BERKELEY HEIGHTS PUBLIC SCHOOLS BERKELEY HEIGHTS, NEW JERSEY. Curriculum Guide. September 2013

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1 BERKELEY HEIGHTS PUBLIC SCHOOLS BERKELEY HEIGHTS, NEW JERSEY GOVERNOR LIVINGSTON HIGH SCHOOL MATHEMATICS DEPARTMENT MATH ANALYSIS/MATH ANALYSIS HONORS #MAY1210/#MAY1220 Curriculum Guide September 2013 Mrs. Judith A. Rattner, Superintendent Ms. Patricia Qualshie, Assistant Superintendent Dr. Mary Ann Kjetsaa, District Supervisor Developed by: William Meaken Jenna Sorrell This curriculum may be modified through varying techniques, strategies, and materials, as per an individual student s Individualized Educational Plan (IEP). Approved by the Berkeley Heights Board of Education at the regular meeting held on 9/26/13.

2 TABLE OF CONTENTS Page Vision Statement... 1 Mission Statement... 2 Course Proficiencies... 3 Course Objectives... 3 Student Proficiencies... 6 Methods of Evaluation... 9 Course Outline/Student Objectives New Jersey Core Curriculum Content Standards Code Sheet Resources/Activities Guide Suggested Audio Visual/Computer Aids Suggested Materials Resources for Students Resources for Teacher... 18

3 VISION STATEMENT Math Analysis and Math Analysis Honors are intended for college- bound students, with above average to high mathematical ability. Both courses expose students to traditional pre- calculus content, with an emphasis on developing their ability, to think in a logical and analytic manner. An emphasis is also placed upon developing arguments and communicating ideas, using appropriate mathematical language. Honors sections will delve, where appropriate, into additional explanations, synthesis, and experimentation, to further prepare students for higher level mathematics. The objectives of this course are: To build upon and enhance skills acquired in Algebra 2 and previous math courses To significantly increase mathematical literacy and enhance problem- solving skills To make mathematical connections with course content in a real- world context To prepare students for more extensive study in calculus, science, and technology Berkeley Heights Public Schools 1

4 MISSION STATEMENT Math Analysis and Math Analysis Honors provide students, with opportunities to extend the concepts they mastered in Algebra 2. Attention is given to theory and application of trigonometric functions, discrete math, and data analysis, as these content areas provide the mathematical framework, for many contemporary mathematical applications. The curriculum also includes rational expressions, vectors, polar coordinates, polar equations, and complex numbers. In addition to further study in these topics, the honors course will explore parametric equations, binomial expansions, partial fraction decomposition, and an introduction to limits. Technology is integrated consistently throughout the course graphing calculators are used extensively at both levels to discover, apply, and reinforce certain concepts. The prerequisite for Math Analysis is successful completion of Algebra 2, along with a teacher s recommendation. The prerequisite for Math Analysis Honors is a B in Algebra 2 Honors, or an A in Algebra 2, along with a teacher s recommendation. Five (5.0) credits are given for successfully completing each of these full year courses. The New Jersey Core Curriculum Content Standards for Technology and 21 st Century Life and Careers, as well as Common Core Standards for Mathematics are integrated throughout the curriculum and are referenced in each section. Berkeley Heights Public Schools 2

5 COURSE PROFICIENCIES COURSE OBJECTIVES 1. Radian measure is introduced and conversions are made between degrees and radians. (F- TF1-4) 2. Calculations for arc length, sector area, angular speed, and linear speed are introduced. (F- TF1, G- C5) 3. Trigonometric functions and their inverses are defined and applied. (F- TF6-7) 4. Trigonometric functions are used in problem- solving. (G- SRT8) 5. Methods for graphing trigonometric functions and their transformations are explored. (F- IF7e) 6. Graphs of inverse trigonometric functions are used to develop the concept of principal values. (F- IF5, F- TF6-7) 7. Trigonometric identities are verified. (G- TF8) 8. The sum and difference formulas are developed and applied. (F- TF9) 9. The double and half- angle formulas are derived and applied. (T- TF8-9) 10. Trigonometric equations are solved, for both principal values and for all solutions. (F- TF8-9) 11. The Law of Sines and the Law of Cosines are derived and applied to word problems. (G- SRT10-11) 12. Formulas for finding the area of a non- right triangle are explored. (G- SRT9-11) 13. Vectors are introduced along with their multiple forms, and are analyzed both geometrically and algebraically. (N- VM1) 14. Methods of performing vector operations are introduced. (Addition, subtraction, division, scalar multiplication, dot products, and angle between vectors. (N- VM1-5) 15. Converting the representation of a vector, from it components to its magnitude/direction and vice- versa. (N- VM4-5) Berkeley Heights Public Schools 3

6 COURSE PROFICIENCIES (continued) 16. Vectors are used in solving word problems involving navigation, sum of forces, and tension. (N- VM3) 17. Polar coordinates are introduced and conversions between polar and rectangular coordinates are explored. (N- CN1-6) 18. Graphing points and simple equations (r = # and θ = #) on polar graph paper. (N- CN1-6) 19. Rectangular equations are converted to polar equations and vice- versa. (N- CN1-6) 20. Polar equations of limacons, rose curves, and lemniscates are identified and graphed, both with and without a graphing calculator. (Honors only) (N- CN1-6) 21. Conic sections are each introduced, as a set of points satisfying a condition (locus). (GPE1-3) 22. Standard form equations, for each conic section are derived. (GPE1-3) 23. Standard form equations of each conic section are found, based on given important characteristics. The reverse is also explored where the important characteristics of a conic section are found, given the standard form equation. (GPE1-3) 24. Sequences and series are defined, both explicitly and recursively, and sigma notation is used to represent a series. (A- SSE3-4; F- BF2; F- IF3, F- LE2, N- RN1-2) 25. Characteristics of arithmetic and geometric sequences/series are explored, and their formulas for the n th term, common difference/ratio, and partial and infinite sums are formulated. (A- SSE3-4; F- BF2; F- IF3, F- LE2; N- RN1-2) 26. Partial fractions are introduced as a method of decomposing rational expressions. (A- APR3,6-7; A- SSE3; F- IF4-8) 27. Factoring higher degree polynomials using grouping, rational root theorem, remainder and factor theorems, and Decscartes rule of signs and bound theorems. (A- APR3,6-7; A- SSE3; F- IF4-8) 28. Distinctions between removable and non- removable discontinuities are made when analyzing the graph s behavior at x values, which are not in the function s domain. (A- APR3,6-7; A- SSE3; F- IF4-8) Berkeley Heights Public Schools 4

7 COURSE PROFICIENCIES (continued) 29. The concept of a limit is informally introduced out of the distinction between removable and non- removable discontinuities. (A- APR3,6-7; A- SSE3; F- IF4-8) 30. Formal notation and evaluation of limits is discussed. (A- APR3,6-7; A- SSE3; F- IF4-8) Berkeley Heights Public Schools 5

8 STUDENT PROFICIENCIES The student will be able to: 1. Define, understand, and evaluate the six trigonometric functions within a right triangle. 2. Extend the understanding of trigonometric functions, from acute angles to angles of any size, positive or negative. 3. Change from radian measure to degree measure and vice- versa. 4. Find complementary, supplementary, co- terminal, and reference angles for a given angle. 5. Apply the appropriate formula for arc length and area of a sector to solve problems, given an angle in either radian or degree measure. 6. Calculate linear and angular speed of a point or object, moving around a circle. 7. Graph the six trigonometric functions and use these graphs, to evaluate the function for different values in the domain. 8. Identify the amplitude, period, and phase shift by looking at a trigonometric equation or its graph. 9. Evaluate and apply inverse trig functions and develop an understanding of principal values. 10. Use trigonometric functions, to model real- world phenomena. 11. Verify and apply trigonometric identities, of varying complexity. 12. Solve trigonometric equations, for one or multiple solutions. 13. Perform simple operations, with vectors algebraically and geometrically. (N- VM2-5) 14. Use inner products to determine whether two vectors are perpendicular. (N- VM2-5) 15. Find a vector s magnitude and direction from its components, and find its components given its magnitude and direction. (N- VM1-5) 16. Use vectors to solve real- world word problems. (N- VM1-5) 17. Calculate the projection of one vector onto another. (Honors only) Berkeley Heights Public Schools 6

9 STUDENT PROFICIENCIES (continued) 18. Write complex numbers in polar form. (Honors only) 19. Perform operations, on complex numbers written in polar form. (Honors only) 20. Apply DeMoivre s theorem, to find powers and roots of complex numbers. (Honors only) 21. Convert rectangular coordinates to polar and vice- versa. 22. Graph polar equations. (Honors only) 23. Define each of the conic sections, as a locus. 24. Use the method of completing the square, to go from general to standard form equations. 25. Determine critical information (center, vertices, focus point(s), asymptotes, directrix) as applicable to each conic section, from a standard form equation. 26. Find the standard form equation of a conic section, given some of its critical information. 27. Identify and graph each of the conic sections accurately, without a calculator. 28. Identify sequences as arithmetic, geometric, or neither. 29. Find a recursive and explicit formula, for the n th term of a sequence. 30. Represent a series, using sigma notation. 31. Calculate partial and infinite sums, as applicable by formula and using a calculator. 32. Introduce a parameter to define a plane curve, as a set of parametric equations. (Honors only) 33. Convert between a set of parametric equations and its rectangular equation. (Honors only) 34. Recognize the parametric equations that govern each conic section. (Honors only) 35. Decompose a rational expression into its partial fractions. (Honors only) 36. Use long division in partial fraction decomposition, for improper fractions. (Honors only) Berkeley Heights Public Schools 7

10 STUDENT PROFICIENCIES (continued) 37. Factor higher degree polynomials, using grouping and/or rational root theorem. (Honors only) 38. Distinguish between removable and non- removable discontinuities in a rational function, and to know the graphical representation of each. (Honors only) 39. Use limits to sketch the graph of a rational function. (Honors only) 40. Evaluate limits numerically and algebraically. (Honors only) 41. Define continuity using limits. (Honors only) Berkeley Heights Public Schools 8

11 METHODS OF EVALUATION 1. Homework and class work. 2. Class participation. 3. Tests and quizzes. 4. Projects. 5. Cooperative learning assignments. 6. Mid- term and final examinations. Berkeley Heights Public Schools 9

12 SCOPE AND SEQUENCE COURSE OUTLINE/STUDENT OBJECTIVE The student will be able to: National & NJ Core Curriculum Standards/ Grade /12 9.1/ /12 9.1/12 Strands & Indicators A- CED4 A- REI1 C- C5 F- TF1-4 G- SRT8-11 A3 A1 A- CED2-3 F- BF3 F- IF1-2,4-6, 7e F- TF5-7 A3 A1 Course Outline/Student Objectives I. Trigonometric Functions (4 Weeks) A. Measure Angles in Degrees and Radians 1. Convert angles between degrees and radians 2. Find complementary, supplementary, and co- terminal angles 3. Find arc length and area of a sector 4. Angular and linear speed B. Right Triangle Definitions of Trigonometric Functions 1. Trigonometric values of special angles without a calculator 2. Use trigonometry to find sides of right triangles C. Find the Values of Trigonometric Functions for an Angle of Any Size in Standard Position 1. Understand the unit circle and its applications 2. Find reference angles for any given angle D. Inverse Trigonometric Functions 1. Use inverse trigonometric notation 2. Use inverse trigonometry to find angles of right triangles II. Graphing Trigonometric Functions (4 Weeks) A. Sketch Graphs of Sine, Cosine, and Tangent Functions 1. By plotting points and recognizing the periodic nature 2. Find domain of these three trigonometric functions and its relevance to the graphs 3. Use period, amplitude, and phase shift to translate the parent graphs of sine, cosine, and tangent (without calculator) 4. Find appropriate viewing windows on graphing calculator B. Sketch Graphs of Reciprocal Trigonometric Functions (Honors) Berkeley Heights Public Schools 10

13 12 8.1/12 9.1/ /12 9.1/12 A- CED2-4 A- REI1 F- BF3 F- IF1-2,4-6, 7e F- TF5-7 G- SRT8-11 A3 A1 A- CED4 A- REI1 A- SSE3 F- IF8 F- TF8-9 A3 A1 II. Graphing Trigonometric Functions (continued) 1. Derive graphs of secant, cosecant, and contangent from their reciprocal function graphs (Honors) 2. Find domain of reciprocal trigonometric functions and its relevance to the graphs (Honors) 3. Graph transformations of reciprocal trigonometric functions (without calculator) (Honors) 4. Find appropriate viewing windows on graphing calculator C. Sketch Graphs of Inverse Trigonometric Functions (Honors) III. Applications Of Trigonometric Functions (4 Weeks) A. Right Triangle Word Problems (Height of Building) B. Using Trigonometric Equations to Model Sinusoidal Motion 1. Derive trigonometric equation from given information/graph 2. Be able to solve for x or y value given the other C. Solving Applications, Using Trigonometric Equations D. Navigational Application Problems 1. Using sea bearings N 30⁰ W 2. Using air bearings 140⁰ (clockwise from north) (Honors) E. Non- Right Triangle Word Problems 1. The Law of Sines 2. The Law of Cosines 3. Find the area of a triangle IV. Trigonometric Identities And Equations (4 Weeks) A. Identities 1. Distinguish between identifies and equations 2. Derive and use basic identities a. reciprocal identities b. Pythagorean identities c. quotient identities d. co- function identities e. even/odd identities 3. Simplify trigonometric expressions using identities 4. Verify trigonometric identities 5. Derive and use sum, difference, and double angle identities B. Solving Trigonometric Equations 1. Solve for principle values 2. Solve for all real solutions Berkeley Heights Public Schools 11

14 12 A- CED1-4 G- SRT1,10 11 N- Q1-3 N- VM /12 9.1/12 A- CED4 A- REI1 F- IF7-9 N- CN1-6 A3 A1 IV. Trigonometric Identities And Equations (continued) 3. Solve for all solutions on any given interval 4. Use algebraic and trigonometric techniques in solving V. Vectors (4 Weeks) A. Vector Notations 1. Component form 2. Sum of unit vectors B. Vector Operations 1. Addition and subtraction 2. Scalar multiplication 3. Dot products 4. Finding the angle between vectors 5. Definition and properties of orthogonal (perpendicular) vectors 6. Finding magnitude and direction angles, given component form 7. Finding component form, given magnitude and direction angle C. Vector Applications 1. Navigation 2. Sum of forces 3. Tension (Honors) D. Vector Projections (Honors) 1. Finding scalar projection 2. Vector projection VI. Polar Coordinates and Complex Number (4 Weeks) A. Conversions 1. Convert rectangular coordinates to polar 2. Convert polar coordinates to rectangular 3. Convert rectangular equations to polar 4. Convert polar equations to rectangular B. Graph Polar Equations 1. Circles centered at pole and lines through the pole 2. Circles, limacons, rose curves, lemniscates (Honors) a. by plotting points b. using the graphing calculator C. Operations with Complex Numbers 1. Addition, subtraction, multiplication, division 2. Simplification to standard form, using cyclic nature of i 3. Finding absolute value of a complex number 4. Represent complex numbers in trigonometric (polar) form Berkeley Heights Public Schools 12

15 12 A- CED4 A- REI4,10 F- IF7-9 F- LE5 G- GPE1-3 VI. Polar Coordinates and Complex Number (continued) D. Find Powers and Roots of Complex Numbers 1. DeMoivre s theorem 2. Finding the n th roots of a complex number VII. Conic Sections (4 Weeks) A. Lines 1. Inclination angles 2. Finding the angle between two lines 3. Finding distance between a point and a line B. Define Conic Sections as a Locus, and Derive the Standard and General Forms of Their Equations 1. Circles 2. Parabolas 3. Ellipses 4. Hyperbolas C. Graphing Conic Sections 1. Circles a. center b. radius 2. Parabolas a. vertex b. focus c. directrix 3. Ellipses a. center b. vertices c. foci d. eccentricity 4. Hyperbolas a. center b. vertices c. foci d. asympototes D. Parametric Equations (Honors Only) 1. Represent a plane curve using parametric equations 2. Convert rectangular equations to a set of parametric equations 3. Convert parametric equations to a single rectangular equation 4. Identify parametric form for each of the conic sections Berkeley Heights Public Schools 13

16 12 A- SSE3-4 F- BF2 F- IF3 F- LE2 N- RN A- APR3,6-7 A- SSE3 F- IF4-8 VIII. Sequences And Series (4 Weeks) A. Finding the Formula for the n th Term of a Sequence 1. Recursive definition (a n in terms of previous term) 2. Explicit definition (a n in terms of n a. for arithmetic sequences b. for geometric sequences c. using pattern recognition for sequences, which are neither B. Identify a sequence as arithmetic or geometric 1. Formula for n th 2. Provide explanation C. Sums of Series 1. n th partial sums for arithmetic and geometric series 2. Sum of infinite geometric series with - 1 < r < 1 3. Represent a series using sigma notation IX. Introduction To Calculus (Honors) (4 Weeks) A. Method of Creating Partial Fractions from Rational Expression 1. Proper Rational Expressions 2. Use of long division for improper rational expressions B. Factoring Higher Degree Polynomials 1. Rational root theorem 2. Remainder and factor theorems 3. Descartes rule of signs 4. Upper and lower bound theorems C. Rational Functions 1. End behavior based on degrees of numerator/ denominator 2. Domain based on setting denominator equal to zero a. vertical asymptotes for non- removable discontinuities b. holes for removable discontinuities D. Notation and Evaluation of Limits 1. Distinguish between limits at asymptotes and holes 2. Use end behavior to introduce limits at infinity E. Derive limit definition of a derivative from instantaneous rate of change of a function Note: The New Jersey Core Curriculum Content Standards can be accessed at Berkeley Heights Public Schools 14

17 NEW JERSEY CORE CURRICULUM CONTENT STANDARDS CODE SHEET Standards: 1. Visual and Performing Arts 2. Comprehensive Health & Physical Ed. 3. Language Arts Literacy 4. Mathematics 5. Science 6. Social Studies 7. World Languages 8. Technological Literacy 9. Career Education and Consumer, Family and Life Skills Strands: A, B, C, D CPI (Cumulative Progress Indicators): 1, 2, 3, 4, 5, etc. Sample: From the Visual and Performing Arts Core Curriculum Content Standards First Standard, then Grade level, then Strand, and last CPI #s 1.1/4A1,2,4 Berkeley Heights Public Schools 15

18 RESOURCES/ACTIVITIES GUIDE Primary Text Larson, Ron, and Robert Hostetler. PreCalculus with Limits. Boston, Mass.: Houghton- Mifflin Company, (Honors Text) Holliday, Berchie, et al. Glencoe Advanced Mathematical Concepts: PreCalculus with Applications. New York, NY: Glencoe/McGraw- Hill Company, (Regular Text) Supplemental Text: Miller, Charles, et. al. Mathematical Ideas. 11 th ed. Boston, Mass.: Pearson Education/ Addison Wesley Company, Bellman, Allan, et. al. Algebra 2. Needham, Mass.: Prentice Hall Company, Aufmann, Richard, et al. College Algebra and Trigonometry. Boston, Mass.: Houghton Mifflin Company, Calculator It is recommended that all students use a Texas Instruments TI- 83, TI- 84, or similar graphing calculator. Berkeley Heights Public Schools 16

19 SUGGESTED AUDIO- VISUAL/COMPUTER AIDS Web Sites: Berkeley Heights Public Schools 17

20 SUGGESTED MATERIALS Resources for Students Primary Text Larson, Ron, and Robert Hostetler. PreCalculus with Limits. Boston, Mass.: Houghton- Mifflin Company, (Honors Text) Holliday, Berchie, et al. Glencoe Advanced Mathematical Concepts: PreCalculus with Applications. New York, NY: Glencoe/McGraw- Hill Company, (Regular Text) Supplemental Text: Miller, Charles, et. al. Mathematical Ideas. 11 th ed. Boston, Mass.: Pearson Education/ Addison Wesley Company, Bellman, Allan, et. al. Algebra 2. Needham, Mass.: Prentice Hall Company, Aufmann, Richard, et al. College Algebra and Trigonometry. Boston, Mass.: Houghton Mifflin Company, Resources for Teacher Primary Text Larson, Ron, and Robert Hostetler. PreCalculus with Limits. Boston, Mass.: Houghton- Mifflin Company, (Honors Text) Holliday, Berchie, et al. Glencoe Advanced Mathematical Concepts: PreCalculus with Applications. New York, NY: Glencoe/McGraw- Hill Company, (Regular Text) Supplemental Text: Miller, Charles, et. al. Mathematical Ideas. 11 th ed. Boston, Mass.: Pearson Education/ Addison Wesley Company, Bellman, Allan, et. al. Algebra 2. Needham, Mass.: Prentice Hall Company, Aufmann, Richard, et al. College Algebra and Trigonometry. Boston, Mass.: Houghton Mifflin Company, Berkeley Heights Public Schools 18