curriculum for college-prep precalculus · interpreting functions interpret functions that arise in...

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This curriculum is part of the Educational Program of Studies of the Rahway Public Schools.

ACKNOWLEDGMENTS

Christine H. Salcito, Director of Curriculum and Instruction Kevin Robinson, Program Supervisor of STEM

The Board acknowledges the following who contributed to the preparation of this curriculum.

Pamela Cook

Subject/Course Title: Date of Board Adoptions:

College Prep Precalculus September 18, 2012 Grade 11

RAHWAY PUBLIC SCHOOLS CURRICULUM

UNIT OVERVIEW

UNIT ONE

Content Area: Algebra

Unit Title: Equations and Inequalities

Target Course/Grade Level: Honors Precalculus/ 11th Grade

Unit Summary: Unit 1 develops students’ understandings and skills related to equations and inequalities. The unit will emphasize the relationship between solutions, roots, zeros, and x-intercepts. Students will use the principles of algebraic equivalence as well as graphical techniques, to solve equations and inequalities. Techniques for solving application problems will be emphasized. Approximate Length of Unit: 23 days

Primary interdisciplinary connections: Science, Geometry, Statistics, Business, Family/Consumer Science, Industrial Arts, Physical Education, Social Studies, Language Arts

LEARNING TARGETS

Common Core State Standards

Content Area Domain Content Area Cluster Standard

Seeing Structure in Expressions Write expressions in equivalent forms to solve problems A-SSE.3

Arithmetic with Polynomials and Rational

Expressions

Use polynomial identities to solve problems A-APR.4, A-APR.5

Creating Equations Create equations that describe numbers or relationships A-CED.1, A-CED.2

Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the reasoning

A-REI.1, A-REI.2

Reasoning with Equations and Inequalities Solve equations and inequalities in one variable A-REI.2, A-REI.3, A-REI.4

Reasoning with Equations and Inequalities Represent and solve equations and inequalities graphically A-REI.10, A-REI.11

English Language Arts Standards: Science and Technical Subjects

Key Ideas and Details RST.11-12.2


Craft and Structure RST.11-12.4, RST.11-12.6

21st Century Workplace Readiness Standards

Standard/Strand Cumulative Progress Indicator CPI#

Personal Financial Literacy/Money Management Compare strategies for saving and investing and the factors that influence how much should be saved or invested to meet financial goals.

9.2.12.B.2

Personal Financial Literacy/Credit and Debt Management

Compare and compute interest and compound interest. 9.2.12.C.2

21st Century Career and Technical Education/Science, Technology, Engineering & Mathematics Career Cluster

Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, engineering, and mathematics problems.

9.4.12.O.(1).1


Develop an understanding of how science and mathematics function to provide results, answers, and algorithms for engineering activities to solve problems and issues in the real world.

9.4.12.O.(2).1

Unit Understandings

Students will understand that… • roots or solutions of equations are represented graphically as zeros or x-intercepts • the quadratic equation can have 0, 1, or 2 solutions, as determined by the discriminant and found with algebraic or graphical methods • inequalities can be expressed graphically or with interval notation or inequality statements • the basic principles of algebraic equivalence can be used to solve equations of varying forms and levels of difficulty • polynomial identities can be applied to solve quadratic or polynomial equations • real life problems can frequently be expressed as mathematical statements and solved in context

Unit Essential Questions

• How are the algebraic solutions to equations or inequalities represented graphically? • How are the principles of algebraic equivalence used to solve equations or inequalities? • How are verbal statements in an application problem translated to mathematical statements in order to solve the problem?

Knowledge and Skills

• Vocabulary – o Intersection method o x-Intercept method o equivalent statements o extraneous solutions o interval notation

Students will be able to…

• solve linear equations including those involving algebraic fractions • solve quadratic equations by taking the square root, completing the square, factoring and the zero-product property, and the quadratic formula • use substitution or polynomial identities to solve polynomial equations • solve radical equations, including those with two radicals, and identify extraneous solutions • solve rational equations and check for domain restrictions • solve linear and quadratic absolute value equations • solve linear and quadratic inequalities and express the solution set in interval notation • use graphical methods, both the x-intercept and the intersection methods, to solve a variety of equations • solve applications problems involving mixtures, distance, work, interest, geometry, and numbers

EVIDENCE OF LEARNING

Assessment What evidence will be collected and deemed acceptable to show that students truly “understand”?

• Unit tests and quizzes • Open-ended problems that involve written responses • Daily student work • Daily Homework • Long term projects

Learning Activities

What differentiated learning experiences and instruction will enable all students to achieve the desired results? • Individualized practice and instruction using Khanacademy.org videos and exercises • Cooperative learning opportunities • Laying The Foundation lesson: Interval Notation

RESOURCES Teacher Resources:

• Precalculus Math Textbook: Teachers’ Edition This unit covers sections 2.1, 2.2, 2.3, 2.4, and 2.5 in Hungerford et al, Precalculus, a Graphing Approach, HRW Publisher, © 2007

• ExamView Pro Assessment Generator • Teacher developed worksheets and activities • Access to www.layingthefoundation.org lessons, assessments, and activities

Equipment Needed:

• Graphing calculators • Internet access for all students


UNIT OVERVIEW

UNIT TWO

Content Area: Functions

Unit Title: Functions and Graphs


Unit Summary: Unit 2 develops students’ understandings and skills related to functions. Students will learn to analyze, combine, evaluate, and transform functions. This unit emphasizes the connection between various representations of functions. Students’ skills related to inverse functions will be strengthened. The concepts of rate of change and the difference quotient will be developed and explored.

Approximate Length of Unit: 27 days


LEARNING TARGET




Apply the concepts, processes, guiding principles, and standards of school mathematics to solve science, engineering, and mathematics problems

9.4.12.O.(1).1


Develop and understanding of how science and mathematics function to provide results, answers, and algorithms for engineering activities to solve problems and issues in the real world.

9.4.12.O.(2).1


Interpreting Functions Understand the concept of a function and use function

notation

F-IF 1, F-IF 2

Interpreting Functions Interpret functions that arise in applications in terms of the context

F-IF.4, 5, 6

Interpreting Functions Analyze functions using different representations F-IF.9

Building Functions Build a function that models a relationship between two

quantities

F-BF.1

Building Functions Build new functions from existing functions F-BF.3, F-BF.4





Unit Understandings

Students will understand that… • a function consists of a set of inputs, called the domain, a rule by which each input determines one and only one output, and a set of outputs,

called the range • a function can be represented in a variety of ways, including but not limited to an equation, a graph, a table, a map, or a verbal description • combined functions can be formed from the addition, subtraction, multiplication, division, or composition of one or more functions • a family of functions consists of a parent function and its transformations • the inverse of a function results from the exchange of input and output values or variables • the average rate of change can be represented by the slope of a secant line • the difference quotient can be used to find the average rate of change


• How do transformations modify the graphs of a parent function, regardless of the function or the representation (equation, graph, or table)? • How are functions and their inverses related? • What is the relationship between the difference quotient of a non-linear function and the slope of a linear function?


Students will know… • Vocabulary –

o extrema o points of inflection o concavity o even/odd behavior o one-to-one functions


• determine if a relation between two variables is a function and, if so, state the domain and range of the function • analyze the graph of a function, including extrema and increasing/decreasing behavior, points of inflection and concavity, and end behavior • write, graph, and evaluate piecewise functions • graph parent functions and their transformations • determine algebraically and graphically if a relation between two variables has x-axis symmetry, y-axis symmetry or origin symmetry, and

whether the function is even or odd • construct, graph, and evaluate combined functions through addition, subtraction, multiplication, division, or composition, and find their domains

and ranges • use tables or graphs to create and evaluate the composition of functions • interpret, graph, and evaluate inverse relations from given functions • determine if a function is one-to-one and restrict the domain appropriately to ensure that the function is one-to-one • prove that two functions are inverses algebraically and graphically • find and interpret the average rate of change of various functions from tables, graphs, and equations • find the difference quotient of a function and use the difference quotient to determine average and instantaneous rates of change at a given point


Assessment

What evidence will be collected and deemed acceptable to show that students truly “understand”? • Unit tests, quizzes, quarterly formative assessments • Open-ended problems that involve written responses • Daily student work • Daily Homework • Long term projects

Learning Activities

What differentiated learning experiences and instruction will enable all students to achieve the desired results? • Individualized practice and instruction using Khanacademy.org videos and exercises • Cooperative learning opportunities • Laying The Foundation lessons: Generic Function, Transformations in Numerical Data, Composition of Functions Graphically

RESOURCES

Teacher Resources:

• Precalculus Math Textbook: Teachers’ Edition. This unit covers sections 3.1, 3.2, 3.4, 3.4A, 3.5, 3.6, and 3.7 in Hungerford et al, Precalculus, a Graphing Approach, HRW Publisher, © 2007


Equipment Needed:


http://www.layingthefoundation.org/


UNIT OVERVIEW

UNIT THREE


Unit Title: Linear, Quadratic, Polynomial, and Rational Functions


Unit Summary: Unit 3 further develops students’ understanding and skills related to linear, polynomial, and rational functions. Students will develop a clear understanding of the impact the degree of a function has on its characteristics. They will use different forms of a quadratic function to analyze and describe that function. Students will strengthen their ability to develop mathematical models and use them to make predictions.



LEARNING TARGETS





9.4.12.O.(1).1



9.4.12.O.(2).1


Interpreting Functions Analyze functions using different representations F-IF.7, F-IF.8

Linear, Quadratic, and Exponential Models Construct and compare linear, quadratic, and exponential models and solve problems

F-LE.1

Linear, Quadratic, and Exponential Models Interpret expressions for functions in terms of the situation they model

F-LE.5





Unit Understandings

Students will understand that… • linear functions grow at a constant rate and that the rate of change of a quadratic, polynomial, or rational function is never constant • the end behavior of a function is described as |x| approaches infinity • quadratic functions exhibit symmetry and the vertex, axis of symmetry, and real zeros of a quadratic function are related • the maximum number of zeros, extrema, and points of inflection are determined by the degree of a function • mathematical models can be used to demonstrate the relationship between input and output variables, and to make predictions • the big-little concept can be used to understand the domain exclusions in rational functions result in a discontinuity represented by either a hole

or a vertical asymptote

Unit Essential Questions • What is the behavior of a function as |x| approaches positive or negative infinity and how is that represented as a horizontal, slant, or parabolic

asymptote? • How can different forms of a quadratic function be useful in analyzing the function, and how can one form be transformed to another? • How do domain restrictions manifest in the graphs of a rational function? • How does the degree of a polynomial influence its shape and the number of zeros, extrema, or points of inflection?



o Big-Little Concept o vertical asymptote o horizontal, slant, parabolic asymptote o Finite differences o Least-squares regression line

Students will be able to… • graph linear, quadratic, polynomial, and rational functions and describe key features of the graph, including intercepts, asymptotes, and lines of

symmetry • identify the complete graphs of polynomial functions • define multiplicity and identify and describe its effect on the graph of a polynomial function • use the big-little concept to describe why and when the domain restrictions of a rational function result in a vertical asymptote or a hole in the

graph • use polynomial division to find slant and parabolic asymptotes • use technology, such as a graphing calculator, to graph and analyze a function • describe the range and domain of a function from its graph or its equation • write a quadratic function in different forms: polynomial, factored, transformation, and intercept; and analyze the function using these different

forms • develop a mathematical model to represent data and make predictions


Assessment What evidence will be collected and deemed acceptable to show that students truly “understand”?

• Unit tests, quizzes, quarterly formative assessments • Open-ended problems that involve written responses • Daily student work • Daily Homework • Long term projects

Learning Activities

What differentiated learning experiences and instruction will enable all students to achieve the desired results? • Individualized practice and instruction using Khanacademy.org videos and exercises • Cooperative learning opportunities • Laying The Foundation lesson: Characteristics of Functions


• Precalculus Math Textbook: Teachers’ Edition. This unit covers sections 3.3, 4.1 (Definition only), 4.3, 4.3A, and 4.4 in Hungerford et al, Precalculus, a Graphing Approach, HRW Publisher, © 2007


Equipment Needed:




UNIT OVERVIEW

UNIT FOUR


Unit Title: Exponential and Logarithmic Functions


Unit Summary: In this unit, students will develop their knowledge and skills related to radicals, rational exponents, and exponential functions. They will then study common and natural logarithms, including their properties and laws. Students will learn to solve exponential and logarithmic equations and use exponential and logarithmic models to describe data.


Primary interdisciplinary connections: Science, Geometry, Business, Family/Consumer Science, Industrial Arts, Physical Education, Social Studies, Language Arts

LEARNING TARGETS



Personal Financial Literacy/Money

Management

Compare strategies for saving and investing and the factors that influence how much should be saved or invested to meet financial goals.

9.2.12.B.2

Personal Financial Literacy/Credit and Debt Management

Compare and compute interest and compound interest. 9.2.12.C.2



9.4.12.O.(1).1



9.4.12.O.(2).1


Interpreting Functions Analyze functions using different representations F-IF.7, F-IF.8

Building functions Build new functions from existing functions F-BF.5

Linear, Quadratic, and Exponential Models Construct and compare linear, quadratic, and exponential models and solve problems

F-LE.1, F-LE.2, F-LE.3, F-LE.4

Linear, Quadratic, and Exponential Models Interpret expressions for functions in terms of the situation they model

F-LE.5





Unit Understandings

Students will understand that… • nth roots can be written as radicals • the rules of exponents apply equally to exponents that are integers or are rational or, irrational, or contain variables • logarithms are a special kind of exponent • the base “e” is used for natural logarithms and the base “10”, for common logarithms • every statement about logarithms is equivalent to a statement about exponents • common and natural logarithms share the same properties • the laws of exponents and the laws of logarithms are related and derive from the inverse nature of the relationship • the graphs of exponential and logarithmic functions reflect the inverse nature of their relationship • exponential and logarithmic graphs can be transformed in the same manner as any other parent function • logarithmic equations can be solved by exponentiation • exponential equations can be solved by taking the logarithm of both sides • exponential and logarithmic functions are used to model many naturally occurring phenomena


• What is the relationship between the domain and range of exponential and logarithmic functions? • How is the algebraic concept of equivalence applied to solve exponential and logarithmic equations? • How are the laws and properties of logarithms used to re-write expressions and solve equations? • How are inverse operations used to solve exponential and logarithmic equations, in the same way that squaring is used to “un-do” a square root?



o nth root o rational exponent o common logarithm o natural base “e” o natural logarithm o exponentiation o radioactive decay o compound interest


• evaluate nth roots and recognize the nth root as a rational exponent • use the laws of exponents to simplify expressions involving rational exponents or nth roots • simplify, evaluate, and perform arithmetic operations on nth roots and expressions with radical exponents • rationalize numerators or denominators to equivalent fractions • graph exponential and logarithmic functions and describe the key features of these graphs, including domain, range, intercepts, and asymptotes • transform the graphs of exponential and logarithmic functions: graph transformed functions and write the functions of transformed graphs • solve exponential equations • evaluate common and natural logarithms and write equivalent exponential and logarithmic equations • use the properties and laws of logarithms to re-write logarithmic expressions:

o combine the sum, difference, or multiple of logarithms into a single logarithm o write a single logarithmic expression into the sum, difference, and multiple of logarithms

• solve common and natural logarithmic equations • solve exponential and logarithmic application problems, including finance, population growth, and radioactive decay


Assessment


Learning Activities

What differentiated learning experiences and instruction will enable all students to achieve the desired results? • Individualized practice and instruction using Khanacademy.org videos and exercises • Cooperative learning opportunities • Laying The Foundation lessons: Exponential and Natural Logarithmic Functions, Exponential Growth and Decay

RESOURCES

Teacher Resources:

• Precalculus Math Textbook: Teachers’ Edition. This unit covers sections 5.1, 5.2, 5.3, 5.4, 5.5, and 5.6 in Hungerford et al, Precalculus, a Graphing Approach, HRW Publisher, © 2007


Equipment Needed:




UNIT OVERVIEW

UNIT FIVE Content Area: Functions and Geometry

Unit Title: Trigonometry


Unit Summary: Unit 5 develops students’ understandings and skills related to trigonometry and trigonometric graphs. The unit will emphasize use of the unit circle, angles of rotation, and the trigonometric functions of angles in the coordinate plane. Students will develop skills involving coteminal and reference angles, arc length and angular motion. Fundamental Trigonometric Identities will be introduced.



LEARNING TARGETS






9.4.12.O.(1).1



9.4.12.O.(2).1


Interpreting Functions Analyze functions using different representations F-IF.7

Trigonometric Functions Extend the domain of trigonometric functions using

the unit circle

F-TF.1, F-TF.2, F-TF.3, F-TF.4

Similarity, Right Triangles, and

Trigonometry

Define trigonometric ratios and solve problems involving right triangles

G-SRT.8





Unit Understandings

Students will understand that… • Similar triangles and trigonometric ratios can be used to solve real-world applications as well as solving right triangles • Angles of rotation in the coordinate plane can be used to determine the trigonometric functions of any real number • The unit circle is the basis for the most useful description of trigonometric functions and is developed with special right triangles • Trigonometric identities describe the relationships between trigonometric functions • New or unproven trigonometric identities can be proven with established trigonometric identities


• How is the unit circle developed with standard right triangles? • How are the trigonometric functions of points in the coordinate plane determined? • How are the trigonometric functions of angles and their reference angles related? • How are the three Pythagorean Identities developed?


• Vocabulary – o Radian o Angular Velocity o Coterminal Angles o Unit Circle o Reference Angle o Pythagorean Identities o Reciprocal Identities o Quotient Identities o Periodicity Identities o Negative Angle Identities


• use trigonometric functions and inverse trigonometric functions to solve a right triangle • solve applications problems involving right triangles • define radian measure and convert between degrees and radians • use a rotating ray to extend the definition of angle measure to negative angles and angles greater than 180o • find positive and negative coterminal angles • use radian measure to solve problems involving arc length or angular velocity • develop the unit circle with special right-triangles • use reference angles to find the trigonometric ratios of any angle in the coordinate plane, including exact values from the unit circle • find the trigonometric functions defined when the terminal side of an angle passes through a given point • find the trigonometric functions defined when the terminal side of an angle is described by a parallel or perpendicular line • use the unit circle and reference angles to find the trigonometric functions of any real number • use the unit circle to find the exact values of the trigonometric functions of standard angles in any quadrant • describe the domain and range of sine, cosine, and tangent functions • develop fundamental trigonometric identities • use fundamental trigonometric identities to prove other identities • use fundamental identities to find one trigonometric function, given another


Assessment

What evidence will be collected and deemed acceptable to show that students truly “understand”? • Unit tests and quizzes • Open-ended problems that involve written responses • Daily student work • Daily Homework • Long term projects

Learning Activities

What differentiated learning experiences and instruction will enable all students to achieve the desired results? • Individualized practice and instruction using Khanacademy.org videos and exercises • Cooperative learning opportunities


• Precalculus Math Textbook: Teachers’ Edition. This unit covers all of Chapter 6 in Hungerford et al, Precalculus, a Graphing Approach, HRW Publisher, © 2007. Chapter 6 is covers sequentially.


Equipment Needed:




UNIT OVERVIEW

UNIT SIX

Content Area: Functions and Geometry

Unit Title: The Graphs of Trigonometric Functions


Unit Summary: Unit 6 develops students’ understandings and skills related to trigonometric graphs.



LEARNING TARGETS






9.4.12.O.(1).1



9.4.12.O.(2).1

Unit Understandings

Students will understand that… • Trigonometric functions are periodic • The periodicity of trigonometric functions results in an infinite number of solutions to a trigonometric equations • Trigonometric graphs can be transformed like the graph of any other parent function


• Why are there an infinite number of values in the domain of a trigonometric function that correspond to a given value of the range? • How are the transformations of trigonometric functions similar to the transformations of other functions? • How can the domain and range of a trigonometric function be recognized from its graph?


Interpreting Functions Analyze functions using different representations F-IF.7, F-TF.8






• Vocabulary – o Periodicity o Amplitude o Frequency o Phase Shift


• use the unit circle to develop graphs of the sine, cosine, and tangent functions • describe the properties of sine, cosine, and tangent functions including domain, range, period, zeros, extrema, and even/odd behavior • state all values of a basic trigonometric function that correspond to a given value of the range • find all the t-values of a sine, cosine, or tangent function, using the function’s periodicity to fully state all the values • graph basic trigonometric functions and write a trigonometric function from its graph • state the amplitude, period, phase shift, and vertical displacement of the graph of a trigonometric function from its equation or graph • use “windowpane pane graphing” to graph a trigonometric function given its amplitude, period, phase shift, and vertical displacement


Assessment

What evidence will be collected and deemed acceptable to show that students truly “understand”? • Unit tests and quizzes • Open-ended problems that involve written responses • Daily student work • Daily Homework • Long term projects

Learning Activities

What differentiated learning experiences and instruction will enable all students to achieve the desired results? • Individualized practice and instruction using Khanacademy.org videos and exercises • Cooperative learning opportunities • Laying The Foundation lesson: Window Pane Graphing of Trigonometric Functions


• Precalculus Math Textbook: Teachers’ Edition. This unit covers sections 7.1and 7.3 in Hungerford et al, Precalculus, a Graphing Approach, HRW Publisher, © 2007. Window Pane Graphing is used to graph transformations of the basic trigonometric graphs


Equipment Needed:




UNIT OVERVIEW

UNIT SEVEN

Content Area: Geometry

Unit Title: Analytical Geometry: Conic Sections


Unit Summary: In this unit, students will define circles, ellipses, hyperbolas, and parabolas in terms of points and distances. Writing equations, translating conics, and graphing conics will be emphasized. Applications of conic sections will be presented.


Primary interdisciplinary connections: Science, Geometry, Business, Family/Consumer Science, Industrial Arts, Physical Education, Social Studies, Language Arts

LEARNING TARGETS





9.4.12.O.(1).1



9.4.12.O.(2).1

Unit Understandings

Students will understand that… • circles, ellipses, hyperbolas, and parabolas all derive from conic sections and have related characteristics • conic sections are described in terms of the coordinate plane and distance. • the standard equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 can be used to describe a point, one or two lines, a circle, ellipse, hyperbola, or

parabola, where B2 – 4AC is the discriminant and identifies the type of conic • the equations for conic sections are used in many real world applications, such as astronomy, mirrors and optics, and location systems • the simplest equation for a conic section has its center at the origin and the translations of conic sections are similar to translations of functions


Express Geometric Properties with

Equations

Translate between the geometric description and the equation for a conic section

G-GPE.1, G-GPE.2, G-GPE.3






• How are circles, ellipses, hyperbolas, and parabolas related? • How are foci, major and minor axis, directrix and asymptotes used to graph conic sections? • How can the shape of a translated conic be determined without graphing?



o conic section, degenerate conic section o ellipse o hyperbola o foci, focal axis o distance difference o verticies o major and minor axis o directrix o auxiliary rectangle


• define, write equations, and graph the following conics: circles, ellipses, parabolas, and hyperbolas o identify a conic from its graph or equation o graph a conic from its equation o write the equation of a conic from its graph or its critical points o write the equation of a translated conic and graph a translated conic from its equation

• solve application problems, such as those encountered in astronomy, optics, and architecture, involving conic sections.


Assessment


Learning Activities

What differentiated learning experiences and instruction will enable all students to achieve the desired results? • Individualized practice and instruction using Khanacademy.org videos and exercises • Cooperative learning opportunities • Laying The Foundation lessons


• Precalculus Math Textbook: Teachers’ Edition. This unit covers sections A.5 (circles only), 11.1, 11.2, 11.3,, and 11.4 in Hungerford et al, Precalculus, a Graphing Approach, HRW Publisher, © 2007.


Equipment Needed:



curriculum for college-prep precalculus · interpreting functions interpret functions that arise in...

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