Algebra 2 Pearson High School SY


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1 Chapters Total School Days Algebra 2 Pearson High School SY Benchmark Cycle1 Benchmark Cycle2 Benchmark Cycle 3 Cycle 4 September 5 October 31 BM Window Opens Nov 1 Term 1 ends Nov 13 November 1 January 26 BM Window Opens Jan 29 Keystone Exam Window Opens Jan 8 Term 2 ends Jan 29 January 29 April 6 BM Window Opens April 9 Term 3 ends Apr 9 April 9 June 12 Keystone Exam Window Opens May 14 Cycle 4 ends Jun Total Days Including 1 Half Day 51 Total Days Including 4 Half Days 46 Total Days Including 3 Half Days 46 Total Days Including 13 Half Day 1 Expressions, Equations, and Inequalities 1 4 Quadratic Functions and Equations 1 (44 to 49) 7 Exponential and Logarithmic Functions 1, 2 10 Quadratic Relations and Conic Sections BEC 2 Functions, Equations, and Graphs 1, 2 3 Linear Systems1, BEC 4 Quadratic Functions and Equations 1 (41 to 43) 5 Polynomials and Polynomial Functions 2 6 Radical Functions and Rational Exponents 1 8 Rational Function 1 9 Sequences and Series 2 11 Probability and Statistics 2, BEC 12 Matrices BEC 13 Periodic Functions and Trigonometry BEC 14 Trigonometric Identities and Equations BEC Notes A Benchmark (BM) Cycle is defined as the time allotted to teach the content on each benchmark, assuming the benchmark is taken on the first day of the window. This means that, even though you may give the benchmark later in the window, you should move into new content as of the first date in the BM Window or you will fall behind the suggested pacing. 1 Contains from Module 1 of the Algebra II Keystone Exam. 2 Contains from Module 2 of the Algebra II Keystone Exam. BEC Contains content that goes beyond the eligible content for the Algebra II Keystone Exam, i.e., material not included on the state blueprint. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
2 Table of s Block Schedule Options... 3 Benchmark Cycle 1 Standards... 4 Benchmark Cycle 1 Scope and Sequence... 7 Benchmark Cycle 2 Standards Benchmark Cycle 2 Scope and Sequence Benchmark Cycle 3 Standards Benchmark Cycle 3 Scope and Sequence Cycle 4 Standards Cycle 4 Scope and Sequence PA Core Standards by Cycle Algebra Taught Document Information Page THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
3 Full Block Spring Full Block Fall A / B Block Full Block Spring Full Block Fall A / B Block Block Schedule Options Note: With block schedules, generally follow the suggested sequence of lessons. Asterisked lessons are approximately 90minutes in length. Nonasterisked lessons (approximately 45minutes in length) may be combined, but consider combining them meaningfully, instead of simply teaching twolessons backtoback. When single nonasterisked lessons precede or follow an asterisked one, consider using Concept Bytes, Mathematical Modeling, or Common Core Performance Tasks to start or complete the lesson (using 45 available minutes to construct a full 90minute lesson). Benchmark Cycle 1 Benchmark Cycle 2 Benchmark Cycle 3 Cycle 4 Pacing Use the suggested pacing, but combine 45minute (nonasterisked) lessons, in accordance with the general note above. 9/5 10/2 Ch. 1 to Ch /3 10/31 Ch. 44 to Ch. 6 11/1 11/9 Ch. 7 11/13 12/8 Ch. 8 to Ch. 9, Ch /11 1/29 Ch. 10, Ch. 12 to Ch. 14 Remediation and Extension 1/30 2/27 Ch. 1 to Ch /28 3/28 Ch. 44 to Ch. 6 4/3 4/10 Ch. 7 Benchmark Schedule 4/11 5/4 Ch. 8 to Ch. 9, Ch. 11 5/7 6/12 Ch. 10, Ch. 12 to Ch. 14 Remediation and Extension Follow Benchmark schedule indicated in the Scope and Sequence document that follows this page. Benchmarks 1 and 2 (No need to complete in one sitting) None (Benchmark 3 not taken) Benchmarks 1 and 2 (No need to complete in one sitting) None (Benchmark 3 not taken) THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
4 Benchmark Cycle 1 Standards PA A Use exponential expressions to represent rational numbers. A Simplify/evaluate expressions involving positive and negative exponents and/or roots (may contain all types of real numbers exponents should not exceed power of 10). A Simplify/evaluate expressions involving multiplying with exponents (e.g., x 6 x 7 = x 13 ), powers of powers (e.g., (x 6 ) 7 = x 42 ), and powers of products (e.g., (2x 2 ) 3 = 8x 6 ). Note: Limit to rational exponents. A Simplify or evaluate expressions involving logarithms and exponents (e.g., log28 = 3 or log42 = ½). A Factor algebraic expressions, including difference of squares and trinomials. Note: Trinomials limited to the form ax 2 +bx+c where a is not equal to 0. A Simplify rational algebraic expressions. Write and/or solve quadratic equations (including factoring and using the Quadratic Formula). Solve equations involving rational and/or radical expressions 2 (e.g., 10/(x + 3) + 12/(x 2) = 1 or x 21x 14. Write and/or solve a simple exponential or logarithmic equation (including common and natural logarithms). PA Core Standards CC.2.1.HS.F.1 Apply and extend the properties of exponents to solve problems with rational exponents. CC.2.2.HS.D.2 Write expressions in equivalent forms to solve problems. CC.2.2.HS.D.1 Interpret the structure of expressions to represent a quantity in terms of its context. CC.2.2.HS.D.2 Write expressions in equivalent forms to solve problems. CC.2.2.HS.D.3 Extend the knowledge of arithmetic operations and apply to polynomials. CC.2.2.HS.D.4 Understand the relationship between zeros and factors of polynomials to make generalizations about functions and their graphs. CC.2.2.HS.D.5 Use polynomial identities to solve problems. CC.2.2.HS.C.2 Graph and analyze functions, and use their properties to make connections between the different representations. CC.2.2.HS.C.3 Write functions or sequences that model relationships between two quantities. CC.2.2.HS.C.4 Interpret the effects transformations have on functions, and find the inverses of functions. CC.2.2.HS.C.5 Construct and compare linear, quadratic, and exponential models to solve problems. CC.2.2.HS.C.6 Interpret functions in terms of the situations they model. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
5 PA PA Core Standards Write, solve, and/or apply linear or exponential growth or decay (including problem situations). A Determine how a change in one variable relates to a change in a second variable (e.g., y = 4/x; if x doubles, what happens to y?). A Use algebraic processes to solve a formula for a given variable (e.g., solve d = rt for r). A Analyze a set of data for the existence of a pattern, and represent the pattern with a rule algebraically and/or graphically. A Identify and/or extend a pattern as either an arithmetic or geometric sequence (e.g., given a geometric sequence, find the 20th term). A Determine the domain, range, or inverse of a relation. A Identify and/or determine the characteristics of an exponential, quadratic, or polynomial function (e.g., intervals of increase/decrease, intercepts, zeros, and asymptotes). CC.2.2.HS.D.5 Use polynomial identities to solve problems. CC.2.2.HS.D.6 Extend the knowledge of rational functions to rewrite in equivalent forms. CC.2.2.HS.D.7 Create and graph equations or inequalities to describe numbers or relationships. CC.2.2.HS.D.8 Apply inverse operations to solve equations or formulas for a given variable. CC.2.2.HS.D.9 Use reasoning to solve equations, and justify the solution method. CC.2.2.HS.D.10 Represent, solve and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically. CC.2.2.HS.C.2 Graph and analyze functions, and use their properties to make connections between the different representations. CC.2.2.HS.C.3 Write functions or sequences that model relationships between two quantities. CC.2.2.HS.C.4 Interpret the effects transformations have on functions, and find the inverses of functions. CC.2.2.HS.D.7 Create and graph equations or inequalities to describe numbers or relationships. CC.2.2.HS.D.8 Apply inverse operations to solve equations or formulas for a given variable. CC.2.2.HS.D.9 Use reasoning to solve equations, and justify the solution method. CC.2.1.HS.F.7 Apply concepts of complex numbers in polynomial identities and quadratic equations to solve problems. CC.2.2.HS.C.1 Use the concept and notation of functions to interpret and apply them in terms of their context. CC.2.2.HS.C.2 Graph and analyze functions, and use their properties to make connections between the different representations. CC.2.2.HS.C.3 Write functions or sequences that model relationships between two quantities. CC.2.2.HS.C.5 Construct and compare linear, quadratic, and exponential models to solve problems. CC.2.2.HS.C.6 Interpret functions in terms of the situations they model. CC.2.3.HS.A.10 Translate between the geometric description and the equation for a conic section. CC.2.4.HS.B.2 Summarize, represent, and interpret data on two categorical and quantitative variables. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
6 PA A Create, interpret, and/or use the equation, graph, or table of a polynomial function (including quadratics). A Create, interpret, and/or use the equation, graph, or table of an exponential or logarithmic function (including common and natural logarithms). A Determine, use, and/or interpret minimum and maximum values over a specified interval of a graph of a polynomial, exponential, or logarithmic function. A Translate a polynomial, exponential, or logarithmic function from one representation of a function to another (graph, table, and equation.) A Identify or describe the effect of changing parameters within a family of functions (e.g., y = x 2 and y = x 2 + 3, or y = x 2 and y = 3x 2 ). A Draw, identify, find, interpret, and/or write an equation for a regression model (lines and curves of best fit) for a scatter plot. A Make predictions using the equations or graphs of regression models (lines and curves of best fit) of scatter plots. PA Core Standards CC.2.1.HS.F.3 Apply quantitative reasoning to choose and interpret units and scales in formulas, graphs, and data displays. CC.2.1.HS.F.4 Use units as a way to understand problems and to guide the solution of multistep problems. CC.2.2.HS.C.3 Write functions or sequences that model relationships between two quantities. CC.2.2.HS.C.4 Interpret the effects transformations have on functions, and find the inverses of functions. CC.2.2.HS.C.5 Construct and compare linear, quadratic, and exponential models to solve problems. CC.2.2.HS.C.6 Interpret functions in terms of the situations they model. CC.2.2.HS.D.7 Create and graph equations or inequalities to describe numbers or relationships. CC.2.3.HS.A.10 Translate between the geometric description and the equation for a conic section. CC.2.2.HS.C.4 Interpret the effects transformations have on functions, and find the inverses of functions. CC.2.2.HS.C.5 Construct and compare linear, quadratic, and exponential models to solve problems. CC.2.2.HS.C.6 Interpret functions in terms of the situations they model. CC.2.1.HS.F.3 Apply quantitative reasoning to choose and interpret units and scales in formulas, graphs, and data displays. CC.2.1.HS.F.5 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. CC.2.4.HS.B.2 Summarize, represent, and interpret data on two categorical and quantitative variables. CC.2.4.HS.B.3 Analyze linear models to make interpretations based on the data. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
7 9/59/20: 1 Per Day + 6 Days 9/59/8 1/302/2 Benchmark Cycle 1 Scope and Sequence Chapter 1: Expressions, Equations, and Inequalities Title Topic(s) 11 Patterns and Expressions Identify and describe patterns. Reviews a A A Properties of Real Numbers Reviews Graph and order real numbers. Identify properties of real numbers. 13 Algebraic Expressions Evaluate algebraic expressions. Simplify algebraic expressions. 14 Solving Equations Solve equations. Solve problems by writing equations. 15* Solving Inequalities Solve and graph inequalities. Write and solve compound inequalities. A A A A A A A A A A A A a A Solve It! begins every lesson, providing students opportunities to strengthen their problemsolving skills. Students draw on prior knowledge and connect with key concepts from the lesson before refining solution methods. The Solve It! is an essential component of the program. * s marked with an asterisk may require two45minute class periods, for which flex days may be used. Note: Consider beginning the year by laying foundations for problemsolving via: WODB.ca, Estimation180.com, SolveMe. Also consider using flex days, on occasion, to complete or present the Common Core Performance Tasks and Mathematical Modeling in Three Acts (on Pearson Realize), which support increased rigor. Review foundational knowledge and check for understanding, as needed, through Do Nows or Bell Ringers, homework, and teachercreated assessments. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
8 (See previous) Chapter 1: Expressions, Equations, and Inequalities (cont.) Title Topic(s) 16* Absolute Value Equations and Write and solve equations and inequalities involving Inequalities absolute value. A A THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
9 9/2510/11: 1 Per Day + 5 Days 9/119/18 2/52/14 Chapter 2: Functions, Equations, and Graphs Title Topic 21* Relations and Functions Graph relations. Reviews Identify functions. A A A A A A Direct Variation Write and interpret direct variation equations. 23 Linear Functions and Slope Intercept Form Graph linear equations. Write equations of lines. 24* More About Linear Equations Write an equation of a line given tits slope and point on the line. 25 Using Linear Models Write linear equations that model realworld data Make predictions from linear models A A A A A A A A A A A A * Families of Functions Analyze transformations of functions 27* Absolute Value Functions and Graphs Graph absolute value functions. 28* TwoVariable Inequalities Graph twovariable inequalities A A A A A A THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
10 10/1210/23: 1 Per Day + 2 Days 9/199/26 2/152/21 Chapter 3:Linear Systems Title Topic 31* Solving Systems Using Tables Solve linear systems using a graph or a table. and Graphs 32* Solving Systems Algebraically Solve linear systems algebraically. 33 Systems of Inequalities Solve systems of linear inequalities. 34 Linear Programming Solve problems using linear programming. 35 BEC Systems with Three Variables Solve systems in three variables using elimination. To solve systems in three variables using substitution. 36 BEC Solving Systems Using Matrices CC Performance Task: Planning a Triathlon Represent a system of linear equations with a matrix. Solve a system of linear equations using matrices. A A A A A A THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
11 10/2010/31: 1 Per Day + 3 Days 9/2710/2 2/222/27 Chapter 4: Quadratic Functions and Equations Title Topic 41* Quadratic Functions and Identify and graph quadratic functions. Transformations 42* Standard Form of a Quadratic Function 43* Modeling With Quadratic Functions Graph quadratic functions written in standard form. Model data with quadratic functions. Benchmark 1 Window: Nov 1 Nov 17 A A A A A A A A A A A A THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
12 Benchmark Cycle 2 Standards PA A Simplify/write square roots in terms of i (e.g., 24 = 2i 6). A Simplify/evaluate expressions involving powers of i (e.g., i6 + i3 = 1 i). A Add and subtract complex numbers (e.g., (7 3i) (2 + i) = 5 4i). A Multiply and divide complex numbers (e.g., (7 3i)(2 + i) = 17 + i). A Use exponential expressions to represent rational numbers. A Simplify/evaluate expressions involving positive and negative exponents and/or roots (may contain all types of real numbers exponents should not exceed power of 10). A Simplify/evaluate expressions involving multiplying with exponents (e.g., x 6 x 7 = x 13 ), powers of powers (e.g., (x 6 ) 7 = x 42 ), and powers of products (e.g., (2x 2 ) 3 = 8x 6 ). Note: Limit to rational exponents. A Simplify or evaluate expressions involving logarithms and exponents (e.g., log28 = 3 or log42 = ½). A Factor algebraic expressions, including difference of squares and trinomials. Note: Trinomials limited to the form ax 2 +bx+c where a is not equal to 0. A Simplify rational algebraic expressions. PA Core Standards CC.2.1.HS.F.6 Extend the knowledge of arithmetic operations and apply to complex numbers. CC.2.1.HS.F.1 Apply and extend the properties of exponents to solve problems with rational exponents. CC.2.2.HS.D.2 Write expressions in equivalent forms to solve problems. CC.2.2.HS.D.1 Interpret the structure of expressions to represent a quantity in terms of its context. CC.2.2.HS.D.2 Write expressions in equivalent forms to solve problems. CC.2.2.HS.D.3 Extend the knowledge of arithmetic operations and apply to polynomials. CC.2.2.HS.D.4 Understand the relationship between zeros and factors of polynomials to make generalizations about functions and their graphs. CC.2.2.HS.D.5 Use polynomial identities to solve problems. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
13 PA Write and/or solve quadratic equations (including factoring and using the Quadratic Formula). Solve equations involving rational and/or radical expressions 2 (e.g., 10/(x + 3) + 12/(x 2) = 1 or x 21x 14. Write and/or solve a simple exponential or logarithmic equation (including common and natural logarithms). Write, solve, and/or apply linear or exponential growth or decay (including problem situations). A Determine how a change in one variable relates to a change in a second variable (e.g., y = 4/x; if x doubles, what happens to y?). A Use algebraic processes to solve a formula for a given variable (e.g., solve d = rt for r). A Analyze a set of data for the existence of a pattern, and represent the pattern with a rule algebraically and/or graphically. PA Core Standards CC.2.2.HS.C.2 Graph and analyze functions, and use their properties to make connections between the different representations. CC.2.2.HS.C.3 Write functions or sequences that model relationships between two quantities. CC.2.2.HS.C.4 Interpret the effects transformations have on functions, and find the inverses of functions. CC.2.2.HS.C.5 Construct and compare linear, quadratic, and exponential models to solve problems. CC.2.2.HS.C.6 Interpret functions in terms of the situations they model. CC.2.2.HS.D.5 Use polynomial identities to solve problems. CC.2.2.HS.D.6 Extend the knowledge of rational functions to rewrite in equivalent forms. CC.2.2.HS.D.7 Create and graph equations or inequalities to describe numbers or relationships. CC.2.2.HS.D.8 Apply inverse operations to solve equations or formulas for a given variable. CC.2.2.HS.D.9 Use reasoning to solve equations, and justify the solution method. CC.2.2.HS.D.10 Represent, solve and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically. CC.2.2.HS.C.2 Graph and analyze functions, and use their properties to make connections between the different representations. CC.2.2.HS.C.3 Write functions or sequences that model relationships between two quantities. CC.2.2.HS.C.4 Interpret the effects transformations have on functions, and find the inverses of functions. CC.2.2.HS.D.7 Create and graph equations or inequalities to describe numbers or relationships. CC.2.2.HS.D.8 Apply inverse operations to solve equations or formulas for a given variable. CC.2.2.HS.D.9 Use reasoning to solve equations, and justify the solution method. CC.2.1.HS.F.7 Apply concepts of complex numbers in polynomial identities and quadratic equations to solve problems. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
14 PA A Identify and/or extend a pattern as either an arithmetic or geometric sequence (e.g., given a geometric sequence, find the 20th term). A Determine the domain, range, or inverse of a relation. A Identify and/or determine the characteristics of an exponential, quadratic, or polynomial function (e.g., intervals of increase/decrease, intercepts, zeros, and asymptotes). A Create, interpret, and/or use the equation, graph, or table of a polynomial function (including quadratics). A Create, interpret, and/or use the equation, graph, or table of an exponential or logarithmic function (including common and natural logarithms). A Determine, use, and/or interpret minimum and maximum values over a specified interval of a graph of a polynomial, exponential, or logarithmic function. A Translate a polynomial, exponential, or logarithmic function from one representation of a function to another (graph, table, and equation.) PA Core Standards CC.2.2.HS.C.1 Use the concept and notation of functions to interpret and apply them in terms of their context. CC.2.2.HS.C.2 Graph and analyze functions, and use their properties to make connections between the different representations. CC.2.2.HS.C.3 Write functions or sequences that model relationships between two quantities. CC.2.2.HS.C.5 Construct and compare linear, quadratic, and exponential models to solve problems. CC.2.2.HS.C.6 Interpret functions in terms of the situations they model. CC.2.3.HS.A.10 Translate between the geometric description and the equation for a conic section. CC.2.4.HS.B.2 Summarize, represent, and interpret data on two categorical and quantitative variables. CC.2.1.HS.F.3 Apply quantitative reasoning to choose and interpret units and scales in formulas, graphs, and data displays. CC.2.1.HS.F.4 Use units as a way to understand problems and to guide the solution of multistep problems. CC.2.2.HS.C.3 Write functions or sequences that model relationships between two quantities. CC.2.2.HS.C.4 Interpret the effects transformations have on functions, and find the inverses of functions. CC.2.2.HS.C.5 Construct and compare linear, quadratic, and exponential models to solve problems. CC.2.2.HS.C.6 Interpret functions in terms of the situations they model. CC.2.2.HS.D.7 Create and graph equations or inequalities to describe numbers or relationships. CC.2.3.HS.A.10 Translate between the geometric description and the equation for a conic section. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
15 11/111/22: 1 Day Per + 8 Days 10/310/10 2/283/7 Mathematical Modeling c Concept Byte d Piles of Tiles Identifying Quadratic Data Benchmark Cycle 2 Scope and Sequence Estimate area. Model data with quadratic functions. 44* Factoring Quadratic Expressions Find common and binomial factors of quadratic expressions. Factor special quadratic expressions. 45* Algebra Review: Square Roots and Radicals Reviews Quadratic Equations Chapter 4: Quadratic Functions and Equations (cont.) Title Topic Solve quadratic equations by factoring Solve quadratic equations by graphing A A A A A A A A A A A A A A c Mathematical Modeling in Three Acts are activities that offer students rich opportunities to engage with the PA Core Standards for Mathematical Practice (SMPs). The focus of these activities is on the process of problemsolving and reasoning. d Concept Bytes are to be used either before or after lessons (as indicated) to build conceptual understanding, support development of problemsolving skills, to engage with the PA Core Standards for Mathematical Practice, and to integrate learning with technology (consider either TI calculators or Desmos for these). THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
16 (see above) Chapter 4: Quadratic Functions and Equations (cont.) Title Topic 46* Completing the Square Solve equations by completing the square Rewrite functions by completing the square 47* The Quadratic Formula Solve quadratic equations using the Quadratic Formula Determine the number of solutions by using the discriminant 48* Complex Numbers Identify, graph, and perform operations with complex numbers Find complex number solutions of quadratic equations 49* Quadratic Systems Solve and graph systems of linear and quadratic equations Solve and graph systems of quadratic inequalities A A A A A A A THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
17 11/2712/22: 1 Day Per + 10 Days 10/1110/19 3/83/16 Chapter 5: Polynomials and Polynomial Functions Title Topic 51* Polynomial Functions Classify polynomials. Graph polynomial functions and describe end behavior. 52* Polynomials, Linear Factors, and Zeros Analyze the factored form of a polynomial. Write a polynomial function from its zeros. 53* Solving Polynomial Equations Solve polynomial equations by factoring. Solve polynomial equations by graphing. 54* Dividing Polynomials Divide polynomials using long division. Divide polynomials using synthetic division. A A A A A A A A A A A A A A A A THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
18 (see above) Title 55* Theorems About Roots of Polynomial Equations 56* The Fundamental Theorem of Algebra Chapter 5: Polynomials and Polynomial Functions (cont.) Topic Solve equations using Rational Root Theorem. Use the Conjugate Root Theorem. Use the Fundamental Theorem to solve polynomial equations with complex solutions. A A A A A A The Binomial Theorem Expand a binomial using Pascal s Triangle. Use the Binomial Theorem. A A THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
19 (see above) Title 58 Polynomial Models in the Real World 59* Transforming Polynomial Functions Chapter 5: Polynomials and Polynomial Functions (cont.) CC Performance Task: Determining the Dimensions of a Diorama Topic Fit data to linear, quadratic, cubic, or quartic models. Apply transformations to graphs of polynomials. A A A A A A A A THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
20 1/31/26: 1 Day Per + 9 Days 10/2010/31 3/193/28 Chapter 6: Radical Functions and Rational Exponents Title Topic 61 Roots and Radical Expressions Find the nth root. 62* Multiplying and Dividing Radical Expressions Multiply and divide radical expressions. 63* Binomial Radical Expressions Add and subtract radical expressions. A A * Rational Exponents Simplify expressions with rational exponents. A A A A * Solving Square Root and Other Radical Equations Solve square root and other radical equations. A A A A A A A A A A THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
21 (see above) Chapter 6: Radical Functions and Rational Exponents (cont.) Title Topic 66* Function Operations Add, subtract, multiply, and divide functions. Find the composite of two functions. 67* Inverse Relations and Functions Find the inverse of a relation or function. 68* Graphing Radical Functions Graph square root and other radical functions. A A A A A A Benchmark 2 Window: Jan 29 Feb 13 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
22 Benchmark Cycle 3 Standards PA A Use exponential expressions to represent rational numbers. A Simplify/evaluate expressions involving positive and negative exponents and/or roots (may contain all types of real numbers exponents should not exceed power of 10). A Simplify/evaluate expressions involving multiplying with exponents (e.g., x 6 x 7 = x 13 ), powers of powers (e.g., (x 6 ) 7 = x 42 ), and powers of products (e.g., (2x 2 ) 3 = 8x 6 ). Note: Limit to rational exponents. A Simplify or evaluate expressions involving logarithms and exponents (e.g., log28 = 3 or log42 = ½). A Factor algebraic expressions, including difference of squares and trinomials. Note: Trinomials limited to the form ax 2 +bx+c where a is not equal to 0. A Simplify rational algebraic expressions. Write and/or solve quadratic equations (including factoring and using the Quadratic Formula). Solve equations involving rational and/or radical expressions 2 (e.g., 10/(x + 3) + 12/(x 2) = 1 or x 21x 14. Write and/or solve a simple exponential or logarithmic equation (including common and natural logarithms). PA Core Standards CC.2.1.HS.F.1 Apply and extend the properties of exponents to solve problems with rational exponents. CC.2.2.HS.D.2 Write expressions in equivalent forms to solve problems. CC.2.2.HS.D.1 Interpret the structure of expressions to represent a quantity in terms of its context. CC.2.2.HS.D.2 Write expressions in equivalent forms to solve problems. CC.2.2.HS.D.3 Extend the knowledge of arithmetic operations and apply to polynomials. CC.2.2.HS.D.4 Understand the relationship between zeros and factors of polynomials to make generalizations about functions and their graphs. CC.2.2.HS.D.5 Use polynomial identities to solve problems. CC.2.2.HS.C.2 Graph and analyze functions, and use their properties to make connections between the different representations. CC.2.2.HS.C.3 Write functions or sequences that model relationships between two quantities. CC.2.2.HS.C.4 Interpret the effects transformations have on functions, and find the inverses of functions. CC.2.2.HS.C.5 Construct and compare linear, quadratic, and exponential models to solve problems. CC.2.2.HS.C.6 Interpret functions in terms of the situations they model. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
23 PA PA Core Standards Write, solve, and/or apply linear or exponential growth or decay (including problem situations). A Determine how a change in one variable relates to a change in a second variable (e.g., y = 4/x; if x doubles, what happens to y?). A Use algebraic processes to solve a formula for a given variable (e.g., solve d = rt for r). A Analyze a set of data for the existence of a pattern, and represent the pattern with a rule algebraically and/or graphically. A Identify and/or extend a pattern as either an arithmetic or geometric sequence (e.g., given a geometric sequence, find the 20th term). A Determine the domain, range, or inverse of a relation. A Identify and/or determine the characteristics of an exponential, quadratic, or polynomial function (e.g., intervals of increase/decrease, intercepts, zeros, and asymptotes). CC.2.2.HS.D.5 Use polynomial identities to solve problems. CC.2.2.HS.D.6 Extend the knowledge of rational functions to rewrite in equivalent forms. CC.2.2.HS.D.7 Create and graph equations or inequalities to describe numbers or relationships. CC.2.2.HS.D.8 Apply inverse operations to solve equations or formulas for a given variable. CC.2.2.HS.D.9 Use reasoning to solve equations, and justify the solution method. CC.2.2.HS.D.10 Represent, solve and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically. CC.2.2.HS.C.2 Graph and analyze functions, and use their properties to make connections between the different representations. CC.2.2.HS.C.3 Write functions or sequences that model relationships between two quantities. CC.2.2.HS.C.4 Interpret the effects transformations have on functions, and find the inverses of functions. CC.2.2.HS.D.7 Create and graph equations or inequalities to describe numbers or relationships. CC.2.2.HS.D.8 Apply inverse operations to solve equations or formulas for a given variable. CC.2.2.HS.D.9 Use reasoning to solve equations, and justify the solution method. CC.2.1.HS.F.7 Apply concepts of complex numbers in polynomial identities and quadratic equations to solve problems. CC.2.2.HS.C.1 Use the concept and notation of functions to interpret and apply them in terms of their context. CC.2.2.HS.C.2 Graph and analyze functions, and use their properties to make connections between the different representations. CC.2.2.HS.C.3 Write functions or sequences that model relationships between two quantities. CC.2.2.HS.C.5 Construct and compare linear, quadratic, and exponential models to solve problems. CC.2.2.HS.C.6 Interpret functions in terms of the situations they model. CC.2.3.HS.A.10 Translate between the geometric description and the equation for a conic section. CC.2.4.HS.B.2 Summarize, represent, and interpret data on two categorical and quantitative variables. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
24 PA A Create, interpret, and/or use the equation, graph, or table of a polynomial function (including quadratics). A Create, interpret, and/or use the equation, graph, or table of an exponential or logarithmic function (including common and natural logarithms). A Determine, use, and/or interpret minimum and maximum values over a specified interval of a graph of a polynomial, exponential, or logarithmic function. A Translate a polynomial, exponential, or logarithmic function from one representation of a function to another (graph, table, and equation.) A Identify or describe the effect of changing parameters within a family of functions (e.g., y = x 2 and y = x 2 + 3, or y = x 2 and y = 3x 2 ). A Draw, identify, find, interpret, and/or write an equation for a regression model (lines and curves of best fit) for a scatter plot. A Make predictions using the equations or graphs of regression models (lines and curves of best fit) of scatter plots. A Use combinations, permutations, and the fundamental counting principle to solve problems involving probability. PA Core Standards CC.2.1.HS.F.3 Apply quantitative reasoning to choose and interpret units and scales in formulas, graphs, and data displays. CC.2.1.HS.F.4 Use units as a way to understand problems and to guide the solution of multistep problems. CC.2.2.HS.C.3 Write functions or sequences that model relationships between two quantities. CC.2.2.HS.C.4 Interpret the effects transformations have on functions, and find the inverses of functions. CC.2.2.HS.C.5 Construct and compare linear, quadratic, and exponential models to solve problems. CC.2.2.HS.C.6 Interpret functions in terms of the situations they model. CC.2.2.HS.D.7 Create and graph equations or inequalities to describe numbers or relationships. CC.2.3.HS.A.10 Translate between the geometric description and the equation for a conic section. CC.2.2.HS.C.4 Interpret the effects transformations have on functions, and find the inverses of functions. CC.2.2.HS.C.5 Construct and compare linear, quadratic, and exponential models to solve problems. CC.2.2.HS.C.6 Interpret functions in terms of the situations they model. CC.2.1.HS.F.3 Apply quantitative reasoning to choose and interpret units and scales in formulas, graphs, and data displays. CC.2.1.HS.F.5 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. CC.2.4.HS.B.2 Summarize, represent, and interpret data on two categorical and quantitative variables. CC.2.4.HS.B.3 Analyze linear models to make interpretations based on the data. CC.2.4.HS.B.4 Recognize and evaluate random processes underlying statistical experiments. CC.2.4.HS.B.5 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
25 PA A Use odds to find probability and/or use probability to find odds. A Use probability for independent, dependent, or compound events to predict outcomes. PA Core Standards Make inferences and justify conclusions based on sample surveys, experiments, and observational studies. CC.2.4.HS.B.6 Use the concepts of independence and conditional probability to interpret data. CC.2.4.HS.B.7 Apply the rules of probability to compute probabilities of compound events in a uniform probability model. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
26 1/292/9: 1 Day Per + 4 Days 11/111/9 4/34/10 Benchmark Cycle 3 Scope and Sequence Chapter 7: Exponential and Logarithmic Functions Title Topic 71 Exploring Exponential Models Model exponential growth and decay. A A A A A A A A A A A A Properties of Exponential Explore the property of functions in the form y = ab x. Functions Graph exponential functions that have base e. A A A A A A THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
27 (see above) Chapter 7: Exponential and Logarithmic Functions (cont.) Title Topic 73* Logarithmic Functions as Write and evaluate logarithmic expressions. Inverses Graph logarithmic functions. A A A A A A A A * Properties of Logarithms Use the properties of logarithms. 75* Exponential and Logarithmic Equations Solve exponential and logarithmic equations. 76* Natural Logarithms Evaluate and simplify natural logarithmic expressions. Solve equations using natural logarithms. A A A A A A A A A THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
28 2/122/28: 1 Day Per + 6 Days 11/1311/17 4/114/17 Chapter 8: Rational Functions Title Topic 81 Inverse Variation Recognize and use inverse variations. Use joint and other variations. 82 The Reciprocal Function Family Graph reciprocal functions. Graph translations of reciprocal functions. 83* Rational Functions and Their Graphs Identify properties of rational functions. Graph rational functions. 84* Rational Expressions Simplify rational expressions. Multiply and divide rational expressions. A A A A A A A A A A A A A A A A A A THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
29 (see above) Chapter 8: Rational Functions (cont.) Title Topic 85* Adding and Subtracting Rational Add and subtract rational expressions. Expressions 86* Solving Rational Equations Solve rational equations. Use rational equations to solve problems. A A A A A A THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
30 3/13/12: 1 Day Per +3 Days 11/2011/29 4/184/25 Chapter 9: Sequences and Series Title Topic 91 Mathematical Patterns Identify mathematical patterns found in a sequence. Use a formula to find the nth term of a sequence. A A A A Arithmetic Sequences To define, identify, and apply arithmetic sequences. 93* Geometric Sequences Define, identify and apply geometric sequences. 94* Arithmetic Series Extends Define arithmetic series and find their sums. 95* Geometric Series Define geometric series and find their sums. A A A A A A A A A A A THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
31 3/134/6: 1 Day Per + 6 Days 11/3012/8 4/265/4 Chapter 11: Probability and Statistics Title Topic 111* Permutations and Combinations Count permutations. Count combinations. 112* Probability Find the probability of an event using theoretical, experimental, and simulation methods. 113* Probability Multiple Events Find the probability of the event A and B. Find the probability of the event A or B Conditional Probability Find conditional probabilities. Use tables and tree diagrams to determine conditional probabilities Probability Models Use probabilities to make fair decisions and analyze decisions Analyzing Data Calculate measures of central tendency. Draw and interpret boxandwhisker plots Standard Deviation Find the standard deviation and variance of a set of values. Apply standard deviation and variance. A A A A A A A THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
32 (see above) Chapter 11: Probability and Statistics (cont.) Title Topic 118 Samples and Surveys Identify sampling methods. Recognize bias in samples and surveys. 119* Binomial Distributions Find binomial probabilities and us binomial distribution * Normal Distributions Use a normal distribution. A A A Beyond Benchmark 3 Window: Apr 9 Apr 25 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
33 Cycle 4 Standards PA No Algebra 2 Alignment No Algebra 2 Alignment No Algebra 2 Alignment PA Core Standards CC.2.2.HS.C.7 Apply radian measure of an angle and the unit circle to analyze the trigonometric functions. CC.2.2.HS.C.8 Choose trigonometric functions to model periodic phenomena and describe the properties of the graphs. CC.2.3.HS.A.10 Translate between the geometric description and the equation for a conic section. No Algebra 2 Alignment No Algebra 2 Alignment CC.2.3.HS.A.7 Apply trigonometric ratios to solve problems involving right triangles. CC.2.4.HS.B.1 Summarize, represent, and interpret data on a single count or measurement variable. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
34 4/94/20: 1 Day Per + 4 Days 12/1112/15 5/75/11 Cycle 4 Scope and Sequence Chapter 10: Quadratic Relations and Conic Sections Title Topic 101* Exploring Conic Sections Graph and identify conic sections Parabolas Write the equations of a parabola and graph parabolas Circles Write and graph the equation of a circle. Find the center and radius of a circle and use them to graph the circle. 104* Ellipses Write the equation of an ellipse. Find the foci of an ellipse. Graph an ellipse. Beyond 105* Hyperbolas Graph hyperbolas. Find and use the foci of a hyperbola. 106* Translating Conic Sections Write the equation of a translated conic section. Identify a translated conic section from an equation. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
35 4/235/8: 1 Day Per + 6 Days 12/181/3 5/145/22 Title 121* Adding and Subtracting Matrices Chapter 12: Matrices Topic Add and subtract matrices and solve matrix equations. 122* Matrix Multiplication Multiply matrices using scalar and matrix multiplication. 123* Determinants and Inverses Find the inverse of a matrix. 124* Inverse Matrices and Systems Solve systems of equations using matrix inverses and multiplication. 125* Geometric Transformations Transform geometric figures using matrix operations. Beyond 126* Vectors Use basic vector operations and the dot product. CC Performance Task: Animal Care THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
36 5/95/29: 1 Day Per + 4 Days 1/41/16 5/236/4 Chapter 13: Periodic Functions and Trigonometry Title Topic 131* Exploring Periodic Data Identify cycles and periods of periodic functions. Find the amplitude of periodic functions. 132* Angles and the Unit Circle Work with angles in standard position. Find coordinates of prints on the unit circle. 133* Radian Measure Use radian measure for angles. Find the length of an arc of a circle. 134* The Sine Function Identify properties of the sine function. Graph sine curves. 135* The Cosine Function Graph and write cosine functions. Solve trigonometric equations The Tangent Function Graph the tangent function. Beyond 137* Translating Sine and Cosine Functions 138* Reciprocal Trigonometric Functions Graph translations of trigonometric functions Write equations of translations. Evaluate reciprocal trigonometric functions. Graph reciprocal trigonometric functions. CC Performance Task: Animating a Game THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
37 5/306/12: 1 Day Per + 3 Day 1/171/26 6/56/12 Chapter 14: Trigonometric Identities and Equations Title Topic 141* Trigonometric Identities Verify trigonometric identities. 142* Solving Trigonometric Equations Using Inverses Evaluate inverse trigonometric functions. Solve trigonometric equations. 143* Right Triangles and Trigonometric Ratios Find lengths of siders in a right triangle. Find measures of angels in a right triangle Area and the Law of Sines Find the area of any triangle. Use the Law of Sines The Law of Cosines Use the Law of Cosines in finding the measures of sides of angles of a triangle. Beyond 146* Angle Identities Extends Verify and use angle identities. Verify and use sum and difference identities. 147* DoubleAngle and HalfAngle Identities Verify and use doubleangle identities. Verify and use halfangle identities. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
38 Remediation and Extension Options Option 1: Revisit Common Core Performance Tasks Option 2: Cover Remaining Option 3: Mathematical Modeling in Three Acts Option 4: Mathematics Assessment Project Option 5: Miscellaneous (Financial Literacy, Quantitative Reasoning, Statistics, etc.) Each chapter contains a Common Core Performance Task (CCPT), intended to deepen students understanding of the content, as they proceed through the lessons. The CCPT may be previewed before beginning a chapter, and students can share their early ideas on how to approach it. Later, as they develop and strengthen tools for solving the CCPT, they can return to it and approach it in a different light. CCPTs can be used as summative assessments for individual chapter or you can return to them to wrapup the school year. Consider teaching the remaining content from the text, which can be a springboard for work in subsequent years. You may also want to check with your colleagues about what content you could preteach or what content they would like you to revisit. Students may remember ThreeAct mathematical modeling activities from Grades 68. During the remainder of the year, consider offering students opportunities to engage authentically in the Standards for Mathematical Practice through the Mathematical Modeling activities available on Pearson Realize. You can find additional threeact modeling activities at: or within this Google Sheet. Classroom Challenges are lessons that support teachers in formative assessment. There are 100 lessons in total, 20 at each grade from 6 to 8 and 40 for Career and College Readiness at High School Grades 9 and above. Some lessons are focused on developing math concepts, others on solving nonroutine problems. The Brief Guide for teachers and administrators (PDF) is recommended reading before using these lessons for the first time. Each lesson is downloadable as an allinone PDF to print out, and a supporting PowerPoint presentation. The Federal Reserve Bank of Philadelphia has posted free lesson plans for high school teachers on financial literacy at: In addition, Census.gov has a number of data analysis activities, using real U.S. Census data, at: Finally, consider other crosscurricular quantitative reasoning activities, such as those available at: THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,
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