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.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.9A 
Determine the domain and range of exponential functions of the form f(x) = ab^{x} and represent the domain and range using inequalities.
Supporting Standard

Determine, Represent
THE DOMAIN AND RANGE OF EXPONENTIAL FUNCTIONS OF THE FORM f(x) = ab^{x} USING INEQUALITIES
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}
 Domain and range of exponential 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 values in realworld problem situations
 Reasonable domain and range for the realworld problem situation
 Comparison of domain and range of a 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 exponential 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 exponential functions.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 VI.B. Functions – Analysis of functions
 VI.B.1. Understand and analyze features of functions.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.

A.9B 
Interpret the meaning of the values of a and b in exponential functions of the form f(x) = ab^{x} in realworld problems.
Supporting Standard

Interpret
THE MEANING OF THE VALUES OF a AND b IN EXPONENTIAL FUNCTIONS OF THE FORM f(x) = ab^{x} IN 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}
 a value is the yintercept, (0, a).
 b value is the successive ratio of range values.
 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 and equations.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 VI.B. Functions – Analysis of functions
 VI.B.1. Understand and analyze features of functions.
 VI.C. Functions – Model realworld situations with functions
 VI.C.1. Apply known functions to model realworld situations.
 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.B. Communication and Representation – Interpretation of mathematical work
 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.

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

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:
 VI.B. Functions – Analysis of functions
 VI.B.1. Understand and analyze features of functions.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.

A.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:
 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.
 VI.C. Functions – Model realworld situations with functions
 VI.C.1. Apply known functions to model realworld situations.
 VI.C.2. Develop a function to model a situation.
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.1. Interpret results of the mathematical problem in terms of the original realworld situation.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.
 IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
