Curriculum Guide - Algebra I Introduction

Transcription

1 Algebra I Curriculum Guide Curriculum Guide - Algebra I Introduction Appropriate Common Core State Standards and Clusters are followed by one of the following symbols. Major Clusters/Standards Supporting Clusters/Standards o Additional Clusters/Standards High school mathematical modeling standards FS Fluency Standard All testable standards (SPIs) from the 'TCAP-EOC Algebra I Framework' have been embedded within this guide. Common Core Mathematical Practice Standards The CCSS for Mathematical Practices are expected to be integrated into every mathematics lesson for all students grades K 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. Last updated on 5/27/2014 Additional resources, CRAs, Instructional Tasks, etc. listed at the bottom of each unit will be found on the sharing server. The file name will be in parentheses after a brief description and will be in: Sharing/Consulting Teachers/Math/Algebra I/Algebra I Resources Questions or comments should be directed to Karl Bittinger, Math Curriculum Consulting Teachers. Page 1 of 56

2 Introduction Clarifications, Evidence, and Assessment PBA - Performance Based Assessment (PBA) evidence statements, clarifications, math practice standards, and calculator usage were taken from the following location: EOY - End of Year (EOY) evidence statements, clarifications, math practice standards, and calculator usage were taken from the following location: PLD - Performance Level Descriptors (PLD) Level 5 descriptors were taken from the following location: Calculator - When we begin Common Core assessments in , students will only be permitted to use the online calculator for state assessments, which will be similar to the TI-84. While this calculator policy will not be enacted for the school year, it has been left in this document to help teachers prepare for the upcoming Common Core changes. Yes indicates a calculator will be accessible through the computer for the indicated assessment No indicates a calculator will not be accessible through the computer for the indicated assessment Neutral indicates a calculator will be accessible through the computer for the indicated assessment but may not be needed Item specific indicates the standard will only have a calculator accessible for certain items on the assessment A balanced use of calculators continues to be encouraged. Limitations - Assessment limits for standards assessed on more than one end-of-course (EOC) test: Algebra I, Geometry, and Algebra II. Limitations can be found on pages 56 through 59 of the Model Content Frameworks, Mathematics, Grades 3-11 at: - For the purposes of CMCSS pacing, Learning Targets are written in teacher friendly language. The Learning Targets in our pacing guides should be completely aligned to the content standards and exhaust the meaning of the standards. The Learning Targets may lead to clear targets. Page 2 of 56

3 Unit Schedule 1st Semester Unit Title Dates Days 1 Numbers and Expressions August 6 - August 20, Linear Reasoning August 21 - September 11, Function Concepts September 12 - September 29, Building Functions September 30 - October 30, Systems October 31 - November 25, Polynomial Operations December 1 - December 9, First Semester Review and Exam December 10 - December 19, Total Days 86 2nd Semester 8 Factoring January 6 - January 27, Quadratic Function Concepts January 28 - February 10, Solving Quadratic Functions February 11 - March 4, Modeling with Functions March 5 - March 24, Data and Statistics March 25 - April 15, Geometry and Probability April 16 - April 28, EOC Review and Assessment April 29 - May 4, Second Semester Review and Exam May 5 - May 21, Total Days 90 Assessments Dates End-of-Course Assessment May 4, 2015 Page 3 of 56

4 Unit 1 Unit 1: Numbers and Expressions 12 Days: August 6 - August 20, 2014 Standard Clarifications, Evidence, and Assessment N.RN.B.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.. o PBA - Base explanations/reasoning on the properties of rational and irrational numbers. - For rational solutions, exact values are required. For irrational solutions, exact or decimal approximations may be required. Simplifying or rewriting radicals is not required for Common Core. - MP 3 Calculator - Yes EOY - Apply properties of rational and irrational numbers to identify rational and irrational numbers. - For rational solutions, exact values are required. For irrational solutions, exact or decimal approximations may be required. Simplifying or rewriting radicals is not required for common core. - MP 6 Calculator - No PLD (PBA and EOY) - Identifies rational and irrational numbers. - Calculates sums and products of two rational and/or irrational numbers and determines whether and generalizes when the sums and products are rational or irrational. - Recognize and understand why the sum of a rational number and an irrational number is irrational. Page 4 of 56

5 Unit 1 A.SSE.A.1 Interpret expressions that represent a quantity in terms of its context. A.SSE.A.1a Interpret parts of an expression, such as terms, factors, and coefficients. EOY - See illustrations at - MP 7 Calculator - Neutral - Interpret parts of an expression, such as terms, factors, and coefficients in terms of the context. EOY - See illustrations at - MP 7 Calculator - Neutral - Define and recognize parts of an expression, such as terms, factors, and coefficients. A.SSE.A.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r) n as the product of P and a factor not depending on P. FS SPI Describe and/or order a given set of real numbers including both rational and irrational numbers SPI Multiply, divide, and square numbers expressed in scientific notation EOY - See illustrations at - MP 7 Calculator - Neutral - Define and recognize parts of a complicated expression, such as terms, factors, and coefficients in terms of the contextual situation. - interpret complicated expressions, in terms of the context, by describing the parts that make up the expression in a variety of ways. Students were taught to add, subtract, multiply, and divide numbers expressed in scientific notation in Math 8 during the school year. Page 5 of 56

6 Unit 1 SPI Operate (add, subtract, multiply, divide, simplify, powers) with radicals and radical expressions including radicands involving rational number and algebraic expressions This concept may be introduced in this unit and taught in context with the Pythagorean Theorem at the end of the year. Additional Resources: Assessments: Unit Vocabulary: Instructional Tasks: STEM Integration Common Student Misconceptions Prerequisite Skills Constructed Response Assessments Page 6 of 56

7 Unit 2 Unit 2: Linear Reasoning 15 Days: August 21 - September 11, 2014 Standard Clarifications, Evidence, and Assessment A.REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. A.REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. EOY - MP 7 Calculator - Item Specific PLD (PBA and EOY) - Algebraically solve linear equations, linear inequalities, and quadratics in one variable (at complexity appropriate to the course), including those with coefficients represented by letters. - Utilize structure and rewriting as strategies for solving. - Identify and correct errors in a given solution. SPI (include equations containing compound inequalities and absolute value) - Solve linear equations with one variable, including equations with coefficients represented by letters (literal equations). - Solve linear inequalities with one variable, including equations with coefficient represented by letters (literal equations). - Solve linear absolute value inequalities and graph solutions on a number line. Limitations - Tasks are limited to quadratic equations. SPI Justify and defend explanations of how to solve simple equations. Page 7 of 56

8 Unit 2 A.CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. PBA - Solve multi-step contextual word problems with degree of difficulty appropriate to the course, require application of courselevel knowledge and skills articulated in A.CED, N.Q, A.SSE.B.3, A.REI.C.6, A.REI.D.12, A.REI.D.11, limited to linear equations and exponential equations with integer exponents. - MP 2 and 4 Calculator - Yes Limitations - Tasks are limited to linear, quadratic, or exponential equations with integer exponents. SPI , SPI (linear) - Create and solve equations and inequalities for a given contextual problem. A.CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. EOY - Solve multi-step contextual problems that require writing and analyzing systems of linear inequalities in two variables to find viable solutions. - Tasks have hallmarks of modeling as a mathematical practice (less defined tasks, more of the modeling cycle, etc.). Scaffolding in tasks may range from substantial to very little or none. - MP 1, 2, and 4 Calculator - Item specific - Write and solve a system of equations and/or inequalities for a given contextual problem. Identify constraints by equations and inequalities. Determine whether or not solutions are viable or nonviable options within a modeling context. Page 8 of 56

9 Unit 2 A.CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. N.Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. PBA and EOY - Tasks have a context. - MP 2, 6, & 7 Calculator - Neutral PLD (PBA and EOY) - Algebraically solve linear equations, linear inequalities, and quadratics in one variable (at complexity appropriate to the course), including those with coefficients represented by letters. - Utilize structure and rewriting as strategies for solving. - Identify and correct errors in a given solution. - Solve equations, including multi-variable formulas or literal equations, for a chosen variable (quantity of interest). - Interpret units within the context of a problem. - Use units to evaluate the appropriateness of the solution to a multi-step problem. - Choose the appropriate units for a specific formula and interpret the meaning of the unit within the context provided. - Choose and interpret both the scale and the origin in graphs and data displays. - Choose and apply appropriate units consistently in formulas. Page 9 of 56

10 Unit 2 N.Q.A.2 Define appropriate quantities for the purpose of descriptive modeling. Limitations - This standard will be assessed in Algebra I by ensuring that some modeling tasks (involving Algebra I content or securely held content from grades 6-8) require the student to create a quantity of interest in the situation being described (i.e., a quantity of interest is not selected for the student by the task). For example, in a situation involving data, the student might autonomously decide that a measure of center is a key variable in a situation, and then choose to work with the mean. - Describe appropriate quantities when using descriptive modeling. N.Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. - Based on the limitations in the context of the situation, determine the accuracy appropriate to limitations on measurement when reporting quantities. Additional Resources: Assessments: Unit Vocabulary: Instructional Tasks: Work with linear equations in a real-world context (A1 Speeding Ticket Problem) Work with solving literal equations in a real-world context (A1 Going Green) Work with linear equations/inequalities in a real-world context (A1 Brandons Band) STEM Integration Common Student Misconceptions Prerequisite Skills Constructed Response Assessments A1 Disc Jockey SG A1 Pauline's Pen SG Page 10 of 56

11 Unit 3 Unit 3: Function Concepts 12 Days: September 12 - September 29, 2014 Standard Clarifications, Evidence, and Assessment F.IF.A.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). PBA and EOY - MP 2 Calculator - Neutral PLD (PBA and EOY) - Determine if a given relation is a function. - Given a context, write and analyze a linear or quadratic function. - For linear and quadratic functions that model contextual relationships, determine and interpret key features, graphs the function, and solve problems. - Determine the domain and relate it to the quantitative relationship it describes for a linear, quadratic, exponential (limited to domains in the integers), square root, cube root, piece-wise, step and absolute value functions. - Define a function. - Identify the domain and range of a function. - Recognize a function given a table, a set of ordered pairs, or a graph. - Evaluate a function for a given x value. SPI (include determining if a relation is a function) Page 11 of 56

12 Unit 3 F.IF.A.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. EOY - See illustrations at - Tasks require students to use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of context. - About a quarter of tasks involve functions defined recursively on a domain in the integers. - MP 2, 6 and 7 Calculator - Item Specific PLD (PBA and EOY) - Evaluates with, uses, and interprets with function notation within a context. - Given a context, write and analyze a linear or quadratic function. - For linear and quadratic functions that model contextual relationships, determine and interpret key features, graphs the function, and solve problems. - Determine the domain and relate it to the quantitative relationship it describes for a linear, quadratic, exponential (limited to domains in the integers), square root, cube root, piece-wise, step and absolute value functions. - Evaluate functions for a given input (domain value). - Interpret and explain statements that use functions in terms of real world situations. SPI , SPI , SPI , SPI Page 12 of 56

13 Unit 3 F.IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n 1. Limitations - This standard is part of the major work in Algebra I and will be assessed accordingly. - - Recognize that sequences are functions. SPI F.IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. - Given a graph, identify and describe the domain of the function. - Relate the domain of the function to its graph. - Given a verbal description of a function (real-life situation), describe the domain of the function. EOY - Limiting to linear functions, square root functions, cube root functions, piecewise-defined functions (including step functions and absolute-value functions), and exponential functions with domains in the integers. - Tasks have a real-world context. - MP 2 PLD (PBA and EOY) - Given a context, write and analyze a linear or quadratic function. - For linear and quadratic functions that model contextual relationships, determine and interpret key features, graphs the function, and solve problems. - Determine the domain and relate it to the quantitative relationship it describes for a linear, quadratic, exponential (limited to domains in the integers), square root, cube root, piece-wise, step and absolute value functions. Calculator - Neutral Page 13 of 56

14 Unit 3 A.REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). - Explain why each point on a graph is a solution to its equation. PBA and EOY - MP 7 Calculator - Item Specific A.CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. - Identify the independent and dependent variable when comparing two quantities. - Create equations in two or more variables to represent relationships between quantities. - Graph equations with two variables on a coordinate axes with appropriate labels and scales. SPI (graph linear equations) Page 14 of 56

15 Unit 3 Additional Resources: STEM Integration Assessments: Common Student Misconceptions Unit Vocabulary: Prerequisite Skills Instructional Tasks: TNCore Instructional Task Arc 1 Constructed Response Assessments Task 1: Create linear functions in two variables to model a graph Task 2: Create linear functions in two variables to model a graph. Analyze rate of change. Task 3: Determine whether a given function correctly models a situation. Analyze domain and range of a function. Task 4: Solidify understanding of rate of change from a graph. Task 5: Create and solve two-variable linear equations; analyze a situation over specific intervals. Task 6: Sketch functions given the rates of change over specific intervals. Task 7: Describe key characteristics of a graph (A1 Task Arc 1 Functions) Page 15 of 56

16 Unit 4 Unit 4: Building Functions 18 Days: September 30 - October 30, 2014 Standard Clarifications, Evidence, and Assessment F.BF.A.1 Write a function that describes a relationship between two quantities. F.BF.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. PBA - Express reasoning about linear and exponential growth. - MP 3 Calculator - Yes Limitations - Tasks have a real-world context - Tasks are limited to linear functions, quadratic functions, and exponential functions with domains in the integers. - Identify the independent and dependent variable when describing the relationship between two quantities. - Write a function that describes a relationship between two quantities. SPI , SPI PBA - Solve multi-step contextual word problems with degree of difficulty appropriate to the course, require application of course-level knowledge and skills articulated inf.bf.a.1a, F.BF.B.3, A.CED.A.1, A.SSE.B.3, F.IF.B, F.IF.C.7, limited to linear functions and exponential functions with domains in the integers. - F.BF.A.1a is the primary content; other listed content elements may be involved in tasks as well. - MP 2 and 4 Calculator - Yes - Define explicit function and recursive process. Page 16 of 56

17 Unit 4 F.IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. EOY - See illustrations at (illustrations/649, 637, and 639) - MP 4 and 6 Calculator - Item Specific Limitations - Tasks have a real-world context. - Tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise-defined functions (including step functions and absolute value functions), and exponential functions with domains in the integers. - Compare the second limitation with standard F.IF.C.7 -- The function types listed here are the same as those listed in Algebra I for standards F.IF.B.6 and F.IF.C.9. - Recognize and define key features in tables and graphs: intercepts; intervals where the function is increasing, decreasing, positive, or negative; and end behavior. - Identify whether the function is linear or nonlinear, given its table or graph. - Sketch graphs showing key features of a function that models a relationship between two quantities from a given verbal description of the relationship. - Interpret key features of graphs and tables of functions in terms of the contextual quantities the function represents. SPI Page 17 of 56

18 F.IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Curriculum Guide - Algebra I Unit 4 - Graph algebraic functions by hand in simple cases and using technology for more complicated cases. - Label the key features of each graph. SPI F.IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. EOY - Functions types should be limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise-defined functions (including step functions and absolute value functions), and exponential functions with domains in the integers. - Tasks may or may not have context. - MP 1, 3, 5, 6, and 8 Calculator - Item specific - Identify types of functions (linear and nonlinear) represented in different formats (algebraically, graphically, numerically in tables, or by verbal descriptions). - Compare functions that are represented in different formats (algebraically, graphically, numerically in tables, or by verbal descriptions). - Compare and contrast properties of two functions using a variety of function representations (algebraically, graphically, numerically in tables, or by verbal descriptions). Page 18 of 56

19 Unit 4 F.IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Limitations - Tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise-defined functions (including step functions and absolute value functions), and exponential functions with domains in the integers - The function types listed here are the same as those listed in the Algebra I standards F.IF.B.4 and F.IF.B.6 PLD (PBA and EOY) - Compare the properties of two functions represented in multiple ways limited to linear, exponential (with domains in the integers), quadratic, square root, absolute value, cube root, piece-wise, and step. SPI , SPI Page 19 of 56

20 Unit 4 F.IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. PBA - Estimate the rate of change from a graph utilizing linear functions and quadratic functions. - Functions are limited to linear, exponential, and quadratic. - Tasks have a context. EOY - Functions limited to square root functions, cube root functions, and piecewise-defined (including step functions and absolute value functions) functions with domains in the integers. - MP 1, 4, 5, and 7 Calculator - Item specific PLD (PBA and EOY) - Calculate and interpret the average rate of change of a linear, exponential, quadratic, square root, cube root and piece-wise-defined functions (presented symbolically or as a table) over a specified interval, and estimate the rate of change from a graph. - Compare rates of change associated with different intervals. Limitations - Tasks are limited to linear functions, quadratic functions, square root functions, cube root functions, piecewise-defined functions (including step functions and absolute value functions), and exponential functions with domains in the integers. - The function types listed here are the same as those listed in Algebra I standards F.IF.B.4 and F.IF.C.9 SPI Page 20 of 56

21 Unit 4 F.LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). EOY - Tasks are limited to constructing linear and exponential functions with domains in the integers, in simple context (not multi-step) - MP 1, 2, and 5 - Solve multi-step contextual problems with degree of difficulty to the course by constructing linear and/or exponential function models, where exponentials are limited to integer exponents. - Prompts describe a scenario using everyday language. Mathematical language such as "function," "exponential," etc. is not used. - Students autonomously choose and apply appropriate mathematical techniques without prompting. For example, in a situation of doubling, they apply techniques of exponential functions. - For some illustrations, see tasks at under F.LE - MP 1, 2, 4, and 6 Calculator - Item specific - Compare tables and graphs of linear and exponential functions to observe that a quantity increasing exponentially exceeds all others to solve mathematical and real-world problems. Page 21 of 56

22 F.LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). F.LE.A.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. Curriculum Guide - Algebra I Unit 4 PLD (PBA and EOY) - Represent linear and exponential (with domain in integers) functions symbolically, in real-life symbolically, in real-life scenarios, graphically, with a verbal description, as a sequence and with input-output pairs to solve mathematical and contextual problems. Limitations - Tasks are limited to constructing linear and exponential functions in simple context (not mutli-step). - Compare tables and graphs of linear and exponential functions to observe that a quantity increasing exponentially exceeds all others to solve mathematical and real-world problems. SPI F.LE.B.5 Interpret the parameters in a linear or exponential function in terms of a context. Limitations - Tasks have a real-world context. - Exponential functions are limited to those with domains in the integers. - Interpret the parameters in a linear or exponential function in terms of the context.b22 Page 22 of 56

23 Unit 4 Additional Resources: Assessments: Unit Vocabulary: Instructional Tasks: TNCore Instructional Task Arc 1 Task 5: Create and solve two-variable linear equations; analyze a situation over specific intervals. Task 6: Sketch functions given the rates of change over specific intervals. Task 7: Describe key characteristics of a graph Task 8: Solidify an understanding of average rate of change. (A1 Task Arc 1 Functions) Rate of change and analyzing graphs (A1 Travels) Represent data from a contextual situation and slope (A1 Pool) STEM Integration Common Student Misconceptions Prerequisite Skills Constructed Response Assessments A1 Buddy Bags SG A1 Downloads SG A1 Gorp SG A1 What's the Point SG Page 23 of 56

24 Unit 5 Unit 5: Systems 16 Days: October 31 - November 25, 2014 Standard Clarifications, Evidence, and Assessment A.REI.C.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. o PBA - Given an equation or system of equations, reason about the number or nature of the solutions. Limited to real solutions only. - MP 3 Calculator - Yes - Solve a system of two linear equations: - Approximately (e.g. with a graph) - Exactly (e.g. linear combination or substitution method) - Determine whether a system of two linear equations describes: - the same line (infinitely many solutions) - parallel lines ( no solution) - intersection lines (exactly 1 solution) Page 24 of 56

25 Unit 5 A.REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. o EOY - Solve multi-step contextual problems that require writing and analyzing systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. - Tasks have hallmarks of modeling as a mathematical practice (less defined tasks, more of the modeling cycle, etc.). - Scaffolding in tasks may range from substantial to very little or none. - MP 1, 2, and 4 Calculator - Item Specific Limitations - Tasks have a real-world connection. - Task have hallmarks of modeling as a mathematical practice (less defined tasks, more of the modeling cycle, etc.). PLD (PBA and EOY) - Write and analyze systems of linear equations in multi-step contextual problems. - Recognize and use properties of equality to maintain equivalent systems of equations. - Justify that replacing one equation in a two-equation system with the sum of that equation and a multiple of the other will yield the same solutions as the original system. SPI Page 25 of 56

26 Unit 5 A.REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. PBA - Given an equation or system of equations, reason about the number or nature of the solutions. Limited to equations of the form f(x) = g(x) where f and g are linear or quadratic. - Base explanations/reasoning on the principle that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane (excluding exponential and logarithmic functions). - MP 1, 3, and 5 Calculator - Item specific EOY - Find the solutions of where the graphs of the equations y=f(x) and y=g(x) intersect, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Limited f(x) and/or g(x) to polynomial functions. - The "explain" part of this standard is assessed on the PBA. - Polynomials are of degree two and higher. - MP 1 and 5 Calculator - Yes Limitations - Tasks that assess conceptual understanding of the indicated concept may involve any of the function types mentioned in the standard except exponential and logarithmic functions. - Finding the solutions approximately is limited to cases where f(x) and g(x) are polynomial functions. Page 26 of 56

27 Unit 5 A.REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. PBA and EOY - MP 1, 5, and 6 Calculator - No - Identify characteristics of a linear inequality and system of linear inequalities, such as: boundary line (where appropriate),shading, and determining appropriate test points to perform test to find a solution set - Graph a line, or boundary line, and shade the appropriate region for a two variable linear inequality. - Graph a system of linear inequalities and shade the appropriate overlapping region for a system of linear inequalities. Additional Resources: STEM Integration Assessments: Common Student Misconceptions Unit Vocabulary: Prerequisite Skills Instructional Tasks: Constructed Response Assessments A1 Knitting Knots SG Page 27 of 56

28 Unit 6 Unit 6: Polynomials 7 Days: December 1 - December 9, 2014 Standard Clarifications, Evidence, and Assessment A.APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. FS PBA - Construct, autonomously, chains of reasoning that will justify or refute algebraic propositions or conjectures. Calculator - Neutral EOY - Add, subtract, and multiply polynomials. - The "understand" part of the standard is not assessed here; it is assessed on the PBA. Calculator - Neutral - Identify that the sum, difference, or product of two polynomials will always be a polynomial, which means that polynomials are closed under the operations of addition, subtraction, and multiplication. - Define closure. - Apply arithmetic operations of addition, subtraction, and multiplication to polynomials. PLD (PBA and EOY) - Write equivalent numerical and polynomial expressions in one variable, using addition, subtraction, multiplication, and factoring including multi-step problems in mathematical and contextual situations. - Interpret parts of complicated exponential and quadratic expressions that represent a quantity in terms of its context. - Evaluate expressions, including for accuracy within context, and justifies the results. SPI , SPI Page 28 of 56

29 Unit 6 Additional Resources: Assessments: Unit Vocabulary: Instructional Tasks: STEM Integration Common Student Misconceptions Prerequisite Skills Constructed Response Assessments Create and work with polynomials (A1 Will and Latisha Tiles) Page 29 of 56

30 Unit 7 Unit 7: First Semester Review and Exam 8 Days: December 10 - December 19, 2014 Additional Resources: Assessments: Page 30 of 56

31 Unit 8 Unit 8: Factoring 15 Days: January 6 - January 27, 2015 Standard Clarifications, Evidence, and Assessment A.SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. A.SSE.B.3a Factor a quadratic expression to reveal the zeroes of the function it defines. PLD (PBA and EOY) - Determine equivalent forms of quadratic and exponential (with integer domain) expressions and functions to reveal and explain their properties. - Given a scenario determine the most appropriate form of a quadratic or exponential (with integer domain) function. - Explain the properties of the quantity represented by the expression. PBA - MP 7 Calculator - Neutral EOY - MP 7 Calculator - Neutral - Factor a quadratic expression to produce an equivalent form of the original expression. SPI , SPI Page 31 of 56

32 Unit 8 A.SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). PBA - Use the structure of numerical expressions and polynomial expression in one variable to identify ways to rewrite it. - Examples: Recognize as a difference of squares and see an opportunity to rewrite it in the easier to evaluate form (53+47)(53-47). See an opportunity to rewrite a 2 +9a+14 as (a+7)(a+2). EOY - Use the structure of numerical expressions or polynomials expression in one variable to rewrite it, in a case where two or more rewriting steps are required. - See examples from PBA. - Example: Factor completely x 2-1+(x-1) 2. (A first iteration might give (x+1)(x-1)+(x-1) 2, which could be rewritten as (x-1)(x+1+x+1) on the way to factoring completely as 2x(x-1). Or the student might first expand as x 2-1+x 2-2x+1, rewriting as 2x 2-2x then factoring as 2x(x-1). -Tasks do not have a context. - MP 1 and 7 Calculator - Neutral PLD (PBA and EOY) - Write equivalent numerical and polynomial expressions in one variable, using addition, subtraction, multiplication, and factoring including multi-step problems in mathematical and contextual situations. - Interpret parts of complicated exponential and quadratic expressions that represent a quantity in terms of its context. - Evaluate expressions, including for accuracy within context, and justifies the results. SPI Page 32 of 56

33 A.SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). SPI Operate with, evaluate, and simplify rational expressions including determining restrictions on the domain of the variables. Curriculum Guide - Algebra I Unit 8 Limitations - Task are limited to numerical expressions and polynomial expressions in one variable - Examples: Recognize as a difference of squares and see an opportunity to rewrite it in the easier to evaluate form (53+47)(53-47). See an opportunity to rewrite a 2 +9a+14 as (a+7)(a+2). Additional Resources: Assessments: Unit Vocabulary: Instructional Tasks: STEM Integration Common Student Misconceptions Prerequisite Skills Constructed Response Assessments Page 33 of 56

34 Unit 9 Unit 9: Quadratic Function Concepts 10 Days: January 28 - February 10, 2015 Standard Clarifications, Evidence, and Assessment A.APR.B.3 Identify zeroes of polynomials when suitable factorizations are available, and use the zeroes to construct a rough graph of the function defined by the polynomial. EOY - For example, find the zeroes of (x-2)(x 2-9) - MP 7 Calculator - No PLD (PBA and EOY) - Graph linear, quadratic, cubic (in which linear and quadratic factors are available), square root, cube root, and piece-wisedefined functions showing key features. - Determine a function given a graph with key features identified. Limitations - Tasks are limited to quadratic and cubic polynomials in which linear and quadratic factors are available. For example, find the zeroes of (x- 2 )(x2-9) SPI Page 34 of 56

35 Unit 9 F.IF.C.7a Graph linear and quadratic functions and show intercepts, maxima, and minima. PBA and EOY - Graph linear functions and show intercepts - Graph quadratic functions and show intercepts, maxima, and minima. - MP 1, 5, and 6 Calculator - Item specific EOY - Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. - Graph linear and quadratic functions, by hand in simple cases or using technology for more complicated cases, and identify intercepts, maxima, and minima of the graph. SPI Page 35 of 56

36 Unit 9 F.IF.C.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. EOY - Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. - MP 1, 5, and 6 Calculator - Item specific PLD (PBA and EOY) - Graph linear, quadratic, cubic (in which linear and quadratic factors are available), square root, cube root, and piece-wisedefined functions showing key features. - Determine a function given a graph with key features identified. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions, by hand in simple cases or using technology for more complicated cases, and identify key features of the graph. Page 36 of 56

37 Unit 9 F.IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. PBA - Construct, autonomously, chains of reasoning that will justify or refute propositions or conjectures about functions. - Tasks involve using algebra to prove properties of given functions. For example, prove algebraically that the functions h(t)=t(t+1) has minimum value 1/4; prove algebraically that the graph of g(x)=x 2 -x+(1/4) is symmetric about the line x=(1/2); prove that x 2 +1 is never less than -2x. - Scaffolding is provided to ensure tasks have appropriate level of difficulty. (For example, the prompt could show the graphs of x 2 +1 and -2x on the same set of axes, and say, "From the graph, it looks as if x 2+ 1 is never less than -2x. In this task you will use algebra to prove it." And so on, perhaps with additional hints or scaffolding. - MP 3 Calculator -Yes - Write a function defined by an expression in different but equivalent forms to identify and explain different properties of the function. Page 37 of 56

38 F.IF.C.8a Use the process of factoring and completing the square in a quadratic function to show zeroes, extreme values, and symmetry of the graph, and interpret these in terms of a context. Curriculum Guide - Algebra I Unit 9 EOY - Tasks have context - MP 2 Calculator - Yes - Apply the process of factoring and completing the square in a quadratic function to determine the vertex, axis of symmetry, direction of opening, and zeroes/roots from the graph of a quadratic function. - Sketch the graph of a quadratic function. - Describe all of the properties of a quadratic function SPI , SPI Additional Resources: Assessments: Unit Vocabulary: Instructional Tasks: Students work with quadratic equations (A1 Pool Patio) TNCore Instructional Task Arc 2 STEM Integration Common Student Misconceptions Prerequisite Skills Constructed Response Assessments A1 Bottle Rocket SG A1 Vegetable Garden SG Task 1: Create and graph a quadratic function in two variables. Task 2: Determine whether values in a table represent a linear or a quadratic function. Graph the function associated with the table. Develop an equivalent equation with linear factors Page 38 of 56

39 Task 3: Graph a quadratic function and develop an equivalent equation that can be used to find the roots of the equation. Interpret roots within the context of problem. Task 4: Solidify understanding of how to create equivalent equations containing linear factors that can be used to find roots for an equation. Solidify understanding of what roots mean in terms of the problem context. Curriculum Guide - Algebra I Unit 9 Task 5: Use a graph to analyze characteristics of a quadratic and understand how a graph shows existence of real roots. Task 6: Create a function and analyze how the context of the problem affects the resulting graph. Task 7: Use characteristics of linear and quadratic functions to analyze graphs. Task 8: Solidify an understanding of the behavior of quadratic functions and strategies for solving them. (A1 Task Arc 2 Quadratics) Page 39 of 56

40 Unit 10 Unit 10: Solving Quadratic Functions 15 Days: February 11 - March 4, 2015 Standard Clarifications, Evidence, and Assessment A.REI.B.4 Solve quadratic equations in one variable. PBA - Given an equation or system of equations, present the solution steps as a logical argument that concludes with the set of solutions (if any). Tasks are limited to quadratic equations with real solutions only. - MP 6 Calculator - Yes PLD (PBA and EOY) - Algebraically solve linear equations, linear inequalities, and quadratics in one variable (at complexity appropriate to the course), including those with coefficients represented by letters. - Utilize structure and rewriting as strategies for solving. - Identify and correct errors in a given solution. - Solve quadratic equations in one variable. A.REI.B.4a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x p) 2 = q that has the same solutions. Derive the quadratic formula from this form. EOY - The derivation part of the standard is not assessed here, it is assessed under PBA - MP 1 and 7 Calculator - Item Specific - Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x-p)2=q that has the same solutions. - Derive the quadratic formula by completing the square on a quadratic equation in x. Page 40 of 56

41 Unit 10 A.REI.B.4b Solve quadratic equations by inspection (e.g., for x 2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. PBA and EOY - Tasks should exhibit variety in initial forms. Examples of quadratic equations with real solutions: t 2 =49, 3a 2 =4, 7=x 2, r 2 =0, (1/2)y 2 =(1/5), y 2-18y+15=0, 2x 2-16x+30=0, 2p=p 2 +1, t 2 =4t, 7x 2 +5x- 3=0, (3/4)c(c-1)=c, (3x-2) 2 =6x-4 - Methods are not explicitly assessed; strategy is assessed indirectly by presenting students with a variety of itial forms. - For rational solutions, exact values are required. For irrational solutions, exact or decimal approximations may be required. Simplifying or rewriting radicals is not required for common core. - Prompts integrate mathematical practices by not indicating that the equation is quadratic. (E.g., "Find all real solutions of the equation t 2 =4t" not, "Solve the quadratic equation t 2 =4t.") - tasks involve recognizing an equation with complex solutions, e.g., "Which of the following equations has no real solutions?" with one of the options being a quadratic equation with non-real solutions. - Writing solutions in the form (a plus or minus bi is not assessed here. (assessed in Algebra 2). - MP 5 and 7 SPI A.REI.B.4b Solve quadratic equations by inspection (e.g., for x 2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Calculator - Item Specific Limitations - Tasks do not require students to write solutions for quadratic equations that have roots with nonzero imaginary parts. However, tasks can require the student to recognize cases in which a quadratic equation has no real solutions. - Note, solving a quadratic equation by factoring relies on the connection between zeroes and factors of polynomials (cluster A.APR.B is formally assessed in Algebra II Page 41 of 56

42 A.SSE.B.3b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Curriculum Guide - Algebra I Unit 10 EOY - MP 7 Calculator - Neutral - Complete the square on a quadratic expression to produce an equivalent form of an expression. - Explain the connection between the completed square form of a quadratic expression and the maximum or minimum value of the function it defines. A.SSE.B.3c Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15 t can be rewritten as (1.15 1/12 ) 12t t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. A.SSE.B.3c Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15 t can be rewritten as (1.15 1/12 ) 12t t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. EOY - Tasks have a context. As described in the standard, there is an interplay between the mathematical structure of the expression and the structure of the situation such that choosing and producing an equivalent form of the expression reveals something about the situation. - MP 1, 2, 4, and 7 Calculator - Neutral - Use the properties of exponents to transform simple expressions for exponential functions. - Choose and produce an equivalent form of an exponential expression to reveal and explain properties of the quantity represented by the original expression. Limitations - Tasks have a real-world context. As described in the standard, there is an interplay between the mathematical structure of the expression and the structure of the situation such that choosing and producing an equivalent form of the expression reveals something about the situation - Tasks are limited to exponential expressions with integer exponents Page 42 of 56

43 Unit 10 Additional Resources: Assessments: Unit Vocabulary: Instructional Tasks: STEM Integration Common Student Misconceptions Prerequisite Skills Constructed Response Assessments Page 43 of 56