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.

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 7^{th} and 8^{th} 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.

- 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.

- 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?*

*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!

-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.

Math MunchCongratulations on your new blog! This is a solid first post. FULL of good stuff. I hope to see more.

nhighstein@gmail.comPost authorThanks for the encouraging words. I have some fun in my math class, and great kids to work with. I’m looking forward to sharing more soon.

-Nat

Fawn NguyenHi Nat,

Talk about a GRAND OPENING, Nat! This is a jam-packed post about an incredibly rich integrated lesson (project really). Thank you for all the resource materials – my goodness, it took me some time to peruse through them. Your recap of the lesson is much appreciated.

Amazing. Congrats and thank you, Nat.

Fawn

nhighstein@gmail.comPost authorHi Fawn,

Thanks so much for your thoughtful and generous words. I hope that it will get even better here soon!

-Nat

Daniel NoelBrilliant and fun. Great post, Nat.

Sarah GlatzNice job, Nat! It looks so professional!

Blogging is always on my list and I’ve succeeded in updating my “weekly blog” almost monthly.

You have an awesome example here.

I am impressed.

Sarah

NatPost authorThanks so much for visiting and for the kind comment, Sarah. I hope that you find ways to use some of this material!

-Nat

Barry LewisWow, Nat! Inspired, inspiring, and massively useful. I believe this is called a triblogfecta. Congrats & thanks! (That’s tri-blo-GFECT-a).

Now then:

(1) I especially love how you ask your students to “brainstorm a list of mathematical questions that they would like to answer.” The road to engagement is often thus paved, I reckon. I’m curious how you “guide them to prioritize” the question that you want them to work with. Presumably, this is the question that will lead to the kinds of tasks that require using knowledge in ways that can eventually demonstrate proficiency with the Learning Target(s) du jour, say, [7.G.B.6] as well as most, if not all, of the Standards for Mathematical Practice [ SMP]. Nicely done.

Have you ever gathered responses and then asked for, and accepted, a consensus on which Question students should address? This is the sort of terrifying classroom democracy that can destabilize a regime, right? And destabilized regimes rarely have all their content packed and shipped by June. Then again, I wonder if: OK. Cool. We did the question that you guys voted for, now here’s one of the others that you didn’t vote for, or even, here’s a different Question that I’d like you to work with. I wonder if following the paths that students make themselves can get us to places where more kids are more interested with the questions we next and really need them to work with. Not, let me add quickly, that the disinclination to engage in good work seems to be taking up much space in your classroom.

(2) I am thrilled to hear about how you are using/requiring writing in your class. How has this idea been received? Have students written in math classes before yours, or were you the first to throw them into that end of the pool?

(3) You suggested a “follow up with some direct instruction about mapping as a math field…” Fun! But have you found the need, either while in progress or as a summarizing process, to provide DI on area techniques or the proportional reasoning that both underlie this Question? [I haven’t broken the seal on your lesson plan materials yet.] Or does enough of all that tend to emerge directly from students during their mad, student-centered escapades?

Man, there sure are a lot of straws here.

NatPost authorBarry, Thanks for the thoughtful comments. I’ll respond to your points in order:

1. I’ve found that it depends on the task. Sometimes, they ask exactly the question I’m hoping for, and sometimes I have to be the one to pose that question… but I’ve also found that it’s worthwhile to honor any questions that they bring up. You’re absolutely right that becoming comfortable with less control has been tough at times, but I’ll take genuine math experiences over plowing through material any day.

2. Writing in math has been natural and easy for me. I think that it’s one of those things I just took for granted as part of the course, without leaving room for argument, so there hasn’t been any.

3. There’s almost always some Direct Instruction involved at some point, but in my experience, it feels very different if students have either asked for instruction, or if we’ve located some gaps in understanding together first.

Also – those sure are a lot of straws! Thanks for sharing the link along with your questions and comments.

-Nat

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