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:

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:
 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 nonexamples, patterns, etc.
 Current knowledge to new learning
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxCCRS:

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 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:
 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.
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:
 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.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:
 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.2I 
Write systems of two linear equations given a table of values, a graph, and a verbal description.
Readiness Standard

Write
SYSTEMS OF TWO LINEAR EQUATIONS 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
 Characteristics of 2 × 2 systems of linear equations
 Two equations
 Two variables
 Given multiple representations
 Table of values
 Graph
 Verbal description
Note(s):
 Grade Level(s):
 Middle School used multiple representations for linear relationships.
 Algebra I formally introduces systems of two linear equations in two variables.
 Algebra II will introduce systems of three linear equations in three variables and systems of one linear equation and one quadratic equation.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 II. Algebraic Reasoning
 A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
 C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
 D1 – Interpret multiple representations of equations and relationships.
 D2 – Translate among multiple representations of equations and relationships.
 III. Geometric Reasoning
 C1 – Make connections between geometry and algebra.
 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.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:
 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 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:
 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.3H 
Graph the solution set of systems of two linear inequalities in two variables on the coordinate plane.
Supporting Standard

Graph
THE SOLUTION SET OF SYSTEMS OF TWO LINEAR INEQUALITIES IN TWO VARIABLES ON THE COORDINATE PLANE
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
 Systems of inequalities
 Two unknowns
 Two inequalities
 Graphical analysis of the system of inequalities
 Graphing of each function
 Shading of inequality region for each
 Representation of the solution as points in the region of intersection
 Justification of solution to systems of inequalities
 Substitution of various points in the solutions region into original functions
Note(s):
 Grade Level(s):
 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:
 I. Numeric reasoning
 C1 – Use estimation to check for errors and reasonableness of solutions.
 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.4 
Linear functions, equations, and inequalities. The student applies the mathematical process standards to formulate statistical relationships and evaluate their reasonableness based on realworld data. The student is expected to:


A.4C 
Write, with and without technology, linear functions that provide a reasonable fit to data to estimate solutions and make predictions for realworld problems.
Supporting Standard

Write
LINEAR FUNCTIONS THAT PROVIDE A REASONABLE FIT TO DATA, WITH AND WITHOUT TECHNOLOGY
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)
 Function models for sets of data
 Trend line by manipulating slope and yintercept
 Regression equation, y = ax + b, using the graphing calculator
To Estimate, To Make
SOLUTIONS AND PREDICTIONS FOR REALWORLD PROBLEMS
Including, but not limited to:
 Function models for sets of data
 Trend line by manipulating slope and yintercept
 Regression equation, y = ax + b, using the graphing calculator
 Correlation coefficient as an indicator of reliability of regression equations
Note(s):
 Grade Level(s):
 Grade 8 graphed scatterplots of bivariate data and used trend lines to analyze the correlation as linear, nonlinear, or no association.
 Algebra I introduces calculation and interpretation of the correlation coefficient between two quantitative variables.
 Algebra I introduces the use of algebraic strategies and regression technology to determine the line of best fit.
 Algebra II will apply regression technology and will determine appropriate models between linear, quadratic, and exponential functions to make predictions and critical judgments.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 VI. Statistical Reasoning
 B1 – Determine types of data.
 B2 – Select and apply appropriate visual representations of data.
 B3 – Compute and describe summary statistics of data.
 B4 – Describe patterns and departure from patterns in a set of data.
 C1 – Make predictions and draw inferences using summary statistics.
 C2 – Analyze data sets using graphs and summary statistics.
 C3 – Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
 C4 – Recognize reliability of statistical results.
 VII. Functions
 C2 – Develop a function to model a situation.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

A.5 
Linear functions, equations, and inequalities. The student applies the mathematical process standards to solve, with and without technology, linear equations and evaluate the reasonableness of their solutions. The student is expected to:


A.5A 
Solve linear equations in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides.
Readiness Standard

Solve
LINEAR EQUATIONS IN ONE VARIABLE, INCLUDING THOSE FOR WHICH THE APPLICATION OF THE DISTRIBUTIVE PROPERTY IS NECESSARY AND FOR WHICH VARIABLES ARE INCLUDED ON BOTH SIDES
Including, but not limited to:
 Linear equation in one variable – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
 Linear equations in one variable including parentheses and variables on both sides of the equation
 Mathematical problem situations
 Realworld problem situations
 Multiple representations of mathematical and realworld problem situations
 Algebraic generalizations
 Missing coordinate of a solution point to a function
 Verbal
 Methods for solving equations
 Concrete and pictorial models (e.g., algebra tiles, etc.)
 Tables and graphs with and without technology
 Transformation of equations using properties of equality
 Distributive property
 Operational properties
 Possible solutions, including special cases
 No solution, empty set, ∅
 Infinite solutions, all real numbers, ℜ
 Relationships and connections between the methods of solution
 Justification of solutions to equations
 Justification of reasonableness of solutions in terms of mathematical and realworld problem situations
Note(s):
 Grade Level(s):
 Grade 5 used equations with variables to represent missing numbers.
 Grade 6 solved onevariable, onestep equations.
 Grade 7 solved onevariable, twostep equations.
 Grade 8 solved onevariable equations with variables on both sides.
 Algebra I introduces solving onevariable equations that include those for which the application of the distributive property is necessary and for which variables are included on both sides.
 Algebra II will introduce solving absolute value linear equations.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 I. Numeric reasoning
 C1 – Use estimation to check for errors and reasonableness of solutions.
 II. Algebraic Reasoning
 A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
 C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
 D1 – Interpret multiple representations of equations and relationships.
 D2 – Translate among multiple representations of equations and relationships.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

A.5C 
Solve systems of two linear equations with two variables for mathematical and realworld problems.
Readiness Standard

Solve
SYSTEMS OF TWO LINEAR EQUATIONS WITH TWO VARIABLES FOR MATHEMATICAL AND REALWORLD PROBLEMS
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
 Systems of 2 × 2 linear equations
 Two equations
 Two variables
 Solutions to systems of equations
 One common point of intersection, (x, y)
 Infinite set of points on a line
 Empty set, Ø
 Methods for solving systems of linear equations with and without technology
 Tables
 Graphs
 Concrete models
 Algebraic methods
 Substitution
 Linear combination (elimination)
 Special cases for empty set, Ø, and all real numbers, ℜ
 Relationships and connections between the methods of solution
 Justification of solutions to systems of equations with and without technology
 Systems of linear equations as models for realworld problem situations
 Interpretation of a solution point in terms of the realworld problem situation
 Justification of reasonableness of solution in terms of the realworld problem situation or data collection
Note(s):
 Grade Level(s):
 Algebra I formally introduces systems of two linear equations in two variables.
 Algebra II will introduce systems of three linear equations in three variables.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 I. Numeric reasoning
 C1 – Use estimation to check for errors and reasonableness of solutions.
 II. Algebraic Reasoning
 A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
 C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
 D1 – Interpret multiple representations of equations and relationships.
 D2 – Translate among multiple representations of equations and relationships.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections
