Unit 1 Introduction to Trigonometry (9 days) First Half of PC.FT.1 PC.FT.2 PC.FT.2a PC.FT.2b PC.FT.3 PC.FT.4 PC.FT.8 PC.GCI.5 Understand that the radian measure of an angle is the length of the arc on the unit circle subtended by the angle. Define sine and cosine as functions of the radian measure of an angle in terms of the x- and y-coordinates of the point on the unit circle corresponding to that angle and explain how these definitions are extensions of the right triangle definitions. Define the tangent, cotangent, secant, and cosecant functions as ratios involving sine and cosine. Write cotangent, secant, and cosecant functions as the reciprocals of tangent, cosine, and sine, respectively. 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, cosine, and tangent for x, + x, and 2 x in terms of their values for x, where x is any real number. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. Justify the Pythagorean, even/odd, and cofunction identities for sine and cosine using their unit circle definitions and symmetries of the unit circle and use the Pythagorean identity to find sin A, cos A, or tan A, given sin A, cos A, or tan A, and the quadrant of the angle. Derive the formulas for the length of an arc and the area of a sector in a circle, and apply these formulas to solve mathematical and real-world problems. Unit 1 - Introduction to Trigonometry (PC.FT.1, PC.FT.3, PC.FT.4, PC.FT.8) Find the positive and negative values of arcs on the unit circle in terms of, arcs between -2pi and 2pi. Write the circular function values of sine and cosine for arcs described above. Use the Pythagorean identity to find sine given cosine or vice-versa. (PC.FT.2, PC.FT.8) Write the definition of circular functions tangent, cotangent, secant and cosecant. Use 3 Pythagorean Identities to find the value of 5 circular functions given one. Name quadrant given signs of circular ratios or arcs of any length. (PC.FT.1, PC.FT.2, PC.FT.2a, PC.FT.2b, PC.FT.4, PC.FT.8, PC.GCI.5) Write the even/odd identities. Determine if a function is even, odd, or neither. Anderson School District Five Page 1 2017-2018
First Half of Unit 1 - Introduction to Trigonometry Draw angles in standard position using both + and degrees values. Relate degree values to central angles and arc length. Change from radians to degrees and degrees to radians. Find the arc length and sector area. Solve problems using latitude. (PC.FT.2, PC.FT.8) Write the definitions for the trigonometric functions. Find the trigonometric values of an angle whose terminal side goes through a point. Use reference angles to solve problems - Name quadrant of any angle measured in degrees or radians. Given 1 trigonometric function, find the other 5. (PC.FT.2, PC.FT.8) Use calculator to find values of trigonometric functions. Use reference angles to solve problems Name quadrant of any angle measured in degrees or radians. Using reference angles, find angles when trigonometric functions are known. Unit Review and Test Anderson School District Five Page 2 2017-2018
Unit 2 Application of Trigonometry to Triangles (7 days) First Half of PC.FT.2 PC.GSRT.9 PC.GSRT.10 PC.GSRT.11 Define sine and cosine as functions of the radian measure of an angle in terms of the x- and y-coordinates of the point on the unit circle corresponding to that angle and explain how these definitions are extensions of the right triangle definitions. Derive the formula A = ½ ab sin C for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Prove the Laws of Sines and Cosines and use them to solve problems. Use the Law of Sines and the Law of Cosines to solve for unknown measures of sides and angles of triangles that arise in mathematical and realworld problems. Unit 2 Application of Trigonometry to Triangles (PC.FT.2) Solve right and isosceles triangles. Solve problems using angles of elevation and depression. (PC.GSRT.9) Find the area of general triangles. Use the Law of Sine s to solve triangles. (PC.GSRT.10) Use the Law of Sine s to solve triangles given ambiguous data. Use the Law of Cosines to solve triangles. (PC.GSRT.10, PC.GSRT.11) Solve real world problems using the Laws of Sine s and Cosines. Review solving triangles. Unit Test Anderson School District Five Page 3 2017-2018
Unit 3 Graphing Trigonometric Functions, Part 1 (4 days) First Half of PC.FBF.4 PC.FIF.7d PC.FT.4 PC.FT.5 Understand that an inverse function can be obtained by expressing the dependent variable of one function as the independent variable of another, as f and g are inverse function if and only if f (x) = y and g (y) = x, for all values of x in the domain of f and all values of y in the domain of g, and find inverse functions for one-to-one function or by restricting the domain. Graph trigonometric functions, showing period, midline, and amplitude. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Unit 3 Graphing Trigonometric Functions, Part 1 (PC.FIF.7d) Graph sine, cosine, secant, and cosecant parent functions and state their domain and range. (PC.FT.4, PC.FT.5, PC.FBF.4) Graph sine, cosine, secant, and cosecant functions in the form y = Asin(Bx + C) + D. Write evaluations of sine and cosine functions when given the graph. Unit Review and Test Anderson School District Five Page 4 2017-2018
Unit 4 Graphing Trigonometric Functions, Part 2 (6 days) First Half of PC.FBF.3 PC.FBF.4 PC.FBF.4a PC.FBF.4b PC.FIF.5 PC.FIF.7d PC.FT.5 PC.FT.6 PC.FT.7 Describe the effect of the transformations k f (x), f (x) + k, f (x + k), and combinations of such transformations on the graph of y = f (x) for any real number k. Find the value of k given the graphs and write the equation of a transformed parent function given its graph. Understand that an inverse function can be obtained by expressing the dependent variable of one function as the independent variable of another, as f and g are inverse function if and only if f (x) = y and g (y) = x, for all values of x in the domain of f and all values of y in the domain of g, and find inverse functions for one-to-one function or by restricting the domain. Use composition to verify one function in an inverse of another. If a function has an inverse, find values of the inverse function from a graph of table. Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes. Graph trigonometric functions, showing period, midline, and amplitude. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Define the six inverse trigonometric functions using domain restrictions for regions where the function is always increasing or always decreasing. 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. Unit 4 Graphing Trigonometric Functions, Part 2 (PC.FBF.3, PC.FIF.5, PC.FIF.7d, PC.FT.5) Graph tangent and cotangent graphs with horizontal and vertical shifts and period changes. (PC.FBF.4, PC.FBF.4a, PC.FBF.4b, PC.FT.6) Graph inverse trigonometric functions and state their domain and range. (PC.FT.7) Evaluate expressions containing the inverse trigonometric functions. Unit Review and Test Anderson School District Five Page 5 2017-2018
Unit 5 Proving Trigonometric Identities (7 days) First Half of PC.FT.8 PC.FT.9 Justify the Pythagorean, even/odd, and cofunction identities for sine and cosine using their unit circle definitions and symmetries of the unit circle and use the Pythagorean identity to find sin A, cos A, or tan A, given sin A, cos A, or tan A, and the quadrant of the angle. Justify the sum and difference formulas for sine, cosine, and tangent and use them to solve problems. Unit 5 Proving Trigonometric Identities (PC.FT.8) Prove Identities and Pythagorean Identities. (PC.FT.9) Prove Identities involving sum, half, and double angle identities. (PC.FT.8) Solve general identities. Unit Test Anderson School District Five Page 6 2017-2018
Unit 6 Solving Trigonometric Equations (9 days) First Half of PC.AAPR.4 PC.FIF.7d PC.FT.7 PC.FT.9 Prove polynomial identities and use them to describe numerical relationships. Graph trigonometric functions, showing period, midline, and amplitude. 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. Justify the sum and difference formulas for sine, cosine, and tangent and use them to solve problems. Unit 6 Solving Trigonometric Equations (PC.FT.7) Solve simple trigonometric equations. (PC.FT.7, PC.FT.9) Solve trigonometric equations using identities. (PC.AAPR.4, PC.FIF.7d, PC.FT.7) Solve trigonometric equations in quadratic form. Unit Review and Test (Solving Equations) MIDTERM EXAM Anderson School District Five Page 7 2017-2018
First Half of Anderson School District Five Page 8 2017-2018
Second Half of Unit 7 Polar System, Parametric Equations, and Vectors (9 days) PC.NCNS.4 PC.NVMQ.1 PC.NVMQ.2 PC.NVMQ.3 PC.NVMQ.4 PC.NVMQ.4a PC.NVMQ.4b PC.NVMQ.5 Graph complex numbers on the complex plane in rectangular and polar form and explain why the rectangular and polar forms of a given complex number represent the same number. 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. Represent and model with vector quantities. Use the coordinates of an initial point and of a terminal point to find the components of a vector. Represent and model with vector quantities. Solve problems involving velocity and other quantities that can be represented by vectors. Perform operations on vectors. Add and subtract vectors using components of the vectors and graphically. Given the magnitude and direction of two vectors, determine the magnitude of their sum and of their difference. Multiply a vector by a scalar, representing the multiplication graphically and computing the magnitude of the scalar multiple. Unit 7 - Polar System, Parametric Equations, and Vectors (PC.NCNS.4) Plot points in a Polar Coordinate System. Write the Polar Coordinate of a point 4 ways. Draw the graphs of Polar Equations with and without a calculator. Change from the polar system to the rectangular system and vice versa. (PC.NCNS.4) Write polar equations as rectangular and vice-versa. (Honors PreCalculus for BC Calculus) Recognize and graph parametric equations over a specific interval for the parameter. Remove the parameter to write a set of parametric equations in y = f(x) form. (PC.NVMQ.1, PC.NVMQ.2, PC.NVMQ.4b) Write vectors in component form and i-j form. Determine the magnitude and direction of vectors. Find the unit vector in the same direction as a given vector. Anderson School District Five Page 9 2017-2018
Second Half of Unit 7 - Polar System, Parametric Equations, and Vectors (PC.NVMQ.4, PC.NVMQ.4a, PC.NVMQ.5) Perform operations with vectors including addition, subtraction, scalar multiplication and dot product. Show the addition of two vectors using the head-to-tail and parallelogram methods. (PC.NVMQ.3) Find the angle between two vectors. Determine if two vectors are orthogonal or parallel. Solve problems involving velocity and other quantities that can be represented with vectors. Unit Review and Test 49 Anderson School District Five Page 10 2017-2018
Unit 8 Conic Sections (4 days) Second Half of PC.GGPE.2 PC.GGPE.3 Use the geometric definition of a parabola to derive its equation given the focus and directrix. Use the geometric definition of an ellipse and of a hyperbola to derive the equation of each given the foci and points whose sum or difference of distance from the foci are constant. Unit 8 Conic Sections (PC.GGPE.3) Write the general and standard forms of the equations of circles and ellipses and graph. (PC.GGPE.3) Write the general and standard forms of the equations of hyperbolas and graph. (PC.GGPE.2) Write the general and standard forms of the equations of parabolas and graph. Unit Test Anderson School District Five Page 11 2017-2018
Unit 9 Polynomial Functions (12 days) Second Half of PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AREI.7 PC.AREI.11 PC.ASE.1 PC.ASE.2 PC.FBF.1 PC.FBF.1b PC.FBF.3 PC.FIF.4 PC.FIF.6 PC.FIF.7 PC.NCNS.2 PC.NCNS.3 PC.NCNS.5 PC.NCNS.6 PC.NCNS.7 PC.NCNS.8 PC.NCNS.9 Know and apply the Division Theorem and the Remainder Theorem for polynomials. Graph polynomials identifying zeros when suitable factorizations are available and indicated end behavior. Write a polynomial function of least degree corresponding to a given graph. Prove polynomial identities and use them to describe numerical relationships. Apply the Binomial Theorem to expand powers of binomials, including those with one and with two variables. Use the Binomial Theorem to factor squares, cubes, and fourth powers of binomials. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. Solve an equation of the form f (x) = g (x) graphically by identifying the x-coordinate(s) of the point(s) of intersection of the graphs of y = f (x) and g = g (x). Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. Analyze the structure of binomials, trinomials, and other polynomials in order to rewire equivalent expressions. Write a function that describes a relationship between two quantities. Combine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations. Describe the effect of the transformations k f (x), f (x) + k, f (x + k), and combinations of such transformations on the graph of y = f (x) for any real number k. Find the value of k given the graphs and write the equation of a transformed parent function given its graph. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. Use the relation i 2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Find the conjugate of a complex number in rectangular and polar forms and use conjugates to find moduli and quotients of complex numbers. Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. Determine the modulus of a complex number by multiplying by its conjugate and determine the distance between two complex numbers by calculating the modulus of their difference. Solve quadratic equations in one variable that have complex solutions. Extend polynomial identities to the complex numbers and use DeMoivre s Theorem to calculate a power of a complex number. Know the Fundamental Theorem of Algebra and explain why complex roots of polynomials with real coefficients must occur in conjugate pairs. Anderson School District Five Page 12 2017-2018
Second Half of Unit 9 Polynomial Functions (PC.ASE.2, PC.FIF.6) Recognize polynomial functions from constant to nth degree. Write the equations for linear functions given various information including lines that are horizontal, vertical, parallel or perpendicular to a given line, etc. (PC.AREI.7, PC.FBF.3, PC.FIF.6) Graph quadratic functions as translations of the parent graph. Write a quadratic function given the zeros and a point, the vertex and a point, 3 arbitrary points or a graph. Write a quadratic function in 2 forms and change from one form to the other. Find the average rate of change of a function over a specific interval. Solve systems of linear and quadratic equations. (PC.ASE.1, PC.FBF.1, PC.FBF.1b) Write a function that describes a relationship between two quantities. Use operations to combine functions that describe the relationship between two quantities. (PC.AAPR.2, PC.AREI.11, PC.FIF.4, PC.NCNS.7) Find the intercepts of any polynomial. Find the real and imaginary zeros of a polynomial. (PC.AAPR.3, PC.ASE.2, PC.FIF.4, PC.FIF.7) Sketch a polynomial given linear factors. Write the polynomial given a graph. (PC.AAPR.4, PC.NCNS.8) Prove polynomial identities and use them to describe numerical relationships. Use DeMoivre s Theorem to calculate the power of a complex number. Extend identities to the complex number system. Anderson School District Five Page 13 2017-2018
Second Half of Unit 9 Polynomial Functions (PC.AAPR.5, PC.NCNS.9) Expand binomials using the Binomial Theorem. Apply the Fundamental Theorem of Algebra. (PC.NCNS.2, PC.NCNS.3) Perform operations with complex numbers. (PC.NCNS.5) Perform operations with complex numbers using graphs in the complex plane. (PC.NCNS.6) Determine the modulus of a complex number and use it to find distance between two complex numbers. (PC.NCNS.7) Solve quadratic equations with complex solutions. Review quadratic and linear functions. Test Anderson School District Five Page 14 2017-2018
Unit 10 Exponential and Logarithmic Functions (9 days) Second Half of PC.ASE.4 PC.FBF.4 PC.FBF.4a PC.FBF.4b PC.FBF.5 PC.FIF.5 PC.FIF.7 PC.FIF.7a PC.FIF.7b PC.FIF.7c PC.FLQE.4 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 including applications to finance. Understand that an inverse function can be obtained by expressing the dependent variable of one function as the independent variable of another, as f and g are inverse function if and only if f (x) = y and g (y) = x, for all values of x in the domain of f and all values of y in the domain of g, and find inverse functions for one-to-one function or by restricting the domain. Use composition to verify one function in an inverse of another. If a function has an inverse, find values of the inverse function from a graph of table. Understand and verify through function composition that exponential and logarithmic functions are inverses of each other and use this relationship to solve problems involving logarithms and exponents. Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes. Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Note: PF.FIF.7a-d are not Graduation Standards.) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Graph radical functions over their domain show end behavior. Graph exponential and logarithmic functions, showing intercepts and end behavior. Express a logarithm as the solution to the exponential equation, abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. Unit 10 Exponential and Logarithmic Function (PC.FIF.7, PC.FIF.7a, PC.FIF.7b, PC.FIF.7c) Graph exponential functions as translations of parent graph. State domain and range of exponential functions. Solve exponential equations using like bases. Solve present and future value of money problems using exponential functions. (PC.FBF.4, PC.FBF.4a, PC.FBF.4b, PC.FBF.5, PC.FIF.5, PC.FLQE.4) Write logarithm functions as inverses of exponential functions using graphs. State domain and range. Simplify simple logarithmic expressions. Solve simple logarithmic equations. Anderson School District Five Page 15 2017-2018
Second Half of Unit 10 Exponential and Logarithmic Function (PC.FLQE.4) Use laws of logarithms to simplify or expand logarithmic expressions. Solve logarithmic and exponential equations with and without calculators. (PC.ASE.4) Derive the formula for the sum of a finite geometric series where the common ratio is not 1 and use the formula to solve problems including applications to finance. Review Unit Test Anderson School District Five Page 16 2017-2018
Unit 11 Rational and Radical Functions and Limits (5 days) Second Half of PC.AAPR.6 PC.AAPR.7 PC.FIF.7 PC.FIF.7a PC.FIF.7b Apply algebraic techniques to rewrite simple rational expressions in different forms; using inspection, long division, or, for the more complicated examples, a computer algebra system. Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Note: PF.FIF.7a-d are not Graduation Standards.) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Graph radical functions over their domain show end behavior. Unit 11 Rational and Radical Functions and Limits (PC.FIF.7, PC.FIF.7a, PC.FIF.7b, PC.AAPR.6, PC.AAPR.7) Sketch rational functions as translations of the parent graph y = 1/x. Sketch more complex rational functions using limits showing both local and end behavior. Graph radical functions over their domain show end behavior. Evaluate limits at infinity and at x = a. Review and Unit Test Anderson School District Five Page 17 2017-2018
Second Half of Unit 12 Matrices (3 days) PC.AREI.8 Represent a system of linear equations as a single matrix equation in a vector variable. PC.AREI.9 Using technology for matrices of dimension 3 x 3 or greater, find the inverse of a matrix if it exists and use it to solve systems of linear equations. PC.NVMQ.6* Use matrices to represent and manipulate data. (Note: This Graduation Standard is covered in Grade 8.) PC.NVMQ.7 Perform operations with matrices of appropriate dimensions including addition, subtraction, and scalar multiplication. PC.NVMQ.8 Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. PC.NVMQ.9 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. PC.NVMQ.10 Multiply a vector by a matrix of appropriate dimension to produce another vector. Work with matrices as transformations of vectors. PC.NVMQ.11 Apply 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area. Unit 12 Rational Functions and Limits (PC.NVMQ.9) Use inverse of matrix to solve system of equations. (PC.NVMQ.6) Represent data with matrices. (PC.NVMQ.7, PC.NVMQ.8, PC.NVMQ.9) Perform operations with matrices including the determinant. (PC.NVMQ.10) Multiply a vector by a matrix. (PC.NVMQ.11) Use matrices to solve area problems with coordinate geometry. (PC.AREI.8) Represent a system of linear equations as a single matrix equation in a vector variable. Anderson School District Five Page 18 2017-2018
Second Half of Unit 12 Rational Functions and Limits (PC.AREI.9) Use matrix equations with vector variables to show a system of linear equations. Review and Unit Test END OF COURSE EXAM Anderson School District Five Page 19 2017-2018