Category Archives: Algebra

Lies and Collaboration

I actually enjoy the puzzle-like aspect of exponent rules, and simplifying radicals.  For me, there is something satisfying about learning ways to manipulate numbers and letters – probably why I love algebra so much.  But I am tuned-in enough to my students to know that many of them don’t find the same satisfaction from doing this work just for the sake of intellectual exercise.   And since calculators came into vogue, it’s been harder to justify the need for rationalizing the denominator or expressing the square root of 50 as 5 times the square root of 2.  But we are tasked to follow standards that often include these kinds of skills and it has been helpful for me to turn this into exploration or game learning as much as possible.

IMG_4135I did some mining of the MTBoS for ideas to teach rules of exponent arithmetic and came across this post, which includes a nice exploratory worksheet from Andrew Stadel.  He describes a similar issue with contextualizing exponent rules for middle schoolers – one of the really great things about our online community are these moments where we are reminded that we are not alone.  He asks his students to find the mistakes in the equations, to explain where the author went wrong, and to find the correct solutions.  He used a bunch of the common misconceptions found on mathmistakes.org to help students to catch themselves in the common errors.  Very nicely done.  This would have been a good lesson as is.

IMG_4147Then I remembered the Bucket O Lies protocol from Nora Oswald at Simplify With Me.  Nora manages to gamify math like no one else that I’ve seen.  She manages to add entertainment even to potentially dry topics like this one.  I combined Andrew’s worksheet with Nora’s idea to make a bucket-o-exponent lies.  I printed the 3 worksheets, cut them out into individual problems, folded them up, and put them into buckets (or baskets).  Voila! Drama and Motivation.  In pairs or threes, learning happened.

IMG_2885Of course, I hammed it up with the students.  There’s nothing like telling teenagers that someone is trying to get one over on them to motivate them.  This has worked well for me in the past, especially when it came from advertisements.  I riled them up by acting outraged that someone had created this whole set of math material, which was full of mistakes!  (Actually, I blamed Andrew :) ) …Lies I tell you… these baskets are FULL OF LIES!  Let’s find the mistakes so we can write a self-righteous set of corrections back to this author who was deliberately spreading bad math.

They quickly saw through my act, but it was enough.  They were already motivated in spite of themselves.  Andrew’s worksheet was just enough for everyone.  I started by coaching the groups who needed help getting started and moved to pairs who were making mistakes with fractional exponents.  For my honors group, I added a few more examples with rational exponents.

Thanks Andrew!  Thanks Nora!  Our generous community is the Best!

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First Week: Building Culture

The start of the school year is one of the most important moments for my classes.  Setting the right tone and attitude right from the beginning can mean buy-in from students right away – and conversely, a bad start can be really tough to recover from.  I had a pretty good start this year in my Algebra 1 and Algebra 2 classes.  I wanted to share some things that worked for me in case someone else might benefit, and to document the week, as I may repeat much of this work next year.

My students have been working on a pseudo-Appolonian gasket on the whiteboard. It makes a nice frame for our learning targets.

My students have been working on a pseudo-Appolonian gasket on the whiteboard. It makes a nice frame for our learning targets.

I have several goals for how I want my classroom to “be,” and the first week is a chance to work on some of the big picture ways that we will be working together this year.

  • It is important to me that as a group, we celebrate scholarship – and the struggles involved in becoming scholars
  • I want to nurture a love of learning and of curiosity
  • Our classroom has to be a safe place to take chances and to make mistakes
  • We need to be able to work collaboratively – even more than in other subjects, I believe that we really need to see how others think in order to understand math
  • To that end, we need to learn to be comfortable talking (and arguing!) about math
  • We need to work independently as well, and to trust and value our own ideas
  • We need to respect each other, and hopefully to love each other at least a little.  Of course I love all of them.

I used a series of activities (all sourced from the MTBoS of course) to try to help establish this culture.

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DAY 1: What does it take to do math?

Very first thing, I assigned each student a “secret partner” for the week, based on this idea from Origins.  Students are to observe their partner throughout the week, and are responsible to report back an acknowledgement of something positive that they observed at the end of the week.  The payoff for this happens on day 5.  Next, I introduced a version of Jasmine’s Tabletop Twitter.  I set up 5 stations around the room.  Each had chart paper with a question/prompt on it.  Students moved around the room in two minute rotations, and were asked to respond silently to each question.  I followed Jasmine’s lead in asking students to take a marker and write their name with that marker, so we could look back and see who had authored each comment.  My five prompts were:

  1. Why do we learn math?
  2. What will make our math class a good learning environment?
  3. What does it take to be a good math student?
  4. Respond to this quote: “Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost.” – W. S. Anglin
  5. Add a song to our class playlists. Write a genre instead of a song if you prefer.

At the last station, I gave out this Capture your thoughts organizer, and asked students to synthesize and summarize the most important points from their station, to add anything they thought was missing, and then report back to the whole group.  We hung our new “posters” in the hall on this year’s “Sweet Wall Of Math,” to help establish that our work will be public this year and we are proud to show our thinking to the world.  I’ll use the ideas they shared to create our learning agreements for the year.  For anyone who would like more detailed plans for day 1, I’ve written them up just for you: Day 1 Plans. :)

Day 2: How can we create the questions?

I am totally convinced of the positive impact that Dan Meyer’s 3-Act format can have an a group of math students, so I was excited to introduce 3-Act math tasks right away on day 2.  Students are so used to arriving in math class, and just imitating the teacher that they often don’t know how to react when they are asked to think of a question themselves, and then asked to figure out what they actually need to do to solve their question.  The first tasks like this can be really tough and even painful, often for some of the “top” students.

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The Super Bear was a nice one for both groups.  The math was easily accessible, which gave us room to learn the structure of how we should approach these kinds of tasks.  I made a new 3 Act handout for students to use, and guided them through the process.  I was strict about keeping silence in the room for the entirety of Act 1, except when they were asked to share their guesses and to establish a high/low range.  I stressed the importance of this “grappling” time, when they get to really think for themselves without the bias of hearing others’ ideas, and promised that they would get to work together for the rest of the task.  This is one of the important routines in my class, and is one of the few rules I impose on the group without their input.  Every group suggested weighing the bears, and several came up with ideas for how to measure volume (displacing in water, melting down the bears…).  Act 3 provided some rich discussion about the discrepancy between their solutions and the revealed answer, and the drama of the reveal of Act 3 can’t be beat!  Even reluctant students can’t look away as the gummy bears are weighed out.

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Day 3: Metaphors for perseverance.

We spent much of our class playing the Game About Squares.  I followed Annie Fetter’s suggestion to try this out with students, and read her post about this several times, so it was fresh in my head. She has a clarity about the importance of these tasks that I wanted to hold on to, emulate, and embody for this lesson.  This is a game that does everything that we want in our math classes.  It meets kids where they are, and little by little gives them slightly more challenging tasks to accomplish.  I (mostly)refused to help them at all, but made it clear that they were expected to figure out what to do.  After some initial discomfort with the whole idea that they were going to be playing an on screen game, and that I wouldn’t help, they dove in.  They grappled, made mistakes, started over, helped each other, groaned, and persevered.   They were competitive and proud when they solved each level.  We used the last 15-20 minutes to debrief the activity, to list the things that helped us succeed, and to respond to a short survey.  We talked about how these skills actually encompass just about everything that they need to be successful math thinkers.  Interesting that the number one thing they did that helped them to be successful in this game was to make mistakes.  I bet that we will be referring back to this often.

Annie Fetter is the best.  Just Saying.

G of Squares Survey results

Survey results after 30 student responses. Notice the top result!

Day 4: Number flexibility: You mean there’s more than one answer?

Day four, we worked on the four 4s.  This has been a favorite of mine since I began working with students.  It allows for multiple approaches and creativity in math thinking.  I’ve written about it further here and here.  This year, I decided to keep it to one class period.  In groups of 3, I challenged Algebra 1 students to create every number from 1-20, and Algebra 2 to shoot for 1-30.  We put their work out on our Sweet Wall, and they may go later to try to fill in any blanks.  Jo Boaler and the Youcubed team put together an excellent week of inspirational math, which began with this activity.  The rest of the inspirational week’s activities were tough to resist.  There are some good ones in there, along with great growth-mindset messages for students.  I may get back to the others later in the year if we have time.  I did play her day one video, and led a short discussion hinting at growth mindset to end Thursday’s class.  I was also especially tempted to jump on the explicit growth mindset work that Julie Reulbach has shared, but we can’t do everything.  I will be following Julie’s reflections closely to see how her implementation goes this year.

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Day 5: Assessing Numeracy + Mathematical Drama.

Trying to do Algebra without a solid understanding of arithmetic is rough.  I’ve seen students suffer through this, and it is not easy for me or for them, and there just isn’t the time during Algebra 2 to work on dividing fractions or operations with negative numbers.  So we’re implementing an after school numeracy workshop this year for grade 8, 9, and 10 students who need more support in this area.  We used this class to assess students’ arithmetic skills, and to identify those who might be most helped by the after school program.

I saved the last 20 minutes of class to follow up on the secret partners activity, and for a read-aloud.  Secret partners takes just a few minutes, but has a nice impact on student attitudes.  They act reluctant to speak nicely about each other, but they are grateful for this opportunity to celebrate each other’s good qualities.  Comments ranged from “I noticed A looking out for the new student at lunch” to “Y worked really hard on the science lab” to “X is really funny and cracked me up in English yesterday.”  I ended the week by reading the introduction to Zero: Biography of a Dangerous Idea.  This is an excellent book, and the introduction is high drama!  And kids just like to be read to.

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Although we didn’t get deep into new content this week, we did some valuable math together.  But equally important is the positive feeling that students left with on Friday afternoon.  With confidence in themselves from their successes, with trust in each other and the knowledge that their peers notice their positive behaviors, and with the assumption that their teacher cares about them, we are set up for the year.  Now we need to hold on to this feeling when the going gets rougher!

 

 

 

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“The UN experts disagree about what the future will hold, so we figured that if we wanted answers to our questions that we would need to become the experts.”best_global_math_big_marker_logo.001

I was fortunate to catch the rerun of an excellent Global Math Department presentation by Kyle Moyer and Zack Miller(@zmill415).  They presented their approach to curriculum and instruction, which focuses on project based learning, and integration in the math classroom.  They included a description of their “Booming Populations” project, designed to study and compare linear and exponential functions by examining population trends and predicting the population of a country in the year 2050.  The materials they designed are well thought out and put together, and I decided to adapt the project for my Algebra 1 students in Cyprus.   This was a rich experience for my students for many reasons.

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I used a gallery walk format to build background knowledge and pique interest, and there was quick and solid engagement.  Students were fascinated by the world population trends, and were especially hooked by the leaps in population size over the last century as compared to the rest of history.  This example of exponential growth was both attention getting and highly understandable.   There was built in choice.  Students were allowed to choose a country – and I can’t overstate how much of a difference this makes for them.  They picked a country that they had some interest in or connection to; a family connection or a place that they had visited or wanted to visit, or just a country that they wanted to learn more about.  Choices ranged from China to Greenland to Peru to North Korea, allowing for deep comparisons of statistical trends, modelling validity, and evaluation of source data.

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The work was easily and naturally differentiated.  Students who were approaching mastery could plug numbers into a slope-intercept equation and into a standard exponential formula, and those who were ready could really push the nuances of their models.  I even had a few students dabble with quadratic models (…and this was before we had covered quadratics in class).  Advanced students could keep on adding complexity and depth to their predictions by taking into account more pieces of information – demographics, political stability, or even global climate change (will the Maldives still be around in 2050 or will the islands be underwater due to rising sea levels leading eventually to a zero population?).  And this was naturally self-paced as well.  Very few students reached a “stuck” point, where they needed to wait for the teacher to tell them where to go next.  Over the four weeks that we worked on this, I used a combination of discovery-based lessons and some direct instruction to help students build skills to be successful in this project.

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Students were asked to examine and compile population data for their country from 1960-1990, and to create linear and exponential models to study this data.  They then created a model to predict the trends that they would expect from 1990-2015.  After comparing this model to the actual population numbers, students committed to one type of model to predict the population of their country in the year 2050.  They were required to complete a written analysis, and to present their analysis and predictions to an audience including a “panel of experts” at our “2015 Population Summit.”  Knowing that they would be presenting this work publicly lent gravitas to most of what they did – they were invested in understanding and being able to explain the math that they used, and to justify the decisions that they made in creating their models.  They learned to harness the power of spreadsheets to help them to organize their data and to create graphs – a really great skill for them to practice.  The public nature of this work forced them to make accurate graphs, and to consider carefully decisions about scale, and how to best communicate data visually.

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This was definitely some of the best learning that I have been able to orchestrate as a teacher.  Every student achieved the basic learning targets, and most exceeded the standards.  Students were comfortably using vocabulary like linear vs. exponential models, initial condition, growth factor vs. growth rate, and I heard many arguments between students who were invested in defending the mathematical choices that they had made.  This project found that sweet spot between just enough structure to keep everyone on track, and enough freedom to allow students to make decisions and to own the work.

While I shamelessly use and reshuffle ideas from books and from the MTBoS, I nearly always have to tweak and remake the materials for my students.  The language, design, or content have to be customized to meet them where they are, and to give them just enough information to succeed without giving them so much that they don’t have the chance to do their own thinking.  The materials that Zack and Kyle have so openly shared (THANK YOU Zack and Kyle!!)  are as close to ready-made as I have found.  I made some minor tweaks to the guidelines and formatting, but used almost all of their work.  Their approach to teaching math is very well articulated, and their Global Math presentation is very much worth watching in its entirety as well.  Their use of “playlists” to help students self-direct is especially interesting.

I am hoping to develop this into a more interdisciplinary and comprehensive project for next year, and perhaps something that could be a staple of the 8th grade curriculum.  My goals for our math program include building inquiry into the math class process, and creating connections between math and other content areas, and I am especially interested in feedback on ways to leverage these things.  Please do throw your ideas in the comments.  If you’d like to see some student work or reflections, just drop me a tweet or an email.  While student presentations were strong this year, I will make sure to add in more rehearsal time for them to practice next time – especially when they request that the panel of experts ask hard questions.

“Hello and welcome to the 2015 AISC population summit. In our 8th grade Algebra class, we have been looking at world population trends, and thinking about what will happen going into the future.  The UN experts disagree about what the future will hold, so we figured that if we wanted answers to our questions that we would need to become the experts.

 Each of us chose one country to study. We examined our country’s population changes since 1960, and created graphs and mathematical models to help us predict what the population of our country will be in the year 2050.

 We compared a linear model and an exponential model, and decided which one we thought would make a better prediction for our specific country. We did some basic research into our country’s history to give some context to our math models.

 We hope that you enjoy yourselves, that you learn something, and that you are willing to ask us hard questions and give us critical feedback.”

BTW: The Desmos Penny Circle is of course a perfect companion/ follow up to this activity.

Quadratics: Mighty Square! (start by completing the square)

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Many texts I’ve seen ask students to solve quadratics by factoring, then by graphing, then memorising the quadratic formula, and then if there’s time left, they are introduced to the idea of completing the square.  I’ve done it before like this myself, and I have seen students struggle mightily and miserably.  For me, beginning with factoring is problematic.  While factoring trinomials can be satisfying, especially for students who like the puzzle solving parts of math class, this technique generally only works with problems that have been contrived by math teachers or textbook writers.  Most quadratics we come across don’t naturally factor to nice clean numbers like y=(x+4)(x-2).

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I was looking for a better approach, and came across James Tanton’s take on quadratics.  He emphasizes symmetry as the key to studying and understanding quadratics, and right from the start, teaches completing the square (Although he relentlessly resists formal vocabulary, and calls it the “box method”) as the way to solve for x.    His course gives a sequence of problems, each of which adds one level of complexity until students can solve just about any quadratic thrown at them.  As he says, “The box method will never let you down.”

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Developing a solid conceptual understanding of completing the square leads naturally to moving back and forth between standard and vertex form, and to the derivation of the quadratic formula – and students know why it works.  And the best part is, only one of my students from this year’s group uses the quadratic formula as her go-to method.  They all go straight for completing the square because they understand why and how it works, and are totally comfortable with the techniques.  Most of them can recite the formula, but they are worried about making arithmetic mistakes, and are not as confident in their results as they are when they use symmetry and completing the square.

 

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I strayed from Tanton’s approach in some ways – we figured out imaginary solutions with the honors group (Tanton advocates to leave out imaginary solutions when studying quadratics with Algebra 1 or 2 classes) – because they asked, and because they were ready to expand their horizons.  And I did eventually teach factoring as a method of solving quadratics.  Tanton suggests that factoring might be better included when we’re working on a discreet study of polynomials – but honestly, it was easy for me to fit in as the last method I taught.  We took less than one full class period to discuss and practice factoring, and by the time we got there, students understood what the factors meant, and how they related to the roots of the quadratic.  They appreciated the quickness of the solutions when factoring worked, and understood that if it didn’t work easily, that they could fall back on methods they know.

Every one of my Algebra students, even those who really struggle with math have had success moving through this sequence.  Although this is only one year, and there are always other variables, I am convinced that this order makes much better intuitive sense.  I’ll report back next year after I’ve had the chance to try this again.

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I love the way that the white boards look when studying quadratics

BTW, some of my favorite resources to go deeply into studying quadratics include

Epic Poodle Bungee Jump

My Algebra students just completed a project using linear equations to predict how many rubber bands it would take to give their poodle a thrilling, yet safe bungee jump from the school balcony.  This activity was adapted from NCTM’s Barbie Bungee activity.  I do appreciate the opportunity to discuss with students the cultural stereotypes and gender assumptions perpetuated by Barbie, but in this case, using these stuffed poodles allowed us to focus on the math and on the fun. IMG_1735

Students were asked to string together one, two, three four, five and six rubber bands, and then make a prediction based on their measurements.  The object was for their poodle to get as close to the ground as possible, without actually landing on its’ head.  They predicted how many bands they would need for the first balcony, and if they survived the first jump, the poodles made the jump from the second balcony.

Check out the students’ reactions at the 1:20 minute mark and the 1:36 minute mark – in my experience, this is not your typical reaction to getting the right answer on a math problem!  And look at the celebration from the winning team at 2:23!  This engagement lent leverage into our reflective learning process.

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I often struggle with how much structure and guidance to give students with a project like this.  I want to set them up for success, but it is important to me that students have the opportunity to think things through for themselves, and to have the opportunity to get the wrong answer.  My students are used to me telling them that the best thing they can do is to make a mistake …and then to go back and figure out where their mistake was (Thanks to Jo Boaler for sharing her convincing evidence of how the brain grows by correcting errors in our thinking!).  My preferred method is to give students a compelling task with very few suggestions, and to share tools and advice only when they are requested.  If an activity is choreographed well, they tend to ask the right questions.  And if they totally screw up their model, it is an opportunity to go back and see why.  If we can convince them to care, then when they make an error, they will want to go back and see where and why they made mistakes.

My students know that we learn math socially in our classroom, and that I always leave room for sharing of methods and thinking, but they are also used to two times when the class needs to be silent – at the start and the completion of tasks.  Everyone needs space to generate their own thinking and ideas  – and the room to come up with their own first guess (I have been convinced by Dan Meyer and others of the engagement generated by asking students for an initial guess).  We also make sure to reflect individually on our own ideas and performance, and this is a regular part of our class as well.  Everyone is expected to look back to see how well our plans and our models worked.

Of course, I include myself in this reflection as well.  I think that this one went well.

Some of my favorite Geoboard activities

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I love using math manipulatives.  For some kids, using visual logic, or a hands-on approach can help to remove barriers to understanding, and can take some of the intimidation and fear out of learning about difficult concepts.  Using Algebra tiles to teach polynomial multiplication and factoring quadratics help to reinforce an area model of multiplication, and completing the square – well it does make more sense if you actually complete a square.  (And if you get the lab gear designed by Picciotto, you can complete the cube too!)  I’ll post about my experience with these sometime soon.  For today, here are some ways that we have used Geoboards in my Algebra classes.

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My Algebra students had a blast using geoboards to find triangles and squares.  The Math Forum has put together a set of very rich ideas for using geoboards, and we worked over a few days to discover line segments, triangle area, and practice with the pythagorean theorem.  I asked students to use the geoboards to find solutions, and then to translate their ideas into pictures on dot paper by drawing all of their solutions.

I had some challenges handy for some students who were ready. Here are some extensions to the challenges posed by the Math Forum, which my students grappled with.

  • How many different squares can you find in a 5 pin x 5 pin board?
  • How many different triangles can you find in a 5 pin x 5 pin board?
  • How many total squares can you find in in a 1 x 1? …2 x 2? …3 x 3? How about nn?
  • How many total rectangles?  IMG_1322

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There was some beautiful and deep thinking going on, and this is a highly differentiable activity.  Students who are at the concrete level could physically count squares, while students who were ready to generalize could find the cubic (I guess that’s not too much of hint), which describes the number of squares in an area.  Teachers, feel free to drop me a note or tweet at me for my solutions.  In an effort to keep the problems from being Google-able, I haven’t included them here.

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Geoboards are excellent to physically see how  slope works as well.  For next year, I’ve ordered the 11 x 11 pin versions.  Stay tuned!  How have you used Geoboards in Algebra class?

 

Who Are You Calling Math Face? (How DO you turn that frown upside down?)

Awesome idea + powerful and GREAT tool + Sweet math wall = this year’s math faces.

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Thank you Fawn, thank you Desmos, and thanks students for your cool ideas.

My students created their own “math faces” through graphing with linear equations, quadratics, conics, and a few trig functions.  They used Desmos.com (an online graphing calculator) to explore different facial expressions, and were asked to articulate the equations behind each feature of their graphs.  They practiced transforming linear and parabolic equations, and learned about restricting domain and range.  Students’ manipulations made subtle differences in their facial features as they figured out how to move eyes up, down or around a line of symmetry, or how to change amplitude or period of a sine function to make a thicker mustache.  I’ve done this project a couple of times now, and I am surprised each time how motivating creating a face can be for some students.

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Desmos has created this beautiful classroom interface, which allows the teacher to see all of the student work at once, or filter students by things like who’s used an inequality or which students have restricted the domain or range.  This gives me an instant formative assessment where at a glance, I can easily target my advice or questions for students.    But even more importantly, students can see each others’ work, and share ideas in real time.  We spent about one hour as a full class on this, and then students were asked to complete a math face of their own over the next week.   I did this project early this past Spring with both Algebra 1 and Algebra 2 students.

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I could see some students struggling with the transformations, but caring enough to work through the struggle.

Screen Shot 2014-03-26 at 9.39.59 AMI am very grateful to have discovered this task.  In my experience as an art teacher, differentiation is often natural and practically effortless.  A student can attempt to draw a portrait whether it is their first try or if they have been practicing for years.  There are different conversations I can have with students depending on his or her experience, but every one of them can approach the task  – and right from the start is set up for success.  Math teachers don’t often have this built in differentiation.  So often, our lessons are targeted to a highly specific set of procedures, for which students must be in exactly the right place in the Algebra sequence.  The low entry, high ceiling aspect of drawing with graphs makes it something we can return to with students again and again.

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.

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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: recursivedrawing.com, 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.”

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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

The Beauty of the Jigsaw

Using a jigsaw is nothing new, but sometimes we hit upon the right format at the right moment.  I’ve had trouble drumming up interest in algebra word problems in the past (to be honest, I sometimes have trouble maintaining my own interest), but this jigsaw worked really beautifully.  My Algebra class includes a number of English language learners, and it seemed important to spend some time discreetly on dissecting, analyzing and solving word problems.

DSC05195We divided into four teams: mixture, work, systems of equations, and distance-rate-time problems.  Within each of these were three levels of difficulty.  Each team was responsible for learning their problems well enough to solve at least level one and level two problems, and for teaching at least level one problem-solving to another team.  For me, the work problems and the systems problems are the most straightforward.  The D-R-T’s are a little more complex and the mixtures seem to cause the most problems — something about moving between ratios and percent just confuses the heck out of them.  I was able to be strategic about who was assigned to each group.

We use a block schedule, so I typically have 60 or 90-minute classes.  We used the first 60-minute block, along with the first 15 minutes of the next block to become experts.  Students then moved back and forth between their “teacher” and “student” roles until they were able to solve level one problems independently from all four areas.  On the quiz, they were expected to solve level one’s from all four areas, and level two’s from at least three areas.  They could exceed the standard by solving level threes.

Screen Shot 2014-03-29 at 3.50.22 PMI did have some related activators, which helped to make connections and drum up initial interest.  For systems, I used some of Don Steward’s Cryptarithms, followed by this Ghost Whisperer Crystal Ball.  I was turned-on to this by Yummy Math, which has a nice lesson HERE.  The cryptarithms were surprisingly engaging and allowed us to practice place value in a way that felt like puzzles.  The Crystal Ball is just a blast.  Kids simply couldn’t believe it and were convinced that the machine must be listening to them, or that I was somehow involved in a conspiracy.  I highly recommend this as an activator or a stand-alone.  It demands a nice little piece of algebraic reasoning.  Dan Meyer’s Playing Catch Up is a 3-Act that goes well with these as well.

You can find these problems in any Algebra text.  I used some free ones from Kutasoftware, along with some I modified/wrote-up myself.  Here is the quiz I put together, along with some CCSS correlations in case they might be useful.

Word Problems Quiz

Word Problems Quiz With Answers

Word Problems Unit CCSS

Mountain Climber Problem: A Nice Follow up to Graphing Stories

I have a new job this year: teaching Algebra 1 and Algebra 2 at an Expeditionary Learning High School.  So far, it is an excellent place; strong and present leadership, dedicated colleagues, and a mission of rigor, relationships, and relevance.  I am very lucky.

Student Graphing StoryLike many Algebra teachers tuned-in to blogland, My students and I worked on some graphing stories to begin the year.  We started with Dan’s Site, and graphed some of the actions (stories) included in the video work there.  Students then created their own actions and short videos (we have 1-1 iPads here this year! Hurray!).  I hope to produce some of these videos with accompanying graphs, and publish them here.  It would be great to have them for next year’s kids to use as well.

After our video analysis, we turned to some work from The Language of Functions and Graphs, an excellent (if somewhat dated) text.  Though the problems are not visually seductive and compelling for students in the way that the graphing videos grab their attention, they are very provocative in terms of the math thinking they demand – and like it or not, we live in a world where students need to practice with traditional looking materials so they’re not caught off guard when taking standardized tests or in their next math class where the teacher may have a more traditional approach.  I found an electronic version of this text HERE, and modified some of the activities – for example, I removed the text but used the graphic from page 42 and asked students to write a “story” depicted by the action in this graph.

The Mountain Climber

As a follow up to this graphing work, I asked them to grapple with a problem involving a mountain climber.  My teaching partner reminded me of this problem, but I’m sure I’ve seen it before.  If you know where it came from, please let me know so I can give credit where it’s due.  Here is the problem:  A climber leaves base camp at 6AM one day, climbs up to the peak and arrives at 6PM.  The next day, she leaves the peak at 6 AM, and begins to climb down.  At a certain point on the trail, she notices that she was at exactly the same spot at exactly the same time on the previous day on the way up.  The question is: What are the chances of this happening?  I asked for initial guesses, and there was unanimous agreement that this would be highly unlikely.  Students talked about this for 5 minutes or so, and as no one was graphing, I suggested that creating a graph might be one way to examine this problem.  After another few minutes, I strongly suggested graphing.  It is so difficult for students to connect yesterday’s work with today’s sometimes – I need to work on this!

Mountain Climber GraphI love this problem because the answer becomes totally clear when you make a time vs. elevation graph – and the answer violates nearly everyone’s expectations and leads to a surprise!  Many students got stuck in their initial guess, and even when we went over together what the intersection of the two lines implied, they tried desperately to draw a version of the graph where the two lines didn’t intersect.  When they figured out that even skydiving down wouldn’t work, some resorted to teleportation.

As a nice reminder, the whole Junior class hiked up a mountain to end the week, and at least 6 students brought the problem up to me during our hike.  It was cool that this one stuck with them.

UPDATE: I added some of the materials I used below in case anyone would like them.

Graphing Stories 2-Create your own

Graphing Story Entrance Ticket

Graphing Story Exit Ticket