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 **integration**, **definite integral**, **area under a curve**, and **Riemann sum** in **Introductory Calculus**.

This lesson plan allows students to understand Riemann sum, calculate the area under a curve using Riemann sum, and explore how this value converges to a definite integral. The activity helps students to apply the Riemann sums method for analysis and comparison of data on CO_{2} emission, which is considered to be a significant contributor to climate change.

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 approximate area under a curve by using the Riemann sums method
- compare the results obtained for left and right Riemann sums for the same curve
- explain how the estimate of the area under a curve (using Riemann sum) converges to the definite integral
- apply the Riemann sum method to analyze and compare CO
_{2}emissions data for the U.S. and China

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.

- 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.
- This activity will help students to analyze CO
_{2}emissions for each country, compare the CO_{2}emissions for the countries by using Riemann sums for the data from 1980 to 2015 and create a proposal for emission reduction by considering past and current CO_{2}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.

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 | Video | 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

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.

through a reading and

exercises

Reading and Associated Activity(30 – 60 min)

- 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)

Micro-lecture (video)(~10 min)

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.

classroom/laboratory

activity

Use the tools and the concepts learned so far to discuss and determine answers to the following questions: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 CO2 data from the Mauna Loa site for the period 1950 to 2017.

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.
- Download the material in the project, “Mauna Loa Yearly Average CO2”, under Calculus I –Differentiation Related Projects.
- Students can plot the time-series graph of atmospheric CO2 by using the data in the Excel file or can directly use the graph provided in the Word file.
- Conduct the exercises 1-6 to predict atmospheric CO2 levels in the future. (Optional: exercises 7and 8).
- Discuss the possible impact of these trends on global temperature and climate.

Suggested questions/assignments for learning evaluation

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

- Plot a graph and find the polynomial equation to model the average yearly atmospheric CO2 levels

from 1950 to 2017 (using data records provided). - Based on observed trends, what will the atmospheric CO2 level be in 2100?

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 CO2 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 CO2 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 Resources | Interactive Mathematics |

Review

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