West Windsor-Plainsboro Regional School District Pre-Calculus Honors Curriculum Grade 11-12
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Unit 1: Trigonometric Functions
Content Area: Mathematics Course & Grade Level: Pre‐Calculus Honors, 11‐12
Summary and Rationale Pre‐Calculus Honors is a continuation of the more advanced concepts of algebra and geometry intended for students who have demonstrated critical thinking skills. Trigonometry is studied along with analytic geometry, incorporating the study of polynomial, exponential and logarithmic functions, conics, and the polar coordinate system. Graphing calculators are used throughout the course. Analysis, problem solving and graphing are stressed.
Recommended Pacing 22 days
NJCCC Standards Standard HSF‐TF.A Extend the domain of trigonometric functions using the unit circle. CPI # 1
Cumulative Progress Indicator (CPI) Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
2
Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
3
Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any real number. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
4 5
Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline
6
Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed
7
Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context
Standard HSG‐C.B Find arc lengths and areas of sectors and circles CPI # 5
Cumulative Progress Indicator (CPI) Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
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Instructional Focus Unit Enduring Understandings Students will apply mathematics to real‐life situations 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 logical, clear sequence of steps. Students will verify the correctness of their solutions through a variety of methods Unit Essential Questions How can change be best represented mathematically? How can patterns, relations, and functions be used as tools to best describe and help explain real‐life situations? How are patterns of change related to the behavior of functions? How can we use mathematical models to describe physical relationships? How can we use physical models to clarify mathematical relationships? What makes an algebraic algorithm both effective and efficient? Objectives Students will know: Angle Measurement Arc length and apparent size. Sectors of Circles The Six Trigonometric Functions ‐ definition ‐ graphs ‐ inverses Students will be able to: Determine the domain, range, zeros, amplitude, phase shift and period of sinusoidal functions Graph sinusoidal functions without a graphing calculator Use transformations to graph Use even‐odd properties to find the exact values of the trigonometric functions Solve real‐world problems applying trigonometric functions Simplify trigonometric expressions Find arc length and area of a sector of a circle
Resources Core Text: Advanced Mathematics – Precalculus with Discrete Mathematics Suggested Resources:
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Unit 2: Trigonometric Equations and Applications
Content Area: Mathematics Course & Grade Level: Pre‐Calculus Honors, 11‐12
Summary and Rationale Pre‐Calculus Honors is a continuation of the more advanced concepts of algebra and geometry intended for students who have demonstrated critical thinking skills. Trigonometry is studied along with analytic geometry, incorporating the study of polynomial, exponential and logarithmic functions, conics, and the polar coordinate system. Graphing calculators are used throughout the course. Analysis, problem solving and graphing are stressed.
Recommended Pacing 25 days
NJCCC Standards Standard HS‐TF.C Prove and apply trig identities CPI # Cumulative Progress Indicator (CPI) 5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given 8 sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
Instructional Focus Unit Enduring Understandings Students will apply mathematics to real life situations 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 Unit Essential Questions How can change be best represented mathematically? How can patterns, relations, and functions be used as tools to best describe and help explain real‐life situations? How are patterns of change related to the behavior of functions? How can we use mathematical models to describe physical relationships? How can we use physical models to clarify mathematical relationships? What makes an algebraic algorithm both effective and efficient? Objectives Students will be able to: Solve simple trig equations and apply them. Find equations of different sine and cosine curves and apply to equations. Use trig functions to model periodic behavior. Prove trigonometric identities. Apply identities, graphs and/or technology to solve more difficult trig equations.
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Resources Core Text: Advanced Mathematics – Precalculus with Discrete Mathematics Suggested Resources:
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Unit 3: Triangle Trigonometry
Content Area: Mathematics Course & Grade Level: Pre‐Calculus Honors, 11‐12
Summary and Rationale Pre‐Calculus Honors is a continuation of the more advanced concepts of algebra and geometry intended for students who have demonstrated critical thinking skills. Trigonometry is studied along with analytic geometry, incorporating the study of polynomial, exponential and logarithmic functions, conics, and the polar coordinate system. Graphing calculators are used throughout the course. Analysis, problem solving and graphing are stressed.
Recommended Pacing 22 days
NJCCC Standards Standard HSG‐SRT.C Define trigonometric ratios and solve problems involving right triangles CPI # Cumulative Progress Indicator (CPI) 8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 9
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
Standard HSG‐SRT.D Apply trigonometry to general triangles CPI # Cumulative Progress Indicator (CPI) 9 Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. 10 11
Prove the Laws of Sines and Cosines and use them to solve problems. Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non‐right triangles (e.g., surveying problems, resultant forces).
Instructional Focus Unit Enduring Understandings Students will apply mathematics to real world situations 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 Unit Essential Questions How can change be best represented mathematically? How can patterns, relations, and functions be used as tools to best describe and help explain real‐life situations? How are patterns of change related to the behavior of functions? How can we use mathematical models to describe physical relationships? How can we use physical models to clarify mathematical relationships? What makes an algebraic algorithm both effective and efficient? Objectives
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Students will be able to: Find the value of trigonometric functions of acute angles in a right triangle Solve and apply problems using right triangle trigonometry Prove Law of Sines and Law of Cosines Use Law of Sines and Law of Cosines to solve triangles Apply Law of Sines and Law of Cosines to real‐world problems Determine when it is appropriate to use Law of Sines (AAS, ASA, SSA Triangles) and Law of Cosines (SAS, SSS Triangles) Derive formulas for the area of a triangle in the SAS (using the sine function) and SSS (Heron’s formula) cases Find the area of a triangle in the SAS (using the sine function) . Apply Law of Sines/Cosines to surveying and navigation problems.
Resources Core Text: Advanced Mathematics – Precalculus with Discrete Mathematics Suggested Resources:
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Unit 4: Trigonometric Formulas
Content Area: Mathematics Course & Grade Level: Pre‐Calculus Honors 11‐12
Summary and Rationale Pre‐Calculus Honors is a continuation of the more advanced concepts of algebra and geometry intended for students who have demonstrated critical thinking skills. Trigonometry is studied along with analytic geometry, incorporating the study of polynomial, exponential and logarithmic functions, conics, and the polar coordinate system. Graphing calculators are used throughout the course. Analysis, problem solving and graphing are stressed.
Recommended Pacing 22 days
NJCCC Standards Standard HS‐TF.C Prove and apply trig identities CPI # Cumulative Progress Indicator (CPI) 8 Prove the Pythagorean identity sin2(θ) + cos2(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 Unit Essential Questions How can change be best represented mathematically? How can patterns, relations, and functions be used as tools to best describe and help explain real‐life situations? How are patterns of change related to the behavior of functions? How can we use mathematical models to describe physical relationships? How can we use physical models to clarify mathematical relationships? What makes an algebraic algorithm both effective and efficient? Objectives Students will know: The sum and difference formulas The double and half angle formulas Students will be able to: Derive and apply the sum and difference formulas. Derive and apply the double and half Angle formulas Use identities to solve more complex trigonometric equations.
Resources Core Text: Advanced Mathematics – Precalculus with Discrete Mathematics Suggested Resources:
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Unit 5: Conic Sections
Content Area: Mathematics Course & Grade Level: Pre‐Calculus Honors, 11‐12
Summary and Rationale Pre‐Calculus Honors is a continuation of the more advanced concepts of algebra and geometry intended for students who have demonstrated critical thinking skills. Trigonometry is studied along with analytic geometry, incorporating the study of polynomial, exponential and logarithmic functions, conics, and the polar coordinate system. Graphing calculators are used throughout the course. Analysis, problem solving and graphing are stressed.
Recommended Pacing 20 days
NJCCC Standards Standard HSG‐GPE.A Translate between the geometric description and the equation for a conic section CPI # Cumulative Progress Indicator (CPI) 1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. 2
Derive the equation of a parabola given a focus and directrix.
3
Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
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 Unit Essential Questions How can change be best represented mathematically? How can patterns, relations, and functions be used as tools to best describe and help explain real‐life situations? How are patterns of change related to the behavior of functions? How can we use mathematical models to describe physical relationships? How can we use physical models to clarify mathematical relationships? What makes an algebraic algorithm both effective and efficient? Objectives Students will know: Equations of circles, ellipses, hyperbolas and parabolas in standard and Cartesian Form. Terms: foci, directrix, asymptotes, vertices, major axis, minor axis, eccentricity Students will be able to: Sketch the conic sections given their equations in standard form. Determine which conic section from the Cartesian equation and subsequently complete the square to place the equation in standard form and sketch. Determine the equation of specific conics given particular characteristics; i.e. foci, equations of asymptotes,
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vertices, etc. Determine the eccentricity of a conic section. Apply the conic sections to real world applications. Solve systems of second degree equations both algebraically and graphically.
Resources Core Text: Advanced Mathematics – Precalculus with Discrete Mathematics Suggested Resources:
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Unit 6: Polar Coordinates
Content Area: Mathematics Course & Grade Level: Pre‐Calculus Honors, 11‐12
Summary and Rationale Pre‐Calculus Honors is a continuation of the more advanced concepts of algebra and geometry intended for students who have demonstrated critical thinking skills. Trigonometry is studied along with analytic geometry, incorporating the study of polynomial, exponential and logarithmic functions, conics, and the polar coordinate system. Graphing calculators are used throughout the course. Analysis, problem solving and graphing are stressed.
Recommended Pacing 25 days
NJCCC Standards Standard HSN‐CN. B Represent complex numbers and their operations on a complex plane. CPI # Cumulative Progress Indicator (CPI) 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. 6
Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
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 Unit Essential Questions How can change be best represented mathematically? How can patterns, relations, and functions be used as tools to best describe and help explain real‐life situations? How are patterns of change related to the behavior of functions? How can we use mathematical models to describe physical relationships? How can we use physical models to clarify mathematical relationships? What makes an algebraic algorithm both effective and efficient? Objectives Students will know: Terms: pole, polar axis, polar coordinates, limacon, rose curves, cardioid, polar form, Students will be able to: Convert points and equations from Polar to Cartestian and in reverse. Graph lines in polar Polar Graphs and writing equations. Systems of polar equations vs points of intersection Express complex numbers in polar form. Find product and quotients of complex numbers in polar form. To use DeMoivre’s Theorem to find powers of complex numbers.
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To find roots of complex numbers.
Resources Core Text: Advanced Mathematics – Precalculus with Discrete Mathematics Suggested Resources:
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Unit 7: Inequalities in One or Two Variables
Content Area: Mathematics Course & Grade Level: Pre‐Calculus Honors, 11‐12
Summary and Rationale Pre‐Calculus Honors is a continuation of the more advanced concepts of algebra and geometry intended for students who have demonstrated critical thinking skills. Trigonometry is studied along with analytic geometry, incorporating the study of polynomial, exponential and logarithmic functions, conics, and the polar coordinate system. Graphing calculators are used throughout the course. Analysis, problem solving and graphing are stressed.
Recommended Pacing 14 days
NJCCC Standards Standard HSA‐CED.A Create equations that describe numbers or relationships CPI # Cumulative Progress Indicator (CPI) 3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Standard HSA‐REI.D Represent and solve equations and inequalities graphically CPI # Cumulative Progress Indicator (CPI) 11 Explain why the x‐coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions 12
Graph the solutions to a linear inequality in two variables as a half‐plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half‐planes
Instructional Focus Unit Enduring Understandings Students will apply mathematics to real world situations 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
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Unit Essential Questions How can change be best represented mathematically? How can patterns, relations, and functions be used as tools to best describe and help explain real‐life situations? How are patterns of change related to the behavior of functions? How can we use mathematical models to describe physical relationships? How can we use physical models to clarify mathematical relationships? What makes an algebraic algorithm both effective and efficient? Objectives Students will be able to: Solve linear and polynomial inequalities Solve polynomial inequalities in two variables Solve and graph absolute value inequalities
Resources Core Text: Advanced Mathematics – Precalculus with Discrete Mathematics Suggested Resources:
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Unit 8: Rational Functions
Content Area: Mathematics Course & Grade Level: Pre‐Calculus Honors, 11‐12
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, and computer software packages 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 7 days
State Standards Standard 4.HSF‐IF.B Interpret functions that arise in applications in terms of the context CPI # Cumulative Progress Indicator (CPI) 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity Standard HSF‐IF.C Analyze functions using different representations CPI # Cumulative Progress Indicator (CPI) 7d Graph rational functions, identifying zeros and asymptotes 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 Unit Essential Questions How can change be best represented mathematically? How can patterns, relations, and functions be used as tools to best describe and help explain real‐life situations? How are patterns of change related to the behavior of functions? How can we use mathematical models to describe physical relationships? How can we use physical models to clarify mathematical relationships? What makes an algebraic algorithm both effective and efficient? Objectives
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Students will know: Synthetic Substitution Factoring Polynomials Relationships between roots and coefficients Rational Functions Students will be able to: Use the Rational Zeros Theorem and find the real zeros of a polynomial function Graph rational functions by finding zeros, asymptotes, y‐intercept and exploring end behavior. Relate rational function graphs to the idea of a limit Write an equation of a given rational function graph.
Resources Core Text: Advanced Mathematics – Precalculus with Discrete Mathematics Suggested Resources:
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Unit 9: Introduction to Calculus
Content Area: Mathematics Course & Grade Level: Pre‐Calculus Honors, 11‐12
Summary and Rationale Pre‐Calculus Honors is a continuation of the more advanced concepts of algebra and geometry intended for students who have demonstrated critical thinking skills. Trigonometry is studied along with analytic geometry, incorporating the study of polynomial, exponential and logarithmic functions, conics, and the polar coordinate system. Graphing calculators are used throughout the course. Analysis, problem solving and graphing are stressed.
Recommended Pacing 15 days
NJCCC Standards Standard 4.HSF‐IF.B Interpret functions that arise in applications in terms of the context CPI # Cumulative Progress Indicator (CPI) 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity
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 Unit Essential Questions How can change be best represented mathematically? How can patterns, relations, and functions be used as tools to best describe and help explain real‐life situations? How are patterns of change related to the behavior of functions? How can we use mathematical models to describe physical relationships? How can we use physical models to clarify mathematical relationships? What makes an algebraic algorithm both effective and efficient? Objectives Students will know: Terms: Limit, Continuity‐removable and non‐removable finding the slope of a tangent line to a curve given the function, Students will be able to: Identify limits by numerical and graphical techniques. To determine the limit of a function using techniques such as cancellation, rationalization, and additional algebraic techniques Determine continuity of a graph and label discontinuities as removable and non‐removable. Determine the slope of a tangent line using the limit definition of a derivative
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Apply some basic differentiation rules.
Resources Core Text: Advanced Mathematics – Precalculus with Discrete Mathematics Suggested Resources:
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