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


A.12C 
Identify terms of arithmetic and geometric sequences when the sequences are given in function form using recursive processes.
Supporting Standard

Identify
TERMS OF ARITHMETIC AND GEOMETRIC SEQUENCES WHEN THE SEQUENCES ARE GIVEN IN FUNCTION FORM USING RECURSIVE PROCESSES
Including, but not limited to:
 Sequence – a list of numbers or a collection of objects written in a specific order that follow a particular pattern. Sequences can be viewed as functions whose domains are the positive integers.
 Domain of a sequence – set of natural numbers; 1, 2, 3, …
 The domain of a sequence represents the position, n, of the term.
 Range of a sequence – terms in the sequence calculated by the sequence rule
 The range of a sequence represents the value of the term at the n^{th} position.
 The range is the actual listed number in a sequence.
 Although a_{0} can be given or determined, it is not part of the sequence.
 Arithmetic sequence – sequence formed by adding or subtracting the same value to calculate each subsequent term
 Ex: 2, 5, 8, 11, 14, … Three is added to the previous term to calculate each subsequent term.
 Ex: 7, 3, –1, –5, ... Four is subtracted from the previous term to calculate each subsequent term.
 Geometric sequence – sequence formed by multiplying or dividing by the same value to calculate each subsequent term
 Ex: 1, 2, 4, 8, 16, ... The previous term is multiplied by two in order to calculate each subsequent term.
 Ex: 81, 27, 9, 3, 1, , ... Three is divided into the previous term to calculate each subsequent term or onethird is multiplied to calculate each subsequent term.
 Recursive notation
 Recursive process – calculation of the next number in a sequence by repeated application of a rule
 Arithmetic
 a_{n} = a_{n}_{–1} + d, where one term of the sequence is given
 a_{n}_{+1 }= a_{n} + d, where one term of the sequence is given
 f(n) = f(n – 1) + d, where one term of the sequence is given
 f(n + 1) = f(n) + d, where one term of the sequence is given
 Geometric
 a_{n} = r • a_{n}_{–1}, where a_{0} = 1
 a_{n}_{ + 1} = r • a_{n}, where a_{0} = 1
 f(n) = r • f(n – 1), where f(0) = 1
 f(n + 1) = r • f(n), where f(0) = 1
 One term in the sequence must be given in order to find the preceding and/or subsequent terms in the sequence
 Initial values are required to ensure uniqueness of the sequence
 Connections between arithmetic sequence and linear functions
 Linear function: f(x) = mx + b
 f(x) represents the x^{th} term of the sequence
 f(1) represents the 1st term of the sequence, a_{1}
 m represents the common difference of an arithmetic sequence, d
 Connections between geometric sequences and exponential functions
 Exponential function: f(x) = ab^{x}
 f(x) represents the x^{th} term of the sequence
 f(1) represents the 1st term of the sequence, a_{1}
 b represents the common ratio of an geometric sequence, r
 Identification of terms of arithmetic and geometric sequences
Note(s):
 Grade Level(s):
 Previous grade levels introduced the concept of sequences.
 Algebra I introduces the concept of arithmetic and geometric sequences.
 Algebra I introduces application of recursive processes in analysis of arithmetic and geometric sequences.
 Algebra II will formulate exponential equations from recursive notation.
 Precalculus will address arithmetic and geometric sequences and series, including the infinite geometric series.
 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).
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

A.12D 
Write a formula for the n^{th} term of arithmetic and geometric sequences, given the value of several of their terms.
Supporting Standard

Write
A FORMULA FOR THE n^{th} TERM OF ARITHMETIC AND GEOMETRIC SEQUENCES, GIVEN THE VALUE OF SEVERAL OF THEIR TERMS
Including, but not limited to:
 Sequence – a list of numbers or a collection of objects written in a specific order that follow a particular pattern. Sequences can be viewed as functions whose domains are the positive integers.
 Arithmetic sequence – sequence formed by adding or subtracting the same value to calculate each subsequent term
 Ex: 2, 5, 8, 11, 14, … Three is added to the previous term to calculate each subsequent term.
 Ex: 7, 3, –1, –5, ... Four is subtracted from the previous term to calculate each subsequent term.
 Geometric sequence – sequence formed by multiplying or dividing by the same value to calculate each subsequent term
 Ex: 1, 2, 4, 8, 16, ... Two is multiplied times the previous term to calculate each subsequent term.
 Ex: 81, 27, 9, 3, 1, , ... Three is divided into the previous term to calculate each subsequent term or onethird is multiplied to calculate each subsequent term.
 Notation for arithmetic sequences
 a_{1} represents the 1^{st} term of the sequence
 a_{n} represents the n^{th} term of the sequence
 a_{n}_{1} represents the term before the n^{th} term of the sequence
 d represents the common difference of an arithmetic sequence
 Notation for geometric sequences
 a_{1} represents the 1^{st} term of the sequence
 a_{n} represents the n^{th} term of the sequence
 a_{n}_{1} represents the term before the n^{th} term of the sequence
 r represents the common ratio of a geometric sequence
 Recursive process – calculation of the next number in a sequence by repeated application of a rule
 Explicit notation
 Arithmetic: a_{n} = a_{1} + d(n– 1)
 Geometric: a_{n} = a_{1 } • r^{n}^{–1}
 Arithmetic and geometric sequences in function form
 Arithmetic: f(n) = a_{1} + d(n – 1)
 Geometric: f(n) = a_{1} • r^{n}^{–1}
 Initial values are required to ensure uniqueness of the sequence
 Connections between arithmetic sequences and linear functions
 Arithmetic sequences can be viewed as linear functions.
 Simplification of the explicit arithmetic notation formula
 y = mx + b, where b is the a_{0} and m is the common differenc
 Domain of the sequence is the set of all positive integers.
 Connections between geometric sequences and exponential functions.
 Geometric sequences can be viewed as exponential functions.
 Simplification of the explicit geometric notation formula
 y = ab^{x}, where a is the a_{0} and b is the common ratio
 Domain of the sequence is the set of all positive integers.
 Representation of sequences by formula or rule
 Application of sequences in realworld problem situations
 Justification of solutions in terms of realworld problem situations
Note(s):
 Grade Level(s):
 Previous grade levels introduced the concept of sequences.
 Algebra I introduces the concept of arithmetic and geometric sequences.
 Algebra I introduces application of recursive processes in analysis of arithmetic and geometric sequences.
 Algebra II will formulate exponential equations from recursive notation.
 Precalculus will address arithmetic and geometric sequences and series, including the
 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).
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections
