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Mathematical and Computational Methods for Planning a Sustainable Future (PS-Future)

About the PS-Future project and its modules
The PS-Future project is a collaboration led by Rutgers University involving the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), the School of Environmental and Biological Sciences, and Heldrich Center for Workforce Development at Rutgers University; the Consortium for Mathematics and its Applications (COMAP); Colorado State University; the Groton School; Hobart and William Smith Colleges; and a number of authors and educators from around the country.

The modules are intended to provide 4-6 days of classroom activities on a variety of topics that apply computational and mathematical methods in sustainability. Each module links to Common Core State Standards (CCSS) in Mathematics and targets content for a particular mathematics course. Each module also addresses disciplinary core concepts required by the Next Generation Science Standards (NGSS), especially targeting concepts taught in Environmental Science and Biology courses. The modules are also particularly suited to address Practice Standards for both mathematics and science.

Each module contains examples of jobs related to the module topic, together with a discussion of the skills and training required, as well as information on the salary and future demand for such jobs. In many cases, specific job titles are mentioned within the modules.

Click here to request access to the teacher material.


Student Material

Teacher Material

Passive Solar
A Module in Planning for Sustainability

About this module
This module is written to be taught in one week of class time and is separated into five
lessons of approximately 45 minutes each. Lessons can be combined to accommodate a
block schedule or lengthened to provide class time for homework assignments. Teacher’s
notes are provided to increase the rigor necessary to suitably challenge students of higher
grade and experience levels. The student activities handouts are in the appendix.
In what types of classes could this module be used?
This module is suitable for students in Environmental Science or Pre-Calculus who have had
right-triangle trigonometry. Teachers may wish to reduce or eliminate parts of Lesson 2,
depending on students’ trigonometry experience.

Prerequisite skills
• Students should understand basic earth science
• Students should understand basic right-triangle trigonometry, that is, they should
be able to apply SOA CAH TOA to basic right triangles.

Required materials
• Big copy paper box (or similar)—1 per group (of 4)
• 12" x 24" sheet of cardboard (approximate)
• Insulated cups
• Materials for thermal mass: sand, water, plaster of Paris, wood shavings or other.
• Graph paper
• Flashlight or height-adjustable desk lamp
• Heat lamp or desk lamp with reflective bulb
• Copies of the relevant student worksheets, 1 per student
• Method to watch online animations and videos as a class, such as a computer and

Summary of lessons
Lesson 1: Students learn about passive solar and passive cooling.
Lesson 2: Students review right-triangle trigonometry, then apply SOA CAH TOA to basic
passive solar problems.
Lesson 3: Students work to understand thermal mass.
Lesson 4: Students learn about declination and how to orient their house due south. They
also learn how the angle of the sun varies throughout the year.
Lesson 5: Students discover how overhangs are used to block the summer sun.


Student Material

Teacher Material

Where Is the Water Going?
A Module in Planning for Sustainability

About this module
Historically water has been taken for granted, especially in humid areas like the southeastern
United States. This module uses math concepts of estimation and dimensional analysis to
understand the discharge rate of water and total volume of runoff. Using geometry concepts of
volume and area, students will explore what it really means to get one inch of rain and how it
differs between various terrains. Students will calculate runoff and determine where the water
goes and the effects it has on the environment.

Summary of module objectives
Unit 1: The student will identify usable and unusable water and calculate volume of various
Unit 2: The student will use dimensional analysis to convert rates.
Unit 3: The student will calculate discharge rates.
Unit 4: The student will describe the differences between runoff and infiltration.
Unit 5: The student will estimate runoff of larger communities.

What are the “Big Ideas”?
Students will explore the reasonableness of their answers and begin to connect their assumptions
with the accuracy of their conclusion.
Students will understand runoff: how to calculate runoff, determine the effects of runoff, and
investigate ways to mitigate runoff.

What types of classes could this module be used in?
This module would be appropriate for an algebra or a geometry class. It would fit into a science
curriculum dealing with ecology, sustainability, or the water cycle.

Prerequisite Skills
Students should be able to use formulas to calculate volume and should have knowledge of the
water cycle.

One tape measure per 2–3 students
Class set of scissors
Cardstock paper for dominoes is best but not required
Internet (Google Maps)
The activity in Topic 4 recommends either a video or a demonstration. If you choose a
demonstration, several materials are needed:
• 4 clear 2-liter soda bottles
• container to measure 8 oz of water
• 2 four- to six-inch plastic tubing about ½ inch diameter (found at hardware store)
• 2 clear containers to catch the runoff
• soil (enough to fill the 2-liter bottles)
• ground cover (grass, dead plants, weeds, etc.)
• Samples of soil types (sandy loam) recommended but not required


Student Material

Teacher Material

Weather Generators
A Module in Planning for Sustainability

About this module
This module is written to be taught in roughly one week to one and one-half weeks of class time, and it is separated into six lessons of approximately 55 minutes each. Lessons can be combined to accommodate a block schedule or lengthened to provide class time for homework assignments. Lessons are intended to be introductory enough to be accessible to 9th-grade students with little to no dataanalysis experience, but they have also shown useful for 11th and 12 th grade college-bound students.

In what types of classes could this module be used?
This module is suitable for students in Earth Science, Environmental Science, and Biology, and it fits well as an application problem after learning the water cycle. The module is also suitable in statistics classes where the focus is on data analysis when summary statistics such as the mean and standard deviation are not the most appropriate summary statistics for the data set. Teachers may wish to reduce or
eliminate either Lesson 1 (introduction to the water cycle) or Lesson 2 (introduction to basic statistics), depending on students’ prior scientific or statistical experience.

Prerequisite skills
• Though it will be reviewed in the first lesson, students should understand the global water cycle
and how it is powered.
• Students should have a basic understanding of the uses of a spreadsheet program (such as Microsoft Excel or Google Spreadsheet) for data analysis, but skills using the spreadsheet are not necessary.

Required Materials
• Standard six-sided dice (1 per student)
• Computers with a Microsoft Excel spreadsheet for Lessons 3, 4, and 5. Other spreadsheet programs may work if they are macros-enabled.
• The weather generator spreadsheet file, available for download in MS Excel format.
• A method to watch online animations and videos as a class, such as a computer and projector.

Optional Materials
• Copies for students are available in two formats:
o An entire Student Workbook that includes places for students to capture their notes and thinking during class, as well as the homework worksheets. Students may use this as a reference as they work through the module.
o Individual handouts for each lesson. These worksheets are shorter and do not provide as much information or note-taking space as the Student Workbook. These may be provided as tools to offer structure during class and homework and may not be necessary for all classrooms.
Teachers use discretion in determining which format, if any, is needed for their group of students.

Summary of Lessons
Lesson 1: Students review the water cycle. Then students work toward understanding the concept of worldwide sustainability and how that relates to the water cycle. Students learn about our current unsustainable patterns of water use.
Lesson 2: Students learn simple statistics, including basic vs. conditional probability, as well as practicing when to apply different mathematical routines based on the situation. Students then relate those concepts to weather.
Lesson 3: Students work to understand statistical persistence, create a flowchart for weather-based statistical persistence, and then use dice to simulate data.
Lesson 4: Students use a spreadsheet program to produce precipitation data. Students relate the computations the spreadsheet program performs to the dice lesson the day before. Students work to understand that typical summary statistics are inappropriate for this data set, and they determine appropriate and relevant methods to summarize the data.
Lesson 5: Students work to understand the difference between natural variability and variability that is the result of hidden, and possibly problematic, variables. Students use the weather generator again to examine and summarize variability that is natural.
Lesson 6: Students use the weather generator again to predict what the future holds. Using output from NCAR models to seed their own weather generator models, students examine data and work to determine how future weather patterns might be different from current patterns. Finally, the module ends with students acknowledging the merits and limitations of building models.


Student Material

Teacher Material

Going Batty
Modeling White Nose Syndrome in Bats

Introduction to the Teacher Edition of “Going Batty”:

Disease looms large in our world today and students are faced with challenging questions about
pandemics, vaccinations, and the reemergence of diseases long thought to be under control. In
this module, students will gain an understanding of the mathematics that underlie the science and
will be able to approach some of these questions in a quantitatively literate way. If vaccines
work, why should parents who vaccinate their children be concerned about those who do not?
What leads to a catastrophic disease outbreak, and what do government organizations do to bring these to an end?

Learning Objectives:
This module will provide students with an understanding of the susceptible-infected-resistant
model of epidemics. To do this, students will take part in an active simulation of an epidemic,
work with an existing agent-based model, and build their own model using a spreadsheet. They
will learn the spreadsheet skills necessary to examine how changes to the inputs to a system will
affect the outcome of a model. Students will learn about the biology of white nose syndrome, a
fungal infection that threatens North American bats.


Student Material

Teacher Material

Invasive Species
A Module in Planning for Sustainability

About This Module:
This module focuses on the role of non-native species in ecosystems. Students explore local
examples of such species, including how and why these species become established in ecosystems.
Then, after working through the concept of population growth and competitive exclusion, students work with a hands-on randomized simulation model to explore how the number of individuals of a nonnative (or an exotic) species population might influence the ability of that species to take hold and become invasive, doing damage to the ecosystem. A second deterministic model raises questions about how the spatial pattern of an invasion can impact the effectiveness of efforts to contain or slow invasive spread. The module concludes with a focus on sustainability: how construction workers, ecologists, farmers, groundskeepers, marine biologists, landscapers, policy makers, urban planners, wildlife biologists, and zoologists study, plan for, respond to, and manage invasive species?

This module is written to be taught in roughly one week to one and one-half weeks of class time, and is separated into 4 lessons of approximately 90 minutes each, though each lesson has a logical break midway through for teachers working with shorter class periods. The fifth lesson is for a conclusion and optional assessment, which could take as little as 20 minutes or as much as a few days. Lessons are intended to be introductory enough to be accessible to 9th-grade students.

Summary of Module Objectives:
Lesson 1: The student will be able to define and describe invasive species and identify a
local example of invasive species. The student will also be able to describe the potential
impacts of an invasive species.
Lesson 2: The student will be able to describe the different phases of logistic population
growth and understand potential causes of these different phases. The student will be aware
that model findings depend on simplification and assumptions.
Lesson 3: The student will develop an understanding of how an increased number of
individuals introduced will correlate with an increased likelihood of success in mate finding
(and thus an increased likelihood of establishment by the invader). Similarly, the student will
understand why some non-native species introductions do not result in a population taking
Lesson 4: The student will consider a simulation based on recursive rules to understand that
once a population is introduced, how that population spreads may be modeled in a variety of
ways that will have different implications for what interventions are useful. The student will
consider how spatial changes in initial conditions affect simulation results. After practicing
the implementation of recursive rules, the student will explore how effectiveness of invasive
removal depends on the spatial pattern of the initial invasion.
Lesson 5: Conclusions and optional project-based assessment. The student will return to the
thinking maps created in Lesson 1 of the module and re-evaluate his or her initial response.
The student will explore a broad decision-making process to understand that each nonnative
species should be considered individually.

What are the "Big Ideas"
Students will learn that ecosystems exist as a delicate balance of abiotic and biotic factors. When one of these factors is disrupted, the ripple effect of this disruption can be significant. We use mathematical models to simplify complex systems with the hope of using the predictions and
qualitative observations that arise from those models to inform decision makers about how to address new ecological problems.

In what types of classes could this module be used?
This module is suitable for students in Biology and Environmental Science, and it fits well as an
application problem after learning about food webs.

Prerequisite skills
Students should have a basic understanding of ecology and ecosystems. The following terms will be
used in the module: ecosystem, food web, species, niche, competition, resources, and carrying
capacity. In addition, students should have basic graphing skills.

Method to watch online animations as a class, such as a computer and projector.
Two different color tokens, two different coins, or two different color sticky notes to use in
Lesson 3.
Student access to a computer, tablet, or smartphone with a web browser.
Student handouts are available in the format of a Lab Notebook. It is designed so that
teachers can produce one copy for each student that will last the duration of the module, but
teachers should use discretion in determining if any pages of the Lab Notebook aren't necessary for their particular group of students.


Student Material

Teacher Material

Water Games
Applying Game Theory to the Sustainable Use of Water (And Other Natural Resources)

Water Games uses concepts drawn from mathematical game theory to model competition for water and other scarce resources and to examine potential strategies available to parties in conflict. Students use game theory to predict outcomes based on play of a game and to change outcomes by changing associated payoffs.


Student Material

Teacher Material

NIMBY (Not in My Backyard)
A Module in Planning for Sustainability

Not in My Backyard explores how to consider social and environmental justice when making decisions like where to put a toxic waste dump. Often such facilities are located in economically depressed areas, based on priorities that further disadvantage those who are already disadvantaged.


Student Material

Teacher Material

Life on the Edge

Sustainability is forward-looking,and natural resources are finite. Planning for sustainability therefore involves making decisions about how we use limited resources in light of the continuing future needs for those resources. Planning for sustainability seeks to identify practices that can continue indefinitely without critically damaging natural resources, people, or economies, especially those that are at risk.

While it is tempting (and simpler) to base decisions purely on their immediate economic impacts, doing so can easily overlook the long-term environmental (and social) consequences that can undermine sustainability. In planning for sustainability, it is important to consider that human economic and social systems exist within and are dependent on the environment. Decision makers need to identify how their choices impact both human and environmental systems, as well as, whether such choices could persist indefinitely into the future.


Student Material

Teacher Material

Ecosystem Services
The Multiple Values of a Forest

Ecosystem Services immerses students in making decisions on forest resource management. The module introduces students to sampling methods to assess forest resources and potential timber values, while also considering the value of ecosystem services like erosion prevention, carbon sequestration, and habitat provision. It illustrates that sustainable management of a natural ecosystem requires a full accounting of all effects of an action--both direct and indirect--over time.


Student Material

Teacher Material

Urban Sustainability

Urban Sustainability introduces basic concepts in the use of geographic information systems (GIS) to explore patterns of greenhouse gas (GHG) emissions both across and within US cities, as well as changes over time. Students also map heat in both urban and rural areas and consider links with land use and GHG emissions.


This unit provides an overview of the urban sustainability concept, importance of urban sustainability, indicators of sustainability and measures for achieving urban sustainability. This unit also serves as an opening for the other units and how each unit contributes to establishing urban sustainability. After an initial class discussion and discussion of a homework prompt, the module provides some background reference summaries on greenhouse gasses, the carbon cycle, and urban heat islands; these can be used as needed to extend the classroom discussions; the core of the module follows these summaries and consists of activities 1.1, 1.2, and 1.3; a suggested module assessment follows the activities.



This material is based in part upon work supported by the National Science Foundation under Grant Number DRL-1503414 DRL-1220022(past). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.