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 Instructional Focus DocumentPrecalculus
 TITLE : Unit 06: Sequences, Series, and Binomial Expansion SUGGESTED DURATION : 10 days

#### Unit Overview

This unit bundles student expectations that address representing arithmetic and geometric sequences and series in various formats and using these representations to solve mathematical and real-world problems. Additionally, the Binomial Theorem is also applied for expanding binomials raised to a positive integer. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this unit, in Algebra I Unit 10, students identified terms of arithmetic and geometric sequences and wrote formulas for the nth term of arithmetic and geometric sequences. In Algebra I Units 01 – 04, students studied linear expressions, functions, and equations. In Algebra I Units 02 and 09 and in Algebra II Units 01 and 09 – 11, students investigated exponential functions and equations in depth. Additionally, in Algebra I Units 01 and 06, students rewrote polynomial expressions of degree one and degree two in equivalent forms using the distributive property, and in Algebra II Unit 4, students multiplied polynomials. In Geometry Unit 10, students used combinations to solve contextual problems.

During this unit, students represent arithmetic sequences using recursive and explicit formulas and use these representations to calculate the nth term of an arithmetic sequence. Students represent arithmetic series using sigma notation and calculate the nth term of the series.  Students calculate nth partial sums and other finite sums written in sigma notation in mathematical and real-world problems. Students represent geometric sequences using recursive and explicit formulas and use these representations to calculate the nth term of a geometric sequence. Students represent geometric series using sigma notation and calculate the nth term of a geometric series. Students calculate nth partial sums of a geometric series and the sum of an infinite geometric series when it exists. Students expand expressions of the form (a + b)n by hand for small, positive, integral values of n and make connections between the coefficients of these expansions, Pascal’s triangle, and combinations. Students 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 solving real-world problems.

After this unit, in subsequent courses in mathematics, students will apply concepts of sequences, series, and binomial expansions as they arise in problem situations.

Function analysis serves as the foundation for college readiness. Analysis of sequences as functions whose domain is the set of whole numbers is emphasized in the Texas College and Career Readiness Standards (TxCCRS): II. Algebraic Reasoning B1, C1; V. Probabilistic Reasoning B1; VII. Functions B2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

According to a 2007 report, published by the Mathematical Association of America, a “fundamental experience” for students entering college mathematics courses is to address problems in real-world situations by creating and interpreting mathematical models. Functions provide a way to quantitatively study the relationships and change in numerous real-world phenomena; this, coupled with the applicability of functions to many mathematical topics, make functions one of the most important topics in high school mathematics (Cooney, Beckmann, & Lloyd, 2010). In Texas, the importance of these skills is emphasized in the TxCCRS, which calls for students to be able to understand and analyze features of a function to model real-world situations. According to Cooney, Beckmann, & Lloyd (2010), “The concept of function is intentionally broad and flexible, allowing it to apply to a wide range of situations” (p. 7). Specifically, sequences represent a form of non-traditional functions: arithmetic sequences can be thought of as linear functions with a domain of the natural numbers, while geometric sequences can be thought of as exponential functions with a domain of the natural numbers. Research suggests that the reasoning we use to analyze sequences is similar to the reasoning we use to analyze continuous functions (Cooney, Beckmann, & Lloyd, 2010). Additionally, research argues that students need both a strong conceptual understanding of functions, as well as procedural fluency; as such, good instruction must include “a conceptual understanding of function, the ability to represent a function in a variety of ways, and fluency in moving among multiple representations of functions” (National Research Council, 2005, p. 353). The Principles and Standards for School Mathematics (2000) from the National Council of Teachers of Mathematics (NCTM) notes the necessity for high school students to generalize patterns using explicitly defined and recursively defined functions and to use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts.

Cooney, T., Beckmann, S., & Lloyd, G. (2010). Developing essential understanding of functions for teaching mathematics in grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc.
Katz, Victor J. (Ed.). (2007). Algebra: Gateway to a technological future. Mathematical Association of America.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Research Council. (2005). How students learn: Mathematics in the classroom. M.S. Donovan & J.D. Bransford (Eds.). Washington, DC: The National Academies Press.

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

Algebraic expressions (numbers, variables, and operational symbols) are the basic tools of algebra.

• Why are algebraic expressions the basic tools of algebra?
• How are algebraic expressions used to express mathematical ideas and model mathematical and real-world situations?
• What operations do algebraic expressions undergo?
• How can two expressions be related?
• Why are algebraic expressions evaluated?

Algebraic relationships can be used to describe mathematical and real-world patterns.

• Why is it important to describe the algebraic relationships found in numeric patterns?
• What algebraic relationships can be found in patterns?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic and Proportional Reasoning

• Evaluate
• Expressions
• Formulas
• Patterns/Rules

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Formulas for sequences and series can be determined by analysis of patterns that can be generalized and modeled using explicitly and recursively defined functions to represent relationships in terms of the problem situation.

• What types of sequences are used in mathematical and real-world problems?
• What are the key attributes of arithmetic and geometric sequences?
• How are arithmetic sequences related to linear functions?
• How are geometric sequences related to exponential functions?
• How are arithmetic and geometric sequences related to arithmetic and geometric series?
• What patterns or situations give rise to arithmetic sequences and series?
• What patterns or situations give rise to geometric sequences and series?
• How can arithmetic and geometric sequences be represented?
• How can arithmetic and geometric series be represented?
• How can the nth term of an arithmetic or geometric series be calculated?
• How can arithmetic and geometric series be evaluated, including when written in sigma notation?
• Under what conditions does the sum of an infinite series exist?
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic and Proportional Reasoning

• Evaluate
• Expressions
• Formulas
• Patterns/Rules

Probability

• Conclusions/Predictions
• Data
• Events
• Combinations
• Simulations/Experiments
• Theoretical Probability

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Binomial expansions of the form (a + b)n yield predictable patterns that can be applied to model real-world problem situations.

• How can binomials of the form (a + b)n be expanded?
• What patterns arise from expansions of the form (a + b)n?
• What is Pascal’s Triangle and how is it related to binomial expansion?
• What is the Binomial Theorem?
• How are combinations, Pascal’s Triangle, and the Binomial Theorem related?
• When can the Binomial Theorem be applied to solve real-world problems?

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that there is no distinction between the term in a sequence (an)and its position in the sequence (n), particularly when using the related formulas. For example, the fifth term in a sequence (a5) is not necessarily a five (even though n = 5).  For students, this can be particularly troublesome in the notation for recursively defined sequences, such as .
• Some students may represent a sequence with a recursive rule without defining the first term. The recursive formula  can describe an infinite number of arithmetic sequences. To describe a specific arithmetic sequence, students must also indicate the value of a term in the sequence, such as a1.
• Some students may overlook the lower bound when computing a sum written in sigma notation. The summation (3n – 4) will have a different sum than the summation (3n – 4).
• Some students may think they do not need parentheses in the summation notation when the formula includes addition and subtraction. The summation (3n – 4) will have a different sum than 3n – 4. In the second summation, the sum is found using the formula 3n and 4 is subtracted from the final summation.
• Some students may choose an incorrect r value when computing nCr for the coefficient of the (r + 1) term in the binomial expansion of (a + b)n. For example, when determining the coefficient of the third term in the binomial expansion of (a + b)7, students should calculate the combination 7C2, not 7C3.
• Some students may make sign errors when they apply the Binomial Theorem to expansions of the form (a – b)n. Students should interpret these expressions as (a + (–b))n.

Underdeveloped Concepts:

• Some students may think that the first term in a sequence is always denoted as a1. Some conventions denote the first term as a0. In such a sequence, a4 would really be the fifth term in the sequence.

#### Unit Vocabulary

• Arithmetic sequences – sequence formed by adding or subtracting the same value to calculate each subsequent term
• Arithmetic series – the sum of an arithmetic sequence
• Convergent Geometric Series – series in which the partial sum approaches a given number or a limit as the term number increases
• Domain of a sequence – set of natural numbers; 1, 2, 3, ...
• Geometric sequences – sequence formed by multiplying or dividing by the same value to calculate each subsequent term
• Geometric series – the sum of a geometric sequence
• Range of a sequence – terms in the sequence calculated by the sequence rule
• Recursive process - calculation of a term in a sequence by the application of a rule to the previous term in the sequence
• 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.
• Series – the sum of the terms of a sequence

Related Vocabulary:

 Binomial expansion Binomial Theorem Combination Common difference Common ratio Divergent Explicit formula Explicit formula Exponential function Factorial Fibonacci sequence Index Infinite sum Linear function Lower bound Partial sum Pascal’s triangle Recursive formula Sigma notation Sum Summation Upper bound
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Creator if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards (select CCRS from Standard Set dropdown menu)

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Precalculus Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TEKS# SE# TEKS Unit Level Specificity

• Bold black text in italics: Knowledge and Skills Statement (TEKS)
• Bold black text: Student Expectation (TEKS)
• Strike-through: 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: Unit-specific 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:
• X. Connections
P.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving 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 non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
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: Sn = (a1 + an)
• Variables
• n = the number of terms in the series
• a1 = the first term in the series
• an = the last (or nth) term in the series
• Sn = the finite sum of the first n terms in the series
• Geometric series
• Summation formulas
• S
• S
• Variables
• n = the number of terms in the series
• a1 = the first term in the series
• r = the common ratio of the terms in the series
• Sn = the nth 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 real-world problems involving finite arithmetic and finite geometric series

Note(s):

• Algebra I wrote linear equations for relationships based on a table of values.
• Algebra I wrote exponential equations of the form y = abx 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 nth position.
• The range is the acutal listed number in a sequence.
• Although a0 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: a1
• Second term: a2
• nth term: an
• 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
• an = an-1 + d, where one term of the sequence is given
• an+1= an + 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
• an = a1 + d(n – 1)
• 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 one-third 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: a1
• Second term: a2
• nth term: an
• 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
• an = rn-1 where a0 = 1
• an+1 = r • an where a0 = 1
• f(n) = r f(n - 1), where f(0) = 1
• f(n + 1) = rf(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
• an = a1rn-1

Note(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 = abx 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 nth term and the nth partial sum of an arithmetic series in mathematical and real-world problems.

Calculate

THE nth TERM AND THE nth PARTIAL SUM OF AN ARITHMETIC SERIES IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Arithmetic series – the sum of an arithmetic sequence
• Finding the nth 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, an = an – 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: an a1 + d(n – 1)
• n = number of the term in the sequence
• an = the nth term
• a1 = the first term
• d = common difference
• Finding the nth partial sum
• Sequence of terms
• Terms of sequence in order
• Calculation of the sum of the sequence of terms
• Formula to find nth partial sum of an arithmetic series
• Sn = (a + an)
• n = number of terms in the series
• an = the nth term
• a1 = the first term
• Sn the nth partial sum (or the sum of the first n terms)

Note(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: an = a1 + d(n – 1)
• Geometric: an = a1(r)n–1
• Using sigma notation with the correct upper and lower bounds
• Formulas to evaluate the sum of a series
• Arithmetic series
• Sn = (a1an)
• Variables
• n = the number of terms in the series
• a1 = the first term in the series
• an = the last (or nth) term in the series
• Sn = the finite sum of the first n terms in the series
• Geometric series
• Sn
• Variables
• n = the number of terms in the series
• a1 = the first term in the series
• r = the common ratio of the terms in the series
• Sn = the nth partial sum of the first n terms in the series
• Infinite geometric series
• S
• Variables
• a1 = 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):

• Algebra I wrote exponential equations of the form yabx 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 nth term of a geometric series, the nth partial sum of a geometric series, and sum of an infinite geometric series when it exists.

Calculate

THE nth TERM OF A GEOMETRIC SERIES, THE nth 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 nth 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, an = (an–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 (a7) can be found by multiplying the fourth term (a4 = 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 nth term in a geometric sequence: an = a1 •(r)n-1
• n = number of the term in the sequence
• an = the nth term
• a1 = the first term
• r = common ratio
• Finding the nth 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 nth partial sum of a geometric series: Sn =
• n = number of terms in the series
• = common ratio
• a1 = the first term
• Snthe nth 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
• = common ratio
• a1 = the first term
• S the sum of the terms

Note(s):

• Algebra I wrote exponential equations of the form yabx 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 (b)n IN POWERS OF AND FOR A POSITIVE INTEGER n, WHERE AND 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 ≤ 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
• C(nr)
• nCr
• General formula to find terms in a binomial expansion
• Formula:
• Solving real-world problems involving binomial expansion

Note(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