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. :)

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.

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.

**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

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.