West Windsor-Plainsboro Regional School District Pre-Calculus Grades 11-12

Transcription

1 West Windsor-Plainsboro Regional School District Pre-Calculus Grades 11-12

2 Unit 1: Solving Equations and Inequalities Algebraically and Graphically Content Area: Mathematics Course & Grade Level: Pre Calculus, Summary and Rationale Pre Calculus is a continuation of the more advanced concepts of algebra and geometry integrated with the study of analytic and triangle trigonometry; it is a segue to calculus from algebra. This course is designed for students with a strong background in mathematics. Functions are explored in great detail including polynomial, rational, logarithmic, trigonometric and inverse trigonometrics. A strong emphasis is placed on technology and the use of graphing calculators to explore the patterns in graphing. Skills in analysis, reasoning and making connections are stressed throughout the course. Probability, sequences and series will also be studied. Real world problem solving and critical thinking will be stressed. Solving problems numerically, graphically and algebraically while utilizing technology appropriately will be the focus of the course. Recommended Pacing 17 days Standard 4.A CED Creating Equations State Standards 3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non viable options in a modeling context. Standard 4.A REI Reasoning with Equations and Inequalities 3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by variables. Instructional Focus Unit Enduring Understandings Students will increase their interest in learning mathematics Students will approach problem solving by focusing on understanding concepts rather than rote use of procedures and formulas Students will justify all problem solutions with a logically clear sequence of steps Students will verify the correctness of their solutions through a variety of methods The symbolic language of algebra is used to communicate and generalize the patterns in mathematics Algebraic representation can be used to generalize patterns and relationships Patterns and relationships can be represented graphically, numerically, symbolically, or verbally Mathematical models can be used to describe and quantify physical relationships Physical models can be used to clarify mathematical relationships Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole Reasoning and/or proof can be used to verify or refute conjectures or theorems in algebra Unit Essential Questions What is a function? How can you use models of functions to solve real life applications? How can you describe the behavior and characteristics of a graph of a function? What is an inverse and how is it related to its parent graph?

3 Objectives Students will be able to: Evaluate functions and determine their domains and ranges. Determine when a function is increasing, decreasing or constant Find relative maximums and minimums using a graphing calculator. Determine if a function is even, odd or neither both graphically and algebraically. Identify and graph shifts, reflections, and non rigid transformations of functions. Find arithmetic combinations and compositions of functions Find inverses of functions graphically and algebraically. Resources Core Text: Pre Calculus with Limits: A Graphing Approach: 3rd Edition, Rolant E Larson, Robert P. Hostetler and Bruce H. Edwards; Houghton Mifflin Company, copyright 2001 Suggested Resources:

4 Unit 2: Polynomial and Rational Functions Content Area: Mathematics Course & Grade Level: Pre Calculus, Summary and Rationale Pre Calculus is a continuation of the more advanced concepts of algebra and geometry integrated with the study of analytic and triangle trigonometry; it is a segue to calculus from algebra. This course is designed for students with a strong background in mathematics. Functions are explored in great detail including polynomial, rational, logarithmic, trigonometric and inverse trigonometrics. A strong emphasis is placed on technology and the use of graphing calculators to explore the patterns in graphing. Skills in analysis, reasoning and making connections are stressed throughout the course. Probability, sequences and series will also be studied. Real world problem solving and critical thinking will be stressed. Solving problems numerically, graphically and algebraically while utilizing technology appropriately will be the focus of the course. Recommended Pacing 18 days Standard 4.A SSE Seeing Structure in Expressions State Standards 3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it describes. Standard 4.A APR Arithmetic with Polynomials and Rational Expressions 3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Standard 4.A APR Arithmetic with Polynomials and Rational Expressions 7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Instructional Focus Unit Enduring Understandings Students will increase their interest in learning mathematics Students will approach problem solving by focusing on understanding concepts rather than rote use of procedures and formulas Students will justify all problem solutions with a logically clear sequence of steps Students will verify the correctness of their solutions through a variety of methods The symbolic language of algebra is used to communicate and generalize the patterns in mathematics Algebraic representation can be used to generalize patterns and relationships

5 Patterns and relationships can be represented graphically, numerically, symbolically, or verbally Mathematical models can be used to describe and quantify physical relationships Physical models can be used to clarify mathematical relationships Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole Reasoning and/or proof can be used to verify or refute conjectures or theorems in algebra Unit Essential Questions How can you use polynomials to solve real life applications? How do the transformations and characteristics of polynomial and rational functions affect their graphs? How do imaginary roots affect the factorization and graph of a polynomial function? What are asymptotes and how do they affect the behavior of the graph? Objectives Students will be able to: Sketch and analyze graphs of quadratic and polynomial functions. Use long division and synthetic division to divide polynomials. Determine the number of rational and real zeros of polynomial functions, and solve for them. Analyze and solve real life applications. Perform operations with complex numbers and plot complex numbers in the complex plane. Determine the domain, find asymptotes, and sketch the graphs of rational functions. Resources Core Text: Pre Calculus with Limits: A Graphing Approach: 3rd Edition, Rolant E Larson, Robert P. Hostetler and Bruce H. Edwards; Houghton Mifflin Company, copyright 2001 Suggested Resources:

6 Unit 3: Exponential and Logarithmic Functions Content Area: Mathematics Course & Grade Level: Pre Calculus, Summary and Rationale Pre Calculus is a continuation of the more advanced concepts of algebra and geometry integrated with the study of analytic and triangle trigonometry; it is a segue to calculus from algebra. This course is designed for students with a strong background in mathematics. Functions are explored in great detail including polynomial, rational, logarithmic, trigonometric and inverse trigonometrics. A strong emphasis is placed on technology and the use of graphing calculators to explore the patterns in graphing. Skills in analysis, reasoning and making connections are stressed throughout the course. Probability, sequences and series will also be studied. Real world problem solving and critical thinking will be stressed. Solving problems numerically, graphically and algebraically while utilizing technology appropriately will be the focus of the course. Recommended Pacing 11 days Standard 4. A SSE Seeing Structure in Expressions State Standards 3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. c. Use the properties of exponents to transform expressions for exponential functions. Standard 4.F IF Interpreting Functions 7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. e. Graph exponentials and logarithmi8c functions, showing intercepts and end behavior, and trigonometric functions, showing periods midline and amplitude. 8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Use the properties of exponents to interpret expressions for exponential functions. Instructional Focus Unit Enduring Understandings Students will increase their interest in learning mathematics Students will approach problem solving by focusing on understanding concepts rather than rote use of procedures and formulas Students will justify all problem solutions with a logically clear sequence of steps Students will verify the correctness of their solutions through a variety of methods The symbolic language of algebra is used to communicate and generalize the patterns in mathematics Algebraic representation can be used to generalize patterns and relationships

7 Patterns and relationships can be represented graphically, numerically, symbolically, or verbally Mathematical models can be used to describe and quantify physical relationships Physical models can be used to clarify mathematical relationships Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole Reasoning and/or proof can be used to verify or refute conjectures or theorems in algebra Unit Essential Questions How can you use exponential and logarithmic models and properties to solve real life applications? How do the transformations and characteristics of logarithmic and exponential functions affect their graphs? Objectives Students will be able to: Recognize, evaluate, and graph exponential and logarithmic functions. Rewrite logarithmic functions with different bases. Use properties of logarithms to evaluate, rewrite, expand, or condense logarithmic expressions. Solve exponential and logarithmic equations. Use exponential growth and decay models to solve real life problems. Use logarithmic and exponential models to explore sets of data. Resources Core Text: Pre Calculus with Limits: A Graphing Approach: 3rd Edition, Rolant E Larson, Robert P. Hostetler and Bruce H. Edwards; Houghton Mifflin Company, copyright 2001 Suggested Resources:

8 Unit 4: Trigonometric Functions Content Area: Mathematics Course & Grade Level: Pre Calculus, Summary and Rationale Pre Calculus is a continuation of the more advanced concepts of algebra and geometry integrated with the study of analytic and triangle trigonometry; it is a segue to calculus from algebra. This course is designed for students with a strong background in mathematics. Functions are explored in great detail including polynomial, rational, logarithmic, trigonometric and inverse trigonometrics. A strong emphasis is placed on technology and the use of graphing calculators to explore the patterns in graphing. Skills in analysis, reasoning and making connections are stressed throughout the course. Probability, sequences and series will also be studied. Real world problem solving and critical thinking will be stressed. Solving problems numerically, graphically and algebraically while utilizing technology appropriately will be the focus of the course. Recommended Pacing 23 days (Including midterm and midterm review) State Standards Standard 4.G SRT Similarity, Right Triangles, and Trigonometry 7 Explain and use the relationship between the sine and cosine of complementary angles. 8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 9 Derive the formula A = ½ ab sin for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. 10 Prove the Laws of Sines and Cosines and use them to solve problems. 11 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non right triangles. Instructional Focus Unit Enduring Understandings Students will increase their interest in learning mathematics Students will approach problem solving by focusing on understanding concepts rather than rote use of procedures and formulas Students will justify all problem solutions with a logically clear sequence of steps Students will verify the correctness of their solutions through a variety of methods The symbolic language of algebra is used to communicate and generalize the patterns in mathematics Algebraic representation can be used to generalize patterns and relationships Patterns and relationships can be represented graphically, numerically, symbolically, or verbally Mathematical models can be used to describe and quantify physical relationships Physical models can be used to clarify mathematical relationships Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole Reasoning and/or proof can be used to verify or refute conjectures or theorems in algebra Unit Essential Questions What are radians and how can we use them to describe the measurement of an angle? How can you use right triangles and the unit circle to find the trigonometric ratios of special angles?

9 How can trigonometric functions be represented graphically? How can you use trigonometric functions to solve real life applications? How do the transformations and characteristics of trigonometric functions affect their graphs? Objectives Students will be able to: Explore and understand radians as a unit of measurement. Describe an angle and convert between degree and radian measures. Identify coterminal angles. Identify a unit circle and its relationship to real numbers. Evaluate the length of an arc and the area of a sector. Evaluate trigonometric functions of any angle using the unit circle or right triangle trigonometry. Sketch graphs of trigonometric functions. Evaluate inverse trigonometric functions and use them to solve right triangles. Evaluate the composition of trigonometric functions. Use trigonometric functions to model and solve real life problems. Resources Core Text: Pre Calculus with Limits: A Graphing Approach: 3rd Edition, Rolant E Larson, Robert P. Hostetler and Bruce H. Edwards; Houghton Mifflin Company, copyright 2001 Suggested Resources:

10 Unit 5: Analytic Trigonometry Content Area: Mathematics Course & Grade Level: Pre Calculus, Summary and Rationale Pre Calculus is a continuation of the more advanced concepts of algebra and geometry integrated with the study of analytic and triangle trigonometry; it is a segue to calculus from algebra. This course is designed for students with a strong background in mathematics. Functions are explored in great detail including polynomial, rational, logarithmic, trigonometric and inverse trigonometrics. A strong emphasis is placed on technology and the use of graphing calculators to explore the patterns in graphing. Skills in analysis, reasoning and making connections are stressed throughout the course. Probability, sequences and series will also be studied. Real world problem solving and critical thinking will be stressed. Solving problems numerically, graphically and algebraically while utilizing technology appropriately will be the focus of the course. Recommended Pacing 11 days Standard 4.F TF Trigonometric Functions State Standards 8 Prove the Pythagorean identity sin 2 (θ) + cos 2 (θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. 9 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Instructional Focus Unit Enduring Understandings Students will increase their interest in learning mathematics Students will approach problem solving by focusing on understanding concepts rather than rote use of procedures and formulas Students will justify all problem solutions with a logically clear sequence of steps Students will verify the correctness of their solutions through a variety of methods The symbolic language of algebra is used to communicate and generalize the patterns in mathematics Algebraic representation can be used to generalize patterns and relationships Patterns and relationships can be represented graphically, numerically, symbolically, or verbally Mathematical models can be used to describe and quantify physical relationships Physical models can be used to clarify mathematical relationships Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole Reasoning and/or proof can be used to verify or refute conjectures or theorems in algebra Unit Essential Questions What is an identity? How can trigonometric identities be used to solve trigonometric equations? Objectives Students will be able to: Use fundamental trigonometric identities to evaluate trigonometric functions and simplify trigonometric expressions.

11 Verify trigonometric identities. Use standard algebraic techniques and inverse trigonometric functions to solve trigonometric equations. Use sum and difference formulas, double angle formulas, and half angle formulas to rewrite and evaluate trigonometric functions. Resources Core Text: Pre Calculus with Limits: A Graphing Approach: 3rd Edition, Rolant E Larson, Robert P. Hostetler and Bruce H. Edwards; Houghton Mifflin Company, copyright 2001 Suggested Resources:

12 Unit 6: Law of Sines + Law of Cosines Content Area: Mathematics Course & Grade Level: Pre Calculus, Summary and Rationale Pre Calculus is a continuation of the more advanced concepts of algebra and geometry integrated with the study of analytic and triangle trigonometry; it is a segue to calculus from algebra. This course is designed for students with a strong background in mathematics. Functions are explored in great detail including polynomial, rational, logarithmic, trigonometric and inverse trigonometrics. A strong emphasis is placed on technology and the use of graphing calculators to explore the patterns in graphing. Skills in analysis, reasoning and making connections are stressed throughout the course. Probability, sequences and series will also be studied. Real world problem solving and critical thinking will be stressed. Solving problems numerically, graphically and algebraically while utilizing technology appropriately will be the focus of the course. Recommended Pacing 6 days Standard 4.A CED Creating Equations State Standards 9 Derive the formula A = ½ ab sin for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. 10 Prove the Laws of Sines and Cosines and use them to solve problems. 11 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non right triangles. Instructional Focus Unit Enduring Understandings Students will increase their interest in learning mathematics Students will approach problem solving by focusing on understanding concepts rather than rote use of procedures and formulas Students will justify all problem solutions with a logically clear sequence of steps Students will verify the correctness of their solutions through a variety of methods The symbolic language of algebra is used to communicate and generalize the patterns in mathematics Algebraic representation can be used to generalize patterns and relationships Patterns and relationships can be represented graphically, numerically, symbolically, or verbally Mathematical models can be used to describe and quantify physical relationships Physical models can be used to clarify mathematical relationships Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole Reasoning and/or proof can be used to verify or refute conjectures or theorems in algebra Unit Essential Questions How can Law of Sines and Law of Cosines be used to solve real life applications? How can you find the area of oblique triangles? Objectives Students will be able to: Use the Law of Sines and the Law of Cosines to solve oblique triangles.

13 Understand when the ambiguous case of the Law of Sines has zero, one, or two solutions. Find the areas of oblique triangles. Determine when it is appropriate to use Law of Sines or Law of Cosines. Use the Law of Sines and the Law of Cosines to solve real life applications. Resources Core Text: Pre Calculus with Limits: A Graphing Approach: 3rd Edition, Rolant E Larson, Robert P. Hostetler and Bruce H. Edwards; Houghton Mifflin Company, copyright 2001 Suggested Resources:

14 Unit 7: Sequences, Series, and Probability (Chapter 9) Content Area: Mathematics Course & Grade Level: Pre Calculus, Summary and Rationale Pre Calculus is a continuation of the more advanced concepts of algebra and geometry integrated with the study of analytic and triangle trigonometry; it is a segue to calculus from algebra. This course is designed for students with a strong background in mathematics. Functions are explored in great detail including polynomial, rational, logarithmic, trigonometric and inverse trigonometrics. A strong emphasis is placed on technology and the use of graphing calculators to explore the patterns in graphing. Skills in analysis, reasoning and making connections are stressed throughout the course. Probability, sequences and series will also be studied. Real world problem solving and critical thinking will be stressed. Solving problems numerically, graphically and algebraically while utilizing technology appropriately will be the focus of the course. Recommended Pacing 14 days Standard 4.A SSE Seeing Structure in Expressions State Standards 4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. Standard 4.F BF 2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Instructional Focus Unit Enduring Understandings Students will increase their interest in learning mathematics Students will approach problem solving by focusing on understanding concepts rather than rote use of procedures and formulas Students will justify all problem solutions with a logically clear sequence of steps Students will verify the correctness of their solutions through a variety of methods The symbolic language of algebra is used to communicate and generalize the patterns in mathematics Algebraic representation can be used to generalize patterns and relationships Patterns and relationships can be represented graphically, numerically, symbolically, or verbally Mathematical models can be used to describe and quantify physical relationships Physical models can be used to clarify mathematical relationships Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole Reasoning and/or proof can be used to verify or refute conjectures or theorems in algebra Unit Essential Questions How can you use counting principles and probability to solve real life applications? How do we use arithmetic and geometric sequences and series to solve real life applications? Objectives Students will be able to:

15 Use sequence, factorial, and summation notation to write the terms and sums of sequences. Recognize, write, and use arithmetic sequences and geometric sequences. Solve counting problems using the Fundamental Counting Principle, permutations, and combinations. Find the probability of events and their compliments. Resources Core Text: Pre Calculus with Limits: A Graphing Approach: 3rd Edition, Rolant E Larson, Robert P. Hostetler and Bruce H. Edwards; Houghton Mifflin Company, copyright 2001 Suggested Resources:

16 Unit 8: Polar Coordinates (Chapter 10) Content Area: Mathematics Course & Grade Level: Pre Calculus, Summary and Rationale Pre Calculus is a continuation of the more advanced concepts of algebra and geometry integrated with the study of analytic and triangle trigonometry; it is a segue to calculus from algebra. This course is designed for students with a strong background in mathematics. Functions are explored in great detail including polynomial, rational, logarithmic, trigonometric and inverse trigonometrics. A strong emphasis is placed on technology and the use of graphing calculators to explore the patterns in graphing. Skills in analysis, reasoning and making connections are stressed throughout the course. Probability, sequences and series will also be studied. Real world problem solving and critical thinking will be stressed. Solving problems numerically, graphically and algebraically while utilizing technology appropriately will be the focus of the course. Recommended Pacing 6 days Standard 4.N CN The Complex Number System State Standards 4 Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. Instructional Focus Unit Enduring Understandings Students will increase their interest in learning mathematics Students will approach problem solving by focusing on understanding concepts rather than rote use of procedures and formulas Students will justify all problem solutions with a logically clear sequence of steps Students will verify the correctness of their solutions through a variety of methods The symbolic language of algebra is used to communicate and generalize the patterns in mathematics Algebraic representation can be used to generalize patterns and relationships Patterns and relationships can be represented graphically, numerically, symbolically, or verbally Mathematical models can be used to describe and quantify physical relationships Physical models can be used to clarify mathematical relationships Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole Reasoning and/or proof can be used to verify or refute conjectures or theorems in algebra Unit Essential Questions What are other ways of representing functions other than on the rectangular coordinate system? How do the characteristics of polar functions affect their graphs? How do you solve equations in the polar coordinate plane? Objectives Students will be able to: Write equations in polar form. Graph polar coordinates and equations and recognize special polar graphs.

17 Convert polar coordinates and equations to rectangular form. Convert rectangular coordinates and equations to polar form. Resources Core Text: Pre Calculus with Limits: A Graphing Approach: 3rd Edition, Rolant E Larson, Robert P. Hostetler and Bruce H. Edwards; Houghton Mifflin Company, copyright 2001 Suggested Resources:

18 Unit 9: Introduction to Limits (Enrichment Unit) Chapter 12 Content Area: Mathematics Course & Grade Level: Pre Calculus, Summary and Rationale Pre Calculus is a continuation of the more advanced concepts of algebra and geometry integrated with the study of analytic and triangle trigonometry; it is a segue to calculus from algebra. This course is designed for students with a strong background in mathematics. Functions are explored in great detail including polynomial, rational, logarithmic, trigonometric and inverse trigonometrics. A strong emphasis is placed on technology and the use of graphing calculators to explore the patterns in graphing. Skills in analysis, reasoning and making connections are stressed throughout the course. Probability, sequences and series will also be studied. Real world problem solving and critical thinking will be stressed. Solving problems numerically, graphically and algebraically while utilizing technology appropriately will be the focus of the course. Recommended Pacing 2 days Standard 4.F IF State Standards 4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. 5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. 6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Instructional Focus Unit Enduring Understandings Students will increase their interest in learning mathematics Students will approach problem solving by focusing on understanding concepts rather than rote use of procedures and formulas Students will justify all problem solutions with a logically clear sequence of steps Students will verify the correctness of their solutions through a variety of methods The symbolic language of algebra is used to communicate and generalize the patterns in mathematics Algebraic representation can be used to generalize patterns and relationships Patterns and relationships can be represented graphically, numerically, symbolically, or verbally Mathematical models can be used to describe and quantify physical relationships Physical models can be used to clarify mathematical relationships Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole Reasoning and/or proof can be used to verify or refute conjectures or theorems in algebra Unit Essential Questions What is a limit? How do we represent limits graphically? What techniques are most appropriate to find a limit?

19 Objectives Students will be able to: Estimate limits and use properties and operations of limits. Find limits by direct substitution and by using the dividing out and rationalizing techniques. Resources Core Text: Pre Calculus with Limits: A Graphing Approach: 3rd Edition, Rolant E Larson, Robert P. Hostetler and Bruce H. Edwards; Houghton Mifflin Company, copyright 2001 Suggested Resources: