District OverviewThe mathematics curriculum provides sequential and comprehensive K-12 instruction in a collaborative, student-centered learning environment that fosters critical thinking, creativity, skillful problem-solving, and effective communication in order to enable all students to adapt to an ever-changing, global society and increase college and career readiness. An emphasis has been placed on conceptual understanding, higher-order thinking, and problem solving skills to prepare students for 21st century careers. This is further embedded through the integrated use of technology into daily lessons. Instruction focuses on meaningful development of mathematical ideas at each grade level where students are given the opportunity to explore, engage, and take risks with content as they build and expand their knowledge and understanding of mathematics. Students will experience mathematics as a coherent and useful subject within the context of real-life situations. In all, the curriculum aims to reach high standards while encouraging curiosity and building confidence in a collaborative atmosphere.

AP Calculus BC DescriptionThis is a second year calculus course comparable to a single semester of college-level calculus and is taught as a follow-up to AP Calculus AB. The course covers single variable differential and integral calculus and is broken up into four big ideas: Limits, Derivatives, Integrals / Fundamental Theorem of Calculus, and Series. Students learn about these big ideas conceptually and then learn how to use these big ideas to solve problems. This course closely follows the College Board’s course description for AP Calculus BC. Multiple representations are used to study calculus problems including tables, graphs, and symbolic representation of functions. The following mathematical practices are used throughout the year as tools for learning about calculus and analyzing problems MPAC1: Reasoning with definitions and theorems, MPAC2: Connecting concepts, MPAC3: Implementing algebraic/computational processes, MPAC4: Connecting multiple representations, MPAC5: Building notational fluency, MPAC6: Communicating. Students are required to work problems with and without the use of a graphing calculator. This is a rigorous course that demands a strong foundation in algebra, geometry, pre-calculus, trigonometry, and AP Calculus AB topics. Students should have a sincere interest in mathematics and the willingness to consistently study the content outside of class.

Units: Unit 0 – Foundation Work Unit 1 – Review of Selected AB Topics 1 Unit 2 – Review of Selected AB Topics 2 Unit 3 – Parametric Functions and Vectors Unit 4 – Volume by Shells, Arc Length, and Polar Coordinates Unit 5 – Differential Equations Unit 6 – Additional Integration Techniques Unit 7 – Series Unit 8 – Review for AP / Final Exam

Subject: AP Calculus BC Grade: 10/11/12 Suggested Timeline: 8 – 42 minute periodsUnit Title: Unit 0 – Foundation WorkUnit Overview/Essential Understanding: In this unit, students warm up by working on constructing a “Mystery Curve” which must meet a list of criteria. This list includes several ordered pairs that the function must pass through, as well as, several extrema. The function must be continuous and differentiable. This opening unit gives students the chance to revisit many calculus topics learned in the previous year and help firm up the foundation for the study of AP Calculus BC.Unit Objectives: Upon completion of this unit, students will demonstrate an understanding of the following topics and/or be able to do the following:

design a piece-wise function that meets a set of criteria (criteria includes calculus ideas, such as continuity and differentiability) Focus Standards Addressed in this Unit:

LO 1.1C – Determine limits of functions LO 1.2A – Analyze functions for intervals of continuity or points of discontinuity LO 1.2B – Determine the applicability of important calculus theorems using continuity LO 2.1C – Calculate derivatives LO 2.2B – Recognize the connection between differentiability and continuity

Important Standards Addressed in this Unit:N/AMisconceptions:N/AConcepts/Content:

Continuity limit derivative piecewise functions differentiability

Competencies/Skills: construct graphs using technology use one-sided limits in an

appropriate manner to force continuity and differentiability

Description of Activities: “Mystery Curve” activity from the book – A

Watched Cup Never Cools

Assessments: informal questioning problem set

Interdisciplinary Connections:N/A

Additional Resources: Foerster textbook (Kendall Hunt) TI-89 graphing calculator, desmos.com College Board’s AP Calculus AB course description

Subject: AP Calculus BC Grade: 10/11/12 Suggested Timeline: 26 – 42 minute periods Unit Title: Unit 1 – Review of Selected AB Topics 1Unit Overview/Essential Understanding: In this unit, students review selected topics from their previous year of calculus. Students are surveyed at the end of AP Calculus AB to determine the topics that presented students with the most difficulty. The list of particular topics is used to build the content for the unit. All of the selected topics are reintroduced in a slow, methodical manner to build a firm foundation. However, after introduction, these topics are studied and mastered by working advanced problems. Unit Objectives: Upon completion of this unit, students will demonstrate an understanding of the following topics and/or be able to do the following:

successfully solve problems involving selected review topics Focus Standards Addressed in this Unit: N/AImportant Standards Addressed in this Unit:

various standards as outlined in the College Board’s course description for AP Calculus ABMisconceptions:N/AConcepts/Content:

limits derivatives integrals

Competencies/Skills: evaluating limits, derivatives, and

integrals in straightforward and applied problems

Description of Activities: teacher led direct instruction question & answer check/review HW warm-up / exit ticket questions quiz problem set end-of-unit test

Assessments: informal questioning quiz problem set end-of-unit test

Interdisciplinary Connections:N/A

Additional Resources: Foerster textbook (Kendall Hunt) – various sections covered in

previous year textbook supplemental materials (explorations) TI-89 graphing calculator, desmos.com

selected MC/FR items from released AP exams College Board’s AP Calculus AB course description

Subject: AP Calculus BC Grade: 10/11/12 Suggested Timeline: 26 – 42 minute periods

Unit Title: Unit 2 – Review of Selected AB Topics 2Unit Overview/Essential Understanding: In this unit, students review selected topics from their previous year of calculus. This is a continuation of Unit 1 and is broken up into two units in order keep the quantity of material manageable. Students are surveyed at the end of AP Calculus AB to determine the topics that presented students with the most difficulty. The list of particular topics is used to build the content for the unit. All of the selected topics are reintroduced in a slow, methodical manner to build a firm foundation. However, after introduction, these topics are studied and mastered by working advanced problems. Unit Objectives: Upon completion of this unit, students will demonstrate an understanding of the following topics and/or be able to do the following:

successfully solve problems involving selected review topics Focus Standards Addressed in this Unit: N/AImportant Standards Addressed in this Unit:

various standards as outlined in the College Board’s course description for AP Calculus ABMisconceptions:N/AConcepts/Content:

limits derivatives integrals

Competencies/Skills: evaluating limits, derivatives, and

integrals in straightforward and applied problems

Description of Activities: teacher led direct instruction question & answer check/review HW warm-up / exit ticket questions quiz problem set end-of-unit test

Assessments: informal questioning quiz problem set end-of-unit test

Interdisciplinary Connections:N/A

Additional Resources: Foerster textbook (Kendall Hunt) – various sections covered in

previous year

textbook supplemental materials (explorations) TI-89 graphing calculator, desmos.com selected MC/FR items from released AP exams College Board’s AP Calculus AB course description

Subject: AP Calculus BC Grade: 10/11/12 Suggested Timeline: 26 – 42 minute periods

Unit Title:

Unit 3 – Parametric Functions and Vectors

Unit Overview/Essential Understanding: In this unit, parametric functions are introduced and analyzed. Parametric functions allow for the expansion of motion in one dimension to motion in two dimensions. An introduction into vectors is also covered as a way to compactly hold information found through the analysis of parametric functions. Several notations used for representing vectors are covered, as well as, some basic vector operations.Unit Objectives: Upon completion of this unit, students will demonstrate an understanding of the following topics and/or be able to do the following:

find position, velocity, and acceleration of a particle moving in two dimensions use the parametric chain rule to determine the slope of a particle’s path find the rate of increase of a particle’s slope (find the 2nd derivative) use unit vectors to represent a particle’s position, velocity, and acceleration as a function of time determine whether an object is speeding up or slowing down by using vector projections

Focus Standards Addressed in this Unit: EK 2.1C7 – Methods for calculating derivatives of real-valued functions can be extended to vector-valued functions, parametric functions,

and functions in polar coordinates EK 2.3C4 – Derivatives can be used to determine velocity, speed, and acceleration for a particle moving along curves given by parametric

or vector-valued functions EK 3.4C2 – The definite integral can be used to determine displacement, distance, and position of a particle moving along a curve given

by parametric or vector-valued functions

Important Standards Addressed in this Unit:N/AMisconceptions:

Care should be taken when students work with the various derivatives in this unit, as students can easily misinterpret the meaning of a

particular derivative. For example, calculating dydx for a set of parametric functions yields the slope of the curve that represents the

particle’s path. Often students believe that dydx represent the speed of the particle, which is not true.

Concepts/Content: parametric equations derivatives with respect to time vectors

Competencies/Skills: calculate derivatives with respect

to time determine the motion of a object represent a particle’s motion with

vectors

Description of Activities: teacher led direct instruction question & answer check/review HW warm-up / exit ticket questions quiz

problem set end-of-unit test specific example of activity “Modeling the

motion of a vinyl record player”Assessments:

informal questioning quiz problem set end-of-unit test

Interdisciplinary Connections:N/A

Additional Resources: Foerster textbook (Kendall Hunt) – sec 4-7, 10-6 textbook supplemental materials (explorations) TI-89 graphing calculator, desmos.com selected MC/FR items from released AP exams College Board’s AP Calculus AB course description

Subject: AP Calculus BC Grade: 10/11/12 Suggested Timeline: 26 – 42 minute periods Unit Title: Unit 4 – Volume by Shells, Arc Length, and Polar Coordinates

Unit Overview/Essential Understanding: In this unit, students are introduced to an alternative method used to calculate volumes by revolution – volume by shells. Then students study how to calculate arc length for functions defined with Cartesian coordinates. The concept of arc length is extended to polar coordinates, as students first learn how to plot points in the polar plane and then learn to calculate area of a region bound by a polar curve. The relationship between polar coordinates and Cartesian coordinates is emphasized. Students learn to calculate arc length with curves defined by polar functions and finally revisit parametric equations to apply the concept of arc length.Unit Objectives: Upon completion of this unit, students will demonstrate an understanding of the following topics and/or be able to do the following:

calculate volumes by revolution with disc/washer method and by shells calculate arc length for functions defined in the Cartesian coordinate plane calculate arc length for functions defined in the polar plane calculate arc length for functions defined parametrically calculate area bounded by a polar curve transform to polar coordinates from Cartesian coordinates and vice versa

Focus Standards Addressed in this Unit: EK 2.1C7 – Methods for calculating derivatives of real-valued functions can be extended to vector-valued functions, parametric functions,

and functions in polar coordinates EK 2.2A4 – For a curve given by a polar equation r=f (θ), derivatives of r, x, and y with respect to θ and first and second derivatives of y

with respect to x can provide information about the curve EK 3.4C2 – The definite integral can be used to determine displacement, distance, and position of a particle moving along a curve given

by parametric or vector-valued functions EK 3.4D1 – Areas of certain regions in the plane can be calculated with definite integrals. Area bounded by polar curves can be

calculated with definite integrals EK 3.4D3 – The length of a planar curve defined by a function or by a parametrically defined curve can be calculated using a definite

integralImportant Standards Addressed in this Unit:N/AMisconceptions:N/A

Concepts/Content: volume by shells polar coordinates

Competencies/Skills: calculating arc length in a variety

of coordinate systems, calculating

Description of Activities: question & answer check/review HW

area arc length

area with polar equations warm-up / exit ticket questions quiz problem set end-of-unit test specific example of activity “Holiday Polar

Coordinates Plot”Assessments:

informal questioning quiz problem set end-of-unit test

Interdisciplinary Connections:N/A

Additional Resources: Foerster textbook (Kendall Hunt) – sec 8-4, 8-5, 8-7 textbook supplemental materials (explorations) TI-89 graphing calculator, desmos.com selected MC/FR items from released AP exams College Board’s AP Calculus AB course description

Subject: AP Calculus BC Grade: 10/11/12 Suggested Timeline: 26 – 42 minute periods

Unit Title:

Unit 5 – Differential Equations Unit Overview/Essential Understanding: In this unit, students revisit a topic studied last year, but with greater depth – differential equations. Graphical, numerical, and algebraic methods are used to solve differential equations. Euler’s method is introduced as a way to approximate solutions to differential equations. And, students learn about the logistic function and its use in modeling systems where there exists a carrying capacity.

Unit Objectives: Upon completion of this unit, students will demonstrate an understanding of the following topics and/or be able to do the following:

use slope fields to analyze differential equations solve differential equations algebraically using separation of variables give approximate solutions to differential equations using Euler’s method work differential equation problems involving the logistic function

Focus Standards Addressed in this Unit: EK 2.3F2 – For differential equations, Euler’s method provides a procedure for approximating a solution or a point on a solution curve EK 3.5B2 – The model for logistic growth that arises from the statement “The rate of change of a quantity is jointly proportional to the

size of the quantity and the difference between the quantity and the carrying capacity” is dydt

=ky (a− y )

Important Standards Addressed in this Unit:N/AMisconceptions:N/AConcepts/Content:

differential equations slope fields separation of variables Euler’s method the logistic function

Competencies/Skills: analyze differential equations

using graphical, numerical, and algebraic approaches

approximate solutions to differential equations using Euler’s method

solve problems involving the logistic function

Description of Activities: question & answer check/review HW warm-up / exit ticket questions quiz problem set end-of-unit test

Assessments: informal questioning quiz problem set

end-of-unit test Interdisciplinary Connections:

biology connection – systems and their carrying capacityAdditional Resources:

Foerster textbook (Kendall Hunt) – sec 7-2, 7-3, 7-4, 7-5, 7-6 textbook supplemental materials (explorations) TI-89 graphing calculator, desmos.com selected MC/FR items from released AP exams College Board’s AP Calculus AB course description

Subject: AP Calculus BC Grade: 10/11/12 Suggested Timeline: 26 – 42 minute periods Unit Title: Unit 6 – Additional Integration TechniquesUnit Overview/Essential Understanding:

In this unit, students build their integration “toolbox” by adding the following additional integration techniques: integration by parts, rapid-repeated integration by parts, and integration with partial fractions. Students also learn about improper integrals, which they will use during their study of power series.Unit Objectives: Upon completion of this unit, students will demonstrate an understanding of the following topics and/or be able to do the following:

apply integration by parts to products apply rapid-repeated integration by parts apply integration by partial fractions where appropriate identify when an integral is considered improper determine whether an improper integral converges or diverges

Focus Standards Addressed in this Unit: EK 3.2D1 – An improper integral is an integral that has one or both limits infinite or has an integrand that is unbounded in the interval of

integration EK 3.2D2 – Improper integrals can be determined using limits of definite integrals EK 3.3B5 – Techniques for finding antiderivatives include algebraic manipulation such as long division and completing the square,

substitution by variables, integration by parts, and nonrepeating linear partial fractionsImportant Standards Addressed in this Unit:N/AMisconceptions:

Students are comfortable working a definite integration problem without paying too much attention to the nature of the function over the given limits of integration. With the introduction of improper integrals, students need to be trained to check and see if there is any unbounded behavior of the integrand within the interval covered by the limits of integration.

Concepts/Content: Improper integrals integration by parts integration with partial fractions

Competencies/Skills: work with improper integrals use appropriate integration

techniques (i.e. identify when to use integration by parts and when to use integration with partial fractions)

Description of Activities: question & answer check/review HW warm-up / exit ticket questions quiz problem set end-of-unit test

Assessments: informal questioning quiz problem set end-of-unit test

Interdisciplinary Connections:N/A

Additional Resources: Foerster textbook (Kendall Hunt) – sec 9-2, 9-3, 9-7, 9-10 textbook supplemental materials (explorations) TI-89 graphing calculator, desmos.com selected MC/FR items from released AP exams College Board’s AP Calculus AB course description

Subject: AP Calculus BC Grade: 10/11/12 Suggested Timeline: 26 – 42 minute periods

Unit Title: Unit 7 – Series

Unit Overview/Essential Understanding: In this unit, students learn how to model functions with power series. This is a powerful tool that allows students to gain some insight into how calculators and computers make computations involving transcendental functions. Students first learn the fundamentals of sequences and series and how to represent series using sigma notation. Then students learn how to apply calculus to these series.Unit Objectives: Upon completion of this unit, students will demonstrate an understanding of the following topics and/or be able to do the following:

represent an infinite series with sigma notation calculate the partial sum of a series determine whether a series can be represented by a geometric series determine if a geometric series converges or diverges and if the series converges, find the precise value to which it converges use the process of equating derivatives to find power series for any function know and use the general formula for a Taylor series and Maclaurin series

memorize and use the following power series: ex , sin (x ) ,cos ( x ) , 11−x

for a given power series, find the open interval of convergence by using the ratio technique use a set of tests to determine whether a series of constants converges or diverges determine and control the error induced by using a series to approximate a function

Focus Standards Addressed in this Unit: EK 4.1A1 – The nth partial sum is defined as the sum of the first n terms of a sequence EK 4.1A2 – An infinite series of numbers converges to a real number S (or has sum S), if and only if the limit of its sequence of partial

sums exists and equals S EK 4.1A3 – Common series of numbers include geometric series, the harmonic series, and p-series EK 4.1A4 – A series may be absolutely convergent, conditionally convergent, or divergent EK 4.1A5 – If a series converges absolutely, then it converges EK 4.1A6 – In addition to examining the limit of the sequence of partial sums of the series, methods for determining whether a series of

numbers converges or diverges are the nth term test, the comparison test, the limit comparison test, the integral test, the ratio test, and the alternating series test

EK4.1B1 – If a is a real number and r is a real number such that |r|<1, then the geometric series ∑n=0

∞

arn= a1−r

EK 4.1B2 – If an alternating series converges by the alternating series test, then the alternating series error bound can be used to estimate how close a partial sum is to the value of the infinite series

EK 4.1B3 – If a series converges absolutely, then any series obtained from it by regrouping or rearranging the terms has the same value

EK 4.2A1 – The coefficient of the nth-degree term in a Taylor polynomial centered at x=a for the function f is f( n)(a)n !

EK 4.2A2 – Taylor polynomials for a function f centered at x=a can be used to approximate function values of f near x=a

EK 4.2A3 – In many cases, as the degree of a Taylor polynomial increases, the nth-degree polynomial will converge to the original function over some interval

EK 4.2A4 – The Lagrange error bound can be used to bound the error of a Taylor polynomial approximation to a function EK 4.2A5 – In some situations where the signs of a Taylor polynomial are alternating, the alternating series error bound can be used to

bound the error of a Taylor polynomial approximation to the function

EK 4.2B1 – A power series is a series of the form ∑n=0

∞

an(x−r )n where n is a non-negative integer, {an } is a sequence of real numbers, and r

is a real number EK 4.2B2 – The Maclaurin series for sin ( x ) ,cos (x ) ,∧ex provide the foundation for constructing the Maclaurin series for other functions

EK 4.2B3 – The Maclaurin series for 11−x is a geometric series

EK 4.2B4 – A Taylor polynomial for f (x) is a partial sum of the Taylor series for f (x) EK 4.2B5 – A power series for a given function can be derived by various methods (e.g., algebraic processes, substitutions, using

properties of geometric series, and operations on known series such as term-by-term integration or term-by-term differentiation) EK 4.2C1 – If a power series converges, it either converges at a single point or has an interval of convergence EK 4.2C2 – The ratio test can be used to determine the radius of convergence of a power series EK 4.2C3 – If a power series has a positive radius of convergence, then the power series is the Taylor series of the function to which it

converges over the open interval EK 4.2C4 – The radius of convergence of a power series obtained by term-by-term differentiation or term-by-term integration is the

same as the radius of convergence of the original power seriesImportant Standards Addressed in this Unit:N/AMisconceptions:N/AConcepts/Content:

sigma notation power series interval of convergence radius of convergence divergence geometric series Maclaurin series Taylor series partial sums Taylor polynomial

Competencies/Skills: expressing power series with

sigma notation finding power series for functions determining the interval of

convergence using power series to model

functions finding the error induced by using

a power series for modeling

Description of Activities: question & answer check/review HW warm-up / exit ticket questions quiz problem set end-of-unit test

error bound Assessments:

informal questioning quiz problem set end-of-unit test direct application to computer science – power series are used

to model transcendental functions in technology and looping mechanisms in programming are modeled with finite series

Additional Resources: Foerster textbook (Kendall Hunt) – sec 12-2, 12-3, 12-4, 12-5,

12-6, 12-7, 12-8 textbook supplemental materials (explorations) TI-89 graphing calculator, desmos.com selected MC/FR items from released AP exams College Board’s AP Calculus AB course description

Subject: AP Calculus BC Grade: 10/11/12 Suggested Timeline: 26 – 42 minute periods

Unit Title: Unit 12 – Review for AP / Final Exam

Unit Overview/Essential Understanding: In this unit, students work on reviewing all of the material learned during the year. The review materials focus on publically released AP Exams, namely the 2012, 2008, 2003, and 1998 exams. Special attention is also paid to recent free response exam questions, which are released 48 hours after the previous year’s exam is administered.Unit Objectives: Upon completion of this unit, students will demonstrate an understanding of the following topics and/or be able to do the following:

students will demonstrate mastery in all topics outlined in the College Board’s course description for AP Calculus BC through working many multiple choice and free response items

Focus Standards Addressed in this Unit: No new standards are introduced during this unit. All focus standards addressed in previous units will be reviewed as students work

through the practice exams.Important Standards Addressed in this Unit:N/AMisconceptions:N/AConcepts/Content:

AP Calculus BC topics as outlined in the College Board’s course description for AP Calculus

Competencies/Skills: Work problems involving limits,

derivatives, integrals, and series

Description of Activities: assignment, review, and discussion of previously

released exam questions

Assessments: informal questioning formula / theorem quiz final exam

Interdisciplinary Connections: N/A

Additional Resources: Foerster textbook (Kendall Hunt) – sections: all relevant

sections textbook supplemental materials (explorations) TI-89 graphing calculator, desmos.com selected MC/FR items from released AP exams College Board’s AP Calculus AB course description