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Algebra I
TITLE : Unit 10: Arithmetic and Geometric Sequences SUGGESTED DURATION : 15 days

Unit Overview

Introduction
This unit bundles student expectations that address defining and writing formulas for arithmetic and geometric sequences and identifying terms of arithmetic and geometric sequences. Connections between arithmetic sequences and linear functions and geometric sequences and exponential functions are also addressed. 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 Grade 4, students were introduced to sequences. Students defined sequences by position, term value, and the sequence rule.

During this Unit
Students define and identify terms of arithmetic and geometric sequences when sequences are given in recursive, explicit, and function notation using recursive processes. Students write a formula for the nth term of arithmetic and geometric sequences in recursive, explicit, and function notation, given the value of several of their terms. Students connect arithmetic sequences to linear functions, graph sequences on the coordinate plane, and compare key attributes of the representative function and sequence in mathematical and real-world problems. Students connect geometric sequences to exponential functions, graph sequences on the coordinate plane, and compare key attributes of the representative function and sequence in mathematical and real-world problems. Students compare and contrast arithmetic and geometric sequences in real-world problems and data collections.

After this Unit
In Precalculus Unit 06, students will extend these concepts to include series and binomial expansions.

Additional Notes
In Algebra I, graphing and identifying key attributes of linear functions are identified as STAAR Readiness Standard A.3C and subsumed under STAAR Reporting Category 2: Describing and Graphing Linear Functions, Equations, and Inequalities. Graphing and identifying key attributes of exponential functions are identified as STAAR Readiness Standard A.9D and subsumed under STAAR Reporting Category 5: Exponential Functions and Equations. Identifying terms of sequences and writing formulas to represent sequences are identified as STAAR Supporting Standards A.12C and A.12D. These are subsumed under STAAR Reporting Category 1: Number and Algebraic Methods. This unit is supporting the development of Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning A1, B1; III. Geometry C1; VII. Functions A2, B1; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

Research
According to the National Council of Teachers of Mathematics (2010), Developing Essential Understanding of Functions for Teaching Mathematics in Grades 9 – 12, students in grades 9 – 12 need to understand functions well if they are to succeed in courses that build on quantitative thinking and relationships. According the National Council of Teachers of Mathematics (2000), Principles and Standards for School Mathematics, students should have an opportunity to build on their earlier experiences, both deepening their understanding of relations and functions and expanding their repertoire of familiar functions. One of the Big Ideas proposed in the National Council of Teachers of Mathematics (2010), Developing Essential Understanding of Functions for Teaching Mathematics in Grades 9 – 12, states that arithmetic and geometric sequences can be viewed as types of functions with domains restricted to the set of positive integers. Furthermore, arithmetic sequences are linked to linear functions and geometric sequences to exponential functions. “Some functions such as sequences are different from the classical functions because they are defined on discrete sets (integers) rather than on intervals and they do not have the kind of “continuous variation” that the classical functions do” (2010, p. 19). High school students’ algebra experience should enable them to create and use tabular, symbolic, graphical, and verbal representations and to analyze and understand patterns, relations, and functions with more sophistication than in the middle grades. High school algebra should provide students with insights into mathematical abstraction and structure for modeling real-world problem situations and making predictions and drawing conclusions.

 

National Council of Teachers of Mathematics. (2010). Developing essential understanding of functions for teaching mathematics grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9


  • Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgments in everyday life.
    • What patterns exist within different types of quantitative relationships and where are they found in everyday life?
    • Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
  • Understanding how two quantities vary together (covariation) builds flexible functional reasoning in order to make predictions and critical judgments about the relationship.
    • What are the strengths and limitations of a particular function model for a problem situation?
    • How can functions be used to model problem situations efficiently?
    • How can it be determined if a relationship between two variables can be represented by a function?
    • How is function notation used to define and describe a function rule?
    • How is function notation used to make predictions and critical judgements about the relationship?
    • How are recursive processes connected to functional reasoning?
  • Different families of functions, each with their own unique characteristics, can be used to model problem situations to make predictions and critical judgments.
    • Linear functions are characterized by a constant rate of change and can be used to describe, model, and make predictions about situations.
    • Arithmetic sequences can be thought of as linear functions whose domains are the positive integers.
      • How can the rate of change of an arithmetic sequence be determined?
      • What kinds of mathematical and real-world situations can be modeled by arithmetic sequences?
      • What graphs, key attributes, and characteristics are unique to arithmetic sequences?
      • What patterns of covariation are associated with arithmetic sequences?
      • What relationships exist between arithmetic sequences and linear functions?
  • Equations can be written, transformed, and solved using various methods to make critical judgments, with different methods being more efficient or informative depending on the structure of the equation.
    • How can equations be used to represent relationships between quantities?
    • Why must solutions be justified in terms of problem situations?
    • What methods can be used to write formulas for arithmetic sequences?
    • How does the given information and/or representation influence the selection of an efficient method for writing formulas for arithmetic sequences?
    • How are properties and operational understandings used to transform formulas for arithmetic sequences?
  • Functions, Equations, and Inequalities
    • Functions
      • Linear
  • Number and Algebraic Methods
    • Relations and Functions
      • Arithmetic sequences
      • Formulas
    • Patterns, Operations, and Properties
  • Associated Mathematical Processes
    • Application
    • Problem Solving Model
    • Tools and Techniques
    • Communication
    • Representations
    • Relationships
    • Justification
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.

  • Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgments in everyday life.
    • What patterns exist within different types of quantitative relationships and where are they found in everyday life?
    • Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
  • Understanding how two quantities vary together (covariation) builds flexible functional reasoning in order to make predictions and critical judgments about the relationship.
    • What are the strengths and limitations of a particular function model for a problem situation?
    • How can functions be used to model problem situations efficiently?
    • How can it be determined if a relationship between two variables can be represented by a function?
    • How is function notation used to define and describe a function rule?
    • How is function notation used to make predictions and critical judgements about the relationship?
    • How are recursive processes connected to functional reasoning?
  • Different families of functions, each with their own unique characteristics, can be used to model problem situations to make predictions and critical judgments.
    • Exponential functions are characterized by a rate of change that is proportional to the value of the function and can be used to describe, model, and make predictions about problem situations.
    • Geometric sequences can be thought of as exponential functions whose domains are the positive integers.
      • How can the rate of change of a geometric sequence be determined?
      • What kinds of mathematical and real-world situations can be modeled by geometric sequences?
      • What graphs, key attributes, and characteristics are unique to geometric sequences?
      • What patterns of covariation are associated with geometric sequences?
      • What relationships exist between geometric sequences and exponential functions?
  • Equations can be written, transformed, and solved using various methods to make critical judgments, with different methods being more efficient or informative depending on the structure of the equation.
    • How can equations be used to represent relationships between quantities?
    • Why must solutions be justified in terms of problem situations?
    • What methods can be used to write formulas for geometric sequences?
    • How does the given information and/or representation influence the selection of an efficient method for writing formulas for geometric sequences?
    • How are properties and operational understandings used to transform formulas for geometric sequences?
  • Functions, Equations, and Inequalities
    • Functions
      • Linear
      • Exponential
  • Number and Algebraic Methods
    • Relations and Functions
      • Arithmetic sequences
      • Geometric sequences
      • Formulas
    • Patterns, Operations, and Properties
  • Associated Mathematical Processes
    • Application
    • Problem Solving Model
    • Tools and Techniques
    • Communication
    • Representations
    • Relationships
    • Justification
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.

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

  • Some students may think that the amount each term increases or decreases by in an arithmetic sequence is added to or subtracted from n in the final formula rather than multiplied. (e.g., Given 2, 6, 10, 14, … → 2 + n + 4 instead of the correct answer 2 + 4n.)
  • Some students may think that the amount each term is multiplied by in a geometric sequence is multiplied with n in the final formula rather than n raised to that power. (e.g., Given 2, 6, 18, 54, … → 2(n • 3) instead of the correct answer 2(3)n.)

Unit Vocabulary

  • Arithmetic sequence – sequence formed by adding or subtracting the same value to calculate each subsequent term
  • Domain of a sequence – set of natural numbers; 1, 2, 3, …
  • Exponential decay – an exponential function where 0 < b < 1 and as x increases, y decreases exponentially
  • Exponential function – a function in which the independent variable, x, is in the exponent, denoted by f(x) = abx
  • Exponential growth – an exponential function where b > 1 and as x increases, y increases exponentially
  • Geometric sequence – sequence formed by multiplying or dividing by the same value to calculate each subsequent term
  • Linear function – a relationship with a constant rate of change represented by a graph that forms a straight line in which each element of the input (x) is paired with exactly one element of the output (y)
  • 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.

Related Vocabulary:

  • Attributes of exponential functions
  • Attributes of linear functions
  • Common difference
  • Common ratio
  • Continuous domain
  • Discrete domain
  • Domain
  • Explicit notation
  • Function (simplified)
  • Range
  • Recursive notation
Unit Assessment Items System Resources Other Resources

Show this message:

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 Center 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

  

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 Algebra I Mathematics TEKS

 

Texas Instruments – Graphing Calculator Tutorials


TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

  • Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
  • Student Expectations (TEKS) identified by TEA are in bolded, black text.
  • Student Expectations (TEKS) are labeled Readiness as identified by TEA of the assessed curriculum.
  • Student Expectations (TEKS) are labeled Supporting as identified by TEA of the assessed curriculum.
  • Student Expectations (TEKS) are labeled Process standards as identified by TEA of the assessed curriculum.
  • Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Specificity Legend:

  • Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
  • Unit-specific clarifications are in italicized, blue text.
  • Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
  • A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
TEKS# SE# TEKS SPECIFICITY
A.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
A.1A Apply mathematics to problems arising in everyday life, society, and the workplace.
Process Standard

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE
Including, but not limited to:

  • Mathematical problem situations within and between disciplines
    • Everyday life
    • Society
    • Workplace

Note(s):    

  • The mathematical process standards may be applied to all content standards as appropriate.
  • TxCCRS:
    • X. Connections
A.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.
Process Standard

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
A.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.
Process Standard

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS
Including, but not limited to:

  • Appropriate selection of tool(s) and techniques to apply in order to solve problems
    • Tools
      • Real objects
      • Manipulatives
      • Paper and pencil
      • Technology
    • Techniques
      • Mental math
      • Estimation
      • Number sense

Note(s):    

  • The mathematical process standards may be applied to all content standards as appropriate.
  • TxCCRS:
    • VIII. Problem Solving and Reasoning
A.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Process Standard

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE
Including, but not limited to:

  • Mathematical ideas, reasoning, and their implications
    • Multiple representations, as appropriate
      • Symbols
      • Diagrams
      • Graphs
      • Language

Note(s):    

  • The mathematical process standards may be applied to all content standards as appropriate.
  • TxCCRS:
    • IX. Communication and Representation
A.1E Create and use representations to organize, record, and communicate mathematical ideas.
Process Standard

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

  • Representations of mathematical ideas
    • Organize
    • Record
    • Communicate
  • Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
  • Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):    

  • The mathematical process standards may be applied to all content standards as appropriate.
  • TxCCRS:
    • IX. Communication and Representation
A.1F Analyze mathematical relationships to connect and communicate mathematical ideas.
Process Standard

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:

  • Mathematical relationships
    • Connect and communicate mathematical ideas
      • Conjectures and generalizations from sets of examples and 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
A.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Process Standard

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION

Including, but not limited to:

  • Mathematical ideas and arguments
    • Validation of conclusions
      • Displays to make work visible to others
        • Diagrams, visual aids, written work, etc.
      • Explanations and justifications
        • Precise mathematical language in written or oral communication

Note(s):    

  • The mathematical process standards may be applied to all content standards as appropriate.
  • TxCCRS:
    • IX. Communication and Representation
A.3 Linear functions, equations, and inequalities. The student applies the mathematical process standards when using graphs of linear functions, key features, and related transformations to represent in multiple ways and solve, with and without technology, equations, inequalities, and systems of equations. The student is expected to:
A.3C

Graph linear functions on the coordinate plane and identify key features, including x-intercept, y-intercept, zeros, and slope, in mathematical and real-world problems.


Readiness Standard

Graph

LINEAR FUNCTIONS ON THE COORDINATE PLANE

Including, but not limited to:

  • Linear function – a relationship with a constant rate of change represented by a graph that forms a straight line in which each element of the input (x) is paired with exactly one element of the output (y)
  • Linear functions in mathematical problem situations
  • Linear functions in real-world problem situations
  • Multiple representations
    • Tabular
    • Graphical
    • Verbal
    • Algebraic generalizations

Note(s):

  • Grade Level(s):
    • Grades 7 and 8 introduced linear relationships using tables of data, graphs, and algebraic generalizations.
    • Grade 8 introduced using tables of data and graphs to determine rate of change or slope and y-intercept.
    • Algebra I introduces key attributes of linear, quadratic, and exponential functions.
    • Algebra II will continue to analyze the key attributes of exponential functions and will introduce the key attributes of square root, cubic, cube root, absolute value, rational, and logarithmic functions.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxCCRS:
    • III. Geometric Reasoning
      • C1 – Make connections between geometry and algebra.
    • VII. Functions
      • B1 – Understand and analyze features of a function.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
A.9 Exponential functions and equations. The student applies the mathematical process standards when using properties of exponential functions and their related transformations to write, graph, and represent in multiple ways exponential equations and evaluate, with and without technology, the reasonableness of their solutions. The student formulates statistical relationships and evaluates their reasonableness based on real-world data. The student is expected to:
A.9D

Graph exponential functions that model growth and decay and identify key features, including y-intercept and asymptote, in mathematical and real-world problems.


Readiness Standard

Graph

EXPONENTIAL FUNCTIONS THAT MODEL GROWTH AND DECAY

Including, but not limited to:

  • Exponential function – a function in which the independent variable, x, is in the exponent, denoted by f(x) = abx
    • a value is the y-intercept, (0, a).
    • b value is the successive ratio of range values.
      • The b value is a rational number greater than 0.
      • Exponential growth – an exponential function where b > 1 and as x increases, y increases exponentially
      • Exponential decay – an exponential function where 0 < b < 1 and as x increases, y decreases exponentially
  • Mathematical problem situations
  • Real-world problem situations

Note(s):

  • Grade Level(s):
    • Algebra I introduces exponential functions.
    • Algebra II will continue to investigate exponential functions, including continuous growth and decay.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxCCRS:
    • III. Geometric Reasoning
      • C1 – Make connections between geometry and algebra.
    • VII. Functions
      • A2 – Recognize and distinguish between different types of functions.
      • B1 – Understand and analyze features of a function.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
A.12 Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to write, solve, analyze, and evaluate equations, relations, and functions. The student is expected to:
A.12C Identify terms of arithmetic and geometric sequences when the sequences are given in function form using recursive processes.
Supporting Standard

Identify

TERMS OF ARITHMETIC AND GEOMETRIC SEQUENCES WHEN THE SEQUENCES ARE GIVEN IN FUNCTION FORM USING RECURSIVE PROCESSES

Including, but not limited to:

  • Sequence – a list of numbers or a collection of objects written in a specific order that follow a particular pattern. Sequences can be viewed as functions whose domains are the positive integers.
    • Domain of a sequence – set of natural numbers; 1, 2, 3, …
      • The domain of a sequence represents the position, n, of the term.
    • Range of a sequence – terms in the sequence calculated by the sequence rule
      • The range of a sequence represents the value of the term at the nth position.
      • The range is the actual listed number in a sequence.
      • Although a0 can be given or determined, it is not part of the sequence.
  • Arithmetic sequence – sequence formed by adding or subtracting the same value to calculate each subsequent term
    • Ex: 2, 5, 8, 11, 14, … Three is added to the previous term to calculate each subsequent term.
    • Ex: 7, 3, –1, –5, ... Four is subtracted from the previous term to calculate each subsequent term.
  • Geometric sequence – sequence formed by multiplying or dividing by the same value to calculate each subsequent term
    • Ex: 1, 2, 4, 8, 16, ... The previous term is multiplied by two in order to calculate each subsequent term.
    • Ex: 81, 27, 9, 3, 1, , ... Three is divided into the previous term to calculate each subsequent term or one-third is multiplied to calculate each subsequent term.
  • Recursive notation
    • Recursive process – calculation of the next number in a sequence by repeated application of a rule
    • Arithmetic
      • 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
    • Geometric
      • an = ran–1, where a0 = 1
      • an + 1 = ran, where a0 = 1
      • f(n) = rf(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
  • Initial values are required to ensure uniqueness of the sequence
  • Connections between arithmetic sequence and linear functions
    • Linear function: f(x) = mx + b
      • f(x) represents the xth term of the sequence
        • f(1) represents the 1st term of the sequence, a1
      • m represents the common difference of an arithmetic sequence, d
  • Connections between geometric sequences and exponential functions
    • Exponential function: f(x) = abx
      • f(x) represents the xth term of the sequence
        • f(1) represents the 1st term of the sequence, a1
      • b represents the common ratio of an geometric sequence, r
  • Identification of terms of arithmetic and geometric sequences

Note(s):

  • Grade Level(s):
    • Previous grade levels introduced the concept of sequences.
    • Algebra I introduces the concept of arithmetic and geometric sequences.
    • Algebra I introduces application of recursive processes in analysis of arithmetic and geometric sequences.
    • Algebra II will formulate exponential equations from recursive notation.
    • Precalculus will address arithmetic and geometric sequences and series, including the infinite geometric series.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxCCRS:
    • I. Numeric Reasoning
      • B1 – Perform computations with real and complex numbers.
    • II. Algebraic Reasoning
      • A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
      • B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g. polynomials, radicals, rational expressions).
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
A.12D Write a formula for the nth term of arithmetic and geometric sequences, given the value of several of their terms.
Supporting Standard

Write

A FORMULA FOR THE nth TERM OF ARITHMETIC AND GEOMETRIC SEQUENCES, GIVEN THE VALUE OF SEVERAL OF THEIR TERMS

Including, but not limited to:

  • Sequence – a list of numbers or a collection of objects written in a specific order that follow a particular pattern. Sequences can be viewed as functions whose domains are the positive integers.
  • Arithmetic sequence – sequence formed by adding or subtracting the same value to calculate each subsequent term
    • Ex: 2, 5, 8, 11, 14, … Three is added to the previous term to calculate each subsequent term.
    • Ex: 7, 3, –1, –5, ... Four is subtracted from the previous term to calculate each subsequent term.
  • Geometric sequence – sequence formed by multiplying or dividing by the same value to calculate each subsequent term
    • Ex: 1, 2, 4, 8, 16, ... Two is multiplied times the previous term to calculate each subsequent term.
    • Ex: 81, 27, 9, 3, 1, , ... Three is divided into the previous term to calculate each subsequent term or one-third is multiplied to calculate each subsequent term.
  • Notation for arithmetic sequences
    • a1 represents the 1st term of the sequence
    • an represents the nth term of the sequence
    • an-1 represents the term before the nth term of the sequence
    • d represents the common difference of an arithmetic sequence
  • Notation for geometric sequences
    • a1 represents the 1st term of the sequence
    • an represents the nth term of the sequence
    • an-1 represents the term before the nth term of the sequence
    • r represents the common ratio of a geometric sequence
  • Recursive process – calculation of the next number in a sequence by repeated application of a rule
    • Explicit notation
      • Arithmetic: an = a1 + d(n– 1)
      • Geometric: an = a1 rn–1
    • Arithmetic and geometric sequences in function form
      • Arithmetic: f(n) = a1 + d(n – 1)
      • Geometric: f(n) = a1rn–1
  • Initial values are required to ensure uniqueness of the sequence
  • Connections between arithmetic sequences and linear functions
    • Arithmetic sequences can be viewed as linear functions.
    • Simplification of the explicit arithmetic notation formula
    • y = mx + b, where b is the a0 and m is the common differenc
    • Domain of the sequence is the set of all positive integers.
  • Connections between geometric sequences and exponential functions.
    • Geometric sequences can be viewed as exponential functions.
    • Simplification of the explicit geometric notation formula
    • y = abx, where a is the a0 and b is the common ratio
    • Domain of the sequence is the set of all positive integers.
  • Representation of sequences by formula or rule
  • Application of sequences in real-world problem situations
  • Justification of solutions in terms of real-world problem situations

Note(s):

  • Grade Level(s):
    • Previous grade levels introduced the concept of sequences.
    • Algebra I introduces the concept of arithmetic and geometric sequences.
    • Algebra I introduces application of recursive processes in analysis of arithmetic and geometric sequences.
    • Algebra II will formulate exponential equations from recursive notation.
    • Precalculus will address arithmetic and geometric sequences and series, including the 
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxCCRS:
    • I. Numeric Reasoning
      • B1 – Perform computations with real and complex numbers.
    • II. Algebraic Reasoning
      • A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
      • B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g. polynomials, radicals, rational expressions).
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
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.
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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 08/01/2018
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