As a **high school** or **undergraduate** **Mathematics** teacher, you can use this set of computer-based tools to help you in teaching topics such as **Differentiation**, **Derivatives of Polynomials**, and **Tangent Line Problems** in **Introductory Calculus**.

This lesson plan allows students to perform polynomial differentiation and solve tangent line problems using climate data such as atmospheric CO_{2} concentrations data since 1950.

Thus, the use of this lesson plan allows you to integrate the teaching of a climate science topic with a core topic in Mathematics.

**The tools in this lesson plan will enable students to:**

- Calculate the derivatives of polynomials
- Interpret and compare the slope of a curve at different points
- Compare and analyze the rate of change of atmospheric CO2 levels by applying polynomial differentiation
- Predict future atmospheric CO2 levels based on current levels, and discuss the corresponding effect on climate

**This activity will help students to:**

- Observe the trend in increasing atmospheric CO2 levels
- Infer the approximate year when atmospheric CO2 levels could cause global temperatures to increase by 2°C (leading to serious climate change-related problems)
- Determine the desired trends in atmospheric CO2 levels that could help in avoiding or mitigating such climate change-related consequences

About

Step-by-Step User Guide

Questions

Additional Resources

Credits

Review

About

Step-by-Step User Guide

Here is a step-by-step guide to using this lesson plan in the classroom/laboratory. We have suggested these steps as a possible plan of action. You may customize the lesson plan according to your preferences and requirements.

- Introduce the topic of differentiation.
- Explain derivatives of polynomials with the help of the reading and exercises, “Derivatives of Polynomials”, from World Web Math, Massachusetts Institute of Technology

(video)

- Next, play this micro-lecture (approx. 10 min), “Differentiating polynomials”, to help students further understand polynomial differentiation through examples and exercises.
- The micro-lecture “Differentiating Polynomials”, from Khan Academy, is available.

- Then, help your students apply the learned concepts through a hands-on classroom/laboratory activity, “Mauna Loa Yearly Average CO2”, by Thomas J. Pfaff at Sustainability Math. This activity uses atmospheric CO
_{2}data from the Mauna Loa site for the period 1950 to 2017. - For the same f(x) (as above) and double the value of n (from above), calculate and compare the left and right Riemann sums.
- This activity will help students to
- observe the trend in increasing atmospheric CO
_{2}levels - infer the approximate year when atmospheric CO
_{2}levels could cause global temperatures to increase by 2°C (leading to serious climate change-related problems) - determine the desired trends in atmospheric CO
_{2}levels that could help in avoiding or mitigating such climate change-related consequences - Download the material in the project, “Mauna Loa Yearly Average CO2”, under Calculus I – Differentiation Related Projects.
- Conduct the exercises 1-6 to predict atmospheric CO
_{2}levels in the future. (Optional: exercises 7 and 8). - Discuss the possible impact of these trends on global temperature and climate.

Questions

**Use this lesson plan to help your students find answers to:**

*Plot a graph and find the polynomial equation to model the average yearly atmospheric CO*_{2}levels from 1950 to 2017 (using data records provided).*Compare and analyze the rate of change of atmospheric CO2 levels by applying Polynomial Differentiation.**Based on observed trends, what will the atmospheric CO*_{2}level be in 2100?

Additional Resources

1 | Visualization | An interactive visualization, “Interactive Graph showing Differentiation of a Polynomial Function” from Interactive Mathematics: This can be accessed here. |

Credits

1 | Reading, “Derivatives of Polynomials” | World Web Math, Massachusetts Institute of Technology |

2 | Micro-lecture (video), “Differentiating polynomials” | Khan Academy |

3 | Classroom/Laboratory Activity , “Mauna Loa Yearly Average CO2” | Thomas J. Pfaff,Sustainability Math |

3 | Additional Resource | Interactive Mathematics |

Review

About

Step-by-Step User Guide

Questions

Additional Resources

Credits

Review

About

Step-by-Step User Guide

Here is a step-by-step guide to using this lesson plan in the classroom/laboratory. We have suggested these steps as a possible plan of action. You may customize the lesson plan according to your preferences and requirements.

Reading and Associated Activity (30 – 60 min)

- Introduce the topic of Riemann sum and the types of Riemann sums by using the reading,“Calculating the area under a curve using Riemann sums” from D. Q. Nykamp at Math Insight.
- Conduct the activities using the applets to further explain how to calculate the area under a curve by using the Riemann sum and how this value can converge to the definite integral.

Classroom/Laboratory activity (~60 min)

Then, help your students apply the learned concepts through a hands-on classroom/laboratory activity, “U.S. and China CO2 Emissions”, by Thomas J. Pfaff at Sustainability Math. This activity uses CO2 emission data and population data for the U.S. and China for the period 1980 to 2015.

- analyze CO2 emissions for each country
- compare the CO2 emissions for the countries by using Riemann sums for the data from 1980 to 2015
- create a proposal for emission reduction by considering past and current CO2 emissions for the two countries
- Download the material in the project, “U.S. and China CO2 Emissions”, under Calculus I – Integration Related Projects.
- Students can perform the exercises described in the Word file by using the data in the Excel file.

Suggested questions/assignments for learning evaluation

Use the tools and the concepts learned so far to discuss and determine answers to the following questions:

- For a given function f(x) and n, calculate the left Riemann sum and right Riemann sum.
- For the same f(x) (as above) and double the value of n (from above), calculate and compare the left and right Riemann sums.
- Using Riemann sums, calculate and compare the total CO2 emissions (data records provided in the activity) for the U.S. and China from 1980 to 2015. What are the possible effects of these CO2 emissions on the Earth’s climate?

Questions

**Use this lesson plan to help your students find answers to:**

- For a given function f(x) and n, calculate the left Riemann sum and right Riemann sum.
- For the same f(x) (as above) and double the value of n (from above), calculate and compare the left and right Riemann sums.
- Using Riemann sums, calculate and compare the total CO2 emissions (data records provided in the activity) for the U.S. and China from 1980 to 2015. What are the possible effects of these CO2 emissions on the Earth’s climate?

Additional Resources

1 | Videos | Video tutorial, “Riemann sums”, from Khan Academy: This can be accessed here. |

Credits

1 | Reading and Associated Activity, “Calculating the area under a curve using Riemann sums” | D. Q. Nykamp; from Math Insight |

2 | Classroom/Laboratory Activity, “U.S. and China CO2 Emissions | Thomas J. Pfaff, Sustainability Math |

3 | Additional Resources | Khan Academy |

Review

TROP ICSU is a project of the International Union of Biological Sciences and Centre for Sustainability, Environment and Climate Change, FLAME University.