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Algebra I
TITLE : Unit 01: Linear Expressions, Equations, and Inequalities (one variable) SUGGESTED DURATION : 15 days

Unit Overview

This unit bundles student expectations that address polynomial expressions of degree one, operations with polynomial expressions of degree one, and solving linear equations and inequalities in one variable. Solving formulas and literal equations for a specified variable are also addressed. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this unit, in Grade 5, students used equations with a variable to represent an unknown quantity. In Grade 6, students wrote and solved one variable, one-step equations and inequalities, representing solutions on a number line. In Grade 7, students wrote and solved one variable, two-step equations and inequalities, representing solutions on a number line. In Grade 8, students wrote, modeled, and solved one variable equations with variables on both sides using rational number coefficients and constants.

During this unit, students define polynomial expressions and perform operations (addition, subtraction, scalar multiplication) with polynomials of degree one, including rewriting a polynomial to an equivalent form when distributing by a rational scale factor. Students determine the quotient of a polynomial of degree one divided by a polynomial of degree one. Students make connections between expressions and equations, and solve linear equations in one variable, including variables on both sides and the application of the distributive property. Students model both mathematical and real-world problem situations using equations. Students solve linear inequalities in one variable, including variables on both sides and the application of the distributive property. Students model both mathematical and real-world problem situations using inequalities. Students solve mathematical formulas (including solving for y), scientific formulas, and other literal equations for a specified variable.

After this unit, in Unit 03 and Unit 04, students will apply their understanding of degree one polynomial operations, linear equations, linear inequalities, and literal equations when studying linear functions and applications of linear functions. The concepts in this unit will also be applied in later units in Algebra I and subsequent mathematics courses.

In Algebra I, solving linear equations in one variable is identified as STAAR Readiness Standard A.5A and part of STAAR Reporting Category 3: Writing and Solving Linear Functions, Equations, and Inequalities. Solving linear inequalities in one variable is STAAR Supporting Standard A.5B. Simplifying polynomial expressions of degree one using all four operations is identified as STAAR Supporting Standards A.10A, A.10C, and A.10D. Solving formulas and literal equations for a specified variable is STAAR Supporting Standard A.12E. 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, C1; II. Algebraic Reasoning A1, B1, C1, C2, D1, D2; III. Geometric Reasoning C1; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

According to the National Council of Teachers of Mathematics (NCTM), Principles and Standards for School Mathematics (2000), 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)

 

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.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9

OVERARCHING UNDERSTANDINGS and QUESTIONS

  • Numeracy requires the ability to work flexibly with quantities in order to recognize, reason, and solve situations of varying contexts in everyday life, society, and the work place.
    • How is numeracy like literacy?
    • What are some examples of numeracy in everyday life, society, and the work place?
    • How does context influence understanding of a quantity?
    • Why is the ability to work flexibly with quantities essential to developing the foundations of numeracy?
  • Understanding and generalizing operational relationships leads to more sophisticated representations and solution strategies in order to investigate or solve problem situations in everyday life.
    • What relationships exist within and between mathematical operations?
    • How does generalizing operational relationships lead to developing more flexible, efficient representations and/or solution strategies?
    • Why is understanding the problem solving process an essential part of learning and working mathematically?
  • Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgments in everyday life.
    • What patterns exist within different types of quantitative relationships and where are they found in everyday life?
    • Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
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.
  • Number and Algebraic Methods
    • Expressions
      • Polynomial
    • Patterns, Operations, and Properties
  • Associated Mathematical Processes
    • Application
    • Problem Solving Model
    • Tools and Techniques
    • Communication
    • Representations
    • Relationships 
    • Justification
  • The ability to represent quantities in various forms develops the understanding of equivalence and allows for working flexibly with algebraic expressions in order to communicate and reason about quantities.
    • How can expressions be used to represent situations?
    • What mathematical conventions are used when representing expressions? Why?
    • How can it be determined if two expressions are equivalent?
    • How are properties and operational understandings used to generate equivalent expressions?
    • Why can it be useful to simplify expressions?
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.
  • Functions, Equations, and Inequalities
    • Equations and Inequalities
      • Linear
    • Patterns, Operations, and Properties
  • Associated Mathematical Processes
    • Application
    • Problem Solving Model
    • Tools and Techniques
    • Communication
    • Representations
    • Relationships
    • Justification
  • Equations and inequalities can be written, transformed, and solved using various methods to make critical judgments, with different methods being more efficient or informative depending on the structure of the equation or inequality.
    • How does knowing more than one solution strategy build mathematical flexibility?
    • How can equations and inequalities be used to represent relationships between quantities?
    • How do solutions to inequalities differ from solutions to equations?
    • Why must solutions be justified in terms of problem situations?
    • What methods can be used to write linear equations and linear inequalities?
    • What methods can be used to solve linear equations and linear inequalities?
    • How does the structure of the equation influence the selection of an efficient method for solving linear equations?
    • How are properties and operational understandings used to transform linear equations and linear inequalities?
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.
  • Number and Algebraic Methods
    • Relations and Functions
      • Formulas
      • Literal equations
    • Patterns, Operations, and Properties
  • Associated Mathematical Processes
    • Application
    • Problem Solving Model
    • Tools and Techniques
    • Communication
    • Representations
    • Relationships
    • Justification
  • Equations can be written, transformed, and solved using various methods to make critical judgments, with different methods being more efficient or informative depending on the structure of the equation.
    • How can equations be used to represent relationships between quantities?
    • How are properties and operational understandings used to transform literal equations?
    • How does the context of the problem situation affect which variable to solve for in a literal equation?
    • What is the purpose for solving for a specific variable in a literal equation?

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

  • Some students may think that a constant term can be combined with a variable term (e.g., 2x + 5 = 7x) rather than constant terms only combining with other constant terms and like-variable terms combining with like-variable terms.
  • Some students may think that in the graph or table method of solving the equation, the y-value is the answer rather than the x-value.
  • Some students may think that the negative in front of the parentheses is distributed only to the first term of the expression in parentheses rather than to all terms of the expression in parentheses.
  • Some students may think that answers to both equations and inequalities are exact answers rather than correctly identifying the solutions to equations as exact answers and the solutions to inequalities as range of answers.
  • Some students may think that whenever a negative is involved, the order of the inequality switches rather than only switching the order of inequality when multiplying or dividing by a negative.

Unit Vocabulary

  • Algebraic expression – a generalization that is a combination of variables, numbers (constants and coefficients), and operators
  • Binomial – two term expression; e.g., 4 – 2y, 3a + 1
  • Degree of polynomial – same as the degree of the term in the polynomial with the highest degree
  • Degree of term – sum of the powers on the variables in the term
  • First degree polynomial – polynomial whose highest degree term contains one variable with power of one
  • Linear equation in one variable – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
  • Linear inequality in one variable – a mathematical statement composed of algebraic and/or numeric expressions set apart by an inequality symbol
  • Literal equations – equations in which all or part of the terms are expressed in variables
  • Monomial – one term expression; e.g., –2.5x,
  • Polynomial expression – monomial or sum of monomials not including variables in the denominator or under a radical
  • Trinomial – three term expression

Related Vocabulary:

  • Associative property
  • Commutative property
  • Distributive property
  • Equation
  • Equivalent
  • Evaluate
  • Expression
  • Graphic solution
  • Inequality
  • Inverse operation
  • Numeric solution
  • Proportions
  • Rates
  • Reciprocal
  • Ratios
  • Simplify
  • Solve
  • Terms
Unit Assessment Items System Resources Other Resources

Show this message:

Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment 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

  

Texas Education Agency – Mathematics Curriculum

 

Texas Education Agency – STAAR Mathematics Resources

 

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

 

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

 

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

 

Texas Education Agency Texas Gateway –Resources Aligned to Algebra I Mathematics TEKS

 

Texas Instruments – Graphing Calculator Tutorials

TEKS# SE# TEKS Unit Level Specificity
 

Legend:

  • 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)

Legend:

  • 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)
A.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
A.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

Apply

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

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

Note(s):    

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

Use

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

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

Note(s):    

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

Select

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

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

Note(s):    

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

Communicate

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

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

Note(s):    

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

Create, Use

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

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

Note(s):    

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

Analyze

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

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

Note(s):    

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

Display, Explain, Justify

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

Including, but not limited to:

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

Note(s):    

  • The mathematical process standards may be applied to all content standards as appropriate.
  • TxCCRS:
    • IX. Communication and Representation
A.5 Linear functions, equations, and inequalities. The student applies the mathematical process standards to solve, with and without technology, linear equations and evaluate the reasonableness of their solutions. The student is expected to:
A.5A Solve linear equations in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides.
Readiness Standard

Solve

LINEAR EQUATIONS IN ONE VARIABLE, INCLUDING THOSE FOR WHICH THE APPLICATION OF THE DISTRIBUTIVE PROPERTY IS NECESSARY AND FOR WHICH VARIABLES ARE INCLUDED ON BOTH SIDES

Including, but not limited to:

  • Linear equation in one variable – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
  • Linear equations in one variable including parentheses and variables on both sides of the equation
  • Mathematical problem situations
  • Real-world problem situations
  • Multiple representations of mathematical and real-world problem situations
    • Algebraic generalizations
    • Missing coordinate of a solution point to a function
    • Verbal
  • Methods for solving equations
    • Concrete and pictorial models (e.g., algebra tiles, etc.)
    • Tables and graphs with and without technology
    • Transformation of equations using properties of equality
      • Distributive property
      • Operational properties
    • Possible solutions, including special cases
      • No solution, empty set, ∅
      • Infinite solutions, all real numbers, ℜ
    • Relationships and connections between the methods of solution
  • Justification of solutions to equations
  • Justification of reasonableness of solutions in terms of mathematical and real-world problem situations

Note(s):

  • Grade Level(s):
    • Grade 5 used equations with variables to represent missing numbers.
    • Grade 6 solved one-variable, one-step equations.
    • Grade 7 solved one-variable, two-step equations.
    • Grade 8 solved one-variable equations with variables on both sides.
    • Algebra I introduces solving one-variable equations that include those for which the application of the distributive property is necessary and for which variables are included on both sides.
    • Algebra II will introduce solving absolute value linear equations.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxCCRS:
    • I. Numeric reasoning
      • C1 – Use estimation to check for errors and reasonableness of solutions.
    • 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
A.5B Solve linear inequalities in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides.
Supporting Standard

Solve

LINEAR INEQUALITIES IN ONE VARIABLE, INCLUDING THOSE FOR WHICH THE APPLICATION OF THE DISTRIBUTIVE PROPERTY IS NECESSARY AND FOR WHICH VARIABLES ARE INCLUDED ON BOTH SIDES

Including, but not limited to:

  • Linear inequality in one variable – a mathematical statement composed of algebraic and/or numeric expressions set apart by an inequality symbol
    • Inequality symbols
      • > (is greater than)
      • < (is less than)
      • ≥ (is greater than or equal to)
      • ≤ (is less than or equal to)
      • ≠ (is not equal to)
  • Linear inequalities including parentheses and variables on both sides of the equation
  • Mathematical problem situations
  • Real-world problem situations
  • Multiple representations of mathematical and real-world problem situations
    • Algebraic generalizations
    • Verbal
  • Solutions to include numeric, graphic, and verbal representations
  • Methods for solving inequalities
    • Concrete and pictorial models (e.g., algebra tiles, etc.)
    • Graphs and tables with and without technology
    • Transformation of inequalities using properties of inequalities
      • Distributive property
      • Operational properties
    • Special cases for empty set, Ø, and all real numbers, ℜ
    • Relationships and connections between the methods of solution
  • Justification of solutions to inequalities
  • Differentiation between solutions of equations and inequalities
  • Justification of reasonableness of solutions in terms of mathematical and real-world problem situations

Note(s):

  • Grade Level(s):
    • Grade 6 solved one-variable, one-step inequalities.
    • Grade 7 solved one-variable, two-step inequalities.
    • Grade 8 wrote one-variable inequalities with variables on both sides.
    • Algebra I introduces solving one-variable inequalities, including those for which the application of the distributive property is necessary and for which variables are included on both sides.
    • Algebra II will introduce solving absolute value linear inequalities.
    • Various mathematical process standards will be applied to this student expectation as appropriate.
  • TxCCRS:
    • I. Numeric reasoning
      • C1 – Use estimation to check for errors and reasonableness of solutions.
    • II. Algebraic Reasoning
      • C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
      • C2 – Explain the difference between the solution set of an equation and the solution set of an inequality.
      • 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
A.10 Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to rewrite in equivalent forms and perform operations on polynomial expressions. The student is expected to:
A.10A

Add and subtract polynomials of degree one and degree two.


Supporting Standard

Add, Subtract

POLYNOMIALS OF DEGREE ONE

Including, but not limited to:

  • Algebraic expression – a generalization that is a combination of variables, numbers (constants and coefficients), and operators
  • Polynomial expression – monomial or sum of monomials not including variables in the denominator or under a radical
    • Monomial – one term expression; e.g., –2.5x,
    • Binomial – two term expression; e.g., 4 – 2y, 3a + 1
    • Trinomial - three term expression
  • Degree
    • Degree of term – sum of the powers on the variables in the term
    • Degree of a polynomial – same as the degree of the term in the polynomial with the highest degree
      • First degree polynomial – polynomial whose highest degree term contains one variable with power of one
        • Ex: 3x + 8; The highest degree term is 3x, and the power on x is one.
        • Ex: –2x – 5y; Both terms are degree one with the power on x and y both equal to one.
  • Simplifying polynomials by addition/subtraction using concrete models
    • Algebra tiles
  • Simplifying polynomials by addition/subtraction algebraically
    • Clear grouping symbols using the distributive property.
    • Combine like terms.
    • Place terms in order
      • Alphabetical order
      • Decreasing degree order
  • Applications of addition/subtraction of polynomials in mathematical problem situations

Note(s):

  • Grade Level(s):
    • Previous grade levels calculated the perimeter of triangles and rectangles.
    • Grade 6 generated and compared equivalent expressions using concrete models, pictorial models, and algebraic properties of operations.
    • Algebra I introduces operations with polynomials of degree two.
    • Algebra II will extend operations with polynomials of degree three and degree four, including division of 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
      • A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
      • B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
    • III. Geometric Reasoning
      • C1 – Make connections between geometry and algebra.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
A.10C

Determine 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.


Supporting Standard

Determine

THE QUOTIENT OF A POLYNOMIAL OF DEGREE ONE WHEN DIVIDED BY A POLYNOMIAL OF DEGREE ONE

Including, but not limited to:

  • Degree one polynomial divided by another degree one polynomial
  • Division of polynomials
    • Division by factoring
      • Cancellation of common factors in the numerator and denominator
    • Array method
    • Long division
      • 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
  • Applications of division of polynomials in mathematical problem situations

Note(s):

  • Grade Level(s):
    • Previous grade levels calculated the area of triangles and rectangles.
    • Algebra I introduces operations with polynomials of degree one and degree two.
    • Algebra II will extend operations with 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).
    • III. Geometric Reasoning
      • C1 – Make connections between geometry and algebra.
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
A.10D

Rewrite polynomial expressions of degree one and degree two in equivalent forms using the distributive property.


Supporting Standard

Rewrite

POLYNOMIAL EXPRESSIONS OF DEGREE ONE IN EQUIVALENT FORMS USING THE DISTRIBUTIVE PROPERTY

Including, but not limited to:

  • Polynomial expression – monomial or sum of monomials not including variables in the denominator or under a radical
    • First degree polynomial – polynomial whose highest degree term contains one variable with power of one
  • Factorization of the greatest common factor (GCF)
  • Operations on polynomials
    • Addition/subtraction
    • Multiplication 

Note(s):

  • Grade Level(s):
    • Algebra I introduces operations with polynomials of degree one and degree two.
    • Algebra II will extend operations with 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
      • A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
      • B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
A.12 Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to write, solve, analyze, and evaluate equations, relations, and functions. The student is expected to:
A.12E Solve mathematic and scientific formulas, and other literal equations, for a specified variable.
Supporting Standard

Solve

MATHEMATIC AND SCIENTIFIC FORMULAS, AND OTHER LITERAL EQUATIONS, FOR A SPECIFIED VARIABLE

Including, but not limited to:

  • Literal equations – equations in which all or part of the terms are expressed in variables
    • Two variable linear equations
    • Mathematical formulas
    • Scientific formulas
  • Transforming literal equations is subsumed within solving
    • Solving for one of the variables in two variable linear equations.
    • Solving formulas for a specified variable
      • Mathematical formulas
      • Scientific formulas

Note(s):

  • Grade Level(s):
    • Algebra I introduces solving mathematical formulas and literal equations.
    • 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
The English Language Proficiency Standards (ELPS), as required by 19 Texas Administrative Code, Chapter 74, Subchapter A, §74.4, outline English language proficiency level descriptors and student expectations for English language learners (ELLs). School districts are required to implement ELPS as an integral part of each subject in the required curriculum.

School districts shall provide instruction in the knowledge and skills of the foundation and enrichment curriculum in a manner that is linguistically accommodated commensurate with the student’s levels of English language proficiency to ensure that the student learns the knowledge and skills in the required curriculum.


School districts shall provide content-based instruction including the cross-curricular second language acquisition essential knowledge and skills in subsection (c) of the ELPS in a manner that is linguistically accommodated to help the student acquire English language proficiency.

http://ritter.tea.state.tx.us/rules/tac/chapter074/ch074a.html#74.4 


Choose appropriate ELPS to support instruction.

ELPS# Subsection C: Cross-curricular second language acquisition essential knowledge and skills.
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ELPS.c.1 The ELL uses language learning strategies to develop an awareness of his or her own learning processes in all content areas. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.1A use prior knowledge and experiences to understand meanings in English
ELPS.c.1B monitor oral and written language production and employ self-corrective techniques or other resources
ELPS.c.1C use strategic learning techniques such as concept mapping, drawing, memorizing, comparing, contrasting, and reviewing to acquire basic and grade-level vocabulary
ELPS.c.1D speak using learning strategies such as requesting assistance, employing non-verbal cues, and using synonyms and circumlocution (conveying ideas by defining or describing when exact English words are not known)
ELPS.c.1E internalize new basic and academic language by using and reusing it in meaningful ways in speaking and writing activities that build concept and language attainment
ELPS.c.1F use accessible language and learn new and essential language in the process
ELPS.c.1G demonstrate an increasing ability to distinguish between formal and informal English and an increasing knowledge of when to use each one commensurate with grade-level learning expectations
ELPS.c.1H develop and expand repertoire of learning strategies such as reasoning inductively or deductively, looking for patterns in language, and analyzing sayings and expressions commensurate with grade-level learning expectations.
ELPS.c.2 The ELL listens to a variety of speakers including teachers, peers, and electronic media to gain an increasing level of comprehension of newly acquired language in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in listening. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.2A distinguish sounds and intonation patterns of English with increasing ease
ELPS.c.2B recognize elements of the English sound system in newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters
ELPS.c.2C learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions
ELPS.c.2D monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed
ELPS.c.2E use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken language
ELPS.c.2F listen to and derive meaning from a variety of media such as audio tape, video, DVD, and CD ROM to build and reinforce concept and language attainment
ELPS.c.2G understand the general meaning, main points, and important details of spoken language ranging from situations in which topics, language, and contexts are familiar to unfamiliar
ELPS.c.2H understand implicit ideas and information in increasingly complex spoken language commensurate with grade-level learning expectations
ELPS.c.2I demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs.
ELPS.c.3 The ELL speaks in a variety of modes for a variety of purposes with an awareness of different language registers (formal/informal) using vocabulary with increasing fluency and accuracy in language arts and all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in speaking. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.3A practice producing sounds of newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters to pronounce English words in a manner that is increasingly comprehensible
ELPS.c.3B expand and internalize initial English vocabulary by learning and using high-frequency English words necessary for identifying and describing people, places, and objects, by retelling simple stories and basic information represented or supported by pictures, and by learning and using routine language needed for classroom communication
ELPS.c.3C speak using a variety of grammatical structures, sentence lengths, sentence types, and connecting words with increasing accuracy and ease as more English is acquired
ELPS.c.3D speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency
ELPS.c.3E share information in cooperative learning interactions
ELPS.c.3F ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete vocabulary, including key words and expressions needed for basic communication in academic and social contexts, to using abstract and content-based vocabulary during extended speaking assignments
ELPS.c.3G express opinions, ideas, and feelings ranging from communicating single words and short phrases to participating in extended discussions on a variety of social and grade-appropriate academic topics
ELPS.c.3H narrate, describe, and explain with increasing specificity and detail as more English is acquired
ELPS.c.3I adapt spoken language appropriately for formal and informal purposes
ELPS.c.3J respond orally to information presented in a wide variety of print, electronic, audio, and visual media to build and reinforce concept and language attainment.
ELPS.c.4 The ELL reads a variety of texts for a variety of purposes with an increasing level of comprehension in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in reading. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations apply to text read aloud for students not yet at the stage of decoding written text. The student is expected to:
ELPS.c.4A learn relationships between sounds and letters of the English language and decode (sound out) words using a combination of skills such as recognizing sound-letter relationships and identifying cognates, affixes, roots, and base words
ELPS.c.4B recognize directionality of English reading such as left to right and top to bottom
ELPS.c.4C develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials
ELPS.c.4D use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary and other prereading activities to enhance comprehension of written text
ELPS.c.4E read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned
ELPS.c.4F use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language
ELPS.c.4G demonstrate comprehension of increasingly complex English by participating in shared reading, retelling or summarizing material, responding to questions, and taking notes commensurate with content area and grade level needs
ELPS.c.4H read silently with increasing ease and comprehension for longer periods
ELPS.c.4I demonstrate English comprehension and expand reading skills by employing basic reading skills such as demonstrating understanding of supporting ideas and details in text and graphic sources, summarizing text, and distinguishing main ideas from details commensurate with content area needs
ELPS.c.4J demonstrate English comprehension and expand reading skills by employing inferential skills such as predicting, making connections between ideas, drawing inferences and conclusions from text and graphic sources, and finding supporting text evidence commensurate with content area needs
ELPS.c.4K demonstrate English comprehension and expand reading skills by employing analytical skills such as evaluating written information and performing critical analyses commensurate with content area and grade-level needs.
ELPS.c.5 The ELL writes in a variety of forms with increasing accuracy to effectively address a specific purpose and audience in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in writing. In order for the ELL to meet grade-level learning expectations across foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations do not apply until the student has reached the stage of generating original written text using a standard writing system. The student is expected to:
ELPS.c.5A learn relationships between sounds and letters of the English language to represent sounds when writing in English
ELPS.c.5B write using newly acquired basic vocabulary and content-based grade-level vocabulary
ELPS.c.5C spell familiar English words with increasing accuracy, and employ English spelling patterns and rules with increasing accuracy as more English is acquired
ELPS.c.5D edit writing for standard grammar and usage, including subject-verb agreement, pronoun agreement, and appropriate verb tenses commensurate with grade-level expectations as more English is acquired
ELPS.c.5E employ increasingly complex grammatical structures in content area writing commensurate with grade-level expectations, such as:
ELPS.c.5F write using a variety of grade-appropriate sentence lengths, patterns, and connecting words to combine phrases, clauses, and sentences in increasingly accurate ways as more English is acquired
ELPS.c.5G narrate, describe, and explain with increasing specificity and detail to fulfill content area writing needs as more English is acquired.
Last Updated 08/01/2018
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