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South Plainfield Public Schools Curriculum Guide Mathematics Honors Algebra 2 Authors: Anthony Emmons John Greco Curriculum Coordinator: Paul C. Rafalowski Board Approved on: June 13, 2012 1 Table of Contents South Plainfield Public Schools Board of Education Members and Administration Page: 3 Recognitions Page: 4 District Mission Statement Page: 5 Index of Courses Page: 6 Curriculum Guides Page: 7-45 Mathematics Practice Standards Page: 46-49 Common Core Standards Page: 50-57 Resources for State Assessments Page: 58 2 Members of the Board of Education Jim Giannakis, President Debbie Boyle, Vice President Carol Byrne John T. Farinella, Jr Christopher Hubner Sharon Miller William Seesselberg Joseph Sorrentino Gary Stevenson Central Office Administration Dr. Stephen Genco, Superintendent of Schools Dr. Frank Cocchiola, Interim Assistant Superintendent of Schools Mr. James Olobardi, Board Secretary/ BA Mrs. Laurie Hall, Supervisor of Student Personnel Services Mr. Vincent Parisi, Supervisor of Math and Science Mrs. Marlene Steele, Supervisor of Transportation Mrs. Annemarie Stoeckel, Supervisor of Technology Ms. Elaine Gallo, Director of Guidance Mr. Al Czech, Director of Athletics Mr. Paul Rafalowski, Curriculum Coordinator 3 Recognitions The following individuals are recognized for their support in developing this Curriculum Guide: Grade/Course Writer(s) Kindergarten: Ms. Joy Czaplinski and Ms. Pat Public Grade 1: Ms. Patti Schenck-Ratti, Ms. Kim Wolfskeil and Ms. Nicole Wrublevski Grade 2: Ms. Cate Bonanno, Ms. Shannon Colucci and Ms. Maureen Wilson Grade 3: Ms. Cate Bonanno and Ms. Theresa Luck Grade 4: Ms. Linda Downey and Ms. Kathy Simpson Grade 5: Mr. John Orfan and Ms. Carolyn White Grade 6: Ms. Joanne Haus and Ms. Cathy Pompilio Grade 7: Ms. Marianne Decker and Ms. Kathy Zoda Grade 8: Ms. Marianne Decker and Ms. Donna Tierney Algebra 1: Ms. Donna Tierney and Ms. Kathy Zoda Geometry: Mr. Anthony Emmons and Ms. Kathy Zoda Algebra 2: Mr. Anthony Emmons and Mr. John Greco Algebra 3/Trigonometry: Ms. Anu Garrison and Mr. David Knarr Senior Math Applications: Mr. John Greco Pre-Calculus: Ms. Anu Garrison and Mr. David Knarr Calculus: Mr. David Knarr Supervisors: Supervisor of Mathematics and Science: Mr. Vince Parisi Curriculum Coordinator: Mr. Paul C. Rafalowski Supervisor of Technology: Ms. Annemarie Stoeckel 4 South Plainfield Public Schools District Mission Statement To ensure that all pupils are equipped with essential skills necessary to acquire a common body of knowledge and understanding; To instill the desire to question and look for truth in order that pupils may become critical thinkers, life-long learners, and contributing members of society in an environment of mutual respect and consideration. It is the expectation of this school district that all pupils achieve the New Jersey Core Curriculum Content Standards at all grade levels. Adopted September, 2008 NOTE: The following pacing guide was developed during the creation of these curriculum units. The actual implementation of each unit may take more or less time. Time should also be dedicated to preparation for benchmark and State assessments, and analysis of student results on the same. A separate document is included at the end of this curriculum guide with suggestions and resources related to State Assessments (if applicable). The material in this document should be integrated throughout the school year, and with an awareness of the State Testing Schedule. It is highly recommended that teachers meet throughout the school year to coordinate their efforts in implementing the curriculum and preparing students for benchmark and State Assessments in consideration of both the School and District calendars. 5 Index of Mathematics Courses Elementary Schools (Franklin, Kennedy, Riley, Roosevelt) Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grant School Grade 5 Grade 6 Honors Grade 6 Middle School Grade 7 Honors Grade 7 Grade 8 Honors Grade 8 Grade 8 Algebra 1 High School Algebra 1 Academic Algebra 1 Honors Algebra 1 Geometry Academic Geometry Honors Geometry Algebra 2 Academic Algebra 2 Honors Algebra 2 Algebra 3/Trigonometry Senior Math Applications Pre-Calculus Honors Pre-Calculus Calculus Calculus AB Calculus BC 6 South Plainfield Public Schools Curriculum Guide Content Area: Mathematics Course Title: Honors Algebra 2 Grade Level: 9-11 Unit 1: Operations and Solving Linear Equations 3 Weeks Unit 2: Linear Functions & Inequalities 3 Weeks Unit 3: Systems of Equations & Inequalities 3 Weeks Unit 4: Quadratic Functions & Equations 3 Weeks Unit 5: Polynomial Functions & Equations 3 Weeks Unit 6: Sequences, Series, & Matrices 3.5 Weeks Board Approved on: June 13, 2012 7 South Plainfield Public Schools Curriculum Guide Content Area: Mathematics Course Title: Honors Algebra 2 Grade Level: 9-11 Unit 7: Probability & Statistics 3.5 Weeks Unit 8: Rational Functions & Equations 3.5 Weeks Unit 9: Powers, Roots, & Radicals 3.5 Weeks Unit 10: Exponential & Logarithmic Functions 3.5 Weeks Unit 11: Trigonometric Functions 3.5 Weeks Board Approved on: June 13, 2012 8 Unit 1 Overview Content Area – Mathematics Unit Title 1: Operations and Solving Linear Equations Target Course/Grade Level – Honors Algebra 2 for Grades 9 through 11 Unit Summary and Rationale – Students will be able to follow the order of operations; identify like- and non-like terms; apply inverse operations to solve linear equations and inequalities algebraically; graph and solve single-variable absolute value equations Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended response questions. Technology standard(s) – Scientific Calculators When Deemed Absolutely Necessary, Overhead, Smart Board 21st Century Themes – Financial Literacy 21st Century Skills – Critical Thinking & Problem Solving Communication & Collaboration Productivity & Accountability Initiative & Self-Direction Learning Targets Practice Standards: MP.1: Make sense of problems and persevere in solving them. MP.2: Reason abstractly and quantitatively. MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision. MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning. Domain Standards: A.CED.1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions A.CED.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. A.SSE.1: Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. 9 A.REI.1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. (Review from Algebra 1) A.REI.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. (Review from Algebra 1) F.IF.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 Unit Essential Questions What is absolute value and how can it be applied to global situations? How does modeling equations facilitate problem solving? Terminology: Variable Inverse Operations Solution Set Unit Enduring Understandings Order of Operations to Evaluate/Simplify Order of Operations in Reverse to Solve Goals/Objectives Students will be able to solve single variable linear equations recognize appropriate use of inverse operations Recommended Learning Activities/Instructional Strategies construct absolute value graphs for simple equations and compound equations and inequalities (single variable) Instructional Strategies Whole Group Lessons Small Group Explorations - In-class group work Independent Practice - Homework Class Discussion Activities 1.Textbook Activities 2.Group Math Tutor – Strong Student Teaching Cooperatively With Students of Varied Levels of Understanding 3.Think Pair Share Evidence of Learning (Formative & Summative) Quiz – Apply Order of Operations Correctly to Simplify Various Algebraic Expression Quiz – Apply Order of Operations and Knowledge of Inverse Operations to Solve Linear Equations Test – Cumulative Assessment of First Two Quizzes Alternative or projectbased assessments will be evaluated using a teacherselected or created rubric or other instrument 10 Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers. Resources - Glencoe McGraw Hill Algebra 2, 2010 11 Unit 2 Overview Content Area – Mathematics Unit 2: Linear Functions & Inequalities Target Course/Grade Level – Honors Algebra 2 for Grades 9 through 11 Unit Summary and Rationale – Students should be able to understand and interpret characteristics of linear functions using a table of values, a graph, and/or function rule. Specific to Linear Functions, students should be able to identify intercepts and slope/rate-of-change, as well as differentiate between vertical & horizontal, positive & negative slopes. Additionally, students will be able to graph linear inequalities and have the understanding that each and every ordered pair in the shaded region represents a solution to the inequality. Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended response questions. Technology standard(s) – Scientific Calculators When Deemed Absolutely Necessary, Overhead, Smart Board 21st Century Themes – Financial Literacy 21st Century Skills – Critical Thinking & Problem Solving Communication & Collaboration Productivity & Accountability Initiative & Self-Direction Learning Targets Practice Standards: MP.1: Make sense of problems and persevere in solving them. MP.2: Reason abstractly and quantitatively. MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision. MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning. 12 Domain Standards: A.SSE.1: Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. A.CED.1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions A.CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A.CED.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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. A.REI.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. (Review from Algebra 1) A.REI.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). (Review from Algebra 1) A.REI.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. (Review from Algebra 1) F.IF.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. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). (Review from Algebra 1) F.IF.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 F.IF.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 F.BF.3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Unit Essential Questions What is slope and what does it represent? How can linear functions be used to model and interpret real-world applications? Unit Enduring Understandings The slope of a linear function is the same between any two points on the line. Slopes of Parallel Lines are Equal; Slopes of Perpendicular Lines are Opposite Reciprocals Key Concepts/Vocabulary/Terminology/Skills: Solution Set, Shaded Region, Slope, Rate of Change, Parallel, Perpendicular, Horizontal, Vertical 13 Goals/Objectives Students will be able to - Recommended Learning Activities/Instructional Strategies create a table of values from a function rule plot ordered pairs on the coordinate plane and sketch a graph of the function 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 Instructional Strategies Whole Group Lessons Small Group Explorations - In-class group work Independent Practice - Homework Class Discussion Activities 1.Textbook Activities 2.Group Math Tutor – Strong Student Teaching Cooperatively With Students of Varied Levels of Understanding 3.Think Pair Share Evidence of Learning (Formative & Summative) Quiz – graphing linear functions, writing equations of lines Quiz – using test points, solving linear inequalities algebraically, graphing linear inequalities Test – Cumulative Assessment on both quizzes Alternative or projectbased assessments will be evaluated using a teacher-selected or created rubric or other instrument graph a function given the rule in slope-intercept form and standard form write the equation of a line in point-slope or slope-intercept form given multiple descriptions of the line graph solutions to a twovariable linear inequality Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers. Resources - Glencoe McGraw Hill Algebra 2, 2010 14 Unit 3 Overview Content Area – Mathematics Unit 3: Systems of Equations & Inequalities Target Course/Grade Level – Honors Algebra 2 for Grades 9 through 11 Unit Summary and Rationale – Students will be able to construct a system of equations and distinguish between appropriate methods of solving systems: develop a cost and income equation using linear programming and apply to global economy: selecting correct coordinates to check inequalities and their graphs; graph and solve two-variable absolute value inequalities Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended response questions. Technology standard(s) – Graphing calculators to illustrate intersecting points: Scientific Calculators When Deemed Absolutely Necessary, Overhead, SMART Board 21st Century Themes –Financial, Economic, Business, and Entrepreneurial Literacy Global Awareness 21st Century Skills – Creativity and Innovation, Critical Thinking and Problem Solving, Communication and Collaboration Productivity & Accountability Initiative & Self-Direction Learning Targets Practice Standards: MP.1: Make sense of problems and persevere in solving them. MP.2: Reason abstractly and quantitatively. MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision. MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning. 15 Domain Standards: N.Q.2: Define appropriate quantities for the purpose of descriptive Modeling (Review from Algebra 1) A.CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales A.CED.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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods A.REI.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 (Review from Algebra 1) A.REI.6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables (Review from Algebra 1) A.REI.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). (Review from Algebra 1) A.REI.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.★ A.REI.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. (Review from Algebra 1) Unit Essential Questions What is the best way to solve a linear system? Why would a business owner need to understand linear programming? Why would inequalities be more useful than equations in illustrating systems of inequalities? Unit Enduring Understandings Ability to solve systems of equations in multiple ways To be able to interpret inequalities and what the graph mean Use linear programming to illustrate business profit and loss Terminology: Systems, Linear Programming, Substitution, Elimination 16 Goals/Objectives Students will be able to: Recommended Learning Activities/Instructional Strategies solve two linear equations with multiple variables Instructional Strategies Whole Group Lessons Small Group Explorations - In-class group work Independent Practice - Homework Class Discussion Activities 1.Textbook Activities 2.Group Math Tutor – Strong Student Teaching Cooperatively With Students of Varied Levels of Understanding 3.Student based lesson- Have students decipher word problem and explain set up of equations to other students evaluate output values in cost/expense ratio formulas sketch inequalities in two variables and appropriately identify solutions Evidence of Learning (Formative & (Summative) Quiz- Apply different methods to solving systems of equations Quiz- Graph systems of inequalities and use knowledge to evaluate linear programming max and min values Test- Cumulative assessment f first two quizzes Alternative or projectbased assessments will be evaluated using a teacher-selected or created rubric or other instrument Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers. Resources - Glencoe McGraw Hill Algebra 2, 2010 17 Unit 4 Overview Content Area – Mathematics Unit 4: Quadratic Functions & Equations Target Course/Grade Level – Honors Algebra 2 for Grades 9 through 11 Unit Summary and Rationale – Students should be able to differentiate between linear and quadratic functions, as well as be able to choose the more appropriate type to model a real world problem. Students will use the vertex of a parabola to determine the maximum/minimum, and interpret its coordinates in the context of a real world problem. Also in this unit, students will extend the domain and range to the set of complex numbers, and apply their understanding of number properties and operations to this set. Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended response questions. Technology standard(s) – Scientific Calculators When Deemed Absolutely Necessary, Overhead, Smart Board, and Graphing Calculators can be used by the teacher, via overhead, to demonstrate the effects that changing the values of a b and c (in the quadratic function) have on the graph. 21st Century Themes – 21st Century Skills – Financial, Economic, Business, and Entrepreneurial Critical Thinking & Problem Solving Literacy, Environmental Literacy Communication & Collaboration Productivity & Accountability Initiative & Self-Direction Learning Targets Practice Standards: MP.1: Make sense of problems and persevere in solving them. MP.2: Reason abstractly and quantitatively. MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision. MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning. 18 Domain Standards: N.CN.1: Know there is a complex number i such that i2 = −1, and every complex number has the form a + bi with a and b real N.CN.2: Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers N.CN.3: Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers N.CN.7: Solve quadratic equations with real coefficients that have complex solutions N.CN.8: Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i) N.CN.9: Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials A.SSE.1: Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. A.SSE.2: Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2) (x2 + y2). (Review from Algebra 1) A.SSE.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. For example, calculate mortgage payments F.IF.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 F.IF.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. b. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. F.IF.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. F.IF.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum F.BF.3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Unit Essential Questions How can the properties of a quadratic function be used to model, analyze, and solve real world problems? Unit Enduring Understandings The graph of a quadratic function is a parabola. A table of values can always be used to graph a function. 19 Why are relations and functions represented in multiple ways? Solution sets to both quadratic equations and inequalities can be displayed using a graph of the function. Key Concepts/Vocabulary/Terminology/Skills: Increasing, Decreasing, Average Rate of Change, Vertex, Axis of Symmetry, Quadratic Formula, Roots, Factors, Zeros, Parabola, Square Root Property Goals/Objectives Students will be able to solve quadratic equations using inverse operations (square root property) solve quadratic equations by factoring and by applying the quadratic formula graph quadratic functions by completing a table of values 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 Recommended Learning Activities/Instructional Strategies Instructional Strategies Whole Group Lessons Small Group Explorations - In-class group work Independent Practice - Homework Class Discussion Activities 1.Textbook Activities 2.Group Math Tutor – Strong Student Teaching Cooperatively With Students of Varied Levels of Understanding 3.Think Pair Share 4. Written Paper detailing the different methods for solving a quadratic equation, discussing the pros/cons, likes/dislikes, etc. simplify and perform operations on complex numbers 20 Evidence of Learning (Formative & Summative) Quiz – solving quadratic equations Quiz – graphing quadratic functions Test – Cumulative Assessment on first two quizzes Alternative or projectbased assessments will be evaluated using a teacher-selected or created rubric or other instrument Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers. Resources- Glencoe McGraw Hill Algebra 2, 2010 21 Unit 5 Overview Content Area – Mathematics Unit 5: Polynomial Functions & Equations Target Course/Grade Level – Honors Algebra 2 for Grades 9 through 11 Unit Summary and Rationale – Students will be able to identify polynomial equations by name and identify specific terms in a polynomial: interpret the fundamental theorem of algebra to solve higher order polynomial equations: factor higher order polynomial functions using known patterns Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended response questions. Technology standard(s) – Scientific Calculators When Deemed Absolutely Necessary, Overhead, SMART Board 21st Century Themes – Global awareness Financial, Economic, Business and Entrepreneurial Literacy 21st Century Skills – Creativity and Innovation Critical thinking and Collaboration Productivity & Accountability Initiative & Self-Direction Learning Targets Practice Standards: MP.1: Make sense of problems and persevere in solving them. MP.2: Reason abstractly and quantitatively. MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision. MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning. Domain Standards: A.SSE.1: Interpret expressions that represent a quantity in terms of its context.★ a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P (1+r)n as the product of P and a factor not depending on P. A.SSE.2: Use the structure of an expression to identify ways to rewrite it. For example: see x4 – y4 22 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2) (x2 + y2). A.APR.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 A.APR.2: Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p (a), so p (a) = 0 if and only if (x – a) is a factor of p(x). A.APR.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. A.APR.4: Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples. A.APR.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 F.BF.1: Write a function that describes a relationship between two quantities.★ a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. c. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h (t) is the height of a weather balloon as a function of time, then T (h (t)) is the temperature at the location of the weather balloon as a function of time. Unit Essential Questions How are operations with polynomials different than operations with real numbers? How do you determine the amount of solutions a given polynomial will have? Unit Enduring Understandings Demonstrate a knowledge of how to solve polynomial equations to different degrees Identify terms and degrees with the appropriate terminology Apply the remainder theorem appropriately and understand the concept behind the theorem Key Concepts/Vocabulary/Terminology/Skills: Polynomial, Fundamental Theorem of Algebra 23 Goals/Objectives Students will be able to organize polynomials in standard form and name by degree and term find possible solutions to polynomial equations and use irrational and imaginary root theorems to understand possibilities Recommended Learning Activities/Instructional Strategies Instructional Strategies Whole Group Lessons Small Group Explorations - In-class group work Independent Practice - Homework Class Discussion Activities solve polynomial equations using remainder theorem, factoring, and previously learned methods students will use end behavior to analyze polynomial functions and determine zeros perform operations on polynomials (addition, subtraction, multiplication, division, composition) determine domain and range of a polynomial function determine extrema of a function identify intervals of the domain in which a function is increasing/decreasing 1.Textbook Activities 2.Group Math Tutor – Strong Student Teaching Cooperatively With Students of Varied Levels of Understanding 3. Think-Pair-Share. Using synthetic division and the remainder theorem each individual pair tries different roots to find first root. Students are able to work independently while also using their partner to check work and find solutions at a quicker pace 4. Telephone – A student begins with a polynomial function, finds all of the roots. The roots are passed on to a second student, who uses the roots to find the ‘original function.’ The function is then passed on to a third student, who finds the roots all over again. This continues on with a set number of students, the object of the activity being to have the same function in the end that you had in the beginning. 5. Communication – A student seated with their back to the board describes a function to a student at the board. The object is for the seated student to give a sufficiently detailed explanation of the function graph so the student at the board is able to accurately sketch the same function. 24 Evidence of Learning (Formative & Summative) Quiz- Identify polynomial names and understanding synthetic division Quiz- Solve polynomial equations using remainder theorem and apply factoring methods to larger order polynomials Test- Cumulative assessment f first two quizzes Alternative or projectbased assessments will be evaluated using a teacher-selected or created rubric or other instrument Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers. Resources - Glencoe McGraw Hill Algebra 2, 2010 25 Unit 6 Overview Content Area – Mathematics Unit 6: Sequence, series, and matrices Target Course/Grade Level – Honors Algebra 2 for Grades 9 through 11 Unit Summary and Rationale – Students will be able to identify patterns in order to develop appropriate formulas and apply to certain problems. They will also be able to compute sums of series and use derived formulas to complete the task. Finally, students will be able to sketch and organize simple matrices and demonstrate an ability to complete simple mathematical operations using matrices Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended response questions. Technology standard(s) – Scientific Calculators When Deemed Absolutely Necessary, Overhead, SMART Board. Graphing calculator can be used by teacher to demonstrate how to enter a matrix into a graphing calculator 21st Century Themes – Financial, Economic, Business, and Entrepreneurial Literacy Environmental literacy 21st Century Skills – Critical Thinking & Problem Solving Communication & Collaboration Productivity & Accountability Initiative & Self-Direction Learning Targets Practice Standards: MP.1: Make sense of problems and persevere in solving them. MP.2: Reason abstractly and quantitatively. MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision. MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning. 26 Domain Standards: A.SSE.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. For example, calculate mortgage payments.★ F.IF.3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f (0) = f (1) = 1, f (n+1) = f (n) + f (n-1) for n ≥ 1. 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.★ N.VM.6: Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network (HSPA) N.VM.7: Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. (HSPA) N.VM.8: Add, subtract, and multiply matrices of appropriate dimensions (HSPA) Unit Essential Questions How do sequences and series model realworld problems and their solutions? How can matrices help visualize and organize data for use with other applications? Unit Enduring Understandings Sequences and series are discrete functions whose domain is the set of whole numbers Matrices are commonly used to organize data while comparing and contrasting information Terminology: Scalars, Commutative property, Sequence, Series, Infinite, Arithmetic, Geometric, Fibonacci Goals/Objectives Students will be able to sketch matrices of different sizes and organize data appropriately use matrix properties to add, subtract, and multiply by a scalar with matrices identify a sequence and design a formula based on the sequence evaluate sum of arithmetic and geometric series perform matrix Recommended Learning Activities/Instructional Strategies Instructional Strategies Whole Group Lessons Small Group Explorations - In-class group work Independent Practice - Homework Class Discussion Activities 1.Textbook Activities 2.Group Math Tutor – Strong Student Teaching Cooperatively With Students of Varied Levels of Understanding 3. Research. A) Find a newspaper article where matrices were used to compare and 27 Evidence of Learning (Formative & Summative) Quiz- Complete simple operations with matrices Quiz- Identify patterns and understand sequence and series Quiz- Recognize sum formulas and apply appropriately Test- Cumulative multiplication, understand it is not commutative, determine whether matrices are inverses (emphasize the recurring idea of inverses and identity elements) contrast data. B) Real-life series (Choose real-life situations which use arithmetic or geometric sequences and series. You must have at least one of each type for this project.) After data is acquired teacher can adapt project to fit idea in lesson and develop and change formulas. Also can use for computer programming if applicable. assessment of first three quizzes Alternative or projectbased assessments will be evaluated using a teacher-selected or created rubric or other instrument Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers. Resources - Glencoe McGraw Hill Algebra 2, 2010 28 Unit 7 Overview Content Area – Mathematics Unit 7: Probability & Statistics Target Course/Grade Level – Honors Algebra 2 for Grades 9 through 11 Unit Summary and Rationale – Students will be able to calculate probabilities, identify types of samples, find measures of central tendency, calculate standard deviation, make statistical decisions based on the normal curve, and use the binomial theorem to expand polynomials and calculate probability Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended response questions. Technology standard(s) – Scientific Calculators When Deemed Absolutely Necessary, Overhead, SMART Board. Graphing calculator can be used to demonstrate how calculator can obtain statistical data in a much simpler way 21st Century Themes – Financial, Economic, Business, and Entrepreneurial Literacy Global awareness 21st Century Skills – Critical Thinking & Problem Solving Communication & Collaboration Creativity and Innovation Productivity & Accountability Initiative & Self-Direction Learning Targets Practice Standards: MP.1: Make sense of problems and persevere in solving them. MP.2: Reason abstractly and quantitatively. MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision. MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning. 29 Domain Standards: A.APR.5: Know and apply the Binomial Theorem for the expansion of (x + y)n 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 S.ID.4: Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. S.IC.1: Understand statistics as a process for making Inferences about population parameters based on a random sample from that population. S.IC.2: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model? S.IC.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. S.IC.4: Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. S.IC.5: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. S.IC.6: Evaluate reports based on data S.MD.6: Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). S.MD.7: Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game Unit Essential Questions How can probability be applied in realworld situations and used to make predictions? How can the curve of best fit help predict trends of data? How can probability help make unemotional decisions? Unit Enduring Understandings More advanced concepts of probability can be used to solve real-word problems. Data sets can be analyzed to form hypotheses and make predictions using different types of statistical models Measures of central tendency can be understood more clearly and used to relate data accordingly Terminology: Mean, median, mode, standard deviation, z-score, probability, conditional probability, normal curve, outlier, Binomial Theorem 30 Goals/Objectives Students will be able to - Recommended Learning Activities/Instructional Strategies evaluate mean, median, and mode Instructional Strategies Whole Group Lessons Small Group Explorations evaluate standard - In-class group work deviation and make Independent Practice decisions using data - Homework calculate probabilities and Class Discussion understand the Activities ramifications of certain 1.Textbook Activities probabilities 2. Think-pair-share. Students should use use standard normal curve partners to check answers when calculating statistical parameters and assigning to approximate range of appropriately. data and identify outliers 3. Statistical experiment. Students should expand polynomials using conduct an experiment with appropriate sampling method to obtain data and then fit Binomial Theorem and data into a normal curve explaining how Pascal’s triangle values were obtained and the meaning of use Binomial Theorem to standard deviation. Students can present findings in a report or as a class project to the calculate binomial entire class. This may also be done as a group probabilities project 4. Group project. Students will use probability identify specific types of to obtain data to help explain why people samples and select make the decisions they do in real world appropriate sampling situations. Simple hypothesis testing may be method to obtain data introduced if time permits. Examples may be blackjack strategy, safety ratings on cars, baseball statistics, etc. Evidence of Learning (Formative & Summative) Quiz- Probability models and simple sampling Quiz- Standard deviation, the standard normal curve, and the Binomial theorem Test- Cumulative assessment of first two quizzes Alternative or projectbased assessments will be evaluated using a teacher-selected or created rubric or other instrument Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers. Resources - Glencoe McGraw Hill Algebra 2, 2010 31 Unit 8 Overview Content Area – Mathematics Unit 8: Rational Functions & Equations Target Course/Grade Level – Honors Algebra 2 for Grades 9 through 11 Unit Summary and Rationale – Building upon their understanding of other function types, students will identify key characteristics of rational functions such as asymptotes, domain, and range. Necessary skills include factoring polynomials as well as performing operations on fractions and polynomials. Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended response questions. Technology standard(s) – Scientific Calculators When Deemed Absolutely Necessary, Overhead, Smart Board. Graphing calculator can be used by the teacher to enhance the presentation and understanding of graphs of functions and the corresponding table of values. 21st Century Themes – Financial, Economic, Business, and Entrepreneurial Literacy Environmental Literacy 21st Century Skills – Critical Thinking & Problem Solving Communication & Collaboration Productivity & Accountability Initiative & Self-Direction Learning Targets Practice Standards: MP.1: Make sense of problems and persevere in solving them. MP.2: Reason abstractly and quantitatively. MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision. MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning. 32 Domain Standards: A.SSE.1: Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2) (x2 + y2). A.APR.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 A.APR.7: Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions A.REI.2: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise A.REI.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 F.IF.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 F.IF.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function F.IF.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 Unit Essential Questions How are the properties of functions and functional operations useful? How are rational functions related to rational numbers? Unit Enduring Understandings Performing Operations on Rational Expressions is Exactly Like Performing Operations on Fractions Not being able to divide by zero is the reason for restrictions in the domain of a rational function. Key Concepts/Vocabulary/Terminology/Skills: Asymptote, Limit, Removable and Non-Removable Discontinuity, Increasing, Decreasing, End Behavior, Domain, Range 33 Goals/Objectives Students will be able to extend knowledge of operating on fractions to operating on rational expressions Recommended Learning Activities/Instructional Strategies identify domain and range of a rational function factor polynomials to identify greatest common factor / common denominator students will reduce complex fractions and understand how a complex fraction is simplified graph rational expressions Instructional Strategies Whole Group Lessons Small Group Explorations - In-class group work Independent Practice - Homework Class Discussion Activities 1.Textbook Activities 2.Group Math Tutor – Strong Student Teaching Cooperatively With Students of Varied Levels of Understanding 3.Think Pair Share 4. Communication – A student seated with their back to the board describes a function to a student at the board. The object is for the seated student to give a sufficiently detailed explanation of the function graph so the student at the board is able to accurately sketch the same function. determine extrema of a function identify intervals of the domain in which a function is increasing/decreasing identify and understand occurrence of asymptotes, points of removable discontinuity (holes) understand why a function can cross a horizontal or slant asymptote but not a 34 Evidence of Learning (Formative & Summative) Quiz – Performing Operations on Rational Expressions, Solving Rational Equations Quiz – Graphing Rational Functions Test – Cumulative Assessment on first two quizzes Alternative or projectbased assessments will be evaluated using a teacher-selected or created rubric or other instrument vertical asymptote Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers. Resources - Glencoe McGraw Hill Algebra 2, 2010 35 Unit 9 Overview Content Area – Mathematics Unit 9: Powers, Roots, & Radicals Target Course/Grade Level – Honors Algebra 2 for Grades 9 through 11 Unit Summary and Rationale – Students will be able to evaluate and simplify expressions involving powers and radicals. They will have the understanding that powers and radicals have an inverse relationship and can be used to solve equations. Building upon their understanding of square roots, students will interpret, simplify, operate on, and solve equations involving radical expressions with indices greater than two. Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended response questions. Technology standard(s) – Scientific Calculators When Deemed Absolutely Necessary, Overhead, SMART Board. Graphing calculator can be used by teacher to demonstrate via graph and table of values the inverse relationship between powers and radicals. 21st Century Themes – Financial, Economic, Business, and Entrepreneurial Literacy Environmental Literacy 21st Century Skills – Critical Thinking & Problem Solving Communication & Collaboration Productivity & Accountability Initiative & Self-Direction Learning Targets Practice Standards: MP.1: Make sense of problems and persevere in solving them. MP.2: Reason abstractly and quantitatively. MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision. MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning. 36 Domain Standards: A.SSE.1: Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. A.SSE.2: Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2) (x2 + y2). A.APR.5: Know and apply the Binomial Theorem for the expansion of (x + y)n 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 A.REI.2: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise F.BF.3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. F.BF.4: Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x-1) for x ≠ 1. b. (+) Verify by composition that one function is the inverse of another. c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse Unit Essential Questions Unit Enduring Understandings What’s the most efficient way to simplify a radical expression? How are expressions and functions involving radicals and exponents related? What are the strengths and weaknesses of radical expressions? of exponential expressions? Power and radical functions add to our arsenal of inverse functions and operations and can be used to model and solve real world applications. Terminology: Prime Factorization, Exponent, Index Number, Radical, Radicand, Rational Exponent, Properties of Exponents 37 Goals/Objectives Students will be able to extend properties of integer exponents to rational exponents convert to and from exponential form/radical form, understand appropriateness and benefits of each form evaluate radical expressions solve radical equations Recommended Learning Activities/Instructional Strategies Instructional Strategies Whole Group Lessons Small Group Explorations - In-class group work Independent Practice - Homework Class Discussion Activities 1.Textbook Activities 2.Group Math Tutor – Strong Student Teaching Cooperatively With Students of Varied Levels of Understanding 3.Think Pair Share perform binomial expansions using Pascal’s Triangle Evidence of Learning (Formative & Summative) Quiz – simplifying exponential and radical expressions, binomial expansion Quiz – solving exponential and radical equations Test – Cumulative Assessment Alternative or projectbased assessments will be evaluated using a teacher-selected or created rubric or other instrument Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers. Resources - Glencoe McGraw Hill Algebra 2, 2010 38 Unit 10 Overview Content Area – Mathematics Unit 10: Exponential and Logarithmic Functions Target Course/Grade Level – Honors Algebra 2 for Grades 9 through 11 Unit Summary and Rationale – Students will be able to extend their knowledge of exponential functions from to include real number powers. Properties and graphs, including transformed graphs, of exponential functions are studied. The number e is explored as the natural base of exponential functions. Applications include exponential growth and decay, as well as modeling of exponential functions from data. Logarithmic functions are introduced as inverses of exponential functions. The properties and graphs of logarithmic functions are studied, including transformed graphs. Students solve exponential and logarithmic equations by equating bases and by using the inverse relationship between logarithms and exponents. Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended response questions. Technology standard(s) – Scientific Calculators When Deemed Absolutely Necessary, Overhead, SMART Board. Graphing calculator can be used by teacher to illustrate exponential and logarithmic graphs 21st Century Themes – Financial, Economic, Business, and Entrepreneurial Literacy Global awareness Health Literacy 21st Century Skills – Critical Thinking & Problem Solving Communication & Collaboration Creativity & Innovation Productivity & Accountability Initiative & Self-Direction Learning Targets Practice Standards: MP.1: Make sense of problems and persevere in solving them. MP.2: Reason abstractly and quantitatively. MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision. MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning. 39 Domain Standards: N.RN.1: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. (Review from Algebra 1) N.RN.2: Rewrite expressions involving radicals and rational exponents using the properties of exponents (Review from Algebra 1) F.IF.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 exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. F.IF.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay. F.LQE.3: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. F.LQE.4: For exponential models, express as a logarithm the solution to a bct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology Unit Essential Questions How do exponential functions model real-world problems and their solutions? How do logarithmic functions model real-world problems and their solutions? How are expressions involving exponents and logarithms related? Unit Enduring Understandings The characteristics of exponential and logarithmic functions and their representations are useful in solving real-world problems Translating equations from logarithmic form to exponential form can be a useful tool for completing appropriate operations Terminology: Logarithmic, Exponential, Interest, Decay, Growth, Asymptote, Natural Logs, Continuously Compounding Interest 40 Goals/Objectives Students will be able to model exponential growth and decay to introduce the number e to apply constants to the equation y=abx to evaluate logarithmic expressions to sketch exponential and logarithmic graphs to understand and apply the properties of logarithms to solve exponential equations to solve logarithmic equations to introduce and evaluate natural logarithms to solve equations using natural logarithms Recommended Learning Activities/Instructional Strategies Instructional Strategies Whole Group Lessons Small Group Explorations - In-class group work Independent Practice - Homework Class Discussion Activities 1.Textbook Activities 2. Think-pair-share. Students should use partners to check answers when calculating statistical parameters and assigning appropriately. 3. Think of one real situation that involves exponential growth and that involves exponential decay. for each example, your project should include the following: * Paragraph - Briefly explain the situation. You may make up your own information, but make it realistic. Include the facts needed to write an equation. * Equation - Model the situation with an exponential equation. * Graph - Make a graph of the exponential function. Be sure to label what each axis represents and use an appropriate scale. to explain inverse variation and use to solve equations solve real-life problems including but not limited to compound interest, half-life, logistic representations of population growth 41 Evidence of Learning (Formative & Summative) Quiz- Exponential growth/decay and the exponential equation Quiz- Log functions and solving exponential/logarithmi c equations Test- Cumulative assessment of first two quizzes Alternative or projectbased assessments will be evaluated using a teacher-selected or created rubric or other instrument Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers. Resources - Glencoe McGraw Hill Algebra 2, 2010 42 Unit 11 Overview Content Area – Mathematics Unit 11: Trigonometric Functions Target Course/Grade Level – Honors Algebra 2 for Grades 9 through 11 Unit Summary and Rationale – Introducing Pre-Calculus concepts, students will be introduced to the unit circle, radian/degree conversions, moving on to the concept of periodic and oscillating functions. The graphs of trigonometric functions will be explored and used to analyze and solve real world problems. They will also use their knowledge of 30-60-90’s and 45-45-90’s from geometry to prove trigonometric identities. Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended response questions. Technology standard(s) – Scientific Calculators When Deemed Absolutely Necessary, Overhead, Smart Board 21st Century Themes – Financial, Economic, Business, and Entrepreneurial Literacy, Environmental Literacy 21st Century Skills – Critical Thinking & Problem Solving Communication & Collaboration Productivity & Accountability Initiative & Self-Direction Learning Targets Content and Practice Standards: MP.1: Make sense of problems and persevere in solving them. MP.2: Reason abstractly and quantitatively. MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision. MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning. 43 Domain Standards: F.TF.1: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. F.TF.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 F.TF.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 π-x, π+x, and 2π-x in terms of their values for x, where x is any real number F.TF.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline F.TF.8: Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant Unit Essential Questions Why are triangles so important? How are trigonometric functions, Pythagorean Theorem, and other properties of triangles related? What natural phenomena can be modeled using trigonometric functions? Unit Enduring Understandings Definition of Sine Cosine and Tangent Reference Angles are based upon 30-60-90’s and 45-45-90’s and their sine and cosine values are reflected over the x- and y- axes to obtain the sine and cosine values in the unit circle. Triangle Similarity is the reason we are able to use trigonometric ratios to solve triangles. Key Concepts/Vocabulary/Terminology/Skills: Reference Angles, Unit Circle, Radian, Co-terminal Angles, Amplitude, Period, Midline, Frequency Goals/Objectives Students will be able to - Recommended Learning Activities/Instructional Strategies recall properties of special right triangles and use them to derive the unit circle Instructional Strategies using coordinates from the unit circle, extend trigonometric functions to all real numbers and graph trigonometric functions Whole Group Lessons Evidence of Learning (Formative & Summative) Quiz – derive the unit circle, identifying trig ratios Small Group Explorations - In-class group work Quiz – graphing trigonometric functions Independent Practice - Homework Test – Cumulative Assessment on Quizzes Class Discussion Alternative or project- 44 interpret key properties of trigonometric function Activities based assessments will be evaluated using a teacherselected or created rubric or other instrument 1.Textbook Activities 2.Group Math Tutor – Strong Student Teaching Cooperatively With Students of Varied Levels of Understanding 3.Think Pair Share Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include, but are not limited to, learning centers and cooperative learning activities in either heterogeneous or homogeneous groups, depending on the learning objectives and the number of students that need further support and scaffolding, versus those that need more challenge and enrichment. Modifications may also be made as they relate to the special needs of students in accordance with their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These may include, but are not limited to, extended time, copies of class notes, refocusing strategies, preferred seating, study guides, and/or suggestions from special education or ELL teachers. Resources - Glencoe McGraw Hill Algebra 2, 2010 45 Mathematics: Standards for Mathematical Practice The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy). 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not 46 just how to compute them; and knowing and flexibly using different properties of operations and objects. 3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, 47 recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. 8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when 48 expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. 49 Common Core Standards (Bold Apply to Algebra 2) The Complex Number System N-CN Perform arithmetic operations with complex numbers. 1. Know there is a complex number i such that i2 = −1, and every complex number has the form a + bi with a and b real. 2. Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 3. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. Represent complex numbers and their operations on the complex plane. 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. 5. (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1 + √3 i)3 = 8 because (-1 + √3 i) has modulus 2 and argument 120°. 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. Use complex numbers in polynomial identities and equations. 7. Solve quadratic equations with real coefficients that have complex solutions. 8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i). 9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Seeing Structure in Expressions A-SSE Interpret the structure of expressions. 1. Interpret expressions that represent a quantity in terms of its context.★ a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P. 2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). Write expressions in equivalent forms to solve problems. 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. 50 Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. 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. For example, calculate mortgage payments.★ b. Arithmetic with Polynomials and Rational Expressions A-APR Perform arithmetic operations on polynomials. 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. Understand the relationship between zeros and factors of polynomials. 2. Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). 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. Use polynomial identities to solve problems. 4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples. 5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n 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.1 Rewrite rational expressions. 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. 7. (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Reasoning with Equations and Inequalities A-REI Understand solving equations as a process of reasoning and explain the reasoning. 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 1 The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument. 51 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Solve equations and inequalities in one variable. 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 4. Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Solve systems of equations. 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. 7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. 8. (+) Represent a system of linear equations as a single matrix equation in a vector variable. 9. (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 3 or greater). Represent and solve equations and inequalities graphically. 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). 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. Trigonometric Functions F-TF Extend the domain of trigonometric functions using the unit circle. 1. 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. 52 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 π-x, π+x, and 2π-x in terms of their values for x, where x is any real number. 4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. Model periodic phenomena with trigonometric functions. 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.★ Prove and apply trigonometric identities. 8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant. 9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Creating Equations★ A-CED Create equations that describe numbers or relationships. 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. 4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. Interpreting Functions F-IF Understand the concept of a function and use function notation. 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. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). 2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. 3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. Interpret functions that arise in applications in terms of the context. 53 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.★ 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★ 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.★ Analyze functions using different representations. 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. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, 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. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay. 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Building Functions F-BF Build a function that models a relationship between two quantities. ★ 1. Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. 54 c. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. 2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.★ Build new functions from existing functions. 3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. 4. Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x-1) for x ≠ 1. b. (+) Verify by composition that one function is the inverse of another. c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain. 5. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Linear, Quadratic, and Exponential Models★ F-LQE Construct and compare linear, quadratic, and exponential models and solve problems. 1. Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. 2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). 3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. 4. For exponential models, express as a logarithm the solution to a bct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. Interpret expressions for functions in terms of the situation they model. 5. Interpret the parameters in a linear, quadratic, or exponential function in terms of a context. 55 Interpreting Categorical and Quantitative Data S-ID Summarize, represent, and interpret data on a single count or measurement variable. 1. Represent data with plots on the real number line (dot plots, histograms, and box plots). 2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. 3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). 4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. Summarize, represent, and interpret data on two categorical and quantitative variables. 5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. b. Informally assess the fit of a function by plotting and analyzing residuals. c. Fit a linear function for a scatter plot that suggests a linear association. Interpret linear models. 7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. 8. Compute (using technology) and interpret the correlation coefficient of a linear fit. 9. Distinguish between correlation and causation. Making Inferences and Justifying Conclusions S-IC Understand and evaluate random processes underlying statistical experiments. 1. Understand statistics as a process for making inferences to be made about population parameters based on a random sample from that population. 2. Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model? Make inferences and justify conclusions from sample surveys, experiments, and observational studies. 3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. 4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. 5. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. 6. Evaluate reports based on data. 56 Using Probability to Make Decisions S-MD Calculate expected values and use them to solve problems. 1. (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions. 2. (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. 3. (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes. 4. (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? Use probability to evaluate outcomes of decisions. 5. (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant. b. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident. 6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). 7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). Resources for State Assessments 57