AB Calculus Course Syllabus
Course overview: The goal of this course is for students to become mathematically proficient all topics included in first semester college Calculus. This mathematical
proficiency includes conceptual understanding, comprehending the concepts, operations, and relations in Calculus, procedural fluency, the ability to accurately and
efficiently complete the procedures involved in Calculus, strategic competence, the ability to formulate, represent, and solve mathematics problems involving Calculus
concepts, adaptive reasoning, the ability to carry out logical thought, reflection, and justification, and productive disposition, seeing mathematics as useful along with
the belief in one’s own efficacy. (Taken from Adding It Up, National Research Council, 2001)
Textbook: Single Variable Calculus, 3rd edition, Hughes-Hallett, et al, John Wiley and Sons Inc. Publishers, 2002.
Class Culture: A large part of the AP Calculus program is dependent on building a classroom culture in which the students feel safe in sharing their ideas and
frustrations. The AP classroom should be a place where students work together using synergy to accomplish higher achievement as a class than they could as
individuals. This culture is intentionally created through several means. Students begin the year by reading an article titled “Building a Mathematical Culture’ from
Connected Geometry, Teacher’s Guide, Everyday Learning Corporation, 2000. They also read a synopsis of The Habits of Mind, Association for Supervision and
Curriculum Development, 2000. The students get involved in weekly before and after school help session aptly named “Calculus Parties.” At these sessions students
bring snack food and work together to complete problems and master concepts. Students also attend a quarterly movie in the evening at a local Arts Theater where they
view and discuss math related movies like Stand and Deliver, A Beautiful Mind, and Good Will Hunting. These things contribute to a sense of community within the AP
Calculus program, encouraging students to work together in learning Calculus.
Required Components
Multiple Representations of Functions: The course goals are accomplished by creating a classroom culture in which students work together to investigate, discuss,
explain, justify, and reflect on the topics of Calculus. The development of each topic includes multiple representations; graphical, verbal, numerical, and analytical.
Most topics are introduced in context verbally and developed graphically first to encourage a conceptual understanding before any analytical approach is applied. This
is usually followed by a numerical approach to the topic which is finally developed into the more abstract analytical approach. Through out the course students are
presented problem situations given in any of the four representations and to connect their understanding of the topic among the different representations.
Communication: Students communicate verbally as they work in groups to investigate new concepts, to solve problems, and to process homework. They also
communicate verbally through problem presentations to their peers as well as a project presentation each semester in which they present a larger problem or present a
new topic to their peers and their parents. Students communicate in written form when they write justification for homework problems, when they are asked to explain
their work as part of a test questions, and when they complete there AP Packet at the end of each unit. Each unit ends with a packet of five or six released, free-response
AP questions that connect current learning with previous learning. All of these packets are scored on the 9 point AP scale and must include justification for complete
credit.
Technology: Almost every lesson involves the incorporation of graphing calculator technology. Our purpose is to use the graphing calculator as a vehicle for
understanding the Calculus. Because of that, graphing calculator techniques for differentiation, integration, and general algebraic analysis is done concurrently or even
prior to paper and pencil analytical approaches. All tests are 50% with a graphing calculator and 50% without a graphing calculator. Computer technology is also
incorporated, although on a less frequent basis. Students use Geometer’s Sketchpad to visualize some of the problems. Students also have the opportunity to use Maple
software in the semester projects.
Content: All AP Calculus AB topics are covered in this course. This can be seen in the unit and lesson syllabus given on the following pages.
AB Calculus Course Syllabus
Unit/
Lesson
Unit 1
1-1
1-2
1-3
1-4
1-5
1-6
1-7
Review
Test
Unit 2
2-1
2-2
2-3
2-4
2-5
2-6
2-7
Review
Test
Unit 3
3-1
3-2
Summary/Topic
A Library of Functions: This unit reviews all of the functions introduced in the Precalculus course. (Functions Modeling Change, Connally)
Lessons should be approached as a quick review with the idea that concepts from these lessons will be reviewed and revisited as they arise in the
Calculus content. Lesson 7 on Continuity is new and is approached from a purely intuitive perspective in the first unit. The note to students on
page xi serves as a good introduction to the rule of four, the format of the book, and the philosophical approach authors. Some time should be
spent during this unit on setting individual and class goals in regards to the course and the AP test. Every test, beginning with unit 1, consists of
four parts divided into calculator and non-calculator as well as multiple choice and free-response.
Functions and Change
Exponential Functions
Transformation of Functions
Logarithmic Functions
Trigonometric Functions
Power, Polynomial, & Rational Functions
Continuity of Functions
Unit 1 Review/ Check Your Understanding
Unit 1 Test
Key Concept: The Derivative: This unit introduces the concept of the derivative as student transition form “average rate of change” used in
Precalculus to “instantaneous rate of change.” Slope as a way of measuring rate of change is introduced as the main theme for the first semester of
the course. Students estimate derivatives graphically and calculate derivatives numerically by finding the limit of the average rates of change for
successively closer points. They also learn the difference between the derivative at a point and the derivative function. The numerical derivative
function on the calculator is introduced in this unit after students see derivatives as the limit of average derivatives. Beginning with this unit, the
review at the end of each chapter will include the chapter review, the true-false conceptual check up in “Check Your Understanding”, and some AP
practice from released free-response questions. The AP practice should be scored using the 9 point AP scoring guide.
Measuring Speed
Limits
The Derivative at a Point
The Derivative Function
Interpretation of the Derivative
The Second Derivative
Continuity and Differentiability
Unit 2 Review/ Check Your Understanding/ AP Review
Chapter Two Test
Short-Cuts to Differentiation: In this unit students expand their conceptual understanding of derivatives to include algebraic procedures for
differentiation. These rules are “discovered” using graphical and numerical methods and, when appropriate, are justified analytically. The
technique of implicit differentiation is also introduced. The tangent line formula in lesson 3-9 is presented as a point-slope form equation rather
than a formula. Students are encouraged to use and appreciate the efficiency of the point slope form. Graphing calculators are used to illustrate
local linearity and the closeness of the fit of the tangent line near the point of tangency.
Derivatives of Polynomials
Exponential Derivatives
Approx
Time
2 weeks
1 day
1 day
1 day
1 day
1 day
1 day
1 day
2 days
1 day
3-4
weeks
2 days
2 days
2 days
2 days
1 day
2 days
1 day
4 day
1 day
4-5
weeks
2 days
2 days
AB Calculus Course Syllabus
3-3
3-4
3-5
3-6
3-7
3-9
3-10
Review
Test
Unit 4
4-1
4-2
4-3
4-5
4-7
Review
Test
Semester
Review
Unit 5
5-1
5-2
5-3
5-4
Review
Test
Unit 6
6-1
6-2
6-3
Derivatives of Products and Quotients
Derivatives by the Chain Rule
Derivatives of Trig Functions
Applications of the Chain Rule
Implicit Functions
Linear Approx & Derivatives
Local Linearity and Limits
Chapter Review/ Check Your Understanding/ AP Review
Chapter 3 Test
Using the Derivative: This unit provides students with practice finding derivatives as they apply them to solve problems. Some applications are
given with parameters that challenge the students in terms of their algebraic skills. Geometer’s Sketchpad is used to investigate or demo the
families of functions in lesson 4-2. Some problems that students have seen in Geometry and Precalculus, like minimizing the surface area of a
Coke can or minimizing the cost of laying a power line under a river, are redone using differentiation techniques.
4-1; Using Derivatives
4-2; Families of Curves
4-3; Optimization
4-5; Optimization & Modeling
4-7; Theorems
Unit 4 Review/ Check Your Understanding/ AP Review
Unit 4 Test
Semester 1 Projects: Students are given a list of possible projects or are invited to propose one of their own. The projects offer an opportunity
to apply first semester Calculus concepts learned in semester in solving more in-depth problems that take more than one class period to complete.
The students, working with a partner, write up their problem and solution on a display board and present it to the class.
Key Concept: The Integral: Students are introduced to the definite integral through using velocity-time graphs to find distance. Dimensional
analysis is used to connect integrals and areas of rectangles. Students work with Riemann Sums as a way of estimating integrals. The Riemann
Sums are used in numerical, graphical, and algebraic contexts. The programs CRIEMANN and DRIEMANN are linked to student graphing
calculators for use in finding Riemann Sums with a large number of intervals. For lesson 5.3, example 1 in the Instructor’s Manual provides an
excellent in-class investigation of a graphical approach in deducing information using functions, derivatives, and integrals. Lesson 5-4
introduces the Fundamental Theorem of Calculus, connecting derivatives (semester 1) with integrals (semester 2). Students are introduced to the
calculator’s definite integral function in 5-2.
Measuring Distance
The Definite Integral
Interpret the Definite Integral
Theorems about Integrals
Unit 5 Review/ Check Your Understanding/ AP Review
Unit 5 Test
Constructing Antiderivatives: Integrals are approaches as “undoing” differentiation with the understanding that there may be a vertical shift in
the final result. The derivative function is also introduced as are differential equations.
Antiderivatives
Antiderivatives Analytically
Differential Equations
2 days
2 days
2 days
3 days
2 days
1 day
1 day
5 days
1 day
3 weeks
2 days
1 day
2 days
2 days
1 day
4 days
1 day
1 week
2 weeks
1 day
1 day
2 days
2 days
3 day
1 day
2 weeks
1day
2 days
1 day
AB Calculus Course Syllabus
6-4
Review
Test
Unit 7
7-1
7-5
Review
Test
Unit 8
8-1
8-2
8-3
8-4
Review
Test
Unit 11
11-1
11-2
11-4
11-5
11-6
Review
Test
AP
Review
7-2
Semester
Review
Second Fundamental Theorem
Unit 6 Review/ Check Your Understanding/ AP Review
Test
Integration: Techniques for integration are introduced and practiced. Integration by Parts (lesson 7.2) is done after the AP test as it is often part
of Calculus 1 in a college course, but is not part of the AP test for AB Calculus. Partial fraction decomposition is not covered in this course. (All
of these are covered in the Calculus BC course.)
Integration by Substitution
Approximating Definite Integrals
Unit 7 Review/ AP Review
Unit 7 Test
Using the Definite Integral: The main focus of this unit is finding volumes of revolution. The approach taken is to find the find the volume of
one slice, write a Riemann sum representing all slices and convert it to integral notation. Lessons 8.5-8.7 offer applications to science, business,
and statistics. These lessons are offered as student project options after the AP exam.
Area and Volume by Integration
Volume of Revolution and Length
Density Using Integrals
Applications to Physics
Unit 8 Review/ Check Your Understanding/ AP Review
Unit 8 Test
Differential Equations: Students are introduced to more involved differential equations with separable variables. Slope fields are introduced as
a graphical solution to a differential equation. Slope fields are introduced by students working together to make a slope field on a large set of
coordinate axes. They then do slopefields by hand and on the calculator.
Differential Equations
Slope Fields
Separation of Variables
Growth and Decay
Applications and Modeling
Chapter 11 Review/ Check Your Understanding/ AP Review
Unit 11 Test
AP Test Practice: Students spend any remaining time before the AP exam working released AP problems and tests, including multiple choice
and free-response. Students spend any remaining time before the AP exam working released AP problems and tests, including multiple choice
and free-response. A schedule of “Calculus Parties” is set before school, after school, and in the evening after extra-curricular practices, for
students to get together and discuss the review problems. About one week prior to the exam, students take and actual full-length released test in
a mock-testing situation. Test results are used to pinpoint individual strengths and weaknesses and to determine focus areas for the final week of
preparation.
Integration by Parts
Semester 2 Projects: Students are given a list of possible projects or are invited to propose one of their own. The projects offer an opportunity
to apply second semester Calculus concepts learned in semester in solving more in-depth problems that take more than one class period to
complete. The students, working with a partner, write up their problem and solution on a display board and present it to the class. Parents are
also invited to the problem presentations.
1 day
3 day
1 day
2 weeks
4 days
1 day
4 days
1 day
2-3 weeks
3 days
2 days
1 day
1 day
4 day
1 day
3 weeks
1 day
2 days
2 days
2 days
1 day
4 day
1 day
2-3 weeks
1 week
1 week