
 Bold black text in italics: Knowledge and Skills Statement (TEKS)
 Bold black text: Student Expectation (TEKS)
 Strikethrough: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit(s)

 Blue text: Supporting information / Clarifications from TCMPC (Specificity)
 Blue text in italics: Unitspecific clarification
 Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS)

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


P.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:

P.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:
 VIII. Problem Solving and Reasoning

P.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:
 VIII. Problem Solving and Reasoning

P.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:
 IX. Communication and Representation

P.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:
 IX. Communication and Representation

P.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:

P.1G 
Display, explain, and 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:
 IX. Communication and Representation

P.5 
Algebraic reasoning. The student uses process standards in mathematics to evaluate expressions, describe patterns, formulate models, and solve equations and inequalities using properties, procedures, or algorithms. The student is expected to:


P.5A 
Evaluate finite sums and geometric series, when possible, written in sigma notation.

Evaluate
FINITE SUMS AND GEOMETRIC SERIES, WHEN POSSIBLE, WRITTEN IN SIGMA NOTATION
Including, but not limited to:
 Series – the sum of the terms of a sequence
 Sigma notation, (3n + 5), to represent a series
 Terms and symbols
 Summation symbol (sigma),
 Formula, 3n + 5
 Index (variable), n
 Lower bound of domain, n = 1
 Upper bound of domain, n = 10
 Processes to determine the sum
 Determination of the sum from a list of all terms
 Application of sum formulas for arithmetic and geometric series
 Formulas to evaluate the sum of a series
 Arithmetic series
 Summation formula: S_{n} = (a_{1} + a_{n})
 Variables
 n = the number of terms in the series
 a_{1} = the first term in the series
 a_{n} = the last (or n^{th}) term in the series
 S_{n} = the finite sum of the first n terms in the series
 Geometric series
 Summation formulas
 S_{n }=
 S_{∞} =
 Variables
 n = the number of terms in the series
 a_{1} = the first term in the series
 r = the common ratio of the terms in the series
 S_{n} = the n^{th} partial sum of the first n terms in the series
 S_{∞ }= the sum of all the terms in an infinite geometric series (with r < 1)
 Solve realworld problems involving finite arithmetic and finite geometric series
Note(s):
 Grade Level(s):
 Algebra I wrote linear equations for relationships based on a table of values.
 Algebra I wrote exponential equations of the form y = ab^{x} for relationships based on a table of values.
 Algebra I wrote representations for arithmetic and geometric sequences and found terms in arithmetic and geometric sequences.
 Precalculus applies skills from writing linear equations in the context of arithmetic sequences by relating the terms in the sequence to the explicit and recursive formulas based on the common difference.
 Precalculus applies skills from writing exponential equations in the context of geometric sequences by relating the terms in the sequence to the explicit and recursive formulas based on the common ratio.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 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).
 C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
 VII. Functions
 B2 – Algebraically construct and analyze new functions.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

P.5B 
Represent arithmetic sequences and geometric sequences using recursive formulas.

Represent
ARITHMETIC SEQUENCES AND GEOMETRIC SEQUENCES USING RECURSIVE FORMULAS
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 acutal listed number in a sequence.
 Although a_{0} can be given or determined, it is not part of the sequence.
 Arithmetic sequences – 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 caculate each subsequent term.
 Ex: 7, 3, –1, –5, ... four is subtracted from the previous term to calculate ach subsequent term.
 Common difference between consecutive terms
 Notation to represent the terms and common difference of an arithmetic sequence
 First term: a_{1}
 Second term: a_{2}
 n^{th} term: a_{n}
 Common difference: d
 Recursive notation
 Recursive process – calculation of a term in a sequence by the application of a rule to the previous term in the sequence
 Arithmetic
 a_{n} = a_{n1} + 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
 Use of technology and the recursive formula to determine a sequence
 Explicit notation for an arithmetic sequence
 Geometric sequences – 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 ach subsequent term.
 Common ratio between consecutive terms
 Notation to represent the terms and common ratio of a geometric sequence
 First term: a_{1}
 Second term: a_{2}
 n^{th} term: a_{n}
 Common ratio: r
 Recursive notation
 Recursive process – calculation of a term in a sequence by the application of a rule to the previous term in the sequence
 Geometric
 a_{n} = r • _{n1} 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
 Use of technology and the recursive formula to determine a sequence
 Explicit notation for a geometric sequence
Note(s):
 Grade Level(s):
 Algebra I wrote linear equations for relationships based on a table of values.
 Algebra I determined terms in arithmetic and geometric sequences.
 Algebra I wrote exponential equations of the form y = ab^{x} for relationships based on a table of values.
 Algebra I wrote representations for arithmetic and geometric sequences and found terms in arithmetic and geometric sequences.
 Precalculus applies skills from writing linear equations in the context of arithmetic sequences by relating the terms in the sequence to the explicit and recursive formulas based on the common difference.
 Precalculus applies skills from writing exponential equations in the context of geometric sequences by relating the terms in the sequence to the explicit and recursive formulas based on the common ratio.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 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).
 C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
 VII. Functions
 B2 – Algebraically construct and analyze new functions
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

P.5C 
Calculate the n^{th} term and the n^{th} partial sum of an arithmetic series in mathematical and realworld problems.

Calculate
THE n^{th} TERM AND THE n^{th} PARTIAL SUM OF AN ARITHMETIC SERIES IN MATHEMATICAL AND REALWORLD PROBLEMS
Including, but not limited to:
 Arithmetic series – the sum of an arithmetic sequence
 Finding the n^{th} term
 Recursive process – calculation of a term in a sequence by the application of a rule to the previous term in the sequence
 Given the first term and recursive formula, a_{n} = a_{n}_{ – 1} + d
 Determination of the common difference
 Repeated addition of the common difference
 Use of technology and the recursive formula to determine a sequence
 Explicit formula process
 Formula to find any term in an arithmetic sequence: a_{n} = a_{1} + d(n – 1)
 n = number of the term in the sequence
 a_{n} = the n^{th} term
 a_{1} = the first term
 d = common difference
 Finding the n^{th} partial sum
 Sequence of terms
 Terms of sequence in order
 Calculation of the sum of the sequence of terms
 Formula to find n^{th} partial sum of an arithmetic series
 S_{n} = (a + a_{n})
 n = number of terms in the series
 a_{n} = the n^{th} term
 a_{1} = the first term
 S_{n} = the n^{th} partial sum (or the sum of the first n terms)
Note(s):
 Grade Level(s):
 Algebra I wrote linear equations for relationships based on a table of values.
 Algebra I determined terms in arithmetic and geometric sequences.
 Precalculus applies skills from writing linear equations in the context of arithmetic sequences by relating the terms in the sequence to the explicit and recursive formulas, based on the common difference.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 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).
 C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
 VII. Functions
 B2 – Algebraically construct and analyze new functions.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

P.5D 
Represent arithmetic series and geometric series using sigma notation.

Represent
ARITHMETIC SERIES AND GEOMETRIC SERIES USING SIGMA NOTATION
Including, but not limited to:
 Arithmetic series – the sum of an arithmetic sequence
 Geometric series – the sum of a geometric sequence
 Sigma notation, (3n + 5), to represent a series
 Terms and symbols
 Summation symbol (sigma),
 Formula, 3n + 5
 Index (variable), n
 Lower bound of domain, n = 1
 Upper bound of domain, n = 10
 Process to determine the sum
 Determination of the sum from a list of all terms
 Application of sum formulas for arithmetic and geometric series
 Writing a series in sigma notation
 Type of sequence
 Arithmetic (consecutive terms have a common difference)
 Geometric (consecutive terms have a common ratio)
 Other types of sequences (e.g., Fibonacci, etc.)
 Explicit formula
 Arithmetic: a_{n} = a_{1} + d(n – 1)
 Geometric: a_{n} = a_{1}(r)^{n}^{–1}
 Using sigma notation with the correct upper and lower bounds
 Formulas to evaluate the sum of a series
 Arithmetic series
 S_{n} = (a_{1} + a_{n})
 Variables
 n = the number of terms in the series
 a_{1} = the first term in the series
 a_{n} = the last (or n^{th}) term in the series
 S_{n} = the finite sum of the first n terms in the series
 Geometric series
 S_{n} =
 Variables
 n = the number of terms in the series
 a_{1} = the first term in the series
 r = the common ratio of the terms in the series
 S_{n} = the n^{th} partial sum of the first n terms in the series
 Infinite geometric series
 S_{∞} =
 Variables
 a_{1} = the first term in the series
 r = the common ratio of the terms in the series
 S_{∞} = the sum of all the terms in an infinite geometric series (with r < 1)
Note(s):
 Grade Level(s):
 Algebra I wrote exponential equations of the form y = ab^{x} for relationships based on a table of values.
 Precalculus applies skills from writing linear equations in the context of arithmetic sequences by relating the terms in the sequence to the explicit and recursive formulas based on the common difference.
 Precalculus applies skills from writing exponential equations in the context of geometric sequences by relating the terms in the sequence to the explicit and recursive formulas based on the common ratio.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 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).
 C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
 VII. Functions
 B2 – Algebraically construct and analyze new functions.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

P.5E 
Calculate the n^{th} term of a geometric series, the n^{th} partial sum of a geometric series, and sum of an infinite geometric series when it exists.

Calculate
THE n^{th} TERM OF A GEOMETRIC SERIES, THE n^{th} PARTIAL SUM OF A GEOMETRIC SERIES, AND SUM OF AN INFINITE GEOMETRIC SERIES WHEN IT EXISTS
Including, but not limited to:
 Geometric series – the sum of a geometric sequence
 Finding the n^{th} term of a geometric series
 Recursive process – calculation of a term in a sequence by the application of a rule to the previous term in the sequence
 Given the first term and recursive formula, a_{n} = (a_{n}_{–1}) • r
 Determination of the common ratio
 Repeated multiplication with the common ratio
 Use of technology and the recursive formula to determine a sequence
 Ex: For the geometric sequence 8, 4, 2, 1, …, the seventh term (a_{7}) can be found by multiplying the fourth term (a_{4} = 1) by the common ratio (r= 0.5) three more times. (1 • 0.5 • 0.5 • 0.5 = 0.125)
 Explicit formula process
 Formula to find the n^{th} term in a geometric sequence: a_{n} = a_{1}_{ }•(r)^{n1}
 n = number of the term in the sequence
 a_{n} = the n^{th} term
 a_{1} = the first term
 r = common ratio
 Finding the n^{th} partial sum of a geometric series
 Writing out the terms
 Sequence of terms
 Terms of sequence in order
 Calculation of the sum of the sequence of terms
 Formula to find n^{th} partial sum of a geometric series: S_{n} =
 n = number of terms in the series
 r = common ratio
 a_{1} = the first term
 S_{n} = the n^{th} partial sum (or the sum of the first n terms)
 Finding the sum of an infinite geometric series (when it exists)
 Identifying when an infinite geometric series converges
 Investigating whether a geometric series is divergent or convergent
 Convergent geometric series – series in which the partial sum approaches a given number or a limit as the term number increases
 For an infinite geometric series to converge, r must be less than 1.
 Formula for the sum of an infinite geometric series: S_{∞} =
 r = common ratio
 a_{1} = the first term
 S_{∞} = the sum of the terms
Note(s):
 Grade Level(s):
 Algebra I wrote exponential equations of the form y = ab^{x} for relationships based on a table of values.
 Precalculus applies skills from writing exponential equations in the context of geometric sequences by relating the terms in the sequence to the explicit and recursive formulas, based on the common ratio.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 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).
 C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
 VII. Functions
 B2 – Algebraically construct and analyze new functions.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

P.5F 
Apply the Binomial Theorem for the expansion of (a + b)^{n} in powers of a and b for a positive integer n, where a and b are any numbers.

Apply
THE BINOMIAL THEOREM FOR THE EXPANSION OF (a + b)^{n} IN POWERS OF a AND b FOR A POSITIVE INTEGER n, WHERE a AND b ARE ANY NUMBERS
Including, but not limited to:
 Development of rules for binomial expansion
 Expansion of expressions of the form (a + b)^{n} algebraically (by hand) for small values of n (n ≤ 4)
 Connection of coefficients to Pascal’s Triangle
 Representation of coefficients using the symbols involving factorials
 Factorials
 n! = (n)(n – 1)(n – 2) ∙ ∙ ∙ 3 ∙ 2 ∙ 1
 Binomial coefficients are represented by combinations.
 , where n and r are whole numbers with n ≥ r
 Other notations for combinations
 General formula to find terms in a binomial expansion
 Formula:
 Solving realworld problems involving binomial expansion
Note(s):
 Grade Level(s):
 Algebra I rewrote polynomial expressions of degree one and degree two in equivalent forms using the distributive property.
 Geometry used combinations to solve problems.
 Algebra II multiplied polynomials.
 Precalculus applies all of these skills to expand binomials to whole number powers.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 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).
 V. Probabilistic Reasoning
 B1 – Compute and interpret the probability of an event and its complement.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections
