Category Archives: Statistics

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 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.


“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.


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.


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.

“First Past The Post” And The Dreaded Disease “Z”

This is a lesson I adapted from a model I first learned about at a math teaching conference that I attended at the Dana Hall School.  This conference was without a doubt, the best professional development workshop I’ve attended.  The faculty was amazing, the workshops were specifically geared by grade level, and the resources were practical and rich.  My teacher for the grade 6-8 workshops was Tom Harding, who is the current math department head at Shady Hill.  He was a wonderful and caring teacher.  I can’t say enough good things about this workshop.

This problem has a fun, if highly unbelievable back-story.  As the kids picked apart the story – and mine spent some time doing this – there was no pretense of the specific situation actually being useful, except perhaps for creative writing, but the ideas are highly transferable.  It is not a real world problem.  Perhaps even better, it falls under the Realistic Math Education (REM) umbrella described by HERE.

I used this as an integrated lesson about forms of government in addition to the math lesson.  I’ll set-up the problem and then explain what I did with it.

Dreaded Z Histogram Image


A group of 5 people are boarding a spaceship to Mars. Their health is one of the following:

  • Normal
  • Mono
  • Allergies
  • Common cold

And one has…

  • The Dreaded Disease Z

Students must identify who has The Dreaded Disease Z, as it is highly contagious – and fatal!  If the infected person boards the ship with everyone else, they are all doomed.


The only way to identify each person’s health is through blood testing; a bag for each person has representative chips for their blood levels (per chart included below).  I used color tiles in brown paper bags for this, and let students take 40 “blood samples” (a sample is taken by blindly taking one chip out of the bag, noting its color, and putting it back in the bag).  With 40 chances, students had to be strategic about which bags to sample from.  Should we take 5 samples from one bag or 1 sample from each bag?  Should we decide on 20 samples and then re-evaluate, or re-evaluate our plan every 5 samples?

Tile Graphic 1

Of course, students need some time to just play with the tiles and make patterns too.

You’ll need to set up at least one set of bags with tiles to match the histogram.  If you want separate groups to choose their own ways of sampling, you’ll need multiple set-ups.  40 samples were just about enough to make students pretty confident in their guesses, but not enough for them to be sure.  For me, the big math idea here is sample size, and given a situation, how many samples do you need.  I had students make individual predictions at 0, 10, 20, and 30 samples, and then come to a group consensus about who to kick off the spaceship.


I was working on forms of government as part of an integrated Social Studies unit, and saw a nice opportunity here.  When I did this at the conference with other teachers, there was a certain amount of strife about how we should decide to take our samples.  We each had ideas about how to go about finding the bag representing the Dreaded “Z,” and it was sometimes hard to let someone else decide.  And we were adults in a situation where we shouldn’t have cared about solving the problem, but instead should have been focused on examining the merits of different strategies, and why a student might choose one or another.  I imagined that for kids, this would be much harder than it had been for us – and I couldn’t resist leveraging these potential conflicts.

I did this in a whole group setting.  Each group of 4-5 students got one bag, and I asked them to give it a name.  In naming their bag, they grew attached to it, and were more invested in having their bag not be “infected.”  We later talked about how statisticians remain impartial (…or not).

I divided the 40 allowed samples into eight groups of five, and gave specific instructions for who got to decide which samples were taken.

  1. Samples 1-5: By Consensus:  Students all had to give a thumbs-up (I’m totally behind this method), thumbs-sideways (I can live with this method), or thumbs-down (I can’t live with this method).  They had to keep talking it out until everyone had at least thumbs-sideways, and then we took the samples.  As you might imagine, there were some strong ideas and this took a while.
  2. Samples 6-10: “First Past The Post” Popular Vote:  Anyone could suggest a method, and we voted on which method to use.  The method that received the most votes was the one we used.
  3. Samples 11-15: Alternative Vote: Students suggested methods, and voted for their top three choices in ranked order.  We tallied votes, and eliminated the lowest ranks until we had a vote of over 50%.
  4. Samples 16-20: Students voted for three classmates by secret ballot.  The three classmates got together and decided how to take the samples.
  5. Samples 21-25: I chose one student to make the decisions for the whole class, and didn’t allow any consultation.
  6. Samples 26-30: I kept the single decision maker, but added two advisors to help with the decision.  The final decision was still up to the one student with the power.
  7. Samples 31-35: I chose four students to make the decisions for everyone.
  8. Samples 36-40: I chose names out of a hat, and each student whose name was picked could choose one sample.

After each set of 5, I asked students to name the decision-making method, and reflect on how it felt in writing (The organizer that I used for this is included below).  At the end, I had each group choose who to kick off the spaceship, dumped the tiles for the big reveal, and debriefed the math.  We talked over statistical analysis, probability, and sample size.  This was an excellent big picture statistics discussion, and worked well as a hook for probability exercises to come.

After the math discussion, we debriefed the decision making process.  As you likely figured out, the decision-making methods roughly correspond to different forms of government: Variations of Democracy, Republic, Dictatorship, Socialism, Oligarchy, and Anarchy (OK, Anarchy analogy sort of falls apart).  The analogies only went so far, but the written reflections gave us a great opportunity to talk over the benefits and liabilities of different systems, and the different perspectives of those in charge vs. those not in charge.  “It was awesome! I got to decide for everyone.”  Or… “It was really hard.  There was a lot of pressure and I had to decide for everyone!”  I referred back to both the mathematical and the philosophical content of this discussion several times over the course of the news few weeks’ studies, and the kids are still talking about this and asking to do it again.



Organizer for Recording the Decision Making Process

Dreaded Disease Z prediction sheet

Dreaded Disease Z Graph

BONUS: Dreaded Disease X Graph for doing this a second time, or for adding a sixth crew member for more complexity