Integrals Lesson: Analyzing NYC Climate Goals
Lesson Plan
2 hours 55 minutesGrades11, 12, APMathematics
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Written by Sol Giesso
About this lesson
Students analyze a logistic model of New York City greenhouse gas emissions using definite integrals to quantify accumulation over time. Students apply integrals to find the area between curves, compare NYC’s trajectory to three policy benchmarks, and write a report evaluating whether the city is doing enough to meet its climate goals.
Inquire: Students examine a graph from the NYC Comptroller’s website to generate questions about whether NYC is doing enough to meet its climate goals.
Investigate: Students evaluate a regression model of NYC’s greenhouse gas emissions, set up and calculate definite integrals of the model to find cumulative totals, and interpret their findings.
Inspire: Students use areas between curves to assess NYC’s climate progress against multiple benchmarks and interpret what this means for meeting its climate goals.
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Suggestions
- This lesson is designed to be completed over two days: Day 1, Inquire and Investigate; Day 2, Inspire, but could also be extended over several days.
- This lesson can be taught as an application of integrals that helps understand their meaning in context. Because it requires skills such as finding areas between integrals and u-substitution, it may be best used as a lesson near the end of an integration unit, after students have had substantial practice with basic antiderivatives and definite integrals.
- This lesson can be used to reinforce the skills of setting up and evaluating area-between-curves problems with correct bounds.
- In addition, this lesson highlights the importance of government accountability and public availability of data, putting students in the role of the NYC Comptroller.
- To reinforce student action and agency in the face of climate change, consider sending the final reports to the NYC Comptroller's Office.
- This lesson is a companion lesson to Derivatives Lesson: Evaluating NYC Climate Trends. Together, the two lessons help students see how derivatives and integrals provide complementary perspectives to understand climate change and climate action.
Prerequisites
- Students should know how to calculate indefinite and definite integrals using u-substitution.
- Students should know how to find the area between two curves using integration.
- Teachers should skim over the NYC’s Roadmap to 80x50 report to become more familiar with the policy.
- Teachers should ensure that each student has access to a calculator capable of evaluating expressions involving the constant e and natural logarithms.
Differentiation
- To support students who are still struggling with integration concepts, solve the indefinite integral as a class. In addition, provide a worked example of the area-between-two-curves analysis.
- To support students in solving the indefinite integral, teachers can consider scaffolding the first step (algebraic rewrite). Depending on students’ skill level, teachers should decide whether to let students attempt integration first, allowing them to discover they need to rewrite the function before supporting them through this algebraic step, or to walk the class through the algebraic rewrite upfront as a scaffold.
- Teachers can also add sentence starters to the written response exercises in the Student Document.
- For extra support, teachers can create groups of four instead of three for the final assessment.
- To reduce the amount of writing required for the Inspire section, teachers can ask students to instead develop a public-facing infographic from the NYC Comptroller's Office.
- To push more advanced students, teachers can extend concepts of derivatives to the logistic growth model. Students can compute and interpret derivatives as the rate of emissions reductions and compare peak reduction years to policy milestones.
- In addition, teachers can ask advanced students to compare the historical trajectory with the baseline scenario and the carbon budget benchmark using the area between the curves.
- To extend the lesson, ask students to explore “what if” scenarios in Desmos. For example, students could shift the logistic curve horizontally or change its steepness to model delayed action. Then, students can recompute the three benchmark comparisons and reason about how these changes affect cumulative emissions.
From the scientist review
This lesson tasks students with using their calculus skills to measure NYC’s progress toward achieving climate goals. Up-to-date GHG emissions data sourced from the NYC Climate Dashboard is fitted to a logistic regression to facilitate students’ analysis. This lesson’s greatest strength is how it illustrates, through students’ application of calculus, the urgency of action to mitigate climate change, focusing not just on emissions reductions but also on carbon budgets. This lesson is recommended for teaching.
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About the teacher writer
Sol Giesso
Content Writer
Sol Giesso, Ed.M., is SubjectToClimate's Content Writer and a middle school math teacher in Boston. She holds an M.S. in Community Climate Adaptation from Worcester Polytechnic Institute and an Ed.M. in Teaching and Teacher Leadership from the Harvard Graduate School of Education, and previously worked at the Smithsonian Environmental Research Center leading hands-on outdoor education field trips for K–12 students. Fun fact: she loves to draw, scuba dive, and turn any recipe into a vegetarian one.
