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Precalculus 1

MONMOUTH COUNTY VOCATIONAL SCHOOL DISTRICT

4000 Kozloski Road PO Box 5033 Freehold, NJ 07728-5033

Precalculus

School: Grade: Credits: Subject Area: Prepared by: Date:

All Career Academies 11th and 12th 5 Mathematics Chrisine Burger, Bernie Groninger, William Shropshire, Scott Stengele, Gene Stoye, Linda Sutton Summer 2004

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Course Philosophy

Mathematics is a tool to enhance the understanding of the world in which we live. It is much more than computation; it is a multi-layered human endeavor providing a way of viewing the world. It contributes to the intellectual development of a student by enhancing his or her problem solving and analytical skills. This course is designed to increase the student's development of mathematical concepts, ability to solve real life problems, communicate mathematically, and make connections and model phenomenon. The course gives preference to developing an understanding of the subject beyond memorizing formulas and performing computation. Mathematics is thinking about The course serves to

numbers, relationships, graphs, change, and creating models.

develop a student's problem solving abilities and opportunities to apply mathematics in the real world and to the related themes of each career academy.

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Course Description and Materials

Course Description: Prerequisite: Students will have successfully completed Algebra I, Geometry and

Algebra II/Trigonometry.

This course is designed to extend the students' previous knowledge of algebraic concepts and trigonometric functions, to help students truly understand the fundamental concepts of algebra, trigonometry and analytic geometry, to build an intuitive foundation for calculus, and to show how algebra and trigonometry can be used to model real ­ life problems. A principle feature is the balance among the algebraic, numerical and verbal methods of representing problems. In addition, the students will continue to develop their analytical and problem solving skills.

The course will integrate the use of the graphing calculator and other technologies to develop a deeper understanding of the mathematical concepts and make connections to real life phenomenon Text: Demana, Waits, Foley and Daniel Kennedy. Pre Calculus: Graphical, Numerical and Algebraic, fifth edition. Reading, Massachusetts: Addition Wesley Longman, Inc, 2001.

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Methodology of Instruction

To learn the essential mathematics needed for the twenty-first century, students need a non - threatening environment in which they are encouraged to ask questions and take risks. The learning climate should incorporate high expectation for all students, regardless of gender, race, handicapping condition, or socio - economic status. Students need to explore mathematics using manipulatives, measuring devices, models, calculators, and computers. They need to have opportunities to talk to each other and to write about mathematics. Students need to have models of instruction that are suitable for the increased emphasis on problem solving, applications, and higher-order thinking skills. For example, cooperative learning allows students to work together in problemsolving situations to pose questions, analyze solutions, try alternative strategies, and check for reasonableness of results. (NCSM, 1988)

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Course Goals

The Precalculus developed for the Academies has the following goals: 1. The student, after completing this course, should understand and have the ability to apply the basic principles of Precalculus. 2. The student should have an understanding of the process behind the development of the mathematical theories. 3. The student, after completing this course, should appreciate the importance and applications of Precalculus concepts in his/her everyday life. 4. The students should recognize the problem solving skills developed in this Precalculus course can be applied to other courses and to real life. 5. The student, after completing this course, should be able to use technology as a tool for developing calculus skills and concepts, solving problems and modeling. 6. The student should have enhanced his/her problem solving skills. 7. The student should be able to use the mathematical skills to describe and analyze physical situations and real world problems.

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Course Proficiencies

Students will be able to: (*** if time permits) 1. Recognize and graph higher-order polynomial functions. 2. Use graphic and algebraic techniques to solve polynomial functions. 3. Identify the asymptotes and end behavior of a function. 4. Apply the Remainder Theorem, Factor Theorem, and Rational Zeroes Theorem. 5. Graph exponential and logarithmic functions. 6. Evaluate exponential and logarithmic functions. 7. Solve exponential and logarithmic functions. 8. Solve application problems involving exponential and logarithmic equations. 9. Evaluate and solve compound interest problems. *** 10. Convert between radian and degree measure. 11. Calculate arc length and sector area. 12. Define, graph and apply trig functions to solve problems. 13. Recognize, graph and apply the inverse trigonometric functions to solve problems. 14. Use trigonometric identities to simplify expressions and solve equations. 15. Validate trigonometric identities. 16. State and apply the addition, power reducing, double and half angle identities. 17. Use trigonometry to determine the area of a triangle and the area of a segment. 18. Describe a vector and use vector operations and properties. *** 19. Formulate and graph parametric equations. *** 20. Simulate motion with a graphing calculator.

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21. Construct a polar coordinate system. 22. Convert between rectangular and polar coordinates. 23. Analyze and graph polar equations. 24. Construct the trigonometric form of a complex number. 25. Use the trigonometric form to multiply complex numbers and to raise them to powers (using DeMoivre's Theorem). *** 26. Decompose partial fractions. *** 27. Find and graph an equation of a parabola and identify the focus, directrix, and focal width. 28. Find and graph an equation of an ellipse and identify the foci, focal axis, center, eccentricity, and vertices. 29. Find and graph an equation of a hyperbola and identify the foci, focal axis, center, vertices, eccentricity, and asymptotes. 30. Translate the coordinate axes to fix the center of a conic section at the origin. 31. Graph on a three-dimensional Cartesian plane and use the distance and midpoint formulas. *** 32. Find the sum of finite and converging infinite geometric series. 33. Formulate proofs using the principle of mathematical induction. 34. Apply the properties of limits. 35. Evaluate limits.

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Evaluative Measures

Students' performance can be assessed through the use of the following: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Tests and quizzes Examinations Homework Notebook Computer and graphing calculator activities Participation Group projects and activities Portfolio Oral presentations In class projects and activities Teacher observation and evaluation Classroom discussions Students questions and answers Journal

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Sequence and Integration of Topics

A. Prerequisites The following topics are a review from Algebra I and Algebra II. They will be covered as part of a summer study package and will be reviewed during the first days of class. Outline: 1. Real numbers 2. Cartesian Coordinate System 3. Linear Equations and Inequalities 4. Lines in the Plane 5. Solving Equations Graphically, Numerically and Algebraically and Graphically Objectives: Upon completion of this unit the student will be able to: 1. Convert between decimals and fractions, write inequalities, apply the basic properties of Algebra, and work with exponents and scientific notation. (CC 4.1.A.1, 4.1.A.2, 4.1.B.1, 4.1.B.2, 4.1.B.4) 2. Graph points, find distances and midpoints on a number line and in a coordinate plane, and write standard-form equations of circles. (CC 4.1.A.2, 4.1.B.1, 4.2.C.1, 4.3.B.1) 3. Solve linear equations and inequalities in one variable. (CC 4.3.D.2) 4. Use the concepts of slope and y-intercept to graph and write linear equations in two variables. (CC 4.3.B.2, 4.3.C.1, 4.3.D.2) 5. Solve equations involving quadratic, absolute value, and fractional expressions, by finding x-intercepts or intersections on graphs, by using algebraic technologies, or by using numerical technologies. (CC 4.3.B.2, 4.3.C.1, 4.3.D.2) 6. Solve inequalities involving absolute value, quadratic polynomials, and expressions involving fractions. (CC 4.3.C.1, 4.3.D.2) Suggested Activities: 1. Complete a package of summer review materials. 2. Investigating graphs of linear equations on a graphing calculator. 3. Application of linear equations in astronomy. 4. Finding real zeros of equations on a graphing calculator. 5. Mathematical History ­ Functional Notation B. Functions and Graphs The following topics are a review from Algebra I and Algebra II. They will be covered as part of a summer study package and will be reviewed during the first days of class. Outline: 1. Modeling and Equation Solving 2. Functions and Their Properties 3. Ten Basic Functions 4. Building Functions from Functions

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5. Graphical Transformations 6. Modeling with Functions Objectives: Upon completion of this unit the student will be able to: 1. Use numerical, algebraic, and graphical models to solve problems and will be able to solve problems and will be able to translate from one model to another. (CC 4.3.B.1, 4.3.C.1) 2. Represent functions numerically, algebraically, and graphically, determine the domain and range for functions, and analyze function characteristics such as extreme values, symmetry, asymptotes, and end behavior. (CC 4.3.B.1, 4.3.B.2) 3. Recognize graphs of the ten basic functions, determine domains of functions related to the ten basic functions and combine the ten basic functions in various ways to create new functions. (CC 4.3.B.4, 4.3.C.2) 4. Build new functions from old functions in several ways: by adding, subtracting, multiplying, dividing, and composing functions, by defining functions parametrically, and by computing inverses of functions. (CC 4.3.B.2) 5. Algebraically and graphically represent translations, reflections, stretches, and shrinks of functions and parametric relations. (CC 4.3.B.3) 6. Identify appropriate basic functions with which to model real world problems and be able to produce specific functions to model data, formulas, graphs, and verbal descriptions. (CC 4.3.B.4, 4.3.C.1) Suggested Activities: 1. Complete a summer review package. 2. Investigate grapher failure and hidden behavior with a graphing calculator. 3. Group activity to explore increasing, decreasing and constant data. (86) 4. Investigate translations, reflections, stretches and shrinks on a graphing calculator. 5. Activity using regression to model the number of diagonals in a regular polygon. 6. Environmental Application: model solid waste growth using linear regression. C. Polynomial, Power and Rational Functions Outline: 1. Linear and Quadratic Functions with Modeling 2. Power Functions with Modeling 3. Polynomial Functions of Higher Degree with Modeling 4. Real Zeros of Polynomial Functions 5. Complex Numbers 6. Rational Functions and Equations Objectives: Upon completion of this unit the student will be able to: 1. Recognize and graph linear and quadratic functions, and model quadratic functions (CC 4.3.C.1, 4.3.C.2) 2. Sketch power functions. (CC 4.3.C.1)

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3. Graph polynomial functions, predict their end behavior, and find their real zeros using a grapher or an algebraic method. (CC 4.3.B.2, 4.3.C.2, 4.3.D.2) 4. Divide polynomials using long division or synthetic division; to apply the Remainder Theorem, Factor Theorem, and Rational Zeros Theorem; and find upper and lower bounds for zeros of polynomials. (CC 4.3.C.2, 4.3.D.2) 5. Add, subtract, multiply, and divide complex numbers; and to find complex zeros of quadratic functions. (CC 4.3.D.2) 6. Describe the graphs of rational functions, identify horizontal and vertical asymptotes, and predict the end behavior of rational functions. (CC 4.3.A.2, 4.3.C.2) Suggested Activities: 1. Group activity to model revenue for small business. 2. Use graphing calculator to compare graphs of monomial functions and power functions. 3. Math History: Modeling Kepler's Planetary Law. 4. Use a graphing calculator to model end behavior of monomial function. 5. Group activity to design a box. 6. Group activity to investigate coefficient changes in cubic equations. 7. Use a graphing calculator to approximate real zeros of a polynomial function. 8. Explore complex solutions in cubic equations graphically. 9. Use a graphing calculator to investigate translations and reflections of a simple rational function. D. Exponential and Logarithmic Functions Outline: 1. Exponential Functions 2. Logarithmic Functions and Their Graphs 3. Properties of Logarithmic Functions 4. Equation Solving and Modeling 5. Mathematics of Finance (***if time allows) Objectives: Upon completion of this unit the student will be able to: 1. Evaluate and graph exponential equations. (CC 4.3.B.3, 4.3.B.4, 4.3.C.2) 2. Convert equations between logarithmic form and exponential form, evaluate common and natural logarithms, and graph common and natural logarithmic functions. (CC 4.3.B.1, 4.3.B.2, 4.3.B.3, 4.3.B.4, 4.3.C.2) 3. Apply the properties of logarithms to evaluate expressions and graph functions, and be able to re-express data. (CC 4.1.B.4, 4.3.B.3) 4. Apply properties of logarithms to solve exponential and logarithmic equations algebraically and solve applications problems using these equations. (CC 4.1.B.4, 4.3.C.1, 4.3.D.1, 4.3.D.2) 5. Use exponential functions and equations to solve business and finance applications related to compound interest and annuities. (CC 4.3.C.1, 4.3.C.3, 4.3.D.1) (***if time allows)

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Suggested Activities: 1. Use graphing calculator to explore the basic characteristics of exponential functions. 2. Biology: Model bacteria growth and carbon dating. 3. Math History: Napier and Computational Tools 4. Use graphing calculators to explore comparison of exponential and logarithmic functions. 5. Application: Noise Level Modeling E. Trigonometric Functions Outline: 1. Angles and Their Measures 2. Trigonometric Functions of Acute Angles 3. Trigonometry Extended: The Circular Functions 4. Graphs of Sine and Cosine 5. Graphs of Tangent, Cotangent, Secant, and Cosecant 6. Inverse Trigonometric Functions 7. Solving Problems with Trigonometry Objectives: Upon completion of this unit the student will be able to: 1. Convert between radians and degrees (CC 4.2.E.1) 2. Find arc lengths (CC 4.2.A.3, 4.2.E.1, 4.2.E.2) 3. Find the area of a sector (CC 4.2.E.1) 4. Define the six trigonometric functions using the lengths of the sides of a right triangle (CC 4.2.D.1, 4.2.E.1, 4.3.B.4, 4.3.C.1, 4.3.D.1, 4.3.D.2) 5. Solve problems involving the trigonometric functions of real numbers and the properties of the sine and cosine as periodic functions (CC 4.2.E.1, 4.3.B.4, 4.3.C.1, 4.3.D.2) 6. Generate the graphs of the sine and cosine functions (CC 4.2.E.1, 4.3.B.2, 4.3.B.3, 4.3.C.2) 7. Generate the graphs for the tangent, cotangent, secant, and cosecant functions (CC 4.2.E.1, 4.3.B.2, 4.3.B.3, 4.3.B.4, 4.3.C.2) 8. Graph transformations of the six trigonometric functions (CC 4.2.E.1, 4.3.B.2, 4.3.B.3, 4.3.B.4, 4.3.C.2) 9. Derive the concept of inverse function to trigonometric functions (CC 4.2.E.1, 4.3.B.2, 4.3.B.3, 4.3.B.4, 4.3.C.2) 10. Apply the concepts of trigonometry to solve real-life problems (CC 4.2.A.1, 4.2.E.1, 4.3.C.1, 4.3.D.2) Activities: 1. Construct a one radian angle 2. Foucault Pendulum problem 3. Latitude and Longitude of U.S. cities 4. Derive the area of a circular sector

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5. 6. 7. 8.

Garden design Mirrors: angle of incidence and angle of reflection Exploring the unit circle Group activity: use body language to illustrate the graphs of the six trigonometric functions with different periods, amplitudes, and phase shifts 9. Television coverage problem 10. Group activity: tuning fork displacement versus time 11. Ferris wheel motion 12. Modeling the motion of a pendulum F. Analytic Trigonometry Outline: 1. Fundamental Identities 2. Proving Trigonometric Identities 3. Sum and Difference identities 4. Multiple-Angle Identities 5. Areas of Triangles and Segments Objectives: Upon completion of this unit the student will be able to: 1. Use the fundamental identities to simplify trigonometric expressions and solve trigonometric equations (CC 4.2.A.4, 4.2.E.1, 4.3.B.1, 4.3.C.1, 4.3.D.1, 4.3.D.2) 2. Decide whether an equation is an identity and to confirm identities analytically (CC 4.2.A.4, 4.3.B.1, 4.3.D.1) 3. Apply the identities for the cosine, sine, and tangent of a difference or sum (CC 4.3.B.1, 4.3.D.1) 4. Apply the double-angle identities, power-reducing identities, and half-angle identities (CC 4.3.B.1, 4.3.D.1) 5. Determine the area of a triangle by three separate formulas: Hero's formula, two sides and an included angle, and two angles and a side (CC 4.2.A.1, 4.2.E.2) 6. Determine the area of a circular segment (CC 4.2.A.1, 4.2.A.3, 4.2.E.2) Activities: 1. Orbit of the moon 2. Cooperative group activity: have students find matching trigonometric expressions 3. Maximizing volume of a water trough 4. Tunnel design problem 5. Estimating acreage of an irregular plot of land G. Vectors, Parametric Equations, and Polar Equations Outline: 1. Vectors in the Plane *** 2. Dot Products of Vectors *** 3. Parametric Equations and Motion 4. Polar Coordinates

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5. Graphs of Polar Equations 6. De Moivre's Theorem and nth Roots *** Objectives: Upon completion of this unit the student will be able to: 1. Apply the arithmetic of vectors and use vectors to solve real-world problems (CC 4.2.A.1, 4.2.C.2, 4.3.C.1)*** 2. Calculate dot products and projections of vectors (CC 4.2.A.1, 4.2.C.2, 4.3.C.1) *** 3. Define parametric equations, graph curves parametrically, and solve application problems using parametric equations (CC 4.2.A.1, 4.2.C.2, 4.3.C.1) 4. Convert points and equations from polar to rectangular coordinates and vice versa (CC 4.2.C.1) 5. Graph polar equations and determine the maximum r-value and the symmetry of a graph (CC 4.3.B.3) 6. Represent complex numbers in trigonometric form and perform operations on them *** Suggested Activities: 1. Combining forces 2. Navigation with different current velocities 3. Calculate braking force with different slopes 4. Throwing a ball at a Ferris wheel 5. Using a graphing calculator to convert between different coordinate systems 6. Draw graphs of a family of polar curves in radian mode with the graphing calculator 7. Visualizing roots of unity H. Systems Outline: 1. Partial Fractions *** Objectives: Upon completion of this unit the student will be able to: 1. Decompose rational expressions into partial fractions (CC 4.1.B.1, 4.1.B.3, 4.3.D.1) *** Suggested Activities: 1. Using graphing calculator with augmented matrix to solve complex partial fraction problems I. Analytic Geometry in Two Dimensions Outline: 1. Conic sections and Parabolas 2. Ellipses 3. Hyperbolas

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4. Translations 5. Three Dimensional Cartesian Coordinate System *** Objectives: Upon completion of this unit the student will be able to: 1. Find the standard form equation, focus, and directrix of a parabola (CC 4.2.A.1, 4.2.B.1, 4.3.B.1, 4.2.B.2, 4.3.B.3, 4.3.B.4, 4.3.C.1, 4.3.C.2) 2. Find the standard form equation, vertices, and foci of an ellipse (CC 4.2.A.1, 4.2.A.3, 4.2.B.1, 4.3.B.1, 4.2.B.2, 4.3.B.3, 4.3.B.4, 4.3.C.1, 4.3.C.2) 3. Find the standard form equation, vertices, and foci of a hyperbola (CC 4.2.A.1, 4.2.B.1, 4.3.B.1, 4.2.B.2, 4.3.B.3, 4.3.B.4, 4.3.C.1, 4.3.C.2) 4. Graph conic sections that have a horizontal and vertical translation (CC 4.2.B.1, 4.3.B.3) 5. Draw three-dimensional figures (CC 4.2.A.1, 4.2.A.2,) Suggested Activities: 1. Construct a parabola 2. Construct an ellipse to understand eccentricity 3. Design a satellite dish 4. Dynamically construct a parabola using Geometer's Sketchpad 5. Degenerate hyperbolas 6. Construct a hyperbola using Geometer's Sketchpad 7. Long-Range Navigation 8. Use computer software to draw three dimensional figures J. Discrete Mathematics Outline: 1. Sequences and Series 2. Mathematical Induction Objectives: Upon the completion of this unit the student will be able to: 1. Find the sum of a finite series (CC 4.3.A.1) 2. Find the sum of a converging infinite geometric series (CC 4.3.A.1, 4.3.A.2, 4.3.B.1) 3. Use the principle of mathematical induction to prove mathematical generalizations (CC 4.3.A.3, 4.3.B.2, 4.3.C.3) Suggested Activities: 1. Summing with sigma notation using the graphing calculator 2. Gauss' Insight 3. Find the sum of a finite number of natural numbers 4. The Tower of Hanoi problem 5. Winning the Game 6. Fibonacci Sequence

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7. Connecting geometry and sequences by inscribing a regular polygon sequence in the unit circle K. An introduction to calculus: Limits Outline: 1. Limits Objectives: Upon the completion of this unit the student will be able to: 1. Use the properties of limits (CC 4.3.A.2) 2. Evaluate one and two sided limits (CC 4.3.A.2, 4.3.B.2) 3. Evaluate limits involving infinity (CC 4.3.A.2, 4.3.B.2) Suggested Activities: 1. "What's the limit?" - Graphing Calculator Exploration 2. Model Rabbit Population Growth

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