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:
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.1. Interpret results of the mathematical problem in terms of the original realworld situation.
 IX.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
 IX.A.2. Connect mathematics to the study of other disciplines.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.
 IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
 IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.

A.1B 
Use a problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process and the reasonableness of the solution.
Process Standard

Use
A PROBLEMSOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEMSOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION Including, but not limited to:
 Problemsolving model
 Analyze given information
 Formulate a plan or strategy
 Determine a solution
 Justify the solution
 Evaluate the problemsolving process and the reasonableness of the solution
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxCCRS:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.1. Use estimation to check for errors and reasonableness of solutions.
 V.A. Statistical Reasoning – Design a study
 V.A.1. Formulate a statistical question, plan an investigation, and collect data.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VII.A.2. Formulate a plan or strategy.
 VII.A.3. Determine a solution.
 VII.A.4. Justify the solution.
 VII.A.5. Evaluate the problemsolving process.
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.2. Evaluate the problemsolving process.

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:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.1. Use estimation to check for errors and reasonableness of solutions.
 V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
 V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.

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:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.

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:
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.

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 nonexamples, patterns, etc.
 Current knowledge to new learning
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxCCRS:
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
 IX.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
 IX.A.2. Connect mathematics to the study of other disciplines.

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:
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.4. Justify the solution.
 VII.B. Problem Solving and Reasoning – Proportional reasoning
 VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
 VII.C. Problem Solving and Reasoning – Logical reasoning
 VII.C.1. Develop and evaluate convincing arguments.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.

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 realworld situations, both continuous and discrete; and represent domain and range using inequalities.
Readiness Standard

Determine
THE DOMAIN AND RANGE OF A LINEAR FUNCTION IN MATHEMATICAL PROBLEMS AND REASONABLE DOMAIN AND RANGE VALUES FOR REALWORLD 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, y ∈ Ζ
 Domain and range of linear functions in realworld problem situations
 Reasonable domain and range for realworld problem situations
 Comparison of domain and range of function model to appropriate domain and range for a realworld problem situation
Note(s):
 Grade Level(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:
 VI.B. Functions – Analysis of functions
 VI.B.1. Understand and analyze features of functions.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.

A.2C 
Write linear equations in two variables given a table of values, a graph, and a verbal description.
Readiness Standard

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
 Slopeintercept form, y = mx + b
 m is the slope.
 b is the yintercept.
 Pointslope form, y – y_{1} = m(x – x_{1})
 m is the slope.
 (x_{1, }y_{1}) is a given point
 Standard form, Ax + By = C; A, B, C ∈ Ζ, 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):
 Grade Level(s):
 Middle School introduced using multiple representations for linear relationships.
 Grade 8 represented linear proportional and nonproportional relationships in tables, graphs, and equations in the form y = mx + b.
 Algebra I introduces the use of standard form and pointslope form to represent linear relationships.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
 VI.C. Functions – Model realworld situations with functions
 VI.C.2. Develop a function to model a situation.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.

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 xaxis
 Parallel to the xaxis, y = #
 Perpendicular to the xaxis, x = #
 Equations of lines parallel or perpendicular to the yaxis
 Parallel to the yaxis, x = #
 Perpendicular to the yaxis, y = #
 Equations of lines parallel or perpendicular to lines that are parallel or perpendicular to the xaxis
 Parallel to a line parallel to the xaxis, y = #
 Parallel to a line perpendicular to the xaxis, x = #
 Perpendicular to a line parallel to the xaxis, x = #
 Perpendicular to a line perpendicular to the xaxis, y = #
 Equations of lines parallel or perpendicular to lines that are parallel or perpendicular to the yaxis
 Parallel to a line parallel to the yaxis, x = #
 Parallel to a line perpendicular to the yaxis, y = #
 Perpendicular to a line parallel to the yaxis, y = #
 Perpendicular to a line perpendicular to the yaxis, x = #
 Determine whether the slope of a parallel or perpendicular line is zero or undefined
 Slope of lines parallel to the xaxis, m = 0
 Slope of lines parallel to the yaxis, m is undefined
 Slope of lines perpendicular to the xaxis, m is undefined
 Slope of lines perpendicular to the yaxis, 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 xaxis.
 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 yaxis.
 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 xaxis.
 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 xaxis.
 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 xaxis.
 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 xaxis.
 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 yaxis.
 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 yaxis.
 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 yaxis.
 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 yaxis.
 Generalizations
 A line parallel to the xaxis and perpendicular to the yaxis has a slope of zero.
 A line parallel to the yaxis and perpendicular to the xaxis has an undefined slope.
Note(s):
 Grade Level(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:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
 VI.C. Functions – Model realworld situations with functions
 VI.C.2. Develop a function to model a situation.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.

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):
 Grade Level(s):
 Middle School used multiple representations for linear relationships.
 Grade 8 solved problems using onevariable 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:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.

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 y – y_{1} = m(x – x_{1}).
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 y – y_{1} = m(x – x_{1})
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 (x_{1}, y_{1}) and (x_{2}, y_{2}), m =
 Slope from multiple representations
 Tables of values
 Graphs
 Two points on a line
 Linear equations in various forms
 Slopeintercept form, y = mx + b
 Pointslope form, y – y_{1} = m(x – x_{1})
 Standard form, Ax + By = C
 m = –
Note(s):
 Grade Level(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:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
 VI.B. Functions – Analysis of functions
 VI.B.1. Understand and analyze features of functions.
 VII.B. Problem Solving and Reasoning – Proportional reasoning
 VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.

A.3B 
Calculate the rate of change of a linear function represented tabularly, graphically, or algebraically in context of mathematical and realworld problems.
Readiness Standard

Calculate
THE RATE OF CHANGE OF A LINEAR FUNCTION REPRESENTED TABULARLY, GRAPHICALLY, OR ALGEBRAICALLY IN CONTEXT OF MATHEMATICAL AND REALWORLD 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 realworld 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 realworld problem situations
 Emphasis on units of rate of change in relation to realworld problem situations
Note(s):
 Grade Level(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:
 II.B. Algebraic Reasoning – Manipulating expressions
 II.B.1. Recognize and use algebraic properties, concepts, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
 II.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
 VI.B. Functions – Analysis of functions
 VI.B.1. Understand and analyze features of functions.
 VII.B. Problem Solving and Reasoning – Proportional reasoning
 VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.

A.3C 
Graph linear functions on the coordinate plane and identify key features, including xintercept, yintercept, zeros, and slope, in mathematical and realworld problems.
Readiness Standard

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 realworld problem situations
 Multiple representations
 Tabular
 Graphical
 Verbal
 Algebraic generalizations
Identify
KEY FEATURES OF LINEAR FUNCTIONS, INCLUDING xINTERCEPT, yINTERCEPT, ZEROS, AND SLOPE, IN MATHEMATICAL AND REALWORLD 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 realworld problem situations
 Multiple representations
 Tabular
 Graphical
 Verbal
 Algebraic generalizations
 Characteristics of linear functions
 xintercept – x coordinate of a point at which the relation crosses the xaxis, meaning the y coordinate equals zero, (x, 0)
 Zeros – the value(s) of x such that the y value of the relation equals zero
 yintercept – y coordinate of a point at which the relation crosses the yaxis, 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):
 Grade Level(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 yintercept.
 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:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
 VI.B. Functions – Analysis of functions
 VI.B.1. Understand and analyze features of functions.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.

A.3D 
Graph the solution set of linear inequalities in two variables on the coordinate plane.
Readiness Standard

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):
 Grade Level(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:
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.

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(x – c), 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)
 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 (xaxis)
 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 yaxis
 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(x – c)
 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 yintercept
 Combined parameter changes
 Effects of parameter changes in realworld problem situations
Note(s):
 Grade Level(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:
 II.B. Algebraic Reasoning – Manipulating expressions
 II.B.1. Recognize and use algebraic properties, concepts, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
 VI.B. Functions – Analysis of functions
 VI.B.2. Algebraically construct and analyze new functions.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
