Course Syllabus: AP Calculus I (AB)

Text: Calculus Of A Single Variable, seventh edition; Larson, Hostetler & Edwards; Houghton Mifflin Company, 2002.

Technology: A graphing calculator is required for this course and for the AP exam. TI-84 calculators will be available for students to borrow for the entire school year. I will regularly use a TI-84 calculator &/or emulator to demonstrate and explore functions and applications of calculus. Students are expected to follow along and then to do further exploration on their own. Students may use the calculators on some but not all assessments. Motion Detectors and other CBL (Calculator-Based Laboratory) instruments will sometimes be used in conjunction with the graphing calculators, in order to provide students with hands-on learning activities. These activities will focus both on using technology and on relating mathematical concepts to real life situations.

Course Plan: The following is an outline of the topics that will be covered, in their expected order, during the course of the school year. Time limits are not set, because many of the activities planned may lead to extra time spent on the discussion and exploration of topics. Lessons are designed to first introduce students to the material, usually with a discovery-based activity or proof, and then to lead into exploration of topics graphically, analytically, numerically and verbally. As time goes on, students should feel confident about providing their own verbal explanations for problems, both in writing and orally.

Unit/Chapter Themes/Objectives/Skills Assessments

Chapter 1 Limits and their Properties, Estimating Problem Sets, “Is There No Limits from Graphs and Tables, Limit to These Labs?” Asymptotes and Activity, Quizzes and a Local Linearity Chapter Test

Chapter 2 Differentiation Graphical, Analytical **Distance vs. Time and Verbal Approaches, Including Activities, Problem Sets, Basic Rules of Differentiation (Chain “What Goes Down, Must Rule, Product Rule, Quotient Rule), Come Up . . .” Lab, Problems Continuity & Differentiability, The Involving Rates and Real- Derivative at a Point, Rates of Change world Applications, Quizzes and Other Interpretations of the and a Chapter Test Derivative, the Extreme Value (This unit takes about 4 Theorem and an Introduction to weeks) Implicit Differentiation Chapter 3 Applications of Differentiation; Problem Sets, Extrema, Rolle’s Theorem, the Mean Maximizing/Minimizing Areas Value Theorem, the First and Second Project, Curve Sketching Derivative Tests, Relative and Packet, “Mystery Curve” Absolute Extrema, Maximizing or Activity, and a Chapter Test Minimizing Functions (Critical Values), Concavity, Inflection Points and Related Rates. Chapter 4 Accumulation Functions, Riemann Problem Sets, Group Work for Sums, The Trapezoidal Method, Area, Riemann Sums and the Integration, The Average Value of a Trapezoidal Method, Drawing Function, The Fundamental Theorem Slope Fields Activity, Quizzes of Calculus, u-Substitution and Slope and a Chapter Test Fields (This unit also takes about 4 weeks.) Chapter 5 Logarithmic, Exponential and Inverse Problem Sets, Trigonometric Functions, Including Inverse Functions Review Worksheet, Trigonometric Functions and “A Watched Cup Never Exponential Growth and Decay Models Cools” Lab, M&M Growth/Decay Activity, Quizzes and a Chapter Test Chapter 6.1 – Applications of Integration; Area, Arc Problem Sets, Activities to 6.5 Length and Surfaces of Revolution, Help Visualize Solids, Volume (focusing on the Disk and Mapping Islands activity, Washer methods as well as Solids with “H2O in the S-K-Y” Activity, Regular Cross-sections), and Work Volume of an Object Projects, (Center of Mass will be covered after Quizzes and a Unit Test the AP Exam.) The Shell Method is studied, but not in great depth.

Chapter 7.1- Integration Techniques, Focusing on Problem Sets, and a Quiz 7.3 Integration By Parts

* Most labs and activities are adapted from the book, A Watched Cup Never Cools, by Ellen Kamischke, 1999.

Teaching Strategies:

On the first day of school, I hand out TI-84 calculators and assign a library of curves for memorization. The remainder of the first day as well as the second class is spent reviewing important algebra and trigonometry concepts. This is the only time that will be dedicated to reviewing past concepts. This will be a fast-paced course, and students are urged to utilize the after school tutoring sessions in order to keep up with all materials.

Throughout the course, and especially during labs and activities, student exploration and discovery is encouraged and fostered. Students work together on a regular basis. Using the technology available to us, the “rule of four” is easily followed; topics are presented graphically, numerically, algebraically, and verbally. This provides students with a real understanding of the material and its many applications. Students often fit data that they collect to various functions using their graphing calculators. Topics such as local linearity, maximums, minimums, concavity, area under a curve and slope fields are also investigated graphically. Computer programs as well as wooden models, clay models and food items are used to help students visualize three-dimensional objects and to better understand arc length, surface area and volume. Fold-out party decorations are used to relate the area found with Riemann Sums to volume with the disk method. This allows students to actually see the object that is formed from revolution about an axis.

I begin to integrate AP multiple-choice questions into quizzes and review sessions as soon as possible. By October, students are tackling some free-response questions, including learning how to provide justification for steps and answers. Students may spend most of a class period working alone or together in small groups on previous free- response questions, and the rest of the period is dedicated to discussion and explanation as a class. Many of the labs and activities require student-generated reports, in which they are asked AP-type questions in order to foster a higher level of thinking. In the middle of April (as long as the class is on track with all material), I begin to concentrate on AP questions, both multiple choice and free-response. I also try to fit in a couple extra activities to reinforce limits, local linearity and related rates.

There will be a comprehensive final exam given at the end of the course.

Daily Expectations For Students:

 Have respect for the teacher and for your fellow students.  Be on time and be prepared for class.  Pay attention during class and participate when appropriate.  Complete homework/assignments by their due date.  Ask questions when you do not understand.  Be curious.  Discover something new.

The main objectives of this course are 1) to prepare students for the Calculus AB exam and 2) to cover all of the material students would encounter in a typical first-semester, college calculus course and 3) to help students really understand Calculus and how it relates to everyday life. This course ends at the close of the school year, not after the AP exam.