2A.1 
Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:


2A.1A 
Apply mathematics to problems arising in everyday life, society, and the workplace.

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.

2A.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.

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.

2A.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.

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.

2A.1D 
Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

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.

2A.1E 
Create and use representations to organize, record, and communicate mathematical ideas.

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.

2A.1F 
Analyze mathematical relationships to connect and communicate mathematical ideas.

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.

2A.1G 
Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

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.

2A.8 
Data. The student applies mathematical processes to analyze data, select appropriate models, write corresponding functions, and make predictions. The student is expected to:


2A.8A 
Analyze data to select the appropriate model from among linear, quadratic, and exponential models.
Supporting Standard

Analyze
DATA
Including, but not limited to:
 Data collected from data collection devices
 Data given in mathematical problem situations
 Types of data
 Linear
 Constant rate of change (slope)
 If independent values change sequentially, the dependent values have a common first difference.
 Quadratic
 Nonconstant rate of change
 If independent values change sequentially the dependent values have a common second difference.
 Exponential
 Nonconstant rate of change
 If independent values change sequentially the dependent values have a common ratio, .
 Comparisons between sequential data as linear, quadratic, and exponential
To Select
THE APPROPRIATE MODEL FROM AMONG LINEAR, QUADRATIC, AND EXPONENTIAL MODELS
Including, but not limited to:
 Data collected from data collection devices
 Data given in realworld problem situations
 Data relationships
 Linear
 Quadratic
 Exponential
 Data representations
 Data tables
 Graphs/scatterplots
 Verbal descriptions
 Algebraic generalizations
Note(s):
 Grade Level(s):
 Algebra I introduced the linear, quadratic, and exponential functions.
 Algebra II expands on transformations and applications of exponential functions.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
 III.C.2. Make connections between geometry, statistics, and probability.
 V.B. Statistical Reasoning – Describe data
 V.B.4. Describe patterns and departure from patterns in the study of data.
 V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
 V.C.1. Analyze data sets using graphs and summary statistics.
 V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
 VI.A. Functions – Recognition and representation of functions
 VI.A.2. Recognize and distinguish between different types of functions.
 VI.C. Functions – Model realworld situations with functions
 VI.C.1. Apply known functions to model realworld situations.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 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.2. Create and use representations to organize, record, and communicate mathematical ideas.
 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.

2A.8B 
Use regression methods available through technology to write a linear function, a quadratic function, and an exponential function from a given set of data.
Supporting Standard

Use
REGRESSION METHODS AVAILABLE THROUGH TECHNOLOGY TO WRITE A LINEAR FUNCTION, A QUADRATIC FUNCTION, AND AN EXPONENTIAL FUNCTION FROM A GIVEN SET OF DATA
Including, but not limited to:
 Data collected from data collection devices
 Data given in mathematical and realworld problem situations
 Data relationships
 Linear
 Quadratic
 Exponential
 Regression equation – line of best fit representing a set of bivariate data
 Correlation coefficient (rvalue) – numeric value that assesses the strength of the linear relationship between two quantitative variables in a set of bivariate data
 When the correlation coefficient, r, is given in regression calculations, it can be used to determine the strength of the regression model as a representation of mathematical and realworld problem situations.
 The correlation coefficient, r, can only be used to analyze linear relationships or relationships that can be linearized such as exponential
 Correlation coefficients closest to ±1 indicate the best model for mathematical and realworld problem situations.
 Value of the correlation coefficient, –1 ≤ r ≤ 1
 Perfect correlation, r = 1 or –1
 Strong correlation, 0.68 < r < 1.00
 Moderate correlation, 0.34 ≤ r ≤ 0.68
 Weak, 0 < r < 0.34
 No correlation, r = 0
 Coefficient of determination (r^{2}value) – representation of the percent of data closest to the regression line and used to measure how well the regression line can be used as a prediction model
 When the coefficient of determination, r^{2}, is given in regression calculations, it can be used to determine the strength of the regression model to represent and make predictions in mathematical and realworld problem situations.
 The coefficient of determination, r^{2}, can be used to analyze and compare all types of relationships such as linear, quadratic, and exponential
Note(s):
 Grade Level(s):
 Algebra I introduced the linear, quadratic, and exponential functions.
 Algebra I calculated, using technology, the correlation coefficient between two quantitative variables and interpreted this quantity as a measure of the strength of association.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
 III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
 III.C.2. Make connections between geometry, statistics, and probability.
 VI.A. Functions – Recognition and representation of functions
 VI.A.2. Recognize and distinguish between different types of functions.
 VI.C. Functions – Model realworld situations with functions
 VI.C.2. Develop a function to model a situation.
 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.
 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.

2A.8C 
Predict and make decisions and critical judgments from a given set of data using linear, quadratic, and exponential models.
Readiness Standard

Predict, Make
DECISIONS AND CRITICAL JUDGMENTS FROM A GIVEN SET OF DATA USING LINEAR, QUADRATIC, AND EXPONENTIAL MODELS
Including, but not limited to:
 Mathematical and realworld problem situations modeled by linear, quadratic, and exponential functions and equations
 Predictions, decisions, and critical judgments from function models
 Justification of reasonableness of solutions in terms of mathematical and realworld problem situations
 Mathematical justification
 Substitution in original problem
 Justification for predictions using the coefficient of determination, r^{2}
Note(s):
 Grade Level(s):
 Algebra I introduced the linear, quadratic, and exponential functions.
 Algebra I introduced the correlation coefficient as a measure of the strength of linear association.
 Algebra I applied linear, quadratic, and exponential functions to model and make predictions in realworld problem situations.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
 III.C.2. Make connections between geometry, statistics, and probability.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.4. Justify the solution.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
 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.
