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Algebra I
TITLE : Unit 02: Introduction to Functions SUGGESTED DURATION : 10 days

Unit Overview

Introduction
This unit bundles student expectations that address an introductory exploration of various representations of relations, including functionality, domain and range, function notation, and evaluation of the function for specific domain values. 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 6, students identified independent and dependent quantities from tables and graphs and wrote equations to represent the relationships. In Grade 8, students were introduced to functions as proportional and non-proportional relationships using sets of ordered pairs, tables, mappings, and graphs.

During this Unit
Students identify relations and determine if relations represented verbally, tabularly, graphically, and symbolically define a function. Students identify domain and range (continuous and discrete) of functions and represent the domain and range using inequality notation and verbal descriptions. Students express functions in function notation. Students evaluate functions in function notation given one or more elements in their domains. Students explore real-world problem situations, identify the domain and range of problem situations, express representative functions for problem situations using function notation, and evaluate functions for specified domains in problem situations. Problems in this unit incorporate linear, quadratic, and exponential functions, since those are the functions studied in Algebra I.

After this Unit
In Algebra I, students will continue to represent functions, identify attributes of functions, and apply function notation in a more in-depth study of linear, quadratic, and exponential functions. The concepts in this unit will also be applied in subsequent mathematics courses.

Additional Notes
In Algebra I, determining domain and range of linear functions is identified as STAAR Readiness Standard A.2A and part of STAAR Reporting Category 3: Writing and Solving Linear Functions, Equations, and Inequalities. Determining domain and range of quadratic functions is identified as STAAR Readiness Standard A.6A and part of STAAR Reporting Category 4: Quadratic Functions and Equations. Determining domain and range of exponential functions is identified as STAAR Supporting Standard A.9A and part of STAAR Reporting Category 5: Exponential Functions and Equations. Deciding whether relations define functions is identified as STAAR Supporting Standards A.12A, and evaluating functions written in function notation is identified as STAAR Supporting Standards A.12B. Both STAAR Supporting Standards A.12A and A.12B 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, D1, D2; III. Geometric Reasoning C1; VII. Functions A1, B1; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

Research
According to the National Council of Teachers of Mathematics (NCTM), Developing Essential Understanding of Functions, Grades 9 – 12 (2010), foundations for functions begin in elementary when studying patterns and using informal notation to represent variables. It continues in middle school by developing rules to represent tabular data with an understanding of relationships in bivariate data. In high school students further analyze these relationships and develop an understanding of families of functions and their characteristics. The National Council of Teachers of Mathematics (NCTM), the Texas Education Agency (TEA) in the math professional development modules and Algebra 1 EOC Success, and other mathematics research Algebra should take a functional approach. According to Navigating through Algebra in Grades 9 – 12 (2002), “much of what has traditionally been Algebra I in secondary schools is expected content for the middle grades. It is imperative then that a broadening and deepening of mathematics content take place in high school. New topics…such as classes of functions and using technology on symbolic expressions are emerging in the high school curriculum” (NCTM, p. v). Additionally, Algebra in a Technological World (1995) states, the high school algebra curriculum should undergo “a shift in perspective from algebra as skills for transforming, simplifying, and solving symbolic expressions to algebra as a way to express and analyze relationships” (NCTM, p. v). By beginning formal algebra with real-life situations that are naturally algebraic, students understand that formal algebra is not only a manipulation of symbols, but also a logical way to approach mathematical situations in an effort to make sense of them. Experiencing real-life functional situations and their characteristics helps build algebraic habits of mind (Driscoll, 1999). Through careful instruction, teachers connect real-life with algebraic representation and build conceptual understanding before delving into algebraic manipulation. If students completely develop solving equations using symbolic manipulation before they develop a solid conceptual foundation for their work, they will be unable to do more than symbolic manipulation (National Research Council, 1998).

 

Driscoll, M. (1999). Fostering algebraic thinking: A guide for teachers grades 6 – 10. Portsmouth, VA: Heinemann.
National Council of Teachers of Mathematics. (1995). Curriculum and evaluation standards for school mathematics: Algebra in a technological world. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2002). Navigating through algebra in grades 9 – 12. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2010). Developing Essential Understanding of Functions, Grades 9 – 12. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Research Council. (1998). High school mathematics at work: Essays and examples for the education of all students. Washington, DC: National Academy Press.
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?
  • Different families of functions, each with their own unique characteristics, can be used to model problem situations to make predictions and critical judgments.
    • What kinds of mathematical and real-world situations can be modeled by …
      • linear functions?
      • quadratic functions?
      • exponential functions?
    • How can the domain and range of linear, quadratic, and exponential functions be …
      • determined?
      • analyzed?
      • described?
    • How can the graph of the function and the domain and range be used to determine if the function is linear, quadratic or exponential?
    • How can domain and range be used to describe the behavior of linear, quadratic, and exponential functions?
    • What relationships exist between the mathematical and real-world meanings of the domain and range of linear, quadratic, and exponential function models?
    • How can domain and range be used to make predictions and critical judgments about the problem situation?
  • Functions can be represented in various ways with different representations of the function highlighting different characteristics and being more useful than other representations depending on the context.
    • How can functions be represented?
    • What is the purpose of representing functions in various ways?
    • How are function characteristics highlighted in different representations of the function?
    • What are the limitations of different function representations?
    • What connections can be made between multiple representations of a function?
  • Functions, Equations, and Inequalities
    • Attributes of Functions
      • Domain and range
      • Continuous or discrete
    • Functions and Equations
      • Linear
      • Quadratic
      • Exponential
    • Relations and Generalizations
  • Number and Algebraic Methods
    • Relations and Functions
  • 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 all relations are functions rather than that relations are functions only if each element of the input is paired with exactly one element of the output.
  • Some students may think functions can only be expressed in function notation, such as f(x) = mx + b or h(x) = ax2 + bx + c, and not recognize equations without function notation, such as y = mx + b or w = ax2 + bx + c, as valid representations of the same functional relationships.
  • Some students may think that variables represented as letters identify an object rather than the number or quantity of objects.

Unit Vocabulary

  • Continuous function – function whose values are continuous or unbroken over the specified domain
  • Discrete function – function whose values are distinct and separate and not connected; values are not continuous. Discrete functions are defined by their domain.
  • Domain – set of input values for the independent variable over which the function is defined
  • Function – relation in which each element of the domain (x) is paired with exactly one element of the range (y)
  • Function notation – notation that describes a specific function such as f(x) = x
  • Inequality notation – notation in which the solution is represented by an inequality statement
  • Range – set of output values for the dependent variable over which the function is defined
  • Relation – a set of ordered pairs (x, y) where the x is associated with a specific y

Related Vocabulary:

  • Decreasing function
  • Dependent variable
  • Exponential function
  • Increasing function
  • Independent variable
  • Linear function
  • Quadratic function
  • Representations
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 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.2 Linear functions, equations, and inequalities. The student applies the mathematical process standards when using properties of linear functions to write and represent in multiple ways, with and without technology, linear equations, inequalities, and systems of equations. The student is expected to:
A.2A Determine the domain and range of a linear function in mathematical problems; determine reasonable domain and range values for real-world situations, both continuous and discrete; and represent domain and range using inequalities.
Readiness Standard

Determine

THE DOMAIN AND RANGE OF A LINEAR FUNCTION IN MATHEMATICAL PROBLEMS AND REASONABLE DOMAIN AND RANGE VALUES FOR REAL-WORLD SITUATIONS, BOTH CONTINUOUS AND DISCRETE 

Represent

THE DOMAIN AND RANGE OF A LINEAR FUNCTION USING INEQUALITIES

Including, but not limited to:

  • Domain and range of linear functions in mathematical problem situations
    • Domain – set of input values for the independent variable over which the function is defined
      • Continuous function – function whose values are continuous or unbroken over the specified domain
      • Discrete function – function whose values are distinct and separate and not connected; values are not continuous. Discrete functions are defined by their domain.
    • Range – set of output values for the dependent variable over which the function is defined
  • Inequality representations
    • Verbal description
      • Ex: x is all real numbers less than five.
      • Ex: x is all real numbers.
      • Ex: y is all real numbers greater than –3 and less than or equal to 6.
      • Ex: y is all integers greater than or equal to zero.
    • Inequality notation – notation in which the solution is represented by an inequality statement
      • Ex: x < 5, x ∈ ℜ
      • Ex: x ∈ ℜ
      • Ex: –3 < y ≤ 6, y ∈ ℜ
      • Ex: y ≥ 0, yΖ
  • Domain and range of linear functions in real-world problem situations
    • Reasonable domain and range for real-world problem situations
    • Comparison of domain and range of function model to appropriate domain and range for a real-world problem situation

Note(s):

  • Grade Level(s):
    • The notation ℜ represents the set of real numbers, and the notation Ζ represents the set of integers.
    • Grade 6 identified independent and dependent quantities.
    • Grade 8 identified functions using sets of ordered pairs, tables, mappings, and graphs.
    • Algebra I introduces the concept of domain and range of a function.
    • Algebra I represents domain and range using inequality verbal descriptions and inequality notation.
    • Algebra II will introduce representing domain and range using interval and set notation.
    • Precalculus will introduce piecewise functions.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxCCRS:
    • VII. Functions
      • B1 – Understand and analyze features of a function.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
A.6 Quadratic functions and equations. The student applies the mathematical process standards when using properties of quadratic functions to write and represent in multiple ways, with and without technology, quadratic equations. The student is expected to:
A.6A Determine the domain and range of quadratic functions and represent the domain and range using inequalities.
Readiness Standard

Determine, Represent

THE DOMAIN AND RANGE OF QUADRATIC FUNCTIONS USING INEQUALITIES

Including, but not limited to:

  • Domain and range of quadratic functions in mathematical problem situations
    • Domain – set of input values for the independent variable over which the function is defined
      • Continuous function – function whose values are continuous or unbroken over the specified domain
      • Discrete function – function whose values are distinct and separate and not connected; values are not continuous. Discrete functions are defined by their domain.
    • Range – set of output values for the dependent variable over which the function is defined
  • Domain and range of quadratic functions in real-world problem situations
    • Reasonable domain and range for the real-world problem situation
    • Comparison of domain and range of function model to appropriate domain and range for real-world problem situation
  • Inequality representations
    • Verbal description
      • Ex: x is all real numbers less than five.
      • Ex: x is all real numbers.
      • Ex: y is all real numbers greater than –3 and less than or equal to 6.
      • Ex: y is all integers greater than or equal to zero.
    • Inequality notation – notation in which the solution is represented by an inequality statement
      • Ex: x < 5
      • Ex: x ∈ ℜ
      • Ex: –3 < y ≤ 6
      • Ex: y ≥ 0, y ∈ Ζ

Note(s):

  • Grade Level(s):
    • Grade 6 identified independent and dependent quantities.
    • Grade 8 identified functions using sets of ordered pairs, tables, mappings, and graphs.
    • Algebra I introduces quadratic functions.
    • Algebra I introduces the concept of domain and range of a function.
    • Algebra I represents domain and range using inequality verbal descriptions and inequality notation.
    • Algebra II will extend the concept of domain and range.
    • Algebra II will introduce representing domain and range using interval and set notation.
    • Algebra II will continue to investigate quadratic functions.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxCCRS:
    • 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.9A Determine the domain and range of exponential functions of the form f(x) = abx and represent the domain and range using inequalities.
Supporting Standard

Determine, Represent

THE DOMAIN AND RANGE OF EXPONENTIAL FUNCTIONS OF THE FORM f(x) = abUSING INEQUALITIES

Including, but not limited to:

  • Domain and range of exponential functions in mathematical problem situations
    • Domain – set of input values for the independent variable over which the function is defined
      • Continuous function – function whose values are continuous or unbroken over the specified domain
      • Discrete function – function whose values are distinct and separate and not connected; values are not continuous. Discrete functions are defined by their domain.
    • Range – set of output values for the dependent variable over which the function is defined
  • Domain and range values in real-world problem situations
    • Reasonable domain and range for the real-world problem situation
    • Comparison of domain and range of a function model to appropriate domain and range for real-world problem situation
  • Inequality representations
    • Verbal description
      • Ex: x is all real numbers less than five.
      • Ex: x is all real numbers.
      • Ex: y is all real numbers greater than –3 and less than or equal to 6.
      • Ex: y is all integers greater than or equal to zero.
    • Inequality notation – notation in which the solution is represented by an inequality statement
      • Ex: x < 5
      • Ex: x ∈ ℜ
      • Ex: –3 < y ≤ 6
      • Ex: y ≥ 0, yΖ

Note(s):

  • Grade Level(s):
    • Grade 6 identified independent and dependent quantities.
    • Grade 8 identified functions using sets of ordered pairs, tables, mappings, and graphs.
    • Algebra I introduces exponential functions.
    • Algebra I introduces the concept of domain and range of a function.
    • Algebra I represents domain and range using inequality verbal descriptions and inequality notation.
    • Algebra II will extend the concept of domain and range.
    • Algebra II will introduce representing domain and range using interval and set notation.
    • Algebra II will continue to investigate exponential functions.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxCCRS:
    • VII. 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.12A Decide whether relations represented verbally, tabularly, graphically, and symbolically define a function.
Supporting Standard

Decide

WHETHER RELATIONS REPRESENTED VERBALLY, TABULARLY, GRAPHICALLY, AND SYMBOLICALLY DEFINE A FUNCTION

Including, but not limited to:

  • Relation – a set of ordered pairs (x, y) where the x is associated with a specific y
  • Function – relation in which each element of the domain (x) is paired with exactly one element of the range (y)
  • Identification of functions
    • Verbally
    • Set of points
    • Tabularly
    • Graphically (Vertical line test – a vertical line must intersect the graph at one and only one point.)
    • Symbolically
  • Applications to mathematical problems
  • Applications to real-world problems
  • Applications to data collection and analysis

Note(s):

  • Grade Level(s):
    • Grade 8 developed the basic foundation of functions.
    • Algebra I extends linear functions and introduces quadratic and exponential functions.
    • Algebra II will extend to inverses of functions and restricting domains as needed to maintain functionality.
    • 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
      • D1 – Interpret multiple representations of equations and relationships.
      • D2 – Translate among multiple representations of equations and relationships.
    • III. Geometric Reasoning
      • C1 – Make connections between geometry and algebra.
    • VII. Functions
      • A1 – Recognize whether a relation is a function.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
A.12B Evaluate functions, expressed in function notation, given one or more elements in their domains.
Supporting Standard

Evaluate

FUNCTIONS, EXPRESSED IN FUNCTION NOTATION, GIVEN ONE OR MORE ELEMENTS IN THEIR DOMAINS

Including, but not limited to:

  • Function – relation in which each element of the domain (x) is paired with exactly one element of the range (y)
  • Function notation – notation that describes a specific function such as f(x) = x
    • Letters are used to name specific functions, such as f, g, h, etc.
    • x is the input or domain.
    • f(x) = 2x + 3 is the rule for the specific function.
    • f(x) is the range at a domain of x.
    • f(x) is the output or range from the rule, when the input is x.
  • Benefits of function notation
    • Naming of function
      • f(x) names function f.
      • g(x) names function g.
    • Designating value to be evaluated in the function
      • h(–5) means to find the rule h, substitute –5 in for the variable, and simplify.
  • Evaluation of functions in function notation
    • Identification of specified function rule
    • Substitution of domain value into the identified rule
    • Simplification of the numeric expression
  • Application of function notation in real-world problem situations

Note(s):

  • Grade Level(s):
    • Algebra I introduces function notation.
    • Algebra II will continue to investigate functions and apply function notation.
    • 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).
    • VII. Functions
      • A1 – Recognize whether a relation is a function.
    • 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 08/01/2018
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