Lesson Plan: Comparing Rates of Change

Objective: Students will be able to compare the rates of change at two points using average rates of change near the points.

Essential Knowledge: 1.2.A.1: The average rate of change of a function over an interval of the function’s domain is the constant rate of change that yields the same change in the output values as the function yielded on that interval of the function’s domain. It is the ratio of the change in the output values to the change in input values over that interval. 1.2.A.2: The rate of change of a function at a point quantifies the rate at which output values would change were the input values to change at that point. The rate of change at a point can be approximated by the average rates of change of the function over small intervals containing the point, if such values exist. 1.2.A.3: The rates of change at two points can be compared using average rate of change approximations over sufficiently small intervals containing each point, if such values exist.

Materials: - Graphing calculators or computers with graphing software - Whiteboard or blackboard - Markers or chalk - Worksheets with practice problems

Procedure:

1. Introduction (5 minutes): - Begin the lesson by asking students if they have ever noticed how things change at different rates. For example, the speed of a car may change as it accelerates or decelerates. - Explain that in mathematics, we can also measure how things change using rates of change. Today, we will be focusing on comparing rates of change at different points on a graph.

2. Average Rate of Change (10 minutes): - Review the concept of average rate of change using the essential knowledge 1.2.A.1. - Provide an example problem and guide students through calculating the average rate of change over a given interval. - Discuss the meaning of the average rate of change in the context of the problem.

3. Rate of Change at a Point (10 minutes): - Introduce the concept of rate of change at a point using the essential knowledge 1.2.A.2. - Provide an example problem and guide students through approximating the rate of change at a specific point using average rates of change over small intervals. - Discuss the meaning of the rate of change at a point in the context of the problem.

4. Comparing Rates of Change (15 minutes): - Explain that we can compare the rates of change at two points using average rate of change approximations over small intervals, as stated in essential knowledge 1.2.A.3. - Provide an example problem with two points and guide students through comparing the rates of change using average rates of change approximations. - Discuss the significance of comparing rates of change and how it can help us understand the behavior of a function.

5. Practice Problems (15 minutes): - Distribute worksheets with practice problems for students to work on individually or in pairs. - Circulate the classroom to provide assistance and answer any questions.

6. Review and Conclusion (5 minutes): - Review the main concepts covered in the lesson: average rate of change, rate of change at a point, and comparing rates of change. - Ask students to share any insights or observations they made during the practice problems. - Summarize the importance of understanding rates of change in analyzing functions.

7. Extension Activity (optional): - For students who finish early or want an additional challenge, provide more complex problems involving rates of change and ask them to compare the rates at different points.

Note: The duration of each section can be adjusted based on the pace and needs of the students.