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Pre-Calculus Pre-AP – Scope and Sequence - Year at a Glance Three Weeks Topics/ Concepts Pre-Calculus Pre-AP - First Semester – Merrill – Advanced Mathematical Concepts 1st 3 weeks 2nd 3 weeks 3rd 3 weeks 4th 3 weeks Polynomials and Rational Functions Quadratic Equations and Inequalities Remainder and Factor Theorems Circle Exponential and Logarithmic Functions Parabola Rational Exponents Ellipse Number e Conics Hyperbola Rational Root Theorem Common Logarithms Transformations Locating Zeros of a Function Rational Equations and Partial Fractions Radical Equations and Inequalities Natural Logarithms Systems of Second Degree Equations and Inequalities Sequences and Series Arithmetic Geometric Infinite 5th 3 weeks 6th 3 weeks Combinatorics and Probability Statistics and Data Analysis Permutations Frequency Distribution Repetitions and Circular Convergent and Divergent Combinations Sigma Notation and the nth Term Probability and Odds Binomial Theorem Mathematical Induction Tangents and Normals to Conic Sections Measures of Central Tendency, Variability Normal Distribution Sample Sets of Data Independent and Dependent Events Scatter Plots and Regression Lines Mutually Exclusive or Inclusive Events Conditional Probability Resource: Merrill Advanced Mathematical Concepts Foerster (f) Chapter 4 Chapter 16(f) Teks - c1C, c1D, c1E, c3A Chapter 10 Chapter 12 & 13 (f) Teks - c1D, c2A, c5A, c5B, c5C Chapter 11 Teks - c1A, c1B, c1D, c1E, c2C, c3A Chapter 12 Teks - c4A, c4B, c4C, c4D Binomial Theorem and Probability Chapter 14 Teks – 111.35, c1B, c3A, c3B, c3C 111.36c 4A, 4B 111.24b 8.11A, 8.11B, 8.11C Chapter 15 Teks – 111.36c, 2A, 2B, 2C, 2D, 3A, 3B, 3C Three Weeks Topics/ Concepts Pre-Calculus Pre-AP - Second Semester – Merrill – Advanced Mathematical Concepts 1st 3 weeks 2nd 3 weeks 3rd 3 weeks 4th 3 weeks Trigonometric Functions Angles and Their Measure Central Angles and Arcs Graphs and Inverses of Trigonometric Functions Amplitude, Period and Phase Shift Principal Values of Inverses Circular Functions Special Angles Simple Harmonic Motion Trigonometric Identities and Equations Sum and Difference Identities Double Angle and Half-Angle Identities Solving Trigonometric Equations Right Triangles Vectors and Parametric Equations Geometric and Algebraic Vectors Perpendicular Vectors Applications with Vectors Resource: Merrill Advanced Mathematical Concepts Foerster (f) Chapter 5 Teks - c1C, c2B, c3A, c3C, c3D Chapter 7 Teks - c2C, c3A, c3C, c3D Applied Problems Connections to Calculus Polar Graphs Polar-Rectangular Coordinates Derivatives Polar Form of Linear Function Polar Form of Complex Numbers Products and Quotients of Complex Numbers Distance form a Point to a Line Chapter 6 Teks - c1A, c1B , c1C, c1D, c2A, c2B, c2C, c3A, c3B, c3C Polar Coordinates and Complex Numbers Motion Modeling Normal Form of a Linear Equation Area of Triangles 6th 3 weeks Limits 3D Space Law of Sines Law of Cosines 5th 3 weeks Chapter 8 Teks - c5A, c5C, c5D, c6A, c6B Powers and Roots of Complex Numbers Chapter 9 Teks - c1A, c1B, c2A, c3D Chapter 17 Various Chapters Foerster and Larson Books Polynomials and Rational Functions Objectives/concepts TEKS 111.35 Describe symmetry of c1C graphs of even and odd functions. (1st 3 weeks) Topics (not in sequential order) Suggested Resources Polynomials and Rational Functions 4.1-4.7 Recognize and use connections among significant points of a function (roots, maximum points, and minimum points), the graph of a function, and the symbolic representation of a function. Remainder and Factor Theorems Investigate continuity, end behavior, vertical and horizontal asymptotes, and limits and connect these characteristics to the graph of a function. Use functions such as logarithmic, exponential, trigonometric, polynomial, etc., to model real life data. Quadratic Equations and Inequalities c1D Rational Root Theorem Locating Zeros of a Function Rational Equations and Partial Fractions Radical Equations and Inequalities c1E c3A Assessments TAKS Objectives 1,2,3,4 Conics (2nd 3 weeks) Objectives/concepts Recognize and use connections among significant points of a function (roots, maximum points, and minimum points), the graph of a function, and the symbolic representation of a function. Apply basic transformations, including a • f(x), f(x) + d, f(x - c), f(b • x), |f(x)|, f(|x|), to the parent functions. TEKS 111.35 c1D Suggested Resources Circle 10.1-10.8 Parabola Ellipse c2A Hyperbola Transformations Systems of Second Degree Equations and Inequalities Use conic sections to model motion, such as the graph of velocity vs. position of a pendulum and motions of planets. c5A Use properties of conic sections to describe physical phenomena such as the reflective properties of light and sound. c5B Convert between parametric and rectangular forms of functions and equations to graph them. Topics (not in sequential order) Tangents and Normals to Conic Sections c5C Assessments TAKS Objectives 1,2,3,4,5,6,7 Exponential and Logarithmic Functions (3rd 3 weeks) Objectives/concepts TEKS Topics (not in sequential order) 111.35 Exponential and Logarithmic Functions Describe parent functions symbolically c1A and graphically, including y = xn , y = Rational Exponents ln x, y = loga x, y = 1/x, y =ex, y = ax, y = sin x, etc. Number e c1B Determine the domain and range of Common Logarithms functions using graphs, tables, and symbols. Natural Logarithms Recognize and use connections among significant points of a function (roots, c1D maximum points, and minimum points), the graph of a function, and the symbolic representation of a function. Investigate continuity, end behavior, vertical and horizontal asymptotes, and limits and connect these characteristics to the graph of a function. Investigate identities graphically and verify them symbolically, including logarithmic properties, trigonometric identities, and exponential properties. Use functions such as logarithmic, exponential, trigonometric, polynomial, etc. to model real-life data. c1E c2C c3A Suggested Resources 11.1-11.7 Assessments TAKS Objectives 1,2,3,4,5 Sequences and Series (4th 3 weeks) Objectives/concepts Represent patterns using arithmetic and geometric sequences and series. Use arithmetic, geometric, and other sequences and series to solve real-life problems. Describe limits of sequences and apply their properties to investigate convergent and divergent series. Apply sequences and series to solve problems including sums and binomial expansion. TEKS 111.35 c4A Topics (not in sequential order) Suggested Resources Sequences and Series 12.1-12.8 Arithmetic Geometric c4B Infinite Convergent and Divergent Sigma Notation and the nth Term c4C Binomial Theorem Mathematical Induction c4D Assessments TAKS Objectives 1,2,3,4,5,6,10 Combinatorics and Probability (5th 3 weeks) Objectives/concepts TEKS Compare theoretical and empirical probability. 111.36c 4A Use experiments to determine the reasonableness of a theoretical model such as binomial, geometric. 4B Find the probabilities of compound events (dependent and independent). 111.24b 8.11A Use theoretical probabilities and experimental results to make predictions and decisions. 8.11B Select and use different models to simulate an event. 8.11C Topics (not in sequential order) Suggested Resources Combinatorics and Probability 14.1-14.8 Permutations Repetitions and Circular Combinations Probability and Odds Independent and Dependent Events Mutually Exclusive or Inclusive Events Conditional Probability Determine the domain and range of functions using graphs, tables, and symbols. Use functions such as logarithmic, exponential, trigonometric, polynomial, etc., to model real life data. Use regression to determine a function to model real life data. Use properties of functions to analyze and solve problems and make predictions Binomial Theorem and Probability 111.35 c1B c3A c3B c3C Assessments TAKS Objectives 1,2,3,4,5,8 Statistics and Data Analysis (6th 3 weeks) Objectives/concepts TEKS 111.36c 2A Interpret information from various graphs, including line graphs, bar graphs, circle graphs, histograms, and scatterplots to draw conclusions from the data. Analyze numerical data using measures of central tendency, variability, and correlation in order to make inferences. Topics (not in sequential order) Suggested Resources Statistics and Data Analysis 15.1-15.6 Frequency Distribution Measures of Central Tendency, Variability Normal Distribution 2B Sample Sets of Data Scatter Plots and Regression Lines Analyze graphs from journals, newspapers, and other sources to determine the validity of stated arguments. Use regression methods available through technology to describe various models for data such as linear, quadratic, exponential, etc., select the most appropriate model, and use the model to interpret information. Formulate a meaningful question, determine the data needed to answer the question, gather the appropriate data, analyze the data, and draw reasonable conclusions. Communicate methods used, analysis conducted, and conclusions drawn for a dataanalysis project by written report, visual display, oral report, or multimedia presentation. Determine the appropriateness of a model for making predictions from a given set of data 2C 2D 3A 3B 3C Assessments TAKS Objectives 1,2,3,4,5,8 Trigonometric Functions (7th 3 weeks) Objectives/concepts TEKS Topics (not in sequential order) 111.35 Trigonometric Functions Describe symmetry of c1C graphs of even and odd Angles and Their Measure functions. Perform operations including composition on functions, find inverses, and describe these procedures and results verbally, numerically, symbolically, and graphically. Central Angles and Arcs c2B Circular Functions Special Angles Right Triangles Law of Sines Use functions such as logarithmic, exponential, trigonometric, polynomial, etc., to model real life data. c3A Use properties of functions to analyze and solve problems and make predictions. c3C Solve problems from physical situations using trigonometry, including the use of Law of Sines, Law of Cosines, and area formulas. c3D Law of Cosines Area of Triangles Suggested Resources 5.1-5.8 Assessments TAKS Objectives 1,2,3,4 Graphs and Inverses of Trigonometric Functions (8th 3 weeks) Objectives/concepts TEKS Topics (not in sequential 111.35 order) Graphs and Inverses of c1A Describe parent functions symbolically and Trigonometric Functions graphically, including y = xn , y = ln x, y = loga x, y = 1/x, y =ex, y = ax, y = sin x, etc. Amplitude, Period and Phase Shift c1B Determine the domain and range of functions using graphs, tables, and symbols. Principal Values of Inverses c1C Describe symmetry of graphs of even and odd Simple Harmonic Motion functions. Recognize and use connections among significant points of a function (roots, maximum points, and minimum points), the graph of a function, and the symbolic representation of a function. c1D Apply basic transformations, including a • f(x), f(x) + d, f(x - c), f(b • x), |f(x)|, f(|x|), to the parent functions. c2A Perform operations including composition on functions, find inverses, and describe these procedures and results verbally, numerically, symbolically, and graphically. c2B Investigate identities graphically and verify them symbolically, including logarithmic properties, trigonometric identities, and exponential properties. c2C Use functions such as logarithmic, exponential, trigonometric, polynomial, etc. to model real-life data. c3A Use regression to determine a function to model real-life data. c3B Use properties of functions to analyze and solve problems and make predictions. c3C Suggested Resources 6.1 – 6.7 Assessments TAKS Objectives 1,2,3,4,5 Trigonometric Identities and Equations (9th 3 weeks) Objectives/concepts TEKS Topics (not in sequential order) 111.35 Trigonometric Identities and Equations c2C Investigate identities graphically and verify them Sum and Difference Identities symbolically, including logarithmic properties, Double Angle and Half-Angle Identities trigonometric identities, and exponential properties. Solving Trigonometric Equations Use functions such as Normal Form of a Linear Equation c3A logarithmic, exponential, trigonometric, polynomial, Distance form a Point to a Line etc. to model real-life data. Use properties of functions to analyze and solve problems and make predictions. Solve problems from physical situations using trigonometry, including the use of Law of Sines, Law of Cosines, and area formulas. c3C c3D Suggested Resources 7.1-7.7 Assessments TAKS Objectives 1,2,3,4,5,6 Vectors and Parametric Equations (10th 3 weeks) Objectives/concepts TEKS Topics (not in sequential order) 111.35 Vectors and Parametric Equations c5A Use conic sections to model motion, such as the Geometric and Algebraic Vectors graph of velocity vs. position of a pendulum and 3D Space motions of planets. Convert between parametric and rectangular forms of functions and equations to graph them. Perpendicular Vectors c5C Applications with Vectors Motion Modeling Use parametric functions to simulate problems involving motion. Use the concept of vectors to model situations defined by magnitude and direction. Analyze and solve vector problems generated by real-life situations. c5D c6A c6B Suggested Resources 8.1-8.7 Assessments TAKS Objectives 1,2,3,4,5,6 Polar Coordinates and Complex Numbers (11th 3 weeks) Objectives/concepts TEKS Topics (not in sequential order) 111.35 Polar Coordinates and Complex Numbers c1A Describe parent functions symbolically and Polar Graphs graphically, including y = xn , y = ln x, y = loga x, y Polar-Rectangular Coordinates = 1/x, y =ex, y = ax, y = sin x, etc. Polar Form of Linear Function Determine the domain Polar Form of Complex Numbers c1B and range of functions using graphs, tables, and Products and Quotients of Complex Numbers symbols. Apply basic transformations, including a • f(x), f(x) + d, f(x - c), f(b • x), |f(x)|, f(|x|), to the parent functions. Solve problems from physical situations using trigonometry, including the use of Law of Sines, Law of Cosines, and area formulas. Powers and Roots of Complex Numbers c2A c3D Suggested Resources 9.1-9.8 Assessments TAKS Objectives 2, 5 Connections to Calculus, Applied Problems (12th 3 weeks) Objectives/concepts TEKS Topics (not in sequential order) Limits of Functions All 111.35 Applied Problems Asymptotic Behavior 111.54b Calculus AB Connections to Calculus Continuity Limits Concept of a Derivative Derivatives Suggested Resources Chapter 17 Merrill Various Chapters Foerster and Larson Books Assessments TAKS Objectives N/A