# 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).

Students 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.

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.

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.

# “First Past The Post” And The Dreaded Disease “Z”

This is a lesson I adapted from a model I first learned about at a math teaching conference that I attended at the Dana Hall School.  This conference was without a doubt, the best professional development workshop I’ve attended.  The faculty was amazing, the workshops were specifically geared by grade level, and the resources were practical and rich.  My teacher for the grade 6-8 workshops was Tom Harding, who is the current math department head at Shady Hill.  He was a wonderful and caring teacher.  I can’t say enough good things about this workshop.

This problem has a fun, if highly unbelievable back-story.  As the kids picked apart the story – and mine spent some time doing this – there was no pretense of the specific situation actually being useful, except perhaps for creative writing, but the ideas are highly transferable.  It is not a real world problem.  Perhaps even better, it falls under the Realistic Math Education (REM) umbrella described by MathEd.net HERE.

I used this as an integrated lesson about forms of government in addition to the math lesson.  I’ll set-up the problem and then explain what I did with it.

THE SET-UP

A group of 5 people are boarding a spaceship to Mars. Their health is one of the following:

• Normal
• Mono
• Allergies
• Common cold

And one has…

Students must identify who has The Dreaded Disease Z, as it is highly contagious – and fatal!  If the infected person boards the ship with everyone else, they are all doomed.

THE PROCEDURE

The only way to identify each person’s health is through blood testing; a bag for each person has representative chips for their blood levels (per chart included below).  I used color tiles in brown paper bags for this, and let students take 40 “blood samples” (a sample is taken by blindly taking one chip out of the bag, noting its color, and putting it back in the bag).  With 40 chances, students had to be strategic about which bags to sample from.  Should we take 5 samples from one bag or 1 sample from each bag?  Should we decide on 20 samples and then re-evaluate, or re-evaluate our plan every 5 samples?

Of course, students need some time to just play with the tiles and make patterns too.

You’ll need to set up at least one set of bags with tiles to match the histogram.  If you want separate groups to choose their own ways of sampling, you’ll need multiple set-ups.  40 samples were just about enough to make students pretty confident in their guesses, but not enough for them to be sure.  For me, the big math idea here is sample size, and given a situation, how many samples do you need.  I had students make individual predictions at 0, 10, 20, and 30 samples, and then come to a group consensus about who to kick off the spaceship.

INTEGRATION

I was working on forms of government as part of an integrated Social Studies unit, and saw a nice opportunity here.  When I did this at the conference with other teachers, there was a certain amount of strife about how we should decide to take our samples.  We each had ideas about how to go about finding the bag representing the Dreaded “Z,” and it was sometimes hard to let someone else decide.  And we were adults in a situation where we shouldn’t have cared about solving the problem, but instead should have been focused on examining the merits of different strategies, and why a student might choose one or another.  I imagined that for kids, this would be much harder than it had been for us – and I couldn’t resist leveraging these potential conflicts.

I did this in a whole group setting.  Each group of 4-5 students got one bag, and I asked them to give it a name.  In naming their bag, they grew attached to it, and were more invested in having their bag not be “infected.”  We later talked about how statisticians remain impartial (…or not).

I divided the 40 allowed samples into eight groups of five, and gave specific instructions for who got to decide which samples were taken.

1. Samples 1-5: By Consensus:  Students all had to give a thumbs-up (I’m totally behind this method), thumbs-sideways (I can live with this method), or thumbs-down (I can’t live with this method).  They had to keep talking it out until everyone had at least thumbs-sideways, and then we took the samples.  As you might imagine, there were some strong ideas and this took a while.
2. Samples 6-10: “First Past The Post” Popular Vote:  Anyone could suggest a method, and we voted on which method to use.  The method that received the most votes was the one we used.
3. Samples 11-15: Alternative Vote: Students suggested methods, and voted for their top three choices in ranked order.  We tallied votes, and eliminated the lowest ranks until we had a vote of over 50%.
4. Samples 16-20: Students voted for three classmates by secret ballot.  The three classmates got together and decided how to take the samples.
5. Samples 21-25: I chose one student to make the decisions for the whole class, and didn’t allow any consultation.
6. Samples 26-30: I kept the single decision maker, but added two advisors to help with the decision.  The final decision was still up to the one student with the power.
7. Samples 31-35: I chose four students to make the decisions for everyone.
8. Samples 36-40: I chose names out of a hat, and each student whose name was picked could choose one sample.

After each set of 5, I asked students to name the decision-making method, and reflect on how it felt in writing (The organizer that I used for this is included below).  At the end, I had each group choose who to kick off the spaceship, dumped the tiles for the big reveal, and debriefed the math.  We talked over statistical analysis, probability, and sample size.  This was an excellent big picture statistics discussion, and worked well as a hook for probability exercises to come.

After the math discussion, we debriefed the decision making process.  As you likely figured out, the decision-making methods roughly correspond to different forms of government: Variations of Democracy, Republic, Dictatorship, Socialism, Oligarchy, and Anarchy (OK, Anarchy analogy sort of falls apart).  The analogies only went so far, but the written reflections gave us a great opportunity to talk over the benefits and liabilities of different systems, and the different perspectives of those in charge vs. those not in charge.  “It was awesome! I got to decide for everyone.”  Or… “It was really hard.  There was a lot of pressure and I had to decide for everyone!”  I referred back to both the mathematical and the philosophical content of this discussion several times over the course of the news few weeks’ studies, and the kids are still talking about this and asking to do it again.

THE MATERIALS

Organizer for Recording the Decision Making Process

BONUS: Dreaded Disease X Graph for doing this a second time, or for adding a sixth crew member for more complexity

# There Can Be Only Five

These Platonic Solids – made out of playing cards – are one of the contributions my 8th graders are creating to exhibit during our Grade 5-8 Math and Art Festival next week.  The templates I used were from Jason at Mathcraft and George Hart.  Thanks to cheesemonkeysf for sending a Tweet out to share this resource.

Students had to articulate the characteristics of the five Platonic Solids, and describe in writing how their cards represented each solid.  Can you believe that there are really only FIVE of these?  It totally seems that there should be more!  My students and I spent some time looking for that elusive Platonic Solid #6.  I asked students to define in their own words each of the Platonic Solids.  These writings will accompany their displays.  I’m looking to challenge them a little further – if you have another idea or resource for a more complex card construction, leave a comment. (Thanks!)

For the few students who were ready for this, I had some reading available about the Duality of Platonic Solids, along with an inductive proof that “there can be only five” possible Platonic Solids (…wait is this where the idea for Highlander came from!?).

# Triangle Centers and Student Empowerment

I’ve always found triangle centers to be highly intriguing.  The initial related mathematical questions are low-entry.  But the ideas can go really deep, and there are fun, and valuable visual and logic skills to exercise.

With my 8th graders, I began by giving out an Entrance Ticket.  We had practiced with the Pythagorean Theorem, finding missing side lengths, missing angles, and area, but we hadn’t explicitly referred to triangle centers before this.

The ticket asks them to plot the center of each triangle, but doesn’t give them any more specific instruction.  Most students wanted to know how to do it.  I was totally non-committal.  I pretended to be busy setting up the projector, and told them to hurry up so we could move on to the lesson.  I gave them a few minutes, and then collected the tickets.

Next, we visited Dan and Dave’s Interactive Ice Cream Shop.  This app asks you to decide which ice cream shop you should patronize depending on your position in the park, to describe regions of the park where you would decide on one shop or another, and then shows how your intuitive work compares to others’.  After getting over the idea of why anyone would possibly shop at a place called “Frozen Flavored Milk,” students definitely enjoyed this process.  (Thanks Dave and Dan for putting it together and sharing it so publicly!).  This is a great, low-pressure way to begin the exploration.  Click on the picture to visit their site.

Many of them now thought that this was the “right” way to find a triangle center, and wanted the entrance tickets back to revise them.  Their requests went totally unrequited.  We reviewed our tools for geometric constructions, including bisecting angles, and drawing perpendicular lines, and then headed for Geogebra.  Students created their own circumcenters (although we didn’t use the word yet – thus far, this is “Elsie’s method”).  Oh, and they had to pass the “drag test.”

Then, we watched Vi Hart’s Infinity Elephants.  Kids love the Apollonian Gaskets especially, and to make these, we just needed to find the incenter.  “Wait a minute.  This is different.  Which one is the center? This one or the Ice Cream one?” I cruelly asked them the same question that they were asking me:  “Which method is right?  Which one is the real center?”  I act busy and move on to someone else.  Kids go to Geogebra, and create an incenter.

Now, I passed around some sheet metal triangles that I had lying around (detritus from my night job).  “Try to balance these on the eraser of your pencil.”  I let kids experiment for a few minutes, and then asked them to pass their triangle to the next group. Before attempting to balance this one, they had to mark a point on the steel.  How did they come to that point?  Back to Paper, pencil, and Geogebra for the centroid.

I’m still looking for an orthocenter hook.  The useful things I could think of are only useful if you’re already into the geometry (leave an idea in the comments if you have one handy), but my kids were into it by then anyway.  I just gave them another rule and asked them to construct bisectors.  As students constructed these, I wondered out loud if there was any difference between this center and the first one.

I asked them to complete the exit ticket.  Several still asked which one was the right one.  “Which one is actually the center?  No, really this time.”  I desperately tried to hand the power back to them by asking them the same question.  “YOU get to decide.”  That’s the whole point.  For me, it’s all about giving them the power to decide on their own definitions – based on their own criteria.

What exactly is the center of a triangle anyway?

Here are some extensions to have handy.  I used some of these over the next few days.

• Given 3 cities, what city is at the circumcenter, centroid, etc?  Middle school kids love to find things like this on maps.  You can ask the question referencing ice-cream shops or balancing the entire state on a pencil eraser of that seems like fun.
• How could you find the centroid of square?  A rectangle?  A pentagon?  Any irregular polygon?
• In which of the centers is it possible that the center is outside the triangle?
• Euler Lines – So cool!
• This problem is from Cynthia Lanius, and was only peripherally related, but it was there and relevant at the right moment.  The Three Evil Dictators: To keep them from fighting, you must separate three evil dictators as far away from each other as possible on the surface of the earth. If you put one in Los Angeles, California, name two other places where the other two could be placed so that all are equidistant, and at maximum distances from each other. (Yes, you can put them in the ocean if you have to.)

MATERIALS

Triangle Entrance and Exit Ticket

# 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.

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.

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.

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.

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?”

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!

-Nat

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

Supplements