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?”
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.