Algebra I Pearson High School SY

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1 Chapters Total School Days Algebra I Pearson High School SY Benchmark Cycle 1 Benchmark Cycle 2 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 Eam 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 Eam 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 2 Solving Equations 1 3 Solving Inequalities 1 4 An Introduction to Functions Linear Functions 1,2 6 Systems of Equations and Inequalities Eponents and Eponential Equations 1 8 Polynomials and Factoring Quadratic Functions and Equations BEC Radical Epressions and Equations Rational Epressions and Functions Linear Functions 2 12 Data Analysis and Probability 2 Keystone Eam Review Eponents and Eponential Equations BEC Quadratic Functions and Equations BEC Radical Epressions and Equations BEC Rational Epressions and Functions BEC Notes A Benchmark 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 Eligible from Module 1 of the Algebra I Keystone Eam. 2 Contains Eligible from Module 2 of the Algebra I Keystone Eam. BEC Contains content that goes beyond the eligible content for the Algebra I Keystone Eam, i.e., material not included on the state assessment. 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... 5 Benchmark Cycle 2 Standards... 9 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 Eligible 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 approimately 90-minutes in length. Non-asterisked lessons (approimately 45-minutes in length) may be combined, but consider combining them meaningfully, instead of simply teaching two-lessons back-toback. When single non-asterisked lessons precede or follow an asterisked one, consider using s, Mathematical Modeling, or Common Core Performance Tasks to start or complete the lesson (using 45 available minutes to construct a full 90-minute lesson). Benchmark Cycle 1 Benchmark Cycle 2 Benchmark Cycle 3 Cycle 4 Pacing Use the suggested pacing, but combine 45-minute (non-asterisked) lessons, in accordance with the general note above. 9/5 10/2 Ch. 2 to Ch. 4 10/4 10/31 Ch. 5-1 to 5-6, Ch. 6, Ch. 7-1 to /1 11/30 Ch. 8, Ch. 9-1 to 9-3, Ch to /1 1/5 Ch to 11-2, Ch. 5-7 to 5-8, Ch. 12, Keystone Eam Review 1/5 1/26 Ch. 7-6 to 7-8, Ch. 9-4 to 9-8, Ch to 10-6, Ch to /30 2/27 Ch. 2 to Ch. 4 Benchmark Schedule 2/28 3/22 Ch. 5-1 to 5-6, Ch. 6, Ch. 7-1 to 7-5 3/23 4/18 Ch. 8, Ch. 9-1 to 9-3, Ch to /19 5/11 Ch to 11-2, Ch. 5-7 to 5-8, Ch. 12, Keystone Eam Review 5/14 6/12 Ch. 7-6 to 7-8, Ch. 9-4 to 9-8, Ch to 10-6, Ch to 11-7 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 Eligible A Use estimation to solve problems. A Write, solve, and/or apply a linear equation (including problem situations). A Use and/or identify an algebraic property to justify any step in an equation-solving process. Note: Linear equations only A Interpret solutions to problems in the contet of the problem situation. Note: Linear equations only. A Write or solve compound inequalities and/or graph their solution sets on a number line (may include absolute value inequalities). A Identify or graph the solution set to a linear inequality on a number line. A Interpret solutions to problems in the contet of the problem situation. Note: Linear inequalities only. PA Core Standards CC B.3 Model and solve real-world and mathematical problems by using and connecting numerical, algebraic, and/or graphical representations. CC.2.2.HS.D.9 Use reasoning to solve equations and justify the solution method. 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.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.2.1.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.D.7 Create and graph equations or inequalities to describe numbers or relationships. CC.2.2.HS.D.9 Use reasoning to solve equations and justify the solution method. CC.2.2.HS.D.7 Create and graph equations or inequalities to describe numbers or relationships. 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. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,

5 9/5-10/4: 1 Day Per + 10 Days 9/5-9/14 1/30-2/8 Benchmark Cycle 1 Scope and Sequence a Chapter 2: Solving Equations Title Topic(s) Eligible Modeling One-Step Equations Use algebra tiles to model and solve singe-variable, one step equations. 2-1 Solving One-Step Equations b Solve one-step equations in one variable. 2-2* Solving Two-Step Equations Solve two-step equations in one variable. 2-3* Solving Multi-Step Equations Solve multi-step equations in one variable. 2-4* Modeling Equations with Variables on Both Sides Solving Equations with Variables on Both Sides Use algebra tiles to model and solve equations with variables on both sides. Solve equations with variables on both sides, and identify equations that are identities or that have no solution. 2-5 Literal Equations and Formulas Rewrite and use literal equations and formulas. A , A A , A , A A , A A , A , A Ratios, Rates, and Conversions Find ratios and rates and convert units and rates. A , A , Unit Analysis A Use unit analysis to solve problems, and determine the reasonableness of solutions. 2-7 Solving Proportions Solve and apply proportions. 2-8 Proportions and Similar Figures Find missing lengths in similar figures, and use similar figures when measuring indirectly. A , A , A , A a s 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). b A Solve It! begins every lesson, providing students opportunities to strengthen their problem-solving 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 two 45-minute class periods, for which fle days may be used. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,

6 (see previous) 2-9* Percents 2-10* Change Epressed as a Percent Chapter 2: Solving Equations (cont.) Title Topic Eligible Solve percent problems using proportions and percent problems using the percent equation. Find percent change and the relative error in linear and nonlinear measurements. A , A , A , A , Focus: Chapter 2 concentrates on three essential questions: Can equations that appear different be equivalent? How can you solve equations and make sense of their solutions? What kinds of relationships are represented by proportions? Coherence: Note that Chapter 1 covers content largely addressed in 8th Grade. While many of these lessons help prepare students for what comes later, there are several lessons that directly support work on Algebra I Eligible, specifically: Chapter 1-1 Chapter 1-7 Chapter 1-2 Chapter 1-8 : Operations with Chapter 1-9 Rational and Irrational Numbers Review material from Chapter 1, as you feel necessary and being mindful of the suggested pacing with fle days, but note that not all of this content appears on the Algebra I Keystone Eam. Rigor: The key idea in Chapter 2 involves understanding equality and how to preserve it. Rigor is elevated, when students have opportunities to make sense of and solve problems involving equality, using their own intuition and eplaining their ideas. Concentrate on encouraging students to use models to represent their thinking throughout this chapter and beyond. Consider using Mathematical Modeling in Three Acts (found on Pearson Realize), algebra tiles (available, free and virtually at or SolveMe Puzzles ( Consider beginning the year by laying foundations for problem-solving via: WODB.ca, Estimation180.com, SolveMe. Also consider using fle days, on occasion, to complete or present the Common Core Performance Tasks or Mathematical Modeling in Three Acts (found on Pearson Realize), which support increased rigor. Review foundational knowledge and check for understanding, as needed, through Do Nows or Bell Ringers, homework, and teacher-created assessments. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,

7 10/5-10/31: 1 Day Per Days 9/15-9/20 2/9-2/14 Title Chapter 3: Solving Inequalities Topic Eligible 3-1 Inequalities and Their Graphs Write, graph, and identify solutions of inequalities. 3-2 Solving Inequalities Using Addition or Subtraction Use addition or subtraction to solve inequalities. 3-3* Solving Inequalities Using Multiplication or Division More Algebraic Properties Use multiplication or division to solve inequalities. Review the refleive, symmetric, and transitive properties of equality and introduce the transitive property of inequality. A , A Modeling Multi-Step Inequalities Use algebra tiles to model and solve multi-step inequalities. 3-4* Solving Multi-Step Inequalities Solve multi-step inequalities. 3-5 Working with Sets Write sets and identify subsets, and find the complement of a set. Beyond Eligible THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,

8 (see previous) Title Chapter 3: Solving Inequalities (cont.) Topic Eligible 3-6* Compound Inequalities Solve and graph inequalities containing the word and and or. A , A , Absolute Value Equations and 3-7 Solve equations and inequalities involving absolute value A Inequalities 3-8* Unions and Intersections of Sets Find the union and intersection of sets. Beyond Eligible Focus: In Chapter 3, students address these questions: How do you represent relationships between quantities that are not equal? Can inequalities that appear different be equivalent? How can you solve inequalities and make sense of their solutions? Coherence: Chapter 3 etends ideas from Chapter 2, investigating inequality. Students will draw on their previous understanding of inequality, as they learned about making comparisons and representing inequalities with symbols throughout elementary and middle school. Chapters 3-5 and 3-8 may support students in making sense of inequalities, and investigations of real numbers in Algebra II, but they contain content that goes beyond the eligible content for the Algebra I Keystone Eam. Through sets, a foundation for classifying geometric objects is also established. Rigor: Throughout this Chapter, students should be encouraged to make sense of algebraic inequalities by substituting the variable with specific values and determining whether the result is true or false. Students can also model their thinking, as with equations. Benchmark 1 Window: Nov 1 Nov 17 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,

9 Benchmark Cycle 2 Standards PA Eligible A Compare and/or order any real numbers. Note: Rational and irrational may be mied. A Simplify square roots (e.g., 24 = 2 6). A Simplify/evaluate epressions involving properties/laws of eponents, roots, and/or absolute values to solve problems. Note: Eponents should be integers from -10 to 10. A Use estimation to solve problems. PA Core Standards CC E.1 Distinguish between rational and irrational numbers using their properties. CC E.4 Estimate irrational numbers by comparing them to rational numbers. CC.2.1.HS.F.1 Apply and etend the properties of eponents to solve problems with rational eponents. CC.2.1.HS.F.2 Apply properties of rational and irrational numbers to solve real-world or mathematical problems. CC E.3 Develop and/or apply number theory concepts to find common factors and multiples. CC.2.1.HS.F.2 Apply properties of rational and irrational numbers to solve real-world or mathematical problems. CC.2.1.HS.F.1 Apply and etend the properties of eponents to solve problems with rational eponents. CC.2.1.HS.F.2 Apply properties of rational and irrational numbers to solve real-world or mathematical problems. CC B.1 Apply concepts of radicals and integer eponents to generate equivalent epressions. CC B.3 Model and solve real-world and mathematical problems by using and connecting numerical, algebraic, and/or graphical representations. CC.2.2.HS.D.9 Use reasoning to solve equations and justify the solution method. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,

10 PA Eligible A Write, solve, and/or apply a linear equation (including problem situations). A Write and/or solve a system of linear equations (including problem situations) using graphing, substitution, and/or elimination. Note: Limit systems to two linear equations. A Interpret solutions to problems in the contet of the problem situation. Note: Limit systems to two linear equations. A Write and/or solve a system of linear inequalities using graphing. Note: Limit systems to two linear inequalities. A Interpret solutions to problems in the contet of the problem situation. Note: Limit systems to two linear inequalities. 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.2.1.HS.F.5 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. CC B.3 Analyze and solve linear equations and pairs of simultaneous linear equations. CC C.1 Define, evaluate, and compare functions. CC C.2 Use concepts of functions to model relationships between quantities. CC.2.2.HS.C.3 Write functions or sequences that model relationships between two quantities. 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.2.1.HS.F.5 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. CC B.3 Analyze and solve linear equations and pairs of simultaneous linear equations. CC.2.2.HS.D.7 Create and graph equations or inequalities to describe numbers or relationships. 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.2.1.HS.F.5 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. CC.2.2.HS.D.7 Create and graph equations or inequalities to describe numbers or relationships. CC.2.2.HS.D.10 Represent, solve, and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,

11 PA Eligible A Analyze a set of data for the eistence of a pattern and represent the pattern algebraically and/or graphically. A Determine whether a relation is a function, given a set of points or a graph. A Identify the domain or range of a relation (may be presented as ordered pairs, a graph, or a table). A Create, interpret, and/or use the equation, graph, or table of a linear function. A Translate from one representation of a linear function to another (i.e., graph, table, and equation). PA Core Standards CC C.1 Define, evaluate, and compare functions. CC C.2 Use concepts of functions to model relationships between quantities. CC.2.2.HS.C.1 Use the concept and notation of functions to interpret and apply them in terms of their contet. 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.2.4.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 B.2 Understand the connections between proportional relationships, lines, and linear equations. CC C.1 Define, evaluate, and compare functions. CC C.2 Use concepts of functions to model relationships between quantities. 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.6 Interpret functions in terms of the situations they model. 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,

12 PA Eligible A Identify, describe, and/or use constant rates of change. A Apply the concept of linear rate of change (slope) to solve problems. A Write or identify a linear equation when given the graph of the line, two points on the line, or the slope and a point on the line. Note: Linear equation may be in point-slope, standard, and/or slope-intercept form. A Determine the slope and/or y-intercept represented by a linear equation or graph. 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 contet. 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.2.4.HS.B.1 Summarize, represent, and interpret data on a single count or measurement variable. CC.2.2.HS.C.1 Use the concept and notation of functions to interpret and apply them in terms of their contet. CC.2.2.HS.C.6 Interpret functions in terms of the situations they model. CC.2.4.HS.B.1 Summarize, represent, and interpret data on a single count or measurement variable. 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 eponential models 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 contet. 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,

13 11/1-11/22: 1 Per Day + 5 Full + 2 Half Days 9/25-10/3 2/15-2/ Title Using Graphs to Relate Two Quantities Benchmark Cycle 2 Scope and Sequence Chapter 4: An Introduction to Functions Topic Represent mathematical relationships using graphs. 4-2 Patterns and Linear Functions Identify and represent patterns that describe linear functions. 4-3 Patterns and Nonlinear Functions Identify and represent patterns that describe nonlinear functions. 4-4 Graphing a Function Rule Graph equations that represent functions. Graphing Functions and Solving Equations Use a graphing calculator to graph functions and solve linear equations. 4-5 Writing a Function Rule Write equations that represent functions. 4-6* Formalizing Relations and Functions 4-7* Arithmetic Sequences Determine whether a relation is a function and find domain, range, and use function notation. Identify and etend patterns in sequences and represent arithmetic sequences using function notation. Eligible A A , A , A A , A A , A , A A , A A , A A , A A , A Focus: In Chapter 4, students investigate modeling and functions. Students develop an understanding of the relationship between function notation, graphs, tables, and verbal epressions. They address: How can you represent and describe functions? How do functions describe realworld-situations? Consider reviewing, as needed, two topics from Grade 8 before beginning this chapter: define, evaluate, and compare functions (CC C.1); and use concepts of functions to model relationships between quantities (CC C.2). Coherence: Ideas underlying the essential questions, above, permeate algebra and higher mathematics. Therefore, as a foundation for studying linear functions, systems of linear equations and inequalities, analyzing bivariate data, and so on, this chapter is crucial. Note, though, that it covers only Module 2 content. (Efforts were made to keep as much of Module 1 and Module 2 content together, respectively, as was practical.) Rigor: Offering students opportunities to make sense of patterns, to grapple with rich problems, to observe connections between mathematics and real-world situations, and to facilitate their use of formal notation all address rigor throughout this chapter. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,

14 11/27-12/12: 1 Day Per Days 10/4-10/11 2/28-3/6 Title Chapter 5: Linear Functions Topic 5-1 Rate of Change and Slope Find rates of change from tables and find slope. 5-2 Direct Variation Write and graph an equation of a direct variation. Investigating y = m + b Use a graphing calculator to investigate the slope-intercept form of a linear equation. 5-3* Slope-Intercept Form Write and graph linear equations using slope-intercept form. 5-4* Point-Slope Form Write and graph linear equation using point-slope form. 5-5* Standard Form Inverse of a Linear Function 5-6 Parallel and Perpendicular Lines Graph linear equations using intercepts, and write linear equations in standard form. Build an understanding of inverse functions. Determine whether lines are parallel, perpendicular, or neither, and write equations of parallel and perpendicular lines. Eligible A , A A , A , A A , A , A Beyond Eligible A , A Focus: Students deepen their understanding of Chapter 4 by looking at one family of related functions: linear functions. They investigate these questions: What information does the equation of a line give you? What does the slope of a line indicate about the line? Before starting Chapter 5, consider strategically reviewing material from Chapter 1 (foundations), as well as topics from Grades 7 and/or 8: model and solve real-world and mathematical problems by using and connection numerical, algebraic, and/or graphical representations (7.CC B.3); and analyze and solve linear equations and pairs of simultaneous linear equations (8.CC B.3) Coherence: Note that this chapter covers Module 2 content. To keep much of Module 1 content together and Module 2 content together, to assist in reviewing for the Keystone Eam, the essential pieces of Chapter 5 are covered here. The after Chapter 5-5 is included, to lay a foundation for roots and radicals and for supporting Geometry and Algebra II content (include as time permits). Chapter 5-6 is an important precursor to understanding and solving systems of linear equations and inequalities (Chapter 6). Chapter 5 concludes, later in the year, with scatterplots (Module 2) and absolute value functions (which goes beyond the eligible content on the Algebra I Keystone Eam). Rigor: This material should be considered more than techniques to master and instead part of a broader theme making sense of phenomena that grow or diminish at constant rates. How do we know whether temperatures are changing, wealth is becoming concentrated, or even whether trains will arrive on time? Problem-solving and the tools of algebra, analyzing linearity, help us defend our thinking. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,

15 12/13-1/10: 1 Day Per + 8 Days 10/12-10/20 3/7-3/14 Title 6-1 Solving Systems by Graphing 6-2* 6-3* Solving Systems Using Tables and Graphs Solving Systems Using Algebra Tiles Solving Systems Using Substitution Solving Systems Using Elimination Matrices and Solving Systems 6-4 Applications of Linear Systems 6-5* Linear Inequalities 6-6 Systems of Linear Inequalities Graphing Linear Inequalities Chapter 6: Systems of Equations and Inequalities Topic Solve systems of equations by graphing, and analyze special systems. Use the table and graphing functions on a graphing calculator to solve a system of linear equations. Use algebra tiles to solve systems of linear equations. Solve systems of equations using substitution. Solve systems by adding or subtracting to eliminate a variable. Use augmented matrices to solve systems of linear equations. Choose the best method for solving a system of linear equations. Graph linear inequalities in two variables, and use linear inequalities when modeling real-world situations. Solve systems of linear inequalities by graphing, and model real-world situations by using systems of linear inequalities. Use a graphing calculator to graph linear inequalities. Eligible A , A Beyond Eligible A , A A , A Focus: In Chapter 6, students eplore systems of linear equations and inequalities, raising such questions as: How can you solve a system of equations or inequalities and make sense of their solutions? How do systems of equations and inequalities model real-world situations? The following Chapter 6-3 may be omitted, but it might engage students interested in the representations behind computer graphics. Coherence: The core ideas of Chapter 5 (containing Module 2 content) establish a foundation for Chapter 6 (linear systems), Chapter 7 (eponents), and Chapter 8 (polynomials, including quadratic epressions) all of which address Module 1 content. Some books proceed from linear functions into quadratics, but many states, like PA, do not epect Algebra I students to study quadratic functions. Therefore, this sequence of chapters focuses on composing and decomposing several linear epressions before turning to quadratics. Rigor: Opportunities to translate among the representations of linear functions, and to solve applied problems involving linear systems, support these Assessment Anchors and promote rigor. As always, students should have regular opportunities to eplain their thinking (SMP3). THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,

16 1/11-1/26: 1 Day Per + 6 Days 10/23-10/31 3/15-3/22 Title Chapter 7: Eponents and Eponential Functions Topic 7-1 Zero and Negative Eponents Simplify epressions involving zero and negative eponents. 7-2* 7-3* Multiplying Powers Multiplying Powers with the Same Base Powers of Powers and Powers of Products More Multiplication Properties of Eponents 7-4* Division Properties of Eponents Relating Radicals to Rational Eponents Understand patterns that facilitate the multiplication of powers. Multiply powers with the same base. Discover rules for powers of powers and powers of products. Raise a power to a power, and raise a product to a power. Divide powers with the same base, and raise a quotient to a power. Understand the relationship between radicals and rational eponents. 7-5 Rational Eponents and Radicals Rewrite epressions involving radicals and rational eponents. Eligible A , A A , A , A A , A Focus: In Chapter 7, students investigate these essential questions: How can you represent very large and very small numbers? How can you simplify epressions involving eponents? In Chapter 7-5, focus on square roots and perfect cube roots to address the Assessment Anchors. Coherence: Chapter 7 plays a crucial role in transitioning from studying linear functions and epressions to studying polynomials. Building on students understanding of eponents and inverse functions (Chapter 5), students will compose and decompose linear epressions in performing operations with and factoring polynomial epressions. The last three sections of Chapter 7 go beyond the eligible content for Algebra I. Some tetbooks choose to eplore properties of eponents (Chapters 7-2 to 7-4) before justifying results about zero and negative eponents (Chapter 7-1). Both approaches are valid, and so the order can be switched, provided care is taken with selecting practice and homework problems. Rigor: This topic is traditionally presented as a set of rules to memorize. To foster deeper understanding, students should draw on their problemsolving and reasoning abilities, including eploring with calculators and whole numbers, to conjecture and justify these rules themselves. Benchmark 2 Window: Jan 29 Feb 13 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,

17 Benchmark Cycle 3 Standards PA Eligible A Simplify square roots (e.g., 24 = 2 6). A Simplify/evaluate epressions involving properties/laws of eponents, roots, and/or absolute values to solve problems. Note: Eponents should be integers from 10 to 10 A Use estimation to solve problems. A Write, solve, and/or apply a linear equation (including problem situations). PA Core Standards CC E.3 Develop and/or apply number theory concepts to find common factors and multiples. CC.2.1.HS.F.2 Apply properties of rational and irrational numbers to solve real-world or mathematical problems. CC.2.1.HS.F.1 Apply and etend the properties of eponents to solve problems with rational eponents. CC.2.1.HS.F.2 Apply properties of rational and irrational numbers to solve real-world or mathematical problems. CC B.1 Apply concepts of radicals and integer eponents to generate equivalent epressions. CC B.3 Model and solve real-world and mathematical problems by using and connecting numerical, algebraic, and/or graphical representations. CC.2.2.HS.D.9 Use reasoning to solve equations and justify the solution method. 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.2.1.HS.F.5 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. CC B.3 Analyze and solve linear equations and pairs of simultaneous linear equations. CC C.1 Define, evaluate, and compare functions. CC C.2 Use concepts of functions to model relationships between quantities. CC.2.2.HS.C.3 Write functions or sequences that model relationships between two quantities. 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. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,

18 PA Eligible A Add, subtract, and/or multiply polynomial epressions (epress answers in simplest form). Note: Nothing larger than a binomial multiplied by a trinomial. A Factor algebraic epressions, including difference of squares and trinomials. Note: Trinomials are limited to the form a 2 + b + c where a is equal to 1 after factoring out all monomial factors. A Simplify/reduce a rational algebraic epression. A Calculate and/or interpret the range, quartiles, and interquartile range of data. A Analyze data, make predictions, and/or answer questions based on displayed data (boand-whisker plots, stem-and-leaf plots, scatter plots, measures of central tendency, or other representations). A Make predictions using the equations or graphs of best-fit lines of scatter plots. PA Core Standards CC.2.2.HS.D.1 Interpret the structure of epressions to represent a quantity in terms of its contet. CC.2.2.HS.D.2 Write epressions in equivalent forms to solve problems. CC.2.2.HS.D.3 Etend the knowledge of arithmetic operations and apply to polynomials. CC.2.2.HS.D.5 Use polynomial identities to solve problems. CC.2.2.HS.D.6 Etend the knowledge of rational functions to rewrite in equivalent forms. CC.2.4.HS.B.1 Summarize, represent, and interpret data on a single count or measurement variable. CC.2.4.HS.B.3 Analyze linear models to make interpretations based on the data. CC.2.4.HS.B.1 Summarize, represent, and interpret data on a single count or measurement variable. CC.2.4.HS.B.3 Analyze linear models to make interpretations based on the data. CC.2.4.HS.B.5 Make inferences and justify conclusions based on sample surveys, eperiments, and observational studies. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,

19 1/29-2/23: 1 Day Per + 9 Full + 2 Half Days 11/1-11/15 3/23-4/9 Benchmark Cycle 3 Scope and Sequence 8-1* Title Adding and Subtracting Polynomials 8-2* Multiplying and Factoring Using Models to Multiply Chapter 8: Polynomials and Factoring Topic Classify, add, and subtract polynomials. Multiply a monomial by a polynomial, and factor a monomial from a polynomial. Use models to multiply binomials. 8-3* Multiplying Binomials Multiply two binomials or a binomial by a trinomial. 8-4* Multiplying Specials Cases Using Models to Factor Find the square of a binomial, and find the product of a sum and difference. Use models to factor trinomials. 8-5* Factoring 2 + b + c Factor trinomials of the form 2 + b + c. 8-6* Factoring a 2 + b + c Factor trinomials of the form a 2 + b + c. 8-7* Factoring Special Cases Factor perfect square trinomials, and factor the difference of two squares. 8-8* Factoring by Grouping Factor higher-degree polynomials by grouping. Eligible A A A , A A A A Focus: Chapter 8 addresses these essential questions: Can two algebraic epressions that appear different be equivalent? How are the properties of real numbers related to polynomials? In Chapter 8-8, concentrate on factoring trinomials where a = 1 after factoring out all monomials. Coherence: Chapter 8 ties together storylines from three previous chapters: linear functions in Chapters 5 and 6, and eponents in Chapter 7. In this chapter, students combine linear epressions to make quadratic epressions, just as they combined prime numbers to make composite numbers in elementary school. Likewise, students connect their understanding of factoring whole numbers with factoring quadratics. Students briefly cover quadratics, to motivate the study of roots and radicals, before returning to polynomials via rational epressions. Rigor: Students may reason incorrectly about algebraic epressions, relying on their understanding of numbers. To promote depth of understanding, ask students to investigate their conjectures about polynomials by substituting a value for the variable and seeing whether their approach yields a true or false result. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,

20 2/26-3/6: 1 Day Per + 4 Days 11/16-11/22 4/10-4/12 Title Chapter 9: Quadratic Functions and Equations Topic Eligible 9-1* Quadratic Graphs and Their Properties Graph quadratic functions of the form y = a 2 and y = a 2 + c. 9-2 Quadratic Functions Graph quadratic functions of the form y = a 2 + b + c Rates of Increase Use functions, tables, and graphs to determine and compare rates of change. Beyond Eligible 9-3* Solving Quadratic Equations Solve quadratic equations by graphing and using square roots. Finding Roots Use graphing calculators to find solutions of quadratic equations. Focus: Chapter 9 lays a foundation for understanding quadratic functions, concentrating on their graphs and properties. Coherence: Students are already familiar with square roots from Grade 8, and Chapter 9-3 refreshes these ideas before transitioning to simplifying radical epressions in Chapter 10. Quadratics, as a non-linear family of functions, will also be featured prominently in Algebra II. While the content in Chapter 9 goes beyond the eligible content for the Algebra I Keystone Eam, this material may help with understanding polynomials in Chapter 8 and motivating the study of roots and radicals in Chapter 10. Therefore, spending a little time on this material may well be worth it, but you may return to these sections later if pressed for time. Rigor: Using technology, to engage students in an investigation of quadratic functions and the influence of various coefficients, is a way to deepen students understanding of non-linearity. Translating among representations also builds a connected understanding of functions, as well. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,

21 3/7-3/16: 1 Day Per Days 11/27-11/30 4/13-4/18 Title 10-1 The Pythagorean Theorem Chapter 10: Radical Epressions and Equations Topic Solve problems using the Pythagorean Theorem, and identify right triangles. 10-2* Simplifying Radicals Simplify radicals involving products and quotients. 10-3* Operations with Radical Epressions 10-4* Solving Radical Equations Simplify sums, differences, products, and quotients of radical epressions. Solve equations containing radicals, and identify etraneous solutions. Eligible M08.C-G.2.1.1, M08.C-G A , A , A Beyond Eligible A , A , A Beyond Eligible Focus: Chapter 10 addresses these essential questions: How are radical epressions represented? Can two radical epressions be equivalent, if they appear different? Note that Chapter 10-1 covers material from Grade 8. Consider reviewing this section, briefly, it may also be omitted provided that students have a strong understanding of the Pythagorean Theorem. In Chapter 10-2, students do not need to rationalize denominators, which is material that goes beyond the eligible content. Likewise, Chapter 10-4 goes beyond the eligible content for the Algebra I Keystone Eam, but it could be included as an etension if time permits. Coherence: Having solved basic quadratic equations in Chapter 9-3, students should see the importance of techniques for simplifying radical epressions. This is because there are many equivalent ways of writing such epressions and so simplifying helps with agreement. The latter part of Chapter 10 goes beyond the eligible content for the Keystone Eam, but this content helps with understanding function families and therefore it may support students later understanding of concepts in Algebra II. Rigor: Chapter 10 includes a number of application problems, as well as problems that require attending to precision (SMP6). Support students in developing strategies for checking the reasonableness of their answers and making sense of word problems. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,

22 3/19-3/28: 1 Day Per + 6 Days 12/1-12/5 4/19-4/20 Chapter 11: Rational Epressions and Functions Title Topic Eligible 11-1* Simplifying Rational Epressions Simplify rational epressions. A * Multiplying and Dividing Rational Epressions Multiply and divide rational epressions, and simplify comple fractions. Dividing Polynomials Using Algebra Tiles Use algebra tiles to divide polynomials. Beyond Eligible Focus: Chapter 11 asks: How are rational epressions related? Simplifying comple fractions is an etension that could be treated, but briefly. Coherence: In Chapter 11, students return to studying polynomials, continuing the thread from Chapter 8. Since much of Chapter 11 contains material that goes beyond the eligible content for the Algebra I Keystone Eam, the remainder of Chapter 11 is covered later in the year. Rigor: Consider including the after Chapter 11-2, modeling the division of polynomials, to support transition to the rest of Ch 11. THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,

23 4/3-4/6: 1 Day Per + 2 Days 12/6-12/7 4/23-4/24 Chapter 5: Linear Functions Title Topic Eligible 5-7 Scatter Plots and Trend Lines Using Residuals Write an equation of a trend line and of a line of best fit, and use a trend line and a line of best fit to make predictions. Build an understanding of scatter plots and lines of best fit. A , A , A * Graphing Absolute Value Functions Characteristics of Absolute Value Graphs Graph an absolute value function, and translate the graph of an absolute value function. Investigate absolute value graphs and their characteristics. Beyond Eligible Focus: The latter part of this chapter deepens understanding of bivariate relationships and functions, laying foundations for Chapter 12. Coherence: Ideas from Chapters 2, 3, and even 6 re-emerge in the latter part of Chapter 5. Consider a strategic review of some of this material. Rigor: Problem-solving activities How could we figure out whether one line predicts better on average than another? help maintain rigor. Benchmark 3 Window: Apr 9 Apr 25 THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,

24 Cycle 4 Standards PA Eligible A Estimate or calculate to make predictions based on a circle, line, bar graph, measure of central tendency, or other representation. A Analyze data, make predictions, and/or answer questions based on displayed data (boand-whisker plots, stem-and-leaf plots, scatter plots, measures of central tendency, or other representations). A Make predictions using the equations or graphs of best-fit lines of scatter plots. A Calculate and/or interpret the range, quartiles, and interquartile range of data. A Find probabilities for compound events (e.g., find probability of red and blue, find probability of red or blue) and represent as a fraction, decimal, or percent. PA Core Standards CC.2.4.HS.B.1 Summarize, represent, and interpret data on a single count or measurement variable. CC.2.4.HS.B.3 Analyze linear models to make interpretations based on the data. CC.2.4.HS.B.5 Make inferences and justify conclusions based on sample surveys, eperiments, and observational studies. CC.2.4.HS.B.1 Summarize, represent, and interpret data on a single count or measurement variable. CC.2.4.HS.B.3 Analyze linear models to make interpretations based on the data. CC B.3 Investigate chance processes and develop, use, and evaluate probability models. CC.2.4.HS.B.4 Recognize and evaluate random processes underlying statistical eperiments. 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,

25 4/9-4/27: 1 Day Per Days 12/8-12/18 4/25-5/2 Cycle 4 Scope and Sequence Chapter 12: Data Analysis and Probability Title Topic Eligible 12-1 Organizing Data Using Matrices Organize data in a matri. Beyond Eligible 12-2 Frequency and Histograms Make and interpret frequency tables and histograms. A Measures of Central Tendency and Dispersion Standard Deviation Find mean, median, mode, and range. Find the standard deviation of a data set Bo-and-whisker plots Make and interpret bo-and-whisker plots. Designing Your Own Survey Eplore methods of collecting data. 12-5* Samples and Surveys Classify data and analyze samples and surveys. Two-Way Frequency Table Use two-way frequency tables to analyze data and to make predictions Permutations and Combinations Find permutations and combinations. 12-7* Theoretical and Eperimental Probability Conducting Simulations Find theoretical and eperimental probability. Learn what a simulation is, why simulations are used, and some of the different methods that can be used to simulate eperiments. A , A Beyond Eligible A , A , A A A Beyond Eligible THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,

26 (see previous) Title 12-8* Probability of Compound Events Normal Distribution Chapter 12: Data Analysis and Probability (cont.) Topic Find probabilities of mutually eclusive and overlapping events, as well as independent and dependent events. Use normal distributions to estimate population percentages, and determine whether it is appropriate to model a data set with a normal distribution. Eligible A Beyond Eligible Focus: In Chapter 12, students investigate these questions: How can collecting and analyzing data help you make decisions or predictions? How can you make and interpret different representations of data? How is probability related to real-world events? Note that several sections of this chapter go beyond the eligible content on the Keystone Eam, but that some of this material may be motivating to students and may help tie the chapter together. Consider including this material, as time permits. Consider previewing this content, especially 12-7, by refreshing students memory on 7.M07.D-S.3: investigate chance processes and develop, use, and evaluate probability models. Coherence: Students should be familiar with many of the ideas underlying this chapter, because data analysis and probability are important threads throughout K-12 mathematics. Chapter 12 deepens students understanding by formalizing statistical approaches and counting methods. This chapter also draws upon set theory, which may have been addressed briefly in Chapter 3. Chapter 12 also builds on concepts developed throughout the year, involving bivariate relationships. In some sense, many of the previous chapters deal with ideal data that eists only in a perfect world, and Chapter 12 offers tools for handling messy, real-world data. Rigor: Because data analysis, counting, and probability problems often yield counter-intuitive results, particular attention should be paid to making conjectures and using reasoning. As always, students should have regular opportunities to eplain their thinking (SMP3). Algebra I Keystone Eam Preparation 4/30-5/11 10 Days Use data to determine which topics/skills to revisit. Consider observation data, formative assessments, other teacher-created assessment data, and benchmark assessment data to refresh students knowledge of Eligible. Consider honing-in on essential content by reviewing the Performance Level Descriptors at And consider using practice problems from THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY,