Pre-Calculus. Course of Study

Transcription

1 Pre-Calculus (Course #341) Course of Study Findlay City Schools 2006

2 Pre-Calculus TABLE OF CONTENTS 1. Findlay City Schools Mission Statement and Beliefs 2. Pre-Calculus Curriculum Map 3. Pre-Calculus Indicators and Course of Study 4. State 11 th and 12 th Indicators Pre-Calculus Course of Study Writing Team Barb Buttermore Barb Zalinski Sandra White

3 Mission Statement The mission of the Findlay City Schools, a community partnership committed to educational excellence, is to instill in each student the knowledge, skills and virtues necessary to be lifelong learners who recognize their unique talents and purpose and use them in pursuit of their dreams and for service to a global society. This is accomplished through a passion for knowledge, discovery and vision shared by students, families, staff and community. Beliefs Our beliefs form the ethical foundation of the Findlay City Schools. We believe. every person has worth every individual can learn family is the most important influence on the development of personal values. attitude is a choice and always affects performance motivation and effort are necessary to achieve full potential honesty and integrity are essential for building trust. people are responsible for the choices they make. performance is directly related to expectations. educated citizens are essential for the survival of the democratic process. personal fulfillment requires the nurturing of mind, body and spirit. every individual has a moral and ethical obligation to contribute to the well-being of society. education is a responsibility shared by students, family, staff and community. the entire community benefits by investing its time, resources and effort in educational excellence.

4 a consistent practice of shared morals and ethics is essential for our community to thrive.

5 PRE-CALCULUS CURRICULUM MAP Week Chapter Topic Indicator 1 1 Real numbers, exponents, 11.8 Radicals and Algebraic expressions 2 1 Algebraic expressions 11.8 rational expressions 3 1 rational expressions and 11.8 equations 4 1 equations and inequalities inequalities Coordinate geometry and 11.8 lines 7 2 functions functions functions 11.25, Polynomial and rational 11.7 functions 11 3 Polynomial and rational functions 12 3 Polynomial and rational 12.8 functions 13 4 Exponential and logarithmic function 14 4 Exponential and logarithmic function 15 4 Exponential and logarithmic function 16 4 Exponential and logarithmic function 17 5 Trigonometric functions of real numbers 18 Exams: 1 st semester 19 5 Trigonometric functions of real numbers 20 5 Trigonometric functions of real numbers 21 6 Trigonometric functions of angles 22 6 Trigonometric functions of angles 23 6 Trigonometric functions of angles 12.4

6 24 6 Trigonometric functions of 12.4 angles 25 7 Analytic trigonometry Analytic trigonometry Analytic trigonometry Analytic trigonometry Systems of equations Analytic geometry Analytic geometry Analytic geometry Analytic geometry Analytic geometry , 9 Polar coordinates and 11.15, systems of equations 36 Exams Textbook: Precalculus Fifth edition Mathematics for Calculus; James Stewart, Lothar Redlin, Saleen Watson, authors; Thompson-Brooks/Cole, publisher; ISBN

7 Grade Eleven Number, Number Sense and Operations Standard Number and Number Systems 1. Determine what properties hold for matrix addition and matrix multiplication; e.g., use examples to show addition is commutative and when multiplication is not commutative. (11.1) 2. Determine what properties hold for vector addition and multiplication, and for scalar multiplication. (11.2) 3. Represent complex numbers on the complex plane. (11.3) Meaning of Operations Computation and Estimation 4. Use matrices to represent given information in a problem situation. (11.4) 5. Model, using the coordinate plane, vector addition and scalar multiplication. (11.5) 6. Compute sums, differences and products of matrices using paper and pencil calculations for simple cases, and technology for more complicated cases. (11.6) 7. Compute sums, differences, products and quotients of complex numbers. (11.7) 8. Use fractional and negative exponents as optional ways of representing and finding solutions for problem situations; e.g., 27 2/3 = (27 1/3 ) 2 = 9. (11.8) 9. Use vector addition and scalar multiplication to solve problems. (11.9) Measurement Standard Measurement Units Use Measurement Techniques and Tools 1. Determine the number of significant digits in a measurement. (11.10) 2. Use radian and degree angle measures to solve problems and perform conversions as needed. (11.11) 3. Derive a formula for the surface area of a cone as a function of its slant height and the circumference of its base. (11.12) 4. Calculate distances, areas, surface areas and volumes of composite three-dimensional objects to a specified number of significant digits.

8 (11.13) 5. Solve real-world problems involving area, surface area, volume and density to a specified degree of precision. (11.14) Geometry and Spatial Sense Standard Spatial Relationships Transformations and Symmetry 1. Use polar coordinates to specify locations on a plane. (11.15) 2. Represent translations using vectors. (11.16) 3. Describe multiplication of a vector and a scalar graphically and algebraically, and apply to problem situations. (11.17) Visualization and Geometric Models 4. Use trigonometric relationships to determine lengths and angle measures; i.e., Law of Sines and Law of Cosines. (11.18) 5. Identify, sketch and classify the cross sections of three-dimensional objects. (11.19) Patterns, Functions and Algebra Standard Use Patterns, Relations and Functions 1. Identify and describe problem situations involving an iterative process that can be represented as a recursive function; e.g., compound interest. (11.20) 2. Translate a recursive function into a closed form expression or formula for the nth term to solve a problem situation involving an iterative process; e.g., find the value of an annuity after 7 years. (11.21) 3. Describe and compare the characteristics of the following families of functions: quadratics with complex roots, polynomials of any degree, logarithms, and rational functions; e.g., general shape, number of roots, domain and range, asymptotic behavior. (11.22) 4. Identify the maximum and minimum points of polynomial, rational and trigonometric functions graphically and with technology. (11.23)

9 Use Algebraic Representations 5. Identify families of functions with graphs that have rotation symmetry or reflection symmetry about the y-axis, x-axis or y = x. (11.24) 6. Represent the inverse of a function symbolically and graphically as a reflection about y = x. (11.25) 7. Model and solve problems with matrices and vectors. (11.26) 8. Solve equations involving radical expressions and complex roots. (11.27) Analyze Change 9. Solve 3 by 3 systems of linear equations by elimination and using technology, and interpret graphically what the solution means (a point, line, plane, or no solution). (11.28) 10. Describe the characteristics of the graphs of conic sections. (11.29) 11. Describe how a change in the value of a constant in an exponential, logarithmic or radical equation affects the graph of the equation. (11.30) Data Analysis and Probability Standard Data Collection 1. Design a statistical experiment, survey or study for a problem; collect data for the problem; and interpret the data with appropriate graphical displays, descriptive statistics, concepts of variability, causation, correlation and standard deviation. (11.31) 2. Describe the role of randomization in a well-designed study, especially as compared to a convenience sample, and the generalization of results from each. (11.32) Statistical Methods 3. Describe how a linear transformation of univariate data affects range, mean, mode and median. (11.33) 4. Create a scatterplot of bivariate data, identify trends, and find a function to model the data. (11.34) 5. Use technology to find the Least Squares Regression Line, the regression coefficient, and the correlation coefficient for bivariate data with a linear trend, and interpret each of these statistics in the context of the problem situation. (11.35) 6. Use technology to compute the standard deviation for a set of data,

10 and interpret standard deviation in relation to the context or problem situation. (11.36) 7. Describe the standard normal curve and its general properties, and answer questions dealing with data assumed to be normal. (11.37) 8. Analyze and interpret univariate and bivariate data to identify patterns, note trends, draw conclusions, and make predictions. (11.38) Probability 9. Evaluate validity of results of a study based on characteristics of the study design, including sampling method, summary statistics and data analysis techniques. (11.39) 10. Understand and use the concept of random variable, and compute and interpret the expected value for a random variable in simple cases. (11.40) 11. Examine statements and decisions involving risk; e.g., insurance rates and medical decisions. (11.41)

11 Grade Twelve Number, Number Sense and Operations Standard Number and Number Systems Computation and Estimation 1. Determine what properties (closure, identity, inverse, commutative and associative) hold for operations with complex numbers. (12.1) 2. Apply combinations as a method to create coefficients for the Binomial Theorem, and make connections to everyday and workplace problem situations. (12.2) Measurement Standard Use Measurement Techniques and Tools 1. Solve problems involving derived measurements; e.g., acceleration and pressure. (12.3) 2. Use radian measures in the solution of problems involving angular velocity and acceleration. (12.4) 3. Apply informal concepts of successive approximation, upper and lower bounds, and limits in measurement situations; e.g., measurement of some quantities, such as volume of a cone, can be determined by sequences of increasingly accurate approximations. (12.5) Geometry and Spatial Sense Standard Transformations and Symmetry 1. Use matrices to represent translations, reflections, rotations, dilations and their compositions. (12.6) 2. Derive and apply the basic trigonometric identities; i.e., angle addition, angle subtraction and double angle. (12.7) Visualization and Geometric Models 3. Relate graphical and algebraic representations of lines, simple curves and conic sections. (12.8) 4. Recognize and compare specific shapes and properties in multiple geometries; e.g., plane, spherical and hyperbolic. (12.9)

12 Patterns, Functions and Algebra Standard Use Patterns, Relations and Functions 1. Analyze the behavior of arithmetic and geometric sequences and series as the number of terms increases. ( Translate between the numeric and symbolic form of a sequence or series. (12.11) 3. Describe and compare the characteristics of transcendental and periodic functions; e.g., general shape, number of roots, domain and range, asymptotic behavior, extrema, local and global behavior. (12.12) 4. Represent the inverse of a transcendental function symbolically. (12.13) Use Algebraic Representations 5. Set up and solve systems of equations using matrices and graphs, with and without technology. (12.14) 6. Make arguments about mathematical properties using mathematical induction. (12.15) 7. Make mathematical arguments using the concepts of limit. (12.16) 8. Compare estimates of the area under a curve over a bounded interval by partitioning the region with rectangles; e.g., make successive estimates using progressively smaller rectangles. (12.17) 9. Translate freely between polar and Cartesian coordinate systems. (12.18) Analyze Change 10. Use the concept of limit to find instantaneous rate of change for a point on a graph as the slope of a tangent at a point. (12.19) Data Analysis and Probability Standard Data Collection Statistical Methods 1. Identify and use various sampling methods (voluntary response, convenience sample, random sample, stratified random sample, census) in a study. (12.20) 2. Transform bivariate data so it can be modeled by a function; e.g., use logarithms to allow nonlinear relationship to be modeled by linear function. (12.21) 3. Describe the shape and find all summary statistics for a set of univariate data, and describe how a linear transformation affects shape, center and spread. (12.22)

13 4. Apply the concept of a random variable to generate and interpret probability distributions, including binomial, normal and uniform. (12.23) 5. Use sampling distributions as the basis for informal inference. (12.24) Probability 6. Use theoretical or experimental probability, including simulations, to determine probabilities in real-world problem situations involving uncertainty, such as mutually exclusive events, complementary events, and conditional probability. (12.25)

14 BENCHMARK: (C) Apply factorials and exponents, including fractional exponents, to solve practical problems. STANDARD: Number, Number Sense and Operations TOPIC/UNIT: Chapter 1 Time Line: 1 week Indicator (#11.8) Use fractional and negative exponents as optional ways of representing and finding solutions for problem situations; e.g., 27 2/3 = (27 1/3 ) 2 = 9. KNOW How to apply fractional exponents DO Be able to simplify an expression with fractional exponents How to apply negative exponents How to apply the meaning of the exponent zero The laws of exponents Be able to simplify an expression with negative exponents Be able to apply negative exponents in the use of scientific notation Be able to simplify an expression with an exponent of zero Be able to apply the laws of exponents

15 PRE-ASSESSMENT: Board work for section 1.2 of text ASSESSMENT: Teacher made quiz GRAPHIC ORGANIZER & OR TECHNOLOGY: Use calculator to evaluate expressions with powers TESTING SKILL(S) & OR SAMPLE OGT TYPE QUESTIONS: Chapter 1 Review in text BEST PRACTICES: Chalkboard Interactive Teacher Presentation Pencil and paper practice RESOURCES: Textbook Teacher made worksheets TESTING VOCABULARY: Exponents Scientific notation HISTORICAL/MODERN LINK: P. 20

16 BENCHMARK: (A) Analyze functions by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior. STANDARD: Patterns, Functions and Algebra TOPIC/UNIT: Functions/Chapter 2 Time Line: Indicator (#11.23) Identify the maximum and minimum points of polynomial, rational and trigonometric functions graphically and with technology. (#11.24) Identify families of functions with graphs that have rotation symmetry or reflection symmetry about the y-axis or y = x. (#11.25) Represent the inverse of a function symbolically and graphically as a reflection about y = x. (#11.30) Describe how a change in the value of a constant in an exponential, logarithmic or radical equation affects the graph of the equation.

17 KNOW The meaning of the reflecting graphs How certain transformations of a function affects its graph How to identify the maximum and minimum values of a polynomial DO Be able to identify graphs which have been reflected about the y-axis or the x-axis and identify the axis of reflection Be able to graph the inverse of a function Be able to reflect a graph about the y- axis, x-axis, or the line y=x Be able to determine if a graph is symmetric with respect the y-axis, x-axis or y=x Be able to graph the inverse of a function Be able to complete a graph if it is symmetric with respect to the y-axis or the x-axis Express the function in standard form Sketch the graph of the function Locate the maximum and minimum values How a change in the value of a constant in an exponential equation affects the graph Graph the function f(x), then graph (c)f(x) for various values of c on the same set of axes Compare the graphs of f(x) and (c)f(x) Make a conclusion regarding the effect of c on f(x)

18 PRE-ASSESSMENT: ASSESSMENT: Homework Teacher made quiz GRAPHIC ORGANIZER & OR TECHNOLOGY: Graphing calculator TESTING SKILL(S) & OR SAMPLE OGT TYPE QUESTIONS: Teacher made test BEST PRACTICES: Cooperative Learning Work practice problems RESOURCES: Textbook Graphing Calculator Teacher made handouts Teacher made worksheets TESTING VOCABULARY: On quiz: Function Symmetry Reflection Rotation HISTORICAL/MODERN LINK: P. 165 P. 178 P. 188

19 BENCHMARK: STANDARD: (E) Represent and compute with complex numbers. Number, Number Sense and Operations; Measurement; Geometry and Spatial Sense TOPIC/UNIT: Chapter 3 Time Line: Indicator (#11.7) Compute sums, differences, products and quotients of complex numbers. (#11.22) Describe and compare the characteristics of the following families of functions: quadratics with complex roots, polynomials of any degree, logarithms, and rational functions; e.g., general shape, number of roots, domain and range, asymptotic behavior. (#12.8) Relate graphical and algebraic representations of lines, simple curves ad conic sections. KNOW How to compute sums, differences, products, and quotients of complex numbers DO Add, subtract, multiply, and divide complex numbers expressing the answer in the form a + bi How to describe and compare the characteristics of quadratics with complex roots How to describe and compare the characteristics of polynomials of any degree How to describe and compare the characteristics of rational functions Find the number of roots of a polynomial of any degree Find the domain and range of a given function Describe the general shape of a given function Find the asymptotes of a rational function The graphical representation of a polynomial Sketch the graphical representation of a polynomial

20 PRE-ASSESSMENT: ASSESSMENT: Homework Teacher made quiz GRAPHIC ORGANIZER & OR TECHNOLOGY: Notes from the following sections: Section 3.1 Section 3.5 Ex. 6 Section 3.6 BEST PRACTICES: Examples presented on chalkboard, overhead, and graphing calculator TESTING SKILL(S) & OR SAMPLE OGT TYPE QUESTIONS: Teacher made test RESOURCES: Textbook Teacher made handouts Discovery Project P. 283 TESTING VOCABULARY: HISTORICAL/MODERN LINK: Historical: P. 273, P. 288, 294, 296 Mathematics in the Modern World: P. 252,256,308

21 BENCHMARK: STANDARD: Patterns, Functions and Algebra TOPIC/UNIT: Chapter 4 Time Line: Indicator (#11.22) Describe and compare the characteristics of the following families of functions: quadratics with complex roots, polynomials of any degree, logarithms, and rational functions; e.g., general shape, number of roots, domain and range, asymptotic behavior. (#11.30) Describe how a change in the value of a constant in an exponential, logarithmic or radical equation affects the graph of the equation. KNOW How to describe and compare the characteristics of logarithms The properties of logarithms How a change in the value of a constant in a logarithmic equation affects the graph of the equation DO Graph logarithmic functions Apply the properties of logarithms Graph f(x)=logx then graph f(x) =(a)log (x+b)

22 PRE-ASSESSMENT: ASSESSMENT: Homework: Textbook P. 351 Teacher made quizzes GRAPHIC ORGANIZER & OR TECHNOLOGY: Chalkboard examples Graphing Calculator TESTING SKILL(S) & OR SAMPLE OGT TYPE QUESTIONS: Teacher made test BEST PRACTICES: Practice Problems and applications Interactive Teacher Presentation RESOURCES: Chapter 4 in Textbook TESTING VOCABULARY: HISTORICAL/MODERN LINK: Historical: Textbook P. 346, Modern Link: Textbook P. 374

23 BENCHMARK: (A) Analyze functions by investigating rates of change, intercepts, zeros, asymptotes, and local global behavior STANDARD: Patterns, Functions and Algebra TOPIC/UNIT: Chapter 5 Time Line: Indicator (#11.23) Identify the maximum and minimum points of polynomial, rational and trigonometric functions graphically and with technology. (#12.12) Describe and compare the characteristics of transcendental and periodic functions; e.g., general shape, number of roots, domain and range, asymptotic behavior, extrema, local and global behavior. KNOW How to describe the characteristics of periodic functions DO Find the amplitude and period of periodic functions Find the amplitude, period, and displacement of trigonometric equations How to compare the characteristics of periodic functions Graph y=sinx and y=(a)sin(bx+c) and compare Graph y=cosx and y=(a)cos(bx+c) and compare Graph y=tanx and y=(a)tan(bx+c) and compare How to find the maximum and minimum points of a trigonometric function Graph, on paper, sine, cosine, and tangent curves and identify the maximum and minimum points Graph, with a graphing calculator, sine, cosine, and tangent curves and identify the maximum and minimum points

24 PRE-ASSESSMENT: ASSESSMENT: Textbook Exercises from Chapter 5 Teacher Made Test GRAPHIC ORGANIZER & OR TECHNOLOGY: Graphing Calculator TESTING SKILL(S) & OR SAMPLE OGT TYPE QUESTIONS: Teacher made test BEST PRACTICES: Graphing Calculator Interactive Teacher Presentation RESOURCES: Chapter 5 in Textbook Graphing Calculator TESTING VOCABULARY: Quiz HISTORICAL/MODERN LINK: Modern Link: P. 436

25 BENCHMARK: (B) Apply various measurement scales to describe phenomena and solve problems. STANDARD: Measurement; Geometry and Spatial Sense TOPIC/UNIT: Chapter 6 Time Line: Indicator (#11.11) Use radian and degree angle measures to solve problems and perform conversions as needed. (#11.18) Use trigonometric relationships to determine lengths and angle measures; i.e., Law of Sines and Law of Cosines. (#12.4) Use radian measures in the solution of problems involving angular velocity and acceleration. KNOW How to convert between radians and degrees DO Convert degrees to radians Convert radians to degrees How to solve problems using degrees or radians Solve right triangles using degrees Solve problems to find arc length, area of a sector, linear velocity, and angular velocity Law of Sines Law of Cosines Use the Law of Sines to find the angles and sides of a triangle Use the Law of Cosines to find the angles and sides of a triangle

26 PRE-ASSESSMENT: ASSESSMENT: Homework Board work Teacher made quiz GRAPHIC ORGANIZER & OR TECHNOLOGY: TESTING SKILL(S) & OR SAMPLE OGT TYPE QUESTIONS: Teacher made test BEST PRACTICES: Examples on CD-ROM Interactive Teacher Presentation RESOURCES: Chapter 6 in textbook TESTING VOCABULARY: Quiz HISTORICAL/MODERN LINK: Historical: Textbook P. 480, 482

27 BENCHMARK: (A) Use trigonometric relationships to verify and determine solutions in problem situations. STANDARD: Geometry and Spatial Sense TOPIC/UNIT: Chapter 7 Time Line: Indicator (#12.7) Derive and apply the basic trigonometric identities; i.e., angle addition, angle subtraction and double angle. KNOW How to derive the fundamental trigonometric identities How to apply the fundamental trigonometric identities How to derive and apply the angle Addition and Subtraction Formulas How to derive and apply the Double- Angle Formulas DO State the fundamental trigonometric identities Derive the fundamental trigonometric identities Apply the fundamental trigonometric identities Derive the identities for trigonometric functions of sums and differences Apply the identities for trigonometric functions of sums and differences Derive and apply the Double-Angle Formulas

28 PRE-ASSESSMENT: ASSESSMENT: Homework Teacher made quizzes GRAPHIC ORGANIZER & OR TECHNOLOGY: Examples on CD-ROM TESTING SKILL(S) & OR SAMPLE OGT TYPE QUESTIONS: Teacher made test BEST PRACTICES: Interactive Teacher Presentation RESOURCES: Chapter 7 in textbook TESTING VOCABULARY: HISTORICAL/MODERN LINK: Historical: P. 532,536 Modern Link: P 560 Where to Sit at the Movies, P 562 Weather Prediction

29 BENCHMARK: (A) Use trigonometric relationships to verify and determine solutions in problem situations. STANDARD: Geometry and Spatial Sense TOPIC/UNIT: Chapter 8 Time Line: Indicator (#11.15) Use polar coordinates to specify locations on a plane. KNOW How to use polar coordinates to specify locations on a plane DO Use polar coordinates to specify locations on a plane Use different polar coordinates for the same point

30 PRE-ASSESSMENT: ASSESSMENT: Homework Exercises GRAPHIC ORGANIZER & OR TECHNOLOGY: Graphing Calculator Polar Coordinate Graph Paper TESTING SKILL(S) & OR SAMPLE OGT TYPE QUESTIONS: Teacher made test BEST PRACTICES: Use Polar graphing paper RESOURCES: Chapter 8 in Textbook TESTING VOCABULARY: HISTORICAL/MODERN LINK: Modern Link: P 630 Focus on Modeling

31 BENCHMARK: Standard: Patterns, Functions and Algebra TOPIC/UNIT: Chapter 9 Time Line: Indicator (#12.14) Set up and solve systems of equations using matrices and graphs, with and without technology. KNOW DO How to set up and solve systems of equations using graphs with and without technology Set up and solve systems of equations using graphs with and without technology

32 PRE-ASSESSMENT: ASSESSMENT: Textbook exercises GRAPHIC ORGANIZER & OR TECHNOLOGY: Calculator CD ROM TESTING SKILL(S) & OR SAMPLE OGT TYPE QUESTIONS: Teacher made test BEST PRACTICES: Chalk board examples CD ROM RESOURCES: textbook TESTING VOCABULARY: HISTORICAL/MODERN LINK: Modern link: p656 - global positioning system p682 fair voting methods

33 BENCHMARK: STANDARD: Patterns, Functions and Algebra TOPIC/UNIT: Chapter 10 Time Line: Indicator (#11.29) Describe the characteristics of the graphs of conic sections. (#12.8) Relate graphical and algebraic representations of lines, simple curves and conic sections. KNOW Know the characteristics of the graphs of conic sections DO Use the graph of a conic section to find its characteristics Know how to relate graphical and algebraic representations of conic sections Use the graph of a(n) circle, ellipse, parabola, or hyperbola to write its equation Use the equation of a(n) circle, ellipse, parabola, or hyperbola to sketch its graph

34 PRE-ASSESSMENT: ASSESSMENT: Textbook Exercises GRAPHIC ORGANIZER & OR TECHNOLOGY: Examples on CD -ROM TESTING SKILL(S) & OR SAMPLE OGT TYPE QUESTIONS: Teacher made test BEST PRACTICES: Use string, cardboard, and tacks to discover the graph of an ellipse RESOURCES: Chapter 10 in textbook Use a flashlight to show conic sections on the blackboard TESTING VOCABULARY: HISTORICAL/MODERN LINK: Historical: P 748, 758, 766, 780, 802 Modern Link: P 771