PreCalculus Curriculum Guide

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1 MOUNT VERNON CITY SCHOOL DISTRICT A World Class Organization PreCalculus Curriculum Guide THIS HANDBOOK IS FOR THE IMPLEMENTATION OF THE NYS MATH b CURRICULUM IN MOUNT VERNON. THIS PROVIDES AN OUTLINE OF THE DISTRICT S EXPECTATIONS AND POLICIES. Developed August 2009

2 Mount Vernon City School District Board of Education Charles Stern President Derrick Claye Vice President Board Trustees Maria Aneiro Adrian Armstrong Carol Ben Reuben Maria Cedano Elias Steven Gootzeit Lynn C. Frazier-McBride Michelle Walker Superintendent of Schools W.L. Tony Sawyer, Ed.D. Deputy Superintendent Maureen Gonzalez Assistant Superintendents Timothy Costello, Business Shelley Jallow, Curriculum and Instruction Pre-Calculus 2 Curriculum Guide

3 TABLE OF CONTENTS I. COVER II. MVCSD BOARD OF EDUCATION III. TABLE OF CONTENTS IV. IMPORTANT DATES V. VISION STATEMENT VI. MVCSD PRECALCULUS PACING GUIDE VII. SECONDARY MATHEMATICS GRADING POLICY...16 VIII. CLASSROOM AESTHETICS IX. WORD WALLS ARE DESIGNED TO...18 X. SYSTEMATIC DESIGN OF A MATHEMATICS LESSON...19 This document was prepared by the Mount Vernon City School District Curriculum and Instruction Department in conjunction with the Mathematics Articulation Committee. Pre-Calculus 3 Curriculum Guide

4 IMPORTANT DATES REPORT CARD 10 WEEK PERIOD Marking Period Marking Period Begins Interim Progress Reports Marking Period Ends Duration Report Card Distribution 1 9/9/09 10/5/09 11/13/09 10 weeks 11/23/ /16/09 12/14/09 1/29/10 10 weeks 2/8/10 3 2/1/10 3/8/10 4/23/10 10 weeks 5/3/10 4 4/26/10 5/24/10 6/18/10 8 weeks 6/25/10 As per MVCSD Board Resolution 06-71, the Parent Notification Policy states Parent(s) / guardian(s) or adult students are to be notified, in writing, at any time during a grading period when it is apparent - that the student may fail or is performing unsatisfactorily in any course or grade level. Parent(s) / guardian(s) are also to be notified, in writing, at any time during the grading period when it becomes evident that the student's conduct or effort grades are unsatisfactory. DISTRICT ASSESSMENTS 1. Quarterly Examination * November 2-6, Midterm Examination * January 19-22, Quarterly Examination * April 12-16, Final Examination * June 7-11, 2010 * Note: These dates are tentative. Pre-Calculus 4 Curriculum Guide

5 VISION STATEMENT True success comes from co-accountability and co-responsibility. In a coherent instructional system, everyone is responsible for student learning and student achievement. The question we need to constantly ask ourselves is, "How are our students doing?" The starting point for an accountability system is a set of standards and benchmarks for student achievement. Standards work best when they are well defined and clearly communicated to students, teachers, administrators, and parents. The focus of a standards-based education system is to provide common goals and a shared vision of what it means to be educated. The purposes of a periodic assessment system are to diagnose student learning needs, guide instruction and align professional development at all levels of the system. The primary purpose of this Instructional Guide is to provide teachers and administrators with a tool for determining what to teach and assess. More specifically, the Instructional Guide provides a "road map" and timeline for teaching and assessing the NYS Mathematics Core Curriculum. I ask for your support in ensuring that this tool is utilized so students are able to benefit from a standards-based system where curriculum, instruction, and assessment are aligned. In this system, curriculum, instruction, and assessment are tightly interwoven to support student learning and ensure ALL students have equal access to a rigorous curriculum. We must all accept responsibility for closing the achievement gap and improving student achievement for all of our students. Satish Jagnandan Administrator for Mathematics and Science (K-12) Pre-Calculus 5 Curriculum Guide

6 PRE-CALCULUS PACING GUIDE This guide using A Graphical Approach to Pre-calculus with Limits (ISBN #: X) was created to provide teachers with a time frame to complete the Pre-Calculus Curriculum. Chapter Topics Sub-Topics Timeframe 1-1 Real Numbers and Sept Coordinate Systems 1. Linear, Equations, and Inequalities 1-2 Introduction to Relations and Sets of Real Numbers Coordinate Systems Viewing Window Roots Distance Formula Midpoint Formula Set-builder notation and Interval Notation Relations, Domains, and Range Tables Function Notation 1-3 Linear Basic Concepts About Linear Slope of a Line Slope-intercept Form of the Equation of a Line 1-4 Equation of Lines and Linear Models 1-5 Linear Equations and Inequalities 1-6 Applications of Linear Point-slope Form of the Equation of a Line Other Forms of the Equation of a Line Parallel and Perpendicular Lines Linear Models and Regression Solving Linear Equations Identities and Contradictions Solving Linear Inequalities Graphical Approaches to Solving Linear Inequalities Three-Part Inequalities Applications of Linear Equations Break-Even Analysis Direct Variation Formulas EXAM #1 Sept Sept Sept Sept Sept Pre-Calculus 6 Curriculum Guide

7 Chapter Topics Sub-Topics Timeframe 2. Analysis of 2-1 Graphs of Basic Continuity Oct Graphs of and Increasing and Decreasing Relations; Symmetry The Identity Function The Squaring Function and Symmetry with Respect to the y-axis The Cubing Function and Symmetry with Respect to the Origin The Square Root and Cube Root The Absolute Value Function The Relation x = y 2 and Symmetry with Respect to the x-axis 3. Polynomial 2-2 Vertical and Horizontal Shifts of Graphs 2-3 Stretching, Shrinking, and Reflecting Graphs 2-4 Absolute Value : Graphs, Equations, Inequalities, and Applications 2-5 Piece-Wise Defined 2-6 Operations and Composition 3-1 Complex Numbers 3-2 Quadratic and Graphs 3-5 Quadratic Equations and Inequalities Even and Odd Vertical Shifts Horizontal Shifts Combinations of Vertical and Horizontal Shifts Effects of Shifts on the Domain and Range Horizontal Shifts Applied to Equations for Modeling Vertical Stretching Vertical Shrinking Reflecting Across and Axis Combining Transformations of Graphs The Graph of y = f(x) Properties of Absolute Value Equations and Inequalities Involving Absolute Value Graphing Piece-Wise Defined The Greatest Integer Function Applications of Piece-Wise Defined Operations on The Difference Quotient Composition of Applications of Operation and Composition EXAM #2 The Number i Operations with Complex Numbers Completing the Square Graphs of Quadratic Vertex Formula Extreme Values Applications and Modeling Zero-Product Property Solving x 2 = k Quadratic Formula and the Discriminant Solving Quadratic Equations Solving Quadratic Inequalities Formulas involving Quadratics Another Quadratic Model QUARTERLY Pre-Calculus 7 Curriculum Guide Oct Oct Oct Oct Oct Oct Oct Oct

8 Chapter Topics Sub-Topics Timeframe Higher-Degree Cubic Nov Polynomial Polynomial Quartic and Graphs Extrema End Behavior x-intercepts (Real Zeros) Comprehensive Graphs 4. Rational, Power, and Root 3-7 Topics in the Theory of Polynomial 3-8 Polynomial Equations and Inequalities 4-1 Rational and Graphs 4-3 Rational Equations, Inequalities, Applications, and Models Curve Fitting and Polynomial Models Intermediate Value Theorem Division of Polynomials Synthetic Division Remainder Theorem Factor Theorem Complex Zeros and the Fundamental Theorem of Algebra Nubmer of Zeros Rational Zeros Theorem Polynomial Equations Polynomial Inequalities Complex nth Roots Applications and Polynomial Models EXAM #3 The Reciprocal Function The Rational Function Defined by f(x) = 1/x 2 Vertical and horizontal asymptotes Oblique Asymptotes Graphs with Points of Discontinuity Solving Rational Equations Solving Rational Inequalities Inverse Variation Combined and Joint Variation Nov Nov Nov Nov Pre-Calculus 8 Curriculum Guide

9 Chapter Topics Sub-Topics Timeframe 4. Rational, 4-4 Defined Power and Root Dec Power, and by Powers and Roots Modeling Using Power Root Graph of f(x)= n ax + b Graphing Circles and Horizontal Parabolas Using Root 4-5 Equations, Equations and Inequalities Dec Inequalities, and Applications Involving Root Applications 5. Inverse, Exponential, and Logarithmic EXAM #4 5-1 Inverse Inverse Operations One-to-One Inverse and their Graphs An Application of Inverse 5-2 Exponential Function 5-3 Logarithms and their Properties 5-4 Logarithmic Real-Number Exponents Graphs of Exponential Exponential Equations (Type 1) The Number e Compound Interest Definition of Logarithm Common Logarithm Natural Logarithms Properties of Logarithms Change-of-Base Rule Graph of Logarithmic Applications of Logarithmic functions Dec Dec Dec Dec Pre-Calculus 9 Curriculum Guide

10 Chapter Topics Sub-Topics Timeframe 5. Inverse, 5-5 Exponential and Exponential Equations Jan Exponential, Logarithmic Equations Exponential Inequalities and and Inequalities Logarithmic Equations Logarithmic Logarithmic Inequalities Equations with Both Exponentials and Logarithms Formulas Involving Exponentials and Logarithms. 6. Analytic Geometry 6-1 Circles and Parabolas 6-2 Ellipses and Hyperbolas 6-4 Parametric Equations EXAM #5 Conic Sections Equations of Circles with Center at the Origin and Off the Origin Graphs of Circles An Application of Circles Equations of Parabolas Graphs of Parabolas Equations of Ellipses Graph of an Ellipse Application of Ellipses Equations of Hyperbolas Graphs of Hyperbolas Parametric Equations and their Graphs Rectangular Equivalents of Parametric Equations Alternative Forms of Parametric Equations. EXAM #6 MIDTERM Jan Jan Jan Pre-Calculus 10 Curriculum Guide

11 Chapter Topics Sub-Topics Timeframe 7-1 Systems of Feb Equations 7. Matrices and Systems of Equations and Inequalities 7-2 Solution of Linear Systems by Echelon Method 7-3 Solution of Linear Systems by Row Transformation 7-4 Matrix Properties and Operations 7-5 Determinants and Cramer s Rule Linear Systems Substitution Method Elimination Method Special Systems Nonlinear Systems Application of Systems Geometric Considerations Analytic Solutions of Systems in Three Variables Matrices and Technology Matrix Row Transformation Row Echelon Method Reduced Row Echelon Method Special Cases Terminology of Matrices Operations on Matrices Applying Matrix Algebra Determinants of 2x2 Matrices Determinants of Larger Matrices Derivation of Cramer s Rule Using Cramer s Rule to Solve Systems 7-7 Partial Fractions Decomposition of Rational Expression Distinct Linear Factors Repeated Linear Factors Distinct Linear and Quadratic Factors Repeated Quadratic Factors EXAM #7 Feb Feb Feb Feb Feb Pre-Calculus 11 Curriculum Guide

12 Chapter Topics Sub-Topics Timeframe Angles and Arcs Basic Terminology March Trigonometric Degree Measure and Standard Position and Co-terminal Applications Angles Radian Measure Arc Lengths and Sectors Angular and Linear Speed 8-2 Unit Circle Circular March Using a Calculator to Find Function Values Exact Function Values for π /4, π /6, π /3 8-3 Graph of Sine and Cosine 8-5 Function of Angles and Fundamental Identities 8-6 Evaluating Trigonometric 8-7 Application of Right Triangles Review of Basic Concepts Periodic Function Graph of the Sine Function Graph of the Cosine Function Graphing Techniques, Amplitude, and Period Translations Graphs of the Secant and Cosecant Graph of the Tangent and Cotangent Addition of Ordinates Trigonometric Quadrantal Angles Reciprocal Identities Signs and Ranges of Function Values Pythagorean Identities Quotient(Ratio) Identities Definitions of the Trigonometric Trigonometric Function Values of Special Angles Co-function Identities Reference Angles Special Angles as Reference Angles Finding Function Values with a Calculator Finding Angle Measures Significant Digits Solving Triangles. Angles of Elevation and Depression Bearing, Heading, and Further Applications EXAM #8 March March March March Pre-Calculus 12 Curriculum Guide

13 Chapter Topics Sub-Topics Timeframe 9-1 Trigonometric Fundamental Identities April Identities Using the Fundamental Identities 9. Trigonometric Identities and Equations 9-2 The Sum and Difference Identities 9-4 Inverse Circular 9-6 Trigonometric Equations and Inequalities Verifying Identities Cosine Sum and Difference Identities Sine and Tangent Sum and Difference Identitiers Double-Angle Identities Product-to-Sum and Sum-to-Product Identities. Half-Angle Identities Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Remaining Inverse Trigonometric Inverse Function Values Equation Solvable by Linear Method Equations Solvable by Factoring Equations Solvable by Quadratic Formula Using Trigonometric Identities to Solve Equations Inequalities Involving Multiple-Angle Identities Inequalities Involving Half-Angle Identities EXAM #9 QUARTERLY April April April 10. Applications of Trigonometry and Vectors 10-1 Law of Sines Congruency and Oblique Triangles Derivation of the Law of Sines Ambiguous Case Applications 10-2 Law Cosines and Area Formulas 10-3 Vectors and Their Applications Derivation of the Law of Cosines and Applications Area Formulas and Applications Basic Terminology Algebraic Interpretation of Vectors Operations with Vectors Dot Product and the Angle Between Vectors April April April Pre-Calculus 13 Curriculum Guide

14 Chapter Topics Sub-Topics Timeframe 10-4 May Trigonometric/Polar Form of Complex Numbers 10. Applications of Trigonometry and Vectors 11. Further Topics in Algebra 10-5 Powers and Roots of Complex Numbers 10-6 Polar Equations and Graphs 11-1 Sequences and Series 11-2 Arithmetic Sequence and Series 11-3 Geometric Sequence and Series 11-4 The Binomial Theorem The Complex Plane and Vector Representation Polar Form Product of Complex Numbers in Trigonometric Form Quotient of Complex Numbers in Polar Form Powers of Complex Numbers (De Moivre s Theorem) Roots of Complex Numbers Polar Coordinate System Graphs of Polar Equations Classifying Polar Equations Converting Equations Parametric Equations with Trigonometric EXAM #10 Sequences Series and Summation Notation Summation Properties Basic Terminology Arithmetic Sequence Arithmetic Series Geometric Sequence Geometric Series Infinite Geometric Series Annuities A Binomial Expansion Pattern Pascal s Triangle n-factorial Binomial Coefficients The Binomial Theorem rth Term of a Binomial Expansion May May May May May May Pre-Calculus 14 Curriculum Guide

15 Chapter Topics Sub-Topics Timeframe 11. Further 11-6 Counting Theory Fundamental Principle of Counting June Topics in Permutations Algebra Combinations Distinguishing Between Permutation and Combination 11-7 Probability Basic Concepts June Complements and Venn Diagrams Odds Union of Two Events Binomial Probability 12. Limits, Derivative, and Definite Integrals 12-1 Introduction to Limits 12-2 Techniques for Calculating Limits EXAM #11 Limit of a Function Finding Limits of Various Tyes of Limits That Do Not Exist Rules for Limits Limits Involving Trigonometric 12-3 One-Sided Limits Right-Hand Limits and Left-Hand Limits Infinity as a Limit Limits as x Approaches Positive or Negative Infinity 12-4 Tangent Lines and Derivatives 12-5 Area and the Definite Integral The Tangent Line as a Limit of the Secant Lines Derivative of a Function Interpretation of the Derivative as a Rate of Change Marginal Concepts in Economics Area by Approximation The Definite Integral EXAM #12 FINAL June June June June June Pre-Calculus 15 Curriculum Guide

16 SECONDARY MATHEMATICS GRADING POLICY This course of study includes different components, each of which are assigned the following percentages to comprise a final grade. I want you--the student--to understand that your grades are not something that I give you, but rather, a reflection of the work that you give to me. COMPONENTS 1. Chapter / Unit Tests 35% 2. Quizzes 15% 3. Homework 20% 4. Notebook and/or Journal 15% 5. Classwork / Class Participation 15% o Class participation will play a significant part in the determination of your grade. Class participation will include the following: attendance, punctuality to class, contributions to the instructional process, effort, contributions during small group activities and attentiveness in class. Important Notice As per MVCSD Board Resolution 06-71, the Parent Notification Policy states Parent(s) / guardian(s) or adult students are to be notified, in writing, at any time during a grading period when it is apparent - that the student may fail or is performing unsatisfactorily in any course or grade level. Parent(s) / guardian(s) are also to be notified, in writing, at any time during the grading period when it becomes evident that the student's conduct or effort grades are unsatisfactory. Pre-Calculus 16 Curriculum Guide

17 CLASSROOM AESTHETICS PRINT RICH ENVIRONMENT CONDUCIVE TO LEARNING TEACHER NAME: PERIOD: ROOM: CHECKLIST Teacher Schedule YES NO Class Lists Seating Charts Code of Conduct / Discipline (Class Rules) Mathematics Grading Policy Mathematics Diagrams, Posters, Displays, etc. Updated Student Work (Projects, Assessments, Writing, etc.) Updated Student Portfolios Updated Grade Level Mathematics Word-Wall Organization of Materials Cleanliness Principal Signature: Date: AP Signature: Date: Pre-Calculus 17 Curriculum Guide

18 WORD WALLS ARE DESIGNED to promote group learning support the teaching of important general principles about words and how they work Foster reading and writing in content area Provide reference support for children during their reading and writing Promote independence on the part of young students as they work with words Provide a visual map to help children remember connections between words and the characteristics that will help them form categories Develop a growing core of words that become part of their vocabulary Important Notice A Mathematics Word Wall must be present in every mathematics classroom. The List of Mathematical Language for Math 4 Math B level instruction must be incorporated into the Mathematics Word Wall. Math Word Wall Create a math word wall Place math words on your current word wall but highlight them in some way. Pre-Calculus 18 Curriculum Guide

19 SYSTEMATIC DESIGN OF A MATHEMATICS LESSON What are the components of a Secondary Mathematics Lesson? PHASE I: ENGAGE AND EXPLAIN 20 minutes Standard: Indicate NYS Performance Indicator Instructional Objective: This is a statement which informs the student what he/she will be able to do by the end of the lesson; the aim of the learning session. Motivation: This is an activity, a problem, idea, or picture, etc. which produces a mental set. The motivation is used to hook the student into the lesson. It should be based on the level of student interest/experiences. Delivery of Instruction: This is the instructional delivery of the teacher, which gives the learner the information needed to complete the instructional objective of the lesson. Activities such as demonstrations, cooperative groups, teacher / peer directed approaches, teacher modeling, etc. may be used. PHASE II: EXPLORE AND ELABORATE 15 minutes Applications: Checking for understanding - a check of the students possession of essential information. 1. Guided Practice: Posing questions that gradually lead students from easy or familiar examples to new understandings 2. Independent Practice: students complete private, individual work based on extending the lesson objective. 3. Cooperative Learning: Small teams may work with sets of data or develop alternatives or a preferred approach to solving a problem. 4. Activity (using manipulative, technology) to development conceptual understanding of content. PHASE III: EVALUATE AND EXTEND 10 minutes Closure: 1. This is the final summary that clarifies the concepts/skills taught. (eg. What new information (skill, theme) did we learn today? ) This phase is used by the teacher to help students connect the taught part of the lesson with the application portion. The time involves questioning, reflection, and reteaching / reconstructing content. Extended Practice: 1. Homework, projects or enrichment activities should be assigned on a daily basis. 2. SPIRALLING OF HOMEWORK - Teacher will also assign problems / questions pertaining to lessons taught in the past. Pre-Calculus 19 Curriculum Guide

20 Important Notice All aims must be numbered with corresponding homework. For example, aim #1 will corresponded to homework #1 and so on. Writing assignments at the end of the lesson (closure) bring great benefits. Not only do they enhance students' general writing ability, but they also increase both the understanding of content while learning the specific vocabulary of the disciplines. Spiraling Homework o Homework is used to reinforce daily learning objectives. The secondary purpose of homework is to reinforce objectives learned earlier in the year. The assessments are cumulative, spiraling homework requires students to review coursework throughout the year. Lesson #66 AIM #66: What is the relationship between the slopes of two parallel lines? Students will be able to: 1. determine that if two lines are parallel, their slopes are equal and the converse 2. given the equation of two lines, determine if they are parallel 3. write the equation of a line parallel to a given line and passing through a given point on the line Writing Exercise / Closure: Explain the relationship between the slopes of two parallel lines. Homework #66 Page 234 #5, 7, 9 Page 228 #4, 13 Page 153 #21, 33 Page 78 #40 Study for Quiz #8 on Wednesday, January 7, 2009 Manipulatives must be incorporated in all lessons. With students actively involved in manipulating materials, interest in mathematics will be aroused. Using manipulative materials in teaching mathematics will help students learn: a. to relate real world situations to mathematics symbolism. b. to work together cooperatively in solving problems. c. to discuss mathematical ideas and concepts. d. to verbalize their mathematics thinking. e. to make presentations in front of a large group. f. that there are many different ways to solve problems. g. that mathematics problems can be symbolized in many different ways. h. that they can solve mathematics problems without just following teachers' directions. Pre-Calculus 20 Curriculum Guide

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