**Title:** The “Perfect” Related Rates Lesson: A lesson study in calculus**Discipline(s) or Field(s):** Mathematics**Authors:** Joy Becker, Christopher Bendel, Petre Ghenciu, Laura Schmidt, Radi Teleb, University of Wisconsin – Stout**Submission Date: **February 28, 2007

**Executive Summary**

The lesson topic is related rates in Calculus I or Calculus & Analytic Geometry I. Related rates problems tend to be difficult for students since they are generally word problems that require setting up equations before solving. This topic is important as one common example of an application of derivatives.

*Learning Goals: *There are two immediate goals for this lesson: 1) Students will understand that related rates problems are applications of implicit differentiation and 2) Students will be able to translate, compile, model, and solve a related rates problem and interpret the meaning of the answer. A longer-term goal is that students’ problem-solving and critical thinking skills will be improved.

*Lesson Design:* The lesson is designed to span two class days. On the first day, students start by working through an introductory worksheet, which extends what they have previously learned to introduce the concept of related rates. Since word problems are often a stumbling block for students, the lesson includes an overview of problem-solving strategies, somewhat specific to related rates, although they can be generalized. A warm-up worksheet reviews necessary material and gives students a chance to set up equations, an essential part of the problem-solving process. On the second day of the lesson, the instructor works through two examples with the class to model the problem-solving process, and students are given a chance to solve problems on their own or in small groups. The examples and worksheet problems were chosen to show students a variety of different types of related rates problems, starting with more straightforward problems and ending with more difficult problems.

*Major Findings about Student Learning: * In terms of our specific lesson goals, by looking at the data we collected, the first two were achieved by most students: 1) Students will understand that related rates problems are applications of implicit differentiation and 2) Students will be able to translate, compile, model, and solve a related rates problem and interpret the meaning of the answer. Since the third goal, “Students’ problem-solving and critical thinking skills will be improved,” is more general, there will need to be a series of lesson studies in order for it to be assessed properly. Is this the “perfect” lesson? The answer is probably no. However, the planned activities did visibly increase student engagement and responsiveness. The lesson developed will help instructors to assemble an excellent lesson, depending on the classroom settings and other institutional factors.

*Below are links related to the lesson plan and the materials used to teach it.*

Basketball Example

Warm-Up to Related Rates Worksheet

In-Class Related Rates Examples

Rales Rates Worksheet

*Below are links related to the study of the lesson.*

Observation Forrm

Student Survey

Related Rates Homework Spring 2006

Related Rates Homework Fall 2006

Student Data