Algebra I Pearson High School 2017-2018 SY · 1 Algebra I Pearson High School 2017-2018 ... 1...

THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY, 2017-2018

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Algebra I Pearson High School 2017-2018 SY

Benchmark Cycle 1 Benchmark Cycle 2 Benchmark Cycle 3 Cycle 4

Dat

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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 12

Tota

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39 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

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2 – Solving Equations1 3 – Solving Inequalities1

4 – An Introduction to Functions2

5-1 – 5-6 – Linear Functions1,2 6 – Systems of Equations and

Inequalities1 7-1 – 7-5 – Exponents and

Exponential Equations1

8 – Polynomials and Factoring1 9-1 – 9-3 – Quadratic Functions

and EquationsBEC 10-1 – 10-4 – Radical

Expressions and Equations1 11-1 – 11-2 – Rational

Expressions and Functions1 5-7 – 5-8 – Linear Functions2

12 – Data Analysis and Probability2

Keystone Exam Review 7-6 – 7-8 – Exponents and

Exponential EquationsBEC 9-4 – 9-8 – Quadratic Functions

and EquationsBEC 10-5 – 10-6 – Radical

Expressions and EquationsBEC 11-3 – 11-7 – Rational

Expressions and FunctionsBEC

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 Content from Module 1 of the Algebra I Keystone Exam.

2 Contains Eligible Content from Module 2 of the Algebra I Keystone Exam.

BEC Contains content that goes beyond the eligible content for the Algebra I Keystone Exam, i.e., material not included on the state assessment.


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Table of Contents

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 ......................................................................................................................................................................... 13

Benchmark Cycle 3 Standards ........................................................................................................................................................................................... 17

Benchmark Cycle 3 Scope and Sequence ......................................................................................................................................................................... 19

Cycle 4 Standards .............................................................................................................................................................................................................. 24

Cycle 4 Scope and Sequence ............................................................................................................................................................................................. 25

PA Core Standards by Cycle .............................................................................................................................................................................................. 31

Algebra Eligible Content Taught ....................................................................................................................................................................................... 33

Document Information Page ............................................................................................................................................................................................ 36


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Block Schedule Options

Note: With block schedules, generally follow the suggested sequence of lessons. Asterisked lessons are approximately 90-minutes in length. Non-asterisked lessons (approximately 45-minutes in length) may be combined, but consider combining them meaningfully, instead of simply teaching two-lessons back-to-back. When single non-asterisked 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 90-minute lesson).

Benchmark Cycle 1 Benchmark Cycle 2 Benchmark Cycle 3 Cycle 4

Suggested Pacing

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Use the suggested pacing, but combine 45-minute (non-asterisked) lessons, in accordance with the general note above.

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Fall

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 7-5

11/1 – 11/30 Ch. 8, Ch. 9-1 to 9-3, Ch. 10-1 to 10-4

12/1 – 1/5 Ch. 11-1 to 11-2, Ch. 5-7 to 5-8, Ch. 12, Keystone Exam Review

1/5 – 1/26 Ch. 7-6 to 7-8, Ch. 9-4 to 9-8, Ch. 10-5 to 10-6, Ch. 11-3 to 11-7

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1/30 – 2/27 Ch. 2 to Ch. 4

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. 10-1 to 10-4

4/19 – 5/11 Ch. 11-1 to 11-2, Ch. 5-7 to 5-8, Ch. 12, Keystone Exam Review

5/14 – 6/12 Ch. 7-6 to 7-8, Ch. 9-4 to 9-8, Ch. 10-5 to 10-6, Ch. 11-3 to 11-7

Benchmark Schedule

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Follow Benchmark schedule indicated in the Scope and Sequence document that follows this page.

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Fall

Benchmarks 1 and 2 (No need to complete

in one sitting)

None (Benchmark 3 not taken)

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Benchmarks 1 and 2

(No need to complete in one sitting)

None (Benchmark 3 not taken)


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Benchmark Cycle 1 Standards

PA Eligible Content PA Core Standards

A1.1.1.4.1 Use estimation to solve problems. CC.2.2.7.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.

A1.1.2.1.1 Write, solve, and/or apply a linear equation (including problem situations).

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.

A1.1.2.1.2 Use and/or identify an algebraic property to justify any step in an equation-solving process. Note: Linear equations only

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.

A1.1.2.1.3 Interpret solutions to problems in the context of the problem situation. Note: Linear equations only.

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 multi-step problems.

A1.1.3.1.1 Write or solve compound inequalities and/or graph their solution sets on a number line (may include absolute value inequalities).

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.

A1.1.3.1.2 Identify or graph the solution set to a linear inequality on a number line.

CC.2.2.HS.D.7 Create and graph equations or inequalities to describe numbers or relationships.

A1.1.3.1.3 Interpret solutions to problems in the context of the problem situation. Note: Linear inequalities only.

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 multi-step problems.


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Benchmark Cycle 1 Scope and Sequence

Chapter 2: Solving Equations

Suggested Dates

Chapter- Lesson

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Concept Bytea

Modeling One-Step Equations Use algebra tiles to model and solve singe-variable, one step equations.

A1.1.2.1.1, A1.1.2.1.2

2-1 Solving One-Step Equationsb Solve one-step equations in one variable. A1.1.2.1.1, A1.1.2.1.2, A1.1.2.1.3

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.

Concept Byte

Modeling Equations with Variables on Both Sides

Use algebra tiles to model and solve equations with variables on both sides.

A1.1.2.1.1, A1.1.2.1.2

2-4* Solving Equations with Variables on Both Sides

Solve equations with variables on both sides, and identify equations that are identities or that have no solution.

A1.1.2.1.1, A1.1.2.1.2, A1.1.2.1.3 2-5 Literal Equations and Formulas Rewrite and use literal equations and formulas.

2-6 Ratios, Rates, and Conversions Find ratios and rates and convert units and rates. A1.1.1.4.1, A1.1.2.1.1, A1.1.2.1.3

Concept Byte

Unit Analysis Use unit analysis to solve problems, and determine the reasonableness of solutions.

2-7 Solving Proportions Solve and apply proportions. A1.1.1.4.1, A1.1.2.1.1, A1.1.2.1.2, A1.1.2.1.3

2-8 Proportions and Similar Figures Find missing lengths in similar figures, and use similar figures when measuring indirectly.

a Concept Bytes are to be used either before or after lessons (as indicated) to build conceptual understanding, support development of problem-solving 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.

* Lessons marked with an asterisk may require two 45-minute class periods, for which flex days may be used.


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Chapter 2: Solving Equations (cont.)

Suggested Dates

Chapter- Lesson

Lesson Title Lesson Topic Eligible Content (s

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2-9* Percents Solve percent problems using proportions and percent problems using the percent equation.

A1.1.1.4.1, A1.1.2.1.1, A1.1.2.1.2, A1.1.2.1.3, 2-10* Change Expressed as a Percent

Find percent change and the relative error in linear and nonlinear measurements.

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 Content, specifically:

Chapter 1-1 Chapter 1-7

Chapter 1-2 Chapter 1-8

Concept Byte: 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 flex days, but note that not all of this content appears on the Algebra I Keystone Exam.

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 explaining 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 http://nlvm.usu.edu/), or SolveMe Puzzles (http://solveme.edc.org/).

Consider beginning the year by laying foundations for problem-solving via: WODB.ca, Estimation180.com, SolveMe. Also consider using flex 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.


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Chapter 3: Solving Inequalities

Suggested Dates

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3-1 Inequalities and Their Graphs Write, graph, and identify solutions of inequalities.

A1.1.3.1.2, A1.1.3.1.3

3-2 Solving Inequalities Using Addition or Subtraction

Use addition or subtraction to solve inequalities.

3-3* Solving Inequalities Using Multiplication or Division

Use multiplication or division to solve inequalities.

Concept Byte

More Algebraic Properties Review the reflexive, symmetric, and transitive properties of equality and introduce the transitive property of inequality.

Concept Byte

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 Content


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Chapter 3: Solving Inequalities (cont.)

Suggested Dates

Chapter- Lesson


(see

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3-6* Compound Inequalities Solve and graph inequalities containing the word and and or. A1.1.3.1.1, A1.1.3.1.2, A1.1.3.1.3 3-7

Absolute Value Equations and Inequalities

Solve equations and inequalities involving absolute value

3-8* Unions and Intersections of Sets Find the union and intersection of sets. Beyond Eligible

Content

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 extends 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 Exam. 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


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A1.1.1.1.1 Compare and/or order any real numbers. Note: Rational and irrational may be mixed.

CC.2.1.8.E.1 Distinguish between rational and irrational numbers using their properties. CC.2.1.8.E.4 Estimate irrational numbers by comparing them to rational numbers. CC.2.1.HS.F.1 Apply and extend the properties of exponents to solve problems with rational exponents. CC.2.1.HS.F.2 Apply properties of rational and irrational numbers to solve real-world or mathematical problems.

A1.1.1.1.2 Simplify square roots (e.g., √24 =

2√6).

CC.2.1.6.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.

A1.1.1.3.1 Simplify/evaluate expressions involving properties/laws of exponents, roots, and/or absolute values to solve problems. Note: Exponents should be integers from -10 to 10.

CC.2.1.HS.F.1 Apply and extend the properties of exponents to solve problems with rational exponents. CC.2.1.HS.F.2 Apply properties of rational and irrational numbers to solve real-world or mathematical problems. CC.2.2.8.B.1 Apply concepts of radicals and integer exponents to generate equivalent expressions.



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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 multi-step problems. CC.2.1.HS.F.5 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. CC.2.2.8.B.3 Analyze and solve linear equations and pairs of simultaneous linear equations. CC.2.2.8.C.1 Define, evaluate, and compare functions. CC.2.2.8.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.

A1.1.2.2.1 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.

CC.2.1.HS.F.5 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. CC.2.2.8.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.

A1.1.2.2.2 Interpret solutions to problems in the context of the problem situation. Note: Limit systems to two linear equations.

A1.1.3.2.1 Write and/or solve a system of linear inequalities using graphing. Note: Limit systems to two linear inequalities.

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.

A1.1.3.2.2 Interpret solutions to problems in the context of the problem situation. Note: Limit systems to two linear inequalities.


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A1.2.1.1.1 Analyze a set of data for the existence of a pattern and represent the pattern algebraically and/or graphically.

CC.2.2.8.C.1 Define, evaluate, and compare functions. CC.2.2.8.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 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.2.4.HS.B.2 Summarize, represent, and interpret data on two categorical and quantitative variables.

A1.2.1.1.2 Determine whether a relation is a function, given a set of points or a graph.

A1.2.1.1.3 Identify the domain or range of a relation (may be presented as ordered pairs, a graph, or a table).

A1.2.1.2.1 Create, interpret, and/or use the equation, graph, or table of a linear function.

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 multi-step problems. CC.2.2.8.B.2 Understand the connections between proportional relationships, lines, and linear equations. CC.2.2.8.C.1 Define, evaluate, and compare functions. CC.2.2.8.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.

A1.2.1.2.2 Translate from one representation of a linear function to another (i.e., graph, table, and equation).


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A1.2.2.1.1 Identify, describe, and/or use constant rates of change.

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.2.4.HS.B.1 Summarize, represent, and interpret data on a single count or measurement variable.

A1.2.2.1.2 Apply the concept of linear rate of change (slope) 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.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.

A1.2.2.1.3 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.

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.

A1.2.2.1.4 Determine the slope and/or y-intercept represented by a linear equation or graph.

CC.2.2.HS.C.1 Use the concept and notation of functions to interpret and apply them in terms of their context. CC.2.4.HS.B.1 Summarize, represent, and interpret data on a single count or measurement variable.


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Focus: In Chapter 4, students investigate modeling and functions. Students develop an understanding of the relationship between function notation, graphs, tables, and verbal expressions. They address: How can you represent and describe functions? How do functions describe real-world-situations? Consider reviewing, as needed, two topics from Grade 8 before beginning this chapter: “define, evaluate, and compare functions” (CC.2.2.8.C.1); and “use concepts of functions to model relationships between quantities” (CC.2.2.8.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.

Chapter 4: An Introduction to Functions

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4-1 Using Graphs to Relate Two Quantities

Represent mathematical relationships using graphs. A1.2.1.1.1.

4-2 Patterns and Linear Functions Identify and represent patterns that describe linear functions. A1.2.1.1.2, A1.2.1.2.1, A1.2.1.2.2

4-3 Patterns and Nonlinear Functions Identify and represent patterns that describe nonlinear functions.

A1.2.1.1.1, A1.2.1.1.2

4-4 Graphing a Function Rule Graph equations that represent functions. A1.2.1.1.1, A1.2.1.2.1, A1.2.1.2.2

Concept Byte

Graphing Functions and Solving Equations

Use a graphing calculator to graph functions and solve linear equations.

A1.2.1.2.2, A1.1.2.1.1

4-5 Writing a Function Rule Write equations that represent functions. A1.2.1.1.1, A1.2.1.2.1

4-6* Formalizing Relations and Functions

Determine whether a relation is a function and find domain, range, and use function notation.

A1.2.1.1.2, A1.2.1.1.3

4-7* Arithmetic Sequences Identify and extend patterns in sequences and represent arithmetic sequences using function notation.

A1.2.1.1.1, A1.2.1.2.1


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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.2.2.7.B.3); and “analyze and solve linear equations and pairs of simultaneous linear equations” (8.CC.2.2.8.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 Exam, the essential pieces of Chapter 5 are covered here. The Concept Byte 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 Exam).

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.

Chapter 5: Linear Functions

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5-1 Rate of Change and Slope Find rates of change from tables and find slope. A1.2.2.1.1, A1.2.2.1.2

5-2 Direct Variation Write and graph an equation of a direct variation. A1.2.2.1.1, A1.2.2.1.2, A1.2.2.1.3

Concept Byte

Investigating y = mx + b Use a graphing calculator to investigate the slope-intercept form of a linear equation.

A1.2.2.1.2, A1.2.2.1.3, A1.2.2.1.4

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 Graph linear equations using intercepts, and write linear equations in standard form.

Concept Byte

Inverse of a Linear Function Build an understanding of inverse functions. Beyond Eligible

Content

5-6 Parallel and Perpendicular Lines Determine whether lines are parallel, perpendicular, or neither, and write equations of parallel and perpendicular lines.

A1.1.2.2.1, A1.1.2.2.2


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Chapter 6: Systems of Equations and Inequalities

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6-1 Solving Systems by Graphing Solve systems of equations by graphing, and analyze special systems.

A1.1.2.2.1, A1.1.2.2.2

Concept Byte

Solving Systems Using Tables and Graphs

Use the table and graphing functions on a graphing calculator to solve a system of linear equations.

Concept Byte

Solving Systems Using Algebra Tiles

Use algebra tiles to solve systems of linear equations.

6-2* Solving Systems Using Substitution

Solve systems of equations using substitution.

6-3* Solving Systems Using Elimination

Solve systems by adding or subtracting to eliminate a variable.

Concept Byte

Matrices and Solving Systems Use augmented matrices to solve systems of linear equations. Beyond Eligible

Content

6-4 Applications of Linear Systems Choose the best method for solving a system of linear equations.

A1.1.2.2.1, A1.1.2.2.2

6-5* Linear Inequalities Graph linear inequalities in two variables, and use linear inequalities when modeling real-world situations.

A1.1.3.2.1, A1.1.3.2.2

6-6 Systems of Linear Inequalities Solve systems of linear inequalities by graphing, and model real-world situations by using systems of linear inequalities.

Concept Byte

Graphing Linear Inequalities Use a graphing calculator to graph linear inequalities.

Focus: In Chapter 6, students explore 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 Concept Byte 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 (exponents), and Chapter 8 (polynomials, including quadratic expressions)—all of which address Module 1 content. Some books proceed from linear functions into quadratics, but many states, like PA, do not expect Algebra I students to study quadratic functions. Therefore, this sequence of chapters focuses on composing and decomposing several linear expressions 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 explain their thinking (SMP3).


16

Chapter 7: Exponents and Exponential Functions

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7-1 Zero and Negative Exponents Simplify expressions involving zero and negative exponents.

A1.1.1.3.1, A1.1.1.4.1

Concept Byte

Multiplying Powers Understand patterns that facilitate the multiplication of powers.

7-2* Multiplying Powers with the Same Base

Multiply powers with the same base.

Concept Byte

Powers of Powers and Powers of Products

Discover rules for powers of powers and powers of products.

7-3* More Multiplication Properties of Exponents

Raise a power to a power, and raise a product to a power.

7-4* Division Properties of Exponents Divide powers with the same base, and raise a quotient to a power.

Concept Byte

Relating Radicals to Rational Exponents

Understand the relationship between radicals and rational exponents.

A1.1.1.1.1, A1.1.1.1.2, A1.1.1.4.1

7-5 Rational Exponents and Radicals Rewrite expressions involving radicals and rational exponents. A1.1.1.1.2, A1.1.1.3.1

Focus: In Chapter 7, students investigate these essential questions: How can you represent very large and very small numbers? How can you simplify expressions involving exponents? 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 expressions to studying polynomials. Building on students understanding of exponents and inverse functions (Chapter 5), students will compose and decompose linear expressions in performing operations with and factoring polynomial expressions. The last three sections of Chapter 7 go beyond the eligible content for Algebra I. Some textbooks choose to explore properties of exponents (Chapters 7-2 to 7-4) before justifying results about zero and negative exponents (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 problem-solving and reasoning abilities, including exploring with calculators and whole numbers, to conjecture and justify these rules themselves.

Benchmark 2 Window: Jan 29 – Feb 13


17



A1.1.1.1.2 Simplify square roots (e.g., √24 =

2√6).

CC.2.1.6.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.

A1.1.1.3.1 Simplify/evaluate expressions involving properties/laws of exponents, roots, and/or absolute values to solve problems. Note: Exponents should be integers from –10 to 10

CC.2.1.HS.F.1 Apply and extend the properties of exponents to solve problems with rational exponents. CC.2.1.HS.F.2 Apply properties of rational and irrational numbers to solve real-world or mathematical problems. CC.2.2.8.B.1 Apply concepts of radicals and integer exponents to generate equivalent expressions.



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 multi-step problems. CC.2.1.HS.F.5 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. CC.2.2.8.B.3 Analyze and solve linear equations and pairs of simultaneous linear equations. CC.2.2.8.C.1 Define, evaluate, and compare functions. CC.2.2.8.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.


18


A1.1.1.5.1 Add, subtract, and/or multiply polynomial expressions (express answers in simplest form). Note: Nothing larger than a binomial multiplied by a trinomial.

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.5 Use polynomial identities to solve problems. CC.2.2.HS.D.6 Extend the knowledge of rational functions to rewrite in equivalent forms.

A1.1.1.5.2 Factor algebraic expressions, including difference of squares and trinomials. Note: Trinomials are limited to the form ax2 + bx + c where a is equal to 1 after factoring out all monomial factors.

A1.1.1.5.3 Simplify/reduce a rational algebraic expression.

A1.2.3.1.1 Calculate and/or interpret the range, quartiles, and interquartile range of 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.

A1.2.3.2.2 Analyze data, make predictions, and/or answer questions based on displayed data (box-and-whisker plots, stem-and-leaf plots, scatter plots, measures of central tendency, or other representations).

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, experiments, and observational studies.

A1.2.3.2.3 Make predictions using the equations or graphs of best-fit lines of scatter plots.


19


Chapter 8: Polynomials and Factoring

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8-1* Adding and Subtracting Polynomials

Classify, add, and subtract polynomials. A1.1.1.5.1

8-2* Multiplying and Factoring Multiply a monomial by a polynomial, and factor a monomial from a polynomial.

A1.1.1.2.1 A1.1.1.5.1, A1.1.1.5.2

Concept Byte

Using Models to Multiply Use models to multiply binomials.

A1.1.1.5.1 8-3* Multiplying Binomials Multiply two binomials or a binomial by a trinomial.

8-4* Multiplying Specials Cases Find the square of a binomial, and find the product of a sum and difference.

Concept Byte

Using Models to Factor Use models to factor trinomials. A1.1.1.5.2

8-5* Factoring x2 + bx + c Factor trinomials of the form x2 + bx + c.

8-6* Factoring ax2 + bx + c Factor trinomials of the form ax2 + bx + c.

A1.1.1.5.2 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.

Focus: Chapter 8 addresses these essential questions: Can two algebraic expressions 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 exponents in Chapter 7. In this chapter, students combine linear expressions to make quadratic expressions, 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 expressions.

Rigor: Students may reason incorrectly about algebraic expressions, 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.


20

Chapter 9: Quadratic Functions and Equations

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9-1*

Quadratic Graphs and Their Properties

Graph quadratic functions of the form y = ax2 and y = ax2 + c.


9-2 Quadratic Functions Graph quadratic functions of the form y = ax2 + bx + c

Concept Byte

Rates of Increase Use functions, tables, and graphs to determine and compare rates of change.

9-3* Solving Quadratic Equations Solve quadratic equations by graphing and using square roots.

Concept Byte

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 expressions 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 Exam, 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.


21

Chapter 10: Radical Expressions and Equations

Suggested Dates

Chapter- Lesson


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10-1 The Pythagorean Theorem

Solve problems using the Pythagorean Theorem, and identify right triangles.

M08.C-G.2.1.1, M08.C-G.2.1.2

10-2* Simplifying Radicals Simplify radicals involving products and quotients.

A1.1.1.1.2, A1.1.1.3.1, A1.1.1.4.1


10-3* Operations with Radical Expressions

Simplify sums, differences, products, and quotients of radical expressions.

A1.1.1.1.2, A1.1.1.3.1, A1.1.1.4.1

10-4* Solving Radical Equations Solve equations containing radicals, and identify extraneous solutions.


Focus: Chapter 10 addresses these essential questions: How are radical expressions represented? Can two radical expressions 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 Exam, but it could be included as an extension if time permits.

Coherence: Having solved basic quadratic equations in Chapter 9-3, students should see the importance of techniques for simplifying radical expressions. This is because there are many equivalent ways of writing such expressions and so simplifying helps with agreement. The latter part of Chapter 10 goes beyond the eligible content for the Keystone Exam, 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.


22

Chapter 11: Rational Expressions and Functions

Suggested Dates

Chapter- Lesson


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11-1* Simplifying Rational Expressions Simplify rational expressions.

A1.1.1.5.3

11-2* Multiplying and Dividing Rational Expressions

Multiply and divide rational expressions, and simplify complex fractions.

Concept Byte

Dividing Polynomials Using Algebra Tiles

Use algebra tiles to divide polynomials. Beyond Eligible

Content

Focus: Chapter 11 asks: How are rational expressions related? Simplifying complex fractions is an extension 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 Exam, the remainder of Chapter 11 is covered later in the year.

Rigor: Consider including the Concept Byte after Chapter 11-2, modeling the division of polynomials, to support transition to the rest of Ch 11.


23

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

Chapter 5: Linear Functions

Suggested Dates

Chapter- Lesson


4/3

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on

+ 2

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5-7 Scatter Plots and Trend Lines

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.

A1.2.3.1.1, A1.2.3.2.2, A1.2.3.2.3

Concept Byte

Using Residuals Build an understanding of scatter plots and lines of best fit.

5-8* Graphing Absolute Value Functions

Graph an absolute value function, and translate the graph of an absolute value function.


Concept Byte

Characteristics of Absolute Value Graphs

Investigate absolute value graphs and their characteristics.


24

Cycle 4 Standards


A1.2.3.2.1 Estimate or calculate to make predictions based on a circle, line, bar graph, measure of central tendency, or other representation.

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, experiments, and observational studies. A1.2.3.2.2 Analyze data, make predictions, and/or

answer questions based on displayed data (box-and-whisker plots, stem-and-leaf plots, scatter plots, measures of central tendency, or other representations).

A1.2.3.2.3 Make predictions using the equations or graphs of best-fit lines of scatter plots.

A1.2.3.1.1 Calculate and/or interpret the range, quartiles, and interquartile range of 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.

A1.2.3.3.1 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.

CC.2.4.7.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 experiments. CC.2.4.HS.B.7 Apply the rules of probability to compute probabilities of compound events in a uniform probability model.


25

Cycle 4 Scope and Sequence

Chapter 12: Data Analysis and Probability

Suggested Dates

Chapter- Lesson


4/9

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12-1 Organizing Data Using Matrices Organize data in a matrix. Beyond Eligible

Content

12-2 Frequency and Histograms Make and interpret frequency tables and histograms. A1.2.3.2.2

12-3 Measures of Central Tendency and Dispersion

Find mean, median, mode, and range. A1.2.3.2.1, A1.2.3.2.2

Concept Byte

Standard Deviation Find the standard deviation of a data set. Beyond Eligible

Content

12-4 Box-and-whisker plots Make and interpret box-and-whisker plots. A1.2.3.1.1, A1.2.3.2.1, A1.2.3.2.2

Concept Byte

Designing Your Own Survey Explore methods of collecting data.

A1.2.3.2.2 12-5* Samples and Surveys Classify data and analyze samples and surveys.

Concept Byte

Two-Way Frequency Table Use two-way frequency tables to analyze data and to make predictions.

12-6 Permutations and Combinations Find permutations and combinations. A1.2.3.3.1

12-7* Theoretical and Experimental Probability

Find theoretical and experimental probability.

Concept Byte

Conducting Simulations Learn what a simulation is, why simulations are used, and some of the different methods that can be used to simulate experiments.



26

Chapter 12: Data Analysis and Probability (cont.)

Suggested Dates

Chapter- Lesson


(see

pre

vio

us)

12-8* Probability of Compound Events Find probabilities of mutually exclusive and overlapping events, as well as independent and dependent events.

A1.2.3.3.1

Concept Byte

Normal Distribution Use normal distributions to estimate population percentages, and determine whether it is appropriate to model a data set with a normal distribution.


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 Exam, 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 exists 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 explain their thinking (SMP3).

Algebra I Keystone Exam 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 Content. Consider honing-in on essential content by reviewing the Performance Level Descriptors at http://static.pdesas.org/content/documents/Keystone_Algebra_I_PLD_021113.pdf. And consider using practice problems from http://www.pdesas.org/.

http://static.pdesas.org/content/documents/Keystone_Algebra_I_PLD_021113.pdf

http://www.pdesas.org/


27

Remediation and Extension Options

5/29-6/12

11 Days

Option 1: Revisit Common

Core Performance

Tasks

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 use them to summarize and wrap-up the school year.

Option 2: Cover Remaining

Content

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 pre-teach or what content they would like you to revisit. On the following pages, you will see several possibilities, related to exponents and exponential functions (Chapter 7), quadratic functions and equations (Chapter 9), radical expressions and equations (Chapter 10), and rational expressions and functions (Chapter 11).

Option 3: Mathematical Modeling in Three Acts

Students may remember Three-Act mathematical modeling activities from Grades 6-8. 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 three-act modeling activities at: https://whenmathhappens.com/3-act-math/, https://gfletchy.com/3-act-lessons/, or within this Google Sheet.

Option 4: Miscellaneous

(Financial Literacy, Quantitative Reasoning,

Statistics, etc.)

The Federal Reserve Bank of Philadelphia has posted free lesson plans for high school teachers on financial literacy at: https://philadelphiafed.org/education/teachers/lesson-plans?tabNum=3. In addition, Census.gov has a number of data analysis activities, using real U.S. Census data, at: https://census.gov/schools/activities/math.html.html. Finally, consider other cross-curricular quantitative reasoning activities, such as those available at: https://ww2.kqed.org/education/.

https://whenmathhappens.com/3-act-math/

https://gfletchy.com/3-act-lessons/

https://docs.google.com/spreadsheets/d/1jXSt_CoDzyDFeJimZxnhgwOVsWkTQEsfqouLWNNC6Z4/edit#gid=0

https://philadelphiafed.org/education/teachers/lesson-plans?tabNum=3

https://census.gov/schools/activities/math.html.html

https://ww2.kqed.org/education/


28

Chapter 7: Exponents and Exponential Functions

Suggested Dates

Chapter- Lesson

Lesson Title Lesson Topic Eligible Content O

ne

45

-m

inu

te

per

iod

per

less

on

7-6 Exponential Functions Evaluate and graph exponential functions. Beyond Eligible

Content 7-7 Exponential Growth and Decay Model exponential growth and decay.

7-8 Geometric Sequences Write and use recursive formulas for geometric sequences.

Focus: The remainder of Chapter 7 builds on properties of exponentiation, to explore the characteristics of exponential functions.

Coherence: These sections develop an understanding of families of functions that is important in Algebra II. In addition, studying geometric sequences supports students’ understanding of functions and modeling real-world situations with mathematics.

Rigor: Looking for and making use of structure, and applying regularity in reasoning, permeate functions and geometric sequences.

Chapter 9: Quadratic Functions and Equations

Suggested Dates

Chapter- Lesson


On

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5-m

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(2

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)

9-4 Factoring to Solve Quadratic Equations

Solve quadratic equations by factoring.


Concept Byte

Writing Quadratic Equations Write quadratic equations and related quadratic functions when the roots of a quadratic equation or the zeros of the related quadratic function are known.

9-5* Completing the Square Solve quadratic equations by completing the square.

9-6* The Quadratic Formula and the Discriminant

Solve quadratic equations using the quadratic formula, and find the number of solutions of a quadratic equation.

9-7* Linear, Quadratic, and Exponential Models

Choose a linear, quadratic, or exponential model for data.


29

Chapter 9: Quadratic Functions and Equations (cont.)

Suggested Dates

Chapter- Lesson


(see

pre

vio

us)

Concept Byte

Analyzing Residual Plots Use residual plots to analyze the fit of quadratic and exponential regression for sets of data. Beyond Eligible

Content 9-8*

Systems of Linear and Quadratic Equations

Solve systems of linear and quadratic equations.

Focus: Chapter 9 concludes by studying quadratic equations and related functions. This content goes beyond the eligible content for the Algebra I Keystone Exam, but it is foundational to Algebra II.

Coherence: If time permits, an exploration of Chapter 9 supports students as they move into Geometry and Algebra II. Many topics of study in Geometry (like area) involve quadratic relationships, and Algebra II devotes significant time to studying non-linearity. This chapter builds on students understanding of roots, radicals, and inverse functions, studied earlier in the year and in less formal ways in prior grades.

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.

Chapter 10: Radical Expressions and Equations

Suggested Dates

Chapter- Lesson


(see

pre

vio

us)

10-5 Graphing Square Root Functions Graph square root functions, and translate graphs of square root functions. Beyond Eligible

Content 10-6* Trigonometric Ratios Find and use trigonometric ratios.

Focus: Chapter 10 concludes with a study of the square root function and trigonometric rations, topics typically covered in Geometry and Algebra II and that are beyond the eligible content on the Algebra I Keystone Exam.

Coherence: These sections extend students’ understanding of inverse functions of quadratics, allowing for connections to Geometry and Algebra II via triangles and function families, respectively.

Rigor: Consider introducing the Common Core Performance Task to students before covering this content and using their ideas on solutions, to lead into work on these sections. Then return to the CCPT after concluding the chapter, using the methods studied throughout Chapter 10.


30

Chapter 11: Rational Expressions and Functions

Suggested Dates

Chapter- Lesson

Lesson Title Lesson Topic Eligible Content O

ne

45

-min

ute

per

iod

per

le

sso

n (

2 p

erio

ds

for

less

on

s

wit

h a

ster

isks

) 11-3* Dividing Polynomials Divide polynomials.


11-4* Adding and Subtracting Rational Expressions

Add and subtract rational expressions.

11-5* Solving Rational Equations Solve rational equations and proportions.

11-6* Inverse Variation Write and graph equations for inverse variation.

11-7* Graphing Rational Functions Compare direct and inverse variation.

Concept Byte

Graphing Rational Functions Graph rational functions.

Focus: Chapter 11 concludes by asking: What are characteristics of rational functions? How can you solve and make sense of solutions of rational equations? This material goes beyond the eligible content for the state assessment.

Coherence: This chapter extends the work done, earlier in Chapter 11, on simplifying rational expressions.

Rigor: Students develop a stronger understanding of families of functions in this chapter. Make connections to working with rational numbers during this chapter, to promote depth of understanding. Consider using the Common Core Performance Task to introduce and wrap-up this material. In doing so, students should share their solution strategies and explain their thinking, as they refine their approach.


31

PA Core Standards by Cycle

CYCLES

PA Core Standards

CC

.2.1

.6.E

.3

CC

.2.1

.8.E

.1

CC

.2.1

.8.E

.4

CC

.2.2

.7.B

.3

CC

.2.2

.8.B

.1

CC

.2.2

.8.B

.2

CC

.2.2

.8.B

.3

CC

.2.2

.8.C

.1

CC

.2.2

.8.C

.2

CC

.2.4

.7.B

.3

CC

.2.1

.HS.

F.1

CC

.2.1

.HS.

F.2

CC

.2.1

.HS.

F.3

CC

.2.1

.HS.

F.4

CC

.2.1

.HS.

F.5

CC

.2.2

.HS.

D.1

CC

.2.2

.HS.

D.2

CC

.2.2

.HS.

D.3

CC

.2.2

.HS.

D.5

CC

.2.2

.HS.

D.6

CC

.2.2

.HS.

D.7

CC

.2.2

.HS.

D.8

CC

.2.2

.HS.

D.9

CC

.2.2

.HS.

D.1

0

CC

.2.2

.HS.

C.1

CC

.2.2

.HS.

C.2

CC

.2.2

.HS.

C.3

CC

.2.2

.HS.

C.4

CC

.2.2

.HS.

C.5

CC

.2.2

.HS.

C.6

CC

.2.4

.HS.

B.1

CC

.2.4

.HS.

B.2

CC

.2.4

.HS.

B.3

CC

.2.4

.HS.

B.4

CC

.2.4

.HS.

B.5

CC

.2.4

.HS.

B.7

1 2 3 4

Eligible Content ↓

MO

DU

LE 1

A1.1.1.1.1 x

A1.1.1.1.2 x x

A1.1.1.2.1 x

A1.1.1.3.1 x x

A1.1.1.4.1 x x x

A1.1.1.5.1 x

A1.1.1.5.2 x

A1.1.1.5.3 x

A1.1.2.1.1 x x

A1.1.2.1.2 x

A1.1.2.1.3 x

A1.1.2.2.1 x

A1.1.2.2.2 x

A1.1.3.1.1 x

A1.1.3.1.2 x

A1.1.3.1.3 x

A1.1.3.2.1 x

A1.1.3.2.2 x


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CYCLES

PA Core Standards

CC

.2.1

.6.E

.3

CC

.2.1

.8.E

.1

CC

.2.1

.8.E

.4

CC

.2.2

.7.B

.3

CC

.2.2

.8.B

.1

CC

.2.2

.8.B

.2

CC

.2.2

.8.B

.3

CC

.2.2

.8.C

.1

CC

.2.2

.8.C

.2

CC

.2.4

.7.B

.3

CC

.2.1

.HS.

F.1

CC

.2.1

.HS.

F.2

CC

.2.1

.HS.

F.3

CC

.2.1

.HS.

F.4

CC

.2.1

.HS.

F.5

CC

.2.2

.HS.

D.1

CC

.2.2

.HS.

D.2

CC

.2.2

.HS.

D.3

CC

.2.2

.HS.

D.5

CC

.2.2

.HS.

D.6

CC

.2.2

.HS.

D.7

CC

.2.2

.HS.

D.8

CC

.2.2

.HS.

D.9

CC

.2.2

.HS.

D.1

0

CC

.2.2

.HS.

C.1

CC

.2.2

.HS.

C.2

CC

.2.2

.HS.

C.3

CC

.2.2

.HS.

C.4

CC

.2.2

.HS.

C.5

CC

.2.2

.HS.

C.6

CC

.2.4

.HS.

B.1

CC

.2.4

.HS.

B.2

CC

.2.4

.HS.

B.3

CC

.2.4

.HS.

B.4

CC

.2.4

.HS.

B.5

CC

.2.4

.HS.

B.7

1 2 3 4

Eligible Content ↓

MO

DU

LE 2

A1.2.1.1.1 x

A1.2.1.1.2 x

A1.2.1.1.3 x

A1.2.1.2.1 x

A1.2.1.2.2 x

A1.2.2.1.1 x

A1.2.2.1.2 x

A1.2.2.1.3 x

A1.2.2.1.4 x

A1.2.2.2.1 ` x

A1.2.3.1.1 x

A1.2.3.2.1 x

A1.2.3.2.2 x x

A1.2.3.2.3 x

A1.2.3.3.1 x


33

Algebra Eligible Content Taught

PA Eligible Content Cycle

1 Cycle

2 Cycle

3 Cycle

4

A1.1.1.1.1 Compare and/or order any real numbers. Note: Rational and irrational may be mixed. x

A1.1.1.1.2 Simplify square roots (e.g., √24 = 2√6). x x

A1.1.1.2.1 Find the Greatest Common Factor (GCF) and/or the Least Common Multiple (LCM) for sets of monomials. x

A1.1.1.3.1 Simplify/evaluate expressions involving properties/laws of exponents, roots, and/or absolute values

to solve problems.Note: Exponents should be integers from –10 to 10. x x

A1.1.1.4.1 Use estimation to solve problems. x x x

A1.1.1.5.1 Add, subtract, and/or multiply polynomial expressions (express answers in simplest form). Note: Nothing larger than a binomial multiplied by a trinomial. x

A1.1.1.5.2 Factor algebraic expressions, including difference of squares and trinomials. Note: Trinomials are

limited to the formax2+ bx + c where a is equal to 1 after factoring out all monomial factors. x

A1.1.1.5.3 Simplify/reduce a rational algebraic expression. x

A1.1.2.1.1 Write, solve, and/or apply a linear equation (including problem situations). x x

A1.1.2.1.2 Use and/or identify an algebraic property to justify any step in an equation-solving process. Note: linear equations only. x

A1.1.2.1.3 Interpret solutions to problems in the context of the problem situation. Note: linear equations only. x

A1.1.2.2.1 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. x

A1.1.2.2.2 Interpret solutions to problems in the context of the problem situation. Note: Limit systems to two linear equations. x


34


1 Cycle

2 Cycle

3 Cycle

4

A1.1.3.1.1 Write or solve compound inequalities and/or graph their solution sets on a number line (may include absolute value inequalities). x

A1.1.3.1.2 Identify or graph the solution set to a linear inequality on a number line. x

A1.1.3.1.3 Interpret solutions to problems in the context of the problem situation. Note: linear inequalities only. x

A1.1.3.2.1 Write and/or solve a system of linear inequalities using graphing. Note: limit systems to two linear inequalities. x

A1.1.3.2.2 Interpret solutions to problems in the context of the problem situation. Note: limit systems to two linear inequalities. x

A1.2.1.1.1 Analyze a set of data for the existence of a pattern and represent the pattern algebraically and/or graphically. x

A1.2.1.1.2 Determine whether a relation is a function, given a set of points or a graph. x

A1.2.1.1.3 Identify the domain or range of a relation (may be presented as ordered pairs, a graph, or a table). x

A1.2.1.2.1 Create, interpret, and/or use the equation, graph, or table of a linear function. x

A1.2.1.2.2 Translate from one representation of a linear function to another (i.e., graph, table, and equation). x

A1.2.2.1.1 Identify, describe, and/or use constant rates of change. x

A1.2.2.1.2 Apply the concept of linear rate of change (slope) to solve problems. x

A1.2.2.1.3 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. x

A1.2.2.1.4 Determine the slope and/or y-intercept represented by a linear equation or graph. x


35


1 Cycle

2 Cycle

3 Cycle

4

A1.2.2.2.1 Draw, identify, find, and/or write an equation for a line of best fit for a scatter plot. x

A1.2.3.1.1 Calculate and/or interpret the range, quartiles, and interquartile range of data. x

A1.2.3.2.1 Estimate or calculate to make predictions based on a circle, line, bar graph, measure of central tendency, or other representation. x

A1.2.3.2.2 Analyze data, make predictions, and/or answer questions based on displayed data (box-and- whisker plots, stem-and-leaf plots, scatter plots, measures of central tendency, or other representations). x x

A1.2.3.2.3 Make predictions using the equations or graphs of best-fit lines of scatter plots. x

A1.2.3.3.1 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. x


36

Document Information Page Overview of Contents of Document

The Cover Page lays out the Topics taught within each Cycle as well as the corresponding dates.

The Benchmark Cycle X Standards pages that precede each cycle outline all of the standards that are taught within that cycle. These are, therefore, all of the standards that may be on that Cycle’s Benchmark.

The Benchmark Cycle X Scope and Sequence pages provide suggested pacing that allows for 1 day per lesson plus some flex days. We recommend proactively using the flex days for: reviewing pre-requisite content, splitting lessons over multiple days, assessing, reteaching, and doing projects. Of course, some of these will also be taken by field trips and other school activities.

The PA Core Standards and Eligible Content by Cycle page lists all of the standards and indicates in which cycle(s) they are taught.

What is a Cycle?

We want to offer clarity on what appears on each benchmark. Additionally, there should be sufficient time to teach that content before it is tested. Because each school administers the benchmark on a different day, not necessarily corresponding with the last day of the Term, we have created Cycles. Each Cycle contains the content that is to be taught and tested on a given benchmark. Please refer to the dates on the Cover Page to ensure you are aware of the beginning and ending dates for each Cycle.

What If I Fall Behind?

We trust you to make decisions about what is best for your students. This pacing will prepare you for the Benchmarks and Algebra 1 Keystone Exam, but it is a suggested, not mandated, pacing. You may also wish to move at a faster pace. Do not feel you should slow down to match this guide.

If you are concerned about content that you may not reach before the Keystone Exam, consider implementing number talks and other short routines and games. For example, a lot of mathematics vocabulary and concepts could be taught through Which Once Doesn’t Belong. Rather than pushing to “cover” content, or using test prep resources, content can be infused through short but meaningful structures.


37

The Standards for Mathematical Practice are:

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

4. Model with mathematics

5. Use appropriate tools strategically

6. Attend to precision

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoning

You should try to infuse these Standards into your lessons regularly. Rather than thinking of them as a box to check, think about how you are providing students with opportunities to develop as mathematical thinkers and doers.

The Common Core has identified the following as the critical areas for Algebra I:

(1) Students analyze and explain the process of solving an equation, developing fluency in writing, interpreting, and translating between various forms of linear equations and inequalities, and using them to solve problems;

(2) Students learn function notation and develop the concepts of domain and range; they interpret functions given graphically, numerically, symbolically, and verbally, translate between representations, and understand the limitations of various representations; they use this understanding to explore systems of linear equations and inequalities;

(3) Students use regression techniques to describe approximately linear relationships between quantities, and they use graphical representations and knowledge of the context to make judgments about the appropriateness of linear models;

(4) They extended the laws of exponents to rational exponents; and (5) Students consider quadratic functions, comparing the key characteristics of quadratic functions to those of linear and exponential functions.

These do not necessarily reflect Keystone Exam content, but they are generally the essential content for preparing students for their studies of higher mathematics, including Geometry and Algebra II.

The School District of Philadelphia’s Vision for Mathematics Teaching and Learning:

All students think mathematically, and they will be empowered to own, share, and do mathematics.

Our Guiding Principles:

Equitable Discourse Rich + Meaningful Tasks Purpose-Driven Work

Questioning and Curiosity Valuing Diverse Thinking

If you have any questions or would like more information, please contact us at [email protected].

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