Teacher+Syllabus+AP+Calculus


 * Syllabus for** **Advanced Placement Calculus AB**


 * Meeting Times:** This course meets for 36 weeks in 85 minute blocks.

AP Calculus AB provides an understanding of the fundamental concepts and methods of differential and integral calculus with an emphasis on their application, and the use of multiple representations incorporating graphic, numeric, analytic, algebraic, and verbal and written responses. Topics of study include: functions, limits, derivatives, and the interpretation and application of integrals. An in-depth study of functions occurs in the course. Technology is an integral part of the course and includes the use of graphing calculators, computers, and data analysis software. On a regular basis, graphing calculators are used to explore, discover, and reinforce concepts of calculus. Though our system has an open enrollment policy, students should understand that this course is designed to be a fourth-year mathematics course, and the equivalent of a year-long, college-level course in single variable calculus. The course requires a solid foundation of advanced topics in algebra, geometry, trigonometry, analytic geometry, and elementary functions. The breadth, pace, and depth of material covered exceeds the standard high school mathematics course, as does the college-level textbook, and time and effort required of students. AP Calculus AB provides the equivalent of the first course in a college calculus sequence, while AP Calculus BC is an extension of AP Calculus AB, and provides the equivalent of a second course in a college calculus sequence. Students are expected to take the AP Calculus AB Exam at the end of this course.
 * Course Description:**

__Philosophy__ Understanding change is the basis of this course. The study of the concept of the derivative in calculus is the formal study of mathematical change. A key component of the course is fluency in the use of multiple representations that include graphic, numeric, analytic, algebraic, and verbal and written responses. Students build an understanding of calculus concepts as they construct relationships and make connections among the various representations. The course is more than a collection of topics; it is a coherent focused curriculum that develops a broad range of calculus concepts and a variety of methods and real-world applications. These include practical applications of integrals to model biological, physical, and economic situations. Although the development of techniques and fluency with algebraic symbolism to represent problems is important, it is not a primary focus of the course. Rather, the course emphasizes differential and integral calculus for functions of a single variable through the Fundamental Theorem of Calculus. Technology is used to enhance students’ understanding of calculus concepts and techniques. The College Board requires the use of graphing calculators for this course. Mathematical problem solving, investigations, and projects require adequate and timely access to technology including graphing calculators, databases, spreadsheets, Internet and on-line resources, and data analysis software. In this course, technology is introduced in the context of real-world problems, incorporates multiple representations, and facilitates connections among mathematics topics. Students use estimation, mental math, calculators, and paper-and-pencil techniques of calculus to conduct investigations and solve problems. According to the National Council of Teachers of Mathematics (2000), “Estimation serves as an important companion to computation. It provides a tool for judging the reasonableness of calculator, mental, and paper-and-pencil computations” (NCTM, p. 155). The standards support the unifying themes of derivatives, integrals, limits, approximation, and applications and modeling in the course. Instruction is designed and sequenced to provide students with learning opportunities in appropriate settings. Teaching strategies include collaborative small-group work, pairs engaged in problem solving, whole-group presentations, peer-to-peer discussions, and an integration of technology when appropriate. In this course, students are often engaged in mathematical investigations that enable them to collaborative with peers in designing mathematical models to solve problems and interpret solutions. They are encouraged to talk about the mathematics of change in calculus, to use the language and symbols of calculus to communicate, and to discuss problems and methods of solution.
 * Course Purpose and Goals:**

__Goals__ //Students should be able to://
 * 1) Understand the major topics of functions, limits, derivatives, and integrals.
 * 2) Incorporate multiple representations of functions using graphic, numeric, analytic, algebraic, and verbal and written responses, and understand the connections among these representations.
 * 3) Construct an understanding of derivatives as an instantaneous rate of change, applications of derivatives as functions, and use various techniques to solve problems.
 * 4) Understand definite integrals as a limit of Riemann sums, and as the net accumulation of sums, and use them to solve a variety of problems.
 * 5) Develop an understanding of the Fundamental Theorem of Calculus as a relationship between derivatives and definite integrals.
 * 6) Use graphing calculators to problem solve, experiment with ‘what if’ hypotheses, display and interpret results, and justify conclusions.
 * 7) Make sense of and determine the reasonableness of solutions.
 * 8) Develop an appreciation for an historical perspective of calculus.

__Conceptual Organization__ The content and level of depth of the material for this course is equivalent to a college-level course. The course content is organized to emphasize major topics in the course to include the following: (1) functions, graphs, and limits; (2) derivatives, and (3) integrals. Building on most students’ prior knowledge, the course begins with a review of a variety of functions using multiple representations: graphic, numeric, algebraic, analytic, and verbal and written responses. Technology enhances students’ constructing an understanding of mathematical relationships among the different representations used in solving problems. Then, this supports and leads to students’ development and visualization of properties of limits and continuity, and rates of change of functions. The concept of a derivative is interpreted as a rate of change and local linearity. Using graphing calculators, numeric derivatives are examined. This is followed with a focus on derivatives of functions—algebraic, trigonometric, logarithmic, and exponential. Applications of the derivative are investigated through velocity, acceleration, and optimization problems. The definite integral is studied as a limit of Riemann sums and the rate of change of a quantity over a specific interval. This sequence of topics naturally leads to students’ introduction to the Fundamental Theorem of Calculus. Applications of definite integrals are also investigated which include summing rates of change, particle motion, areas in a plane, and volumes of solids. This order of topics within the course, not only provides a logical and systemic study to calculus, but also accommodates the frequent transfer of students within the schools of the system, so that transfer students can maintain a consistent flow of learning.

In order for students to get as much as possible out of this or any course, regular active participation and engagement with the ideas discussed are necessary. And working through questions in a group setting is a good way to develop a deeper understanding of the mathematics involved, if students approach the enterprise as a truly //collaborative effort//. That means asking questions of fellow students when something is not understood, and being willing to explain ideas carefully to someone else. Collaboration is not letting someone else do the work and give others the answer. Looking to the future as well, working effectively in a group is also a valuable skill to have. So with these points in mind, regularly throughout the school year the class will break down into groups of 3 or 4 students for one or more days, and each group will work individually for most of those class periods on a group discussion exercise. I will be responsible for designing and preparing these exercises using material from previous AP Calculus Exams, and I will be available for questions and other help during these periods. At the conclusion of each discussion, the class as a whole may reconvene to talk about what has been done, to sum up the results, to hear short oral reports from each group, etc. Each group will keep a written record of their observations, results, questions, etc. which will be handed in. I will make copies of these, and return them with comments, for all members of the group. The other meetings of the class will be structured as lectures when that seems appropriate. Questioning is encouraged during these time so that it is a flow of information and not just the teacher talking. Techniques such as ‘dip sticking’ for understanding will be employed. Regular short homework quizzes will be given to check for understanding and comprehension. Calculus topics build on each other. A good foundation is essential for success on the exam. Students are encouraged to seek help outside of classroom time. All quizzes are corrected and kept for review. Students are expected to be prompt, prepared, and polite. Parents are contacted after two tardies. Students have one day for each day absent to make up missed work. Attendance is essential for success in AP Calculus. A good work ethic is also essential for success. Students should come to class, do assigned work on a regular basis, read the textbook, take notes in class and use them, have a regular study time, and get help as soon as problems arise. Students do not have to be ‘math geniuses’ to succeed. They just need to be consistent in studying and keeping up with the course. Students are expected to do their own work. Students will be asked to put specific homework problems on the board and explain them to the rest of the class. Copying others work will result in a zero. Collaboration is encouraged, cheating is not! Quarter grades for the course will be based on Semester grades for the course will be based on Quarter Grades (40% each) and a Semester Exam (20%). Weighted grades are calculated for students completing and taking the requisite exam of an AP course.
 * Course Format and Policies:**
 * Grading Policy **
 * 1) Short quizzes(10 to 15 minutes) on homework due the previous class.
 * 2) Quizzes (30 – 45 minutes) on specific topics
 * 3) Unit Tests
 * 4) Quarterly Assessments
 * 5) Presentations and written reports from small group projects -- one report from each group
 * 6) Individual homework assignments, given out in class. //No credit will be given for late homework, except in the case of an excused absence, or with my permission.//
 * 7) Graded Homework assignments done on computer.

Unweighted Scale A=4 Weighted Scale A=5 Unweighted Scale B=3 Weighted Scale B=4 Unweighted Scale C=2 Weighted Scale C=3 Unweighted Scale D=1 Weighted Scale D=2 Unweighted Scale F=0 Weighted Scale F=0
 * Textbook, Materials and Other Resources:**

__Required Textbook__ __Supplemental Textbooks and Readings__
 * Finney, R. L., Demana, F.D., Waits, B.K., and Kennedy, D. (2003). //Calculus: Graphical, numerical, algebraic//. Upper Saddle River, NJ: Pearson Education-Prentice Hall.
 * Finney, R. L., Demana, F.D., Waits, B.K., and Kennedy, D. (2003). //Advanced placement correlations and preparation for// //calculus: Graphical, numerical, algebraic//. Upper Saddle River, NJ: Pearson Education-Prentice Hall.
 * Finney, R. L., Demana, F.D., Waits, B.K., and Kennedy, D. (2003). //Technology resources manual for// //calculus: Graphical, numerical, algebraic//. Upper Saddle River, NJ: Pearson Education-Prentice Hall.
 * Barton, Brunsting, Diehl, Hill, Tyler, and Wilson (2006). //Preparing for the Calculus AP* Exam with Calculus: Graphica, Numberical, Algebraic.// Boston, MA: Pearson Education, Inc.
 * Kahn, David S. (2005). //Cracking the AP* Calculus AB & BC Exams, 2006-2007 Edition.// New York, NY: The Princeton Review, Inc.
 * Hockett, S.O., and Bock, D. (2002). //Barron’s How to Prepare for the AP* Calculus Advanced Placement Examination, 7th edition.// Hauppauge, NY: Barron’s Educational Series, Inc.

__Other Resources__
 * Computers: The mathematics department has a COW(computer on wheels) with 15 laptop available and a smart board for presentations.
 * Software: Testworks 3.3 Test generator for//Calculus: Graphical, numerical, algebraic//
 * Graphing calculators: There is a classroom set of TI 84 Silver Edition. Students are trained on these and encouraged to purchase their own.
 * Internet access and online resources.
 * Math Tools Website: __http://www.mathforum.org/mathtools/cell.html?&new_co=c__
 * Math Archives: Calculus Resources On-Line Website: __[]__
 * __Webassign__ – a computer based homework site


 * Course Content Outline:**

Pre-Calculus Review of Lines and Functions ||  1   ||  1 - 2  || Graphing and analyzing relations and functions; Equations of lines and applications; Properties of functions: absolute, composite, and piecewise; Function types: exponential, logarithmic, and trigonometric || Homework Quizzes Graded Homework Test I: Unit 1 Multiple Choice, Problems, Constructed Response || Functions and Limits ||  1   ||  3 - 4  || Rates of change; Intuitive introduction of limits; Definition of a limit; Properties of limits at a point; One-sided and two-sided limits; Sandwich Theorem; Limits related to infinity and description of asymptotic behavior; Visualizing and properties of limits || Homework Quizzes Graded Homework
 * Unit ||  Quarter  ||  Week  ||  Topics  ||  Assessments  ||
 * 1
 * 2

Quiz 1: Properties of limits

Group work || Properties of continuous functions; Discontinuous functions: removable, jump, and infinite || Quiz 2: Discontinuous functions || Tangent and Normal lines to a curve; Slope of a curve || Test 2: Unit 2 Multiple Choice, Problems, Constructed Response || Concept of the Derivative ||  1   ||  6 - 8  || Definition of the derivative; Instantaneous rate of change; Differentiability: local linearity, numerical, and relationship to continuity; Intermediate Value Theorem; Fluency with differentiation techniques: power, sums, products, and the quotients rule || Homework Quizzes Graded Homework Quiz 3: Using Definition of the Derivative to find instantaneous rate of change
 * Continuity of Functions ||  1   ||  5  || Continuous functions;
 * Rates of Change of Functions ||  1   ||  5  || Comparing rates of change for different functions;
 * 3

Quarter I Assessment Multiple Choice, Problems, Constructed Response || Derivatives of functions: algebraic, trigonometric, inverse trigonometric, logarithmic, and exponential; Chain rule for composite functions; Implicit differentiation: differential and y’ techniques || Homework Quizzes Group Project: velocity, speed and acceleration applications. Quiz 4: Derivatives (power rule, quotient rule, product rule, trig.rules) Quiz 5: Chain Rule Quiz 6: Implicit Differentiation Quiz 7: Derivatives (Inverse trigonometric, logarithmic, and exponential) Test 3: Unit 3 Multiple Choice, Problems, Constructed Response || Applications of Derivatives ||  2   || 13-16 || Extreme values of a function: absolute and relative extrema; Characteristics of increasing and decreasing functions; Mean Value theorem and Rolle’s theorem; Analysis of graphs using 1st and 2nd derivatives: Relative and absolute maxima and minima; Concavity and points of inflection || Homework Quizzes Graded Homework
 * Derivatives and Functions ||  2   ||  9 -12  || Rates of change: velocity, speed, and acceleration applications;
 * 4

Quiz 8: Applications (Extreme values, Mean Value theorem and Rolle’s theorem) Quiz 9: Application (Graphing- max/min – concavity- inflection points)

Group Project: Applications of Derivatives || Linearization models; Modeling related rates problems || Graded Homework Quiz 10: Optimization Quiz 11: Related Rates Test 4: Unit 4 Multiple Choice, Problems, Constructed Response Quarter 2 Assessment Multiple Choice, Problems, Constructed Response || Definite Integral: Approximating Areas ||  3   ||  19-20  || Areas under curves; Riemann Sums: partition and subintervals; Trapezoidal Rule; Definite Integrals; Integration terminology and notation || Semester I Assessment Mock AP Exam – Derivatives only - Multiple Choice, Problems, Constructed Response Homework Quizzes Graded Homework || Graded Homework
 * Business and Industry Applications ||  2   ||  17-18  || Optimization problems;
 * 5
 * Fundamental Theorem of Calculus (part 1) ||  3   ||  21-22  || Differential and Integral calculus: Connections between slopes of tangent lines and areas under curves || Homework Quizzes

Quiz 12: Approximating Area || Upper and lower bounds; Evaluate definite integrals; Average Value Theorem || Homework Quizzes Graded Homework Quiz 13: Evaluating Definite Integrals Group Project: Area || Using anti-derivatives to find area || Graded Homework Quiz 14: Area Test 5: Unit 5 Multiple Choice, Problems, Constructed Response Quarter 3 Assessment Multiple Choice, Problems, Constructed Response || Differential Equations ||  4   ||  26  || Antiderivatives; Properties of Indefinite Integrals; Applied differential equations; Substitution method of integration; Integration by parts || Homework Quizzes Graded Homework Quiz 15: Differential Equations Quiz 16: Integration by substitution Quiz 17 : Integration by parts || Test 6: Unit 6 Multiple Choice, Problems, Constructed Response Group Project: Applications || Definite Integral Applications ||  4   ||  29-32  || Summing rates of change; Particle motion; Areas in a Plane;
 * Definite Integrals and Antiderivatives ||  3   ||  23-24  || Properties of definite integrals;
 * Fundamental Theorem of Calculus (part 2) ||  3   ||  25  || Integral Evaluation Theorem;
 * 6
 * Mathematical Modeling ||  4   ||  27-28  || Separable differential equations: Growth and decay, slope fields, and other general differential equations || Graded Homework
 * 7

Volumes: Washers; Solids with known cross-sections and Solids of revolution: Disk method and Shell method; Integration by parts || Graded Homework Quiz 18: Volume Group Project: Volume Quarter 4 Assessment Multiple Choice, Problems, Constructed Response

Semester 2 Assessment AP Mock Exam ||
 * ||  4   ||  33-36  || Review and AP Exam || AP Exam ||

Assessment and evaluation are essential to learning and teaching.Ongoing assessment and evaluation are significant in supporting student achievement, motivating student performance and providing the basis upon which teachers make meaningful instructional decisions. All aspects of progress in mathematics are measured using multiple methods such as authentic, performance, observational, and formative assessments; group and individual projects, student presentations, and conventional summative assessments. Student understanding is evaluated using an assessment cycle that includes pre-, formative, and summative assessments. Pre-assessments are used to determine where the student understanding level is, as the unit is begun. The pre-assessment is used by a teacher to plan instruction. Formative assessments are used to check student understanding while learning is occurring, and provide students and teachers with learning progress information. Pre- and formative assessments are not used to determine grades. Summative assessments, such as unit and semester tests, evaluate student achievement, and along with other measures such as student presentations and project work are data points used to determine the level of student performance.
 * Assessment:**


 * Assessment Type ||  Goal  ||  Description  ||
 * Homework Quizzes || To assess understanding of concepts, principles, and techniques of calculus. || One to five questions from the homework due the previous class (10-15 minutes) ||
 * Quizzes || To assess understanding of concepts, principles, and techniques of calculus. || 30 – 45 minute quizes containing multiple-choice items, problems to solve, and constructed responseitems. ||
 * Unit Tests || To assess understanding of concepts, principles, and techniques of calculus. || 75 -85 minute testscontaining multiple-choice items, problems to solve, and constructed responseitems. ||
 * Semester Assessments || To assess understanding of concepts, principles, applications, and techniques of calculus for several units. To prepare students for the AP Exam. || 90 – 120 minute tests in the format of the AP exam with Section I, part A and part B Multiple Choice questions and Section II, part A and part B free response items. ||
 * Group Projects || To provide students with an opportunity to examine a calculus topic in-depth within a group and demonstrate the processes and skills of a well-designed investigation || A short-term project, in which students work in a small group to research a calculus topic. It includes an oral presentation and written paper. ||


 * Supporting Services:**
 * 1. There is a math workshop offered two days a week after school for an hour.**
 * 2. Students come for help during seminar time.**
 * 3. Appointments can be made for help before or after school or during lunch.**
 * 4. The base personnel volunteer as tutors for students.**
 * 5. National Honor Society students who took AP Calculus as Juniors tutor other students.**