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.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:
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 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.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.

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

Solve
LINEAR INEQUALITIES 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 inequality in one variable – a mathematical statement composed of algebraic and/or numeric expressions set apart by an inequality symbol
 Inequality symbols
 > (is greater than)
 < (is less than)
 ≥ (is greater than or equal to)
 ≤ (is less than or equal to)
 ≠ (is not equal to)
 Linear inequalities 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
 Verbal
 Solutions to include numeric, graphic, and verbal representations
 Methods for solving inequalities
 Concrete and pictorial models (e.g., algebra tiles, etc.)
 Graphs and tables with and without technology
 Transformation of inequalities using properties of inequalities
 Distributive property
 Operational properties
 Special cases for empty set, Ø, and all real numbers, ℜ
 Relationships and connections between the methods of solution
 Justification of solutions to inequalities
 Differentiation between solutions of equations and inequalities
 Justification of reasonableness of solutions in terms of mathematical and realworld problem situations
Note(s):
 Grade Level(s):
 Grade 6 solved onevariable, onestep inequalities.
 Grade 7 solved onevariable, twostep inequalities.
 Grade 8 wrote onevariable inequalities with variables on both sides.
 Algebra I introduces solving onevariable inequalities, including 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 inequalities.
 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.

A.10 
Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to rewrite in equivalent forms and perform operations on polynomial expressions. The student is expected to:


A.10A 
Add and subtract polynomials of degree one and degree two.
Supporting Standard

Add, Subtract
POLYNOMIALS OF DEGREE ONE
Including, but not limited to:
 Algebraic expression – a generalization that is a combination of variables, numbers (constants and coefficients), and operators
 Polynomial expression – monomial or sum of monomials not including variables in the denominator or under a radical
 Monomial – one term expression; e.g., –2.5x,
 Binomial – two term expression; e.g., 4 – 2y, 3a + 1
 Trinomial – three term expression
 Degree
 Degree of term – sum of the powers on the variables in the term
 Degree of a polynomial – same as the degree of the term in the polynomial with the highest degree
 First degree polynomial – polynomial whose highest degree term contains one variable with power of one
 Ex: 3x + 8; The highest degree term is 3x, and the power on x is one.
 Ex: –2x – 5y; Both terms are degree one with the power on x and y both equal to one.
 Simplifying polynomials by addition/subtraction using concrete models
 Simplifying polynomials by addition/subtraction algebraically
 Clear grouping symbols using the distributive property.
 Combine like terms.
 Place terms in order
 Alphabetical order
 Decreasing degree order
 Applications of addition/subtraction of polynomials in mathematical problem situations
Note(s):
 Grade Level(s):
 Previous grade levels calculated the perimeter of triangles and rectangles.
 Grade 6 generated and compared equivalent expressions using concrete models, pictorial models, and algebraic properties of operations.
 Algebra I introduces operations with polynomials of degree two.
 Algebra II will extend operations with polynomials of degree three and degree four, including division of polynomials.
 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.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).

A.10C 
Determine the quotient of a polynomial of degree one and polynomial of degree two when divided by a polynomial of degree one and polynomial of degree two when the degree of the divisor does not exceed the degree of the dividend.
Supporting Standard

Determine
THE QUOTIENT OF A POLYNOMIAL OF DEGREE ONE WHEN DIVIDED BY A POLYNOMIAL OF DEGREE ONE
Including, but not limited to:
 Degree one polynomial divided by another degree one polynomial
 Division of polynomials
 Division by factoring
 Cancellation of common factors in the numerator and denominator
 Array method
 Long division
 Long division format with divisor outside division box, dividend inside the division box, and quotient on top of division box
 Missing terms in series represented by adding a zero term
 Applications of division of polynomials in mathematical problem situations
Note(s):
 Grade Level(s):
 Previous grade levels calculated the area of triangles and rectangles.
 Algebra I introduces operations with polynomials of degree one and degree two.
 Algebra II will extend operations with polynomials of degree three and degree four.
 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.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).

A.10D 
Rewrite polynomial expressions of degree one and degree two in equivalent forms using the distributive property.
Supporting Standard

Rewrite
POLYNOMIAL EXPRESSIONS OF DEGREE ONE IN EQUIVALENT FORMS USING THE DISTRIBUTIVE PROPERTY
Including, but not limited to:
 Polynomial expression – monomial or sum of monomials not including variables in the denominator or under a radical
 First degree polynomial – polynomial whose highest degree term contains one variable with power of one
 Factorization of the greatest common factor (GCF)
 Operations on polynomials
 Addition/subtraction
 Multiplication
Note(s):
 Grade Level(s):
 Algebra I introduces operations with polynomials of degree one and degree two.
 Algebra II will extend operations with polynomials of degree three and degree four.
 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.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).

A.12 
Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to write, solve, analyze, and evaluate equations, relations, and functions. The student is expected to:


A.12E 
Solve mathematic and scientific formulas, and other literal equations, for a specified variable.
Supporting Standard

Solve
MATHEMATIC AND SCIENTIFIC FORMULAS, AND OTHER LITERAL EQUATIONS, FOR A SPECIFIED VARIABLE
Including, but not limited to:
 Literal equations – equations in which all or part of the terms are expressed in variables
 Two variable linear equations
 Mathematical formulas
 Scientific formulas
 Transforming literal equations is subsumed within solving
 Solving for one of the variables in two variable linear equations.
 Solving formulas for a specified variable
 Mathematical formulas
 Scientific formulas
Note(s):
 Grade Level(s):
 Algebra I introduces solving mathematical formulas and literal equations.
 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.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.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
 II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
