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Precalculus
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.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/collegereadiness/crs.pdf

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 Instruments – Graphing Calculator Tutorials

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency – Texas Response to Curriculum Focal Points for K-8 Mathematics Revised 2013

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – Mathematics TEKS: Supporting Information

Texas Education Agency – Interactive Mathematics Glossary

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):

  • 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 = 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):

  • 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 = 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):

  • 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: 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):

  • Grade Level(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):

  • Grade Level(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):

  • 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 
The English Language Proficiency Standards (ELPS), as required by 19 Texas Administrative Code, Chapter 74, Subchapter A, §74.4, outline English language proficiency level descriptors and student expectations for English language learners (ELLs). School districts are required to implement ELPS as an integral part of each subject in the required curriculum.

School districts shall provide instruction in the knowledge and skills of the foundation and enrichment curriculum in a manner that is linguistically accommodated commensurate with the student’s levels of English language proficiency to ensure that the student learns the knowledge and skills in the required curriculum.


School districts shall provide content-based instruction including the cross-curricular second language acquisition essential knowledge and skills in subsection (c) of the ELPS in a manner that is linguistically accommodated to help the student acquire English language proficiency.

http://ritter.tea.state.tx.us/rules/tac/chapter074/ch074a.html#74.4 


Choose appropriate ELPS to support instruction.

ELPS# Subsection C: Cross-curricular second language acquisition essential knowledge and skills.
Click here to collapse or expand this section.
ELPS.c.1 The ELL uses language learning strategies to develop an awareness of his or her own learning processes in all content areas. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.1A use prior knowledge and experiences to understand meanings in English
ELPS.c.1B monitor oral and written language production and employ self-corrective techniques or other resources
ELPS.c.1C use strategic learning techniques such as concept mapping, drawing, memorizing, comparing, contrasting, and reviewing to acquire basic and grade-level vocabulary
ELPS.c.1D speak using learning strategies such as requesting assistance, employing non-verbal cues, and using synonyms and circumlocution (conveying ideas by defining or describing when exact English words are not known)
ELPS.c.1E internalize new basic and academic language by using and reusing it in meaningful ways in speaking and writing activities that build concept and language attainment
ELPS.c.1F use accessible language and learn new and essential language in the process
ELPS.c.1G demonstrate an increasing ability to distinguish between formal and informal English and an increasing knowledge of when to use each one commensurate with grade-level learning expectations
ELPS.c.1H develop and expand repertoire of learning strategies such as reasoning inductively or deductively, looking for patterns in language, and analyzing sayings and expressions commensurate with grade-level learning expectations.
ELPS.c.2 The ELL listens to a variety of speakers including teachers, peers, and electronic media to gain an increasing level of comprehension of newly acquired language in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in listening. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.2A distinguish sounds and intonation patterns of English with increasing ease
ELPS.c.2B recognize elements of the English sound system in newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters
ELPS.c.2C learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions
ELPS.c.2D monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed
ELPS.c.2E use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken language
ELPS.c.2F listen to and derive meaning from a variety of media such as audio tape, video, DVD, and CD ROM to build and reinforce concept and language attainment
ELPS.c.2G understand the general meaning, main points, and important details of spoken language ranging from situations in which topics, language, and contexts are familiar to unfamiliar
ELPS.c.2H understand implicit ideas and information in increasingly complex spoken language commensurate with grade-level learning expectations
ELPS.c.2I demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs.
ELPS.c.3 The ELL speaks in a variety of modes for a variety of purposes with an awareness of different language registers (formal/informal) using vocabulary with increasing fluency and accuracy in language arts and all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in speaking. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.3A practice producing sounds of newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters to pronounce English words in a manner that is increasingly comprehensible
ELPS.c.3B expand and internalize initial English vocabulary by learning and using high-frequency English words necessary for identifying and describing people, places, and objects, by retelling simple stories and basic information represented or supported by pictures, and by learning and using routine language needed for classroom communication
ELPS.c.3C speak using a variety of grammatical structures, sentence lengths, sentence types, and connecting words with increasing accuracy and ease as more English is acquired
ELPS.c.3D speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency
ELPS.c.3E share information in cooperative learning interactions
ELPS.c.3F ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete vocabulary, including key words and expressions needed for basic communication in academic and social contexts, to using abstract and content-based vocabulary during extended speaking assignments
ELPS.c.3G express opinions, ideas, and feelings ranging from communicating single words and short phrases to participating in extended discussions on a variety of social and grade-appropriate academic topics
ELPS.c.3H narrate, describe, and explain with increasing specificity and detail as more English is acquired
ELPS.c.3I adapt spoken language appropriately for formal and informal purposes
ELPS.c.3J respond orally to information presented in a wide variety of print, electronic, audio, and visual media to build and reinforce concept and language attainment.
ELPS.c.4 The ELL reads a variety of texts for a variety of purposes with an increasing level of comprehension in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in reading. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations apply to text read aloud for students not yet at the stage of decoding written text. The student is expected to:
ELPS.c.4A learn relationships between sounds and letters of the English language and decode (sound out) words using a combination of skills such as recognizing sound-letter relationships and identifying cognates, affixes, roots, and base words
ELPS.c.4B recognize directionality of English reading such as left to right and top to bottom
ELPS.c.4C develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials
ELPS.c.4D use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary and other prereading activities to enhance comprehension of written text
ELPS.c.4E read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned
ELPS.c.4F use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language
ELPS.c.4G demonstrate comprehension of increasingly complex English by participating in shared reading, retelling or summarizing material, responding to questions, and taking notes commensurate with content area and grade level needs
ELPS.c.4H read silently with increasing ease and comprehension for longer periods
ELPS.c.4I demonstrate English comprehension and expand reading skills by employing basic reading skills such as demonstrating understanding of supporting ideas and details in text and graphic sources, summarizing text, and distinguishing main ideas from details commensurate with content area needs
ELPS.c.4J demonstrate English comprehension and expand reading skills by employing inferential skills such as predicting, making connections between ideas, drawing inferences and conclusions from text and graphic sources, and finding supporting text evidence commensurate with content area needs
ELPS.c.4K demonstrate English comprehension and expand reading skills by employing analytical skills such as evaluating written information and performing critical analyses commensurate with content area and grade-level needs.
ELPS.c.5 The ELL writes in a variety of forms with increasing accuracy to effectively address a specific purpose and audience in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in writing. In order for the ELL to meet grade-level learning expectations across foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations do not apply until the student has reached the stage of generating original written text using a standard writing system. The student is expected to:
ELPS.c.5A learn relationships between sounds and letters of the English language to represent sounds when writing in English
ELPS.c.5B write using newly acquired basic vocabulary and content-based grade-level vocabulary
ELPS.c.5C spell familiar English words with increasing accuracy, and employ English spelling patterns and rules with increasing accuracy as more English is acquired
ELPS.c.5D edit writing for standard grammar and usage, including subject-verb agreement, pronoun agreement, and appropriate verb tenses commensurate with grade-level expectations as more English is acquired
ELPS.c.5E employ increasingly complex grammatical structures in content area writing commensurate with grade-level expectations, such as:
ELPS.c.5F write using a variety of grade-appropriate sentence lengths, patterns, and connecting words to combine phrases, clauses, and sentences in increasingly accurate ways as more English is acquired
ELPS.c.5G narrate, describe, and explain with increasing specificity and detail to fulfill content area writing needs as more English is acquired.
Last Updated 08/24/2016
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