Analytic Geometry/ Trigonometry

Transcription

1 Analytic Geometry/ Trigonometry Course Numbers , Lake County School Curriculum Map Released Page 1 of 33

2 PREFACE Teams of Lake County teachers created the curriculum maps in order to ensure that all students throughout the district receive a common curriculum. The maps help ensure that all state requirements are taught and that the content is divided into teachable segments with appropriate pacing. The curriculum maps will guide your instruction but provide flexibility based on the individual needs of students. The maps are living documents and feedback is requested of teachers to ensure continuous improvement. All teachers are expected to use the curriculum maps, in conjunction with data, to drive instruction. The maps were designed for the instruction to take place by quarter. There is some flexibility within the quarters for mastery and re-teaching. The expectation is that teachers will finish the content within each quarter in its entirety. The maps have been structured in such a way as to scaffold student learning. Listed below are a few of the new or updated features common to all curriculum maps: Essential Question(s): o Provide application of the skills/concepts o Have more than one right answer which promotes student discourse o Increase the rigor in the classroom, by changing from teacher-centered to student-centered learning o Are referred to at the beginning, middle, and end of the lesson o Require you to make a decision o Promote critical thinking and problem solving o Encourage interdependence o Are open-ended Academic Vocabulary are: o Unfamiliar vocabulary that are essential to understanding new content within explicit instruction o Not necessarily the bold words in the chapter. o Cumulative and continuously used throughout the year. o Integrated into word walls, a research-based strategy that will facilitate vocabulary acquisition. Common Board Configuration Elements (specific layouts may vary by sites, but must include each of these): Purpose: For the student to know what is being taught and what the student will learn o Date o Benchmark o Measurable, student-friendly objective o Essential Question o Bell work o Agenda (Specific daily schedule) o Homework o Exit Strategy/Card Page 2 of 33

3 Lessons that infuse reading, writing, and discussion are imperative components of every subject area. There should be daily: o Teacher to student and student to student discourse utilizing academic vocabulary. o Reading and authentic writing o Writing that includes higher-order thinking o Incorporation of effective reading and writing instructional strategies Maps are organized to include the following: o o o o Pacing Objective Essential questions, content and understanding, benchmarks, and assessment Appendix/ resources Page 3 of 33

4 Next Generation Sunshine State Standards Math Benchmark Coding Scheme MA. 5. A Subject Grade Level Body of Knowledge Big Idea / Supporting Idea Benchmark Body of Knowledge Key A ~ Algebra G ~ Geometry C ~ Calculus P ~ Probability D ~ Discrete Mathematics S ~ Statistics F ~ Financial Literacy T ~ Trigonometry Language Arts Benchmarks LA The student will use new vocabulary that is introduced and taught directly. LA The student will use background knowledge of subject and related content areas, prereading strategies (e.g., previewing, discussing, generating questions), text features, and text structure to make and confirm complex predictions of content, purpose, and organization of a reading selection. LA The student will identify cause-and-effect relationships in text. LA The student will prewrite by making a plan for writing that addresses purpose, audience, a controlling idea, logical sequence, and time frame for completion. LA The student will prewrite by using organizational strategies and tools (e.g., technology, spreadsheet, outline, chart, table, graph, Venn Diagram, web story map, plot pyramid) to develop a personal organizational style. LA The student will draft writing by establishing a logical organizational pattern with supporting details that are substantial, specific, and relevant. Page 4 of 33

5 Differentiated Instruction Strategies The following differentiated instruction strategies should be incorporated throughout the entire course: Cooperative Groups Computer Assisted Instruction Tiered Assignments Centers Flexible Grouping Curriculum Compacting/Contracts Learning Stations Scaffolding Hands-on Instruction Leveled Texts/Resources Teacher Led Small Groups Web Quest Page 5 of 33

6 Math Pacing Guide for ANALYTIC GEOMETRY/TRIGONOMETRY First Quarter I. Functions and Graphs (4 weeks) A. Solve various types of equations by factoring, completing the square, and by using powers and roots B. Recognize and graph various types of functions, including polynomial, rational, radical, and absolute value functions (with and without calculators) C. Determine sums, differences, products, and quotients of functions D. Determine composites of functions E. Describe the end behavior of polynomial functions First Quarter Second Quarter II. Logarithmic and Exponential Functions (2 weeks) A. Sketch the graphs for exponential and logarithmic functions B. Identify key features of graphs C. Use the rules of logarithms D. Solve exponential and logarithmic equations E. Solve word problems involving applications of logarithmic and exponential functions III. Conic Sections (2 weeks) A. Sketch graphs and determine equations for conic sections (with and without calculators) B. Write the equations of conic sections in standard form (completing the square and using translations as necessary). C. Determine the type of conic section and its geometric properties (foci, asymptotes, eccentricity, etc.). D. Solve applications involving conic sections Second Quarter IV. Parametric Graphs and Equations (1 week) A. Use parametric equations to sketch graphs, indicating the direction of motion. B. Convert from parametric form to rectangular equations C. Solve applications for parametric equations (with and without calculators) Page 6 of 33

7 Math Pacing Guide for ANALYTIC GEOMETRY/TRIGONOMETRY Third Quarter Third Quarter Fourth Quarter Fourth Quarter V. Trigonometric definitions (2 weeks) A. Sketch angles in standard position using protractors B. Identify direction for angles C. Convert between degree and radian measures. D. Identify reference angles and reference triangles and their quadrants E. Determine sine and cosine using the unit circle. F. Find and use exact values of trigonometric functions for special angles (degree and radian measures). G. Evaluate inverse trigonometric functions H. Determine values for trigonometric ratios using a calculator. VI. Trigonometry and Triangles (1 week) A. Use trigonometric ratios to solve right and oblique triangles B. Solve word problems involving right and oblique triangles, including areas. VII. Trigonometric functions (2 weeks) A. Define and graph trigonometric functions using domain, range, intercepts, period, amplitude, phase shift, vertical shift, and asymptotes. B. Sketch graphs of trigonometric functions using graphing calculator C. Solve word problems involving applications of trigonometric functions D. Sketch graphs for inverse trigonometric functions VIII. Trigonometric Identities and Equations (2 weeks) A. Use the reciprocal, ratio, Pythagorean, sum and difference, and double angle identities. B. Use the basic trigonometric identities to verify other identities and simplify expressions C. Solve trigonometric equations and word problems involving trigonometric equations IX. Polar Coordinates and Complex Numbers (1 week) A. Define polar coordinates and relate polar coordinates to Cartesian coordinates B. Represent equations given in rectangular coordinates in terms of polar coordinates. C. Graph equations in the polar coordinate plane. D. Define complex numbers, convert complex numbers to trigonometric form, and multiply complex numbers in trigonometric form. E. Apply DeMoivre s Theorem to operations with complex numbers. X. Vectors (1 week) A. Sketch vectors and perform vector operations B. Solve applications involving vectors Page 7 of 33

8 I. Functions and Graphs 4 weeks Objectives: 1. Analyze features in the graphs for selected types of functions by identifying key features and their locations on the graphs. 2. Organize operations with functions. Vocabulary: domain, polynomial, composite, inverse function, intercept, symmetry, asymptote, end behavior Essential Questions Essential Content & Understandings Essential Skills & Benchmarks Assessment What are the techniques commonly used to solve equations? How can one predict the key features of a graph based on a function s algebraic form? What math operations can be applied to functions? A. Solve various types of equations 1. equation balancing 2. factoring and the multiplication property of zero 3. completing the square 4. using roots and powers B. Model and solve A. Recognize and graph various types of functions, including polynomial, rational, radical, and absolute value (with and without calculators) 1. Identify domain and range 2. Locate x and y intercepts 3. Locate vertical and horizontal asymptotes 4. Identify discontinuities in graphs 5. Describe symmetry for the graphs of functions 6. Determine whether functions are even, odd, or neither. 7. Apply transformations to graphing functions. 8. Define, find, and check inverse functions MA.912.A.4.5 MA.912.A.4.9 MA.912.A.4.8 MA.912.A.5.6 Formal: Chapter Test Vocabulary Test Mini-quizzes Informal: Check for understanding with the use of 5-minute checks FCAT style bell ringers Word Wall activity: Use writing strategies to display the connection between the various vocabulary terms. Page 8 of 33

9 Essential Questions Essential Content & Understandings Essential Skills & Benchmarks Assessment C. Determine sums, differences, products, and quotients of functions D. Determine composites of functions E. Describe the end behavior of polynomial functions Page 9 of 33

10 II. Exponential and Logarithmic Functions 2 weeks Objectives: 1. Use exponential and logarithmic patterns to simplifying expressions 2. Illustrate the graphs for exponential and logarithmic functions 3. Calculate the solutions for exponential and logarithmic equations 4. Analyze real world problems using exponential and logarithmic functions and graphs Vocabulary: expression, representation, model, base e, natural logarithm Essential Questions Essential Content & Understandings Essential Skills & Benchmarks Assessment Formal: What are the key features for exponential and logarithmic functions and graphs? What real world issues can be analyzed using exponential and logarithmic functions and graphs? A. Sketch the graphs for exponential and logarithmic functions B. Identify key features of graphs 1. domain and range 2. x and y intercepts 3. asymptotes 4. growth and decay C. Use the rules of logarithms 1. simplify expressions 2. combine logarithms 3. expand expressions D. Solve exponential and logarithmic equations E. Solve word problems involving applications of logarithmic and exponential functions MA.912.A.8.7 Chapter Test Vocabulary Test Mini-quizzes Informal: Check for understanding with the use of 5-minute checks FCAT style bell ringers Word Wall activity: Use writing strategies to display the connection between the various vocabulary terms. Page 10 of 33

11 III. Conic Sections 2 weeks Objectives: 1.Write equations for the conic sections when given key features. 2.Draw graphs for the conics when given the equations. Vocabulary: conic section, eccentricity, focus, directrix, asymptote Essential Questions Essential Content & Understandings Essential Skills & Benchmarks Assessment What are the shapes known as the conic sections? How can equations be used to represent the conic sections? What real-world problems can be modeled using the conics? A. Sketch graphs and determine equations for conic sections (with and without calculators) B. Write the equations of conic sections in standard form (completing the square and using translations as necessary). C. Determine the type of conic section and its geometric properties (foci, asymptotes, eccentricity, etc.). D. Solve applications involving conic sections. MA.912.A.9.2 MA.912.A.9.1 MA.912.D.6.4 MA.912.A.9.3 Formal: Chapter Test Vocabulary Test Mini-quizzes Informal: Check for understanding with the use of 5-minute checks FCAT style bell ringers Word Wall activity: Use writing strategies to display the connection between the various vocabulary terms. Page 11 of 33

12 IV. Parametric Equations and Graphs 1 week Objectives: 1. Identify key features for parametric functions using graph paper, straightedges, and pencils as well as graphing calculators. 2. Convert from parametric to rectangular form 3. Draw conclusions for applications of parametric equations. Vocabulary: parametric equation, angle, scalar, reflection, arc Essential Questions Essential Content & Understandings Essential Skills & Benchmarks Assessment What alternate methods for graphing exist in mathematics? How do you transform equations to reflect alternate graphical representations? What kinds of applications can be modeled with parametric equations? A. Create a table of values and sketch graphs for parametric relationships (with and without calculators) 1. Identify direction with arrow B. Eliminate parameters to determine rectangular equations for parametric forms and then graph in rectangular form. C. Determine parametric equations for conic sections. D. Solve application problems involving parametric equations. MA.912.D.10.1 MA.912.D.10.2 MA.912.D.10.3 Formal: Chapter Test Vocabulary Test Mini-quizzes Informal: Check for understanding with the use of 5-minute checks FCAT style bell ringers Word Wall activity: Use writing strategies to display the connection between the various vocabulary terms. Page 12 of 33

13 V. Trigonometric Definitions 2 weeks Objectives: 1. Illustrate trigonometric ratios in the rectangular coordinate system 2. Calculate trigonometric ratios and their inverses using technology Vocabulary: sine, cosine, tangent, cotangent, secant, cosecant, reference angle Essential Questions Essential Content & Understandings Essential Skills & Benchmarks Assessment What are trigonometric ratios? How can you draw diagrams to determine the values for trigonometric ratios? A. Sketch angles in standard position using protractors B. Identify direction for angles C. Convert between degree and radian measures. D. Identify reference angles and reference triangles and their quadrants E. Determine sine and cosine using the unit circle. F. Find and use exact values of trigonometric functions for special angles (degree and radian measures). G. Evaluate inverse trigonometric functions H. Determine values for trigonometric ratios using a calculator MA.912.T.1.1 MA.912.T.1.5 MA.912.T.1.2 MA.912.T.1.3 MA.912.T.1.4 Formal: Chapter Test Vocabulary Test Mini-quizzes Informal: Check for understanding with the use of 5-minute checks FCAT style bell ringers Word Wall activity: Using writing strategies to display the connection between the various vocabulary terms. Page 13 of 33

14 VI. Triangles and Trigonometric Ratios - 1 week Objectives: 1. Formulate strategies for determining sides and angles of right and oblique triangles 2. Analyze real-world problems using triangles and trigonometric ratios Vocabulary: angle of depression, angle of elevation, oblique, representations Essential Questions Essential Content & Understandings Essential Skills & Benchmarks Assessment What is the least amount of information needed to determine all sides and angles for a given triangle? How can trigonometry be used to solve real-world problems? A. Use trigonometric ratios to solve triangles 1. right triangles a. opposite, adjacent, hypotenuse patterns 2. oblique triangles a. Law of Sines b. Law of Cosines B. Solve word problems involving right and oblique triangles, including areas. MA.912.T.2.1 MA.912.T.2.3 MA.912.T.2.2 MA.912.T.2.4 Formal: Chapter Test Vocabulary Test Mini-quizzes Informal: Check for understanding with the use of 5-minute checks FCAT style bell ringers Word Wall activity: Use writing strategies to display the connection between the various vocabulary terms. Page 14 of 33

15 VII. Trigonometric Functions and Graphs 2weeks Objectives: 1. Sketch graphs for all six trigonometric functions with and without calculators Vocabulary: amplitude, period, phase shift, vertical shift, scale, oscillate Essential Questions Essential Content & Understandings Essential Skills & Benchmarks Assessment What are the key features of the graphs for trigonometric functions? What real-world problems can be modeled by trigonometric functions? A. Graph trigonometric functions on graph paper and identify key features: 1. domain and range 2. intercepts 3. period 4. amplitude 5. phase shift 6. vertical shift 7. vertical asymptotes B. Sketch graphs of trigonometric functions using graphing calculator 1. set appropriate window 2. identify key features C. Solve word problems involving applications of trigonometric functions D. Sketch graphs for inverse trigonometric functions MA.912.T.1.6 MA.912.T.1.8 MA.912.T.1.5 MA.912.T.1.7 Formal: Chapter Test Vocabulary Test Mini-quizzes Informal: Check for understanding with the use of 5-minute checks FCAT style bell ringers Word Wall activity: Use writing strategies to display the connection between the various vocabulary terms. Page 15 of 33

16 VIII. Trigonometric Identities and Equations 2 weeks Objectives: 1. Create logically valid algebraic steps for verifying trigonometric identities 2. Collect and display solutions for trigonometric equations Vocabulary: equivalent expressions, substitution, identity Essential Questions Essential Content & Understandings Essential Skills & Benchmarks Assessment What are the important relationships among the six trigonometric ratios? How can one use logic to verify when arguments are valid? How can one solve equations involving trigonometric functions? A. Use the reciprocal, ratio, Pythagorean, sum and difference, and double angle identities. 1. find all ratios given one ratio and a quadrant 2. identify equivalent expressions 3. simplify expressions B. Use the basic trigonometric identities to verify other identities 1. rewrite as equivalent expressions 2. construct proofs for identities C. Solve trigonometric equations and word problems involving trigonometric equations MA.912.T.3.1 MA.912.T.3.3 MA.912.T.3.2 MA. 912.T.3.4 Formal: Chapter Test Vocabulary Test Mini-quizzes Informal: Check for understanding with the use of 5-minute checks FCAT style bell ringers Word Wall activity: Use writing strategies to display the connection between the various vocabulary terms. Page 16 of 33

17 IX. Polar Coordinates and Complex Numbers 1 week Objectives: 1.Sketch graphs in polar coordinates 2.Show rectangular equations in polar form. 3.Organize operations with complex numbers 4. Convert complex numbers to trigonometric form. Vocabulary: complex number, axes, plane, scalar Essential Questions Essential Content & Understandings Essential Skills & Benchmarks Assessment Formal: MA.912.T.4.1 How is graphing in polar coordinates different from graphing in the rectangular system? What are complex numbers and how do you perform operations in the complex number system? A. Define polar coordinates and relate polar coordinates to Cartesian coordinates B. Represent equations given in rectangular coordinates in terms of polar coordinates. C. Graph equations in the polar coordinate plane. D. Define complex numbers, convert complex numbers to trigonometric form, and multiply complex numbers in trigonometric form. E. Apply DeMoivre s Theorem to operations with complex numbers. MA.912.T.4.2 MA.912.T.4.3 MA.912.T.4.4 MA.912.T.4.5 Chapter Test Vocabulary Test Mini-quizzes Informal: Check for understanding with the use of 5-minute checks FCAT style bell ringers Word Wall activity: Use writing strategies to display the connection between the various vocabulary terms. Page 17 of 33

18 X. Vectors 1 week Objective: 1. Sketch graphs for vectors 2. Calculate magnitude for vectors 3. Organize operations involving vectors 4. Investigate real-world problems involving vectors. Vocabulary: vector, magnitude, length, slope Essential Questions Essential Content & Understandings Essential Skills & Benchmarks Assessment How are vectors written and graphed? What kinds of problems can be solved using vectors? A. Sketch vectors and perform vector operations B. Solve applications involving vectors MA.912.D.9.1 MA.912.D.9.2 MA.912.D.9.3 Formal: Chapter Test Vocabulary Test Mini-quizzes Informal: Check for understanding with the use of 5-minute checks FCAT style bell ringers Word Wall activity: Use writing strategies display the connection between the various vocabulary terms. Page 18 of 33

19 Curriculum Map Appendix for High Schools Scope and Sequence Correlated to Textbook Pages State Approved Course Descriptions Page 19 of 33

20 Analytic Geometry Scope and Sequence Correlated to Blitzer, Algebra and Trigonometry 4thed. and NGSSS Topic Chapter Section, resources NGSSS I. Functions and Graphs LA A.Solve various type 1-2 equations by factoring, 1-3 completing the square, and 1-5 by using powers and roots 1-6 B.Recognize and graph various MA.912.A.4.5 types of functions, including MA.912.A.4.8 polynomial, rational, radical, and MA.912.A.5.6 absolute value functions (with and without calculators) 1. Identify domain and range Locate x and y intercepts 2-2 MA.912.A Locate vertical, horizontal, 3-5 and slant asymptotes 4. Identify removable and non-removable discontinuities in graphs 5. Describe symmetry for the graphs of functions Determine whether functions are even, odd, or neither Apply transformations to graphing functions. 2-5 Page 20 of 33

21 Analytic Geometry 8. Define, find, and check inverse functions 2-7 C. Determine sums, differences, products, and quotients of functions 2-.6 D. Determine composites of functions E. Describe the end behavior of polynomial functions II. Logarithmic and Exponential Functions A. Sketch the graphs for exponential and logarithmic functions B. Identify key features of gr domain and range x and y intercepts 3. asymptotes 4. growth and decay C. Use the rules of logarithms simplify expressions 2. combine logarithms 3. expand expressions D. Solve exponential and logarithmic equations 4-4 E. Solve word problems involving applications 4-5 LA of logarithmic and exponential functions MA.912.A.8.7 III. Conic Sections LA A. Sketch graphs and determine equations for 2-8 MA.912.A.9.2 conic sections (with and without calculators) B. Write the equations of conic 10-1 MA.912.A.9.1 Page 21 of 33

22 Analytic Geometry sections in standard form (completing the square and using translations as necessary). C. Determine the type of conic 10-2 MA.912.D.6.4 section and its geometric properties (foci, asymptotes, eccentricity, etc.). D. Solve applications involving conic sections MA.912.A.9.3 IV. Parametric Graphs and Equations 10.5 LA A. Use parametric equations to sketch graphs,. LA indicating the direction of motion. MA.912.D.10.1 B. Convert from parametric form to rectangular equations LA rectangular equations MA.912.D.10.2 C. Solve applications for parametric equations (with and without calculators) MA.912.D.10.3 Page 22 of 33

23 Scope and Sequence Correlated to Blitzer, Algebra and Trigonometry 4thed. and NGSSS Topic Chapter Section, resources NGSSS V. Trigonometric definitions LA A. Convert between decimal degree and DMS supplement B. Convert between degree and MA.912.T.1.1 radian measures. 5-1 C. Sketch angles in standard position using protractors positive and negative rotation 2. degree and radian measure 3. identify quadrants D. Identify reference angles and reference triangles 5-3 E. Determine all six trigonometric MA.912.T.1.2 ratios given: a quadrantal angle 2. one trig ratio 3. a point on the terminal side. F. Determine exact values of MA.912.T.1.3 trigonometric ratios using special MA.912.T.1.5 right triangle patterns in all quadrants 5-3 Page 23 of 33

24 1. degree and radian measure G. Find exact values for inverse trigonometric functions 5-7 H. Determine values for MA.912.T.1.4 trigonometric ratios using a calculator VI. Trigonometry and Triangles A. Use trigonometric ratios to solve triangles right triangles MA.912.T.2.1 a. opposite, adjacent, hypotenuse patterns 2. oblique triangles MA.912.T.2.3 a. Law of Sines 7-1 b. Law of Cosines 7-2 B. Solve word problems MA.912.T.2.2 involving right and oblique MA.912.T.2.4 triangles VII. Trigonometric functions A. Graph trigonometric functions on graph paper and MA.912.T.1.6 identify key features: domain and range 2. intercepts 3. period Page 24 of 33

25 4. amplitude 5. phase shift 6. vertical shift 7. vertical asymptotes B. Sketch graphs of trigonometric functions using graphing calculator set appropriate window 2. identify key features C. Solve word problems 5-4 MA.912.T.1.8 involving applications of 5-5 trigonometric functions 5-6 D. Sketch graphs for inverse MA.912.T.1.7 trigonometric functions 5.7 VIII. Trigonometric Identities and Equations A. Use the reciprocal, ratio, Pythagorean, negative-angle, 5-2 MA.912.T.3.1 and as time permits, the sum and difference, 6-1 MA.912.T.3.3 and double angle identities find all ratios given one ratio and a quadrant 2. identify equivalent expressions 3. simplify expressions B. Use the basic trigonometric identities 6-3 MA.912.T.3.2 to verify other identities 1. rewrite as equivalent expressions 2. construct proofs for identities C. Solve trigonometric equations and word MA.912.T.3.4 problems involving trigonometric equations 6.5 Page 25 of 33

26 IX. Polar coordinates Polar Coordinates and Complex Numbers A. Define polar coordinates and LA relate polar coordinates to MA.912.T.4.1 Cartesian coordinates B. Represent equations given MA.912.T.4.2 in rectangular coordinates 7-4 LA in terms of polar coordinates. Graph equations in the MA.912.T.4.3 polar coordinate plane. C. Define complex numbers, 7-5 MA.912.T.4.4 convert complex numbers to trigonometric form, and multiply complex numbers in trigonometric form. E. Apply DeMoivre s MA.912.T.4.5 Theorem to operations with complex numbers. X. Vectors A. Sketch vectors and LA perform vector operations 7-6 LA B. Solve applications LA involving vectors MA.912.D.9.1 MA.912.D.9.2 MA.912.D.9.3 Page 26 of 33

27 State Approved Analytic Geometry Course Description Course Code Course Category 6-12 Subject Area Mathematics Course Type Core Course Title Analytic Geometry Course Level 3 Course Length One Semester Credit Description 0.5 Abbreviated Title Analytic Geometry RELATED BENCHMARKS (18) : Scheme Descriptor LA The student will use new vocabulary that is introduced and taught directly; LA LA LA The student will use background knowledge of subject and related content areas, prereading strategies (e.g., previewing, discussing, generating questions), text features, and text structure to make and confirm complex predictions of content, purpose, and organization of a reading selection; The student will identify cause-and-effect relationships in text; The student will prewrite by making a plan for writing that addresses purpose, audience, a controlling idea, logical sequence, and time frame for completion; and Page 27 of 33

28 LA LA MA.912.A.4.5 MA.912.A.4.8 MA.912.A.4.9 MA.912.A.5.6 MA.912.A.8.7 MA.912.A.9.1 MA.912.A.9.2 MA.912.A.9.3 MA.912.D.6.4 The student will prewrite by using organizational strategies and tools (e.g., technology, spreadsheet, outline, chart, table, graph, Venn Diagram, web, story map, plot pyramid) to develop a personal organizational style. The student will draft writing by establishing a logical organizational pattern with supporting details that are substantial, specific, and relevant; and Graph polynomial functions with and without technology and describe end behavior. Describe the relationships 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. Use graphing technology to find approximate solutions for polynomial equations. 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 function. Solve applications of exponential growth and decay. Write the equations of conic sections in standard form and general form, in order to identify the conic section and to find its geometric properties (foci, asymptotes, eccentricity, etc.). Graph conic sections with and without using graphing technology. Solve real-world problems involving conic sections Use methods of direct and indirect proof and determine whether a short proof is logically valid. Page 28 of 33

29 MA.912.D.10.1 Sketch the graph of a curve in the plane represented parametrically, indicating the direction of motion. MA.912.D.10.2 Convert from a parametric representation of a plane curve to a rectangular equation, and vice-versa. MA.912.D.10.3 Use parametric equations to model applications of motion in the plane. Page 29 of 33

30 State Approved Trigonometry Course Description Course Code Course Category 6-12 Subject Area Mathematics Course Type Core Course Title Trigonometry Course Level 3 Course Length One Semester Credit Description 0.5 Abbreviated Title Trigonometry RELATED BENCHMARKS (30) : Scheme Descriptor LA The student will use new vocabulary that is introduced and taught directly; LA LA LA The student will use background knowledge of subject and related content areas, prereading strategies (e.g., previewing, discussing, generating questions), text features, and text structure to make and confirm complex predictions of content, purpose, and organization of a reading selection; The student will identify cause-and-effect relationships in text; The student will prewrite by making a plan for writing that addresses purpose, audience, a controlling idea, logical sequence, and time frame for completion; and Page 30 of 33

31 LA LA MA.912.D.9.1 MA.912.D.9.2 MA.912.D.9.3 MA.912.T.1.1 MA.912.T.1.2 The student will prewrite by using organizational strategies and tools (e.g., technology, spreadsheet, outline, chart, table, graph, Venn Diagram, web, story map, plot pyramid) to develop a personal organizational style. The student will draft writing by establishing a logical organizational pattern with supporting details that are substantial, specific, and relevant; and Demonstrate an understanding of the geometric interpretation of vectors and vector operations including addition, scalar multiplication, dot product and cross product in the plane and in three-dimensional space. Demonstrate an understanding of the algebraic interpretation of vectors and vector operations including addition, scalar multiplication, dot product and cross product in the plane and in three-dimensional space. Use vectors to model and solve application problems. Convert between degree and radian measures. Define and determine sine and cosine using the unit circle. MA.912.T.1.3 State and use exact values of trigonometric functions for special angles, i.e. multiples of and (degree and radian measures) MA.912.T.1.4 MA.912.T.1.5 Find approximate values of trigonometric and inverse trigonometric functions using appropriate technology. Make connections between right triangle ratios, trigonometric functions, and circular functions. Page 31 of 33

32 MA.912.T.1.6 MA.912.T.1.7 MA.912.T.1.8 MA.912.T.2.1 MA.912.T.2.2 MA.912.T.2.3 MA.912.T.2.4 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. Solve real-world problems involving applications of trigonometric functions using graphing technology when appropriate. Define and use the trigonometric ratios (sine, cosine, tangent, cotangent, secant, cosecant) in terms of angles of right triangles. Solve real-world problems involving right triangles using technology when appropriate. Apply the laws of sines and cosines to solve real-world problems using technology. Use the area of triangles given two sides and an angle or three sides to solve real-world problems. MA.912.T.3.1 Verify the basic Pythagorean identities, e.g., Theorem., and show they are equivalent to the Pythagorean MA.912.T.3.2 MA.912.T.3.3 MA.912.T.3.4 Use basic trigonometric identities to verify other identities and simplify expressions. Use 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 Page 32 of 33

33 technology when appropriate. MA.912.T.4.1 MA.912.T.4.2 MA.912.T.4.3 MA.912.T.4.4 MA.912.T.4.5 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. Define the trigonometric form of complex numbers, convert complex numbers to trigonometric form, and multiply complex numbers in trigonometric form. Apply DeMoivre's Theorem to perform operations with complex numbers. Page 33 of 33