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 Instructional Focus DocumentPrecalculus
 TITLE : Unit 04: Rational Functions, Equations, and Inequalities SUGGESTED DURATION : 12 days

#### Unit Overview

This unit bundles student expectations that address graphs, attributes, and transformations of rational functions and application of rational functions in mathematical and real-world problem situations. Discontinuities and asymptotic behavior are analyzed and described. Rational equations and inequalities are also addressed. These topics are studied using multiple representations, including graphical, tabular, verbal, and algebraic methods. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this unit, in Algebra I Unit 06, students determined the quotient of polynomial expressions of degree one and two. In Algebra II Unit 04, students determined the quotient of a polynomial of degree three and of degree four when divided by a polynomial of degree one and of degree two. In Unit 08, students determined the sum, difference, product, and quotient of rational expressions with numerators and denominators of degree one and two. Additionally, in Algebra II Unit 08, students studied rational functions extensively, including their graphs, key attributes, transformations, equations, and real-world applications, as well as inverse variation.

During this unit, students graph rational functions, including f(x) =  and f(x) =  , and their transformations, including af(x), f(x) + d, f(x – c), and f(bx) for specific values of a, b, c, and d, in mathematical and real-world problems. Students analyze the behavior of rational functions around their horizontal asymptotes and analyze and describe the end behavior of rational functions using infinity notation based on their equations. Students determine various types of discontinuities in rational functions in the interval (–∞, ∞), including infinite discontinuities (vertical asymptotes) and removable discontinuities, using verbal, symbolic, tabular, and graphical representations. Students describe the left-sided and right-sided behavior of the graph of the function around the discontinuities, including limitations of the graphing calculator as it relates to the behavior of the function around the discontinuities. Students analyze graphs of rational functions that contain oblique asymptotes, describe the behavior of the function around these asymptotes (using verbal, symbolic, tabular, and graphical methods), and determine the equations of these oblique asymptotes using polynomial division. Students determine and analyze the key features of rational functions (including domain, range, symmetry, relative maximum, relative minimum, zeros, asymptotes, intervals over which the function is increasing or decreasing, end behavior, and discontinuities) in mathematical and real-world problems. Students also make connections between interrelated representations to algebraically construct a model for the function. Students solve rational equations using algebraic methods (factoring, quadratic formula, etc.), graphs, and tables in both mathematical and real-world problems. Students solve rational inequalities with real coefficients by solving the related rational equation, determining discontinuities in the related function, and testing the intervals between the solutions and points of discontinuity (numerically, graphically, and/or with tables) in both mathematical and real-world problems. Students write the solution set of rational inequalities in interval notation.

After this unit, in Precalculus Unit 05 and Unit 08, students will identify discontinuities and asymptotic behavior in exponential, logarithmic, and trigonometric functions. In subsequent courses in mathematics, these concepts will continue to be applied in problem situations involving rational functions, equations and inequalities.

Function analysis serves as the foundation for college readiness. Analyzing, representing, and modeling with functions are emphasized in the Texas College and Career Readiness Standards (TxCCRS): II. Algebraic Reasoning C1, D1, D2; VII. Functions A1, B1, B2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

According to a 2007 report published by the Mathematical Association of America, a “fundamental experience” for students entering college mathematics courses is to address problems in real-world situations by creating and interpreting mathematical models. Functions provide a way to quantitatively study the relationships and change in numerous real-world phenomena; this, coupled with the applicability of functions to many mathematical topics, make functions one of the most important topics in high school mathematics (Cooney, Beckmann, & Lloyd, 2010). In Texas, the importance of these skills is emphasized in the Texas College and Career Readiness Standards (2009), which call for students to be able to understand and analyze features of a function to model real-world situations. Algebraic models allow us to efficiently visualize and analyze the vast amount of interconnected information that is contained in a functional relationship; these tools are particularly helpful as the mathematical models become increasingly complex (National Research Council, 2005). Additionally, research argues that students need both a strong conceptual understanding of functions, as well as procedural fluency; as such, good instruction must include “a conceptual understanding of function, the ability to represent a function in a variety of ways, and fluency in moving among multiple representations of functions” (NRC, 2005, p. 353). Lastly, students need to be involved in metacognitive engagement in mathematics as they problem solve and reflect on their solutions and strategies; this is particularly important as students transition into more abstract mathematics, where fewer “clues” may exist warning students of a mathematical misstep (NRC, 2005). In Principles and Standards for School Mathematics, the National Council of Teachers of Mathematics (2000) notes the necessity for high school students to understand and compare the properties and classes of functions, including rational functions. In the AP Calculus Course Description, the College Board (2012) states that mathematics designed for college-bound students should involve analysis and understanding of elementary functions, including rational functions. Students must be familiar with the properties, algebra, graphs, and language of these functions. More specifically, students should have a basic understanding of asymptotes in terms of graphical behavior so that this knowledge can later be extended to describing asymptotic behavior in terms of limits involving infinity.

College Board.  (2012). AP calculus course description. Retrieved from http://media.collegeboard.com/digitalServices/pdf/ap/ap-calculus-course-description.pdf.
Cooney, T., Beckmann, S., & Lloyd, G. (2010). Developing essential understanding of functions for teaching mathematics in grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc.
Katz, Victor J. (Ed.). (2007). Algebra: Gateway to a technological future. Mathematical Association of America.
National Council of Teachers of Mathematics.  (2000). Principles and standards for school mathematics.  Reston, VA.
National Research Council. (2005). How Students Learn: Mathematics in the Classroom. M.S. Donovan & J.D. Bransford (Eds.). Washington, DC: The National Academies Press.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/collegereadiness/crs.pdf

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

Functions can be classified into different families with each function family having its own unique graphs, attributes, and relationships.

• Why are functions classified into families of functions?
• How are functions classified as a family of functions?
• What graphs, key attributes, and characteristics are unique to each family of functions?
• What patterns of covariation are associated with the different families of functions?
• How are the parent functions and their families used to model real-world situations?

Function models for problem situations can be determined by collecting and analyzing data using a variety of representations and applied to make predictions and critical judgments in terms of the problem situation.

• Why is it important to determine and apply function models for problem situations?
• What representations can be used to analyze collected data and how are the representations interrelated?
• Why is it important to analyze various representations of data when determining appropriate function models for problem situations?
• How can function models be used to evaluate one or more elements in their domains?
• How do the key attributes and characteristics of the function differ from the key attributes and characteristics of the function model for the problem situation?
• How does technology aid in the analysis and application of modeling and solving problem situations?

Transformations of a parent function create a new function within that family of functions.

• Why are transformations of parent functions necessary?
• How do transformations affect a function?
• How can transformations be interpreted from various representations?
• Why does a transformation of a function create a new function?
• How do the attributes of an original function compare to the attributes of a transformed function?

Equations and inequalities can model problem situations and be solved using various methods.

• Why are equations and inequalities used to model problem situations?
• How are equations and inequalities used to model problem situations?
• What methods can be used to solve equations and inequalities?
• Why is it essential to solve equations and inequalities using various methods?
• How can solutions to equations and inequalities be represented?
• How do the representations of solutions to equations and solutions to inequalities compare?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

• Equations
• Expressions
• Patterns/Rules
• Solve

Functions

• Attributes of Functions
• Non-Linear Functions

Associated Mathematical Processes

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

Transformations of rational functions can be used to determine graphs and equations of representative functions in problem situations.

• What are the effects of changes on the graph of rational functions when f(x) is replaced by af(x), for specific values of a?
• What are the effects of changes on the graph of rational functions when f(x) is replaced by f(bx), for specific values of b?
• What are the effects of changes on the graph of rational functions when f(x) is replaced by f(x - c) for specific values of c?
• What are the effects of changes on the graph of rational functions when f(x) is replaced by f(x) + d, for specific values of d?
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

• Equations
• Expressions
• Inequalities
• Multiple Representations
• Patterns/Rules
• Solve

Functions

• Attributes of Functions
• Non-Linear Functions

Associated Mathematical Processes

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

Rational functions have key attributes, including domain, range, symmetry, relative extrema, zeros, asymptotes, intervals over which the function is increasing or decreasing, end behavior, discontinuities, and left- and right-sided behavior of the graph around discontinuities.

• What are the key attributes of rational functions?
• How can the key attributes of a rational function be determined from multiple representations of the function?
• How can knowledge of the key attributes of a rational function be used to sketch the graph of the function?
• How can the domain and range of a rational function be determined and described?
• How can the relative extrema of a rational function be determined and described?
• How can the zeros of a rational function be determined and described?
• What is an asymptote and what types of asymptotes can exist in a rational function?
• How can the asymptotes (horizontal, vertical, and slant) of a rational function be determined and described?
• How can the intervals where the rational function is increasing and decreasing be determined and described?
• How can the end behavior of a rational function be determined and described using infinity notation?
• How do even and odd functions compare graphically and symbolically?
• What is a discontinuity, and what types of discontinuities can exist in a rational function?
• How can the discontinuities of a rational function and the left- and right-side behavior of the function near these discontinuities be determined and described?
• Under what conditions does a rational function have a vertical asymptote, horizontal asymptote, oblique asymptote, or removable discontinuity?
• What are the limitations of using the graphing calculator to analyze the behaviors of a rational function around discontinuities?

Transformations of rational functions can be used to determine graphs and equations of representative functions in problem situations.

• What are the effects of changes on the graph of rational functions when f(x) is replaced by af(x), for specific values of a?
• What are the effects of changes on the graph of rational functions when f(x) is replaced by f(bx), for specific values of b?
• What are the effects of changes on the graph of rational functions when f(x) is replaced by f(x - c) for specific values of c?
• What are the effects of changes on the graph of rational functions when f(x) is replaced by f(x) + d, for specific values of d?
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

• Equations
• Expressions
• Inequalities
• Multiple Representations
• Solve

Functions

• Attributes of Functions
• Non-Linear Functions

Associated Mathematical Processes

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

Rational equation and inequalities can be used to model and solve mathematical and real-world problem situations.

• How are rational inequalities used to model problem situations?
• What methods can be used to solve rational equations and inequalities?
• What are the advantages and disadvantages of various methods used to solve rational equations and inequalities?
• What methods can be used to justify the reasonableness of solutions?
• How are the solutions of rational inequalities related to the solutions of their corresponding rational equations?
• How do the discontinuities of a rational function affect the solution of a rational equation or inequality?
• How can solutions to rational inequalities be represented?
• What causes extraneous solutions in rational equations?
• How can extraneous solutions be identified in graphs, tables, and algebraic calculations?

Rational functions can be used to model real-world problem situations by analyzing collected data, key attributes, and various representations in order to interpret and make predictions and critical judgments.

• What representations can be used to display rational function models?
• What key attributes identify a rational function model?
• How does the domain and range of the function compare to the domain and range of the problem situation?
• What are the connections between the key attributes of a rational function model and the real-world problem situation?
• What are the limitations of the various representations of a rational function in regards to discontinuities?
• How do asymptotic behaviors and discontinuities of a rational function affect the domain and range of the function?
• How can rational function representations be used to interpret and make predictions and critical judgments in terms of the problem situation?

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think an asymptote is a line that can never be crossed instead of a line that is approached. Although a vertical asymptote cannot be crossed, a horizontal asymptote can be crossed and approached in another section of the graph.
• Some students may have misconceptions about the nature of the graph of a rational function when using a graphing calculator, where the calculator obscures the details or hidden behavior of a function.
• Some students may think that rational functions must have a vertical asymptote. Although many rational functions do have vertical asymptotes, some may only have one or more removable discontinuities.
• Some students may forget to consider the undefined values in a rational equation when solving a rational inequality. They may only consider the zeros of the corresponding rational equation when determining test intervals.

Underdeveloped Concepts:

• Some students may think that when zeros of an expression occur in the denominator of a function, they always produce vertical asymptotes. However, if an x-value makes both the numerator and denominator equal to zero, this value indicates a removable discontinuity and not a vertical asymptote.
• Some students may forget to consider the presence of extraneous solutions when solving rational equations.

#### Unit Vocabulary

• Asymptotic behavior – behavior such that as x approaches infinity, f(x) approaches a given value
• Covariation – pattern of related change between two variables in a function
• Infinite discontinuities – values of x where vertical asymptotes occur. Specifically, if a function has an infinite discontinuity at x = c, as x → cf(x) → ±∞
• Jump discontinuities – values or intervals of x where a function “jumps” (or skips, or disconnects). If a function has a jump discontinuity at x = c, then the function approaches a specific y-value on the left of x = c (or, when x < c), but approaches a different y-value on the right side of x = c (or, when x > c).
• Removable discontinuities – values or intervals of x where a function has a “hole” in the graph. If a function has a removable discontinuity at x = c, then the function approaches the same specific y-value on both the left and right of x = c, even though f(c) is not the same (or undefined).

Related Vocabulary:

 compression decreasing degree denominator discontinuity domain end behavior equation even function extraneous solution factor factoring fraction horizontal asymptote increasing inequality notation infinity notation interval notation leading coefficient least common denominator left-sided behavior maximum minimum numerator oblique asymptote odd function polynomial polynomial division quotient range rational equation rational function rational inequality reflection right-sided behavior root set notation solution solution set stretch symmetry transformation translation undefined vertical asymptote x-intercept zero
Unit Assessment Items System Resources Other Resources

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

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

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

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

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

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

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

Texas Instruments – Graphing Calculator Tutorials

TEKS# SE# TEKS Unit Level Specificity

• Bold black text in italics: Knowledge and Skills Statement (TEKS)
• Bold black text: Student Expectation (TEKS)
• Strike-through: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit(s)
• Blue text: Supporting information / Clarifications from TCMPC (Specificity)
• Blue text in italics: Unit-specific clarification
• Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS)
P.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
P.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

Apply

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

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
P.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

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

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
P.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select

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

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

Note(s):

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

Communicate

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

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

Note(s):

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

Create, Use

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

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

Note(s):

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

Analyze

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

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
P.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Display, Explain, Justify

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

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
P.2 Functions. The student uses process standards in mathematics to explore, describe, and analyze the attributes of functions. The student makes connections between multiple representations of functions and algebraically constructs new functions. The student analyzes and uses functions to model real-world problems. The student is expected to:
P.2F

Graph exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions.

Graph

RATIONAL FUNCTIONS

Including, but not limited to:

• Graphs of the parent functions
• Graphs of both parent functions and other forms of the identified functions from their respective algebraic representations
• Various methods for graphing
• Curve sketching
• Plotting points from a table of values
• Transformations of parent functions (parameter changes abc, and d)
• Using graphing technology

Note(s):

• Grade Level(s):
• Algebra II graphed various types of functions, including square root, cube root, absolute value, and rational functions.
• Precalculus extends the analysis of functions to include other types, such as trigonometric, power, piecewise-defined, and others.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II. Algebraic Reasoning
• D2 – Translate among multiple representations of equations and relationships.
• VII. Functions
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
P.2G

Graph functions, including exponential, logarithmic, sine, cosine, rational, polynomial, and power functions and their transformations, including af(x), f(x) + d, f(x – c), f(bx) for specific values of a, b, c, and d, in mathematical and real-world problems.

Graph

FUNCTIONS, INCLUDING RATIONAL AND THEIR TRANSFORMATIONS, INCLUDING af(x), f(x) + d, f(x – c), f(bx) FOR SPECIFIC VALUES OF abc, AND d, IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• General form of parent function
• Rational functions: f(x) =
• Representations with and without technology
• Graphs
• Verbal descriptions
• Algebraic generalizations (including equation and function notation)
• Changes in parameters abc, and d on graphs
• Effects of a on f(x) in af(x)
• a ≠ 0
• |a| > 1, the graph stretches vertically
• 0 < |a| < 1, the graph compresses vertically
• Opposite of a reflects vertically over the horizontal axis (x-axis)
• Effects of d on f(x) in f(x) + d
• d = 0, no vertical shift
• Translation, vertical shift up or down by |d| units
• Effects of c on f(x) in f(x – c)
• c = 0, no horizontal shift
• Translation, horizontal shift left or right by |c| units
• Effects of b on f(x) in f(bx)
• b ≠ 0
• |b| > 1, the graph compresses horizontally
• 0 < |b| < 1, the graph stretches horizontally
• Opposite of b reflects horizontally over the vertical axis or y-axis
• Combined transformations of parent functions
• Transforming a portion of a graph
• Illustrating the results of transformations of the stated functions in mathematical problems using a variety of representations
• Mathematical problem situations
• Real-world problem situations

Note(s):

• Grade Level(s):
• Algebra II graphed transformations of various types of functions, including square root, cube, cube root, absolute value, rational, exponential, and logarithmic functions.
• Precalculus extends the analysis of functions to include other types, such as trigonometric, power, piecewise-defined, and others.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II. Algebraic Reasoning
• D1 – Interpret multiple representations of equations and relationships.
• VII. Functions
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
P.2I

Determine and analyze the key features of exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions such as domain, range, symmetry, relative maximum, relative minimum, zeros, asymptotes, and intervals over which the function is increasing or decreasing.

Determine, Analyze

THE KEY FEATURES OF RATIONAL FUNCTIONS SUCH AS DOMAIN, RANGE, SYMMETRY, RELATIVE MAXIMUM, RELATIVE MINIMUM, ZEROS, ASYMPTOTES, AND INTERVALS OVER WHICH THE FUNCTION IS INCREASING OR DECREASING

Including, but not limited to:

• Covariation – pattern of related change between two variables in a function
• Multiplicative patterns
• Rational functions
• Domain and range
• Represented as a set of values
• {0, 1, 2, 3, 4}
• Represented verbally
• All real numbers greater than or equal to zero
• All real numbers less than one
• Represented with inequality notation
• x ≥ 0
• y < 1
• Represented with set notation
• {x| x ≥ 0}
• {y| y < 1}
• Represented with interval notation
• [0, ∞)
• (–∞, 1)
• Symmetry
• Reflectional
• Rotational
• Symmetric with respect to the origin (180° rotational symmetry)
• Relative extrema
• Relative maximum
• Relative minimum
• Zeros
• Roots/solutions
• x-intercepts
• Asymptotes
• Vertical asymptotes (x = h)
• Horizontal asymptotes (y = k)
• Slant asymptotes (y = mx + b)
• Intervals where the function is increasing or decreasing
• Represented with inequality notation, –1 <  ≤ 3
• Represented with set notation, {x|x  , –1 < x ≤ 3}
• Represented with interval notation, (–1, 3]
• Connections among multiple representations of key features
• Graphs
• Tables
• Algebraic
• Verbal

Note(s):

• Grade Level(s):
• Algebra II analyzed functions according to key attributes, such as domain, range, intercepts, symmetries, asymptotic behavior, and maximum and minimum values over an interval.
• Precalculus extends the analysis of key attributes of functions to include zeros and intervals where the function is increasing or decreasing.
• 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
P.2J

Analyze and describe end behavior of functions, including exponential, logarithmic, rational, polynomial, and power functions, using infinity notation to communicate this characteristic in mathematical and real-world problems.

Analyze, Describe

END BEHAVIOR OF FUNCTIONS, INCLUDING RATIONAL FUNCTIONS, USING INFINITY NOTATION IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Describing end behavior with infinity notation
• Right end behavior
• As x → ∞ (or as x approaches infinity) the function becomes infinitely large; f(x) → ∞.
• As x → ∞ (or as x approaches infinity) the function becomes infinitely small; f(x) → –∞.
• As x → ∞ (or as x approaches infinity) the function approaches a constant value, cf(x) → c.
• Left end behavior
• As x → –∞ (or as x approaches negative infinity) the function becomes infinitely large; f(x) → ∞.
• As x → –∞ (or as x approaches negative infinity) the function becomes infinitely small; f(x) → –∞.
• As x → –∞ (or as x approaches negative infinity) the function approaches a constant value, cf(x) → c.
• Determining end behavior from multiple representations
• Tables: evaluating the function for extreme negative (left end) and positive (right end) values of x
• Graphs: analyzing behavior on the left and right sides of the graph
• Determining end behavior from analysis of the function type and the constants used
• Rational: f(x) = , where p(x) and q(x) are polynomials in terms of xq(x) ≠ 0
• Ex: If the degree of p(x) is greater than the degree of q(x), as x → –∞, f(x) → ±∞ on the left, and as x → ∞,  f(x) → ∞ on the right.
• Ex: If the degree of p(x) is less than the degree of q(x), as x → –∞, f(x) → 0 on the left, and as x → ∞, f(x) → 0 on the right.
• Ex: If the degree of p(x) and q(x) are the same, as x → –∞, f(x) → k on the left, and as x → ∞, f(x) → k on the right, where k is a constant determined by the leading coefficients of p(x) and q(x).
• Interpreting end behavior in real-world situations

Note(s):

• Grade Level(s):
• Algebra II analyzed the domains and ranges of quadratic, square root, exponential, logarithmic, and rational functions.
• Algebra II determined any asymptotic restrictions on the domain of a rational function.
• Precalculus extends analysis of domain, range, and asymptotic restrictions to determine the end behavior of functions and describes this behavior using infinity notation.
• Precalculus lays the foundation for understanding the concept of limit even though the term limit is not included in the standard.
• 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
P.2K Analyze characteristics of rational functions and the behavior of the function around the asymptotes, including horizontal, vertical, and oblique asymptotes.

Analyze

CHARACTERISTICS OF RATIONAL FUNCTIONS

Including, but not limited to:

• Analyzing algebraically rational functions of the form f(x) =, where p(x) and q(x) are polynomials in x, q(x) ≠ 0
• Degrees (highest power of x) of p(x) and q(x)
• Leading coefficients of p(x) and q(x)
• Factors of p(x) and q(x)
• Zeros of p(x) and q(x) (or where p(x) = 0 and q(x) = 0)
• Quotient of p(x) and q(x) (or p(x) ÷ q(x), obtained through long division of polynomials)
• Determining discontinuities algebraically, for rational functions of the form f(x) = , where p(x) and q(x) are polynomials in x, q(x) ≠ 0
• Behavior of function around discontinuities
• Determining whether a discontinuity is a removable discontinuity or a non-removable discontinuity
• Non-removable discontinuities
• Vertical asymptotes
• Vertical asymptotes occur at values of x where q(x) = 0, but p(x) ≠ 0.
• Removable discontinuities
• Removable discontinuities occur at values of x where both p(x) = 0 and q(x) = 0.
• Determining end behavior algebraically, for rational functions of the form f(x) = , where p(x) and q(x) are polynomials in terms of xq(x) ≠ 0,/span>
• Horizontal asymptotes
• If the degree of p(x) is less than the degree of q(x), f(x) has a horizontal asymptote at y = 0.
• If the degrees of p(x) and q(x) are the same, f(x) has a horizontal asymptote at y = k where k is a constant determined by the leading coefficients of p(x) and q(x).
• Oblique (or slant) asymptotes
• If the degree of p(x) is one more than the degree of q(x), then f(x) has an oblique asymptote of the form y = mx + b determined by the quotient of p(x) and q(x) through long division.

Analyze

THE BEHAVIOR OF THE RATIONAL FUNCTION AROUND THE ASYMPTOTES, INCLUDING HORIZONTAL, VERTICAL, AND OBLIQUE ASYMPTOTES

Including, but not limited to:

• Behavior around horizontal asymptotes
• Verbal and symbolic
• If a rational function f(x) has a horizontal asymptote at k, then as the x-values increase or decrease without bound (or as x → ±∞), the y-values of the function approach k (or f(x) → k).
• Tabular
• Graphical
• Behavior around vertical asymptotes
• Verbal and symbolic
• If a rational function f(x) has a vertical asymptote at h, then as x-values approach h (or as x → h), the y-values of the function either increase or decrease without bound (or f(x) → ±∞).
• Tabular
• Graphical
• Behavior around oblique (or slant) asymptotes
• Verbal and symbolic
• If a rational function f(x) has an oblique (or slant) linear asymptote at mx + b, then as x-values increase or decrease without bound (or as x → ±∞), the y-values of the function approach the line y = mx + b (or f(x) → mx + b).
• Tabular
• Graphical

Note(s):

• Grade Level(s):
• Algebra II determined any asymptotic restrictions on the domain of a rational function.
• Algebra II determined the quotient of polynomials using algebraic methods.
• Precalculus analyzes the behavior of rational functions and describes this behavior around asymptotes.
• Precalculus uses the quotient of polynomials to determine oblique (or other) asymptotes of rational 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
P.2L Determine various types of discontinuities in the interval (-∞, ∞) as they relate to functions and explore the limitations of the graphing calculator as it relates to the behavior of the function around discontinuities.

Determine

VARIOUS TYPES OF DISCONTINUITIES IN THE INTERVAL (–∞, ∞) AS THEY RELATE TO FUNCTIONS

Including, but not limited to:

• Determining whether a discontinuity is a removable discontinuity or a non-removable discontinuity
• Behavior of function around discontinuities
• Non-removable discontinuities
• Jump discontinuities – values or intervals of x where a function “jumps” (or skips, or disconnects). If a function has a jump discontinuity at x = c, then the function approaches a specific y-value on the left of x = c (or when x < c), but approaches a different y-value on the right side of x = (or when x > c).
• Graphical
• Tabular
• Algebraic
• Piecewise defined functions
• Evaluate both parts of the function to the left and right at breaks in the domain, then check to see if the values agree.
• General functions
• Jump discontinuities can occur at values of x where the function is not defined due to limits on the domain.
• Infinite discontinuities – values of x where vertical asymptotes occur, function has an infinite discontinuity at x = c, as x → cf(x) → ±∞
• Graphical
• Tabular
• Algebraic
• Rational functions
• For rational functions of the form f(x) = , where p(x) and q(x) are polynomials in terms of x, vertical asymptotes (or infinite discontinuities) occur at values of x where q(x) = 0 but p(x) ≠ 0.
• Trigonometric functions
• For trigonometric functions, vertical asymptotes can occur at values of x where the function is undefined.
• Removable discontinuities – values or intervals of x where a function has a “hole” in the graph. If a function has a removable discontinuity at x = c, then the function approaches the same specific y-value on both the left and right of x = c, even though f(c) is not the same (or undefined).
• Graphical
• Tabular
• Algebraic
• Rational functions
• For rational functions of the form f(x) = , where p(x) and q(x) are polynomials in terms of x, removable discontinuities occur at values of x where both p(x) = 0 and q(x) = 0.

Explore

THE LIMITATIONS OF THE GRAPHING CALCULATOR AS IT RELATES TO THE BEHAVIOR OF THE FUNCTION AROUND DISCONTINUITIES

Including, but not limited to:

• Tables
• Hidden behavior of a function
• Because tables show only discrete values of x and y, the tables often do not fully describe the behavior of a function.
• Values of x that get skipped
• Because tables default to integer values of x and y, the tables often skip important features of a function that occur at the rational (decimal) values in between.
• Values of where a function is undefined
• While tables can locate values of x where a function is undefined, tables do not identify the type of discontinuity that has occurred.
• Graphing functions with graphing calculators
• Evaluating functions at specific x-values
• Setting a window
• Screen width = (maximum x-value) – (minimum x-value)
• Resolution = number of pixels in the screen width
• x = (screen width) ÷ (resolution)
• Behavior of calculator graphs around discontinuities
• Jump discontinuities
• Infinite discontinuities
• Removable discontinuities

Note(s):

• Grade Level(s):
• Algebra II determined any asymptotic restrictions on the domain of a rational function.
• Precalculus extends the idea of domain restrictions to include various types of discontinuities: removable, infinite, and jump.
• 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
P.2M Describe the left-sided behavior and the right-sided behavior of the graph of a function around discontinuities.

Describe

THE LEFT-SIDED BEHAVIOR AND THE RIGHT-SIDED BEHAVIOR OF THE GRAPH OF A FUNCTION AROUND DISCONTINUITIES

Including, but not limited to:

• Verbal and symbolic
• Left-sided behavior near a discontinuity at x = c
• Words: As x approaches c from the left
• Symbols: x → c
• Right-sided behavior near a discontinuity at x = c
• Words: As x approaches c from the right
• Symbols: x → c+
• Function behavior near a discontinuity at x = c
• As x approaches c (from the left or right), the function values can approach a constant, k.
• Words: As x approaches c, the function approaches k.
• Symbols: As x → c, f(x) → k (or y → k)
• As x approaches c (from the left or right), the function values can continue to increase without limit.
• Words: As x approaches c, the function approaches infinity.
• Symbols:  As x → c,  f(x) → ∞ (or y → ∞)
• As x approaches c (from the left or right), the function values can continue to decrease without limit.
• Words: As x approaches c, the function approaches negative infinity.
• Symbols:  As x → c, f(x) → –∞ (or y → –∞)
• Graphical
• Left-sided behavior near a discontinuity at x = c
• Move along the graph on the interval x < c from left to right
• Right-sided behavior near a discontinuity at x = c
• Move along the graph on the interval x > c from right to left
• Tabular
• Left-sided behavior near a discontinuity at x = c
• Consider values in the table where x < c, such as c – 0.1, c – 0.01, c – 0.001, etc.
• Right-sided behavior near a discontinuity at x = c
• Consider values in the table where x > c, such as c + 0.1, c + 0.01, c + 0.001, etc.
• Use left- and right-sided behavior of a function to determine whether a discontinuity is a removable discontinuity or a non-removable discontinuity

Note(s):

• Grade Level(s):
• Algebra II determined any asymptotic restrictions on the domain of a rational function.
• Precalculus extends the concept of domain restrictions around asymptotes to include other types of discontinuities and analyzes the left-sided and right-sided behavior of functions near these discontinuities.
• Precalculus lays the foundation for understanding the concept of limit even though the term limit is not included in the standard.
• 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
P.2N

Analyze situations modeled by functions, including exponential, logarithmic, rational, polynomial, and power functions, to solve real-world problems.

Analyze, To Solve

SITUATIONS MODELED BY FUNCTIONS, INCLUDING RATIONAL FUNCTIONS

Including, but not limited to:

• Models that represent problem situations
• Understanding the meaning of the variables (both independent and dependent)
• Evaluating the function when independent quantities (x-values) are given
• Solving equations when dependent quantities (y-values) are given
• Appropriateness of given models for a situation
• Analyzing the attributes of a problem situation
• Determining which type of function models the situation
• Determining a function to model the situation
• Using transformations
• Using attributes of functions
• Using technology
• Describing the reasonable domain and range values
• Comparing the behavior of the function and the real-world relationship
• Rational functions (e.g., averages, temperature/pressure/volume relationships (Boyle’s Law), etc.)
• Asymptotic behavior – behavior such that as x approaches infinity, f(x) approaches a given value

Note(s):

• Grade Level(s):
• Algebra II analyzed situations involving exponential, logarithmic, and rational functions.
• Precalculus extends function analysis to include polynomial and power functions and expects students to solve real-world problems and interpret solutions to those problems.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II. Algebraic Reasoning
• D2 – Translate among multiple representations of equations and relationships.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
P.5 Algebraic reasoning. The student uses process standards in mathematics to evaluate expressions, describe patterns, formulate models, and solve equations and inequalities using properties, procedures, or algorithms. The student is expected to:
P.5L Solve rational inequalities with real coefficients by applying a variety of techniques and write the solution set of the rational inequality in interval notation in mathematical and real-world problems.

Solve

RATIONAL INEQUALITIES WITH REAL COEFFICIENTS BY APPLYING A VARIETY OF TECHNIQUES IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Relating solutions of rational inequalities to the solutions of the related rational equations
• Solving related rational equations using algebraic methods
• Factoring
• Quadratic formula
• Solving related rational equations with technology
• Graphs
• Tables
• Relating solutions of rational inequalities to the discontinuities of the related rational functions
• Identifying types of discontinuities
• Vertical asymptotes
• Removable discontinuities
• Locating the discontinuities of rational functions using algebraic methods
• Finding values where the denominator of a rational expression is equal to zero
• Factoring
• Quadratic formula
• Checking to see if these values are also zeros of the numerator of the rational expression
• Locating discontinuities of rational functions with technology
• Graphs
• Tables
• Testing the intervals between the solutions and points of discontinuity
• Evaluating the expression to determine whether values satisfy the inequality
• Analyzing graphs to determine whether values satisfy the inequality
• Using tables to determine whether values satisfy the inequality

Write

THE SOLUTION SET OF THE RATIONAL INEQUALITY IN INTERVAL NOTATION IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Using brackets for closed intervals
• Using parentheses for open intervals
• Using parentheses and the infinity symbol for boundless intervals
• Using the symbol for set union to describe solution sets with more than one interval

Note(s):

• Grade Level(s):
• Algebra II determined the sum, difference, product, and quotient of rational expressions.
• Algebra II formulated and solved rational equations with real solutions.
• Precalculus extends these skills to solve rational inequalities.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II. Algebraic Reasoning
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• VII. Functions
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
ELPS# Subsection C: Cross-curricular second language acquisition essential knowledge and skills. 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. Choose appropriate ELPS to support instruction. Click here to collapse or expand this section. ELPS.c.1 The ELL uses language learning strategies to develop an awareness of his or her own learning processes in all content areas. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to: ELPS.c.1A use prior knowledge and experiences to understand meanings in English ELPS.c.1B monitor oral and written language production and employ self-corrective techniques or other resources ELPS.c.1C use strategic learning techniques such as concept mapping, drawing, memorizing, comparing, contrasting, and reviewing to acquire basic and grade-level vocabulary ELPS.c.1D speak using learning strategies such as requesting assistance, employing non-verbal cues, and using synonyms and circumlocution (conveying ideas by defining or describing when exact English words are not known) ELPS.c.1E internalize new basic and academic language by using and reusing it in meaningful ways in speaking and writing activities that build concept and language attainment ELPS.c.1F use accessible language and learn new and essential language in the process ELPS.c.1G demonstrate an increasing ability to distinguish between formal and informal English and an increasing knowledge of when to use each one commensurate with grade-level learning expectations ELPS.c.1H develop and expand repertoire of learning strategies such as reasoning inductively or deductively, looking for patterns in language, and analyzing sayings and expressions commensurate with grade-level learning expectations. ELPS.c.2 The ELL listens to a variety of speakers including teachers, peers, and electronic media to gain an increasing level of comprehension of newly acquired language in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in listening. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to: ELPS.c.2A distinguish sounds and intonation patterns of English with increasing ease ELPS.c.2B recognize elements of the English sound system in newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters ELPS.c.2C learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions ELPS.c.2D monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed ELPS.c.2E use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken language ELPS.c.2F listen to and derive meaning from a variety of media such as audio tape, video, DVD, and CD ROM to build and reinforce concept and language attainment ELPS.c.2G understand the general meaning, main points, and important details of spoken language ranging from situations in which topics, language, and contexts are familiar to unfamiliar ELPS.c.2H understand implicit ideas and information in increasingly complex spoken language commensurate with grade-level learning expectations ELPS.c.2I demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs. ELPS.c.3 The ELL speaks in a variety of modes for a variety of purposes with an awareness of different language registers (formal/informal) using vocabulary with increasing fluency and accuracy in language arts and all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in speaking. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to: ELPS.c.3A practice producing sounds of newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters to pronounce English words in a manner that is increasingly comprehensible ELPS.c.3B expand and internalize initial English vocabulary by learning and using high-frequency English words necessary for identifying and describing people, places, and objects, by retelling simple stories and basic information represented or supported by pictures, and by learning and using routine language needed for classroom communication ELPS.c.3C speak using a variety of grammatical structures, sentence lengths, sentence types, and connecting words with increasing accuracy and ease as more English is acquired ELPS.c.3D speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency ELPS.c.3E share information in cooperative learning interactions ELPS.c.3F ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete vocabulary, including key words and expressions needed for basic communication in academic and social contexts, to using abstract and content-based vocabulary during extended speaking assignments ELPS.c.3G express opinions, ideas, and feelings ranging from communicating single words and short phrases to participating in extended discussions on a variety of social and grade-appropriate academic topics ELPS.c.3H narrate, describe, and explain with increasing specificity and detail as more English is acquired ELPS.c.3I adapt spoken language appropriately for formal and informal purposes ELPS.c.3J respond orally to information presented in a wide variety of print, electronic, audio, and visual media to build and reinforce concept and language attainment. ELPS.c.4 The ELL reads a variety of texts for a variety of purposes with an increasing level of comprehension in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in reading. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations apply to text read aloud for students not yet at the stage of decoding written text. The student is expected to: ELPS.c.4A learn relationships between sounds and letters of the English language and decode (sound out) words using a combination of skills such as recognizing sound-letter relationships and identifying cognates, affixes, roots, and base words ELPS.c.4B recognize directionality of English reading such as left to right and top to bottom ELPS.c.4C develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials ELPS.c.4D use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary and other prereading activities to enhance comprehension of written text ELPS.c.4E read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned ELPS.c.4F use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language ELPS.c.4G demonstrate comprehension of increasingly complex English by participating in shared reading, retelling or summarizing material, responding to questions, and taking notes commensurate with content area and grade level needs ELPS.c.4H read silently with increasing ease and comprehension for longer periods ELPS.c.4I demonstrate English comprehension and expand reading skills by employing basic reading skills such as demonstrating understanding of supporting ideas and details in text and graphic sources, summarizing text, and distinguishing main ideas from details commensurate with content area needs ELPS.c.4J demonstrate English comprehension and expand reading skills by employing inferential skills such as predicting, making connections between ideas, drawing inferences and conclusions from text and graphic sources, and finding supporting text evidence commensurate with content area needs ELPS.c.4K demonstrate English comprehension and expand reading skills by employing analytical skills such as evaluating written information and performing critical analyses commensurate with content area and grade-level needs. ELPS.c.5 The ELL writes in a variety of forms with increasing accuracy to effectively address a specific purpose and audience in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in writing. In order for the ELL to meet grade-level learning expectations across foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations do not apply until the student has reached the stage of generating original written text using a standard writing system. The student is expected to: ELPS.c.5A learn relationships between sounds and letters of the English language to represent sounds when writing in English ELPS.c.5B write using newly acquired basic vocabulary and content-based grade-level vocabulary ELPS.c.5C spell familiar English words with increasing accuracy, and employ English spelling patterns and rules with increasing accuracy as more English is acquired ELPS.c.5D edit writing for standard grammar and usage, including subject-verb agreement, pronoun agreement, and appropriate verb tenses commensurate with grade-level expectations as more English is acquired ELPS.c.5E employ increasingly complex grammatical structures in content area writing commensurate with grade-level expectations, such as: ELPS.c.5F write using a variety of grade-appropriate sentence lengths, patterns, and connecting words to combine phrases, clauses, and sentences in increasingly accurate ways as more English is acquired ELPS.c.5G narrate, describe, and explain with increasing specificity and detail to fulfill content area writing needs as more English is acquired.
Last Updated 08/24/2016

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