Course Outline, Mathematics - Algebra 2-Trigonometry -

Transcription

1 Course Outline, Mathematics - Algebra 2-Trigonometry - Course Description: Algebra 2 is the study of equations of second degree, which include such diverse geometric manifestations as parabolas, ellipses, hyperbolas, and circles. We learn techniques to solve quadratic equations, as well as further techniques to solve systems of linear equations. Equations of the second degree have applications from astrophysics (planetary motion) to geometry (Pythagorean Theorem) to sports (how a basketball travels through the air). In addition, we study polynomial, rational, algebraic, and transcendental functions and important theorems such as the Fundamental Theorem of Algebra. The connection to circles and right triangles marks the transition to trigonometry, a subject that introduces angle measurement to the algebraic context of functions, equations, and graphs. Algebra 2 with Trigonometry will also be used to develop general skills of effective thinking, including imagination, abstraction, persistence, and active learning. All of these skills are increasingly necessary in our modern mathematical world, but are also timelessly valuable in understanding what the scientist Galileo called this grand book the universe, which stands continually open to our gaze it is written in the language of mathematics. Course Objectives: This course is intended to: 1. Help students truly understand the fundamental concepts of algebra, geometry, trigonometry, exponential and logarithmic functions, statistics and probability. 2. Foreshadow important ideas of calculus. 3. Illustrate to students how algebra and trigonometry can be used to model real-life problems. 4. Prepare students for the Geometry and Algebra 2/Trigonometry Regents exam. 5. Prepare students for upper level mathematics. Essential questions: 1. How do you determine the differences and similarities between the families of functions, (Exponential, Logarithmic, Quadratic, Polynomial, Rational, Trigonometric)? 2. Which problem-solving strategy is most appropriate in this situation? 3. How do I use my mathematical tools to create models for real-world situations and then solve them efficiently for a given set of conditions? 4. How does the unit circle relate to the trigonometric functions?

2 Instructor: Mr. Goncalves Classroom: Room 401 Schedule: Refer to your program Extra help hours: Tuesday 5 th period Thursday 1 st period Contact Information: Phone: Office (212) ggoncalves@erhsnyc.net Materials: 1. Primary textbook: PRECALCULUS graphical, numerical, algebraic by Demana, Waits, Foley, and Kennedy. Pearson Education, Inc Secondary textbook: AMSCO s Math B 3. Graphing Calculator: TI-83 is recommended 4. Pencils, an eraser, a ruler (preferably a protractor), and a compass 5. Graph Paper 6. Notebook Composition book recommended 7. Folder or binder to keep quizzes, exams, and hand-outs. Course Requirements: 1. Notebook: Students strongly encouraged to take notes during class. Extra credit will be awarded to students who capture complete notes in an orderly manner. 2. Homework: Students will be assigned daily homework and encouraged to use the composition book. Homework assignments submitted past due date will not be accepted unless under extreme circumstances. Homework quizzes will be given weekly to ensure learning and homework completion. 3. Classwork: Participation in class assignments and discussions Group work/ projects 4. Tests and Quizzes: At least one-week notice will be given for tests and quizzes. There will be no make up quizzes or tests unless the instructor is informed in advance and an acceptable written note is provided with a justification for the absence. There will be no test revisions. 5. Pencil and eraser: Ink is not allowed on any assignment or test.

3 Classroom Expectations: 1. Respect yourself, your peers, your teacher, and the learning environment. 2. Timeliness and punctual attendance is expected and rewarded 3. Complete classroom activities within the allotted time to better prepare you for exams 4. Follow the homework assignments in the recommended format provided to you by your instructor 5. Be prepared for class, i.e. bring in all your learning tools daily. Grading Policy: Criteria for computing grades: Weight Exams 40% Quizzes 30% Homework Quizzes/ NEAT Notebook 20% Class Participation/Group Activity 10% Suggestions and Resources: 1. One-on-one with teacher: Students are encouraged to approach the instructor either immediately after school or via Textbook reading assignments: Read each section before class discussion, then re-read as homework activities are assigned, and use this material to study for quizzes and exams. Inclass activities leverage textbook examples. 3. Homework assignments: Make an effort to figure out even the more challenging homework problems, try multiple times, even consider a break in between turns. Often the brain works it out while doing other things. ERHS Academic Honesty Requirement: Academic Dishonesty will not be tolerated and will result in automatic failure of Exam/Quiz/Project and disciplinary action will be taken I have read and am aware of the grading policy for Mr. Gonçalves class. Student name: Student signature: Date: / / Parent/Guardian name: Parent/Guardian signature: Date: / /

4 Tentative Course Outline: Chapter Time Topic P.1 3 or Representing Real numbers (Natural numbers, whole numbers, integers, rational, irrational) A P or 3 or Order and interval notation Radicals Simplifying radical expressions Solving radical equations Rationalizing the denominator Integer and Rational exponents Definition of Absolute-value (algebraically pg. 14 and geometrically pg. 15) Solving Absolute-value equations Absolute-value inequalities P.3 3 or Solving equations (algebraic and graphically) 1. Linear 2. Quadratic various methods - later 3. Cubic - later 4. Fractional - later 5. Algebraic - later 6. Absolute-value - later 7. Exponential - later 8. Logarithmic - later 9. Trigonometric - later Linear inequalities Solving linear inequalities P.4 5 or 6 days P days days Slope of a line Point-slope form equation Slope-intercept form Graphing linear equations Parallel and Perpendicular lines Linear equations in two variables (solving them graphically and algebraically) Linear modeling and correlation coefficient Solving equations graphically Graphing quadratic equations Solving quadratic equations factoring, square roots, completing the square, and quadratic formula Optimization Motion problems Ex: Calculate the maximum height of a rocket Discriminant Solving system of non-linear equations algebraically (line and parabola, line and circle) Using completing the square to write an equation for a circle Imaginary numbers Complex numbers Addition and subtraction of complex numbers Multiplication and division of complex numbers Solving quadratic equation with imaginary roots

5 A A days days 6-10 days days 2 days 2 6 days The nature of the roots of any quadratic equation Using given conditions to write a quadratic equation Solution of system of equations Quadratic inequalities Adding, subtracting, and multiplying polynomials Special products Factoring polynomials using special products Factoring trinomials (1) Factoring by grouping (2) Factoring the sum and difference of two cubes Domain of an algebraic Expression Domain of rational expression (1 st commandment of math) Reducing Rational expressions Multiplying and dividing rational expressions Adding or subtracting rational expressions Simplifying complex fractions Solving Rational equations Extraneous Solutions Applications Function definition and notation Domain and Range Continuity Increasing and Decreasing Functions Local and Absolute Extrema (Extreme Value Theorem EVT) Symmetry (even and odd functions) Asymptotes (horizontal and vertical) End Behavior Twelve basic functions Analyzing functions graphically Piecewise functions Composition of Functions Relations and Implicity Defined functions Relations defined Parametrically (Section 6.3 pg. 522) Inverse functions Graphical transformations Vertical and Horizontal translations Reflections Vertical and Horizontal stretches and shrinks Composition of transformations Functions from formulas Functions from graphs Functions from data page 149 (Regression Types) Polynomial functions Linear functions and their graphs Average rate of change Linear correlation and modeling Quadratic functions and their graphs Power functions and variation (direct and inverse) Graphs of power functions ( x 3 3, x, x ) Modeling with power functions

6 & & days 8 days 6 days Graphs of Polynomials functions End behavior of polynomial functions Zeros of polynomials functions Intermediate Value Theorem Modeling Long division Remainder and Factor Theorems Synthetic Division Rational Zeros Theorem Upper and Lower bounds Fundamental Theorem of Algebra Linear Factorization Theorem Complex Conjugate Zeros Rational functions (y = 1/x) Limits and Asymptotes Analyzing Graphs of Rational Functions 3 days Polynomial inequalities Rational inequalities Applications 3 days zes 10 days 10 days days 2-3 quizzes 8 12 days 1- Exponential functions and their graphs The Natural Base e Transformations Constant percentage rate Exponential growth and decay Modeling Inverses of exponential functions Logarithmic functions and their graphs Common log and natural log Properties of logarithmic functions Change of base Solving exponential equations Solving logarithmic equations Regression models (page 328) Mathematics of finance Interest Compounded Annually Interest Compounded k times per year Interest Compounded Continuously The right triangle (sine, cosine, tangent, and their reciprocals) Angles as rotations Sine and cosine as coordinates The sine and cosine functions: Sinusoidal Functions The tangent function Function values of special angles Finding reference angles Radian measure (at this point students should start thinking in radian) Trigonometric functions involving radian measure The Pythagorean identities Cofunctions The wrapping function Graph of y = sin x and y = cos x Amplitude, Frequency, and Period

7 5 Analytic Trigonometry 9 Discrete Mathematics 10 An Introduction to Calculus (optional) 7 System and Matrices (optional) 20 days 3 quizzes 12 days Sketching sine and cosine curves Transformations of sine and cosine curves 3 days Graph of y = tan x Graphs of inverse trig functions arcsine, arccosine, and arctangent Solving Problems with trigonometry Fundamental Identities Solving Trigonometric equations Proving Trigonometric Identities Sum and Difference Identities Multiple Angle Identities The Law of Sines The Law of Cosines Basic Combinatorics The Binomial Theorem Probability Sequences and Series Mathematical Induction - maybe Statistics and Data (Graphical and Algebraic) Limits and Motion: The tangent Problem Limits and Motion: The Area Problem More on Limits Numerical Derivatives and Integrals Solving System of two equations Matrix Algebra Multivariate Linear Systems and Row Operations Partial Fractions