Category Archives: 3-Act Math

Generating questions

MTBoS blogging initiative, week 3!  This week’s prompt focuses on questioning.

betterquestionsMy colleague, who teaches the grade 6 and grade 7 math courses at my school is in training to run a marathon.  He has put together a training program for himself, which includes a schedule of endurance-building, and he has been collecting data with a GPS watch.  As he examined the data, he thought that this might make a rich exploration for his students and we have been working together to set up a project  for them.

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Here is what the raw data looks like

We started with the driving question: How long will Mr. Feutz take to complete the Limassol marathon? and then we began by brainstorming questions together.

  • How long will Mr. Feutz take to complete the marathon?
  • How many steps will he take to complete the marathon?
  • How many calories will he burn during the run?
  • What percentage of his overall time will be spent moving?  (Compared to taking breaks)
  • What will his average heart rate be during the marathon (In B.P.M.)?
  • What will be the shortest/longest mile time, and what is the range between these?

We tried to analyse which questions are actually interesting, and what might we be able to ask kids to do with them.  While running, he found that he was constantly doing math of one sort or another.  How much further will I run today? When will I arrive back at home?  Things that he had genuine curiosity about, and questions that math gives us the power to answer.

We set up a graphic organizer, and decided to ask our driving question directly.  Here are some of the kids’ initial responses.

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They always manage to think of something that we haven’t anticipated (…which is why we love kids!).  “Will you be listening to music, because if it’s like… Taylor Swift, you wouldn’t be as inspired as if it was like… hard rock.”  Students were given a graphic organizer and asked to write their first guesses.   As they acquire more information, they will refine their estimates.  Part of the beauty of this work is that they will get to actually test their prediction, and compare their answer to what actually happens.

Students will revisit this project over the next weeks, and will be asked to refine their work.  They have already studied unit rates, and are moving into work on ratio and proportion next.  We are hoping that more questions will arise as we continue this work.  My favorite so far is, “Given a start time, time spent running so far, and a map of the run, can you figure out where Mr. Feutz is now?”

Here is our Graphic Organizer – totally open to your critique and suggestions.  What questions can you add to our list, and how do you come up with your project questions?  We would be most grateful if you share your curiosities or strategies in the comments.

Related:

The Thirsty Crow

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Not sure which one I saw first, but I got the idea for this lesson hook from at least two teachers: Jensilvermath, and Pam Wilson. Both are creative educators, and generous online colleagues, who share their ideas, resources, and materials.

Screen Shot 2015-12-03 at 7.33.57 PMOne of Aesop’s Fables tells the story of a crow who comes across a half-full pitcher of water in the desert. He cannot reach the water until he figures out that by dropping pebbles into the vessel, the displacement causes the water level to rise until he can quench his thirst. Using this narrative as our lesson hook, students were given a cup full of marbles, and a graduated cylinder partially filled with water. They were asked to predict how many marbles they would need to reach 2000 mL, and then how many more until the water overflowed.

I have found that giving too much structure can take some of the life out of a task, but not enough structure, and students flounder. In this case, I asked them a direct question, but did not suggest any methods at first.  As we were right in the midst of linear equations, my assumption was that they would jump right to dropping their marbles into the cylinder, creating a scatterplot, find an average rate of change and line of best fit.  But students always surprise me.  They asked for an extra graduated cylinder to do some experiIMG_4298mentation, and pulled out the scale to start weighing marbles.  They traced the cylinder base to see how many marbles fit in that circle.  As we had more than one color, it was important to them to see if the lighter blue marbles were consistent with the dark blue – something I hadn’t even considered.  One group even qualified their prediction with the caveat, “…if the ratio of light blue to dark blue marbles is consistent with our sample, then
this prediction should hold.” What a nice expression of understanding. Reminder to self: always give students as much freedom as possible. Let them run until they really need help.

The students who dropped their marbles into the cylinder one at a time collectIMG_4312ed data points as the water level rose. They created scatterplots of this data, and calculated an average rate of change. Next, they used this information to find an equation for a line of best fit, which helped them to make a confident prediction about how many marbles they would need to bring the water all the way to the top. We took out enough marbles to test their predictions, and added them to the cylinder until the water level reached 2000 mL and then until it overflowed. Cheers and groans for the accuracy of their predictions.

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Creating ways for students to create mathematical models and make predictions is one of the most important opportunities that I can set up for them.  These types of tasks help students to connect the math from their classroom to questions that they will come across in the real world. Even if they will not need to calculate the number of marbles to overflow a cylinder, they will almost certainly need to use similar problem-solving skills, and equally importantly, they will have to decide what math skills they need to apply to novel situations.  Students react very strongly when they see the “answer” to this type of task – very different from how most students react when looking up the answer in the back of the math textbook.  Even reluctant mathematicians couldn’t help but look closely as we counted the last few marbles out!

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I used this video to introduce the Crow and the Pitcher. It’s short but gets the point across. I shot a video of our cylinder, and edited it into a 3-Act format while I had the supplies out.  I think that if you can get your hands on some marbles and a vessel, you may as well do this hands-on, but in case you don’t have a bunch of marbles handy, or if video is your preferred medium, I’ve published the materials below for you to use.  Did I give enough information in act 2 or did I forget something?  Please do let me know if you use any of this, and how it goes …and don’t forget to check out the Action Version…

Thirsty Crow Act 1

Thirsty Crow Act 2

Thirsty Crow Act 3

Thirsty Crow Act 3 Extended (Includes Action Version!)

Student Created 3-Act Math

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When will the world population reach 8 billion?

I integrated a bunch of 3-Act math tasks into my Algebra classes this year, and I love the spirit in which these can be presented.  3-Acts give math teachers the language of drama and storytelling, language often reserved for writing or drama class, revealing information bit by bit to students, and keeping them hungry for more through a regular dramatic format.  In my experience, this has been motivating for students, and this motivation has led to leverage for convincing students to care about mathematical rigor.  Thank you to everyone who creates and shares the work to make this possible, including 3-Acts, for the benefit of myself and my students.  MTBoS rocks, and it’s a verdant time to be a math teacher.

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How do slow and fast speeds compare in “Piano Tiles”?

For one of our final projects this year in my Algebra classes, I asked students to design their own 3-Act tasks.  Even though they had seen a number of these, and were familiar with this format, there was mixed reaction to this assignment from students (…and mixed results).  Some created really great work, but some still resent the idea that they are expected to be creative in math class.  Lots of students have a pretty narrow definition of math, and it’s really hard for many of them to shift in attitude – in spite of  my persistence this year in presenting tasks that required critical and creative thinking.  Lots of them have long since defined themselves as a certain kind of math student and have become accustomed to being taught procedures, and repeating them back on demand.  Just for context, while my school doesn’t totally track classes, in general the highest achievers are not in my sections, and I work with many concrete thinkers.

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What are Leroy’s chances of survival?

3-Acts are hard.  They are demanding on students, and they require rigor and precision, synthesis and critical thinking.  And this is a tough job for the teacher as well.  We need to craft the lesson in such a way that students actually feel a need for the math skills we want them to practice, and then make the right tools available and accessible at the right moment. I think that overall, I did have some success in shifting student attitudes in general towards math this year, and I think we created some good work together.  I think that just like me, students would get better at creative thinking in math with practice.  It would be good to try this assignment mid year, and then again at the end of the year.  During our share, they definitely enjoyed viewing and solving each others’ 3-Acts (some in spite of themselves).  And there is value in the final math class experience of the school year being so positive. Their 3-Act subjects ranged from estimation to frisbee to World of Warcraft.  Check out a few of their projects HERE or at the top of the page, and let me know what you think.

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How many mattresses?

BTW, I asked the following question on my end of year student survey: “What project/lesson/assessment have you learned the most from this year? Why?”  48/62 student responses included positive references to the 3-Act problems.  Here are a few excerpts:

  • “3 part problems- more realistic then normal word problems- feels like the math I know will be useful…”
  • “The 3 act problems. We did a lot of them and each time I could learn more of it. Culminating with creating one myself really helped to practice it even more.”
  • “3 act work because we never know what it’s going to be when you walk in class,”
  • “…you found something I was Passionate about and taught me how to make it in to a fun learning Experience.”

An aside: I’m heading out for an adventure, and will be teaching at an International School in Cyprus next year!  Even though I am excited about my new position, I am super sad about leaving my AWESOME school in Portland Maine: Casco Bay High School.  My colleagues were inspirational, demanding.  My math colleagues, and the junior team (BTW, check out the amazing 2014 Junior Documentary Work HERE) are world class educators, every one of them.  Derek Pierce, the school principal is a truly exceptional leader; supportive, inspiring, and kind.  I have been incredibly lucky to work for and with him.  Derek and my colleagues at CBHS encouraged me to take risks, and to push myself as an educator and a person, and helped me to encourage students to take risks.  Without their support, this kind of work would not be possible.  Thank you to everyone at CBHS! CBHS Staff Further Reading: I’m pretty sure that the 3-Act rubric I found HERE came from a Math Forum problem solving session.  I wonder how their students did?

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.

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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? 
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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!

-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