COURSE TITLE

Advanced Placement Calculus

LENGTH

Full Year

DEPARTMENT

STEM Department

SCHOOL

Rutherford High School

Primary Content

Mathematics

 Initial Board of Education Approval Date (Born on): 5/13/2024

Revisions:  

Embedded Content

 Career Readiness, Life Literacies and Key Skills

 Initial Board of Education Approval Date (Born on): 5/13/2024

Computer Science and Design Thinking

 Initial Board of Education Approval Date (Born on): 5/13/2024

Advanced Placement Calculus

I.        Introduction/Overview/Philosophy

AP Calculus consists of a full high school academic year that is comparable to calculus courses in colleges and universities.  Upon completing Honors Precalculus as juniors, the students who take this course will continue their study of differential and integral calculus and be prepared to take the College Entrance Examination Board’s Advanced Placement Calculus Examination  (AB version) in the spring of their senior year.  

This course is concerned with developing the students’ understanding of the concepts of calculus and providing experience with its methods and applications.  It provides a multi-representational approach to calculus, with concepts, results, and problems being expressed geometrically, numerically, analytically, and verbally.  The connections among these methods of representation are also emphasized.

II.        Objectives

Course Outline:

1. Functions, Graphs, and Limits

1. Analysis of Graphs

1. Produce graphs of functions with and without the use of technology.
2. Relate geometric and analytic information from graphs.
3. Use calculus to predict and to explain observed local and global behavior of functions.

2. Limits of Functions (incl. one-sided limits)

1. Demonstrate an intuitive understanding of the limiting process.
2. Calculate limits using algebra.
3. Estimate limits from graphs or tables of data.

3. Asymptotic and Unbounded Behavior

1. Understand asymptotes in terms of graphical behavior.
2. Describe asymptotic behavior in terms of limits involving infinity.
3. Compare relative magnitudes of functions and their rates of change. (For example, contrasting exponential growth, polynomial growth, and logarithmic growth.)

4. Continuity as a Property of Functions

1. Develop an intuitive understanding of continuity. (The function values can be made as close as desired by taking sufficiently close values of the domain.)
2. Understand continuity in terms of limits.
3. Demonstrate a geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem).

2. Derivatives

1. Concept of the Derivative

1. Present and work with derivatives graphically, numerically, and analytically.
2. Interpret the derivative as an instantaneous rate of change.
3. Define the derivative as the limit of the difference quotient.
4. Understand the relationship between differentiability and continuity.

2. Derivative at a Point

1. Calculate the slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.
2. Find the tangent line to a curve at a point and local linear approximation.
3. Estimate the instantaneous rate of change as the limit of average rate of change.
4. Approximate rate of change from graphs and tables of values.

3. Derivative as a Function

1. Observe corresponding characteristics of graphs of f and f '.
2. Determine the relationship between the increasing and decreasing behavior of f and   the sign of f '.
3. Understand the Mean Value Theorem and its geometric consequences.
4. Solve equations involving derivatives and translate verbal descriptions into equations involving derivatives and vice versa.

4. Second Derivatives

1. Observe corresponding characteristics of the graphs of f, f ', and f ".
2. Understand the relationship between the concavity of f and the sign of f ".
3. Recognize points of inflection as places where concavity changes.

5. Applications of Derivatives

1. Analyze curves, including the notions of monotonicity and concavity.
2. Understand optimization, both absolute (global) and relative (local) extrema.
3. Model rates of change, including related rates problems.
4. Use implicit differentiation to find the derivative of an inverse function.
5. Interpret the derivative as a rate of change in varied applied contexts, including  velocity, speed, and acceleration.
6. Give a geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations.

6. Computation of Derivatives

1. Demonstrate knowledge of derivatives of basic functions, including power,    exponential, logarithmic, trigonometric, and inverse trigonometric functions.
2. Use the basic rules for the derivative of sums, products, and quotients of functions.
3. Use the chain rule and implicit differentiation.

3. Integrals

1. Interpretations and Properties of Definite Integrals

1. Define the definite integral as a limit of Riemann sums.
2. Understand the integral of the rate of change of a quantity over an interval  interpreted as the change of the quantity over the interval.
3. Utilize the basic properties of definite integrals. (Examples include additivity and linearity.)

2. Applications of Integrals

1. Appropriately use integrals in a variety of applications to model physical, biological, or economic situations.
2. Be able to adapt their knowledge and techniques to solve other similar application problems using the integral of a rate of change to give accumulated change or using the method of setting up an approximating Riemann sum and representing its limit as a definite integral.
3. Find the area of a region, the volume of a solid with known cross sections, the average value of a function, and the distance traveled by a particle along a line.

3. Fundamental Theorem of Calculus

1. Use the Fundamental Theorem to evaluate definite integrals.
2. Use the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analyses of functions so defined.

4. Techniques of Antidifferentiation

1. Use antiderivatives that follow directly from derivatives of basic functions.
2. Evaluate antiderivatives by substitution of variables (including change of limits for definite integrals).

5. Applications of Antidifferentiation

1. Find specific antiderivatives using initial conditions, including applications to motion along a line.
2. Solve separable differential equations and use them in modeling. In particular, study the equation y' = ky and exponential growth.

6. Numerical Approximations to Definite Integrals

1. Use Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values.

Student Outcomes:

After successfully completing this course, the student will:

• Represent functions in a variety of ways: graphical, numerical, analytical, or verbal and understand the connections among these representations.
• Understand the meaning of the derivative in terms of a rate of change and local linear approximation and be able to use derivatives to solve a variety of problems.
• Understand the meaning of the definite integral both as the limit of Riemann sums and as the net accumulation of a rate of change and should be able to use  integrals to solve a variety of problems.
• Understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus.
• Communicate mathematics both orally and in well-written sentences and explain solutions to problems.
• Model a written description of a physical situation with a function, a differential equation, or an integral.
• Use technology to help solve problems, experiment, interpret results, and verify conclusions.
• Determine the reasonableness of solutions, including sign, size, relative  accuracy, and units of measurement.
• Develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment.

New Jersey Student Learning Standards

Career Readiness, Life Literacies, and Key Skills Practices

CRLLKSP 1   Act as a responsible and contributing community members and employee.

Students understand the obligations and responsibilities of being a member of a community, and they demonstrate this understanding every day through their interactions with others. They are conscientious of the impacts of their decisions on others and the environment around them. They think about the near-term and long-term consequences of their actions and seek to act in ways that contribute to the betterment of their teams, families, community and workplace. They are reliable and consistent in going beyond the minimum expectation and in participating in activities that serve the greater good.

CRLLKSP 2   Attend to financial well-being.

Students take regular action to contribute to their personal financial well-being, understanding that personal financial security provides the peace of mind required to contribute more fully to their own career success.

CRLLKSP 3   Consider the environmental, social and economic impacts of decisions.

Students understand the interrelated nature of their actions and regularly make decisions that positively impact and/or mitigate negative impact on other people, organization, and the environment. They are aware of and utilize new technologies, understandings, procedures, materials, and regulations affecting the nature of their work as it relates to the impact on the social condition, the environment and the profitability of the organization.

CRLLKSP 4   Demonstrate creativity and innovation.

Students regularly think of ideas that solve problems in new and different ways, and they contribute those ideas in a useful and productive manner to improve their organization. They can consider unconventional ideas and suggestions as solutions to issues, tasks or problems, and they discern which ideas and suggestions will add greatest value. They seek new methods, practices, and ideas from a variety of sources and seek to apply those ideas to their own workplace. They take action on their ideas and understand how to bring innovation to an organization.

CRLLKSP 5   Utilize critical thinking to make sense of problems and persevere in solving them.

Students readily recognize problems in the workplace, understand the nature of the problem, and devise effective plans to solve the problem. They are aware of problems when they occur and take action quickly to address the problem; they thoughtfully investigate the root cause of the problem prior to introducing solutions. They carefully consider the options to solve the problem. Once a solution is agreed upon, they follow through to ensure the problem is solved, whether through their own actions or the actions of others.

CRLLKSP 6   Model integrity, ethical leadership and effective management.

Students consistently act in ways that align personal and community-held ideals and principles while employing strategies to positively influence others in the workplace. They have a clear understanding of integrity and act on this understanding in every decision. They use a variety of means to positively impact the directions and actions of a team or organization, and they apply insights into human behavior to change others’ action, attitudes and/or beliefs. They recognize the near-term and long-term effects that management’s actions and attitudes can have on productivity, morals and organizational culture.

CRLLKSP 7   Plan education and career paths aligned to personal goals.

Students take personal ownership of their own education and career goals, and they regularly act on a plan to attain these goals. They understand their own career interests, preferences, goals, and requirements. They have perspective regarding the pathways available to them and the time, effort, experience and other requirements to pursue each, including a path of entrepreneurship. They recognize the value of each step in the education and experiential process, and they recognize that nearly all career paths require ongoing education and experience. They seek counselors, mentors, and other experts to assist in the planning and execution of career and personal goals.

CRLLKSP 8   Use technology to enhance productivity, increase collaboration and communicate effectively.

Students find and maximize the productive value of existing and new technology to accomplish workplace tasks and solve workplace problems. They are flexible and adaptive in acquiring new technology. They are proficient with ubiquitous technology applications. They understand the inherent risks-personal and organizational-of technology applications, and they take actions to prevent or mitigate these risks.

CRLLKSP 9   Work productively in teams while using cultural/global competence.

Students positively contribute to every team, whether formal or informal. They apply an awareness of cultural difference to avoid barriers to productive and positive interaction. They find ways to increase the engagement and contribution of all team members. They plan and facilitate effective team meetings.

Career Readiness, Life Literacies, and Key Skills 

9.1.12.CDM.6: Compute and assess the accumulating effect of interest paid over time when using a variety of sources of credit.

9.1.12.CDM.8: Compare and compute interest and compound interest and develop an amortization table using business tools.

9.1.12.PB.1: Explain the difference between saving and investing.

9.4.12.TL.1: Assess digital tools based on features such as accessibility options, capacities, and utility for accomplishing a specified task.

Computer Science and Design Thinking

8.2.12.NT.1: Explain how different groups can contribute to the overall design of a product.

8.2.12.NT.2: Redesign an existing product to improve form or function.

Mathematics

A.APR.B.2. Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

A.APR.B.3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

A.APR.C.4. (+)Prove polynomial identities and use them to describe numerical relationships.

A.APR.D.6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

A.CED.A.1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

A.CED.A.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A.CED.A.3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

A.CED.A.4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

A.REI.B.4. Solve quadratic equations in one variable.

A.REI.D.11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★

A.SSE.A.1b. Interpret complicated expressions by viewing one or more of their parts as a single entity.

A.SSE.A.2. Use the structure of an expression to identify ways to rewrite it.

A.SSE.B.3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. ★

F.BF.A.1. Write a function that describes a relationship between two quantities.

F.BF.B.3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

F.BF.B.4. Find inverse functions.

F.BF.B.5. (+) Use the inverse relationship between exponents and logarithms to solve problems involving logarithms and exponents.

F.IF.B.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F.IF.B.5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

F.IF.C.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

F.IF.C.7a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

F.IF.C.7b. Graph square root, cube root, and piecewise.defined functions, including step functions and absolute value functions.

F.If.C.7c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

F.IF.C.7d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

F.IF.C.8a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

F.IF.C.8b. Use the properties of exponents to interpret expressions for exponential functions.

F.LE.B.5. Interpret the parameters in a linear or exponential function in terms of a context.

F.TF.A.1.(+) Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

F.TF.A.2.(+) Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

F.TF.A.3. (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for /3,  /4 and  /6, and use the unit circle to express the values of sine, cosines, and tangent for . x, + x, and 2 –x in terms of their values for x, where x is any real number.

F.TF.A.4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

F.TF.B.5. (+)Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

F.TF.B.6. (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

F.TF.B.7. (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

F.TF.C.8. (+)Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

F.TF.C.9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

G.GMD.A.3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

G.GMD.B.4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

G.MG.A.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

G.SRT.C.7. Explain and use the relationship between the sine and cosine of complementary angles.

G.SRT.C.8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

N.CN.C.9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

N.Q.A.1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

N.Q.A.2. Define appropriate quantities for the purpose of descriptive modeling.

N.Q.A.3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

N.VM.A.3. (+) Solve problems involving velocity and other quantities that can be represented by vectors.

Mathematical Practices

1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

English Language Arts

SL.PE.11–12.1. Initiate and participate effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with peers on grades 11–12 topics, texts, and issues, building on others’ ideas and expressing their own clearly and persuasively.

A. Come to discussions prepared, having read and researched material under study; explicitly draw on that preparation by referring to evidence from texts and other research on the topic or issue to stimulate a thoughtful, well-reasoned exchange of ideas.

B. Collaborate with peers to promote civil, democratic discussions and decision-making, set clear goals and assessments (e.g., student developed rubrics), and establish individual roles as needed.

C. Propel conversations by posing and responding to questions that probe reasoning and evidence; ensure a hearing for a full range of positions on a topic or issue; clarify, verify, or challenge ideas and conclusions; and promote divergent and creative perspectives.

D. Respond thoughtfully to diverse perspectives; synthesize comments, claims, and evidence made on all sides of an issue; resolve contradictions when possible; and determine what additional information or research is required to deepen the investigation or complete the task.

Science

HS-PS1-2 Construct and revise an explanation for the outcome of a simple chemical reaction based on the outermost electron states of atoms, trends in the periodic table, and knowledge of the patterns of chemical properties.

HS-PS1-5 Apply scientific principles and evidence to provide an explanation about the effects of changing the temperature or concentration of the reacting particles on the rate at which a reaction occurs.

HS-PS1-7 Use mathematical representations to support the claim that atoms, and therefore mass, are conserved during a chemical reaction.

HS-LS2-4 Use mathematical representations to support claims for the cycling of matter and flow of energy among organisms in an ecosystem.

HS-ESS2-6 Develop a quantitative model to describe the cycling of carbon among the hydrosphere, atmosphere, geosphere, and biosphere.

HS-ESS3-6 Use a computational representation to illustrate the relationships among Earth systems and how those relationships are being modified due to human activity (i.e., climate change).

III.         Proficiency Levels

Advanced Placement Calculus is the fourth course in the high school AP/Honors track.  Senior students with teacher recommendation who have successfully completed Honors PreCalculus may choose to take this course.  

IV.        Methods of Assessment

Student Assessment

The teacher will provide a variety of assessments during the course of the year. The assessment may include but is not limited to: chapter and unit tests and quizzes, teacher observations, open-ended problems, cooperative work, and homework.

Curriculum/Teacher Assessment

The teacher will provide the subject area supervisor with suggestions for changes on an ongoing basis.

V.        Grouping

Advanced Placement Calculus is a homogeneously grouped course at the twelfth-grade level.

VI.        Articulation/Scope & Sequence/Time Frame

Course length is one year.

VII.        Resources

Texts/Supplemental Reading/References

Larson, Ron and Battaglia, Paul. Calculus for AP.  Cengage Learning, 2021.

VIII.        Suggested Activities

Appropriate activities are listed in the curriculum map.

IX.        Methodologies

Students in this course will use technology on a daily basis in the form of a graphing calculator.  Appropriate use of the calculator is emphasized throughout the course, particularly in preparation for the calculator active and non-active portions of the AP Exam.   Through discovery exercises and laboratory explorations, they will discover the concepts for themselves.  They will take an active part in using various algebraic manipulatives in cooperative learning situations, thus applying teamwork to the learning process.  

X.        Interdisciplinary Connections

Connections are made to science, particularly physics and chemistry, by means of collaborative projects coordinating topics in the two subject areas.  Connections are also made by the use of spreadsheets to collect, interpret and graph data.  Writing assignments and portfolios strengthen the connection between mathematics and language arts literacy and fine arts.

XI.         Differentiating Instruction for Students with Special Needs: Students with Disabilities, Students at Risk, Students with 504s, English Language Learners, and Gifted & Talented Students

Differentiating instruction is a flexible process that includes the planning and design of instruction, how that instruction is delivered, and how student progress is measured. Teachers recognize that students can learn in multiple ways as they celebrate students’ prior knowledge. By providing appropriately challenging learning, teachers can maximize success for all students.

Differentiating in this course includes but is not limited to:

Differentiation for Support (ELL, Special Education, Students at Risk, Students with 504s)

• Peer mentoring on problems
• Differentiated teacher feedback on assignments
• Modeling out problems on whiteboard
• Visual aids as we project problems on whiteboard
• Study guides
• Tiered assignments
• Scaffolding of materials and assignments
• Re-teaching and review
• Guided note taking
• Exemplars of varied performance levels
• Multi-media approach to accommodating various learning styles

• Supplemental reading material for independent study
• Flexible grouping
• Tiered assignments
• Topic selection by interest
• Enhanced expectations for independent study
• Elevated questioning techniques using Webb's Depth of Knowledge matrix

XII.        Professional Development

The teacher will continue to improve expertise through participation in a variety of professional development opportunities.

XII.        Curriculum Map/Pacing Guide