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Algebra II
TITLE : Unit 09: Exponential Functions and Equations SUGGESTED DURATION : 10 days

Unit Overview

This unit bundles student expectations that address transformations, characteristics, and applications of exponential functions. Student expectations also address formulating, solving, and justifying solutions for exponential equations and equations involving rational exponents. 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, Unit 09 and Unit 11, students analyzed exponential functions and applied laws of exponents in mathematical and real-world problem situations. In Algebra II Unit 04, students applied laws of exponents. In Units 06 – 08, students solved equations involving rational exponents.

During this unit, students graph the exponential function, f(x) = bx, where b is 2, 10, and e, and, when applicable, analyze the key attributes such as domain, range, intercepts, and asymptotic behavior. Students determine the effects on the attributes on the graph of f(x) = bx when f(x) is replaced by af(x), f(x) + d, and f(x - c) for specific positive and negative values of a, c, and d, and investigate parameter changes and key attributes in terms of real-world problem situations. Students solve equations involving rational exponents, and conclude with an equation with a variable in the exponent that cannot be solved by the same methods. Students solve exponential equations of the form y = abx where a is a nonzero real number and b is greater than zero and not equal to one, where solutions can be determined without the use of logarithms. Students formulate exponential equations that model real-world situations, including exponential relationships written in recursive notation, solve the exponential equations without the use of logarithms, and justify the solution in terms of the problem situation.

After this unit, in Algebra II Unit 10, students will make connections between exponential and logarithmic functions. In Algebra II Unit 11, students will distinguish between linear, quadratic, and exponential functions to model problem situations. In Algebra II Unit 12, students will review concepts of exponential functions. In subsequent courses in mathematics, these concepts will continue to be applied to problem situations involving exponential functions and equations.

In Algebra II, graphing, transforming, and analyzing key attributes of exponential functions are identified in STAAR Readiness Standards 2A.2A and 2A.5A. Solving exponential equations and equations with rational exponents are identified in STAAR Readiness Standards 2A.5D and 2A.7H. These Readiness Standards are subsumed under STAAR Reporting Category 1: Number and Algebraic Methods, STAAR Reporting Category 2: Describing and Graphing Functions and Their Inverses, and STAAR Reporting Category 5: Exponential and Logarithmic Functions and Equations. Formulating exponential functions to model problem situations is identified in STAAR Supporting Standard 2A.5B, and is subsumed under STAAR Reporting Category 5: Exponential and Logarithmic Functions and Equations. This unit is supporting the development of Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning A1, C1, D1, D2; III. Geometric Reasoning B1, C1; VII. Functions A1, A2, B1, B2, C2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

According to the National Council of Teachers of Mathematics (2000), Principles and Standards for School Mathematics, students should develop an understanding of the algebraic properties that govern manipulation of symbols in expressions, equations, and inequalities. According to Navigating through Algebra in Grades 9 – 12, “High school students continue to develop fluency with mathematical symbols and become proficient in operating on algebraic expressions in solving problems. Their facility with representation expands to include equations, inequalities, systems of equations, graphs, matrices, and functions, and they recognize and describe the advantages and disadvantages of various representations for a particular situation. Such facility with symbols and alternative representations enables them to analyze a mathematical situation, choose an appropriate model, select an appropriate solution method, and evaluate the plausibility of their solutions” (NCTM, 2002, p. 3). Research found in National Council of Teachers of Mathematics also states, “Using a variety of representations can help make functions more understandable to a wider range of students than can be accomplished by working with symbolic representations alone” (as cited by NCTM, 2009, p. 41). This unit places particular emphasis on multiple representations. State and national mathematics standards support such an approach. The Texas Essential Knowledge and Skills repeatedly require students to relate representations of functions, such as algebraic, tabular, graphical, and verbal descriptions. This skill is mirrored in the Principles and Standards for School Mathematics (NCTM, 1989). Specifically, this work calls for instructional programs that enable all students to understand relations and functions and select, convert flexibly among, and use various representations for them. More recently, the importance of multiple representations has been highlighted in Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics (NCTM, 2007). According to this resource, students should be able to translate among verbal, tabular, graphical, and algebraic representations of functions and describe how aspects of a function appear in different representations as early as Grade 8. Also, in research summaries such as Classroom Instruction That Works: Research-Based Strategies for Increasing Student Achievement such concept development is even cited among strategies that increase student achievement. Specifically, classroom use of multiple representations, referred to as nonlinguistic representations, and identifying similarities and differences has been statistically shown to improve student performance on standardized measures of progress (Marzano, Pickering & Pollock) (2001),.

Marzano, R. J., Pickering, D. J., & Pollock, J. E. (2001). Classroom instruction that works: Research-based strategies for increasing student achievement. Alexandria, VA: Association for Supervision and Curriculum Development.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. 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. (2007). Curriculum focal points for prekindergarten through grade 8 mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2009). Focus in high school mathematics: Reasoning and sense making. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/collegereadiness/crs.pdf

OVERARCHING UNDERSTANDINGS and QUESTIONS

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

  • Why are equations used to model problem situations?
  • How are equations used to model problem situations?
  • What methods can be used to solve equations?
  • Why is it essential to solve equations using various methods?
  • How can solutions to equations be represented?

 

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?

 

Transformation(s) 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?

 

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?
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

  • Multiple Representations

 

Functions

  • Attributes of Functions
  • Non-Linear Functions

 

Geometric Reasoning

  • Transformations

 

Associated Mathematical Processes

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

Exponential functions have unique graphs and attributes.

  • What representations can be used to represent an exponential function?
  • How do the different bases of 2, 10, and e, affect the representations of the exponential function?
  • What are the key attributes of an exponential function and how can they be determined?

 

Transformations of the exponential parent function, f(x) = bx, can be used to determine graphs and equations of representative exponential functions in problem situations.

  • What are the effects of changes on the graph of f(x) = bx when f(x) is replaced by af(x), for specific positive and negative values of a?
  • What are the effects of changes on the graph of f(x) = bx  when f(x) is replaced by f(x) + d, for specific positive and negative values of d?
  • What are the effects of changes on the graph of f(x) = bx when f(x) is replaced by f(x - c) for specific positive and negative values of c?
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.

Numeric Reasoning

  • Exponents

 

Algebraic Reasoning

  • Equations
  • Expressions
  • Multiple Representations
  • Solve

 

Functions

  • Attributes of Functions
  • Non-Linear Functions

 

Associated Mathematical Processes

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

Equations can be used to model and solve mathematical and real-world problem situations.

  • How are real-world problem situations identified as ones that can be modeled by exponential equations?
  • How are exponential equations used to model problem situations?
  • What methods can be used to solve exponential equations?
  • What are the advantages and disadvantages of various methods used to solve exponential equations?
  • What methods can be used to justify the reasonableness of solutions to exponential equations?

 

Exponential 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 exponential function models?
  • What key attributes identify an exponential 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 an exponential function model and the real-world problem situation?
  • How can exponential function representations be used to interpret and make predictions and critical judgments in terms of the problem situation?
  • How are exponential models represented in recursive notation?
  • How can exponential models represented in recursive notation be analyzed and represented in function notation?

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

  • Some students may think there is not an exponent on a variable such as x rather than understanding it means x1.
  • Some students may think that every sequence or series starts with an index of n = 1 such as a1 rather than understanding that it can also be n = 0 like a0, depending on context.

 

Underdeveloped Concepts:

  • Students may not understand the connections between geometric sequences and exponential functions.

Unit Vocabulary

  • Asymptotic behavior –behavior such that as x approaches infinity, f(x) approaches a given value
  • 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
  • Exponential decay –an exponential function where 0 < b < 1 and as x increases, y decreases exponentially
  • Exponential growth –an exponential function where b > 1 and as x increases, y increases exponentially
  • Horizontal asymptote –horizontal line approached by the curve as the function approaches positive or negative infinity. Horizontal asymptotes may be crossed by the curve.
  • Inequality notation –notation in which the solution is represented by an inequality statement
  • Interval notation –notation in which the solution is represented by a continuous interval
  • Range –set of output values for the dependent variable over which the function is defined
  • Set notation –notation in which the solution is represented by a set of values
  • x-intercept(s)x coordinate of a point at which the relation crosses the x-axis, meaning the y coordinate equals zero, (x, 0)
  • y-intercept(s)y coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0, y)
  • Zeros – the value(s) of x such that the y value of the relation equals zero

 

Related Vocabulary:

  • Base
  • Decreasing
  • Depreciate
  • Evaluate
  • Exponent
  • Exponential regression
  • Function notation
  • Horizontal compression
  • Horizontal shift
  • Horizontal stretch
  • Increasing
  • Interest
  • Parameter change
  • Parent function
  • Vertical compression
  • Vertical shift
  • Vertical stretch
Unit Assessment Items System Resources Other Resources

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

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

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

Texas Instruments – Graphing Calculator Tutorials

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency – Revised Mathematics TEKS: Vertical Alignment Charts

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

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – Mathematics TEKS: Supporting Information

Texas Education Agency – Interactive Mathematics Glossary

TEKS# SE# TEKS Unit Level Specificity
 
  • Bold black text in italics: Knowledge and Skills Statement (TEKS)
  • Bold black text: Student Expectation (TEKS)
  • Bold red text in italics:  Student Expectation identified by TEA as a Readiness Standard for STAAR
  • Bold green text in italics: Student Expectation identified by TEA as a Supporting Standard for STAAR
  • 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)
2A.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
2A.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
2A.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
2A.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
2A.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
2A.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
2A.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
2A.1G Display, explain, or 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
2A.2 Attributes of functions and their inverses. The student applies mathematical processes to understand that functions have distinct key attributes and understand the relationship between a function and its inverse. The student is expected to:
2A.2A

Graph the functions f(x)=, f(x)=1/x, f(x)=x3, f(x)=, f(x)=bx, f(x)=|x|, and f(x)=logb (x) where b is 2, 10, and e, and, when applicable, analyze the key attributes such as domain, range, intercepts, symmetries, asymptotic behavior, and maximum and minimum given an interval.


Readiness Standard

Graph

THE FUNCTIONS f(x) = bx, WHERE b IS 2, 10, AND e

Including, but not limited to:

  • Representations of functions, including graphs, tables, and algebraic generalizations
    • Exponential, f(x) = bx, where b is 2, 10, and e
  • Connections between representations of families of functions
  • Comparison of similarities and differences of families of functions

Analyze

THE KEY ATTRIBUTES OF THE FUNCTIONS SUCH AS DOMAIN, RANGE, INTERCEPTS, AND ASYMPTOTIC BEHAVIOR, WHEN APPLICABLE

Including, but not limited to:

  • Domain and range of the function
    • 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
    • Representation for domain and range
      • 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  
        • Ex: x  
        • Ex: –3 < y ≤ 6, x  
        • Ex: y ≥ 0, y  
      • Set notation – notation in which the solution is represented by a set of values
        • Braces are used to enclose the set.
        • Solution is read as “The set of x such that x is an element of …”
        • Ex: {x|x  x < 5}
        • Ex: {x|x  }
        • Ex: {y|y  , –3 < y ≤ 6}
        • Ex: {y|y  y ≥ 0}
      • Interval notation – notation in which the solution is represented by a continuous interval
        • Parentheses indicate that the endpoints are open, meaning the endpoints are excluded from the interval.
        • Brackets indicate that the endpoints are closed, meaning the endpoints are included in the interval.
        • Ex: (–, 5)
        • Ex: (–)
        • Ex: (–3, 6]
    • Domain and range of the function versus domain and range of the contextual situation
  • Key attributes of functions
    • Intercepts/Zeros
      • x-intercept(s) – x coordinate of a point at which the relation crosses the x-axis, meaning the y coordinate equals zero, (x, 0)
      • Zeros – the value(s) of x such that the y value of the relation equals zero
      • y-intercept(s) – y coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0, y)
    • Asymptotic behavior – behavior such that as x approaches infinity, f(x) approaches a given value
      • Horizontal asymptote – horizontal line approached by the curve as the function approaches positive or negative infinity. Horizontal asymptotes may be crossed by the curve.
  • Use key attributes to recognize and sketch graphs
  • Application of key attributes to real-world problem situations

Note(s):

  • Grade Level(s):
    • The notation represents the set of real numbers, and the notation represents the set of integers.
    • Algebra I studied parent functions f(x) = x, f(x) = x2, and f(x) = bx and their key attributes.
    • Precalculus will study polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxCCRS:
    • III. Geometric Reasoning
      • B1 – Identify and apply transformations to figures.
      • C1 – Make connections between geometry and algebra.
    • VII. Functions
      • A1 – Recognize whether a relation is a function.
      • A2 – Recognize and distinguish between different types of functions.
      • B1 – Understand and analyze features of a function.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
2A.5 Exponential and logarithmic functions and equations. The student applies mathematical processes to understand that exponential and logarithmic functions can be used to model situations and solve problems. The student is expected to:
2A.5A

Determine the effects on the key attributes on the graphs of f(x) = bx and f(x) = logb(x) where b is 2, 10, and e when f(x) is replaced by af(x), f(x) + d, and f(x - c) for specific positive and negative real values of a, c, and d.


Readiness Standard

Determine

THE EFFECTS ON THE KEY ATTRIBUTES ON THE GRAPHS OF f(x) = bx WHERE b IS 2, 10, AND e WHEN f(x) IS REPLACED BY af(x), f(x) + d, and f(x – c) FOR SPECIFIC POSITIVE AND NEGATIVE REAL VALUES OF a, c, AND d

Including, but not limited to:

  • General form of the square root function
    • Exponential functions, f(x) = bx, where b is 2, 10, and e
      • f(x) = 2x; f(x) = 10xf(x) = ex
  • Representations with and without technology
    • Graphs
    • Tables
    • Verbal descriptions
    • Algebraic generalizations
  • Effects on the graphs of f(x) = bx when parameters ab, c, and d are changed in f(x) = a • b(x – c) + d 
    • Effects on the graphs of f(x) = 2x when f(x) is replaced by af(x) with and without technology
      • a ≠ 0
      • |a| > 1, the graph stretches vertically
      • 0 < |a| < 1, the graph compresses vertically
      • Opposite of a reflects vertically over the x-axis
    • Effects on the graphs of f(x) = 10x, when f(x) is replaced by f(x – c) with and without technology
      • c = 0, no horizontal shift
      • Horizontal shift left for values of c < 0 by |c| units
        • For f(+ 2) → f(– (–2)), c = –2, and the function moves to the left two units.
      • Horizontal shift right for values of c > 0 by |c| units
        • For f(– 2), c = 2, and the function moves to the right two units.
    • Effects on the graphs of f(x) = ex, when f(x) is replaced by f(x) + d with and without technology
      • d = 0, no vertical shift
      • Vertical shift down for values of d < 0 by |d| units
      • Vertical shift up for values of d > 0 by |d| units
  • Connections between the critical attributes of transformed functions and f(x) = bx
    • Determination of parameter changes given a graphical or algebraic representation
    • Determination of a graphical representation given the algebraic representation or parameter changes
    • Determination of an algebraic representation given the graphical representation or parameter changes
  • Descriptions of the effects on the domain and range by the parameter changes
  • Effects of multiple parameter changes
    • Mathematical problem situation
    • Effects of parameter changes in real-world problem situations

Note(s):

  • Grade Level(s):
    • Algebra I determined effects on the graphs of the parent functions, f(x) = x and f(x) = x2 when f(x) is replaced by af(x), f(x) + d, f(xc), f(bx) for specific values of a, b, c, and d.
    • Algebra II continues to investigate the exponential parent function and transformations.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxCCRS:
    • III. Geometric Reasoning
      • B1 – Identify and apply transformations to figures.
      • C1 – Make connections between geometry and algebra.
    • VII. Functions
      • A1 – Recognize whether a relation is a function.
      • A2 – Recognize and distinguish between different types of functions.
      • B1 – Understand and analyze features of a function.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
2A.5B

Formulate exponential and logarithmic equations that model real-world situations, including exponential relationships written in recursive notation.


Supporting Standard

Formulate

EXPONENTIAL EQUATIONS THAT MODEL REAL-WORLD SITUATIONS, INCLUDING EXPONENTIAL RELATIONSHIPS WRITTEN IN RECURSIVE NOTATION

Including, but not limited to:

  • Data collection activities with and without technology
    • Data modeled by exponential functions
      • Exponential growth – an exponential function where b > 1 and as x increases, y increases exponentially
      • Exponential decay – an exponential function where 0 < b < 1 and as x increases, y decreases exponentially
  • Real-world problem situations
    • Real-world problem situations modeled by exponential functions
      • Exponential growth
      • Exponential decay
  • Representations of exponential equations
    • Tables/graphs
    • Verbal descriptions
    • Recursive notation for exponential relationships
      • Recursive formula: starting value and recursion equation
      • Exponential recursive formulas
        • Exponential growth: a1 = 5, an = 4an – 1
        • Exponential decay: a1 = 64, an = an – 1
  • Technology methods
    • Transformations of f(x) = bx 
    • Exponential regression

 

Note(s):

  • Grade Level(s):
    • Algebra I determined formulas and terms for geometric sequences given in recursive and function notation.
    • Algebra II introduces formulating exponential equations.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxCCRS:
    • VII. Functions
      • B1 – Understand and analyze features of a function.
      • B2 – Algebraically construct and analyze new functions.
      • C2 – Develop a function to model a situation.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
2A.5D

Solve exponential equations of the form y = abx where a is a nonzero real number and b is greater than zero and not equal to one and single logarithmic equations having real solutions.


Readiness Standard

Solve

EXPONENTIAL EQUATIONS OF THE FORM y = abx WHERE a IS A NONZERO REAL NUMBER AND b IS GREATER THAN ZERO AND NOT EQUAL TO ONE

Including, but not limited to:

  • Exponential equation, y = abx
    • a – initial value at x = 0
    • b – common ratio
  • Solving exponential equations
    • Application of laws (properties) of exponents
  • Real-world problem situations modeled by exponential functions
    • Exponential growth
      • f(x) = abx, where b > 1
      • f(x) = aekx, where k > 0
    • Exponential decay
      • f(x) = abx, where 0 < b < 1
      • f(x) = aekx, where k < 0

Note(s):

  • Grade Level(s):
    • Algebra I applied exponential functions to problem situations using tables, graphs, and the algebraic generalization, f(x) = a • bx.
    • Algebra II solves exponential equations algebraically.
    • 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.”
      • C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
      • 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.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
2A.7 Number and algebraic methods. The student applies mathematical processes to simplify and perform operations on expressions and to solve equations. The student is expected to:
2A.7H Solve equations involving rational exponents.
Readiness Standard

Solve

EQUATIONS INVOLVING RATIONAL EXPONENTS

Including, but not limited to:

  • Laws (properties) of exponents
    • Product of powers (multiplication when bases are the same): am • an = am+n
    • Quotient of powers (division when bases are the same): = am-n
    • Power to a power: (am)n = amn
    • Negative exponent: a-n =
    • Rational exponent:
    • Equations when bases are the same: am = anm = n
  • Solving equations with rational exponents
    • Isolation of base and power using properties of algebra
    • Exponentiation of both sides by reciprocal of power of base
      • If the denominator of the reciprocal power is even, then the variable must be represented using absolute value.
    • Simplification to obtain solution
    • Verification of solution
  • Real-world problem situations modeled by equations involving rational exponents
  • Justification of reasonableness of solutions in terms of real-world problem situations

Note(s):

  • Grade Level(s):
    • Prior grade levels simplified numeric expressions, including integral and rational exponents.
    • Algebra II introduces equations involving rational exponents.
    • 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.”
      • C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
      • D1 – Interpret multiple representations of equations and relationships.
      • D2 – Translate among multiple representations of equations and relationships.
    • 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 08/17/2016
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