PreCalculus CP/Honors Curriculum Pacing Guide

Transcription

1 The mathematical processes provide the framework for teaching, learning, and assessing in all high school mathematics core courses. Instructional programs should be built around these processes. PC-6: The student will demonstrate through the mathematical processes an understanding of the behavior of conic sections both geometrically and algebraically. s: PC-1.1 Communicate knowledge of algebraic and trigonometric relationships by using mathematical terminology appropriately. PC-1.2 Connect algebra and trigonometry with other branches of mathematics. PC-1.3 Apply algebraic methods to solve problems in real-world contexts. PC-1.4 Judge the reasonableness of mathematical solutions. PC-1.5 PC-1.6 PC-1.7 Demonstrate an understanding of algebraic and trigonometric relationships by using a variety of representations (including verbal, graphic, numerical, and symbolic). Understand how algebraic and trigonometric relationships can be represented in concrete models, pictorial models, and diagrams. Understand how to represent algebraic and trigonometric relationships by using tools such as handheld computing devices, spreadsheets, and computer algebra systems (CASs). Anderson School District Five Page 1 Copyright July 1, 2012

2 Anderson School District Five Page 2 Copyright July 1, 2012

3 Unit 1 Introduction to Trigonometry 11 days First Nine Weeks PC-5.1 Understand how angles are measured in either degrees or radians. 1 Draw and recognize positive and negative angles in standard position using radian measure and give the quadrant of the terminal side. F-TF.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, cosine, and tangent for x, + x, and 2 x in terms of their values for x, where x is any real number. F-TF.4 (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. PC-5.1 Understand how angles are measured in either degrees or radians. PC-5.2 Carry out a procedure to convert between degree and radian measures. 2 Convert radian measure to degree measure and vice versa. F-TF.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, cosine, and tangent for x, + x, and 2 x in terms of their values for x, where x is any real number. F-TF.4 (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. Anderson School District Five Page 3 Copyright July 1, 2012

4 Unit 1 Introduction to Trigonometry 11 days First Nine Weeks PC-5.1 Understand how angles are measured in either degrees or radians. 3 Find the length of an arc and the area of a sector of a circle F-TF.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, cosine, and tangent for x, + x, and 2 x in terms of their values for x, where x is any real number Identify the angles, in degrees & radians, on the unit circle and find the exact values of all 6 trigonometric functions for each angle. Use reference angles to determine the terminal point P(x,y) on the unit circle for a given angle. F-TF.4 PC-5.6 F-TF.3 (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. Apply a procedure to evaluate trigonometric expressions. (+) 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. 9 Use a calculator to find trigonometric values of any angle F-TF.4 (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. 10 Unit Review 11 Unit Test Anderson School District Five Page 4 Copyright July 1, 2012

5 Unit 2 Applications of Trigonometric Functions on Right Triangles 7 days First Nine Weeks Solve right and isosceles triangles using trigonometric ratios. PC-5.8 Analyze given information to write a trigonometric equation that models a given problem situation involving right triangles Solve word problems using the angles of elevation and depression. PC-5.8 Analyze given information to write a trigonometric equation that models a given problem situation involving right triangles Find the trigonometric function values of angles whose terminal side passes through a certain coordinate. PC-5.6 Apply a procedure to evaluate trigonometric expressions Given a trigonometric value, find the quadrant and values of the other five trigonometric functions. PC-5.6 Apply a procedure to evaluate trigonometric expressions. Unit Review Unit Test Anderson School District Five Page 5 Copyright July 1, 2012

6 Unit 3 Applications of Trigonometric Functions on Oblique Triangles 11 days First Nine Weeks Use the Law of Sines to solve oblique triangles. Include the ambiguous case. PC-5.12 Apply the laws of sines and cosines to solve problems. 22 Use the Law of Sines to solve word problems. PC-5.7 PC-5.12 Analyze given information to write a trigonometric function that models a given problem situation involving periodic phenomena. Apply the laws of sines and cosines to solve problems Use the Law of Cosines to solve general oblique triangles. PC-5.12 Apply the laws of sines and cosines to solve problems. 25 Use the Law of Cosines to solve word problems PC-5.7 PC-5.12 Analyze given information to write a trigonometric function that models a given problem situation involving periodic phenomena. Apply the laws of sines and cosines to solve problems Find the area of triangles given various data sets. PC-5.9 Carry out a procedure to calculate the area of a triangle when given the lengths of two sides and the measure of the included angle. 28 Unit Review 29 Unit Test Anderson School District Five Page 6 Copyright July 1, 2012

7 Unit 4 Graph Trigonometric Functions 16 days First Nine Weeks PC-5.4 F-TF.6 Carry out a procedure to graph trigonometric functions by analyzing intercepts, periodic behavior, and graphs of reciprocal functions. (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows it inverse to be constructed 30 Sketch the graphs of the sine and cosine functions and state their domain and range. F-TF.7 PC-5.5 (+) 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. Carry out procedures to determine the characteristics of trigonometric functions (including domain, range, intercepts, and asymptotes.) F-TF.6 (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows it inverse to be constructed F-TF.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. Anderson School District Five Page 7 Copyright July 1, 2012

8 Unit 4 Graph Trigonometric Functions 16 days First Nine Weeks PC-2.1 F-TF.6 Carry out a procedure to graph parent functions (including y = sin x, y = cos x). (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows it inverse to be constructed 31 Find the amplitude, period, and phase shifts of sine and cosine graphs to sketch the graph. F-TF.7 PC-5.4 (+) 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. Carry out a procedure to graph trigonometric functions by analyzing intercepts, periodic behavior, and graphs of reciprocal functions. F-TF.6 (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows it inverse to be constructed F-TF.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. Anderson School District Five Page 8 Copyright July 1, 2012

9 Unit 4 Graph Trigonometric Functions 16 days First Nine Weeks PC-5.4 F-TF.6 Carry out a procedure to graph trigonometric functions by analyzing intercepts, periodic behavior, and graphs of reciprocal functions. (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows it inverse to be constructed 32 Write equations of sine and cosine functions when given the graph. F-TF.7 PC-5.5 (+) 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. Carry out procedures to determine the characteristics of trigonometric functions (including domain, range, intercepts, and asymptotes.) F-TF.6 (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows it inverse to be constructed F-TF.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. Anderson School District Five Page 9 Copyright July 1, 2012

10 Unit 4 Graph Trigonometric Functions 16 days First Nine Weeks 33 Sketch the graph of secant and cosecant functions and state their domain and range. PC-2.1 PC-5.4 PC-5.5 Carry out a procedure to graph parent functions (including y = sin x, y = cos x). Carry out a procedure to graph trigonometric functions by analyzing intercepts, periodic behavior, and graphs of reciprocal functions. Carry out procedures to determine the characteristics of trigonometric functions (including domain, range, intercepts, and asymptotes.) 34 Sketch the graph of tangent and cotangent functions and state their domain and range. F-TF.6 (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows it inverse to be constructed F-TF.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. 35 Graph the inverse trigonometric functions and state their domain and range. Write the restricted ranges for all inverse trigonometric functions. PC-5.13 Apply a procedure to graph the inverse functions of sine, cosine, and tangent. 36 Evaluate expressions using inverse trigonometric functions. PC-5.6 Apply a procedure to evaluate trigonometric expressions. Anderson School District Five Page 10 Copyright July 1, 2012

11 Unit 4 Graph Trigonometric Functions 16 days First Nine Weeks 37 Solve problems using the inverse trigonometric functions. (ex: Arcsin(cos(5/13)) PC-5.6 Apply a procedure to evaluate trigonometric expressions. 38 Unit Review 39 Unit Test Exam Review 45 1 st Nine Weeks Exam Anderson School District Five Page 11 Copyright July 1, 2012

12 First Nine Weeks Anderson School District Five Page 12 Copyright July 1, 2012

13 Unit 5 Trigonometric Identities 13 days Second Nine Weeks Prove identities involving reciprocal and Pythagorean identities. PC-5.14 Apply trigonometric relationships (including reciprocal identities; Pythagorean identities; even and odd identities; addition and subtraction formulas of sine, cosine, and tangent; and double angle formulas) to verify other trigonometric identities Proving more difficult trigonometric identities. F-TF.9 (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems Solve problems and prove identities involving addition formulas Solve problems and prove identities involving double and half-angle formulas. 57 Unit Review 58 Unit Test Anderson School District Five Page 13 Copyright July 1, 2012

14 Unit 6 Solving Trig Equations 11 days Second Nine Weeks PC-5.10 Carry out a procedure to solve trigonometric equations algebraically Solving basic trigonometric equations. PC-5.11 Carry out a procedure to solve trigonometric equations graphically. PC-5.10 Carry out a procedure to solve trigonometric equations algebraically Solve trigonometric equations that involve the use of identities. PC-5.11 Carry out a procedure to solve trigonometric equations graphically. PC-5.14 Apply trigonometric relationships (including reciprocal identities; Pythagorean identities; even and odd identities; addition and subtraction formulas of sine, cosine, and tangent; and double angle formulas) to verify other trigonometric identities. Anderson School District Five Page 14 Copyright July 1, 2012

15 Unit 6 Solving Trig Equations 11 days Second Nine Weeks Solve trigonometric equations that involve the use of the quadratic equation. PC-5.10 PC-5.11 Carry out a procedure to solve trigonometric equations algebraically. Carry out a procedure to solve trigonometric equations graphically. 67 Solve trigonometric equations graphically with the graphing calculator. PC-5.10 PC-5.11 Carry out a procedure to solve trigonometric equations algebraically. Carry out a procedure to solve trigonometric equations graphically 68 Unit Review 69 Unit Test 70 Exam Review 71 Second Nine Weeks Exam Anderson School District Five Page 15 Copyright July 1, 2012

16 Unit 7 Vectors and Parametrics 9 days Second Nine Weeks 72 Graph parametric equations over a specified interval for the parameter Write x-y equation for the set of parametric equations by removing the parameter Determine the magnitude and direction of vectors. N-VM.1 (+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitude (e.g., v, v, v, v) Write vectors in both component form and as the linear combination of the standard unit vector. N-VM.2 (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. N-VM.3 (+) Solve problems involving velocity and other quantities that can be represented by vectors Find the slope of the line given the angle of inclination. PC-5.15 Carry out a procedure to compute the slope of a line when given the angle of inclination of the line. 81 Unit Review 82 Unit Test Anderson School District Five Page 16 Copyright July 1, 2012

17 Unit 8 Polar Coordinates 10 days Second Nine Weeks 83 Plot points in the polar coordinate system Use the polar coordinate system to draw the graphs of polar equations. Change polar coordinates to rectangular coordinates and vice-versa. PC-5.3 Carry out a procedure to plot points in the polar coordinate system Write polar equations as rectangular equations and vice-versa. Unit Review 90 Unit Test Anderson School District Five Page 17 Copyright July 1, 2012

18 Second Nine Weeks Anderson School District Five Page 18 Copyright July 1, 2012

19 Unit 9 Conic Sections 10 days Third Nine Weeks PC-6.1 Carry out a procedure to graph the circle whose equation is of the form (x-h) 2 + (y-k) 2 = r Write the general and standard forms of the equations of circles and graph it. PC-6.2 G-GPE.3 Analyze given information about the center and the radius or the center and the diameter to write an equation of a circle. (+) 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 Write the general and standard forms of the equations of ellipses and graph it. PC-6.4 G-GPE.3 Carry out a procedure to graph the ellipse whose 2 2 x h y k equation is of the form a b (+) 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 Write the general and standard forms of the equations of hyperbolas and graph it. PC-6.5 G-GPE.3 Carry out a procedure to graph the hyperbola whose 2 2 x h y k equation is of the form a b (+) 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. Anderson School District Five Page 19 Copyright July 1, 2012

20 Unit 9 Conic Sections 10 days Third Nine Weeks PC-6.6 Carry out a procedure to graph the parabola whose equation is of the form y k a x h Write the general and standard forms of the equations of parabolas and graph it. G-GPE.3 (+) 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. PC-6.1 Carry out a procedure to graph the circle whose equation is of the form (x-h) 2 + (y-k) 2 = r Recognize the graphs & equations of all conic sections PC-6.4 PC-6.5 Carry out a procedure to graph the ellipse whose 2 2 x h y k equation is of the form a b Carry out a procedure to graph the hyperbola whose 2 2 x h y k equation is of the form a. b PC-6.6 Carry out a procedure to graph the parabola whose equation is of the form y k a x h 2. G-GPE.3 (+) 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. Anderson School District Five Page 20 Copyright July 1, 2012

21 Unit 9 Conic Sections 10 days Third Nine Weeks PC-6.3 Apply a procedure to calculate the coordinates of points where a line intersects a circle. 98 Find points of intersection between quadratics, lines, and conics. A-REI.8 (+) Represent a system of linear equations as a single matrix equation in a vector variable. A-REI.9 (+) 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 x 3 or greater). 99 Unit Review 100 Unit Test Anderson School District Five Page 21 Copyright July 1, 2012

22 Unit 10 Linear and Quadratic Functions 10 days Third Nine Weeks 101 Write equations of and graph linear functions, including parallel, perpendicular, horizontal and vertical lines Graph quadratic functions given various forms and write a quadratic function given a graph PC-3.1 Carry out a procedure to graph quadratic and higherorder polynomial functions by analyzing intercepts and end behavior. 104 Factoring Quadratics 105 Solve Quadratics by Factoring 106 Computing and simplifying imaginary numbers. 107 Solve quadratic equations (including imaginary roots) by quadratic formula PC-3.6 PC-3.7 Carry out a procedure to solve polynomial equations algebraically. Carry out a procedure to solve polynomial equations graphically. 108 Solve quadratic word problems PC-3.5 Analyze given information to write a polynomial function that models a given problem situation. 109 Unit Review 110 Unit Test Anderson School District Five Page 22 Copyright July 1, 2012

23 Unit 11 Polynomial Functions 12 days Third Nine Weeks Recognize higher order polynomial functions, sketch parent graphs to show patterns of graphs of other polynomials and their end behavior. PC-2.1 PC-3.1 Carry out a procedure to graph parent functions (including y = x n ). Carry out a procedure to graph quadratic and higherorder polynomial functions by analyzing intercepts and end behavior. 114 Dividing Polynomials PC-3.2 Apply the rational root theorem to determine a set of possible rational roots of a polynomial equation Use synthetic division/substitution to find the zeros of a polynomial. Use these and patterns to sketch the polynomial. To write a polynomial function given linear factors or a graph. (Rational Root Theorem) PC-3.3 PC-3.6 Carry out a procedure to calculate the zeros of polynomial functions when given a set of possible zeros. Carry out a procedure to solve polynomial equations algebraically. PC-3.7 Carry out a procedure to solve polynomial equations algebraically. Anderson School District Five Page 23 Copyright July 1, 2012

24 Unit 11 Polynomial Functions 12 days Third Nine Weeks Solve polynomial inequalities PC-3.10 PC-3.11 Carry out a procedure to solve polynomial inequalities algebraically. Carry out a procedure to solve polynomial inequalities graphically. 121 Unit Review 122 Unit Test Anderson School District Five Page 24 Copyright July 1, 2012

25 Unit 12 - Exponents 13 days Third Nine Weeks 123 Laws of exponents 124 Simplifying rational exponents Advanced simplifying include factoring with rational exponents PC-4.5 Apply the laws of exponents to solve problems involving rational exponents. 127 Solve exponential equations with like bases PC-4.8 PC-4.9 Carry out a procedure to solve exponential equations algebraically. Carry out a procedure to sole exponential equations graphically. PC-2.1 Carry out a procedure to graph parent functions (including y=a x and y=e x ). 128 Graphs of exponential equations determine the domain & range. PC-2.5 Analyze graphs, tables, and equations to determine the domain and range of parent functions or transformations of parent functions (including y=a x and y=e x ). 129 Unit Review 130 Unit Test Exam Review rd Nine Weeks Exam PC-4.3 Carry out procedures to determine characteristics of exponential functions (including domain, range, intercepts, and asymptotes). Anderson School District Five Page 25 Copyright July 1, 2012

26 Third Nine Weeks Anderson School District Five Page 26 Copyright July 1, 2012

27 Unit 13 Logarithmic Functions 12 days Fourth Nine Weeks PC-2.1 Carry out a procedure to graph parent functions (including y log a x, y ln x). 136 Find the inverse of a function, sketch and give its domain and range. PC-2.5 Analyze graphs, tables, and equations to determine the domain and range of parent functions or transformations of parent functions (including y log a x, y ln x). PC-2.8 Carry out a procedure to determine whether the inverse of a function exists. PC-2.1 Carry out a procedure to graph parent functions (including y=a x and y=e x ). PC-2.5 Analyze graphs, tables, and equations to determine the domain and range of parent functions or transformations of parent functions (including y=a x and y=e x ). 137 Introduce logarithmic function as the inverse of the exponential function. PC-2.8 Carry out a procedure to determine whether the inverse of a function exists. PC-4.4 Carry out procedures to determine characteristics of exponential functions (including domain, range, intercepts, and asymptotes). F-IF.7d (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Anderson School District Five Page 27 Copyright July 1, 2012

28 Unit 13 Logarithmic Functions 12 days Fourth Nine Weeks 138 Write logarithmic expressions as exponential and vice versa PC-2.8 PC-4.5 Carry out a procedure to determine whether the inverse of a function exists. Apply the laws of exponents to solve problems involving rational exponents Use the laws of logarithms to simplify or expand logarithmic expressions. PC-4.5 Apply the laws of exponents to solve problems involving rational exponents. PC-4.7 Apply the laws of logarithms to solve problems. PC-4.8 Carry out a procedure to solve exponential equations algebraically Solve logarithmic and exponential equations with and without calculators including natural logs. PC-4.9 PC-4.10 Carry out a procedure to sole exponential equations graphically. Carry out a procedure to solve logarithmic equations algebraically. PC-4.11 Carry out a procedure to solve logarithmic equations graphically. Anderson School District Five Page 28 Copyright July 1, 2012

29 Unit 13 Logarithmic Functions 12 days Fourth Nine Weeks PC-4.6 Analyze given information to write an exponential function that models a given problem situation Solve exponential growth/decay problems and compound interest problems. PC-4.8 Carry out a procedure to solve exponential equations algebraically. PC-4.10 Carry out a procedure to solve logarithmic equations algebraically. 146 Unit Review 147 Unit Test Anderson School District Five Page 29 Copyright July 1, 2012

30 Unit 14 Rational Functions and Limits 15 days Fourth Nine Weeks 148 Identify and sketch graphs of transformations of absolute value and square root functions from the parent graphs and determine the domain and range of each function. PC-2.1 PC-2.5 Carry out a procedure to graph parent functions. Analyze graphs, tables, and equations to determine the domain and range of parent functions or transformations of parent functions. 149 Identify and sketch parent graphs of transformations of the greatest integer function and determine the domain and range of the function. PC-2.1 PC-2.5 Carry out a procedure to graph parent functions. Analyze graphs, tables, and equations to determine the domain and range of parent functions or transformations of parent functions. 150 Identify and sketch parent graphs of transformations of piecewise functions and determine the domain and range of each function. PC-2.1 PC-2.5 Carry out a procedure to graph parent functions. Analyze graphs, tables, and equations to determine the domain and range of parent functions or transformations of parent functions. 151 Evaluate limits as x approaches infinity, describing the end behavior. PC-3.4 Carry out procedures to determine characteristics of rational functions (including domain, range, intercepts, asymptotes, and discontinuities). Anderson School District Five Page 30 Copyright July 1, 2012

31 Unit 14 Rational Functions and Limits 15 days Fourth Nine Weeks 152 Evaluate limits as x approaches a, describing local behavior. PC-3.4 Carry out procedures to determine characteristics of rational functions (including domain, range, intercepts, asymptotes, and discontinuities) Sketch the graph of rational functions using limits. PC-3.4 Carry out procedures to determine characteristics of rational functions (including domain, range, intercepts, asymptotes, and discontinuities). PC-3.8 Carry out a procedure to solve rational equations algebraically Solve rational equations. PC-3.9 Carry out a procedure to solve rational equations graphically Evaluate limits, use properties of limits ATP 161 Unit Review 162 Unit Test Anderson School District Five Page 31 Copyright July 1, 2012

32 Unit 15 Sequences and Series 18 days Fourth Nine Weeks 163 Introduction to sequences and series looking at patterns inductively 164 Work with factorials and sigma notation. 165 Find the general term of an arithmetic sequence Find the partial sum of an arithmetic series. Include sigma notation. 168 Find the general term of a geometric Sequence Find the partial sum of a geometric series. Include sigma notation. 171 Find the sum of an infinite series 172 Use the Binomial Theorem or Pascal s triangle to expand a binomial to a power and find specific terms of an expansion. 173 Unit Review 174 Unit Test End of Review th Nine Weeks Exam Anderson School District Five Page 32 Copyright July 1, 2012