Verona Public School District Curriculum Overview Precalculus/Trigonometry

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1 Verona Public School District Curriculum Overview Precalculus/Trigonometry Curriculum Committee Members: Marc Russo Danielle Mutovic Rich Wertz Supervisor: Glen Stevenson Curriculum Developed: 2011 Board Approval Date: 9/27/11 Verona Public Schools 121 Fairview Ave., Verona, NJ Verona Public Schools Mission Statement: The mission of the Verona Public Schools, the center of an engaged and supportive community, is to empower students to achieve their potential as active learners and productive citizens through rigorous curricula and meaningful, enriching experiences. Course Description: This course is designed to enhance students preparation for Calculus in high school or college. The course will focus on improving students knowledge of trigonometric and other types of functions, including polynomial, rational, exponential and logarithmic functions. Other topics include: Analytic Trigonometry, Applications of Trigonometry, Sequences and Series, Conic Sections, Parametric Equations, Polar Coordinates, Limits, Probability and Statistics. Much of this course involves real world applications and mathematical modeling. Student must provide their own TI NSpire or TI NSpire CAS graphing calculator; these will be an important tool that will routinely be used in instruction. Prerequisite(s): Algebra II

2 8.1: Educational Technology: All students will use digital tools to access, manage, evaluate, and synthesize information in order to solve problems individually and collaborate and to create and communicate knowledge. X A. Technology Operations and Concepts B. Creativity and Innovation C. Communication and Collaboration D. Digital Citizenship E. Research and Information Fluency F. Critical thinking, problem solving, and decision making Standard 8: Technology Standards 8.2: Technology Education, Engineering, Design, and Computational Thinking Programming: All students will develop an understanding of the nature and impact of technology, engineering, technological design, computational thinking and the designed world as they relate to the individual, global society, and the environment. X A. The Nature of Technology: Creativity and Innovation B. Technology and Society C. Design X D. Abilities for a Technological World E. Computational Thinking: Programming Social and Emotional Learning Core Competencies: These competencies are identified as five interrelated sets of cognitive, affective, and behavioral capabilities Self awareness: The ability to accurately recognize one s emotions and thoughts and their influence on behavior. This includes accurately assessing one s strengths and limitations and possessing a well grounded sense of confidence and optimism. Self management: The ability to regulate one s emotions, thoughts, and behaviors effectively in different situations. This includes managing stress, controlling impulses, motivating oneself, and setting and working toward achieving personal and academic goals. Social awareness: The ability to take the perspective of and empathize with others from diverse backgrounds and cultures, to understand social and ethical norms for behavior, and to recognize family, school, and community resources and supports. Relationship skills: The ability to establish and maintain healthy and rewarding relationships with diverse individuals and groups. This includes communicating clearly, listening actively, cooperating, resisting inappropriate social pressure, negotiating conflict constructively, and seeking and offering help when needed. Responsible decision making: The ability to make constructive and respectful choices about personal behavior and social interactions based on consideration of ethical standards, safety concerns, social norms, the realistic evaluation of consequences of various actions, and the well being of self and others. 9.1: Personal Financial Literacy: This standard outlines the important fiscal knowledge, habits, and skills that must be mastered in order for students to make informed decisions about personal finance. Financial literacy is an integral component of a student's college and career readiness, enabling students to achieve fulfilling, financially secure, and successful careers. SEL Competencies and Career Ready Practices Career Ready Practices: These practices outline the skills that all individuals need to have to truly be adaptable, reflective, and proactive in life and careers. These are researched practices that are essential to career readiness. CRP2. Apply appropriate academic and technical skills. CRP9. Model integrity, ethical leadership, and effective management. CRP10. Plan education and career paths aligned to personal goals. CRP3. Attend to personal health and financial well being. CRP6. Demonstrate creativity and innovation. X CRP8. Utilize critical thinking to make sense of problems and persevere in solving them. CRP11. Use technology to enhance productivity. CRP1. Act as a responsible and contributing citizen and employee. CRP9. Model integrity, ethical leadership, and effective management. X CRP4. Communicate clearly and effectively and with reason. CRP9. Model integrity, ethical leadership, and effective management. CRP12. Work productively in teams while using cultural global competence. X CRP5. CRP7. CRP8. CRP9. Standard 9: 21 st Century Life and Careers 9.2: Career Awareness, Exploration & Preparation: This standard outlines the importance of being knowledgeable about one's interests and talents, and being well informed about postsecondary and career options, career planning, and career requirements. Consider the environmental, social, and economic impact of decisions. Employ valid and reliable research strategies. Utilize critical thinking to make sense of problems and persevere in solving them. Model integrity, ethical leadership, and effective management. 9.3: Career and Technical Education: This standard outlines what students should know and be able to do upon completion of a CTE Program of Study. A. Income and Careers B. Money Management C. Credit and Debt Management D. Planning, Saving, and Investing E. Becoming a Critical Consumer F. Civic Financial Responsibility G. Insuring and Protecting X A. Career Awareness (K 4) B. Career Exploration (5 8) C. Career Preparation (9 12) X A. Agriculture, Food & Natural Res. B. Architecture & Construction C. Arts, A/V Technology & Comm. D. Business Management & Admin. E. Education & Training F. Finance G. Government & Public Admin. H. Health Science I. Hospital & Tourism J. Human Services K. Information Technology L. Law, Public, Safety, Corrections & Security M. Manufacturing N. Marketing O. Science, Technology, Engineering & Math P. Transportation, Distribution & Log. Core Instructional Materials : These are the board adopted and approved materials to support the curriculum, instruction, and assessment of this course. Course Materials Differentiated Resources : These are teacher and department found materials, and also approved support materials that facilitate differentiation of curriculum, instruction, and assessment of this course. Precalculus with Limits: A Graphing Approach Delta Math NJCTL Khan Academy Precalculus Eureka Math: A Story of Functions Modules Various online sources

3 BOE APPROVED 9/27/11 Curriculum Scope & Sequence Subject/Grade Level: MATHEMATICS/HIGH SCHOOL Course: PRE-CALCULUS CP/HONORS *The goals and standards addressed are the same for both levels of Pre-Calculus; however, the depth at which they are explored will vary based on the level of the course. Unit Duration Common Core Standards/ Unit 1 8 days Unit : Algebra Review Polynomial Identities Exponents & Radicals Factoring Rational Expressions Solving Equations A-APR4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x 2 + y 2 ) 2 = (x 2 y 2 ) 2 + (2xy) 2 can be used to generate Pythagorean triples. learning to justify their approaches and their steps when solving problems. - Solving algebraic equations is a basic exercise in their knowledge of the order of operations. - It can be determined (for the most part) if an expression is (or is not) completely simplified. - Basic rules of equality are the key to ensure accuracy in mathematics. - How do we solve an unfamiliar equation? - How do we know if an expression is completely simplified? - Besides checking our work, what are some strategies to help us gain confidence that we are solving a problem correctly? A-SSE2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 y4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). 2 Functions 9 days Unit : Analyzing Graphs Parent functions Transformations learning to analyze the graph of a real-life - Transformations of functions can be represented graphically or - Can all transformations of basic functions be represented in a predictable algebraic

4 Composition Inverses F-IF4. 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. function and use their insights to help them make informed decisions about that function. algebraically. - Graphical representations of functions can provide additional insight into a function s behavior. way? - When is it more appropriate to analyze a function algebraically? Graphically? F-IF7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. F-BF1. Write a function that describes a relationship between two quantities. c. (+) Compose functions. For example, if T(y) is the temperature in

5 the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. F-BF3. 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-BF4. Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2x+3 or f(x) = (x+1)/(x 1) for x 1. b. (+) Verify by composition that one function is the inverse of another. c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain.

6 3 9 days Unit : Fundamental Quadratics Theorem of Polynomial Division Algebra & End-Behavior Rational Zeros Functions Complex Numbers (and vectors) Fundamental Theorem of Algebra Rational Functions N-CN4. (+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. learning to determine what an accurate graph of a polynomial function would look like. - Polynomial and rational functions can be sketched quickly and accurately using nothing but algebraic skills. - All zeros of a polynomial function can be found by using division to break the function down into the product of linear and quadratic factors. - What makes an accurate sketch of a polynomial function? - What makes a good window for the graph of a polynomial function? - What is the best way to find the zeroes of a polynomial function? N-CN5. (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, ( i) 3 = 8 because ( i) has modulus 2 and argument 120. N-CN6. (+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

7 N-VM1. (+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, v, v, v). N-VM4. (+) Add and subtract vectors. a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. c. Understand vector subtraction v w as v + ( w), where w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise. N-CN7. Solve quadratic equations with real coefficients that have complex solutions. N-CN8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x as (x + 2i)(x 2i).

8 N-CN9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. A-APR2. 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-APR3. 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-APR6. 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. F-IF7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and

9 showing end behavior. d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. 4 Exponential and Logarithmic Functions 9 days Unit : Exponential functions Log functions Properties of logs Log Equations Models F-IF7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. learning to investigate important real-world phenomena from a graphical and an algebraic perspective. - Logarithmic functions are the inverse of exponential functions. - Exponential and logarithmic functions model real-world phenomena. - Inverse functions allow us to solve equations algebraically. - Why do we need the logarithm function? - What real-world phenomena are modeled by exponential or logarithmic functions? F-IF8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02) t, y = (0.97) t, y = (1.01) 12t, y = (1.2) t/10, and classify them as representing exponential

10 growth or decay. F-BF5. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. L-FE4. For exponential models, express as a logarithm the solution to ab ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. 5 Right Triangle & Unit Circle Trig 12 days Unit : SohCahToa Co-functions Identities Unit Circle Radians Reference Angles F-TF8. Prove the Pythagorean identity sin 2 (θ) + cos 2 (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-TF1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. F-TF2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions learning to understand the implications of right triangle trigonometry in the unit circle. - Trig functions are ratios of sides of right triangles. - Trig functions evaluated at a certain angle are constant, regardless of the size of the triangle. - Trig identities allow us to express trig expressions in different, but equivalent ways. - The Unit Circle allows us to evaluate trig functions at quadrantal angles. - Reference angles allow us to evaluate trig functions at angles greater than - What does evaluating a trig function at a given angle mean in real life? - How does changing the size of a right triangle affect the sine, cosine and tangent of its angles? - What is the value in knowing trig identities? - How does the unit circle solidify (and enhance) our understanding of trig functions? - What is the purpose of measuring angles in radians?

11 to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. F-TF3. (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π /4 and π /6, and use the unit circle to express the values of sine, cosine, and tangent for π x, π +x, and 2π x in terms of their values for x, where x is any real number. 90 degrees. - Radians are often a more appropriate way of describing angles. - Tangent and secant are named after their relationship to the unit circle. F-TF4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. 6 Graphing Trig Functions 10 days Unit : Sine and Cosine Waves Amplitude, Frequency, Period Transformations Tangent and Cotangent Secant and Cosecant Inverse Trig Functions F-IF7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and knowledge of periodic functions to analyze reallife periodic phenomena. - Changes to the algebraic equation of a function cause predictable changes to the function s graph. - Basic properties of trig graphs can be linked to a deep understanding of the unit circle and fundamental trig identities. - What is the relationship between a trig equation and the trig function s graph? - How can the unit circle begin to explain the graphs of sin, cos, tan, cot, sec and csc?

12 amplitude. F-TF5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. F-TF6. (+) 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-TF7. (+) 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. 7 Trig Equations 8 Triangle Trig 5 days Unit : Linear Trig Equations Quadratic Trig Equations Trig Equations with more than one Function A.REI2. Understand solving equations as a process of reasoning and explain the reasoning. 10 days Unit : Area Law of Sines Law of Cosines Ambiguous Case Forces learning to explain the relationship between trig waves, trig equations and the unit circle. learning of trigonometric formulas to solve reallife surveying problems. - The symmetry of the unit circle allows trig equations to often have multiple answers. - Trig equations can be treated like much more basic equations by using the concept of substitution. - The Law of Cosines, Law of Sines and area formulas are generalized versions of Pythagorean Theorem, - How can we determine if an arc trig function will yield multiple angles in a given interval? - What is the algebraic and graphical relationship between trig equations and quadratic/linear equations? - What is the relationship of the Law of Cosines, Law of Sines and area formula to theorems you previously learned? - What is the utility of the

13 G-SRT6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. SohCahToa and A=1/2bh. - Trigonometry allows you to find distances or angles that you could not have measured. Law of Sines and the Law of Cosines? G-SRT7. Explain and use the relationship between the sine and cosine of complementary angles. G-SRT8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. G-SRT9. (+) Derive the formula A = 1/2 ab sin(c) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. G-SRT10. (+) Prove the Laws of Sines and Cosines and use them to solve problems. G-SRT11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). 9 Combined Sinusoids 7 days Unit : Sum and Difference Formulas Multiple Angle Formulas learning to perform - The relationship between a function and an argument is - Why can t you distribute a function to corresponding pieces of

14 F-TF9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. harmonic analysis. not multiplication. - The composition of ordinates is merely the sum or product of the ordinates of a function. its argument? - What is the algebraic and graphical interpretation of function composition? 10 Sequences, Series & the Binomial Theorem 8 days Unit : Arithmetic Sequences and Series Geometric Sequences and Series Induction Binomial Theorem A-SSE4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. learning to evaluate salary offers using the concepts of sequences and series. - Mathematical induction is a valid form of proof for sequence-type problems. - Sequence and series formulas are common-sense shortcuts. - Some infinite series converge. - What is proof? - How could the sum of an infinite number of numbers possibly converge? A-APR5. (+) Know and apply the Binomial Theorem for the expansion of (x+ y) n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal s Triangle. F-IF3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined

15 recursively by f(0) = f(1) = 1, f(n+1) = f(n) +f(n-1) for n Conic Sections, Parametric Equations & Polar Coordinates F-BF2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. 12 days Unit : Conics Parametric Equations Polar Coordinates G-GPE1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. G-GPE2. Derive the equation of a parabola given a focus and directrix. learning to model a reallife phenomenon using conic sections. - Parametric equations allow an independent time (or angle) variable to dictate both x and y position. - Polar coordinates are an alternative way to specify location on the plane. - Equations of conic sections can be reorganized and analyzed to produce adequate graphs. - What is the benefit of using parametric equations? Rectangular equations? - What is the benefit of using polar coordinates? Rectangular coordinates? - What is the importance of conic sections? G-GPE3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. 12 Matrices 8 days Unit : Systems of Equations Operations with matrices Inverses Determinants learning to perform data analysis using matrices. - Matrices allow for a streamlined method for solving systems. - Matrices have properties similar to, but not the same as - What is the purpose of learning matrices? - What is the relationship between operations with matrices and operations with real numbers?

16 N-VM6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. those of real numbers. N-VM7. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. N-VM8. (+) Add, subtract, and multiply matrices of appropriate dimensions. N-VM9. (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. N-VM10. (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. N-VM11. (+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as

17 transformations of vectors. N-VM12. (+) Work with 2 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area. A-REI8. (+) Represent a system of linear equations as a single matrix equation in a vector variable. A-REI9. (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 3 or greater). 13 Limits 11 days Unit : Evaluating Limits Properties of Limits Continuity F.IF.7d. Analyze functions using different representations. learning to determine if a function has a limit at a certain value. - A function does not have to be defined at a point for its limit to exist at that point. - Limits only exist if the two one-sided limits exist and are equal. - What is a limit? - How do we know if a limit exists? - What is the importance of the concept of continuity? The Intermediate Value Theorem? 14 Families of Functions 8 days Unit : Regressions Interpolations and Extrapolations Families of Functions S-ID6. Represent data on two quantitative variables on a scatter plot, and describe how the variables learning to discuss implications and make suggestions based on data analysis. - The normal curve only applies to certain data sets. - Regressions help us make predictions based on trends and relationships. - How can the normal curve help us make predictions? - Why are some data normal and others not? - How do we find a curve of best fit?

18 are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. b. Informally assess the fit of a function by plotting and analyzing residuals. c. Fit a linear function for a scatter plot that suggests a linear association. S-ID7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. S-ID8. Compute (using technology) and interpret the correlation coefficient of a linear fit. S-ID9. Distinguish between correlation and causation.