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.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 AND DEGREE TWO
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, , 4x^{2}, 2mn
 Binomial – two term expression; e.g., 4 – 2y, 3a + 1, 5x^{2} – 2x, mn – pq
 Trinomial – three term expression; e.g., x^{2} + 2x + 1, a^{2} – 2ab – 8b^{2}
 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.
 Second degree polynomial – polynomial whose highest degree term contains one variable with a power of two, or two variables each having a power of one
 Ex: 3x^{2} + x – 5; The highest degree term is 3x^{2} and the one variable has a power of two.
 Ex: x^{2} – xy + 12y^{2}; All three terms are degree two. The single variable terms, x^{2} and y^{2} both have degree two. For the xy term, both have a power of one which adds to a degree of two for the term.
 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.10B 
Multiply polynomials of degree one and degree two.
Supporting Standard

Multiply
POLYNOMIALS OF DEGREE ONE AND DEGREE TWO
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, , 4x^{2}, 2mn
 Binomial – two term expression; e.g., 4 – 2y, 3a + 1, 5x^{2} – 2x, mn – pq
 Trinomial – three term expression; e.g., x^{2} + 2x + 1, a^{2} – 2ab – 8b^{2}
 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.
 Second degree polynomial – polynomial whose highest degree term contains one variable with a power of two, or two variables each having a power of one
 Ex: 3x^{2} + x – 5; The highest degree term is 3x^{2} and the one variable has a power of two.
 Ex: x^{2} – xy + 12y^{2}; All three terms are degree two. The single variable terms, x^{2} and y^{2} both have degree two. For the xy term, both have a power of one which adds to a degree of two for the term.
 Simplifying polynomials by multiplication using concrete models
 Simplifying polynomials by multiplication algebraically
 Box method
 Distributive method
 Clear grouping symbols using the distributive property.
 Combine like terms.
 Place terms in order
 Alphabetical order
 Decreasing degree order
 Applications of multiplication of polynomials in mathematical problem situations
 Applications of multiple operations 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.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 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
Including, but not limited to:
 Degree one polynomial divided by another degree one polynomial
 Degree two polynomial divided by a degree one polynomial
 Degree two polynomial divided by another degree two polynomial
 Degree of divisor not to exceed degree of dividend
 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 AND DEGREE TWO 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
 Second degree polynomial – polynomial whose highest degree term contains one variable with a power of two, or two variables each having a 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.10E 
Factor, if possible, trinomials with real factors in the form ax^{2} + bx + c, including perfect square trinomials of degree two.
Readiness Standard

Factor
TRINOMIALS WITH REAL FACTORS IN THE FORM ax^{2} + bx + c, INCLUDING PERFECT SQUARE TRINOMIALS OF DEGREE TWO, IF POSSIBLE
Including, but not limited to:
 Trinomial – three term expression
 Factorization of trinomials
 Form ax^{2} + bx + c
 a and b – coefficients of the variables
 c – the constant term
 Terms in descending order alphabetically and by degree
 First check for a greatest common factor (GCF).
 Leading coefficient, a, equal to 1
 Ex: x^{2} – 2x – 63; x^{2} + 5x + 25; p^{2} + 13pq + 40q^{2}
 Leading coefficient, a, real number other than 1
 Ex: 3x^{2} – 24x + 36; 2x^{2} – 9x – 5; 15a^{2} + 11ab + 2b^{2}
 Perfect square trinomial – first term a perfect square, third term a perfect square, middle term double the product of the square roots of the first and last terms
 Ex: 4x^{2} – 12x + 9;
 Identification of factorable trinomials and nonfactorable (prime) trinomials
 Factorization of factorable trinomials
 Leading coefficient of 1
 Leading coefficient other than 1
 Box method
 Grouping
 Multiplication/division method (Bottoms Up)
Note(s):
 Grade Level(s):
 Algebra I introduces factorization of polynomials of degree one and degree two.
 Algebra II will extend factorization to polynomials of degree three and degree four, including factoring by grouping.
 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).

A.10F 
Decide if a binomial can be written as the difference of two squares and, if possible, use the structure of a difference of two squares to rewrite the binomial.
Supporting Standard

Decide
IF A BINOMIAL CAN BE WRITTEN AS THE DIFFERENCE OF TWO SQUARES
Including, but not limited to:
 Binomial – two term expression
 Binomial whose terms are the difference of two squares
 Both terms are perfect squares.
 The two terms have opposite signs.
 Difference of squares written as a^{2} – b^{2}
 Ex: x^{2} – 9; 81x^{2} – 121y^{2}; 25a^{2} – 9b^{2};
Use
THE STRUCTURE OF A DIFFERENCE OF TWO SQUARES TO REWRITE THE BINOMIAL, IF POSSIBLE
Including, but not limited to:
 First check for a greatest common factor (GCF).
 Difference of squares written as a^{2} – b^{2}
 Factorization of binomial that is the difference of two squares
 Factors into two binomials
 (square root of first term + square root of second term)(square root of first term – square root of second term)
 a^{2} – b^{2} = (a + b)(a – b)
 Identification of factorable trinomials and nonfactorable (prime) binomials
 Factorization of factorable trinomials
Note(s):
 Grade Level(s):
 Algebra I introduces factorization of polynomials of degree one and degree two.
 Algebra II will extend factorization to polynomials of degree three and degree four, including sum and difference of two cubes and factoring by grouping.
 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).
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.

A.11 
Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to rewrite algebraic expressions into equivalent forms. The student is expected to:


A.11B 
Simplify numeric and algebraic expressions using the laws of exponents, including integral and rational exponents.
Readiness Standard

Simplify
NUMERIC AND ALGEBRAIC EXPRESSIONS USING THE LAWS OF EXPONENTS, INCLUDING INTEGRAL AND RATIONAL EXPONENTS
Including, but not limited to:
 Algebraic expression – a generalization that is a combination of variables, numbers (constants and coefficients), and operators
 Laws (properties) of exponents
 Product of powers (multiplication when bases are the same): a^{m} • a^{n} = a^{m+n}
 Quotient of powers (division when bases are the same): = a^{m–n}
 Power to a power: (a^{m})^{n} = a^{mn}
 Negative exponent: a^{–n} =
 Zero exponent: a^{0} = 1
 Rational exponent:
 Simplification of expressions using laws (properties) of exponents
 Numeric expressions, including scientific notation
 Algebraic expressions
 Variables can appear as either the base or the exponent, but in either case must be rational numbers.
 Applications of algebraic expressions involving exponents
Note(s):
 Grade Level(s):
 Prior grade levels simplified numeric expressions involving whole number exponents.
 Grade 8 introduced scientific notation.
 Algebra I introduces exponential functions.
 Algebra I applies laws (properties) of exponents to simplify numeric and algebraic expressions.
 Algebra II will introduce equations involving rational exponents.
 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).
