AP CALCULUS BCSyllabus

COURSE DESCRIPTION

AP Calculus BC is a full year course offered to students in grades 11-12. AP Calculus BC emphasizes the concepts of calculus graphically, algebraically, and with tables of values. The course teaches all topics associated with functions, graphs, limits, derivatives, integrals, and polynomial approximations as well as series as outlined in the AP Calculus Course Description.

COURSE MATERIALS

Major Text:Larson,Roland E., Robert P. Hostetler, and Bruce H. Edwards. Calculus. Sixth Edition. Houghton Mifflin Company, 1998.

Supplementary Materials and Additional Resources:Sallee, Tom, Judy Kysh, and Brian Hoey. College Preparatory Mathematics Calculus, First Edition: CPM Educational Program, 2003

apcentral.collegeboard.com

AP Calculus Course Description. The College Board.

AP Calculus Exam free-response questions from 1997 to present.

AP Calculus Exam multiple-choice questions from1993, 1997, 1998, and 2003.

Texas Instrument TI-83, TI-83 Plus, or TI-89 graphing calculator.

COURSE OUTLINE

Chapter 1: Limits (9 days) 1 test

• Find limits graphically and numerically • Evaluate limits analytically• Estimating limits from both graphs and tables• Continuity with one-sided limits and graphs• Intermediate Value Theorem• Infinite limits and asymptotic behavior• Limits at infinity and asymptotic behavior

• Students will work with limits graphically, numerically, analytically, and verbally.

o Graphically, students will have to determine that the graph approaches the same value from the left and the right.

o Numerically, students will describe the value of the limito Analytically, students will find the value of the limito Verbally, students will discuss both orally and written how the above

methods are related.

Chapter 2: Differentiation (16 days) 2 Quizzes, 1 Test

• Examining rates of change for different types of graphs• Definition of derivative• Power Rule and derivatives of a sum and difference• Derivative and Tangent Line (local linearity)• Instantaneous rate of change vs. average rate of change• Approximating rates of change from graphs and tables• Differentiability and continuity• Basic differentiation rules and rates of change• Product and Quotient Rules• Higher Order Derivatives• Chain Rule• Implicit Differentiation• Related Rates• Students will work with differentiation graphically, numerically, analytically, and

verbally. o Graphically, be determining two points on a line students can estimate the

derivative by calculating the slope between the two pointso Numerically, students will use their calculators to verify the derivative at

an exact point.o Analytically, students will understand and be able to use the definition of

derivative to calculate the value of the derivative at a pointo Verbally, students will be able to discuss the meaning of the derivative;

specifically that the derivative is the slope at an exact point

Chapter 3: Applications of Differentiation (17 days) 1 Quiz, 1 Test

• Analysis of functions• Increasing and decreasing functions and their derivatives• Examining the relationship graphically between a function and its derivatives• Provide written explanations of the relationship between graphs• First derivative test

• Concavity, points of inflection, and second derivative test• Extrema on an interval• Rolle’s Theorem and Mean Value Theorem• Word problems using AVT and MVT require students to express, in written form,

the differences and similarities between the two concepts• Curve Sketching and curve analysis• Interpreting first and second derivatives in context (including position, vel., acc.)• Provide written explanations of relationship between velocity and position

functions• Optimization problems• Inverse functions and implicit differentiation• Newton’s Method• Differentials • Slope fields and Euler’s Method• Students will work with differentiation graphically, numerically, analytically, and

verbally. For example, when learning optimization:o Graphically, students will plot various data points in order to visually

determine the approximate value at which the maximum occurso Numerically, students will calculate the volume of a given boxo Analytically, students will determine the maximum value of the boxo Verbally, students will discuss the use of the derivative in identifying

max/min. Chapter 4: Integration (13 days) 1 Test

• Anti-derivatives and indefinite integration• Area• Students will be able to provide written explanations of the relationship between

the anti-derivative and area of under a curve• Riemann sums: left, right, midpoint, trapezoids• Definite integrals• Average value of a function• Fundamental Theorem of Calculus• Word problems require students to use written explanations to verify outcomes• Integration by substitution• Numerical integration• Applications of integration• Integration as an accumulation function• Students will work with integration graphically, numerically, analytically, and

verbally.o Graphically, students will be required to draw rectangles and trapezoids to

estimate the area under a curve.

o Numerically, students will use their calculator to graph and shade the region under a curve, and finally develop the tool to evaluate a definite integral

o Analytically, students will be able to use integration techniques to solve various problems

o Verbally, students will discuss the meaning of the integral and will be able to explain the relationship between Riemann Sums and the value of an integral; i.e. was the estimate above or below the actual value of the integral

Chapter 5, Sections 1-5.7: Logarithmic and Exponential Functions (10 days) 1 Test

• Natural logarithm function and differentiation• Natural logarithmic function and integration• Inverse functions• Exponential functions differentiation and integration• Bases other than e and applications• Differential equations growth and decay • Separable differential equations

Chapter 5.8-5.9: Inverse Functions (6 days) 1 Quiz

• Inverse trigonometric functions and differentiation• Inverse trigonometric functions and integration

Chapter 6: Applications of Integration (12 days) 1 Test

• Area of region between two curves• Volumes by disc, washers, shells and cross sections• Arc Length • Students will work with specific applications of integration graphically,

numerically, analytically, and verbally.o Graphically, students will be able to use their calculators to sketch the two

curves, and thus shade the region between the curveso Numerically, students must be able to differentiate the different methods

of finding volume and must determine the most appropriate method to use.o Analytically, students will be able to use already learned integration

techniques to determine the exact area between two curves, as well as the volume of a rotated region.

o Verbally, students will discuss the various methods (discs, washers, etc.) used and the relationship between the methods.

Midterm (Review 2 days)

Chapter 7: Integration Techniques (10 days) 1 Test

• Basic integration rules• Integration by parts• Partial fractions• Indeterminate forms and L’Hopital’s rule• Improper integrals

Chapter 9: Parametric Equations and Polar Coordinates (10 days ) 1 Test

• Analysis of plane curves and parametric equations• Derivative of parametric functions• Polar coordinates and polar graphs• Derivatives of polar functions• Area in polar coordinates• Analysis and derivatives of vectors• Students will work with polar functions graphically, numerically, analytically, and

verbally.o Graphically, students must be able to graph polar curves o Numerically and analytically, students will be able to find the area

between two polar curves.o Verbally, students must be able to interpret results involving motion of a

particle, given information about one or more of the functions.

Chapter 8: Infinite Series ( 20 days) 2 tests

• Sequences and partial sums• Series and convergence• Geometric Series• Power Series• Representation of functions using power series• Taylor and MacLaurin Series

• Using the calculator to view and examine the relationship between ex, sin x, cos x, and 1/(1-x) and their respective MacLaurin polynomials• Derivatives and anti-derivatives of Taylor and MacLaurin Series

• Nth term test• Relationship between series and improper integrals• Integral and p series test• Direct and limit comparison tests• Alternating series test

• Ratio test• Lagrange error for Taylor polynomials• Students will work with Taylor series graphically, numerically, analytically, and

verbally.o Numerically, given a function, students must be able to write terms of a

series about 0=xo Analytically, students will be able to estimate the value of an integral

using terms of a serieso Verbally, students must be to explain the differences between estimated

values and actual values of an integral involving error bounds

Logistics (5 days) 1 Quiz

• Modeling the spread of disease using the graphing calculator and randomly generated numbers.

• Analyzing and solving logistic differential equations• Checking to see how the experimental model reflects the theoretical differential

equation

CalculatorStudents will use a graphing calculator for the following:

• Explore functions and tangent lines• Explore and interpret how transformations, compositions, and inverses affect a

graph and its domain and range• Input data and find a graph of best fit• Tool to develop derivative functions• Find the derivative of a function• Graph a function, its first, and second derivatives• Graph functions and verify extrema found algebraically • Verify optimization results• Develop integration tools

AP Exam Review (16 days)

• 2 full multiple-choice tests (1998 for HW, 2003 for a grade done in-class)• Various open-ended problems

After the AP Exam:

Chapter 7: Techniques of Integration ( 14 days) 1 Quiz and 1 Test

• Partial Fractions (Repeated Linear Factors and Quadratic Factors)• Trigonometric Substitution• Trigonometric Integrals• Completing the square and trigonometric substitution

Chapter 5: Hyperbolic Functions (5 days) 1 Test

• Hyperbolic functions • Derivatives and integrals of hyperbolic functions

Second Semester Exam (Review 2 days)

EHT Special Education Interventions

ABA strategies and Interventions for both behavior and Instruction (for autistic programs

only)

Picture Exchange Communication System (for non-verbal students)

Ipad with Proloquo2Go (for non-verbal students)

Go Talk device (for non-verbal students)

Mike and Molly Program for language skills and reading (mostly for very low

functioning students)

Edmark Reading Program (mostly for very low functioning students)

Reading Milestones (for deaf students)

Menu Math (mostly for very low functioning students)

Phonics First

Wilson Reading (Miller and Alder only)

Read 180 (High School only)

Edmentum Math Labs (High School only)

IXL Reading & Labs (High School only)

Use of paraprofessionals and 1:1 nurses to assist students

Weighted Vests (for calming agitated students)

Specialized sensory equipment (for student who need sensory stimulation or students

who have tactile or other sensory aversions)

Learning Ally (audio version for textbooks and other published materials) – Also

available for 504 students

Think Through Math

Apex Online Learning – Bridge students only

Accommodations and Modifications

Use of specialized equipment such as beeping balls, text to speech and speech to text

software, special seats or desks

Use of hands-on materials for problem solving

Shopping trips on a regular basis with specific goals and objectives

Cooking to teach math, measurement and functional reading and life skills

Visual supports

Extended time to complete tests and assignments

Graphic Organizers

Mnemonic tricks to improve memory

Reducing work load

Adjusting accountability for standards by focusing only on essential standards

Practice materials unique to particular students in all subjects (ie workbooks for spelling

at grades where it no longer is taught or for basic foundational math skills, telling time,

etc.)

Use of IPads or laptops for students with motor issues that make writing difficult

Study guides

Special paper for writing tasks

Use of sand, sand paper and shaving cream for learning letters

Individual behavior plans

Use of special grips or other fine motor accommodation equipment

Use agenda book for assignments

Provide a posted daily schedule

Provide breaks during the day

Use of classroom behavior management system

Use of rewards for students

Use of “quiet room” for students struggling to maintain self-control

Use of tangible rewards (certificates, small toys, etc. per behavior plan)

Use prompts and model directions

Use task analysis to break down activities and lessons into each individual step needed to

complete the task

Use concrete examples to teach concepts

Have student repeat/rephrase written directions

Provide multi-sensory, hands-on materials for instruction

Break all instruction and verbal directions into small segments

Focus on student’s goals and objectives from IEP instead of pacing guides

Tests read to student

Modify all fine motor tasks for example: (fat crayons, pencil grip, adaptive scissors)

Functional or practical emphasis

Differentiated Instruction Resources for Gifted & Struggling Students

“Differentiating the Lesson” is located in Big Ideas on-line resources and is available for

most sections.

“Additional Topics” can be selected from Big Ideas on-line resources to find a wide

variety of challenging problems for students.

See Big Ideas MATH Pyramid of Tiered Interventions for additional resoruces.

hmhco.com • 800.225.5425

Through print and digital resources, the Big Ideas Math

program completely supports the 3-tier model. Using

research-based instructional strategies, teachers can reach,

challenge, and motivate students with relevant, high-quality

instruction targeted to their individual needs.

Learn more at hmhco.com/bigideasmath

HMHCo

BigIdeasMath BigIdeasLearningBig Ideas Math® and Big Ideas Learning® are registered trademarks of Larson Texts, Inc. Houghton Mifflin Harcourt™ is a trademark of

Houghton Mifflin Harcourt. © Houghton Mifflin Harcourt. All rights reserved. Printed in the U.S.A. 02/15 MS132368

Distributed exclusively byHoughton Mifflin Harcourt

Tier 3: Customized Learning Intervention

Tier 2: Strategic Intervention

Tier 1: Daily Intervention• Record and Practice Journal• Fair Game Review• Graphic Organizers• Vocabulary Support

• Mini Assessments• Game Closet• Lesson Tutorials• On Your Own

• Activities

• Intensive Intervention Lessons

• Lesson Tutorials• Basic Skills Handbook• Skills Review Handbook• Differentiated Instruction• Game Closet

ELL Resources for Students within Big Ideas Math Program

Big Ideas Math Student Editions are available online in Spanish

Letters to Parents are available in the Resources by Chapter book to assist in guiding parents

through each chapter and offer helpful suggestions they can use to demonstrate mathematical

concepts for their child in daily activities. These letters are editable so teachers can customize

them.

Vocabulary Flash Cards

Student Dynamic eBook Audio has the option to be read in English or Spanish

Multi-Language Glossary for new Math vocabulary is available in 14 different languages.

o Audio version is available in English or Spanish.

Game Closet can be accessed in English or Spanish, while also allowing for all students to play

and understand these educational games.

ELL Notes included in Teacher Edition to help teachers overcome obstacles.

Record & Practice Journal available in Spanish.

Student Journal available in Spanish.

Chapter Reviews available in English and Spanish.

Mathematics ELL Resources

TODOS: Mathematics for ALL – Excellence and Equity in Mathematics

o http://www.todos-math.org/

o TODOS: Mathematics for ALL is an international professional organization that

advocates for equity and excellence in mathematics education for ALL students - in

particular, Latina/o students. As articulated in the mission and goals, TODOS advances

educators' knowledge, develops and supports education leaders, generates and

disseminates knowledge, informs the public, influences educational policies, and

informs families about education policies and learning strategies. All of these goals

ultimately result in providing access to high quality and rigorous mathematics

for ALL students.

FABRIC – A Learning Paradigm for ELLs (NJDOE resource)

o http://www.state.nj.us/education/bilingual/pd/fabric/

o NJDOE ELL Resource: This paradigm allows teachers and administrators to provide

diverse groups of ELLs with access to classroom content while they acclimate to an

English learning environment. The six learning threads of the FABRIC paradigm provide a

structure that teachers can use to address the needs of ELLs. Each section contains

research-based recommendations, a classroom example, and application questions.

FABRIC can be utilized during sheltered instruction training, professional learning

community meetings, pre-service teacher education, etc.

For the given level of English language proficiency, with support, English language learners can:

Beginning

Point to stated pictures, words, phrasesFollow one-step oral directionsMatch oral statements to objects, figures or illustrations

Sort pictures, objects according to oral instructionsFollow two-step oral directions Match information from oral descriptions to objects, illustrations

Locate, select, order information from oral descriptionsFollow multi-step oral directionsCategorize or sequence oral information using pictures, objects

Compare/contrast functions, relationships from oral informationAnalyze and apply oral informationIdentify cause and effect from oral discourse

Draw conclusions from oral informationConstruct models based on oral discourseMake connections from oral discourse

Name objects, people, picturesAnswer WH- (who, what, when, where, which) questions

Ask WH- questionsDescribe pictures, events, objects, peopleRestate facts

Formulate hypotheses, make predictionsDescribe processes, proceduresRetell stories or events

Discuss stories, issues, conceptsGive speeches, oral reportsOffer creative solutions to issues, problems

Engage in debatesExplain phenomena, give examples and justify responsesExpress and defend points of view

Match icons and symbols to words, phrases or environmental print Identify concepts about print and text features

Locate and classify informationIdentify facts and explicit messagesSelect language patterns associated with facts

Sequence pictures, events, processesIdentify main ideasUse context clues to determine meaning of words

Interpret information or dataFind details that support main ideasIdentify word families, figures of speech

Conduct research to glean information from multiple sourcesDraw conclusions from explicit and implicit text

Label objects, pictures, diagramsDraw in response to a promptProduce icons, symbols, words, phrases to convey messages

Make listsProduce drawings, phrases, short sentences, notesGive information requested from oral or written directions

Produce bare-bones expository or narrative textsCompare/contrast informationDescribe events, people, processes, procedures

Summarize information from graphics or notesEdit and revise writingCreate original ideas or detailed responses

Apply information to new contextsReact to multiple genres and discoursesAuthor multiple forms/genres of writing

considered in using this information.

En cada nivel de capacidad en el lenguaje inglés, con apoyo, un estudiante de inglés puede hacer lo siguiente:

Señalar dibujos, palabras o frases indicadosSeguir instrucciones orales de un pasoEmparejar declaraciones orales con objetos, figuras o ilustraciones

Clasificar dibujos u objetos siguiendo las instrucciones verbalesSeguir instrucciones verbales de dos pasosEmparejar declaraciones verbales con objetos, figuras o ilustraciones

Localizar, seleccionar y orde-nar información que provi-ene de descripciones oralesSeguir instrucciones verbales de paso múltiplesClasificar o secuenciarinformación oral usando dibujos u objetos

Comparar y contrastar funciones y relaciones de acuerdo a información oralAnalizar y aplicar información oralIdentificar causa y efecto en discurso oral

Sacar una conclusión de información oralConstruir modelos basados en discurso oralHacer conexiones en información oral

Nombrar objetos, personas y dibujosContestar preguntas (quién, qué, cuándo, dónde, cuál)

PreguntarDescribir dibujos, eventos, objetos y personasReformular y decir hechos

Formular hipótesis y hacer prediccionesDescribir procesosRecontar cuentos o eventos

Discutir cuentos, cuestiones, y conceptosHacer presentaciones oralesOfrecer soluciones creativas a cuestiones o problemas

Participar en debatesExplicar fenómenos, dar ejemplos y justificar respuestasExpresar y defender puntos de vista

Emparejar símbolos y dibujos con palabras, frases o letras en la escritura en el medioambienteIdentificar conceptos de la organización de letras y elementos de textos

Localizar y clasificar informaciónIdentificar hechos y mensajes directosSeleccionar patrones de lenguaje asociados con hechos

Secuenciar dibujos, eventos y procesosIdentificar ideas principalesUsar pistas del contexto para determinar el significado de palabras

Interpretar información o datosEncontrar detalles que apoyan las ideas principalesIdentificar figuras retóricas y relaciones entre palabras

Realizar investigaciones para reunir información de fuentes múltiplesSacar una conclusión de texto explícito e implícito

Etiquetar objetos, dibujos, diagramasDibujar respuestas a instruccionesProducir íconos, símbolos, palabras y frases para comunicar un mensaje

Hacer listasProducir dibujos, frases, oraciones cortas y apuntesDar información pedida por instrucciones orales o escritas

Producir textos básicos de estilo narrativo o informativoComparar y contrastar informaciónDescribir eventos, personas, procesos

Resumir información de representaciones gráficas o apuntesCorregir y revisar escrituraCrear ideas originales o respuestas detalladas

Aplicar información a contextos nuevosReaccionar a múltiples géneros y discursosRedactar varias formas/géneros de composiciones

Translated by (Traducido por) Elizabeth J. Hartung, Monona Grove, WI; revised by (revisado por) Andrea Cammilleri, Mariana Castro and Stephanie Herrera, WIDA, Wisconsin Center for Education Research

Esto se debe considerar al usar ésta información.