Category Archives: Math/Art

Math + Art + Desmos… Connections.

“I love math and art, and I’m glad that I was introduced to Desmos, a way to use both subjects at the same time.” – Marianna, Grade 10


Callisto, Grade 10

Drawing with graphs has been a powerful way to motivate students’ interest in understanding how equations relate to functions, and how manipulations of equations lead to transformations from a parent function.  I jumped on to Fawn Nguyen’s Des-Man project as soon as I saw the idea, and have done some incarnation of this work each year.  Each time I’ve guided students through this process, it’s gotten better and deeper, both through the development of my own approach, and from improved tools like the Desmos Des-Man interface (…which I’ve heard is currently “in the shop” undergoing some improvements) and more recently tweaks to this idea like the “Winking Boy” challenge, created by Chris Shore (@MathProjects), and posted on the Desmos Activity Builder by Andrew Stadel.


Amit, Grade 10

This year’s work was definitely the strongest yet, and I owe the major improvements to my reading of  Nat Banting’s post, which extended this project to another level for my students. In the past, I have asked students to create a graph, which had features of a face or a building or a plant.  This year, I asked my grade 10 students to choose a graphic, photo, or work of art, which they had to replicate using only equations.  I asked that they choose an image that was meaningful to them for some reason, and then helped to guide them to something that was challenging, but that they could accomplish – a natural moment for differentiation, built in to the process.  In the earlier versions of this project, students had been motivated by trying to make their face look angry or happy or sad, but they didn’t have a specific place where their equations had to end up. Asking kids to commit to re-creating something forced them to be purposeful and deliberate in every choice.


Ilyas, Grade 10

They took the responsibility of recreating their chosen image seriously, and honestly, their work exceeded my expectations. There were regular exclamations of satisfaction echoing around the room as we worked on this. They persevered. They definitely attended to precision. They argued with each other about the best equations to use. They reflected about how to make the best use of Desmos. They practiced the habits of mind of successful mathematicians.


Anastasia, Grade 10

When we shared the in-progress work for some peer feedback, kids were actually applauding each other when their work came up on the screen.  Not because I reminded them to be a supportive audience, but spontaneously.  Seriously.  And when they saw the staff creative picks at Desmos, they asked me whether they might be able to submit their work.  The whole class was taking pride in creative math work.


Karim, Grade 10

I asked that students reflect in writing on their learning during and at the end of the project.  I haven’t asked for students to do enough writing in math so far this year, so when they seemed to be really struggling with this, I made a fill-in-the-blank “reflection assistant” to scaffold their thinking and writing, and to give them some ideas about what to include in their written analysis.

A few highlights from their reflections:

  • “I was quite surprised that I could replicate a drawing by using graphing.  If somebody asked me to do it last year, I would say that it is a “mission impossible.”   However I was able to do it.”
  • “As my piece of art, I chose the logo of the football club Barcelona because I am a big football fan and FC Barcelona is a club worthy to be recreated through the use of quadratic equations in vertex form. In addition, the logo was an appropriate challenge for me, containing easy and smooth curves but also difficult shapes, like letters or circles. When the project was assigned, I was skeptical that it was possible to recreate an artwork, just by using equations. But now that I am done and a proud owner of a recreated art piece, I strongly believe that it is possible (obviously).”
  • “I found out that desmos is a really good tool to practice and sharpen your understanding on any equation and in my case it was the vertex form of a quadratic. Desmos allows you to experiment and find new ways to fix the problems or even work more efficient in order to surpass the problems in the first place. I am proud of the detail and sharpness of my work in general. I tried really hard to make the whole piece smooth and detailed. In order to do so, I zoomed in a lot and by doing so, I identified minor mistakes and was able to fix them.”
  • I chose “Pumpkin Pepe” as the subject of my project because it provided the right level of challenge for me and it was really fun to do. Overall, I really liked this project because it solidified my knowledge of graphing equations and has made me more comfortable using parabolas. I found that my understanding of quadratic equations really improved while I worked on this project because before, I wasn’t sure which variable shifted the parabola which way, but now I understand.
  • “I found that my understanding of parabolas and linear equations really helped me improve, and made me more confident during my work on this project. At first parabolas seemed to not make any sense to me, but now I feel like I really understand the way they work. Now I have the capability make connections with all these equations in the real world.”

Marianna, Grade 10

Here are the project guidelines, the rubric, some peer editing forms, the “reflection assistant,” and a .pdf, which has a range of student work.  My rubric borrows from the I.B. Math Internal Assessment Guidelines, as one of my tasks as a grade 10 teacher at my school is to do some specific preparation for the I.B. program in grade 11.  Thanks in advance for any feedback on this project, and on the guidelines and rubric.

Desmos Drawing Project Guidelines and Rubric

Desmos Peer Feedback

Desmos Drawing Project Reflection Assistant



There Can Be Only One (Marker)

Observing a student working on a whiteboard is the best way that I’ve found to get immediate insight into his or her thought processes.  Perhaps because of the impermanence of the medium, students act much more freely than when working on paper.  They are more willing to take risks and to potentially make mistakes.  Even when writing in pencil on paper, the act of erasing is slower than it is on a whiteboard – it seems like they can think more quickly and freely on the whiteboard, which leads to a more fluid thought process, and less barriers between their thinking and their writing.


I’ve been a huge advocate of students’ use of whiteboards since I began teaching.  One of the very first things I do when I move to a new classroom, is to cover as many surfaces as possible with whiteboards.  Asking students to stand up and work in a visible way has the immediate effect of increasing sharing of ideas and showing thinking in a public way.   And it’s fun and they just really like it.


John Orr’s whiteboarding protocol in his recent “My Favorite” post (Week 2 of the MTBoS 2016 blogging initiative), has filled in a piece that was missing from the work in my classroom.  When he groups kids at the whiteboard, he gives each group only one marker.  Every few minutes, he calls “marker switch” and whoever has the marker has to give it up to someone else in their group.  Sounds simple, and I know that I have read about this somewhere before (I think maybe in Henri Picciotto‘s blog but I couldn’t find the reference), but I never thought that it would have the profound effect that I observed when we tried this.  When the kids all had markers, some would inevitably be drawing instead of  mathing,  most would be doing their own thing, and they would occasionally talk to each other.  Providing only one marker forced communication and collaboration in a way that I just hadn’t anticipated.  If they wanted to express their idea but it wasn’t their turn to draw, they had to argue for it verbally.  I can’t recommend this strategy highly enough, and it has had a very positive effect in my class.


A related, but maybe non-mathy aside: I was an art teacher, both at the college and the community level for years before I got into math teaching, and have continued that work along with teaching math.  Most of my life as an artist has been focused on making sculpture, but I did some animation and installation work for a couple of years, which involved a technique I learned from studying the South African artist, William Kentridge.  The process involved making a charcoal drawing and taking a photograph of the drawing.  The drawing would then be erased and/or altered slightly, and then photographed again.  This process was iterated again and again and again (this must be related to my interest in fractals…).  The photographs could be played in a sequence, which gave the illusion of motion.  Here is an example of an animation of some flying bats, which I used as a projected component of an installation piece.

For me, this process of animation was extremely freeing.  I was not afraid to make marks on the page because I knew that whatever I did would be erased soon.  There would be a record of the act of making the drawing, and all of the pieces would come together to form a whole, but each individual drawing would only be seen for fraction of a second.  I think that kids experience something similar to this when white-boarding.  They are more inclined to take risks and just try things because there is no danger of permanence.

Try giving them only one marker!  And let me know how it goes.

Intensives Week: Fractal Geometry!

Thank you Math establishment… for designing a curriculum that requires us to keep plowing through material.  For connecting teaching salaries to student test results and for keeping so much pressure on our class time that it feels like all we can do is skill and drill.   For creating a culture in which students think that math is only a series of formulas that they need to memorize, repeat, use on the test, and promptly forget.

Thanks!  No really… Thanks!  Because you’ve set the stage for kids’ minds to be blown when they are given real opportunities to study interesting mathematical questions.  This is a long post, but I had such a good experience, that I wanted to share details.  Skip straight to the bottom if you want to take the materials and run.  I promise I won’t judge.

DSC05551In January, and again in March, my Expeditionary-Learning High School dedicates a full school week to “intensives.”  Students choose from a list of course options and engage in in-depth study for five days straight.  This January, I led a study of Fractal Geometry.   A few kids signed up because they were interested in the subject already; more had math credits to recover.  But in spite of themselves, they had a blast.

Although this was far from a skill and drill study, it was rigorous work.  I’m all about learning targets.  This study had to be differentiated, as I had a group of students from 9th -12th grade all working together.  I teach college art as well as middle and high school math (I’m thinking of getting my 7-12 art certification), and as an art teacher, differentiation is often easy and natural.  Ideas and techniques become more sophisticated, but a beginner or a master can engage with creating a portrait.  The umbrella of Fractal Geometry cast a wide shadow, and allowed for many points of entry.

I’ve been especially interested in the current conversation in math education about what makes a problem “real world.”  Dan Meyer has distilled some of the more interesting arguments HERE.  I think that Fractals make for “real world” math study within multiple definitions.  Self-similarity is everywhere around us and is easy to spot – it’s actually hard NOT to see it once you’ve identified the property, and kids were pointing all over the place “…look a fractal!”  In his book, Fractals Everywhere (2000), Michael Barnsley puts it well:

Fractal geometry will make you see everything differently. There is a danger in reading further. You risk the loss of your childhood vision of clouds, forests, flowers, galaxies, leaves, feathers, rocks, mountains, torrents of water, carpet, bricks, and much else besides. Never again will your interpretation of these things be quite the same.

Kids were seeing fractals in nature, and in special effects, but they were also highly engaged with the abstractions that only exist within the math.  I had multiple kids emailing me with discoveries late into the evenings.

I’ve included all of the materials and plans below in case they might be useful for you.  Here are some (not so brief) highlights from the week:

DSC05585We’re in Portland, Maine, so I began with a local version of the coastline problem: How long is the coast of Peaks Island?  I printed maps of Peaks, and asked kids in pairs to use progressively smaller and smaller rulers and to try to answer the question.  We collected our data, recorded it on a chart and examined the results together.  “Wait a minute.  The coast gets longer and longer depending on the ruler!”  I let this sit in the air, and just moved the discussion forward by asking questions without adding my two cents.  Showing is so much cooler than telling!  It felt poetic to begin our study with the problem that brought Mandelbrot into the spotlight.  We visited his famous paper later in the week.


You can see the care put into this visual exploration of iteration!

Next, we began an informal study of the Koch snowflake.  Students drew the first few iterations, and we generated a whole bunch of questions.  I nudged the kids who were ready to examine what happens to the perimeter and area as this shape is iterated.  Again, I was deliberate at this point about not doing any direct teaching.  I left their questions unanswered, which helped to build drama and curiosity.

We spent the afternoon of the first day with our anchor “text:” the excellent NOVA special, Hunting the Hidden Dimension.  Even for NOVA, this is a really exceptional resource; challenging, but highly understandable.  It’s worth purchasing a copy for the better resolution, but you can find it on youtube HERE.  I let kids know that their knowledge would be assessed through a piece of writing about one aspect of fractals, and gave them a note-catcher to record observations and questions.  I let them know ahead of time that everyone would be expected to share some learning, and I’ve found that a graphic organizer like this helps kids to remember to record their thinking while they watch a video.  We stopped the film about halfway through to share our first key learnings, and repeated this process at the end.

I heard some really beautiful comments amongst kids at the end of the day.  “I see math totally differently than I did yesterday.”  This is why I am really grateful to all those who tried to ruin math for these kids.

DSC05531On the second day, we delved more deeply into the Koch snowflake.  We learned about sums of infinite series, and the paradox of an object with finite area but infinite perimeter.  Wait.  WHAT?  Yup!  Minds blown again.  I introduced Sierpinski’s triangle, along with some more “mathematical monsters,” (they loved this historical context), and we made some charts to examine some of the properties of these classical fractals.  Work ranged from concrete counting of triangles to algebraic generalizations of nth terms to summing infinite series.  Differentiation in math can be so tough, but this was a place where it really came together, and I felt like most students were really working toward their personal best.  After a really rigorous morning, we spent the afternoon creating original works with two excellent sites:, and Fractal Maker Express.

Day three, we looked through a “fractal library” I’d put together (I had a range of books; Fractals: A Graphic Guide was the one that really resonated in a broad way), and then played the chaos game.  HERE is a very nice electronic version, but I think that this is very effective if you print triangles on transparencies, pass out dice, and let kids play manually.  The chaos game has a truly unexpected result, and this comes together well when you stack the transparencies.  I asked kids to make predictions (in writing) every few minutes, and monitored carefully so that when each one got it, they didn’t spoil it for the others.  It was entertaining to see kids noticing the drawings of Sierpinski’s triangle all over the room.  “Wait!  It’s that triangle!”

Clint Fulkerson's Division-1

Clint Fulkerson’s Division-1

I made a connection with a local artist, Clint Fulkerson, who uses the logic of fractals to create some really cool art.  He was generous enough to let a herd of kids into his studio, and he was articulate about how his work uses properties of self-similarity and recursion.  He rode a nice line between a casual attitude, and rigor in his work; one thing I hope the kids take away from this week.  Check out his work HERE.  Clint has a show at the Portland Museum of Art’s Family Spaces, up through this July.

Kids were really inspired by Clint’s work, as well as his work ethic.  We spent the fourth day working on some individual and group projects.  Some kids created a paper version of the dragon fractal, while others studied strange attractors and the Mandelbrot Set.  We watched some excerpts from The Colors of Infinity, Arthur C. Clark’s 1995 fractal documentary.  Wow, documentaries have come a long way in 20 years (…and special effects, thanks to fractals!).

DSC05607Friday was for finishing up loose ends, publishing, individual reflections, group debrief, student write-up for the newsletter, and presenting our work to the school.  One of the best moments for me was when one of the students opened up our presentation by saying, “I thought that fractal geometry was going to be dry and boring, but it was actually ridiculously interesting.”


Honestly, it’s hard to say how much of the week we spent doing rigorous math – it depends on how you define this.  And I had the advantage that this week did not have to stick to the confines of “Algebra 2.”  But I truly believe that any time spent on what some might consider “not mathy,” was more than made up for by their contagious interest and excitement about the topic, which gave me the leverage to dig deep.

Here are some materials and resources I used or created.  Send me a note if you’d like a more comprehensive resource list.  We use standards based grading, so you’ll see that the language matches “meets” and “exceeds.”  I’m hoping that the school runs this intensive every year, so please leave your thoughts or additional resource ideas in the comments.  Thanks in advance!

Fractal Entrance Ticket Day 2;  Fractal Entrance Ticket Day 3;   Fractal Entrance Ticket Day 4;   Fractal Exit Ticket Day 1;   Fractal Exit Ticket Day 2;   Fractal Exit Ticket Day 4;   Fractal Intensive Evaluation Form;   Fractal Intensive Learning Targets;   Fractal Vocabulary Blank;   Hunting the Hidden DimensionNotecatcher;   Peaks Island Tabloid;   Sierpinski’s Triangle;   Student_Newsletter_Writeup;   The Chaos Game;   The Koch Snowflake

Making Math Public and Visible

I had a very visual and enthusiastic group of grade 7/8 math students last year.  They were only too glad for an excuse to do some visual math like our super fun math and art show.

This year has been a little different.  My group is not naturally enthusiastic about math, and has needed some encouragement – and 11th graders require a different kind of nurturing than do middle-schoolers.  I’m definitely still learning.   I’m hoping that making some math visible will help to push things forward for my students, and help them to take their own work more seriously.

Math Wall

Here is the start of our Math Wall.  I put up 4 x 8 sheets of soundboard ($12/sheet), and covered it with paper to create a 14’ wide x 8’ tall bulletin board.  This is in the hallway outside of the classroom in which I teach 2 of my 3 sections of Algebra.

Fly or Drive

I put up our current project (…something I adapted from Dan’s Travel Lesson, and from Mr. Ward’s excellent essential question), with some teacher notes.  But I hope that my voice is eclipsed soon by student work.  Once there’s student work published on the wall, I hope that they get some public feedback.

Guess the Number

I’m also going to include a regular estimation or problem of the week, or post an interactive game on the Math Wall.  I’m hoping to nurture a culture of scholarship and connections with people outside of math class.  I’m starting with a great number guessing game I originally found HERE.  Hopefully it will get competitive.  Stay tuned for a report!

I’d love more ideas on how to leverage public space into student engagement and investment.  Thanks in advance for sharing your thoughts.

Math, Art + Design


Our Grade 5-8 artists/mathematicians created a museum of work showcasing some remarkable visual representations of mathematical ideas and mathematical representations of artistic expressions.  We worked on platonic solids made of playing cards, some tessellation work, apollonian gaskets, Mandalas exploring radial symmetry, Notan designs, and “Function Faces,” inspired by Fawn’s “Des-Man”  (Some of my kids’ works were created with Desmos and some with Geogebra – Ah the joys of supporting multiple platforms, each of which cooperates better or worse with different software).  I have a group of students this year who are natural visual thinkers.  We created way too much work for the show, and had to make difficult choices about what to include—and they were brutal curators!  During the “gallery walk” and opening for our show, we created a couple of gigantic dodecahedrons (…thanks to George Hart for the inspiration).

Math Art Pics.001Students were highly invested in this work, and you could hear the thinking and synthesis in their conversations and commentary: “…adding a fractional slope made mine look much more sinister,” and, “…mine looked much more sarcastic when I restricted the domain,” and, “…well, there are 12 faces, and each one is a pentagon, so we need 60 cards – no wait, each edge is SHARED by two pentagons, so we only need 60 divided by 2…30 cards total.”  They really stepped up for this work, both as designers and as thinkers.

Math Art Pics.003

I was inspired by the “MArTh Madness” work written about HERE at Lost in Recursion.  I’ve been chomping at the bit to participate in a festival like this with students since I found the idea, and planned with colleagues from the math and art department to make this valuable, fun, and integrated.

Math Art Pics.002

BTW, this article was well timed to complement our work here.  I like to think that Patrick Honner and the New York Times are tuned in to what we’re doing in math classes in Maine.

Math Art Pics.004

Math, Art Education, and Risk-Taking

I was listening to the news Sunday morning and heard this story about the importance of the arts in education. I love both art and math, but in my observations, art and math are approached very differently both by students and by their teachers.  The realm of art is generally perceived as a place for tolerance and exploration, while math is too often regarded as rigid and fixed.  Art is living personal expression, while math is stereotyped as impersonal and static.

But I’m not sure that things have to be that way.  I think that math teachers can learn a lot from the approach of art teachers – and vice versa.  The culture of exploration, which is so natural to art class, can be emulated in the modern math class.  The culture of rigor, which is expected in math class, can be useful for artists.  This does over-simplify a complex set of values, but I think that there is some truth to the comparison.  Like painting or sculpture, mathematics is a specific and beautiful way in which we can express ideas to each other, a language that allows us to communicate ideas precisely – and as such can be thought of as a branch of the arts.  Math can be uniquely impartial and intensely personal, and by fostering enthusiasm and alternative perspectives, a math teacher can open the possibilities of creativity.  Possibilities, which are natural for an artist, can become available to students of math as well.

What this looks like for me

I tend to collect intriguing ideas, images, videos, images or descriptions of art, etc.  These will often sit around for a while – sometimes years – before any connection occurs for how to use them in the classroom.  But if something is interesting enough, I’ll keep thinking of it, and when there is the right coincidence, I like to leverage that interest into a lesson hook, and hopefully more.

I was lucky enough to go to take an amazing trip to Croatia earlier this year.  I found a unique landscape, warm people, and great food – and it was cheap – they’re still on the Croatian Kuna! Go now, before they switch to the Euro.  Teachers won’t be able to afford it anymore after the switch.

In Split, I came across this wonderful Mestrovic sculpture.  This striking piece, with a cool back-story, is something I’ve been itching for an excuse to bring this up in class.

Mestrovic Gregory

The statue of Gregory of Nin by Ivan Meštrović

I didn’t have the foresight to take a series of proportion images while I was there, but luckily Google and Flickr came to the rescue for something close to what I have in mind:


BAM.  Enter a ratio and proportion math lesson: How Tall Is Gregory?  I’m planning to develop this idea further, and share it here soon.  But this seemed like a good place to think out loud about my process, and about how to leverage a compelling image like this into something relevant (Hello 7th grade CCSS: 7.RP.A) and useful for my class.  Some mathy ideas around this that come to mind might include: ratio and proportion, percent increase, percent error, writing an equation for a proportional relationship, etc.  This could be extended to How Big Is The Statue Of Liberty,


and this lesson might be great in conjunction with Dan Meyer’s Bone Collector, which Mr. Miller extended in an excellent way HERE

For me, art is based in communication and in ideas – ideas that are made manifest by learning techniques such as drawing or clay modeling, printmaking or performing.  Artists explore techniques and ideas, but also practice skills, a model that readily translates to the math classroom.  Students of math regularly work to develop their technical skill sets and explore conventional techniques, but they are rarely encouraged to take risks or explore ideas in math.  I’m not saying that it’s easy to take risks as an artist either, but it seems like art teachers are more comfortable encouraging mistakes.  Math students should examine and then re-create techniques and methods in order to foster relationships both to tradition and to their own imagination, becoming creators of math rather than passive learners.  And as a math teacher, I get to combine my own interests in and love for the mathematics with ways of making the material both relevant and empowering for students.

Further reading: HERE’s a post at Curiouser and Curiouser that explores this relationship further including some ideas about the contribution of the CCSS to this subject.

HERE’s the link to the recent Weekend Edition story mentioned above.

Image Links

Link to Flickr image of Mestrovic’s sculpture

Link to the Statue of Liberty Toe

Pencil Mapping

Time to give back to the blogosphere, which has been so generous to me.  For my first entry, I thought I’d start by sharing a fun resource and lesson.

To support my teaching habit, I spend my summers doing contracting work, stage carpentry, and art fabrication.  I have done a fair amount of fabrication and installation work for Portland artist and longtime friend, Aaron Stephan, including a recent public art project called Becoming, which is a mural-sized world map made of pencils and recently installed at The Hampden Academy in Maine.


Rendering for Aaron Stephan’s Becoming, made entirely of pencils.

Images like this are like gold to me as math teacher – a compelling and provocative resource with a low entry point, but multiple interesting mathematical directions to go.  You’ll have to figure out the best way to exploit its power depending on your students’ experience and interest.  EVERYONE wants to know the answer to the obvious question: How many pencils will that take?  This leads us happily down the road of number sense and estimation, and keeps us busy and content at problem-solving, calculating, and exploring area.

This worked well as an estimation and geometry project for my 7th and 8th graders.  Related materials I’ve created and accumulated are shared below, including a more complete lesson plan.  But for now, here’s a summary of how I’ve used this, with some notes about my kids’ reactions.

  • Introduce an Essential Question (eg. How does math help us describe the world?) and a Learning Target (eg. I can come up with strategies for estimating area.)  My kids are pretty mature (…did I really just say that about 13-year-olds!?), and typically like to have the big picture in mind – though they sometimes get lost just talking over these big issues.
  • Show the image of the rendering (above image).  Give them a little time to figure out what exactly they’re looking at.  My students experienced a slightly uncomfortable, but wonderful time of confusion, when they were not exactly sure what was going on, or what the image was showing.  I like to begin with the rendering, rather than images of the finished piece, because the first time I did this, the piece hadn’t actually been created yet, which added some authenticity to the exploration.
  • Tell them that it’s made of pencils, and then ask students to brainstorm a list of the mathematical questions that they would like to answer (…a la Dan Meyer’s 3-Act format).  “Turn and Talk” or “Pair and Share” are always good strategies for giving quieter students room to add their voice.   On the slim chance that no one asks about how many pencils, you can offer this concept for discussion(or another question that relates specifically to your learning target, which might include everything from pencil counting to Riemann sums).  As you might imagine, I’ve gotten some gems (Personally, I love the snotty “Why would anyone do that?” type the most, as I know from the passion in the question that they’re hooked).  I honor all questions by publicly celebrating them, and then guide them to prioritize the one I want to answer.
  • Pass out Entrance/ Exit tickets, and ask students to record their first guess.  I like to require about a minute of silence here, both for lesson pacing, and for individual accountability.  Make sure that everyone has a guess written down, and collect these.
  • Group students in pairs or fours.  Give them the strategies organizer and give them time to decide what information they need to solve the problem.   Take this time to look over the guesses.  Bring the group back together; share the highest and lowest guesses, along with a rough median.  Ask what information they’d like, and share what they ask for.  I’ve had students ask for fabrication details, location, different kids of graph paper, and more.  Have as many pencils as you can available.  I’ve included all this stuff below (no pencils… sorry).
  • Give groups time to work through an estimate.  Circulate and offer support/ scaffolding where needed.  Ask lots of questions!  I make sure that they know that I expect everyone at the table to be able to explain their methods and the final estimate.  My experience has been that students are competitive and engaged with this task for at least 30 minutes.  Some need a lot of help right at the beginning to get started, but most hit the ground running.  I try to encourage any techniques, even ones (or especially ones) that seem like the wrong direction to me.  I’ve had students ask for a scale to weigh the pencils… A scale?  Uh… OK, here you go.  Keep a couple of small extensions in your pocket – some of my ideas are below, and there are more in the lesson plan, but some kids are definitely done with this, while some are passionate about needing “…just a few more minutes PLEASE”… OK.  I guess I’ll LET you do some more math.
Pencil Mural Detail 1

Close-up detail of the finished pencil map.

  • Bring the group back together.  Show them some images of the finished artwork.  Give them a little time to revise their work, and finalize their estimate if necessary.
Pencil Mural

Finished Pencil Mural installed at Hampden Academy.

  • The Big Reveal: Of course, it’s best if you can Skype or FaceTime with the artist at the end of the estimation for the big reveal, but maybe the next best thing is a video with Aaron explaining how he came to his materials estimate (included below).  I’ve had students cheer triumphantly or groan in pain at this reveal.  That’s right, math is supposed to hurt.  Gotcha!
  • Pass back and ask students to complete the exit ticket, which asks them to compare their original guess to the answer, and then reflect on the accuracy of their methods, and where they might improve.  Again, for me this works best as quiet, individual reflection.  I require 3-5 minutes of writing – just keep writing even if you don’t have more to say.  This time gives us a quiet pause before the next part of the lesson, or the next block, and forces a little individual reflection and accountability.
  • You can follow up with some direct instruction about mapping as a math field, and the basics of projection, along with a discussion of what goes into public art making decisions or the cultural implications of different map projections if there’s time or inclination.


Map-making and projection has both great math and broad social implications, and I can never resist going there.  Here are a few questions for teenagers to wrap their heads around that have been successful for me:

  • Any idea why the Mercator projection from 1569 became the standard map projection, so much so that it was the only one in any classroom until the late 1960s?
  • What was happening in the world in that time period that made this map so damn useful?
  • What started to change in the 1960’s that changed that?
  • How did you make a cylindrical projection in the 1500’s?  How would you do you do it now?
  • Is the Peters or the Dymaxion projection a more equitable way of looking at the world? 

Buckminster Fuller’s Dymaxion Projection.

  • What do you think Bucky is trying to show with this projection?
  • Does it change the way you see the world? Is his projection “right”?

I have found this to be a rich mathematical excursion. Not because anyone will ever have to estimate the number of pencils in other giant art maps.  In terms of so called real-world relevancy, that’s silly.  But their questions are real and relevant to them in the moment, and many of them deal with right and wrong, which kids love to talk about.  Maps are a statement of values, and what gets put front and center says a lot about what is important. Would Aaron’s piece have said something very different if he had put the Pacific Ocean in the center? What would change? What would it look like? What would it mean?  Which would be “right?” 

All maps are measured against their purpose, and without knowing his purpose, we can’t choose a “right” map, but we can ask, “What do we think the purpose should be?” Push those ideas—subjective ideas that depend on math—and leave them that way—all fired up.  You can give them the bookmarks (below) for a little direction as they leave.

BTW – HERE is a link to the Phoenix review of another of Aaron’s pieces that I fabricated.  I’ll take full credit for the “finely crafted” attribution (with humility of course).  You should head to his website to see more cool work, along with more opportunities for classroom resources.  I’m also responsible for the metal casting and fabrication of the Rock-Paper-Scissors monument. (Yeah, I get to have some fun in the summer!)

A few handy extensions (more included below):

  • If your goal was to pack as many as possible, how many pencils could you fit in the same square footage, while still maintaining the shapes of the continents?
  • What if Aaron wants to make a 3-D version (like a globe)? Can you help him figure out how many pencils he’ll need for this? How many for a 5-foot diameter globe?  How about a 10-footer?
  • What if you tried to make the piece topographically accurate?  Whoa!
  • What if the budget for the mural commission gets cut in half?  How much smaller should Aaron make the mural?  ½ size?  He needs to maintain his idea as an artist, so he doesn’t want to make it too much smaller; how much smaller do you think he should make it, and how many pencils will it take?
  • How many would fit in the footprint of your school?
  • There’s always the Four Color Theorem for a cool direction.

Cynthia Lanius has some nice mapping extensions HERE

And HERE is a link to another nice mapping lesson called “How Big is Chicago?”

And HERE is another link to a judgemental comic about your preferred projection.


I’ve categorized the materials in Dan Meyer’s 3-Act format for ease of use.  Please use them, and let me know how it goes, or if you have ideas to improve or extend this lesson.

… and since this is my FIRST public blog posting, feel free to leave a pat on the back (or even better a brutally honest critique) in the comments, so I know if anyone’s out there.  Thanks!


THE MATERIALS: Please let me know if these don’t work, as I’m new at this, and don’t hesitate to write if you need higher quality files, or with suggestions.

Act 1

Act 2

Act 3