Volusia County Mathematics Curriculum Map Pre-Calculus Course Number 1202340/IOD Mathematics Department Volusia County Schools Revised June 9, 2012 Pre- Calculus Curriculum Map 120234/IOD
COMPONENTS OF THE CURRICULUM MAP Unit/Organizing Principle: the overarching organizational structure used to group content and concepts within the curriculum map Pacing: the recommended time period within the year for instruction related to the essential questions to occur Essential Questions: the overarching question(s) that will serve to guide instruction and push students to higher levels of thinking; essential questions should guide students to the heart of the content Measurement Topics: a list of the major underlying concepts covered in the development of the essential questions Learning Targets/Skills: the content knowledge, processes and enabling skills that will ensure successful mastery of the essential questions Benchmarks: the Next Generation Sunshine State Standards Academic Language: the content vocabulary and other key terms and phrases with which students should be familiar and that support mastery of the learning targets, skills and essential questions Activities and Resources: a listing of available, high quality and appropriate materials, strategies, lessons, textbooks, videos and other media sources that are aligned with the learning targets, skills and essential questions; developed to save teachers time when planning for instruction Assessment: a list of required formative assessments as well as suggested assessments that are available to use as formative or summative assessments
UNIT/ORGANIZING PRINCIPLE: Functions and their Graphs PACING: 1 st 9 weeks ESSENTIAL QUESTIONS: MEASUREMENT TOPICS Function Families Graphs Function Families Modeling Function Families Make sense of problems and persevere in solving them. Can students identify the domain and range from a relation, equation, or graph? Can students manipulate functions through transformations, operations, and compositions? The student will: LEARNING TARGETS/SKILLS describe the concept of a function, use function notation, determine whether a given relation is a function, and link equations to functions. Prior Knowledge determine the composition of functions. (Algebra 2) MA.912.A.2.8 determine the domain and range of a relation. Prior Knowledge identify and graph common functions (including but not, limited to linear, rational, quadratic, cubic, radical, and absolute value). BENCHMARKS ACADEMIC LANGUAGE (Algebra 2) MA.912.A.2.6 describe and graph transformations of functions. (Algebra 2) MA.912.A.2.10 perform operations (addition, subtraction, division, and multiplication) of functions algebraically, numerically, and graphically. (Algebra 2) MA.912.A.2.7 recognize, interpret, and graph functions defined piece-wise with and without technology. (Algebra 2) MA.912.A.2.9 solve problems involving functions and their inverses. (Algebra 2) MA.912.A.2.11 solve real-world problems involving relations and function. (Algebra 2) MA.912.A.2.1 Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Relation Function notation Types of Functions: Polynomial radical rational exponential Look for and make use of structure. logarithmic greatest integer function Piece-wise functions Combination of functions Composition of functions Even and Odd Functions Inverse Functions Increasing and Decreasing One to one functions Horizontal Line Test Look for and express regularity in repeated reasoning. MACC.K12.MP.1 MACC.K12.MP.2 MACC.K12.MP.3 MACC.K12.MP.4 MACC.K12.MP.5 MACC.K12.MP.6 MACC.K12.MP.7 MACC.K12.MP.8 Mathematics Department Volusia County Schools Revised June 9, 2012 Pre- Calculus Curriculum Map 120234/IOD
ESSENTIAL QUESTIONS: Can students identify the domain and range from a relation, equation, or graph? Can students manipulate functions through transformations, operations, and compositions Activities and Resources Assessment Pre Calculus with Limits (Larson, Hostetler, and Edwards): P.4 Solving Equations Algebraically and Graphically Chapter 1: Functions and their graphs
UNIT/ORGANIZING PRINCIPLE: Polynomials and Rational Functions PACING: 1 st 9 weeks Can students find the roots of a polynomial equation using different methods? ESSENTIAL QUESTIONS: Can students graph a rational functions with and without technology? Can students perform operations on complex numbers? MEASUREMENT LEARNING TARGETS/SKILLS TOPICS The student will: Polynomial Functions graph polynomial functions with and without technology and describe end Rational Functions Make sense of problems and persevere in solving them. behavior. understand and apply the Intermediate Value Theorem on a function over a closed interval. apply the theorem of polynomial behavior (including but not limited to the Fundamental Theorem of Algebra, Remainder Theorem, and the Rational Root Theorem, Descartes Rule of Signs, and the Conjugate Root Theorem) to find the zeros of a polynomial function. write a polynomial equation for a given set of real and/or complex roots. dthe relationship among the solutions of an equation, the zeros of a function, the x-intercepts of a graph, and the factors of a polynomial expression with and without technology. identify removable and non-removable discontinuities, and vertical, horizontal, and oblique asymptotes of a graph of a rational function, find the zeros, and graph the. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. BENCHMARKS ACADEMIC LANGUAGE MA.912.A.4.5 MA.912.C.1.12 MA.912.A.4.6 MA.912.A.4.7 MA.912.A.4.8 MA.912.A.5.6 Look for and make use of structure. `Polynomial Equations Quadratic equations Cubic Equations Rational Functions Roots, Solutions, Zeros, and x intercepts Vertex of a quadratic Completing the square General and vertex form Vertical and horizontal asymptotes Oblique (slant) asymptotes Fundamental Theorem of Algebra Rational Root Test Descartes Rule of Signs Complex numbers Conjugate root theorem Intermediate Value Theorem Long Division of Polynomials Division Algorithm Synthetic Division Multiplicity of a solution Look for and express regularity in repeated reasoning. MACC.K12.MP.1 MACC.K12.MP.2 MACC.K12.MP.3 MACC.K12.MP.4 MACC.K12.MP.5 MACC.K12.MP.6 MACC.K12.MP.7 MACC.K12.MP.8
ESSENTIAL QUESTIONS: Can students find the roots of a polynomial equation using different methods? Can students graph a rational functions with and without technology? Can students perform operations on complex numbers? Activities and Resources Assessment Pre Calculus with Limits (Larson, Hostetler, and Edwards): Chapter 2 Polynomials and Rational Functions
UNIT/ORGANIZING PRINCIPLE: Exponential and Logarithm Functions PACING: 1 st /2 nd 9 weeks ESSENTIAL QUESTIONS: MEASUREMENT TOPICS Exponential and Logarithmic Functions Graphs of Exponential and Logarithmic Functions Modeling of Exponential and Logarithmic Functions Make sense of problems and persevere in solving them. Can student s graph, manipulate, and determine the domain and range of an exponential and logarithmic function? Can students solve exponential and logarithmic equations using the properties of logarithms? The student will: LEARNING TARGETS/SKILLS define exponential and logarithmic functions and determine their relationship. define and use the properties of logarithms to simplify logarithmic expressions and to find their approximate values. Reason abstractly and quantitatively. prove laws of logarithms. solve logarithmic and exponential equations. BENCHMARKS (Algebra 2) MA.912.A.8.1 (Algebra 2) MA.912.A.8.2 (Algebra 2) MA.912.A.8.4 (Algebra 2) MA.912.A.8.5 apply the change of base formula. (Algebra 2) MA.912.A.8.6 graph exponential and logarithmic functions. (Algebra 2) MA.912.A.8.3 graph applications of exponential growth and decay. (Algebra 2) MA.912.A.8.7 solve applications of exponential growth and decay. (Algebra 2) MA.912.A.8.7 Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. ACADEMIC LANGUAGE Exponential function Logarithmic function Natural base Natural exponential function Common logarithmic function Natural logarithmic function Change of base formula Exponential growth and decay models Look for and express regularity in repeated reasoning. MACC.K12.MP.1 MACC.K12.MP.2 MACC.K12.MP.3 MACC.K12.MP.4 MACC.K12.MP.5 MACC.K12.MP.6 MACC.K12.MP.7 MACC.K12.MP.8
ESSENTIAL QUESTIONS: Can students graph, manipulate, and determine the domain and range of an exponential and logarithmic function? Can students solve exponential and logarithmic equations using the properties of logarithms? Activities and Resources Assessment Pre Calculus with Limits (Larson, Hostetler, and Edwards): Chapter 3 Exponential and Logarithmic Functions
UNIT/ORGANIZING PRINCIPLE: Trigonometric Functions PACING: 2 nd 9 weeks ESSENTIAL QUESTIONS: Can students use the unit circle to evaluate a trigonometric expression of any angle? Can students convert between degree and radian measure? Do students understand the connection between right triangle trigonometry and the unit circle? Can students solve problems involving right triangle trigonometry? Can students define and graph trigonometric functions with and without technology? Can students use trigonometric functions to model and solve real-life problems? MEASUREMENT LEARNING TARGETS/SKILLS BENCHMARKS ACADEMIC LANGUAGE TOPICS Trigonometric Functions Graphs Trigonometric Functions The student will: convert between degree and radian measures. MA.912.T.1.1 Trigonometry define and determine sine and cosine using the unit circle. state and apply the exact values of trigonometric functions for special angles in degrees and radians. calculate approximate values of trigonometric and inverse trigonometric functions using appropriate technology. create connections between right triangle ratios, trigonometric functions, and circular functions. define and use the trigonometric ratios (sine, cosine, tangent, cotangent, secant, cosecant) in terms of angles of right triangles. define and graph trigonometric functions using domain, range, intercepts, period, amplitude, phase shift, vertical shift, and asymptotes with and without the use of graphing technology. define and graph inverse trigonometric relations and functions. MA.912.T.1.2 MA.912.T.1.3 MA.912.T.1.4 MA.912.T.1.5 MA.912.T.2.1 MA.912.T.1.6 MA.912.T.1.7 Angle Initial side Terminal side Vertex Positive angles Negative angles Coterminal angles Central angle Radian Complementary angles Supplementary angles Degree Unit circle Sine Cosine Tangent Cotangent Secant Cosecant Period Amplitude Phase shift Reference angle Hypotenuse Opposite side Adjacent side Inverse sine function Inverse cosine function Inverse tangent function Modeling Trigonometric Functions solve real world problems involving applications of trigonometric functions using graphing technology when appropriate. solve real world problems involving right triangles using technology when appropriate. MA.912.T.1.8 MA.912.T.2.2. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. MACC.K12.MP.1 MACC.K12.MP.2 MACC.K12.MP.3 MACC.K12.MP.4 MACC.K12.MP.5 MACC.K12.MP.6 MACC.K12.MP.7 MACC.K12.MP.8
UNIT/ORGANIZING PRINCIPLE: Analytic Trigonometry PACING: 2 nd /3 rd 9 weeks ESSENTIAL QUESTIONS: Can students use the trigonometric identities to simplify trigonometric expression, verify trigonometric identities, and solve trigonometric equations? Can students use the sum and difference formula, multiple and half angle formulas to rewrite and evaluate trigonometric functions? MEASUREMENT TOPICS The student will: LEARNING TARGETS/SKILLS BENCHMARKS ACADEMIC LANGUAGE Trigonometric Identities verify the basic Pythagorean identities, such as i) sin 2 x + cos 2 x = 1, and show they are equivalent to the Pythagorean Theorem. apply basic trigonometric identities to verify other identities and simplify expressions. MA.912.T.3.1 MA.912.T.3.2 Pythagorean identities Sum and difference formulas Double angle formulas Half-angle formulas Modeling Trigonometric Identities apply the sum and difference, half-angle and double-angle formulas for sine, cosine, and tangent, when formulas are provided. solve trigonometric equations and real world problems involving applications of trigonometric equations using technology when appropriate. MA.912.T.3.3 MA.912.T.3.4 Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. MACC.K12.MP.1 MACC.K12.MP.2 MACC.K12.MP.3 MACC.K12.MP.4 MACC.K12.MP.5 MACC.K12.MP.6 MACC.K12.MP.7 MACC.K12.MP.8
ESSENTIAL QUESTIONS: Can students use the trigonometric identities to simplify trigonometric expression, verify trigonometric identities, and solve trigonometric equations? Can students use the sum and difference formula, multiple and half angle formulas to rewrite and evaluate trigonometric functions Activities and Resources Assessment Pre Calculus with Limits (Larson, Hostetler, and Edwards): Chapter 5 Analytic Trigonometry
UNIT/ORGANIZING PRINCIPLE: Additional Topics in Trigonometry PACING: 3 rd 9 weeks ESSENTIAL QUESTIONS: MEASUREMENT TOPICS Laws of Trigonometric Functions The student will: Can students use the Law of Sines and Cosines to solve real world problems? Can students connect trigonometry to the use of vectors and their application in the real world? Can students do basic operations with vectors LEARNING TARGETS/SKILLS ACADEMIC BENCHMARKS apply the law of sines and cosines to solve real world problems using technology. MA.912.T.2.3 apply the area of triangles given two sides and an angle or three MA.912.T.2.4 sides to solve real world problems. Vectors demonstrate an understanding of the geometric interpretation of MA.912.D.9.1 vectors and vector operations including addition, scalar multiplication, dot product, and cross products in the plane and in three-dimensional space. demonstrate an understanding of the algebraic interpretation of MA.912.D.9.2 vectors and vector operations including addition, scalar multiplication, dot product, and cross products in the plane and in three-dimensional space. apply vectors to model and solve application problems. MA.912.D.9.3 Complex Numbers define the trigonometric form of complex numbers, convert MA.912.T.4.4 with Trigonometry complex numbers to trigonometric form, and multiply complex numbers in trigonometric form. apply DeMoivre s theorem to perform operations with complex numbers. MA.912.T.4.5 LANGUAGE Law of Sines Law of Cosines Vector Zero Vector Linear Combination Vector Components DeMoivre s Theorem Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. MACC.K12.MP.1 MACC.K12.MP.2 MACC.K12.MP.3 MACC.K12.MP.4 MACC.K12.MP.5 MACC.K12.MP.6 MACC.K12.MP.7 MACC.K12.MP.8
ESSENTIAL QUESTIONS: Can students use the Law of Sines and Cosines to solve real world problems? Can students connect trigonometry to the use of vectors and their application in the real world? Can students do basic operations with vectors Activities and Resources Assessment Pre Calculus with Limits (Larson, Hostetler, and Edwards): Chapter 6 Additional Topics in Trigonometry Chapter 11 sections 1 3 Analytic Geometry in Three Dimensions
UNIT/ORGANIZING PRINCIPLE: Sequences and Series PACING: 3 rd 9 weeks ESSENTIAL QUESTIONS: MEASUREMENT TOPICS Arithmetic & Geometric Sequences Mathematical Induction Can students write a rule for arithmetic and geometric sequences? Can students find the finite or infinite sum of an arithmetic or geometric series? Can students use proof by induction to verify the sum formula for series? Can students use the Binomial Theorem to expand binomial expressions? LEARNING TARGETS/SKILLS BENCHMARKS ACADEMIC LANGUAGE The student will: define arithmetic and geometric sequences and series. (Algebra 2) MA.912.D.11.1 identify specified terms of arithmetic and geometric sequences. (Algebra 2) MA.912.D.11.3 apply sigma notation to describe series. (Algebra 2) MA.912.D.11.2 identify partial sums of arithmetic and geometric series, and find sums of infinite convergent series. Use sigma notation where applicable. MA.912.D.11.4 apply mathematical induction to prove various concepts in MA.912.D.1.3 number theory (such as sums of infinite integer series, divisibility statements, and parity statements), recurrence relations, and other applications. apply the Binomial Theorem to expand binomial expressions. (Algebra 2) MA.912.A.4.12 Infinite sequence Finite sequence Recursive Factorial Summation or sigma notation Infinite series Finite series or nth partial sum Arithmetic sequences Geometric sequences Infinite geometric series Mathematical induction Binomial coefficients Binomial theorem Pascal s triangle Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. MACC.K12.MP.1 MACC.K12.MP.2 MACC.K12.MP.3 MACC.K12.MP.4 MACC.K12.MP.5 MACC.K12.MP.6 MACC.K12.MP.7 MACC.K12.MP.8
ESSENTIAL QUESTIONS: Can students write a rule for arithmetic and geometric sequences? Can students find the finite or infinite sum of an arithmetic or geometric series? Can students use proof by induction to verify the sum formula for series? Can students use the Binomial Theorem to expand binomial expressions? Activities and Resources Assessment Pre Calculus with Limits (Larson, Hostetler, and Edwards): Chapter 9 Sequences, Series, and Probability
UNIT/ORGANIZING PRINCIPLE: Conic Sections PACING: 3 rd /4 th 9 weeks ESSENTIAL QUESTIONS: Can students change conics in general form to standard form? Can students graph conics using the center, vertices, foci, directrix (parabolas), asymptotes (hyperbolas)? Can students write the equation of a conic in standard form when given information about the graph? MEASUREMENT LEARNING TARGETS/SKILLS TOPICS The student will: Conic Euqations write the equations of conic sections in standard form and general form, in order to identify the conic section and to find it geometric properties (foci, asymptotes, eccentricity, etc). BENCHMARKS ACADEMIC LANGUAGE MA.912.A.9.1 Graphing Conics graph conic sections with and without using graphing technology. MA.912.A.9.2 Modeling Conics solve real world problems involving conic sections. MA.912.A.9.3 Conic section Parabola Directrix Focus or foci Ellipse Vertices Major axis Minor axis Center Eccentricity Hyperbola Transverse axis Conjugate axis Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. MACC.K12.MP.1 MACC.K12.MP.2 MACC.K12.MP.3 MACC.K12.MP.4 MACC.K12.MP.5 MACC.K12.MP.6 MACC.K12.MP.7 MACC.K12.MP.8
ESSENTIAL QUESTIONS: Can students change conics in general form to standard form? Can students graph conics using the center, vertices, foci, directrix (parabolas), asymptotes (hyperbolas)? Can students write the equation of a conic in standard form when given information about the graph? Activities and Resources Assessment Pre Calculus with Limits (Larson, Hostetler, and Edwards) Chapter 10 Topics in Analytic Geometry Sections 1 3
UNIT/ORGANIZING PRINCIPLE: Parametric and Polar Equations PACING: 4 th 9 weeks ESSENTIAL QUESTIONS: Can students convert rectangular equations to parametric and polar equations and vice versa? Can students convert between rectangular and polar coordinates? Can students identify special polar graphs (circle, rose curve, limacon, lemniscates) from the graph and equations? Can students identify the conic section in polar form? MEASUREMENT LEARNING TARGETS/SKILLS TOPICS The student will: Parametric Equations sketch the graph of a curve in the plane represented parametrically, indicating the direction of motion. convert from a parametric representation of a plane curve to a rectangular equation and vice versa. apply parametric equations to model applications of motion in the plane. Polar Equations define polar coordinates and relate polar coordinates to Cartesian coordinates with and without the use of technology. represent equations given in rectangular coordinates in terms of polar coordinates. graph equations in the polar coordinate plane with and without the use of graphing technology. BENCHMARKS ACADEMIC LANGUAGE MA.912.D.10.1 Parameter Parametric equations Plane curve MA.912.D.10.2 Orientation Polar coordinate system Pole or origin MA.912.D.10.3 Polar axis Polar coordinates MA.912.T.4.1 MA.912.T.4.2 MA.912.T.4.3 Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. MACC.K12.MP.1 MACC.K12.MP.2 MACC.K12.MP.3 MACC.K12.MP.4 MACC.K12.MP.5 MACC.K12.MP.6 MACC.K12.MP.7 MACC.K12.MP.8
ESSENTIAL QUESTIONS: Can students convert rectangular equations to parametric and polar equations and vice versa? Can students convert between rectangular and polar coordinates? Can students identify special polar graphs (circle, rose curve, limacon, lemniscates) from the graph and equations? Can students identify the conic section in polar form? Activities and Resources Assessment Pre Calculus with Limits (Larson, Hostetler, and Edwards): Chapter 10 Topics in Analytic Geometry Sections 5-8
UNIT/ORGANIZING PRINCIPLE: Limits PACING: 4 th 9 weeks ESSENTIAL QUESTIONS: MEASUREMENT TOPICS The student will: Can the students evaluate the limit of a function, including one-sided limits, using graphs, tables, technology, and direct substitution? Can the students use the properties of limits to evaluate limits? Do students understand when a function is continuous or discontinuous, and the types of discontinuities? LEARNING TARGETS/SKILLS BENCHMARKS Limits understand the concept of limit and estimate limits from graphs and tables of values. MA.912.C.1.1 identify limits by substitution. MA.912.C.1.2 identify limits of sums, differences, products, and quotients. MA.912.C.1.3 Make sense of problems and persevere in solving them. identify limits of rational functions that are undefined at a point. MA.912.C.1.4 identify one-sided limits. MA.912.C.1.5 understand continuity in terms of limits. MA.912.C.1.9 decide if a function is continuous at a point. MA.912.C.1.10 identify the types of discontinuities of a function. MA.912.C.1.11 understand and apply the intermediate value theorem on a function MA.912.C.1.12 over a closed interval. understand and apply the extreme value theorem: If f(x) is continuous over a closed interval, then f has a maximum and a minimum on the interval. MA.912.C.1.13 identify limits at infinity. (Calculus benchmark) MA.912.C.1.6 Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. ACADEMIC LANGUAGE Limit Indeterminate form One-sided Limit Tangent Line Secant Line Slope of the tangent line Limits at infinity Look for and express regularity in repeated reasoning. MACC.K12.MP.1 MACC.K12.MP.2 MACC.K12.MP.3 MACC.K12.MP.4 MACC.K12.MP.5 MACC.K12.MP.6 MACC.K12.MP.7 MACC.K12.MP.8