COURSE NAME: HONORS PRE-CALCULUS GRADE: 10,11,12

Transcription

1 COURSE NAME: HONORS PRE-CALCULUS GRADE: 10,11,12 Prerequisite: HONORS ALGEBRA 2 Credits: 5 ABSTRACT This full year, five credit course is to prepare students for Advanced Placement (AP) Calculus at the high school level or a college Calculus course appropriate for engineering, science or mathematics majors. This course is designed to serve as a comprehensive and in depth review of skills previously acquired and to both refine and extend those abilities. The study of trigonometry, analytic geometry and elementary functions constitute most of this curriculum. Attention is also given to mathematical rigor, career opportunities, and the attitude with which mathematics must be approached in order to experience success at higher levels. There is an intense focus on word problems and real world application of higher mathematics. Pre-Calculus Honors is of great assistance in preparing students for the Scholastic Aptitude Test (SAT). Extensive use is made of graphing calculators and computer laboratory time is assigned. Various projects must be completed and significant time and independent effort are expected outside the classroom. This course is a prerequisite for Advanced Placement (AP) Calculus AB. The prerequisite for the course is Algebra 2

2 Honors or Algebra 2 with teacher recommendation. In preparation for the state assessments, students will be given formative and summative assessments throughout the course. UNIT: STAGE 1: DESIRED RESULTS ESTABLISHED GOALS: (NJ CCCS and/or CCS) CCS F-IF.1-9 F-LE.1-5 F-BF.1-5 CCS F-TF.1-9 CCS N-RN.1-3 N-Q.1-3 N-CN.1-9 ENDURING UNDERSTANDINGS: (Students will Understand that...) Technology st Century Life and Careers A B C D.1-4 They can use algebraic and graphing techniques to explain and analyze the general properties and behaviors of Technology st Century Life and Careers A B C D.1-4 Identities are used to evaluate, simplify, and solve trigonometric expressions and equations. Technology st Century Life and Careers A B C D.1-4 There are real-world uses for different types of numbers. Understanding proportions is important 2

3 functions or relations. They can use functions to model real-world phenomena and solve problems of varying quantities. There are appropriate methods to solve equations and inequalities. They can judge the meaning, utility, and reasonableness of the results of symbol manipulations, including technological. It is important to recognize that data can be displayed in different formats. are used in real-life situations, such as cellphone bills. Equations and The law of cosines and the law of sines can be used to find missing measures. The characteristics of trigonometric and circular functions and their representations are useful in solving realworld problems. Connections among the six trigonometric and circular functions are a result of their properties. for life skills such as reading maps. Understanding the laws of exponents will help them simplify expressions involving numbers raised to powers. There are limitations in using estimation. 3

4 inequalities are important tools in many different fields. They must use inductive reasoning to form generalizations ESSENTIAL QUESTIONS: (What provocative questions will foster inquiry, understanding, and transfer of learning?) How can algebraic equations be used to solve real-life problems? How do exponential functions model realworld problems and their solutions? How do logarithmic functions model realworld problems and their solutions? How do you graph curves parametrically (by hand and with appropriate technology) How do you successfully eliminate How do you graph transformations and combinations of transformations for all basic trigonometric functions? Can you solve problems using the fact that trigonometric ratios (sine, cosine, and tangent) stay constant in similar triangles? How do you use the definitions of the six trigonometric ratios as ratios of sides in a right triangle to solve What methods can be used to recognize and extend patterns? How are the different sets of numbers similar and different? How does changing the order of operations affect the results of mathematical expression? How can algebraic equations be used to solve real-life problems? When is estimation an appropriate method? What are the rules for using estimation? 4

5 parameters by rewriting parametric equations as a single equation? What is the function of this graph? problems about lengths of sides and measures of angles? Can you successfully match a trigonometric equation with its graph? Do you know that the six trigonometric functions can be extended to periodic functions on the real number line? Can you convert from radians to degrees and from degrees to radians? How do you determine the difference made by choice of units for angle measurement when graphing a trigonometric function? How do you find values of inverse trigonometric functions, applying appropriate domain and Are you able to correctly use summation notation as well as expand and collect expressions in both finite and infinite settings? Can you successfully demonstrate an understanding of sequences by representing them recursively and explicitly? Can you use Sigma notation to represent a series? How do you find the sum of a given geometric series (both infinite and finite)? How do you find the sum of a finite arithmetic series? 5

6 6 range restrictions? Can you graph the inverse trigonometric functions and identify their key characteristics? Can you graph the six trigonometric functions and identify characteristics such as period, amplitude, phase shift, and asymptotes? How do you determine the appropriate domains for each of the inverse trigonometric functions? Do you know how to use the following trigonometric identities in verifying other identities: Pythagorean, Reciprocal, Quotient, Sum/Difference, Double Angle? Do you know how to use

7 the following trigonometric identities in solving trigonometric equations: Pythagorean, Reciprocal, Quotient, Sum/Difference, Double Angle? Can you apply the Pythagorean and Reciprocal Identities to verify identities and solve equations? STAGE 2: ASSESSMENT EVIDENCE PERFORMANCE TASKS: (Through what authentic performance tasks will students demonstrate the desired understandings?) (By what criteria will performances of understanding be judged?) Independent assignments Written responses Real-world problems Non-routine problems Collaborative problemsolving Independent assignments Written responses Real-world problems Non-routine problems Collaborative problemsolving Independent assignments Written responses Real-world problems Non-routine problems Collaborative problemsolving 7

8 OTHER EVIDENCE: (Through what other evidence (e.g. quizzes, tests, academic prompts, observations, homework, journals) will students demonstrate achievement of the desired results?) (How will students self-assess their learning?) RESOURCES: Write own problems and assessments Pre/Post-assessments Practice HSPA tests Quizzes Unit tests Multiple choice problems Teacher observations Math journals Think-Alouds Homework Discussions Open-ended problems Pre-calculus textbook (Contemporary Pre-calculus by Hungerford & Shaw Thomson Brooks/Cole, 2009) Online tutorials Educational websites Educational software Research in the Media Center Overhead TI-83 8 Write own problems and assessments Pre/Post-assessments Practice HSPA tests Quizzes Unit tests Multiple choice problems Teacher observations Math journals Think-Alouds Homework Discussions Open-ended problems Pre-calculus textbook (Contemporary Pre-calculus by Hungerford & Shaw Thomson Brooks/Cole, 2009) Online tutorials Educational websites Educational software Research in the Media Center Overhead TI-83 Write own problems and assessments Pre/Post-assessments Practice HSPA tests Quizzes Unit tests Multiple choice problems Teacher observations Math journals Think-Alouds Homework Discussions Open-ended problems Pre-calculus textbook (Contemporary Pre-calculus by Hungerford & Shaw Thomson Brooks/Cole, 2009) Online tutorials Educational websites Educational software Research in the Media Center Overhead TI-83 calculator

9 calculator for use by teacher Overhead projector LCD projector Computer/Computer lab availability Graphing Calculator (TI83 or equivalent) HSPA resource materials com/student/testing/ap/cal culus_ab/samp.html Computer Applications a. Microsoft b. PowerPoint c. Microsoft Excel d. Microsoft Word calculator for use by teacher Overhead projector LCD projector Computer/Computer lab availability Graphing Calculator (TI83 or equivalent) HSPA resource materials com/student/testing/ap/cal culus_ab/samp.html Computer Applications a. Microsoft b. PowerPoint c. Microsoft Excel d. Microsoft Word for use by teacher Overhead projector LCD projector Computer/Computer lab availability Graphing Calculator (TI83 or equivalent) HSPA resource materials om/student/testing/ap/calcu lus_ab/samp.html Computer Applications a. Microsoft b. PowerPoint c. Microsoft Excel d. Microsoft Word STAGE 3: LEARNING PLAN 9

10 SKILLS AND TOPICS: (What specific activities will students do and what skills will students know as a result of the unit?) Determine whether a relation represents a function and find the value of a function. Represent functions numerically, algebraically and graphically. Determine the sum, difference, product and quotient of functions. Determine the domain and range of functions. Identify the graph of a function and obtain information from the graph of a function. Identify the graph of a function and obtain information from the graph of a function. Determine continuity, increasing-decreasing Convert between radians and degrees. Find arc length, angular and linear speed, and area of a sector of a circle. Define and evaluate the six trigonometric functions in terms of the lengths of the sides of a right triangle, the rotation of a ray in standard position, and a point on a unit circle. Find exact values of trigonometric functions and use the calculator to approximate values. Determine the range, domain, and period of trigonometric functions. Graph the six trigonometric functions, Distinguish between a rational and irrational answer Express solutions as approximations Express solutions as numerical expressions Raise number to powers. Assess the amount of error resulting from estimation. Convert to and from scientific notation, multiply numbers in scientific notation, and solve problems in scientific notation. Understand types of numbers, our numeration system, and the ways they are used and applied in real-world situations. Identify rational and irrational numbers. 10

11 behavior, local minima and maxima, symmetry, asymptotes and end behavior of a function both graphically and algebraically. Find the average rate of change of a function. Recognize the characteristics of the functions Graph piecewise functions. Graph functions using vertical and horizontal shifts, compressions and stretches, and reflections about the x and y-axis. Build and analyze functions. Recognize and graph linear and quadratic functions. and transformations of these graphs. Apply the concepts of trigonometry to solve real world problems. Find an exact value of an inverse sine, cosine or tangent function. Find an approximate value of an inverse sine, cosine or tangent function. Find the exact value of composite functions. Find the inverse function of a trigonometric function and solve equations involving inverse functions. Order rational and irrational numbers on a number line. Apply ratios, proportions, and percents to a variety of situations. 11

12 Draw and interpret Scatter Diagrams and find the Line of Best Fit. Graph quadratic functions using transformations, symmetry and intercepts. Build quadratic models from verbal descriptions and data. Know the definitions of the inverse secant, cosecant and cotangent functions and use the calculator to evaluate sec -1 x, csc -1 x, cot -1 x. Use algebra to simplify trigonometric expressions. Identify and graph polynomial functions, predict their end behavior and find their real zeros algebraically and graphically. Identify and graph power functions of the form Use Reciprocal Trigonometric Identities, Quotient Identities, Pythagorean Identities, Co-Function Identities and Odd-Even Identities to simplify trigonometric expressions and solve trigonometric equations. f(x) = ax n. Establish identities. Find the domain and Apply the identities for 12

13 asymptotes of rational functions, and analyze and construct graphs of rational functions. Apply the Remainder Theorem, Factor Theorem, theorems for bounds on zeros, and the Intermediate Value Theorem. Solve polynomial equations. the cosine, sine and tangent of a difference or sum. Apply the Sum and Difference Formulas, Double-angle Formulas, and Half-angle Formulas. Use trigonometric concepts to solve equations and real world problems. Determine the complex zeros of polynomial equations, and determine the polynomial with the specified zeros. Find the value of trigonometric functions of acute angles using right triangles. Apply polynomial, power and rational function to real world Solve right triangles. Solve applied problems. 13

14 problems. Form composite functions and find their domain. Determine if a function is one-to-one, and find the inverse of a function Evaluate exponential expressions. Apply the Law of Sines and Law of Cosines to solve triangles. Find the area of any triangle. Analyze and solve simple harmonic motion problems. Identify and graph exponential and logistic functions. Use exponential growth, decay and regression to model real life problems. Convert equations between logarithmic form and exponential 14

15 form. Evaluate common and natural logarithms. Graph common and natural logarithmic functions. Apply the properties of logarithms to evaluate expressions Solve exponential and logarithmic equations algebraically and graphically. Use exponential and logarithmic equations to solve real life problems CROSS-CURRICULAR / Individual Instruction Individual Instruction Individual Instruction 15

16 DIFFERENTIATION: (What cross-curricular (e.g. writing, literacy, math, science, history, 21 st century life and careers, technology) learning activities are included in this unit that will help achieve the desired results?) (What type of differentiated instruction will be used for ELL, SP.ED. and G&T students?) Plans Current-event problems Manipulatives Tiered Lessons Learning style adaptation R.A.F.T Project Based learning Compacting Multimedia presentations Open-ended responses Conclusions and analysis of exploratory activities Plans Current-event problems Manipulatives Tiered Lessons Learning style adaptation R.A.F.T Project Based learning Compacting Multimedia presentations Open-ended responses Conclusions and analysis of exploratory activities Plans Current-event problems Manipulatives Tiered Lessons Learning style adaptation R.A.F.T Project Based learning Compacting Multimedia presentations Open-ended responses Conclusions and analysis of exploratory activities UNIT: UNIT: 4 Vectors and Matrices (May, 30) UNIT: 5 Algebra Unit: 6 An Introduction to Calculus (June, 20) STAGE 1: DESIRED RESULTS ESTABLISHED GOALS: CCS CCS CCS 16

17 UNIT: 4 Vectors and Matrices (May, 30) UNIT: 5 Algebra Unit: 6 An Introduction to Calculus (June, 20) (NJ CCCS and/or CCS) N-VM.1-12 A-REI.5-12 A-SSE.1-4 A-APR.1-7 A-CED.1-4 A-REI.1-4 A-REI.2 A-CED.4 A-APR.5 Technology st Century Life and Careers A B C D.1-4 Technology st Century Life and Careers A B C D.1-4 Technology st Century Life and Careers A B.1-2,4, C D B C D.1 ENDURING UNDERSTANDINGS: (Students will Understand that...) Vectors are added and multiplied and this helps with mathematical calculations for engineering and physics. Linear and nonlinear Performs operations on expressions containing complex numbers, rational exponents and complex fractions Recognizes classes of The concept of limits can be applied to sequences and to the asymptotic behavior of functions. The limit of a function is the value approached by 17

18 UNIT: 4 Vectors and Matrices (May, 30) UNIT: 5 Algebra Unit: 6 An Introduction to Calculus (June, 20) systems of equations and inequalities symbolically and graphically and performs operations on matrices If the determinant of a matrix is zero, then the inverse of the matrix does not exist. Large systems of linear equations can be solved using technology functions including linear, polynomial, absolute value, step, rational, and exponential from multiple representations such as graphical, tabular, and symbolic and converts between these representations Performs operations on and finds solutions for various types of functions including linear, polynomial, absolute value, and rational Solves linear and nonlinear systems of equations and inequalities symbolically and graphically Uses the language of mathematics to express f(x) as x approaches a given value or infinity. Basic differentiation rules can be used to find the derivative of a function. The instantaneous rate of change represents derivative 18

19 UNIT: 4 Vectors and Matrices (May, 30) UNIT: 5 Algebra Unit: 6 An Introduction to Calculus (June, 20) ESSENTIAL QUESTIONS: (What provocative questions will foster inquiry, understanding, and transfer of learning?) Can you multiply a vector by a scalar both algebraically and graphically? Can you add vectors both algebraically and graphically? Can you calculate magnitude and direction of a vector? Can you calculate and interpret the dot product of two vectors? Do you understand that vectors are determined by the coordinates of their initial and terminal points, or by their components? Can you use vectors to ideas precisely through reasoning, representations, and communication What is the next number in the pattern? How can we use patterns to solve problems? What is a limit? What is the formula for the nth term? What is the difference between a function and a relation? What is the domain and range of a function? What is the maximum? Minimum? What does the solution of a system mean? What is the equation of this graph? What will the graph of this equation look like? How do you construct the difference quotient for a given function and simplify the resulting expression? How do you determine whether a given arithmetic or geometric series converges or diverges? How do you approximate the area under a curve geometrically by constructing a finite number of rectangles and calculating the total area in those rectangles?, How do you compare two different approximations of area under a curve by using a different number 19

20 UNIT: 4 Vectors and Matrices (May, 30) UNIT: 5 Algebra Unit: 6 An Introduction to Calculus (June, 20) model velocity and direction to solve problems? Why can t we add x 3 and X 2? What is (a + b)(a b)? What is it called? What is the solution of the equation? System? of rectangles? How do you use the lengths of curves and areas of curved regions can be defined using the informal notion of limit? STAGE 2: ASSESSMENT EVIDENCE PERFORMANCE TASKS: (Through what authentic performance tasks will students demonstrate the desired understandings?) (By what criteria will performances of understanding be judged?) OTHER EVIDENCE: (Through what other evidence (e.g. quizzes, tests, academic prompts, observations, homework, journals) will students demonstrate achievement of the desired results?) Independent assignments Written responses Real-world problems Non-routine problems Collaborative problemsolving Write own problems and assessments Pre/Post-assessments Practice HSPA tests Quizzes Unit tests Multiple choice problems Independent assignments Written responses Real-world problems Non-routine problems Collaborative problemsolving Write own problems and assessments Pre/Post-assessments Practice HSPA tests Quizzes Unit tests Multiple choice problems Independent assignments Written responses Real-world problems Non-routine problems Collaborative problemsolving Write own problems and assessments Pre/Post-assessments Practice HSPA tests Quizzes Unit tests Multiple choice problems Teacher observations 20

21 UNIT: 4 Vectors and Matrices (May, 30) UNIT: 5 Algebra Unit: 6 An Introduction to Calculus (June, 20) (How will students self-assess their learning?) Teacher observations Math journals Think-Alouds Homework Discussions Open-ended problems RESOURCES: Pre-calculus textbook (Contemporary Precalculus by Hungerford & Shaw Thomson Brooks/Cole, 2009) Online tutorials Educational websites Educational software Research in the Media Center Overhead TI-83 calculator for use by teacher Overhead projector LCD projector Computer/Computer lab availability Graphing Calculator 21 Teacher observations Math journals Think-Alouds Homework Discussions Open-ended problems Pre-calculus textbook (Contemporary Precalculus by Hungerford & Shaw Thomson Brooks/Cole, 2009) Online tutorials Educational websites Educational software Research in the Media Center Overhead TI-83 calculator for use by teacher Overhead projector LCD projector Computer/Computer lab availability Graphing Calculator Math journals Think-Alouds Homework Discussions Open-ended problems Pre-calculus textbook (Contemporary Precalculus by Hungerford & Shaw Thomson Brooks/Cole, 2009) Online tutorials Educational websites Educational software Research in the Media Center Overhead TI-83 calculator for use by teacher Overhead projector LCD projector Computer/Computer lab availability Graphing Calculator (TI83 or equivalent)

22 UNIT: 4 Vectors and Matrices (May, 30) UNIT: 5 Algebra Unit: 6 An Introduction to Calculus (June, 20) (TI83 or equivalent) HSPA resource materials com/student/testing/ap/cal culus_ab/samp.html Computer Applications a. Microsoft b. PowerPoint c. Microsoft Excel d. Microsoft Word (TI83 or equivalent) HSPA resource materials com/student/testing/ap/cal culus_ab/samp.html Computer Applications a. Microsoft b. PowerPoint c. Microsoft Excel d. Microsoft Word HSPA resource materials om/student/testing/ap/calcu lus_ab/samp.html Computer Applications a. Microsoft b. PowerPoint c. Microsoft Excel d. Microsoft Word STAGE 3: LEARNING PLAN SKILLS AND TOPICS: (What specific activities will students do and what skills will students know as a result of the unit?) Convert points and equations from polar to rectangular form and vice versa. Transform equations from polar to rectangular form. Graph polar equations and determine the maximum r-value and the symmetry of the Use iterative and recursive patterns and processes to model a variety of practical situations and solve problems. Show the relationship between equations and Use various types of functions to represent Solve word problems Calculate instantaneous velocities and derivatives using limits. Use the properties of limits to evaluate onesided limits, two sided limits and limits involving infinity. 22

23 UNIT: 4 Vectors and Matrices (May, 30) UNIT: 5 Algebra Unit: 6 An Introduction to Calculus (June, 20) CROSS-CURRICULAR: (What cross-curricular (e.g. writing, literacy, math, science, history, 21 st century life and careers, technology) learning activities are included in this equation s graph. Represent complex numbers in trigonometric form and perform operations on them. Use De Moivre s Theorem Perform operations with vectors and use vectors to solve real world problems. Find dot products and projections of vectors and apply to real world problems. Add and subtract matrices. Organize and interpret data in matrices. Individual Instruction Plans Current-event problems Manipulatives Tiered Lessons that functions. Add, subtract, multiply, and divide polynomials and monomials. Solve equations and inequalities Solve equations and inequalities with variables on both sides of the equal sign. Judge the meaning, utility, and reasonableness of the results of symbol manipulation, including technological. Individual Instruction Plans Current-event problems Manipulatives Tiered Lessons Learning Estimate derivatives and integrals using numerical techniques. Language Arts (solving word problems, translating, explanations during problem solving) Science (Problem solving, 23

24 UNIT: 4 Vectors and Matrices (May, 30) UNIT: 5 Algebra Unit: 6 An Introduction to Calculus (June, 20) unit that will help achieve the desired results?) Learning style adaptation R.A.F.T Project Based learning Compacting Multimedia presentations Open-ended responses Conclusions and analysis of exploratory activities style adaptation R.A.F.T Project Based learning Compacting Multimedia presentations Open-ended responses Conclusions and analysis of exploratory activities scientific notation and applications) Social Studies (reading and interpreting graphs, economic applications Individual Instruction Plans Current-event problems Manipulatives Tiered Lessons Learning style adaptation R.A.F.T Project Based learning Compacting Multimedia presentations Open-ended responses Conclusions and analysis of exploratory activities 24