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 Instructional Focus DocumentAlgebra I
 TITLE : Unit 03: Investigation of Linear Functions and Inequalities (two variables) SUGGESTED DURATION : 10 days

Unit Overview

Introduction
This unit bundles student expectations that address investigating characteristics of linear functions in various representations, including domain, range, slope, intercepts, and forms of linear equations. Student expectations also address the effects of parameter changes on the graph of the linear parent function written in function notation, and writing and graphing the solution set of linear inequalities given various representations. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this Unit
In Grade 8, students investigated relationships in various representations, including tables, graphs, and algebraic generalizations. Students were introduced to the concept of slope as a rate of change, including using the slope formula. Students also investigated arithmetic sequences with constant rates of change and compared proportional and non-proportional linear relationships. In Algebra I Unit 02, students explored the development of the concept of a function, representations of functions, and characteristics of functions.

During this Unit
Students graph linear functions on the coordinate plane given tables, verbal descriptions, and algebraic generalizations. Students determine domain (continuous and discrete) and range of linear functions representing domain and range using inequality notation and verbal descriptions for mathematical problems. Students determine the reasonableness of domain (continuous and discrete) and range in real-world situations. Students also calculate rate of change for a linear function in mathematical and real world problems from tables, graphs, and algebraic methods. Students determine the slope of a line given a table, graph, two points on the line, and an equation written in various forms, including y = mx + b, Ax + By = C, and yy1 = m(x – x1). Students make connections between rate of change and slope of the line. Students determine the rate of change for various intervals on a given graph of a piecewise function. Students write linear equations in two variables given a table of values, a graph, and a verbal description. Students write an equation of a line that is parallel or perpendicular to the x- or y-axis and determine whether the slope of the line is zero or undefined. Students graph linear functions in two variables, identifying key features, including x-intercept, y-intercept, zeros, and slope, in mathematical and real-world problems. Students determine the effects on the graph of the parent function f(x) = x when f(x) is replaced by af(x), f(x) + d, f(xc), f(bx) for specific values of a, b, c, and d, including multiple parameter changes within one linear function. Students write linear inequalities in two variables given a table of values, a graph, and a verbal description; and graph the solution set of linear inequalities in two variables on the coordinate plane.

After this Unit
In Unit 04, students will use these concepts to write equations of lines and analyze real world situations by exploring scatterplots, trend lines, and linear correlation. Precalculus will introduce piecewise functions and their characteristics. Students will continue to apply these concepts in subsequent courses in mathematics.

In Algebra I, determining the domain and range and representing it in inequality notation are identified as STAAR Readiness Standard A.2A and subsumed under STAAR Reporting Category 3: Writing and Solving Linear Functions, Equations, and Inequalities. Identifying key attributes of linear functions such as rate of change, x- and y-intercepts, zeros, and slope are identified as STAAR Readiness Standards A.3B and A.3C. Graphing solution sets of linear inequalities in two variables is identified as STAAR Readiness Standard A.3D. These Readiness Standards are subsumed under STAAR Reporting Category 2: Describing and Graphing Linear Functions, Equations, and Inequalities. Determining zero, slope, undefined slope, parallel slope, and perpendicular slope are identified as STAAR Supporting Standard A.2G. Writing inequalities in two variables is identified as STAAR Supporting Standard A.2H. These are subsumed under STAAR Reporting Category 3: Writing and Solving Linear Functions, Equations, and Inequalities. Determining the effects of parameter changes on linear functions are identified as STAAR Supporting Standard A.3E and is subsumed under STAAR Reporting Category 2: Describing and Graphing Linear Functions, Equations, and Inequalities. This unit is supporting the development of Texas College and Career Readiness Standards (TxCCRS): II. Algebraic Reasoning C1; III. Geometry B1, B3, C1; VII. Functions A1, A2, B1, B2, C1, C2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

Research
According to the National Council of Teachers of Mathematics (NCTM), Developing Essential Understanding of Functions for Teaching Mathematics in Grades 9 – 12 (2010), students in grades 9–12 need to understand functions well if they are to succeed in courses that build on quantitative thinking and relationships. According the National Council of Teachers of Mathematics (2000), students should have an opportunity to build on their earlier experiences, both deepening their understanding of relations and functions and expanding their repertoire of familiar functions. In middle school through the study of direct variation (proportional) and non-proportional linear situations, students explore the patterns that relate to linear functions, building a foundational understanding for development of slope and intercepts and writing equations of lines in high school. High school students’ algebra experience should enable them to create and use tabular, symbolic, graphical, and verbal representations and to analyze and understand patterns, relations, and functions with more sophistication than in the middle grades. High school algebra should provide students with insights into mathematical abstraction and structure for modeling real-world problem situations and making predictions and drawing conclusions.

National Council of Teachers of Mathematics. (2000). Developing essential understanding of functions for teaching mathematics grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgments in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding how two quantities vary together (covariation) builds flexible functional reasoning in order to make predictions and critical judgments about the relationship.
• How is function notation used to define and describe a function rule?
• How is function notation used to make predictions and critical judgements about the relationship?
• Different families of functions, each with their own unique characteristics, can be used to model problem situations to make predictions and critical judgments.
• Linear functions are characterized by a constant rate of change and can be used to describe, model, and make predictions about situations.
• How can the rate of change of a linear function be determined?
• What kinds of mathematical and real-world situations can be modeled by linear functions?
• What graphs, key attributes, and characteristics are unique to linear functions?
• What pattern of covariation is associated with linear functions?
• How can the key attributes of linear functions be …
• determined?
• analyzed?
• described?
• How can key attributes be used to describe the behavior of a linear function?
• What are the real-world meanings of the key attributes of a linear function model?
• How can the key attributes of a linear function be used to make predictions and critical judgments?
• Functions can be represented in various ways with different representations of the function highlighting different characteristics and being more useful than other representations depending on the context.
• How can functions be represented?
• What is the purpose of representing functions in various ways?
• How are function characteristics highlighted in different representations of the function?
• What are the limitations of different function representations?
• What connections can be made between multiple representations of a function?
• Functions can be combined and transformed in predictable ways to create new functions that can be used to describe, model, and make predictions about situations.
• How are functions …
• shifted?
• scaled?
• reflected?
• How do transformations affect the …
• representations
• key attributes
… of a function?
• Equations and inequalities can be written, transformed, and solved using various methods to make critical judgments, with different methods being more efficient or informative depending on the structure of the equation or inequality.
• How can equations and inequalities be used to represent relationships between quantities?
• How do solutions to inequalities differ from solutions to equations?
• Why must solutions be justified in terms of problem situations?
• What methods can be used to write linear equations and linear inequalities?
• How does the given information and/or representation influence the selection of an efficient method for writing linear equations and linear inequalities?
• How can the solutions to linear equations and linear inequalities be determined and represented?
• How are properties and operational understandings used to transform linear equations and linear inequalities?
• Functions, Equations, and Inequalities
• Attributes of Functions
• Domain and range
• Continuous or discrete
• Rate of change/Slope
• x- and y-intercept(s)
• Zeros
• Functions, Equations, and Inequalities
• Linear
• Relations and Generalizations
• Transformations
• Parent functions
• Transformation effects
• Associated Mathematical Processes
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgments in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding how two quantities vary together (covariation) builds flexible functional reasoning in order to make predictions and critical judgments about the relationship.
• What are the strengths and limitations of a particular function model for a problem situation?
• How can functions be used to model problem situations efficiently?
• How can it be determined if a relationship between two variables can be represented by a function?
• How is function notation used to define and describe a function rule?
• How is function notation used to make predictions and critical judgements about the relationships?
• Different families of functions, each with their own unique characteristics, can be used to model problem situations to make predictions and critical judgments.
• Linear functions are characterized by a constant rate of change and can be used to describe, model, and make predictions about situations.
• How can the rate of change of a linear function be determined?
• What kinds of mathematical and real-world situations can be modeled by linear functions?
• What graphs, key attributes, and characteristics are unique to linear functions?
• What pattern of covariation is associated with linear functions?
• How can the key attributes of linear functions be …
• determined?
• analyzed?
• described?
• How can key attributes be used to describe the behavior of a linear function?
• What are the real-world meanings of the key attributes of a linear function model?
• How can the key attributes of a linear function be used to make predictions and critical judgments?
• Functions can be represented in various ways with different representations of the function highlighting different characteristics and being more useful than other representations depending on the context.
• How can functions be represented?
• What is the purpose of representing functions in various ways?
• How are function characteristics highlighted in different representations of the function?
• What are the limitations of different function representations?
• What connections can be made between multiple representations of a function?
• Equations and inequalities can be written, transformed, and solved using various methods to make critical judgments, with different methods being more efficient or informative depending on the structure of the equation or inequality.
• How can equations and inequalities be used to represent relationships between quantities?
• How do solutions to inequalities differ from solutions to equations?
• Why must solutions be justified in terms of problem situations?
• What methods can be used to write linear equations and linear inequalities?
• How does the given information and/or representation influence the selection of an efficient method for writing linear equations and linear inequalities?
• How can the solutions to linear equations and linear inequalities be determined and represented?
• How are properties and operational understandings used to transform linear equations and linear inequalities?
• Functions, Equations, and Inequalities
• Attributes of Functions
• Domain and range
• Continuous or discrete
• Rate of change/Slope
• x- and y-intercept(s)
• Zeros
• Functions, Equations, and Inequalities
• Linear
• Relations and Generalizations
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that all linear functions with fractional slopes are represented with lines whose steepness will be less than the parent function. Students should recognize that a fractional slope whose numerator is greater than the denominator is greater than 1 and will have a slope greater than the parent function.
• Some students may think that the ratio for rate of change (slope) in a linear function is , since the x variable (horizontal) always comes before the y variable (vertical), instead of the correct representation that rate of change (slope) in a linear function is .
• Some students may think that the intercept coordinate is the zero term instead of the non-zero term, since intercepts are associated with zeros. In other words, students may think (0, 4) would be the x-intercept because the 0 is in the x coordinate.

Underdeveloped Concept:

• Although some students may understand that slope is the same along all segments of a line because of the proportionality of similar triangles formed by the vertical distance and horizontal distance, others may need to compare and contrast the slope of various line segments using vertical and horizontal distances.

Unit Vocabulary

• Continuous function – function whose values are continuous or unbroken over the specified domain
• Discrete function – function whose values are distinct and separate and not connected; values are not continuous. Discrete functions are defined by their domain.
• Domain – set of input values for the independent variable over which the function is defined
• Inequality notation – notation in which the solution is represented by an inequality statement
• Linear equation in two variables – a relationship with a constand rate of change represented by a graph that forms a straight line
• Linear function – a relationship with a constant rate of change represented by a graph that forms a straight line in which each element of the input (x) is paired with exactly one element of the output (y)
• Linear inequality in two variables – a relationship with a constant rate of change represented by a solution set denoted by the graph of a line, that may or may be not included in the solution, and the set of points above or below the line
• Parent functions – set of basic functions from which related functions are derived by transformations
• Range – set of output values for the dependent variable over which the function is defined
• Slope of a line – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or or , denoted as m in y = mx + b
• x-intercept(s)x coordinate of a point at which the relation crosses the x-axis, meaning the y coordinate equals zero, (x, 0)
• y-intercept(s)y coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0, y)
• Zeros – the value(s) of x such that the y value of the relation equals zero

Related Vocabulary:

 Decreasing Dependent Equation notation Function notation Horizontal change Increasing Independent Parameter change Parent function Point slope form, y – y1 = m(x – x1) Rate of change Slope-intercept form, y = mx + b Standard form, ax + by = c Vertical change
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Center if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway –Resources Aligned to Algebra I Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

• Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
• Student Expectations (TEKS) identified by TEA are in bolded, black text.
• Student Expectations (TEKS) are labeled Readiness as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Supporting as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Process standards as identified by TEA of the assessed curriculum.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
TEKS# SE# TEKS SPECIFICITY
A.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
A.1A Apply mathematics to problems arising in everyday life, society, and the workplace.
Process Standard

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE
Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
A.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.
Process Standard

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
A.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.
Process Standard

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS
Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
A.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Process Standard

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE
Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
A.1E Create and use representations to organize, record, and communicate mathematical ideas.
Process Standard

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
A.1F Analyze mathematical relationships to connect and communicate mathematical ideas.
Process Standard

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
A.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Process Standard

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION

Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
A.2 Linear functions, equations, and inequalities. The student applies the mathematical process standards when using properties of linear functions to write and represent in multiple ways, with and without technology, linear equations, inequalities, and systems of equations. The student is expected to:
A.2A Determine the domain and range of a linear function in mathematical problems; determine reasonable domain and range values for real-world situations, both continuous and discrete; and represent domain and range using inequalities.

Determine

THE DOMAIN AND RANGE OF A LINEAR FUNCTION IN MATHEMATICAL PROBLEMS AND REASONABLE DOMAIN AND RANGE VALUES FOR REAL-WORLD SITUATIONS, BOTH CONTINUOUS AND DISCRETE

Represent

THE DOMAIN AND RANGE OF A LINEAR FUNCTION USING INEQUALITIES

Including, but not limited to:

• Linear function – a relationship with a constant rate of change represented by a graph that forms a straight line in which each element of the input (x) is paired with exactly one element of the output (y)
• Domain and range of linear functions in mathematical problem situations
• Domain – set of input values for the independent variable over which the function is defined
• Continuous function – function whose values are continuous or unbroken over the specified domain
• Discrete function – function whose values are distinct and separate and not connected; values are not continuous. Discrete functions are defined by their domain.
• Range – set of output values for the dependent variable over which the function is defined
• Inequality representations
• Verbal description
• Ex: x is all real numbers less than five.
• Ex: x is all real numbers.
• Ex: y is all real numbers greater than –3 and less than or equal to 6.
• Ex: y is all integers greater than or equal to zero.
• Inequality notation – notation in which the solution is represented by an inequality statement
• Ex: x < 5, x ∈ ℜ
• Ex: x ∈ ℜ
• Ex: –3 < y ≤ 6, y ∈ ℜ
• Ex: y ≥ 0, yΖ
• Domain and range of linear functions in real-world problem situations
• Reasonable domain and range for real-world problem situations
• Comparison of domain and range of function model to appropriate domain and range for a real-world problem situation

Note(s):

• The notation ℜ represents the set of real numbers, and the notation Ζ represents the set of integers.
• Grade 6 identified independent and dependent quantities.
• Grade 8 identified functions using sets of ordered pairs, tables, mappings, and graphs.
• Algebra I introduces the concept of domain and range of a function.
• Algebra I represents domain and range using inequality verbal descriptions and inequality notation.
• Algebra II will introduce representing domain and range using interval and set notation.
• Precalculus will introduce piecewise functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VII. Functions
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.2C Write linear equations in two variables given a table of values, a graph, and a verbal description.

Write

LINEAR EQUATIONS IN TWO VARIABLES GIVEN A TABLE OF VALUES, A GRAPH, AND A VERBAL DESCRIPTION

Including, but not limited to:

• Linear equation in two variables – a relationship with a constant rate of change represented by a graph that forms a straight line
• Various forms linear equations in two variables
• Slope-intercept form, y = mx + b
• m is the slope.
• b is the y-intercept.
• Point-slope form, y – y1 = m(xx1)
• m is the slope.
• (x1, y1) is a given point
• Standard form, Ax + By = C; A, B, CΖ, A ≥ 0
• x and y terms are on one side of the equation and the constant is on the other side.
• Given multiple representations
• Table of values
• Graph
• Verbal description

Note(s):

• Middle School introduced using multiple representations for linear relationships.
• Grade 8 represented linear proportional and non-proportional relationships in tables, graphs, and equations in the form y = mx + b.
• Algebra I introduces the use of standard form and point-slope form to represent linear relationships.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• B2 – Algebraically construct and analyze new functions.
• C1 – Apply known function models.
• C2 – Develop a function to model a situation.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.2G Write an equation of a line that is parallel or perpendicular to the X or Y axis and determine whether the slope of the line is zero or undefined.
Supporting Standard

Write

AN EQUATION OF A LINE THAT IS PARALLEL OR PERPENDICULAR TO THE X OR Y AXIS

Determine

WHETHER THE SLOPE OF A LINE PARALLEL OR PERPENDICULAR TO THE X OR Y AXIS IS ZERO OR UNDEFINED

Including, but not limited to:

• Write equations for parallel or perpendicular lines
• Equations of lines parallel or perpendicular to the x-axis
• Parallel to the x-axis, y = #
• Perpendicular to the x-axis, x = #
• Equations of lines parallel or perpendicular to the y-axis
• Parallel to the y-axis, x = #
• Perpendicular to the y-axis, y = #
• Equations of lines parallel or perpendicular to lines that are parallel or perpendicular to the x-axis
• Parallel to a line parallel to the x-axis, y = #
• Parallel to a line perpendicular to the x-axis, x = #
• Perpendicular to a line parallel to the x-axis, x = #
• Perpendicular to a line perpendicular to the x-axis, y = #
• Equations of lines parallel or perpendicular to lines that are parallel or perpendicular to the y-axis
• Parallel to a line parallel to the y-axis, x = #
• Parallel to a line perpendicular to the y-axis, y = #
• Perpendicular to a line parallel to the y-axis, y = #
• Perpendicular to a line perpendicular to the y-axis, x = #
• Determine whether the slope of a parallel or perpendicular line is zero or undefined
• Slope of lines parallel to the x-axis, m = 0
• Slope of lines parallel to the y-axis, m is undefined
• Slope of lines perpendicular to the x-axis, m is undefined
• Slope of lines perpendicular to the y-axis, m = 0
• Write the equation of a line and determine the slope of the line that passes through a point and is parallel or perpendicular to the x-axis.
• Write the equation of a line and determine the slope of the line that passes through a point and is parallel or perpendicular to the y-axis.
• Write the equation of a line and determine the slope of the line that passes through a point and is parallel to a line that is parallel to the x-axis.
• Write the equation of a line and determine the slope of the line that passes through a point and is parallel to a line that is perpendicular to the x-axis.
• Write the equation of a line and determine the slope of the line that passes through a point and is perpendicular to a line that is parallel to the x-axis.
• Write the equation of a line and determine the slope of the line that passes through a point and is perpendicular to a line that is perpendicular to the x-axis.
• Write the equation of a line and determine the slope of the line that passes through a point and is parallel to a line that is parallel to the y-axis.
• Write the equation of a line and determine the slope of the line that passes through a point and is parallel to a line that is perpendicular to the y-axis.
• Write the equation of a line and determine the slope of the line that passes through a point and is perpendicular to a line that is parallel to the y-axis.
• Write the equation of a line and determine the slope of the line that passes through a point and is perpendicular to a line that is perpendicular to the y-axis.
• Generalizations
• A line parallel to the x-axis and perpendicular to the y-axis has a slope of zero.
• A line parallel to the y-axis and perpendicular to the x-axis has an undefined slope.

Note(s):

• Previous grade levels introduced slope and meaning of parallel and perpendicular separately.
• Algebra I introduces the concepts of parallel and perpendicular lines in terms of slope.
• Geometry will write the equation of a line parallel or perpendicular to a given line passing through a given point to determine geometric relationships on a coordinate plane.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.2H Write linear inequalities in two variables given a table of values, a graph, and a verbal description.
Supporting Standard

Write

LINEAR INEQUALITIES IN TWO VARIABLES GIVEN A TABLE OF VALUES, A GRAPH, AND A VERBAL DESCRIPTION

Including, but not limited to:

• Linear inequality in two variables – a relationship with a constant rate of change represented by a solution set denoted by the graph of a line, that may or may not be included in the solution, and the set of points above or below the line
• Inequality notation
• Less than, <, dashed line with shading below the graph of the line
• Greater than, >, dashed line with shading above the graph of the line
• Less than or equal to, ≤, solid line with shading below the graph of the line
• Greater than or equal to, ≥, solid line with shading above the graph of the line
• For vertical lines, greater than shades the right side of the graph and less than shades the left side of the graph.
• Given multiple representations
• Table of values
• Graph
• Verbal description

Note(s):

• Middle School used multiple representations for linear relationships.
• Grade 8 solved problems using one-variable inequalities.
• Algebra I introduces linear inequalities in two variables given various representations.
• Algebra II will continue systems of two linear inequalities in two variables and extend to systems of more than two linear inequalities in two variables.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.3 Linear functions, equations, and inequalities. The student applies the mathematical process standards when using graphs of linear functions, key features, and related transformations to represent in multiple ways and solve, with and without technology, equations, inequalities, and systems of equations. The student is expected to:
A.3A Determine the slope of a line given a table of values, a graph, two points on the line, and an equation written in various forms, including y = mx + b, Ax + By = C, and yy1 = m(xx1).
Supporting Standard

Determine

THE SLOPE OF A LINE GIVEN A TABLE OF VALUES, A GRAPH, TWO POINTS ON THE LINE, AND AN EQUATION WRITTEN IN VARIOUS FORMS, INCLUDING y = mx + b, Ax + By = C, and yy1 = m(xx1)

Including, but not limited to:

• Slope of a line – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or or or • Slope by various methods
• Equation method denoted as m in y = mx + b
• Table method by analyzing change in x and y values: m = or • Graph method by analyzing vertical and horizontal change: slope = • Formula method: For two points (x1, y1) and (x2, y2), m= • Slope from multiple representations
• Tables of values
• Graphs
• Two points on a line
• Linear equations in various forms
• Slope-intercept form, y = mx + b
• m is the slope.
• Point-slope form, yy1 = m(xx1)
• m is the slope.
• Standard form, Ax + By = C
• m = – Note(s):

• Grade 8 introduced the concept of slope through the use of proportionality using similar triangles, making connections between slope and proportional relationships, and determining slope from tables and graphs.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• B3 – Use congruence transformations and dilations to investigate congruence, similarity, and symmetries of plane figures.
• C1 – Make connections between geometry and algebra.
• VII. Functions
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.3B Calculate the rate of change of a linear function represented tabularly, graphically, or algebraically in context of mathematical and real-world problems.

Calculate

THE RATE OF CHANGE OF A LINEAR FUNCTION REPRESENTED TABULARLY, GRAPHICALLY, OR ALGEBRAICALLY IN CONTEXT OF MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Linear function – a relationship with a constant rate of change represented by a graph that forms a straight line in which each element of the input (x) is paired with exactly one element of the output (y)
• Linear functions in mathematical problem situations
• Linear functions in real-world problem situations
• Connections between slope and rate of change
• Rate of change by various methods
• Tabular method by analyzing rate of change in x and y values: m = = or m = • Graphical method by analyzing vertical and horizontal change: slope = • Algebraic method by analyzing m in y = mx + b form
• Solve equation for y
• Slope is represented by m
• Rate of change from multiple representations
• Tabular
• Graphical
• Algebraic
• Calculuation and comparison of the rate of change over specified intervals of a graph
• Meaning of rate of change in the context of real-world problem situations
• Emphasis on units of rate of change in relation to real-world problem situations

Note(s):

• Grade 8 introduced the concept of slope as a rate of change, including using the slope formula.
• Precalculus will introduce piecewise functions and their characteristics.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.3C Graph linear functions on the coordinate plane and identify key features, including x-intercept, y-intercept, zeros, and slope, in mathematical and real-world problems.

Graph

LINEAR FUNCTIONS ON THE COORDINATE PLANE

Including, but not limited to:

• Linear function – a relationship with a constant rate of change represented by a graph that forms a straight line in which each element of the input (x) is paired with exactly one element of the output (y)
• Linear functions in mathematical problem situations
• Linear functions in real-world problem situations
• Multiple representations
• Tabular
• Graphical
• Verbal
• Algebraic generalizations

Identify

KEY FEATURES OF LINEAR FUNCTIONS, INCLUDING x-INTERCEPT, y-INTERCEPT, ZEROS, AND SLOPE, IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Linear function – a relationship with a constant rate of change represented by a graph that forms a straight line in which each element of the input (x) is paired with exactly one element of the output (y)
• Linear functions in mathematical problem situations
• Linear functions in real-world problem situations
• Multiple representations
• Tabular
• Graphical
• Verbal
• Algebraic generalizations
• Characteristics of linear functions
• x-intercept – x coordinate of a point at which the relation crosses the x-axis, meaning the y coordinate equals zero, (x, 0)
• Zeros – the value(s) of x such that the y value of the relation equals zero
• y-intercept – y coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0, y)
• Slope of a line – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or = or • denoted as m in y = mx + b
• denoted as m in f(x) = mx + b
• Notation of linear functions
• Equation notation: y= mx + b
• Function notation: f(x) = mx + b

Note(s):

• Grades 7 and 8 introduced linear relationships using tables of data, graphs, and algebraic generalizations.
• Grade 8 introduced using tables of data and graphs to determine rate of change or slope and y-intercept.
• Algebra I introduces key attributes of linear, quadratic, and exponential functions.
• Algebra II will continue to analyze the key attributes of exponential functions and will introduce the key attributes of square root, cubic, cube root, absolute value, rational, and logarithmic functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.3D Graph the solution set of linear inequalities in two variables on the coordinate plane.

Graph

THE SOLUTION SET OF LINEAR INEQUALITIES IN TWO VARIABLES ON THE COORDINATE PLANE

Including, but not limited to:

• Inequality notation
• Less than, <, dashed line with shading below the graph of the line
• Greater than, >, dashed line with shading above the graph of the line
• Less than or equal to, ≤, solid line with shading below the graph of the line
• Greater than or equal to, ≥, solid line with shading above the graph of the line
• For vertical lines, greater than shades the right side of the graph and less than shades the left side of the graph.
• Given multiple representations
• Table of values
• Algebraic generalization
• Verbal description

Note(s):

• Algebra I introduces linear inequalities in two variables.
• Algebra II will continue systems of two linear inequalities in two variables and extend to systems of more than two linear inequalities in two variables.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II. Algebraic Reasoning
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• C1 – Apply known function models.
• C2 – Develop a function to model a situation.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.3E Determine the effects on the graph of the parent function f(x) = x when f(x) is replaced by af(x), f(x) + d, f(x - c), f(bx) for specific values of a, b, c, and d.
Supporting Standard

Determine

THE EFFECTS ON THE GRAPH OF THE PARENT FUNCTION f(x) = x WHEN f(x) IS REPLACED BY af(x), f(x) + d, f(xc), f(bx) FOR SPECIFIC VALUES OF a, b, c, AND d

Including, but not limited to:

• Parent functions – set of basic functions from which related functions are derived by transformations
• General form of linear parent function (including equation and function notation)
• y = x
• f(x) = x
• Multiple representations
• Graphs
• Tables
• Verbal descriptions
• Algebraic generalizations (including equation and function notation)
• Changes in parameters a, b, c, and d on the graph of the parent function f(x) = x
• Effects on the graph of the parent function f(x) = x when f(x) is replaced by af(x)
• a ≠ 0
• |a| > 1, stretches the graph vertically or makes the graph steeper
• 0 < |a| < 1, compresses the graph vertically or makes the graph less steep
• Opposite of a reflects the graph vertically over the horizontal axis (x-axis)
• af(x) effects the slope of the parent function
• Effects on the graph of the parent function f(x) = x when f(x) is replaced by f(bx)
• b ≠ 0
• |b| > 1, the graph compresses horizontally or makes the graph steeper
• 0 < |b| < 1, the graph stretches horizontally or makes the graph less steep
• b < 0, reflects horizontally over the y-axis
• f(bx) effects the slope of the parent function
• Effects on the graph of the parent function f(x) = x when f(x) is replaced by f(xc)
• c = 0, no horizontal shift or translation
• Horizontal shift or translation left or right by |c| units
• Left shift or translation when c < 0
• For f(x + 2) → f(x – (–2)), c = –2, and the function moves to the left or shifts left or translates left two units.
• Right shift or translation when c > 0
• For f(x – 2), c = 2, and the function moves to the right or shifts right or translates right two units.
• Effects on the graph of the parent function f(x) = x when f(x) is replaced by f(x) + d
• d = 0, no vertical shift or translation
• Vertical shift or translation up or down
• Shift or translation down when d < 0
• For f(x) – 2, d = –2, and the function moves down or shifts down or translates down two units.
• Shift or translation up when d > 0
• For f(x) + 2, d = 2, and the function moves up or shifts up or translates up two units.
• Graphical representation given the algebraic representation or parameter changes
• Algebraic representation given the graphical representation or parameter changes
• Descriptions of the effects of the parameter changes on the domain and range
• Descriptions of the effects of the parameter changes on the slope and y-intercept
• Combined parameter changes
• Effects of parameter changes in real-world problem situations

Note(s):

• Algebra I introduces effects of parameter changes a, b, c, and d on the linear parent function.
• Algebra II will extend effects of parameter changes to other parent functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• B1 – Identify and apply transformations to figures.
• C1 – Make connections between geometry and algebra.
• VII. Functions
• A1 – Recognize whether a relation is a function.
• A2 – Recognize and distinguish between different types of functions.
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections 