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Algebra II
TITLE : Unit 04: Expressions, Factoring, and Equations with Rational Exponents SUGGESTED DURATION : 13 days

Unit Overview

This unit bundles student expectations that address characteristics of polynomials, simplification of polynomials using rules of exponents, and application of polynomial operations to problem situations. Student expectations also address equations with 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.

This unit bundles student expectations that address characteristics of polynomials, simplification of polynomials using rules of exponents, and application of polynomial operations to problem situations. Student expectations also address equations with 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 01 and Unit 06, students applied the laws of exponents to simplify and factor polynomials of degree one and two. Also, students performed operations with polynomials, including addition, subtraction, multiplication, and division (a polynomial of degree one and polynomial of degree two when divided by a polynomial of degree one and polynomial of degree two when the degree of the divisor does not exceed the degree of the dividend). In Algebra I Unit 07, students simplified numerical radical expressions involving square roots when solving quadratic equations.

During this unit, students define polynomial expressions, review the rules of exponents, and perform operations (addition, subtraction, multiplication, division) with polynomials. Students determine the quotient of a polynomial of degree three and of degree four when divided by a polynomial of degree one and of degree two. Students rewrite radical expressions containing variables to equivalent forms and solve equations involving rational exponents, using the laws of exponents. Students factor polynomials using patterns, including difference of squares, sum and difference of cubes, trinomials, and grouping. Students also use algebraic methods to determine factors of polynomials of degree three and degree four by applying the Rational Root Theorem, the Remainder Theorem, the Factor Theorem, and synthetic division.

After this unit, in Algebra II Units 05 – 08, students will apply these concepts to simplify expressions and solve equations, including complex, imaginary roots. In subsequent mathematics courses, students will continue to apply these concepts as necessary when simplifying and factoring expressions and solving equations, including those with rational exponents, that arise in problem situations.

In Algebra II, determining linear and quadratic factors of polynomial expressions is identified as STAAR Readiness Standard 2A.7E. Solving equations involving rational exponents is identified as STAAR Readiness Standard 2A.7H. All STAAR Readiness Standards are subsumed under STAAR Reporting Category 1: Number and Algebraic Methods. Simplifying, factoring, and performing operations with polynomial expressions are identified as STAAR Supporting Standards 2A.7B, 2A.7C, and 2A.7D. Rewriting radical expressions is identified under STAAR Supporting Standard 2A.7G. All STAAR Supporting Standards are subsumed under STAAR Reporting Category 1: Number and Algebraic Methods. This unit is supporting the development of Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning A1, B1, C1, D1, D2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; Connections.

According to the National Council of Teachers of Mathematics (NCTM), Developing Essential Understanding of Expressions, Equations, and Functions, Grades 6 – 8, understanding of expressions is essential to a good foundation in algebra, since expressions are the building blocks for equations and functions. According to research from the National Council of Teachers of Mathematics (2000), high school algebra should provide students with insights into mathematical abstraction and structure. Students should develop an understanding of the algebraic properties that govern manipulation of symbols in expressions, equations, and inequalities.

 

National Council of Teachers of Mathematics. (2010). Developing essential understanding of expressions, equations, and functions, grades 6-8. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/collegereadiness/crs.pdf

OVERARCHING UNDERSTANDINGS and QUESTIONS

Algebraic expressions (numbers, variables, and operational symbols) are the basic tools of algebra.

  • Why are algebraic expressions the basic tools of algebra?
  • How are algebraic expressions used to express mathematical ideas and model mathematical and real-world situations?
  • What operations do algebraic expressions undergo?
  • How can two expressions be related?
  • Why are algebraic expressions evaluated?

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?
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.

Numeric Reasoning

  • Exponents

Algebraic Reasoning

  • Equivalence
  • Expressions
  • Simplify

Associated Mathematical Processes

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

Polynomial expressions undergo various operations, including addition, subtraction, multiplication, division, and exponentiation.

  • What processes are used to combine polynomials when adding and subtracting?
  • What processes are used to multiply polynomials?
  • What processes are used to divide polynomials by a polynomial of lesser degree?
  • What rules of exponents are used to simply and perform operations with polynomials?
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
  • Radicals

Algebraic Reasoning

  • Equations
  • Equivalence
  • Expressions
  • Simplify
  • Solve

Functions

  • Attributes of Functions
  • Non-Linear

Associated Mathematical Processes

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

Radical expressions can be rewritten using rational exponents.

  • How is the index number on a radical related to an exponent?
  • How can radicals be rewritten as equivalent exponential expressions?
  • Why are radical expressions sometimes rewritten in exponential form?

Equations involving rational exponents can be used to represent mathematical and real-world problem situations and can be solved using rules of exponents.

  • Why are equations involving rational exponents sometimes used to represent problem situations?
  • How are equations with rational exponents used to represent problem situations?
  • How are the rules of exponents used to solve equations involving rational exponents?
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

  • Equivalence
  • Expressions
  • Simplify

Associated Mathematical Processes

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

Some higher degree polynomials expressions can be broken down into linear and quadratic factors.

  • What patterns can be used to factor polynomial expressions?
  • What algebraic methods can be used to factor polynomial expressions?
  • What method can be used to test the validity of factors determined for a polynomial expression?

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

  • Some students may think that for a fractional exponent, the numerator is the index of the radical and the denominator is the power of the radicand, rather than that the denominator is the index and the numerator is the power of the radicand.

Underdeveloped Concepts:

  • Some students may not correctly implement the laws of exponents.
  • Some students may not factor out the greatest common factor as the first step in the factorization process.

Unit Vocabulary

  • Binomial – two term expression, 5x3y – 3x2z3                     
  • Monomial – one term expression, x3y5
  • Polynomial expression – monomial or sum of monomials not including variables in the denominator or under a radical
  • Radical expression – expression that contains a radical symbol
  • Trinomial – three term expression, 2x2 + 3x + 1

Related Vocabulary:

  • Coefficient
  • Constant
  • Degree of a monomial
  • Degree of a polynomial
  • Exponential expression
  • Factoring
  • Greatest common factor (GCF)
  • Index number
  • Laws of exponents
  • Properties of algebra
  • Radical symbol

 

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.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.7B Add, subtract, and multiply polynomials.
Supporting Standard

Add, Subtract, Multiply

POLYNOMIALS

Including, but not limited to:

  • Polynomial expression – monomial or sum of monomials not including variables in the denominator or under a radical
    • Monomial – one term expression, x3y5
    • Binomial – two term expression, 5x3y – 3x2z3
    • Trinomial – three term expression, 2x2 + 3x + 1
  • Operations with polynomials: addition, subtraction, multiplication
    • Application of properties of algebra to perform operations
      • Distribution to clear grouping symbols
      • Commutative and associative properties to combine like terms
    • Application of laws (properties) of exponents

Note(s):

  • Grade Level(s):
    • Algebra I performed operations on polynomials of degree one and two.
    • Algebra II extends operations on polynomials to polynomials of degree three and four.
    • 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
      • B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g. polynomials, radicals, rational expressions).
      • D1 – Interpret multiple representations of equations and relationships.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
2A.7C Determine the quotient of a polynomial of degree three and of degree four when divided by a polynomial of degree one and of degree two.
Supporting Standard

Determine

THE QUOTIENT OF A POLYNOMIAL OF DEGREE THREE AND OF DEGREE FOUR WHEN DIVIDED BY A POLYNOMIAL OF DEGREE ONE AND OF DEGREE TWO

Including, but not limited to:

  • Division by factoring
    • Cancellation of common factors in numerator and denominator
  • Long division
    • Degree three and four polynomials by degree one and degree two polynomials
      • Long division format with divisor outside division box, dividend inside the division box, and quotient on top of division box
      • Missing terms in series represented by adding a zero term
  • Synthetic division
    • Degree three and four polynomials by degree one polynomials in the form x – c
      • Applications of synthetic division
        • Remainder Theorem: If the polynomial function P(x) is divided by (x – c), then the remainder of the division is P(c).
        • Evaluation of polynomial functions at f(c)
        • Depression of polynomial equations to determine factors and roots

Note(s):

  • Grade Level(s):
    • Algebra I determined the quotient of a polynomial of degree one and polynomial of degree two when divided by a polynomial of degree one and polynomial of degree two when the degree of the divisor does not exceed the degree of the dividend.
    • Algebra II introduces division of degree three and four polynomials by degree one and degree two polynomials.
    • 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
      • B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g. polynomials, radicals, rational expressions).
      • D1 – Interpret multiple representations of equations and relationships.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
2A.7D Determine the linear factors of a polynomial function of degree three and of degree four using algebraic methods.
Supporting Standard

Determine

THE LINEAR FACTORS OF A POLYNOMIAL FUNCTION OF DEGREE THREE AND OF DEGREE FOUR USING ALGEBRAIC METHODS

Including, but not limited to:

  • Connections between roots and factors
    • If x = c is a root of a polynomial, then (xc) is a factor of the polynomial.
  • Determination of linear and quadratic factors from tables
    • Identification of roots from a table, x values where y values equal zero
    • Writing roots as factors
  • Determination of linear and quadratic factors from graphs

    • Identification of roots from a graph, x-intercepts or zeros
    • Writing roots as factors
  • Determination of linear and quadratic factors by depressing polynomials
    • Rational root theorem to determine possible roots
      • For polynomial equation a0xn + a1xn-1 + ... + an-1x + an = 0 with integral coefficients of degree n in which a0 is the coefficient of xn, and an is the constant term, then possible rational roots are  where p is a factor of the leading coefficient, an, and q is a factor of the constant term, a0.
    • Analysis of possible rational roots by synthetic division
    • Remainder Theorem
      • If the remainder is zero, x = c is an actual root of the polynomial.
      • When the polynomial is depressed to a quadratic expression, remaining roots can be determined by factoring or solving using the quadratic formula.
      • The calculated rational roots must be a part of the set of possible rational roots, .

Note(s):

  • Grade Level(s):
    • Algebra I introduced factorization of polynomials of degree one and degree two.
    • Algebra II introduces synthetic division of degree three and four polynomials by degree one polynomials.
    • Algebra II introduces depression of polynomials to determine roots and factors of the polynomial.
    • 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
      • B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g. polynomials, radicals, rational expressions).
      • D1 – Interpret multiple representations of equations and relationships
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
2A.7E Determine linear and quadratic factors of a polynomial expression of degree three and of degree four, including factoring the sum and difference of two cubes and factoring by grouping.
Readiness Standard

Determine

LINEAR AND QUADRATIC FACTORS OF A POLYNOMIAL EXPRESSION OF DEGREE THREE AND OF DEGREE FOUR, INCLUDING FACTORING THE SUM AND DIFFERENCE OF TWO CUBES AND FACTORING BY GROUPING

Including, but not limited to:

  • Determination of linear and quadratic factors by factorization
    • Greatest common factor
    • Difference of squares: a2 – b2 = (a + b)(a – b)
    • Trinomials
    • Sum of cubes: a3 + b3 = (a + b)(a– ab + b2)
    • Difference of cubes: a3 – b3 = (a  b)(a2 + ab + b2)
    • Grouping methods
    • Verify factorization by re-multiplying the factors.

Note(s):

  • Grade Level(s):
    • Algebra I introduced factorization of polynomials of degree one and degree two.
    • Algebra II introduces factorization of polynomials of degree three and degree four.
    • 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
      • B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g. polynomials, radicals, rational expressions).
      • D1 – Interpret multiple representations of equations and relationships.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
2A.7G Rewrite radical expressions that contain variables to equivalent forms.
Supporting Standard

Rewrite

RADICAL EXPRESSIONS THAT CONTAIN VARIABLES TO EQUIVALENT FORMS

Including, but not limited to:

  • Laws (properties) of exponents
  • Radical expression – expression that contains a radical symbol
    • The symbol  is called a radical.
    • The root number in the bend of the radical symbol is called the index. For , the index is n.
    • The expression under the radical symbol is called the radicand. For , the radicand is p.
    • If no index is indicated on the radical symbol, it is understood to be a square root.
    • Sample radical expressions
      •   
    • The radical symbol represents a fractional exponent and the expression can be rewritten with the fractional exponent.
  • Simplification of radical expressions
    • All coefficients and variables should be written as factors in power form, 56xy = 23 • 71x3y5
    • The root is taken by removing groups from the radicand according to the index value. (Hint: Divide the index into the power to determine power on the number or variable taken out of the radicand. Any remainder will be left in the radicand.)
    • The root may also be taken by writing the expression with a fractional exponent and simplifying using the power rule of exponents, (ab)n = anbn.
    • If the index of a radical expression is even and the power of the variable simplified out of the radic and is odd, then the variable, including its power, must be represented using absolute value.
    • A factor of negative one in radicands with even indices, 2, 4, 6, …, have no real solutions.
      •  no real solution
      •  no real solution
    • A factor of negative one in radicands with odd indices, 1, 3, 5, …, have a factor of negative one in the simplified answer.

Note(s):

  • Grade Level(s):
    • Algebra I simplified numerical radical expressions involving square roots.
    • Algebra II simplifies radical expressions involving variables.
    • Algebra II simplifies radical expressions involving various indices.
    • 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
      • B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g. polynomials, radicals, rational expressions).
      • D1 – Interpret multiple representations of equations and relationships.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
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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