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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 – Pre-calculus with Limits; Larson/Hostetler 1st 3 weeks 2nd 3 weeks 3rd 3 weeks 4th 3 weeks Rectangular Coordinates Linear and NonLinear Systems of Equation Graphs of Equations Linear Equations in Two Variables Two-Variable Linear Systems Polynomials and Rational Functions Quadratic Equations and Inequalities Functions Multivariable Linear Systems Remainder and Factor Theorems Parent functions Partial Fractions Rational Root Theorem Transformations Systems of Inequalities Inverse Functions Analyzing graphs Matrices and Systems of Equations 5th 3 weeks 6th 3 weeks Sequences and Series Circle Exponential and Logarithmic Functions Parabola Rational Exponents Ellipse Number e Hyperbola Common Logarithms Conics Transformations Natural Logarithms Locating Zeros of a Function Systems of Second Degree Equations and Inequalities Rational Equations and Partial Fractions Tangents and Normals to Conic Sections Exponential Growth Chapter 1, Appendix A Chapter 16(f) Teks – P.1A, P.1B, P.1C, P.1D, P.1E, P.2A, P.2B, P.2C, P.3A Chapter 7, Chapter 8 P.1B, P.2B, P.3A, P.3B, P.3C, Chapter 2 Chapter 16(f) Teks – P.1C, P.1D, P.1E, P.3A Geometric Infinite Convergent and Divergent Sigma Notation and the nth Term Binomial Theorem Permutations Probability and Odds Combinations Radical Equations and Inequalities Resource: Precalculus with Limits; Larson/Hostetler, Foerster (f) Arithmetic Chapter 10 Chapter 12 & 13 (f) Teks – P.1D, P.2A, P.5A, P.5B, P.5C Chapter 3 Teks – P.1A, P.1B, P.1D, P.1E, P.2C, P.3A Chapter 9 Teks – P.1B, P.3A, P.3B, P.3C, P.4A, P.4B, P.4C, P.4D Three Weeks Topics/ Concepts Pre-Calculus Pre-AP - Second Semester – Pre-calculus with Limits; Larson/Hostetler 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: Precalculus with Limits; Larson/Hostetler Foerster (f) Chapters 4, 5, 6 Teks – P.1C, P.2B, P.3A, P.3C, P.3D, P.3E Chapters 4, 5, 6 Teks – P.2C, P.3A, P.3C, P.3D 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 Chapters 4, 5, 6 Teks – P.1A, P.1B , P.1C, P.1D, P.2A, P.2B, P.2C, P.3A, P.3B, P.3C 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 6 Teks – P.5A, P.5C, P.5D, P.6A, P.6B Powers and Roots of Complex Numbers Chapter 10 Teks – P.1A, P.1B, P.2A, P.3D Chapter 12 Chapter 16(f) *Teks 111.54b Calculus AB * These Teks, 111.54b Calculus AB, have been included for cross reference due to the lack of in-depth explanation with-in the PreCalculus Teks. Functions and Their Graphs (1st 3 weeks) Essential Learning Outcomes TEKS 111.35 The Student Will: P.1A, P.1B, P.1C, P.1D, • Identify functions by definition and graph and state their domain and range P.1E, P.2A, • Discuss continuity; determine intervals P.2B, P.2C, P.3A of increase, decrease or constant; describe end behavior • Find the sum, difference, product, and quotient of two functions • Find the composition of two functions given their equations and be able to determine functions given their composition. • Be able to reflect graphs of functions over x-axis, y-axis and line y = x. • Be able to test a function for symmetry and determine if it is odd, even or neither algebraically and from the graph. • Given the graph of y = f(x), be able to graph a f(x), f(x) + d, f(x - c), f(bx), f( x ) , and f(|x|). • • • • Given a graph that has a transformation, write the equation for the graph. (Optional to address this in this unit) Determine if a function is periodic and, if so, give its period and amplitude and graph it using shifts and stretches. Find the inverse of a function and, given two functions, determine if they are inverses of each other algebraically and graphically. Solve application problems using composite functions. Topics (not in sequential order) Polynomials and Rational Functions Quadratic Equations and Inequalities Remainder and Factor Theorems Rational Root Theorem Locating Zeros of a Function Rational Equations and Partial Fractions Radical Equations and Inequalities Suggested Resources Chapter 1, Appendix A Assessments TAKS Objectives 1,2,3,4 Conics (2nd 3 weeks) Essential Learning Outcomes The Student Will: • • • • • • • solve 2 X 2 systems − linear combination − substitution − graphing (with a graphing calculator) intersecting lines parallel lines lines that coincide − table (graphing calculator) − matrices define a matrix and use a matrix to represent data inverse matrices by hand (2X2 only) inverse matrices with calculator use operations of matrices to solve problems systems of inequalities − graphing calculator − graph by hand − Test a point Solve 3 X 3 systems − Matrices (calculator) Perform matrix multiplication to show multiplication is not commutative − By hand (demo only) − With technology determine what a solution to a system of equations/inequalities means in relationship to the problem determine if the solution to a system of equations/inequalities is reasonable for given contexts connect algebraic solutions to graphical and tabular solutions TEKS 111.35 P.1B, P.2B, P.3A, P.3B, P.3C, Topics (not in sequential order) Circle Parabola Ellipse Hyperbola Transformations Systems of Second Degree Equations and Inequalities Tangents and Normals to Conic Sections Suggested Resources Chapter 7, Chapter 8 Assessments TAKS Objectives 1,2,3,4,5,6,7 Polynomials and Rational Functions (3rd 3 weeks) Essential Learning Outcomes The Student Will: • Identify and classify polynomials by degree • Find zeros (roots) and factors of polynomial functions (equations) • Use synthetic substitution/division • Apply Theorems: Remainder, Factor, Rational Root, Complex Conjugate, and Fundamental Theorem of Algebra • Graph a polynomial function given a factorable equation • Write the equation for a polynomial function given its graph or a set of data • Find maximum and minimum values with or without a graphing calculator and apply to situations (area, volume, cost, distance, etc.) • Solve and graph (on a number line) polynomial inequalities with answer in interval notation • Sketch the graph of a polynomial function using properties of end behavior, zeros, and multiplicity of zeros • Graph rational functions, including those containing more than one vertical asymptote, an oblique asymptote, removable discontinuities, and a crossover of the horizontal or oblique asymptote by locating the asymptotes, “holes”, and intercepts as transformations of one of the parent functions • f ( x) = 1 1 or f ( x ) = 2 . x x Manipulate the function rule algebraically to obtain graphing form. (Example: Use long division to rewrite the • • • • • x+3 1 as 1 + x+2 x+2 Describe the continuity and limits of the function. State the domain and range of the function and write the equations of its asymptotes. Write a rational function given the graph, or description of characteristics such as a verbal description of the transformations of the parent function, the asymptote equations, or function notation showing the transformations. Make connections between the graphical, tabular, verbal, and symbolic representations of the function. Describe the symmetry of the function by identifying it as odd or even. TEKS 111.35 P.1C, P.1D, P.1E, P.3A Topics (not in sequential order) Polynomials and Rational Functions Quadratic Equations and Inequalities Remainder and Factor Theorems Rational Root Theorem Locating Zeros of a Function Rational Equations and Partial Fractions Radical Equations and Inequalities Suggested Resources Chapter 2 Assessments TAKS Objectives 1,2,3,4,5 Conics (4th 3 weeks) Essential Learning Outcomes The Student Will: • Know the geometric and algebraic descriptions of circle, parabola, hyperbola, and ellipse • Represent conic sections in standard form (graphing form) and in parametric form and graph • Determine the equation of a conic given the graph • Determine the equation and graph of a conic from given information • Apply properties of conic sections to solve real-life problems (both as parametric and rectangular) ¾ orbits of planets ¾ LORAN • Convert from general to standard form • Describe how properties of conics are used in the physical world, such as ¾ reflective properties (flashlight, whispering room, lithotripsy. etc.) ¾ elliptipool ¾ cooling towers for nuclear reactors • (optional) graphs of rotated conics TEKS 111.35 P.5A, P.5C, P.5D, P.6A, P.6B Topics (not in sequential order) Conics Circle Parabola Ellipse Hyperbola Transformations Systems of Second Degree Equations and Inequalities Tangents and Normals to Conic Sections Suggested Resources Chapter 10, Chapter 12 & 13(f) Assessments TAKS Objectives 1,2,3,4,5,6,10 Exponential and Logarithmic Functions (5th 3 weeks) Essential Learning Outcomes TEKS Topics (not in sequential order) The Student Will: • Know the definition of an exponential function, identify restrictions and graph • Define the logarithmic function as the inverse of an exponential function and graph • Know and apply laws of integral and rational exponents • Apply transformations to exponential and logarithmic functions • Know restrictions on the log function • Identify intervals where graph of an exponential function is increasing or decreasing, as well as its domain, range, and zeros • Use exponential growth and decay formulas to solve application problems: A = A0 (1 + r ) t t A = A0 b k r⎞ ⎛ A = A0 ⎜ 1 + ⎟ n⎠ ⎝ nt A = A0 e rt • • • Use rational exponents to simplify expressions and solve equations Write an exponential function from a set of data Know the definition of e and the function f(x) = ex and use it in application problem. P.1A, P.1B, P.1D, P.1E, P.2C, P.3A Exponential and Logarithmic Functions Rational Exponents Number e Common Logarithms Natural Logarithms Exponential Growth Suggested Resources Chapter 3 Assessments TAKS Objectives 1,2,3,4,5,8 Sequences and Series (6th 3 weeks) Essential Learning Outcomes TEKS The Student Will: • Identify a sequence as arithmetic (linear), geometric (exponential when r > 0), or other • Define a sequence both explicitly and recursively • Explore representations of a sequence on a graph and table • Use sequences defined recursively to solve problems • Find the sum of the first n terms and the value of a specific term of an arithmetic and geometric series • Find or estimate the limit of an infinite sequence or determine that the limit does not exist • Find the sum of an infinite geometric series when possible • Use sigma notation to represent series • Expand a binomial of the form (a + b ) • n and find the nth term of a binomial expansion Use sequences and series to solve real-life problems P.1B, P.3A, P.3B, P.3C, P.4A, P.4B, P.4C, P.4D Topics (not in sequential order) Suggested Resources Sequences and Series Chapter 9 Arithmetic Geometric Infinite Convergent and Divergent Sigma Notation and the nth Term Binomial Theorem Permutations Probability and Odds Combinations Assessments TAKS Objectives 1,2,3,4,5,8 Trigonometric Functions (1st 3 weeks, 2nd Semester) Essential Learning Outcomes The Student Will: • Unit Circle: know definition of radians, degree measure, standard position, coterminal angles, quadrantal angles, coordinates, and the six trig functions • Understand the concept of radian measure as a location on the Unit Circle) • Define sine, cosine and tangent in terms of x, y and r • Convert between radian measure and degree measure of special angles (multiples π π π π • , , , ) 6 4 3 2 Know the sine, cosine and tangent values of , , , ) 6 4 3 2 Find the values of cotangent, secant, and cosecant from sine, cosine, and tangent Find the reference angle of any given angle Recognize equivalent fractions created by rationalized denominators Find arc length and area of a sector (may be taught in another unit) Solve right triangles using sin, cos, and tan Find area of triangles Use Law of Sines and Law of Cosines Solve application problems involving navigation and surveying, including angles of elevation and depression, area of parallelogram using area of triangle. special angles (multiples • • • • • • • • π π π π TEKS 111.35 P.1C, P.2B, P.3A, P.3C, P.3D, P.3E Topics (not in sequential order) Suggested Resources Trigonometric Functions Chapter 4, 5, 6 Angles and Their Measure Central Angles and Arcs Circular Functions Special Angles Right Triangles Law of Sines Law of Cosines Area of Triangles Assessments TAKS Objectives 1,2,3,4 Graphs and Inverses of Trigonometric Functions (2nd 3 weeks, 2nd Semester) Essential Learning Outcomes TEKS Topics (not in sequential 111.35 order) The Student Will: P.1A, Graphs and Inverses of • Graph all six trig functions and identify their P.1B , Trigonometric Functions P.1C, period, amplitude, domain, range and zeros P.1D, Amplitude, Period and Phase • Graph all six trig functions with parameter P.2A, Shift changes (i.e. P.2B, y = a sin b(x – c) + d; f( x ) , and f(|x|) are P.2C, Principal Values of Inverses optional) P.3A, • Write/identify the equation of a trig function P.3B, Simple Harmonic Motion from its graph P.3C • State whether a trig function graph is odd or even • Investigate continuity, end behavior, vertical asymptotes, increasing/decreasing (especially tan, cot, sec, and csc) • Model real-life data using sine and cosine functions with and without a regression equation • Define and graph principle trig functions (1 to 1 functions) and their inverses (sine, cosine, and tangent). • Make connections between graphs of inverse trig functions and the Unit Circle). • Find values of inverse trig functions with and without the calculator (sine, cosine, and tangent) • Sketch the graphs of the functions Sin-1, Cos1 , Tan-1 and state the domain and range. (Emphasize the range of inverse sine and tangent) Suggested Resources Chapter 4, 5, 6 Assessments TAKS Objectives 1,2,3,4,5 Trigonometric Identities and Equations (3rd 3 weeks, 2nd Semester) Essential Learning Outcomes TEKS Topics (not in sequential order) 111.35 The Student Will: Trigonometric Identities and Equations P.2C, P.3A, • Know and apply the following Sum and Difference Identities P.3C, identities: P.3D → Pythagorean Double Angle and Half-Angle Identities → Reciprocal → Quotient Solving Trigonometric Equations → Negative angle Normal Form of a Linear Equation → Cofunctions → Sum and difference Distance form a Point to a Line → Double angle → Half angle (optional) • Prove trig identities • Simplify and expand trig expressions • Solve trig equations Suggested Resources Chapter 4, 5, 6 Assessments TAKS Objectives 1,2,3,4,5,6 Vectors and Parametric Equations (4th 3 weeks, 2nd Semester) Essential Learning Outcomes TEKS Topics (not in sequential order) 111.35 P.5A, Vectors and Parametric Equations The Student Will: P.5C, • Using geometric and algebraic P.5D, Geometric and Algebraic Vectors representation of vectors, add, P.6A, subtract, and multiply by a scalar P.6B 3D Space and find the magnitude (length) and direction Perpendicular Vectors • Show, analyze, and solve navigation and force problems Applications with Vectors using vectors and trig • Use vectors, vector equations, and Motion Modeling parametric equations to show an object’s motion. • Use polar coordinates to plot points on a graph • Determine if two vectors are parallel or perpendicular using dot product • Calculate the angle between two vectors • Work with 3-dimensional vectors to find their magnitude and midpoint • Find equations of planes parallel and perpendicular to another plane • Write complex numbers and find products in polar form • Find powers and roots of complex numbers • Convert polar coordinates to rectangular coordinates and vice versa Suggested Resources Chapter 6 Assessments TAKS Objectives 1,2,3,4,5,6 Polar Coordinates and Complex Numbers (5th 3 weeks, 2nd Semester) Essential Learning Outcomes TEKS Topics (not in sequential order) 111.35 Polar Coordinates and Complex Numbers P.1A, The Student Will: P.1B, • Determine if two vectors are Polar Graphs P.2A, parallel or perpendicular using P.3D dot product Polar-Rectangular Coordinates • Calculate the angle between two vectors Polar Form of Linear Function • Work with 3-dimensional vectors to find their magnitude Polar Form of Complex Numbers and midpoint • Find equations of planes parallel Products and Quotients of Complex Numbers and perpendicular to another plane Powers and Roots of Complex Numbers • Write complex numbers and find products in polar form • Find powers and roots of complex numbers • Convert polar coordinates to rectangular coordinates and vice versa Suggested Resources Chapter 10 Assessments TAKS Objectives 2, 5 Limits and an introduction to Calculus (16th 3 weeks, 2nd Semester) Essential Learning Outcomes TEKS Topics (not in sequential order) The student will • Estimate limits using graphical and numerical approaches • Learn different ways a limit can fail to exist • Evaluate limits using properties of limits • Develop and use a strategy for finding limits • Evaluate a limit using division and rationalization techniques • Use properties of continuity • Determine infinite limits from the left and right • Find and sketch the vertical asymptotes of the graph of a function • Use limits to justify vertical asymptotes • Find limits as x approaches infinity • Find and sketch the horizontal asymptotes of the graph of a function • Find the slope of the tangent line to a curve at a point • Use the limit definition to find the derivative of a function • Understand the relationship between differentiability and continuity • Find the derivative of a function using the basic differentiation rules All 111.35 Applied Problems *111.54b Calculus AB Connections to Calculus Limits Derivatives Suggested Resources Chapter 12 Chapter 16(f) Assessments TAKS Objectives N/A