways to describe that section in math symbols. We also walked through the process of setting up an account in Desmos, opening a new graph and bringing in the image, and saving the graph so they could access it again.

Double Double
Making ’em hungry before lunch. Double Double, coming up.

By Day Two, we were ready to start getting serious about making some math art.


 

They were pretty excited about this project when they were googling around for images, finding their favorite characters or sports teams. They were less excited about this project when it came time to start writing equations.

A couple wanted to straight-up quit. I’m gonna use all my powers of persuasion to try to convince them otherwise. That, plus walking through the process, step by-step, of writing a general equation, then adding sliders and tweaking values until the curve matched up. I’m not sure it helped.

I did notice that very few of my students actually completed the reference sheet. And (in a related story) almost none had any recall of any function equations except y = mx + b. That is definitely part of the issue – a huge disconnect between a shape on a screen and the math symbols that represent it. And truth be told, that’s part of what I wanted this assignment to do – to cement that relationship.

Best-laid plans, right? I’ve got some work to do.

showtime


 

The morning of Day Three, the putative due date, one of my struggling students came in for extra help on the project. She left with a smile on her face, having made serious progress. Plus she agreed to act as a “resident expert” in class, helping out her tablemates when they got stuck. We made some halting progress as a class, but no one is close to done. Several of my students did say that they understood how to write an equation for a line or curve, and restrict the domain, just that it was going to take a long time and a lot of tedious work. So, similar to last year, with about 10 minutes left in class I offered a reprieve, shifting the due date to Monday. Then I’ll accept whatever they have and go from there. I set up the grading rubric in such a way that the points are weighted toward planning and less on the finished product, so the kids who laid down a foundation can still get a reasonable grade even if their final product is…. incomplete.

But I also want to be able to show them what their project could look like, with a little bit of persistence:

 

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Just a little something I threw together over the weekend. 44 equations later…


 

The breakthrough for many came when they started to use vertex or intercept form for their parabolas. The ones who completed the functions reference sheet caught that first. I showed everyone on Monday, which of course was too late for many folks. Next year I’ll highlight that option earlier.

So, they begrudgingly turned in their paper/pencil planning work, along with a link to their Desmos creation, on Monday. Just like last year, some bit off way more than they could chew. Some got frustrated and quit. Some gave me a half-finished product. But the ones who stuck with it were able to turn in some pretty cool stuff:

 

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Oh, yeah, and this from a student as she turned in the assignment thru Canvas:

Desmos Student Comment
Yeah….

My big takeaways:

  1. I need to steer them towards reasonable images to duplicate. Avoid frustration and shutdown right from the jump.
  2. I need to encourage my students to use the vertex form of quadratics. Anything that makes the movement of the curve more intuitive is good. I think eventually that will help cement translation of functions.
  3. I need to enforce the preparation steps that I built in: the reference sheet, the paper sketch, and the four function equations by hand. I need to help them draw the connection between curves on a screen and the associated math symbols.

The assignment is is a keeper. But I bet you it won’t look exactly the same three years from now as it did this week. In fact, I’m counting on it.

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College of Arts and Sciences

Growing up, every Tom Cruise character was that super-confident, super-cool guy that could bluff his way through any situation with quick wit and a smile. Who didn’t want to be Joel Goodson or Brian Flanagan or Maverick?

But I definitely also had an appreciation for people who planned every move with military precision. Who could see the downstream consequences to actions that everybody else missed. See: Jane Craig in Broadcast News. So: going by the book, or flying by the seat of our pants? Painting by numbers, or just making some happy little trees?

Image result for bob ross memes

Is teaching an art, or a science? If you’ve been around the game for awhile, you’ve probably concluded it’s both.

Joshua Eyler of Rice University turns the question on its head in a 2015 blog post, proposing that “the most effective teaching is that which helps students learn to the greatest extent possible”.

So how might we change the art vs. science question to reflect this positioning of learning?  Though we’ll have to sacrifice the nicely compact nature of the original, a new version of this question might ask whether achieving a deep understanding of how our students learn (both in general and about our fields) is more of an art or a science.

The sorts of collaborations with students that might reveal this knowledge could certainly be called creative and even artistic.  I also think there is something of an art to being attuned to students’ individual approaches to learning (or their Zones of Proximal Development) and adjusting our strategies and techniques accordingly in order to ensure we are helping as many students as possible.

What about science?  I have to admit I’m biased here.  As someone who is writing a book on the science of learning, I lean more heavily in this direction.  Because learning has its basis in the neurobiological mechanisms of the body, I think science has much to teach us about learning.  Learning is also rooted in the social world as well, so the fields of sociology and psychology provide further opportunities for understanding.

Brain science and psychology and making adjustments on the fly for what our students (collectively or individually) need at the moment? Yeah, that sounds exactly like what teaching is. “All Of The Above”.


That was us a couple of weeks ago. I know the look I saw on my kids’ faces after the logs quiz. It’s never a good sign, but that “I don’t get this and math is stupid and I quit” feeling in February makes for a long last 13 weeks for everybody involved.

So I called an audible.

I’m hardly the first to roll out this activity. My favorite instructional coach was doing Barbie Bungee before I was even teaching, long before Twitter and Desmos had even been thought of. The great Fawn Nguyen and Matt Vaudrey have raised it to an art form.

But I gambled that it would be just the antidote for the Math Plague that was threatening to decimate my classroom. Plus, worst-case scenario, I could justify it (at least to myself) by saying that the linear concepts and DOK 3 activity would be ideal for my students in the weeks leading up to ISTEP re-testing season.


 

I leaned heavily on Mr. Vaudrey, who is kind enough to post his materials for anyone to use, and to reflect on his own lessons so that folks downstream might be able to anticipate the stumbling blocks for their students. I teach in the new STEM wing of my school, in what eventually will be a combo computer lab and build/makerspace. So I had some essential ingredients on hand: measuring tools, lots of space, and plenty of surfaces at a variety of heights. What I didn’t have on hand, I sought out: eight bags of #32 rubber bands at WalMart, and 8 WWE wrestling figures from my son’s collection.

Day One I tried to hook them in with an insane missile silo bungee jump, then set them up with a figure, a bundle of ten rubber bands, a data collection sheet, and let them go about the business of jumping.

Perfect world: each group of three or four students would have had about 8-10 data points. Reality: most got 4-5. Several got only 3, and one group managed to record only one distance. Those guys are gonna need some extra support.

BB Blog 2

Day Two, time for some estimates backed up by math: How many bungees would be needed to jump off the top of my projector? How far a jump could their figure make with 25 bands?

And in one of those glorious moments of teaching, I had set the hook. Students were madly pouring over their data, trying to use it to give legit estimates to the questions.

BB Blog 1
Teamwork, baby. Teamwork.

(It was about this moment that I decided that I would honor their efforts at thinking and reasoning and doing actual math on their own by entering some points for the three-day project as a quiz grade. By department policy quizzes and tests account for 75% of a student’s grade, so a good quiz grade is like finding a hundred-dollar bill on the ground outside your classroom.)

So we dumped data into a Desmos graph, let some groups with few data points share some numbers from other groups (that’s that extra support we talked about), made a trend line, set a horizontal line at 533 cm on their graph, and talked about how many bands they’d need to safely make a jump from the top of our two-story Robot/Quadcopter Arena.

BB Hlog 3
Letting Desmos do the heavy lifting to free up brain power for thinking.

Quick group huddle to compare numbers, then after a few minutes of table talk I stopped to see each group, ask about how they came up with their number, and (this is key) have them agree on one number, write it down on their page, and circle it.

Day Three, the Tournament Selection Committee has announced the pairings, and the teams are ready to jump.

BB Blog 4.jpg
Not that I’m craving attention or anything, but yeah, I totally posted the brackets on the window of the arena that faces a heavily traveled hallway.

I pre-assembled strands of ten bands to accelerate the assembly process, then students built their bungees and gathered, two teams at a time, on the second floor. We quickly found out that everyone in my 2nd hour class had seriously miscalculated the number of bands they needed. Fig after fig crashed to the floor. Lacking other options, and wanting to avoid the buzzkill of a six-way tie for last, we finally decided the “less dead” fig would move on.

The afternoon class seemed to have had some better estimates and we had some competetive matchups, as well as some gamesmanship as some teams attempted to scrunch two or three bands together in their hand on the railing to avoid a figurative skull fracture (high school kids, right?). The extra-long bungees in 2nd hour made a great math conversation starter (“what happened, you guys?”). I used Matt Vaudrey’s feedback form, and found out that Barbie Bungee was a near-unanimous hit.

Barbie Bumgee Feedback

Would this three-day activity had made more sense back in September when we were doing linear stuff? Probably. Would I have had the confidence to step back from the curriculum map for a minute when my students needed a breather if I hadn’t been hanging out on the periphery of the #MTBoS with its brilliant minds and fantastic lessons and activities? No way. Would I have tried Barbie Bungee without being able to follow a well-worn path? Not sure. I’m down with taking chances in the classroom, but I’m not sure I’d have been wise enough to add the Desmos piece if Vaudrey hadn’t blogged about it. And that made the whole project. We’d have been dead in the water, guessing a number of rubber bands for the Big Jump without it. Which means we would have missed the math altogether.

What I do know is: my students bought it, real learning happened, we all got the stress relief we needed, and I came out looking like an improv artist taking a prompt and making comedy gold.

Brian Flanagan would have been proud. Jane Craig too.

Art. And Science. It’s a Both/And.

 

Piece By Piece

Image via giphy.

Last time we talked math in this space, I was trying to figure out a way to squeeze way too much content into the last five weeks of school, while still giving my students a chance to practice the skills and giving me a chance to assess their understanding, all while keeping a tiny sliver of their available brain cells focused on math stuff. Because it’s another fantastically gorgeous early May in The Region.

It's May In The Region
“Road Conditions: Wet”.  No kidding…

This week, I needed a performance assessment idea for Conic Sections. I also need to overlay final exam prep with new material in the finite time remaining before June 2.

And, I want to play with Desmos. Or rather, I want my students to play with Desmos.

Put all those ingredients in a blender, hit “Smoothie”, and you’ve got Piecewise Function Art!

Desmos piecewise staff picks

See everything up there labeled “Conics Project”? This project plan of mine is not a new idea, obviously.  I first came across it when Amy Gruen posted about her pencil/paper project back in the day. My co-teacher and I modified it for our Algebra II course that included several students with IEPs.

And then it sat in my back pocket for years until I changed schools and was assigned to Algebra II again this year.

The #MTBoS Search Engine tells me there are some awesome teachers getting cool stuff from their kids regarding this type of project. Check out Lisa Winer and Jessie Hester, to name two.

So I used their work as a starting point, customized it for my students, made up a packet with some sample art, my expectations for the project and the points scale, annnnnd away we go….

I insisted they did the pencil/paper planning first. I want them to make some fun & cool pics, yeah, but first and foremost I want them to get good at moving between representations of functions, and to get some reps on writing and graphing conics. I gave them two days to roll it around and plan at home, maybe sketch a quick picture or two. Then I planned for a pencil/paper Work Day in class Thursday, with the expectation (slightly unrealistic, it turns out) that they walk into class the next day with a list of equations. Then input equations to Desmos on Friday, with the project submitted via Canvas by the end of class.

Docs here:

Alg II (3) Conics Performance Assessment

Alg II (3) Functions one-pager


The initial reaction was… lukewarm: “Ugh”. “I’m taking the L.” “I can’t do this.”

Come on now. Don’t give up before you even try.

Most of them didn’t pick up a pencil before classtime Thursday, putting them in a hole to start. Fortunately I built in support, posting a Desmos Activity (via Stefan Fritz) to our page for them to play with, so they could see how to fine-tune an equation, and to restrict the domain. But the best progress was made in class on Thursday, when I convened some small groups, answered questions, walked through a couple of quick examples of drawing a graph and working backwards to its function rule, and also showing them how to translate a graph.

Next thing you know…

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Guys, for real. In my least interested class, I had 26 kids engaged, helping each other out, graphing, writing, struggling through the rough spots, cheering for each other and squealing with delight at themselves.

If they aren’t at home right now high-fiving themselves, they should be.

Then Friday, the Big Finish:


OK, in reality, my students needed a lot of support to bring this project in for a landing. A lot of them made a pencil/paper design that was way too ambitious to finish even with two days to work in class. Many were asking questions Friday that they should have brought to me on Wednesday or Thursday. Most got down to business in class on Friday, because it was the due date. But almost no one was remotely close to being done.

There’s two ways to handle that: 1) “Too bad, so sad, I told you guys to get started on Tuesday and you didn’t so now you’re out of time and out of luck. F.”

Or: 2) “Look, I can see you guys are making progress. How many of you are happy with your picture as it is right now? Not many, right? But you’re making good progress and probably could turn in something really fantastic with a little more time? Cool. The due date in Canvas is today, but with a time of midnight. Go home, finish it up, turn it in before you go to bed and we’ll call it good.”

In his autobiography “My American Journey“, General Colin Powell stated often one of his life’s guiding principles: “Never step on another man’s enthusiasm”. Good advice from a great man. I’m in, all the way. Why crush my students’ spirit just when they are hitting their groove with Desmos and putting together the equations for a whole big mess of functions? Math is happening here, people. I’d rather ride that wave, let them finish and give me something they can be proud of.

So, midnight it is. And we all get better, together, at teaching and learning.

Piece by piece.