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.2E 
Write the equation of a line that contains a given point and is parallel to a given line.
Supporting Standard

Write
THE EQUATION OF A LINE THAT CONTAINS A GIVEN POINT AND IS PARALLEL TO A GIVEN LINE
Including, but not limited to:
 Multiple representations
 Characteristics of parallel lines
 In the same plane
 Do not intersect
 Same distance apart
 Slopes are equal, m_{y}_{1} = m_{y}_{2} , where m_{y}_{1 }is the slope of line 1 and m_{y}_{2} is the slope of line 2.
 m = 0, y = #
 m = undefined, x = #
 Various forms of the equation of a line
 Slopeintercept form, y = mx + b
 Pointslope form, y – y_{1} = m(x – x_{1})
 Standard form, Ax + By = C
Note(s):
 Grade Level(s):
 Previous grade levels introduced slope and meaning of parallel separately.
 Algebra I introduces the concept of parallel lines in terms of slope.
 Geometry will write the equation of a line parallel to a given line passing through a given point to determine geometric relationships on a coordinate plane.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 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.2F 
Write the equation of a line that contains a given point and is perpendicular to a given line.
Supporting Standard

Write
THE EQUATION OF A LINE THAT CONTAINS A GIVEN POINT AND IS PERPENDICULAR TO A GIVEN LINE
Including, but not limited to:
 Multiple representations
 Characteristics of perpendicular lines
 In the same plane
 Intersecting lines
 Intersect to form four 90^{o} angles
 Slopes are negated (opposite) reciprocals, m_{y}_{2} =
 Given line y = #, where m = 0, the perpendicular line is x = # with an undefined slope.
 Given line x = #, where m = undefined, the perpendicular line is y = # with m = 0.
 Various forms of the equation of a line
 Slopeintercept form, y = mx + b
 Pointslope form, y – y_{1} = m(x – x_{1})
 Standard form, Ax + By = C
Note(s):
 Grade Level(s):
 Previous grade levels introduced slope and meaning of perpendicular separately.
 Algebra I introduces the concept of perpendicular lines in terms of slope.
 Geometry will write the equation of a line perpendicular to a given line passing through a given point to determine geometric relationships on a coordinate plane.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 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.2G 
Write an equation of a line that is parallel or perpendicular to the X or Y axis and determine whether the slope of the line is zero or undefined.
Supporting Standard

Write
AN EQUATION OF A LINE THAT IS PARALLEL OR PERPENDICULAR TO THE X OR Y AXIS
Determine
WHETHER THE SLOPE OF A LINE PARALLEL OR PERPENDICULAR TO THE X OR Y AXIS IS ZERO OR UNDEFINED
Including, but not limited to:
 Write equations for parallel or perpendicular lines
 Equations of lines parallel or perpendicular to the xaxis
 Parallel to the xaxis, y = #
 Perpendicular to the xaxis, x = #
 Equations of lines parallel or perpendicular to the yaxis
 Parallel to the yaxis, x = #
 Perpendicular to the yaxis, y = #
 Equations of lines parallel or perpendicular to lines that are parallel or perpendicular to the xaxis
 Parallel to a line parallel to the xaxis, y = #
 Parallel to a line perpendicular to the xaxis, x = #
 Perpendicular to a line parallel to the xaxis, x = #
 Perpendicular to a line perpendicular to the xaxis, y = #
 Equations of lines parallel or perpendicular to lines that are parallel or perpendicular to the yaxis
 Parallel to a line parallel to the yaxis, x = #
 Parallel to a line perpendicular to the yaxis, y = #
 Perpendicular to a line parallel to the yaxis, y = #
 Perpendicular to a line perpendicular to the yaxis, x = #
 Determine whether the slope of a parallel or perpendicular line is zero or undefined
 Slope of lines parallel to the xaxis, m = 0
 Slope of lines parallel to the yaxis, m is undefined
 Slope of lines perpendicular to the xaxis, m is undefined
 Slope of lines perpendicular to the yaxis, m = 0
 Write the equation of a line and determine the slope of the line that passes through a point and is parallel or perpendicular to the xaxis.
 Write the equation of a line and determine the slope of the line that passes through a point and is parallel or perpendicular to the yaxis.
 Write the equation of a line and determine the slope of the line that passes through a point and is parallel to a line that is parallel to the xaxis.
 Write the equation of a line and determine the slope of the line that passes through a point and is parallel to a line that is perpendicular to the xaxis.
 Write the equation of a line and determine the slope of the line that passes through a point and is perpendicular to a line that is parallel to the xaxis.
 Write the equation of a line and determine the slope of the line that passes through a point and is perpendicular to a line that is perpendicular to the xaxis.
 Write the equation of a line and determine the slope of the line that passes through a point and is parallel to a line that is parallel to the yaxis.
 Write the equation of a line and determine the slope of the line that passes through a point and is parallel to a line that is perpendicular to the yaxis.
 Write the equation of a line and determine the slope of the line that passes through a point and is perpendicular to a line that is parallel to the yaxis.
 Write the equation of a line and determine the slope of the line that passes through a point and is perpendicular to a line that is perpendicular to the yaxis.
 Generalizations
 A line parallel to the xaxis and perpendicular to the yaxis has a slope of zero.
 A line parallel to the yaxis and perpendicular to the xaxis has an undefined slope.
Note(s):
 Grade Level(s):
 Previous grade levels introduced slope and meaning of parallel and perpendicular separately.
 Algebra I introduces the concepts of parallel and perpendicular lines in terms of slope.
 Geometry will write the equation of a line parallel or perpendicular to a given line passing through a given point to determine geometric relationships on a coordinate plane.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 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.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.3D 
Graph the solution set of linear inequalities in two variables on the coordinate plane.
Readiness Standard

Graph
THE SOLUTION SET OF LINEAR INEQUALITIES IN TWO VARIABLES ON THE COORDINATE PLANE
Including, but not limited to:
 Inequality notation
 Less than, <, dashed line with shading below the graph of the line
 Greater than, >, dashed line with shading above the graph of the line
 Less than or equal to, ≤, solid line with shading below the graph of the line
 Greater than or equal to, ≥, solid line with shading above the graph of the line
 For vertical lines, greater than shades the right side of the graph and less than shades the left side of the graph.
 Given multiple representations
 Table of values
 Algebraic generalization
 Verbal description
Note(s):
 Grade Level(s):
 Algebra I introduces linear inequalities in two variables.
 Algebra II will continue systems of two linear inequalities in two variables and extend to systems of more than two linear inequalities in two variables.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 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.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.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:
 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.
 C2 – Explain the difference between the solution set of an equation and the solution set of an inequality.
 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

A.6 
Quadratic functions and equations. The student applies the mathematical process standards when using properties of quadratic functions to write and represent in multiple ways, with and without technology, quadratic equations. The student is expected to:


A.6A 
Determine the domain and range of quadratic functions and represent the domain and range using inequalities.
Readiness Standard

Determine, Represent
THE DOMAIN AND RANGE OF QUADRATIC FUNCTIONS USING INEQUALITIES
Including, but not limited to:
 Quadratic function – a seconddegree polynomial function that can be described by f(x) = ax^{2 }+ bx + c, where a ≠ 0 and the graph of the function is always parabolic or Ushaped
 Domain and range of quadratic 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
 Domain and range of quadratic functions in realworld problem situations
 Reasonable domain and range for the realworld problem situation
 Comparison of domain and range of function model to appropriate domain and range for realworld problem situation
 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
 Ex: x ∈ ℜ
 Ex: –3 < y ≤ 6
 Ex: y ≥ 0, y ∈ Ζ
Note(s):
 Grade Level(s):
 Grade 6 identified independent and dependent quantities.
 Grade 8 identified functions using sets of ordered pairs, tables, mappings, and graphs.
 Algebra I introduces quadratic functions.
 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 extend the concept of domain and range.
 Algebra II will introduce representing domain and range using interval and set notation.
 Algebra II will continue to investigate quadratic 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.6B 
Write equations of quadratic functions given the vertex and another point on the graph, write the equation in vertex form (f(x) = a(x  h)^{2}+ k), and rewrite the equation from vertex form to standard form (f(x) = ax^{2}+ bx + c).
Supporting Standard

Write
EQUATIONS OF QUADRATIC FUNCTIONS GIVEN THE VERTEX AND ANOTHER POINT ON THE GRAPH IN VERTEX FORM (f(x) = a(x – h)^{2} + k)
Including, but not limited to:
 Quadratic function – a seconddegree polynomial function that can be described by f(x) = ax^{2} + bx + c, where a ≠ 0 , and the graph of the function is always parabolic or Ushaped
 Vertex – highest (maximum) or lowest (minimum) point on the graph of a parabola
 Determination of an algebraic representation for a quadratic function in vertex form, y = a(x – h)^{2} + k
 Given vertex (h, k)
 Given point (x, y) or (x, f(x))
Rewrite
THE EQUATION FROM VERTEX FORM TO STANDARD FORM (f(x) = ax^{2 }+ bx + c)
Including, but not limited to:
 Vertex form: f(x) = a(x – h)^{2} + k
 Standard form: f(x) = ax^{2 }+ bx + c
 Mathematical problem situations
 Realworld problem situations
Note(s):
 Grade Level(s):
 Algebra I introduces writing quadratic functions in vertex and standard form.
 Algebra II will transform quadratic functions from standard to vertex form.
 Algebra II will write equations of parabolas given the vertex and other attributes.
 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.
 B2 – Algebraically construct and analyze new functions.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

A.6C 
Write quadratic functions when given real solutions and graphs of their related equations.
Supporting Standard

Write
QUADRATIC FUNCTIONS WHEN GIVEN REAL SOLUTIONS AND GRAPHS OF THEIR RELATED EQUATIONS
Including, but not limited to:
 Quadratic function – a seconddegree polynomial function that can be described by f(x) = ax^{2} + bx + c, where a ≠ 0 and the graph of the function is always parabolic or Ushaped
 Representations of quadratic functions
 Graphs
 Algebraic generalizations
 Comparisons of quadratic equations (0 = ax^{2} + bx + c) and quadratic functions (y = ax^{2} + bx + c)
 Comparisons of zeros/xintercepts of quadratic functions and solutions/roots of quadratic equations
 Comparisons of solutions/roots and factors of the quadratic equation
 Solutions/roots: r_{1} and r_{2}
 Factors: (x – r_{1})(x – r_{2})
 Quadratic equation: 0 = (x – r_{1})(x – r_{2})
 Zeros/xintercepts and factors of the quadratic function
 Zeros/xintercepts: z_{1} and z_{2} or (z_{1}, 0) and (z_{2}, 0)
 Factors: (x – z_{1})(x – z_{2})
 Quadratic function: f(x) = (x – z_{1})(x – z_{2})
 Multiple functions with the same solutions are possible depending on scalar multiples or the “a” value.
Note(s):
 Grade Level(s):
 Algebra I introduces writing quadratic functions from solutions/roots and zeros/xintercepts.
 Algebra II will write quadratic functions given three points on the parabola.
 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.
 B2 – Algebraically construct and analyze new functions.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

A.7 
Quadratic functions and equations. The student applies the mathematical process standards when using graphs of quadratic functions and their related transformations to represent in multiple ways and determine, with and without technology, the solutions to equations. The student is expected to:


A.7A 
Graph quadratic functions on the coordinate plane and use the graph to identify key attributes, if possible, including xintercept, yintercept, zeros, maximum value, minimum values, vertex, and the equation of the axis of symmetry.
Readiness Standard

Graph
QUADRATIC FUNCTIONS ON THE COORDINATE PLANE
Including, but not limited to:
 Quadratic function – a seconddegree polynomial function that can be described by f(x) = ax^{2} + bx + c, where a ≠ 0 and the graph of the function is always parabolic or Ushaped
 Graphs of quadratic functions with and without technology
 Algebraic generalizations
 Realworld problem situations involving quadratic functions
Use
THE GRAPH OF A QUADRATIC FUNCTION TO IDENTIFY KEY ATTRIBUTES, IF POSSIBLE, INCLUDING xINTERCEPT, yINTERCEPT, ZEROS, MAXIMUM VALUE, MINIMUM VALUES, VERTEX, AND THE EQUATION OF THE AXIS OF SYMMETRY
Including, but not limited to:
 Representation of quadratic functions
 Standard form: f(x) = ax^{2} + bx + c
 Vertex form: f(x) = a(x – h)^{2} + k
 Characteristics of quadratic functions
 Intercepts/Zeros
 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)
 Denoted as c in f(x) = ax^{2} + bx + c
 Denoted as ah^{2} + k in f(x) = a(x – h)^{2} + k
 Vertex – highest (maximum) or lowest (minimum) point on the graph of a parabola
 Graphically, the maximum or minimum point of the parabola
 Algebraically x = and solving for y
 From standard form: x = and solving for y
 From vertex form: (h, k)
 Maximum – graph opens downward, negative a value
 Minimum – graph opens upward, positive a value
 Axis of symmetry – line passing through the vertex of a parabola that divides the parabola into two congruent halves
 Equation of the axis of symmetry
 From standard form: x =
 Vertex form: x = h
 Symmetric points – the image and preimage points reflected across the axis of symmetry of the parabola
 Realworld problem situations involving quadratic functions
 Analysis and conclusions for realworld problem situations using key attributes
Note(s):
 Grade Level(s):
 Algebra I introduces quadratic functions.
 Algebra I introduces the key attributes of a quadratic function.
 Algebra II will continue to investigate and apply quadratic functions and equations and write the equation of a parabola from given attributes, including direction of opening.
 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.7C 
Determine the effects on the graph of the parent function f(x) = x^{2} when f(x) is replaced by af(x), f(x) + d, f(x  c), f(bx) for specific values of a, b, c, and d.
Readiness Standard

Determine
THE EFFECTS ON THE GRAPH OF THE PARENT FUNCTION f(x) = x^{2} WHEN f(x) IS REPLACED BY af(x), f(x) + d, f(x – c), f(bx) FOR SPECIFIC VALUES OF a, b, c, AND d
Including, but not limited to:
 Parent functions – set of basic functions from which related functions are derived by transformations
 General form of quadratic parent function (including equation and function notation)
 Multiple representations
 Graphs
 Tables
 Verbal descriptions
 Algebraic generalizations (including equation and function notation)
 Changes in parameters a, b, c, and d on the graph of the parent function f(x) = x^{2}
 Effects on the graph of the quadratic parent function f(x) = x^{2} when f(x) is replaced by af(x)
 a ≠ 0
 a > 1, stretches the graph vertically or makes the graph more narrow
 0 < a < 1, compresses the graph vertically or makes the graph wider
 Opposite of a reflects the graph vertically over the horizontal axis (xaxis)
 Effects on the graph of the quadratic parent function f(x) = x^{2} when f(x) is replaced by f(bx)
 b ≠ 0
 b > 1, the graph compresses horizontally or makes the graph more narrow
 0 < b < 1, the graph stretches horizontally or makes the graph wider
 b < 0, reflects horizontally over the yaxis
 Effects on the graph of the quadratic parent function f(x) = x^{2} when f(x) is replaced by f(x – c)
 c = 0, no horizontal shift or translation
 Horizontal shift or translation left or right by c units
 Left shift or translation when c < 0
 For f(x + 2) → f(x – (–2)), c = –2, and the function moves to the left or shifts left or translates left two units.
 Right shift or translation when c > 0
 For f(x – 2), c = 2, and the function moves to the right or shifts right or translates right two units
 Effects on the graph of the quadratic parent function f(x) = x^{2} when f(x) is replaced by f(x) + d
 d = 0, no vertical shift or translation
 Vertical shift or translation up or down
 Shift or translation down when d < 0
 For f(x) – 2, d = –2, and the function moves down or shifts down or translates down two units.
 Shift or translation up when d > 0
 For f(x) + 2, d = 2, and the function moves up or shifts up or translates up two units.
 Generalizations of parameter changes to f(x) = x on the xand yintercepts
 For af(x) and f(bx), there are no changes to the xand yintercepts..
 For f(x – c),
 If c < 0, then the xintercept shifts or translates left by c units, (c, 0).
 If c > 0, then the xintercept shifts or translates right by c units (c, 0).
 For f(x) + d,
 If d < 0, then the yintercept shifts or translates up by d units (0, d).
 If d > 0, then the yintercept shifts or translates down by d units (0, d).
 Graphical representation given the algebraic representation or parameter changes
 Algebraic representation given the graphical representation or parameter changes
 Descriptions of the effects on the domain and range by the parameter changes
 Combined parameter changes
Note(s):
 Grade Level(s):
 Algebra I introduces effects of parameter changes a, b, c, and d on the quadratic parent function.
 Algebra II will extend effects of parameter changes to other parent functions.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 III. Geometric Reasoning
 B1 – Identify and apply transformations to figures.
 C1 – Make connections between geometry and algebra.
 VII. Functions
 A1 – Recognize whether a relation is a function.
 A2 – Recognize and distinguish between different types of functions.
 B1 – Understand and analyze features of a function.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

A.8 
Quadratic functions and equations. The student applies the mathematical process standards to solve, with and without technology, quadratic equations and evaluate the reasonableness of their solutions. The student formulates statistical relationships and evaluates their reasonableness based on realworld data. The student is expected to:


A.8A 
Solve quadratic equations having real solutions by factoring, taking square roots, completing the square, and applying the quadratic formula.
Readiness Standard

Solve
QUADRATIC EQUATIONS HAVING REAL SOLUTIONS BY FACTORING, TAKING SQUARE ROOTS, COMPLETING THE SQUARE, AND APPLYING THE QUADRATIC FORMULA
Including, but not limited to:
 Quadratic equation in one variable – a seconddegree polynomial function that can be described in standard form by 0 = ax^{2} + bx + c, where a ≠ 0
 Methods for solving quadratic equations with and without technology
 Concrete models
 Applicable only with quadratic equations that when set equal to zero the expression can be factored
 Algebraic methods
 Factoring
 Square roots
 Completing the square
 Quadratic formula,
 Solution sets of quadratic equations
 Two solutions
 One solution (double root)
 No real solutions, Ø
 Realworld problem situations and/or data collection activity involving a quadratic function with and without technology
 Quadratic equation to represent the realworld problem situation
 Method of choice to solve
Note(s):
 Grade Level(s):
 Algebra I introduces solving quadratic equations.
 Algebra II will introduce solving equations involving absolute value (e.g., x^{2} = 25, = , x = 5; therefore, x = ±5) .
 Algebra II will continue to solve and apply quadratic equations, including imaginary solutions.
 Algebra II will solve quadratic inequalities.
 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.
 III. Geometric Reasoning
 C1 – Make connections between geometry and algebra.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

A.8B 
Write, using technology, quadratic functions that provide a reasonable fit to data to estimate solutions and make predictions for realworld problems.
Supporting Standard

Write
QUADRATIC FUNCTIONS THAT PROVIDE A REASONABLE FIT TO DATA, USING TECHNOLOGY
Including, but not limited to:
 Quadratic function – a seconddegree polynomial function that can be described by f(x) = ax^{2} + bx + c, where a ≠ 0 and the graph of the function is always parabolic or Ushaped
 Data representations
 Realworld problem situations
 Data collections using technology
 Technology to determine a function model using quadratic regression
To Estimate, To Make
SOLUTIONS AND PREDICTIONS FOR REALWORLD PROBLEMS
Including, but not limited to:
 Data representations
 Realworld problem situations
 Data collections using technology
 Technology to determine a function model using quadratic regression
 Technology to determine key attributes of functions specific to the realworld problem situations
 Vertex (maximum, minimum)
 Intercepts (yintercepts, xintercepts, zeros)
Note(s):
 Grade Level(s):
 Algebra I introduces writing quadratic functions using technology to reasonably fit data.
 Algebra II will continue to investigate and apply quadratic equations.
 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:
 I. Numeric reasoning
 C1 – Use estimation to check for errors and reasonableness of solutions.
 III. Geometric Reasoning
 C1 – Make connections between geometry and algebra.
 VII. Functions
 B1 – Understand and analyze features of a function.
 B2 – Algebraically construct and analyze new functions.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

A.9 
Exponential functions and equations. The student applies the mathematical process standards when using properties of exponential functions and their related transformations to write, graph, and represent in multiple ways exponential equations and evaluate, with and without technology, the reasonableness of their solutions. The student formulates statistical relationships and evaluates their reasonableness based on realworld data. The student is expected to:


A.9C 
Write exponential functions in the form f(x) = ab^{x} (where b is a rational number) to describe problems arising from mathematical and realworld situations, including growth and decay.
Readiness Standard

Write
EXPONENTIAL FUNCTIONS IN THE FORM f(x) = ab^{x} (WHERE b IS A RATIONAL NUMBER) TO DESCRIBE PROBLEMS ARISING FROM MATHEMATICAL AND REALWORLD SITUATIONS, INCLUDING GROWTH AND DECAY
Including, but not limited to:
 Exponential function – a function in which the independent variable, x, is in the exponent, denoted by f(x) = ab^{x}
 a value is the yintercept, (0, a).
 b value is the successive ratio of range values.
 The b value is a rational number greater than 0.
 Exponential growth – an exponential function where b > 1 and as x increases, y increases exponentially
 Exponential decay – an exponential function where 0 < b < 1 and as x increases, y decreases exponentially
 Exponential functions in realworld problem situations.
 Representative exponential function for the realworld problem situation
 Meaning of a and b in terms of the realworld problem situation
 Growth rate, r, is b – 1
 b = 1 + r, where r is in decimal form
 Rate of decay, r, is 1 – b
 b = 1 – r, where r is in decimal form
Note(s):
 Grade Level(s):
 Algebra I introduces exponential functions.
 Algebra II will continue to investigate exponential functions, including continuous growth and decay.
 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
 A2 – Recognize and distinguish between different types of functions.
 B1 – Understand and analyze features of a function.
 B2 – Algebraically construct and analyze new functions.
 C2 – Develop a function to model a situation.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

A.9D 
Graph exponential functions that model growth and decay and identify key features, including yintercept and asymptote, in mathematical and realworld problems.
Readiness Standard

Graph
EXPONENTIAL FUNCTIONS THAT MODEL GROWTH AND DECAY
Including, but not limited to:
 Exponential function – a function in which the independent variable, x, is in the exponent, denoted by f(x) = ab^{x}
 a value is the yintercept, (0, a).
 b value is the successive ratio of range values.
 The b value is a rational number greater than 0.
 Exponential growth – an exponential function where b > 1 and as x increases, y increases exponentially
 Exponential decay – an exponential function where 0 < b < 1 and as x increases, y decreases exponentially
 Mathematical problem situations
 Realworld problem situations
Identify
KEY FEATURES, INCLUDING yINTERCEPT AND ASYMPTOTE, IN MATHEMATICAL AND REALWORLD PROBLEMS
Including, but not limited to:
 Exponential function – a function in which the independent variable, x, is in the exponent, denoted by f(x) = ab^{x}
 Key attributes
 yintercept(s) – y coordinate of a point at which the relation crosses the yaxis, meaning the x coordinate equals zero, (0, y)
 yintercept in an exponential function: (0, a) where a is the a value in f(x) = ab^{x}
 Asymptote – a line that is approached and may or may not be crossed
 Horizontal asymptote – horizontal line approached by the curve as the function approaches positive or negative infinity. Horizontal asymptotes may be crossed by the curve.
 Mathematical problem situations
 Realworld problem situations
Note(s):
 Grade Level(s):
 Algebra I introduces exponential functions.
 Algebra II will continue to investigate exponential functions, including continuous growth and decay.
 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
 A2 – Recognize and distinguish between different types of functions.
 B1 – Understand and analyze features of a function.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

A.9E 
Write, using technology, exponential functions that provide a reasonable fit to data and make predictions for realworld problems.
Supporting Standard

Write
EXPONENTIAL FUNCTIONS THAT PROVIDE A REASONABLE FIT TO DATA, USING TECHNOLOGY
Including, but not limited to:
 Exponential function – a function in which the independent variable, x, is in the exponent, denoted by f(x) = ab^{x}
 a value is the yintercept, (0, a).
 b value is the successive ratio of range values.
 The b value is a rational number greater than 0.
 Exponential growth – an exponential function where b > 1 and as x increases, y increases exponentially
 Exponential decay – an exponential function where 0 < b < 1 and as x increases, y decreases exponentially
 Concrete models
 Data representations
 Realworld problem situations
 Data collections using technology
 Technology to determine a function model using exponential regression
Make
PREDICTIONS FOR REALWORLD PROBLEMS
Including, but not limited to:
 Realworld problem situations represented by exponential functions
 Exponential functions in the form f(x) = ab^{x}
 Predictions in terms of the realworld problem situation
 Reasonableness of predictions in terms of the realworld problem situation
Note(s):
 Grade Level(s):
 Algebra I introduces exponential functions.
 Algebra I predicts solutions for exponential functions.
 Algebra II will continue to investigate exponential functions, including continuous growth and decay.
 Aglebra II will formulate and solve exponential functions and equations.
 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:
 III. Geometric Reasoning
 C1 – Make connections between geometry and algebra.
 VII. Functions
 A2 – Recognize and distinguish between different types of functions.
 B1 – Understand and analyze features of a function.
 B2 – Algebraically construct and analyze new functions.
 C2 – Develop a function to model a situation.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

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.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:
 I. Numeric Reasoning
 B1 – Perform computations with real and complex numbers.
 II. Algebraic Reasoning
 B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

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:
 I. Numeric Reasoning
 B1 – Perform computations with real and complex numbers.
 II. Algebraic Reasoning
 B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

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:
 I. Numeric Reasoning
 B1 – Perform computations with real and complex numbers.
 II. Algebraic Reasoning
 A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
 B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g. polynomials, radicals, rational expressions).
 III. Geometric Reasoning
 C1 – Make connections between geometry and algebra.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections
