Content Area: Mathematics Course: AP Calculus AB

SHORE REGIONAL HIGH SCHOOL DISTRICT West Long Branch, New Jersey Content Area: Mathematics Course: AP Calculus AB Mr. Leonard Schnappauf, Superintendent/Principal Dr. Robert McGarry, Director of Curriculum and Instruction BOARD OF EDUCATION Anthony F. Moro, Jr., President Tadeusz Ted Szczurek, Vice President Nancy DeScenza David Baker Elizabeth Garrigal Diane Merla Russell T. Olivadotti Ronald O Neill Frank J. Pingitore Paul Rolleri Date of Last Revision and Board Adoption: 8/27/2009

Mathematics AP Calculus AB REVISION PREPARED BY Nicole Andreasi 2

Table of Contents Mathematics Program Mission Statement....... 4 Course Description and Big Ideas.... 4 Essential Questions.. 4-5 Primary (P) Content Area and Secondary (S) Areas of Focus.5 Benchmark Objectives. 5 Scope and Sequence 6 Learning Resources..6 Grading Procedures. 6-7 Course Evaluation 7 New Jersey Core Curriculum Content Standards/Cumulative Progress Indicators Addressed in the Course... 8-23 Units of Study...20-27 3

Mathematics Program Mission Statement The mission of the Mathematics program at Shore Regional High School is to ensure that students develop an understanding of mathematical concepts and processes leading to individual mastery while logic and problem-solving skills assist students in becoming successful global citizens. Course Description and Big Ideas Students study the topics outlined in the College Board's current syllabus for AB Calculus. This includes limits, derivatives, definite integrals, anti-differentiation, slope fields, areas and volumes using integration, related rates, optimization, and extrema. Students learn to apply graphical, numerical, analytical, and verbal approaches to solving problems. Students also learn to use their graphing calculators to find complete graphs of functions, identify roots of equations, and calculate numerical derivatives and integrals. Students prepare throughout the year for the AP exam by solving open-ended questions and multiple-choice problems from previous exams. Essential Questions Throughout this course and in the sequence of courses in this content area, students are consistently guided to consider the following essential questions: 1. Number Sense and Concepts a. How can a tangent line be used to approximate the value of a function near a given point? How can the error be reduced? 2. Geometry and Measurement a. How can slopes be used to identify parallel and perpendicular lines? b. Given the three vertices of a triangle, how can the area of the triangle be found using calculus methods? c. A shaded region is revolved around both axes. Under what circumstances will the resulting volumes be the same? 3. Patterns and Algebra a. What makes a function a relation? b. What is a piecewise function? c. How can you distinguish between various functions, including polynomial, exponential, and trigonometric? 4

d. What are the methods for finding limits and when should each be used? 4. Math Processes a. How do you find the derivative of polynomial, exponential, logarithmic, and trigonometric functions? How does this process change if the functions are combined by various operations? b. How could you use the graph of the derivative of a function f to graph f? c. How can the concept of a derivative affect a company s profit and production cost? d. How can the first and second derivatives of a function f, what can you determine about f and its behavior? e. How can you determine the rate at which a ladder slides down a wall? f. How do you find the anti-derivative of polynomial, exponential, logarithmic, and trigonometric functions? How does this process change if the functions are combined by various operations? g. How you use the graph of the anti-derivative of a function f to graph f? Primary (P) Content Area and Secondary (S) Areas of Focus NJCCC Standard NJCCC Standard NJCCCS Standard 1. Visual and Performing Arts 5. Science 9. Career Education and Consumer/ Family/ Life Skills 2. Health and Physical Education 6. Social Studies 3. Language Arts Literacy 7. World Languages 4. Mathematics P 8. Technology Literacy Benchmark Objectives These objectives focus on the achievement of the Standards/Big Ideas as they pertain to the specific course content and are listed in the units of study found within this document. Summative assessment of these objectives may occur at the point in the course when instruction of the components parts is completed (typically at the end of a unit), at the end of a marking period, end of the year, or in areas tested by the State when the tests are scheduled. 5

Scope and Sequence This represents the order in which units or the big ideas of the course are taught. The specific unit content, CPI s addressed, time frame for instruction and how proficiency will be addressed is included in the units that follow. This list serves the teacher as an overview of course implementation and administrators as a basis for review of lesson plans and orientation for classroom observation. The Units included in this course include: 1. Change 2. Derivatives 3. Applications of Derivatives 4. Anti-Derivatives Learning Resources 1. Calculus: Graphical, Numerical, Algebraic by Finny, Demana, Waits, and Kennedy. 2003. 2. Workbook associated with the above text. 3. AP Central (apcentral.collegeboard.com) 4. Past AP exams released from the Collegeboard 5. Teacher created PowerPoint presentations 6. Cracking the AP Calculus AB & BC Exams. The Princeton Review. 2005. Grading Procedures The final course proficiency grade will be the average of the four marking period grades and the department prepared mid-year and final examinations aligned with NJCCCS/CPI and benchmarks for the content studied in the course. Marking period grades will be based on the average of unit grades and any special cross-unit projects. 6

Unit assessments, delineated for each unit, will include such measures as: 1. Quizzes, tests, and homework checks 2. Review packets distributed after every two chapters 3. Graphical analysis 4. Multiple choice and open-ended reviews, given started one month before the exam 5. Post-exam project Course Evaluation Course achievement will be evaluated as the percent of all pupils who achieve the minimum level of proficiency (final average grade) in the course. Student achievement levels above minimum proficiency will also be reported. Final grades, and where relevant midterm and final exams, will be analyzed by staff for the total cohort and for sub-groups of students to determine course areas requiring greater support or modification). Course evaluation requires the pursuit of answers to the following questions: 1. To what extent is the course content, instruction and assessments aligned with the required NJCCS? 2. Are content, instruction and assessments sufficient to demonstrate student mastery of the Standards/CPI s? 3. Do all students achieve the set proficiencies/benchmarks set for the course, including CPI s designated to be reinforced, introduced, and developed? In this course, the goal is that a minimum of 95% of the pupil s will meet at least the minimum proficiency level (D or better) set for the course. The department will analyze the achievement of students on Unit Assessments, Mid-term and Final Exams and Final Course Grades, with specific attention to the achievement of sub-groups identified by the state to determine if modifications in the curriculum and instructional methods are needed. 7

New Jersey Core Curriculum Content Standards/Cumulative Progress Indicators Addressed in the Course Primary: Mathematics 4.1. Number and Numerical Operations Cumulative Progress Indicator Addressed in this course? A. Number Sense 1. Extend understanding of the number system to all real numbers 2. Compare and order rational and irrational numbers. 3. Develop conjectures and informal proofs of properties of number systems and sets of numbers. B. Numerical Operations 1. Extend understanding and use of operations to real numbers and algebraic procedures. 2. Develop, apply, and explain methods for solving problems involving rational and negative exponents. 3. Perform operations on matrices a. addition and subtraction b. scalar multiplication 4. Understand and apply the laws of exponents to simplify expressions involving numbers raised to 8

powers. C. Estimation 1. Recognize the limitations of estimation, assess the amount of error resulting from estimation, and determine whether the error is within acceptable tolerance limits. 4.2 (Geometry and measurement) All students will develop spatial sense and the ability to use geometric properties, relationships, and measurement to model, describe and analyze phenomena. Cumulative Progress Indicator Addressed in this course? A. Geometric Properties 1. Use geometric models to represent real-world situations and objects and to solve problems using those models (e.g., use Pythagorean Theorem to decide whether an object can fit through a doorway). 2. Draw perspective views of 3D objects on isometric dot paper, given 2D representations (e.g., nets or projective views). 9

3. Apply the properties of geometric shapes. Parallel lines transversal, alternate interior angles, corresponding angles Triangles a. Conditions for congruence b. Segment joining midpoints of two sides is parallel to and half the length of the third side c. Triangle Inequality d. Special right triangles Minimal conditions for a shape to be a special quadrilateral Circles arcs, central and inscribed angles, chords, tangents Self-similarity Counterexamples to incorrect conjectures 4. Use reasoning and some form of proof to verify or refute conjectures and theorems. Verification or refutation of proposed proofs Simple proofs involving congruent triangles Counterexamples to incorrect conjectures B. Transforming Shapes 1. Determine, describe, and draw the effect of a transformation, or a sequence of transformations, on a geometric or algebraic representation, and, conversely, determine whether and how one representation can be transformed to another by a transformation or a sequence of transformations. 10

2. Recognize three-dimensional figures obtained through transformations of two-dimensional figures (e.g., cone as rotating an isosceles triangle about an altitude), using software as an aid to visualization. 3. Determine whether two or more given shapes can be used to generate a tessellation 4. Generate and analyze iterative geometric patterns. Fractals (e.g., Sierpinski s Triangle) Patterns in areas and perimeters of self-similar figures Outcome of extending iterative process indefinitely C. Coordinate Geometry 1. Use coordinate geometry to represent and verify properties of lines and line segments. Distance between two points Midpoint and slope of a line segment Finding the intersection of two lines Lines with the same slope are parallel Lines that are perpendicular have slopes whose product is 1 2. Show position and represent motion in the coordinate plane using vectors. Addition and subtraction of vectors D. Units of Measurement 1. Understand and use the concept of significant digits. 11

2. Choose appropriate tools and techniques to achieve the specified degree of precision and error needed in a situation. Degree of accuracy of a given measurement tool Finding the interval in which a computed measure (e.g., area or volume) lies, given the degree of precision of linear measurements E. Measuring Geometric Objects 1. Use techniques of indirect measurement to represent and solve problems. Similar triangles Pythagorean theorem Right triangle trigonometry (sine, cosine, tangent) 2. Use a variety of strategies to determine perimeter and area of plane figures and surface area and volume of 3D figures. Approximation of area using grids of different sizes Finding which shape has minimal (or maximal) area, perimeter, volume, or surface area under given conditions using graphing calculators, dynamic geometric software, and/or spreadsheets Estimation of area, perimeter, volume, and surface area 12

4.3 (Patterns and algebra) All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes. Cumulative Progress Indicator Addressed in this course? A. Patterns 1. Use models and algebraic formulas to represent and analyze sequences and series. Explicit formulas for n th terms Sums of finite arithmetic series Sums of finite and infinite geometric series 2. Develop an informal notion of limit. 3. Use inductive reasoning to form generalizations. B. Functions and Relationships 1. Understand relations and functions and select, convert flexibly among, and use various representations for them, including equations or inequalities, tables, and graphs. 13

2. Analyze and explain the general properties and behavior of functions or relations, using algebraic and graphing techniques. Slope of a line Domain and range Intercepts Continuity Maximum/minimum Estimating roots of equations Solutions of systems of equations Solutions of systems of linear inequalities using graphing techniques Rates of change 3. Understand and perform transformations on commonly-used functions. Translations, reflections, dilations Effects on linear and quadratic graphs of parameter changes in equations Using graphing calculators or computers for more complex functions 4. Understand and compare the properties of classes of functions, including exponential, polynomial, rational, and trigonometric functions. C. Modeling Linear vs. non-linear Symmetry Increasing/decreasing on an interval 1. Use functions to model real-world phenomena and solve problems that involve varying quantities. 14

Linear, quadratic, exponential, periodic (sine and cosine), and step functions (e.g., price of mailing a first-class letter over the past 200 years) Direct and inverse variation Absolute value Expressions, equations and inequalities Same function can model variety of phenomena Growth/decay and change in the natural world Applications in mathematics, biology, and economics (including compound interest) 2. Analyze and describe how a change in an independent variable leads to change in a dependent one. 3. Convert recursive formulas to linear or exponential functions (e.g., Tower of Hanoi and doubling). D. Procedures 1. Evaluate and simplify expressions. Add and subtract polynomials Multiply a polynomial by a monomial or binomial Divide a polynomial by a monomial Perform simple operations with rational expressions Evaluate polynomial and rational expressions 2. Select and use appropriate methods to solve equations and inequalities. Linear equations and inequalities algebraically Quadratic equations factoring (including trinomials when the coefficient of x2 is 1) and using the quadratic formula Literal equations 15

All types of equations and inequalities using graphing, computer, and graphing calculator techniques 3. Judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology. 4.5 (Mathematical processes) All students will use mathematical processes of problem solving, communication, connections, reasoning, representations, and technology to solve problems and communicate mathematical ideas. Cumulative Progress Indicator Addressed in this course? A. Problem Solving 1. Learn mathematics through problem solving, inquiry, and discovery. 2. Solve problems that arise in mathematics and in other contexts (cf. workplace readiness standard 8.3). Open-ended problems Non-routine problems Problems with multiple solutions Problems that can be solved in several ways 3. Select and apply a variety of appropriate problem-solving strategies (e.g., "try a simpler problem" or "make a diagram") to solve problems. 4. Pose problems of various types and levels of difficulty. 5. Monitor their progress and reflect on the process of their problem solving activity. B. Communication 16

1. Use communication to organize and clarify their mathematical thinking. Reading and writing Discussion, listening, and questioning 2. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing. 3. Analyze and evaluate the mathematical thinking and strategies of others. 4. Use the language of mathematics to express mathematical ideas precisely. C. Connections 1. Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry). 2. Use connections among mathematical ideas to explain concepts (e.g., two linear equations have a unique solution because the lines they represent intersect at a single point). 3. Recognize that mathematics is used in a variety of contexts outside of mathematics. 4. Apply mathematics in practical situations and in other disciplines. 5. Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards).use the language of mathematics to express mathematical ideas precisely. 17

6. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. D. Reasoning 1. Recognize that mathematical facts, procedures, and claims must be justified. 2. Use reasoning to support their mathematical conclusions and problem solutions. 3. Select and use various types of reasoning and methods of proof. 4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions. 5. Make and investigate mathematical conjectures. Counterexamples as a means of disproving conjectures Verifying conjectures using informal reasoning or proofs. 6. Evaluate examples of mathematical reasoning and determine whether they are valid. E. Representations 1. Create and use representations to organize, record, and communicate mathematical ideas. Concrete representations (e.g., base-ten blocks or algebra tiles) Pictorial representations (e.g., diagrams, charts, or tables) 18

Symbolic representations (e.g., a formula) Graphical representations (e.g., a line graph) 2. Select, apply, and translate among mathematical representations to solve problems. 3. Use representations to model and interpret physical, social, and mathematical phenomena. F. Technology 1. Use technology to gather, analyze, and communicate mathematical information. 2. Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information (cf. workplace readiness standard 8.4-D). 3. Use graphing calculators and computer software to investigate properties of functions and their graphs. 4. Use calculators as problem-solving tools (e.g., to explore patterns, to validate solutions). 5. Use computer software to make and verify conjectures about geometric objects. 6. Use computer-based laboratory technology for mathematical applications in the sciences (cf. science standards). 19

Unit 1: Change Unit Question(s) Objectives Resources Formative Assessment Strategies Pacing Guide Marking Period 1. How can slopes be used to identify parallel and perpendicular lines? 2. What makes a relation a function? 3. What is a piecewise function? 4. What are the characteristics of various functions, including polynomial, exponential, and trigonometric? 5. What are the methods of finding a limit and when should you use each? 6. What is the difference between an average rate of change and an instantaneous rate of change and how would you find each? Students will be able to: 1. Solve polynomial, rational, exponential, trigonometric and logarithmic equations. 2. Determine a function s domain and range 3. Find an average rate of change and an instantaneous rate of change graphically 4. Find an average rate of change and an instantaneous rate of change algebraically 5. Determine whether the limit of a function at a certain x-value exists 6. Determine whether a function is continuous and identify points and types of discontinuity 7. Determine the limit of a function as it approaches positive or negative infinity 1. Calculus: Graphical, Numerical, Algebraic 2. Workbook associated with the above text. 3. AP Central (apcentral.collegeboard. com) 4. Past AP exams released from the Collegeboard 5. Teacher created PowerPoint presentations 1. Teacher questioning 2. Class discussions 3. Homework checks 4. Classwork checks 15 days 1 6. Cracking the AP Calculus AB & BC Exams. 20

Standards Instructional Activities, Methods, and Assignments Unit Summative Assessment(s) 4.1.A.1, 4.1.A.2, 4.1.A.3, 4.1.C.1, 4.1.D.2, 4.3.A.2, 4.3.A.3, 4.3.B.1, 4.3.C.2, 4.3.D.1, 4.3.D.2, 4.5.A.1, 4.5.A.2, 4.5.A.3, 4.5.A.4, 4.5.A.5, 4.5.B.1, 4.5.B.2, 4.5.B.3, 4.5.B.4, 4.5.C.1, 4.5.C.3, 4.5.C.4, 4.5.D.1, 4.5.D.2, 4.5.D.3, 4.5.D.4, 4.5.D.6, 4.5.E.1, 4.5.E.2, 4.5.E.3, 4.5.F.1, 4.5.F.2, 4.5.F.3, 4.5.F.4, 1. AP test questions from previous exams 2. Notes 3. Vocabulary 4. Practice Problems: main white board, individual notes 1. Chapter tests 2. AP review packet containing sample AP problems from previous years exams 21

Unit 2: Derivatives Unit Question(s) Objectives Resources Formative Assessment Strategies Pacing Guide Marking Period 1. How do you find the derivative of polynomial, exponential, logarithmic, and trigonometric functions? How does this process change if the functions are combined by various operations? 2. How could you use the graph of the derivative of a function f to graph f (and vice versa)? Students will be able to: 1. Determine the derivative of polynomial, exponential, logarithmic, and trigonometric functions 2. Determine the derivative of a product or quotient of functions 3. Determine the derivative of a composite function 4. Apply the concept of a derivative to solve real world problems dealing with position, velocity, and acceleration. 1. Calculus: Graphical, Numerical, Algebraic 2. Workbook associated with the above text. 3. AP Central (apcentral.collegeboard. com) 4. Past AP exams released from the Collegeboard 1. Teacher questioning 2. Class discussions 3. Homework checks 4. Classwork checks 20-25 days 1 5. Teacher created PowerPoint presentations 6. Cracking the AP Calculus AB & BC Exams. 22

Standards Instructional Activities, Methods, and Assignments Unit Summative Assessment(s) 4.1.B.1, 4.1.B.2, 4.1.B.4, 4.2.A.3, 4.2.C.1, 4.3.A.2, 4.3.B.1, 4.3.B.2, 4.3.B.4, 4.3.D.1, 4.3.D.2, 4.3.D.3, 4.5.A.1, 4.5.A.2, 4.5.A.3, 4.5.A.4, 4.5.A.5, 4.5.B.1, 4.5.B.2, 4.5.B.3, 4.5.B.4, 4.5.C.1, 4.5.C.2, 4.5.C.3, 4.5.C.4, 4.5.C.5, 4.5.C.6, 4.5.D.4, 4.5.D.5, 4.5.D.6, 4.5.E.1, 4.5.E.2, 4.5.E.3, 4.5.F.1, 4.5.F.2, 4.5.F.3, 4.5.F.4 1. AP test questions from previous exams 2. Notes 1. Vocabulary 2. Practice Problems: main white board, individual notes 1. Chapter tests 2. AP review packet containing sample AP problems from previous years exams 23

Unit 3: Applications of Derivatives Unit Question(s) Objectives Resources Formative Assessment Strategies Pacing Guide Marking Period 1. How do you find the derivative of polynomial, exponential, logarithmic, and trigonometric functions? How does this process change if the functions are combined by various operations? 3. How could you use the graph of the first and second derivatives of a function f to graph f? 4. How can the concept of a derivative affect a company s profit and cost? 5. How can a tangent line be used to approximate the value of a function near a given point? How can the error be reduced? 6. How can you determine the rate at which a ladder slides down a wall? Students will be able to: 1. Determine the derivative of polynomial, exponential, logarithmic, and trigonometric functions 2. Determine the derivative of a product or quotient of functions 3. Determine the derivative of a composite function 4. Apply the concept of a derivative to solve real world problems dealing with position, velocity, and acceleration. 5. Use the derivative to solve other real-world application problems 6. Apply the Mean Value Theorem for derivatives 1. Calculus: Graphical, Numerical, Algebraic 2. Workbook associated with the above text. 3. AP Central (apcentral.collegeboard. com) 4. Past AP exams released from the Collegeboard 5. Teacher created PowerPoint presentations 6. Cracking the AP Calculus AB & BC Exams. 1. Teacher questioning 2. Class discussions 3. Homework checks 4. Classwork checks 25 days 2 24

Standards Instructional Activities, Methods, and Assignments Unit Summative Assessment(s) 4.1.C.1, 4.2.A.1, 4.2.A.3, 4.2.A.4, 4.2.C.1, 4.2.D.2, 4.2.E.1, 4.2.E.2, 4.3.A.3, 4.3.B.1, 4.3.B.2, 4.3.B.4, 4.3.C.1, 4.3.D.1, 4.3.D.2, 4.3.D.3, 4.5.A.1, 4.5.A.2, 4.5.A.3, 4.5.A.4, 4.5.A.5, 4.5.B.1, 4.5.B.2, 4.5.B.3, 4.5.B.4, 4.5.C.1, 4.5.C.2, 4.5.C.3, 4.5.C.4, 4..5.C.5, 4.5.C.6, 4.5.D.1, 4.5.D.2, 4.5.D.3, 4.5.D.4, 4.5.D.6, 4.5.E.1, 4.5.E.2, 4.5.E.3, 4.5.F.1, 4.5.F.2, 4.5.F.3, 4.5.F.4 1. AP test questions from previous exams 2. Notes 3. Vocabulary 4. Practice Problems: main white board, individual notes 5. Graphical Analysis 1. Chapter tests 2. AP review packet containing sample AP problems from previous years exams 25

Unit 4: Antiderivatives Unit Question(s) Objectives Resources Formative Assessment Strategies Pacing Guide Marking Period 1. How do you find the antiderivative of polynomial, exponential, logarithmic, and trigonometric functions? How does this process change if the functions are combined by various operations? 2. How could you use the graph of the antiderivative of a function f to graph f? 3. Given three vertices of a triangle, how could you calculate the area of the triangle using only calculus? 4. A shaded region is revolved around both axes. Under what circumstances will the resulting volumes be the same Students will be able to: 1. Find anti-derivatives of functions analytically and graphically 2. Approximate the area below or between curves using rectangular and trapezoidal approximation methods. 3. Solve differential equations using separation of variables. 4. Find the volume of a solid of revolution. 5. Apply the Fundamental Theorem of calculus 6. Apply the Mean Value Theorem for Integrals. 7. Apply the concept of a derivative to solve real world problems dealing with position, velocity, and acceleration. 1. Calculus: Graphical, Numerical, Algebraic 2. Workbook associated with the above text. 3. AP Central (apcentral.collegeboard. com) 4. Past AP exams released from the Collegeboard 5. Teacher created PowerPoint presentations 6. Cracking the AP Calculus AB & BC Exams. 1. Teacher questioning 2. Class discussions 3. Homework checks 4. Classwork checks 50 days 2 and 3 8. Use the derivative to solve other real-world application problems 26

Standards Instructional Activities, Methods, and Assignments Unit Summative Assessment(s) 4.1.B.1, 4.1.B.4, 4.2.D.2, 4.2.E.2, 4.3.B.1, 4.3.B.2, 4.3.B.4, 4.3.C.1, 4.3.D.1, 4.3.D.2, 4.3.D.3, 4.5.A.1, 4.5.A.2, 4.5.A.3, 4.5.A.4, 4.5.A.5, 4.5.B.1, 4.5.B.2, 4.5.B.3, 4.5.B.4, 4.5.C.1, 4.5.C.2, 4.5.C.3, 4.5.C.4, 4..5.C.5, 4.5.C.6, 4.5.D.1, 4.5.D.2, 4.5.D.3, 4.5.D.4, 4.5.D.6, 4.5.E.1, 4.5.E.2, 4.5.E.3, 4.5.F.1, 4.5.F.2, 4.5.F.3, 4.5.F.4 1. AP test questions from previous exams 2. Notes 3. Vocabulary 4. Practice Problems: main white board, individual notes 1. Chapter tests 2. AP review packet containing sample AP problems from previous years exams 27