|
|
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.
|
|
|
Course Description Calculus I (MATH 153) or Calculus & Analytic Geometry I (MATH 156) is the first course in a sequence of calculus courses required for students in various majors including Computer Science, Engineering, Mathematics, Science, and Technology. The main objective of the course is building the essential skills, mastery, and understanding of the applications of several topics including analytic geometry of plane functions, limits, continuity, derivatives of functions and its applications; exponential, logarithmic, trigonometric and inverse functions; indefinite and definite integrals; and the fundamental theorem of calculus. The prerequisite of this course is the completion of MATH 121, a trigonometric functions course, or an appropriate placement test score. The class size has increased over the last few semesters and averages about 40-45 students per class during the fall semester for MATH 153 and about 30-35 students for MATH 156. During the spring semester when Calculus I is off sequence, the class size decreases. The students are split depending on their major into the two versions of the course. MATH 156 contains mainly students majoring in Computer Science and Mathematics, and MATH 153 contains all other majors. The two versions are almost identical with the version for the Computer Science and Mathematics majors requiring more proofs and theory. All the students in either course have laptops and graphing calculators, although these technologies are not crucial to this lesson. The classroom setting varies among the sections of the course from long tables to individual desks, making it difficult to generalize student interactions in the lesson plan. The lesson follows content on general derivative rules and implicit differentiation. It is the first major application section using derivatives and usually takes 2-3 days to complete.
|
|
|
|
