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.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 x 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.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.3F 
Graph systems of two linear equations in two variables on the coordinate plane and determine the solutions if they exist.
Supporting Standard

Graph
SYSTEMS OF TWO LINEAR EQUATIONS IN TWO VARIABLES ON THE COORDINATE PLANE
Determine
THE SOLUTIONS OF SYSTEMS OF TWO LINEAR EQUATIONS IF THEY EXIST
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 x 2 systems of linear equations
 Two equations
 Two variables
 Multiple representations
 Table of values
 Verbal descriptions
 Algebraic generalizations
 Graphical analysis of the system of equations
 Conversion to slopeintercept form
 Graph of each equation
 Identification of possible solutions
 (x, y), one point of intersection
 Ø, no points of intersection (parallel lines)
 Infinite number of points on the line y = mx + b, (coinciding lines)
 Application of systems of equations in realworld problem situations
 Justification of solution to systems of equations
 Substitution of point(s) in the solution into original functions
Note(s):
 Grade Level(s):
 Grade 8 introduced graphing two linear equations simultaneously to identify the values of x and y of the intersection point that satisfy the two linear equations.
 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.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
 II.C.1. Describe and interpret solution sets of equalities and inequalities.
 II.C.2. Explain the difference between the solution set of an equation and the solution set of an inequality.
 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.3G 
Estimate graphically the solutions to systems of two linear equations with two variables in realworld problems.
Supporting Standard

Estimate
GRAPHICALLY THE SOLUTIONS TO SYSTEMS OF TWO LINEAR EQUATIONS WITH TWO VARIABLES IN 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 equations to model realworld problem situations
 Two unknown variables
 Two equations
 Graphical analysis of the system of equations
 Graph of each equation
 Identification of solution
 (x, y), one point of intersection
 Ø, no points of intersection (parallel lines)
 Infinite number of points on the line y = mx + b, (coinciding lines)
 Justification of solution in terms of realworld problem situations
Note(s):
 Grade Level(s):
 Grade 8 introduced graphing two linear equations simultaneously to identify the values of x and y of the intersection point that satisfy the two linear equations.
 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:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.1. Use estimation to check for errors and reasonableness of solutions.
 II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
 II.C.1. Describe and interpret solution sets of equalities and inequalities.
 II.C.2. Explain the difference between the solution set of an equation and the solution set of an inequality.
 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.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:
 II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
 II.C.1. Describe and interpret solution sets of equalities and inequalities.
 II.C.2. Explain the difference between the solution set of an equation and the solution set of an inequality.
 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.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.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 x 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:
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
 II.C.2. Explain the difference between the solution set of an equation and the solution set of an inequality.
 II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.3. Determine a solution.
 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.
 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.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
