Category Archives: Lessons

Shifting Populations

I’ve been wanting to write and think about this for a while now, and with the end of the school year, I finally have a little time to reflect on and share some more work from this year’s Algebra classes.  Population shifts make for potentially compelling and authentic math modeling tasks.  Last year, I had great success  with grade 8 students, who compared linear and exponential models of the population growth of developing countries, and made predictions for the population of their chosen country in the year 2050.  This work was adapted from Kyle Moyer and Zack Miller(@zmill415)‘s Booming Populations project.  I wrote up some reflections on that project last year.  This year, I collaborated with some colleagues to adapt this work for my current grade 8 students, and to extend this work for my grade 10s.  The rubrics and guidelines are at the bottom if you can adapt them for your use. :)

Computer Work

My grade 8s studied the populations of two groups of snails, one in a tank with no predators, and one in a tank with some fish (…who apparently find snail eggs to be tasty).  My colleague from the science department, Heather Charalambous was kind enough to host this study, and to use science class time to support some of the conceptual thinking around how and why the snail populations changed (…and to count the snails!).  Kids used spreadsheets to create linear and exponential models, compared their two models, and made predictions about what would happen to the snail populations.  We checked their predictions against the actual number of snails at benchmark dates, and examined discrepancies between their predictions and the actual outcomes.  Materials are below.

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For grade 10, I collaborated with two excellent colleagues, Julie Jonsson, and Rachel Iannacone to create another permutation of this project, looking at the question: What events have the most significant impacts on the populations of cities?  In grade 10, students study U.S. History, so we tweaked this to fit into their coursework.  Students were asked to choose one American City and  to examine their city’s population from 1850-1940.  As with the 8th graders, they created linear and exponential models to help them to analyze and make “predictions,” about what they thought would happen between 1940 and 1960.  They then compared their predictions to historical data, and made arguments about the reasons for any differences.  Students who were ready created some polynomial models as well – although these models potentially fit the data better, they are complex, and challenging to defend the contextual choices.  The culmination of the work asked students to look at their city’s population changes through 2014, and to make a future prediction for what they think will happen to the population over the next 35 years.

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Improvements from last year…

  • Collaborating with teachers from science and social studies really helped to make this work deeper for students.  Especially with the grade 10 project, students were forced to look beyond the math to examine why populations shifted.
  • The math felt like it was in service of the compelling questions, and I think that students really felt like their math skills helped them to quantify and analyze an interesting problem.
  • The grade 10 project guidelines and rubric were carefully honed down for clarity and depth, and were designed as a precursor to and preparation for the I.B. Extended Essay, which most of our students will complete in grades 11-12.  This improvement was largely due to the collaborative efforts of my two partners, Julie and Rachel, who were just awesome to work with.  We were able to really hash out our different opinions and priorities, without anyone feeling threatened or marginalized, and to keep working until the project met all of our standards — this was one of the best professional collaborations I have experienced.

Better last year…

  • We did not have a culminating event for either of these projects this year.  Last year, we organized a “population summit,” where students presented their findings to a panel of “experts.”  Having to present their work publicly in this way really made students up their game.  This year, we did put up their work on the math wall, but somehow it wasn’t quite the same as public presentation.  Although presenting takes time, I really want to build this into the project if we can in the future.
  • Although the students did get some choice in their cities, there were a few who did not get cities that they were that interested in.  This made for less engagement, and I want to figure out how to really make them feel like they have some control next year (…even if it is just the illusion of choice).

The Materials…

SHIFTING POPULATIONS FINAL ASSIGNMENT + RUBRIC

SHIFTING Essential Questions

SNAIL Project Guidelines

Let me know if you use or adapt this work for your classes, or if you have ideas for how to improve or deepen this work, and please send me a note or find me on twitter if you’d like to see some student exemplars.  Happy to share.

Bullseye

This is a short reflection from a lesson focused on solidifying understanding of linear and absolute value equations with Grade 8 Algebra 1 Students.

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I created a game, based on the Green Globs software.  I’ve never actually used the original materials, but it looked like it would be a highly motivating activity, and being on a tight school budget, I decided that since I wouldn’t be able to make the purchase, next best thing would be to use Geogebra to make my own materials.  I called my game “Bullseye.”  I bet that the original version is slicker and more complex than mine, but it worked pretty well for us.

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Here is what a “game board” looks like.   The basic idea of the game is that you need to write equations which, when graphed, hit the green dots.  Your team scores points based on how many green “orbs” your graph hits.

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I grouped students in pairs and gave them whiteboards.  I handed out the rules, and projected the game board.  Students had 2 minutes to decide on their best two equations.  At the two minute mark, we called “markers down,” and students held their equations in the air.  We entered them into Geogebra and calculated their scores for the round.  I also stole the scoring from the Green Globs people: for each equation, 2 points for the first orb, 4 for the second, 8 for the third, etc. doubling for each additional orb.  Asking them to work in pairs was key.  They were forced to talk and argue about the best two equations to choose.

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Students who spotted the absolute value equation in this one ran the table!

The “Expert” games included “Shot Absorbers.”  If your graph hits a shot absorber, you don’t score any points.  When these were on the board, I also allowed inequalities, but you might want to allow piecewise functions or Domain or Range restrictions if that’s where you’re at.

My 8th grade group this year is by far the most competitive group with whom I’ve worked.  They are just dialed-in when they are competing against each other (There is a total ruckus in the room when we play Grudgeball!).  I have to admit that I am not much of a gamer.  I don’t really play games, and I’m not a very competitive person.  But we need to adapt to the group that we have.  These kids are really pretty good sports.  They desperately want to win, but they are also good losers.  Even though Nathan Kraft has decided that it is potentially destructive to his classroom culture, it just works for my kids.  And as long as I have them playing in pairs or groups, at least there’s collaboration in addition to competition.

Here are about 12 game boards along with my instructions.  These could be very easily modified to work for quadratics or whatever functions you’re studying.  Let me know how it goes if you try this out, or if you have ideas for improving the game.

Bullseye Game Files and Instructions

UPDATE (2/7/2016): Of course several better versions of this activity surfaced quickly.

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!)

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!

 

 

 

BOOM

“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.

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?

 

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