West Windsor-Plainsboro Regional School District Algebra and Trigonometry Grades 11-12

Transcription

1 West Windsor-Plainsboro Regional School District Algebra and Trigonometry Grades 11-12

2 Unit 1: Linear Relationships & Functions Content Area: Mathematics Course & Grade Level: Algebra & Trigonometry, Summary and Rationale Algebra and Trigonometry is a course designed for those students who have successfully completed Algebra II or have experienced difficulty in Advanced Algebra. This course provides the essential concepts and skills of algebra, trigonometry, and the study of functions, which are needed for further study of mathematics. Algebra topics include: advanced factoring, complex numbers, and quadratic relations and systems. In addition to a study of general functions, linear, polynomial, rational, logarithmic, exponential and trigonometric functions are also studied. Problem solving and graphing techniques are stressed throughout the course. Calculators and computers will be used throughout the course. Recommended Pacing 15 days Standard 4.F IF Interpreting Functions 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 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 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

3 Students will know: Terms: domain, range, vertical line test, zero of a function, composition of functions, coinciding lines, scatter plot, regression line Students will be able to: Determine whether a relation is a function Identify the domain and range Evaluate outputs of a function Perform operations and composition of functions Graph various forms of linear functions Write linear functions using points, slopes, and parallel & perpendicular relationships Draw and analyze scatter plots, and write prediction equations Identify and graph piecewise functions, including absolute value functions Graph inequalities Resources Core Text: Advanced Mathematical Concepts: Pre calculus with Applications, Holiday, Cuevas, McClure, (2001) McGraw Hill/Glencoe Suggested Resources:

4 Unit 2: Systems of Linear Equations & Inequalities Content Area: Mathematics Course & Grade Level: Algebra & Trigonometry, Summary and Rationale This unit involves the study of systems of linear equations and inequalities. By learning to solve the systems both algebraically and graphically and to appropriately interpret their solutions students will be able to apply systems to model real world situations. 18 days Standard 4.F IF Interpreting Functions Recommended Pacing State Standards 7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. Standard 4.A REI Reasoning with Equations and Inequalities 5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. 6 Solve systems of linear equations exactly and approximately (e.g. with graphs), focusing on pairs of linear equations in two variables. 10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). 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 Utilize technology (graphing calculator) to investigate and extend applications of functions. Develop a problem solving repertoire and be able to choose the appropriate method to solve real world problems. Analyze data to find patterns in order to make decisions based on the data. Unit Essential Questions How can we find an optimal real world solution to a system of complex linear constraints? How well can mathematics model complex real world situations? When is the mathematical solution not the real solution? Objectives Students will know: Terms: system of linear equations, substitution, elimination, optimization, linear programming, linear

5 inequalities, objective function, constraints, feasible region, ordered triple Substitution Method Elimination Method Students will be able to: Solve linear systems by graphing Solve linear systems by substitution and elimination method Solve linear inequalities graphically Solve linear programming problems Solve systems of linear equations in three variables Solve application problems involving two or three variables Resources Core Text: Advanced Mathematical Concepts: Pre calculus with Applications, Holiday, Cuevas, McClure, (2001) McGraw Hill/Glencoe Suggested Resources:

6 Unit 3: The Nature of Graphs Content Area: Mathematics Course & Grade Level: Algebra & Trigonometry, Summary and Rationale Algebra and Trigonometry is a course designed for those students who have successfully completed Algebra II or have experienced difficulty in Advanced Algebra. This course provides the essential concepts and skills of algebra, trigonometry, and the study of functions, which are needed for further study of mathematics. Algebra topics include: advanced factoring, complex numbers, and quadratic relations and systems. In addition to a study of general functions, linear, polynomial, rational, logarithmic, exponential and trigonometric functions are also studied. Problem solving and graphing techniques are stressed throughout the course. Calculators and computers will be used throughout the course. 25 days Standard 4.F IF Interpreting Functions Recommended Pacing State Standards 7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. b. Graph linear and quadratic functions and show intercepts, maxima, and minima. Standard 4.A REI Reasoning with Equations and Inequalities 5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. 6 Solve systems of linear equations exactly and approximately (e.g. with graphs), focusing onpairs of linear equations in two variables. 10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). 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 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

7 Unit Essential Questions How do you graph functions, relations, inverses and inequalities? How do you analyze families of graphs? How do you use symmetry, continuity, and end behavior to help you graph? How do you find asymptotes and extrema of functions? Objectives Students will know: Terms: absolute maximum/minimum, relative maximum/minimum, asymptote, horizontal/vertical/slant asymptote, continuous, discontinuous, critical point, decreasing, increasing function, end behavior, horizontal line test, even function, odd function, parent graph, point of inflection Horizontal Line Test Students will be able to: Algebraic tests to determine whether a graph of a relation is symmetrical Classify functions as even or odd Identify transformations of simple graphs Sketch graphs of related functions Graph polynomial, absolute value, and radical inequalities in two variables Solve absolute value inequalities Determine inverses of relations and functions Graph functions and their inverses Determine whether a function is continuous or discontinuous Identify the end behavior of functions Determine whether a function is increasing or decreasing on an interval Construct and graph functions with gap discontinuities Find the extreme of a function Graph rational functions Determine vertical, horizontal, and slant asymptotes Resources Core Text: Advanced Mathematical Concepts: Precalculus with applications, Holiday, Cuevas, McClure, (2001) McGraw Suggested Resources: www. amc.glencoe.com

8 Unit 4: Polynomial and Rational Functions Content Area: Mathematics Course & Grade Level: Algebra & Trigonometry, Summary and Rationale Algebra and Trigonometry is a course designed for those students who have successfully completed Algebra II or have experienced difficulty in Advanced Algebra. This course provides the essential concepts and skills of algebra, trigonometry, and the study of functions, which are needed for further study of mathematics. Algebra topics include: advanced factoring, complex numbers, and quadratic relations and systems. In addition to a study of general functions, linear, polynomial, rational, logarithmic, exponential and trigonometric functions are also studied. Problem solving and graphing techniques are stressed throughout the course. Calculators and computers will be used throughout the course. 20 days Standard 4.F IF Interpreting Functions Recommended Pacing State Standards 1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. 2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. 7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Standard 4.A APR Arithmetic with Polynomials and Rational Expressions 1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. 4 Prove polynomial identities and use them to describe numerical relationships. 5 Know and apply the Binomial Theorem for the expansion of (x + y)squared in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal s Triangle. 6 Rewrite simple rational expressions in different forms: write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. Instructional Focus Unit Enduring Understandings 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

9 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 The concept of zeros, intercepts, and solutions to equations all reflect the same mathematical idea 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? How does the degree of the polynomial impact the family of graphs? Objectives Students will know: Terms: complex number, conjugate, degree, depressed polynomial, discriminant, extraneous solution, Fundamental Theorem of Algebra, leading coefficient, quadratic formula, radical equations and inequalities, rational equations, synthetic division, zero Students will be able to: Determine roots of polynomial equations Solve quadratic, rational, and radical equations, and rational and radical inequalities. Find the factors of polynomials Approximate real zeros of polynomial functions Resources Core Text: Advanced Mathematical Concepts: Pre calculus with Applications, Holiday, Cuevas, McClure, (2001) McGraw Hill/Glencoe Suggested Resources:

10 Unit 5: Trigonometric Functions Content Area: Mathematics Course & Grade Level: Algebra & Trigonometry, Summary and Rationale Algebra and Trigonometry is a course designed for those students who have successfully completed Algebra II or have experienced difficulty in Advanced Algebra. This course provides the essential concepts and skills of algebra, trigonometry, and the study of functions, which are needed for further study of mathematics. Algebra topics include: advanced factoring, complex numbers, and quadratic relations and systems. In addition to a study of general functions, linear, polynomial, rational, logarithmic, exponential and trigonometric functions are also studied. Problem solving and graphing techniques are stressed throughout the course. Calculators and computers will be used throughout the course. 25 days Recommended Pacing 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 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: angle of elevation and depression, coterminal angle, sine, cosine, tangent, cosecant, secant,

11 cotangent, Law of Sines and Cosines, reference angles, standard position, unit circle Students will be able to: Convert decimal degree measures to degrees, minutes, and seconds and vice versa Identify angles that are coterminal with a given angle Find values of trigonometric functions Solve triangles Find the area of triangles Resources Core Text: Advanced Mathematical Concepts: Pre calculus with Applications, Holiday, Cuevas, McClure, (2001) McGraw Hill/Glencoe Suggested Resources:

12 Unit 6: Graphs of Trigonometric Functions Content Area: Mathematics Course & Grade Level: Algebra & Trigonometry, Summary and Rationale Algebra and Trigonometry is a course designed for those students who have successfully completed Algebra II or have experienced difficulty in Advanced Algebra. This course provides the essential concepts and skills of algebra, trigonometry, and the study of functions, which are needed for further study of mathematics. Algebra topics include: advanced factoring, complex numbers, and quadratic relations and systems. In addition to a study of general functions, linear, polynomial, rational, logarithmic, exponential and trigonometric functions are also studied. Problem solving and graphing techniques are stressed throughout the course. Calculators and computers will be used throughout the course. 15 days Standard 4.F TF Trigonometric Functions Recommended Pacing State Standards 5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. 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. 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. 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: amplititude, central angle, midline, period, periodic, phase shift, radian, sector Students will be able to:

13 Change from radian measure to degree measure, and vice versa Use and draw graphs of trigonometric functions Find the amplitude, the period, the phase shift, and the vertical shift for trigonometric functions Resources Core Text: Advanced Mathematical Concepts: Pre calculus with Applications, Holiday, Cuevas, McClure, (2001) McGraw Hill/Glencoe Suggested Resources:

14 Unit 7: Trigonometric Identities and Equations Content Area: Mathematics Course & Grade Level: Algebra & Trigonometry, Summary and Rationale Algebra and Trigonometry is a course designed for those students who have successfully completed Algebra II or have experienced difficulty in Advanced Algebra. This course provides the essential concepts and skills of algebra, trigonometry, and the study of functions, which are needed for further study of mathematics. Algebra topics include: advanced factoring, complex numbers, and quadratic relations and systems. In addition to a study of general functions, linear, polynomial, rational, logarithmic, exponential and trigonometric functions are also studied. Problem solving and graphing techniques are stressed throughout the course. Calculators and computers will be used throughout the course. 20 days Standard 4.F TF Trigonometric Functions Recommended Pacing State Standards 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. 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. 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. Instructional Focus Unit Enduring Understandings 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 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?

15 Objectives Students will know: Terms: identity, trigonometric identity Students will be able to: Use reciprocal, quotient, and Pythagorean identities Verify trigonometric identities Solve trigonometric equations Resources Core Text: Advanced Mathematical Concepts: Precalculus with applications, Holiday, Cuevas, McClure, (2001) McGraw Hill/Glencoe Suggested Resources:

16 Unit 8: Exponential & Logarithmic Functions Content Area: Mathematics Course & Grade Level: Algebra & Trigonometry, Summary and Rationale Algebra and Trigonometry is a course designed for those students who have successfully completed Algebra II or have experienced difficulty in Advanced Algebra. This course provides the essential concepts and skills of algebra, trigonometry, and the study of functions, which are needed for further study of mathematics. Algebra topics include: advanced factoring, complex numbers, and quadratic relations and systems. In addition to a study of general functions, linear, polynomial, rational, logarithmic, exponential and trigonometric functions are also studied. Problem solving and graphing techniques are stressed throughout the course. Calculators and computers will be used throughout the course. 22 days Standard 4. A SSE Seeing Structure in Expressions Recommended Pacing 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 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. b. Use the properties of exponents to interpret expressions for exponential functions. Instructional Focus Unit Enduring Understandings 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

17 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: common logarithm, exponential growth and decay, logarithm, natural logarithm 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 Apply exponential growth models, decay models, and logarithmic models to solve real life problems Fit exponential and logarithmic models to sets of data Resources Core Text: Advanced Mathematical Concepts: Pre calculus with Applications, Holiday, Cuevas, McClure, (2001) McGraw Hill/Glencoe Suggested Resources:

18 Unit 9: Sequences and Series Content Area: Mathematics Course & Grade Level: Algebra & Trigonometry, Summary and Rationale Algebra and Trigonometry is a course designed for those students who have successfully completed Algebra II or have experienced difficulty in Advanced Algebra. This course provides the essential concepts and skills of algebra, trigonometry, and the study of functions, which are needed for further study of mathematics. Algebra topics include: advanced factoring, complex numbers, and quadratic relations and systems. In addition to a study of general functions, linear, polynomial, rational, logarithmic, exponential and trigonometric functions are also studied. Problem solving and graphing techniques are stressed throughout the course. Calculators and computers will be used throughout the course. 20 days Standard 4.A SSE Seeing Structure in Expressions Recommended Pacing 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 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 Applications of quadratic functions include acceleration, trajectory, and normal distributions 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? How are quadratic functions different from linear functions?

19 Objectives Students will know: Terms: arithmetic mean, arithmetic sequences and series, common difference and ratio, geometric mean, geometric sequences and series Students will be able to: Identify and find nth terms of arithmetic and geometric sequences. Find the sums of arithmetic and geometric series Resources Core Text: Advanced Mathematical Concepts: Pre calculus with Applications, Holiday, Cuevas, McClure, (2001) McGraw Hill/Glencoe Suggested Resources: