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 Instructional Focus DocumentAlgebra I
 TITLE : Unit 06: Laws of Exponents, Expressions, and Factoring SUGGESTED DURATION : 10 days

#### Unit Overview

Introduction
This unit bundles student expectations that address simplification of numeric and algebraic expressions using the laws of exponents, operations with degree one and degree two polynomial expressions, and factoring difference of square binomials and trinomials. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this Unit
In Grade 6, students were introduced to factors and exponential notation using whole number exponents. In Algebra I Unit 01, first degree polynomial expressions were added, subtracted, multiplied, and divided in mathematical and real-world problem situations.

During this Unit
Students simplify numeric and algebraic expressions and solve equations using the laws of exponents, including integral and rational exponents. Students perform operations (addition, subtraction, multiplication) with polynomials of degree one and degree two, including rewriting a polynomial to an equivalent form using the distributive property. Students 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, and justify the answer by multiplication. Students apply the distributive property to factor out the greatest common factor of the terms in a polynomial expression. Students also factor binomials (difference of two squares) and factor trinomials (ax2 + bx + c) having real roots, including perfect square trinomials of degree two, and justify the results by multiplication.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool HS Algebra I

After this Unit
Rules of exponents, polynomial operations, and factoring will be applied to simplify expressions and solve equations in later units in Algebra I and subsequent mathematics courses.

In Algebra I, generating and applying rules of exponents and polynomial operations are identified in STAAR Readiness Standard A.11B and STAAR Supporting Standards A.10A, A.10B, A.10C, and A.10D; and subsumed under STAAR Reporting Category 1: Number and Algebraic Methods. Factoring polynomial expressions is identified in STAAR Readiness Standard A.10E and STAAR Supporting Standard A.10F; and subsumed under STAAR Reporting Category 1: Number and Algebraic Methods. This unit is supporting the development of Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning A1, B1, D1, D2; V. Statistical Reasoning A1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
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

 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?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• 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?
• Number and Algebraic Methods
• Expressions
• Polynomial
• Rational exponents
• Patterns, Operations, and Properties
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 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?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• 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 factor expressions?
• How does the structure of the expression influence the selection of an efficient method for factoring polynomial expressions?
• Number and Algebraic Methods
• Expressions
• Polynomial
• Patterns, Operations, and Properties
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that that a second degree term can be combined with a first degree term rather than like-variable terms combining with other like-variable terms.
• Some students may think that the exponents combine when adding or subtracting two polynomials (e.g. (2x2 – 3x) + (3x2 + 9x) = (5x4 + 6x2)).
• Some students may think that the coefficient 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 when dividing polynomials with missing term(s) that they do not have to add zero place holders for the missing term(s) (e.g., 3x2 – 5 → 3x2 + 0x – 5).

#### 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, 5x2 – 2x, mnpq
• Degree of a 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
• Monomial – one term expression; e.g., –2.5x, , 4x2, 2mn
• Perfect square trinomial – first term a perfect square, third term a perfect square, middle term double the product of the square roots of the first and last terms
• Polynomial expression – monomial or sum of monomials not including variables in the denominator or under a radical
• Second degree polynomial – polynomial whose highest degree term contains one variable with a power of two, or two variables each having a power of one
• Trinomial – three term expression; e.g., x2 + 2x + 1, a2 – 2ab – 8b2

Related Vocabulary:

 Associative property Commutative property Distributive property Equation Equivalent Evaluate Expression Factoring Graphic solution Greatest common factor (GCF) Inequality Inverse operation Laws of exponents Numeric solution Polynomial division (long division) Reciprocal Simplify Solve Terms
Unit Assessment Items System Resources Other Resources

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

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

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

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

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

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

Texas Instruments – Graphing Calculator Tutorials

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

• Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
• Student Expectations (TEKS) identified by TEA are in bolded, black text.
• Student Expectations (TEKS) are labeled Readiness as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Supporting as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Process standards as identified by TEA of the assessed curriculum.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
TEKS# SE# TEKS SPECIFICITY
A.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
A.1A Apply mathematics to problems arising in everyday life, society, and the workplace.
Process Standard

Apply

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

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world situation.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
• IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.
A.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.
Process Standard

Use

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

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.A. Statistical Reasoning – Design a study
• V.A.1. Formulate a statistical question, plan an investigation, and collect data.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.2. Formulate a plan or strategy.
• VII.A.3. Determine a solution.
• VII.A.4. Justify the solution.
• VII.A.5. Evaluate the problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving process.
A.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.
Process Standard

Select

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

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
• V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
A.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Process Standard

Communicate

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

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
A.1E Create and use representations to organize, record, and communicate mathematical ideas.
Process Standard

Create, Use

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

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
A.1F Analyze mathematical relationships to connect and communicate mathematical ideas.
Process Standard

Analyze

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

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
•  VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
A.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Process Standard

Display, Explain, Justify

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

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.4. Justify the solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
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

POLYNOMIALS OF DEGREE ONE AND DEGREE TWO

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, , 4x2, 2mn
• Binomial – two term expression; e.g., 4 – 2y, 3a + 1, 5x2 – 2x, mnpq
• Trinomial – three term expression; e.g., x2 + 2x + 1, a2 – 2ab – 8b2
• 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.
• Second degree polynomial – polynomial whose highest degree term contains one variable with a power of two, or two variables each having a power of one
• Ex: 3x2 + x – 5; The highest degree term is 3x2 and the one variable has a power of two.
• Ex: x2xy + 12y2; All three terms are degree two. The single variable terms, x2 and y2 both have degree two. For the xy term, both have a power of one which adds to a degree of two for the term.
• 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):

• 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:
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• II.B. Algebraic Reasoning – Manipulating expressions
• II.B.1. Recognize and use algebraic properties, concepts, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
A.10B Multiply polynomials of degree one and degree two.
Supporting Standard

Multiply

POLYNOMIALS OF DEGREE ONE AND DEGREE TWO

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, , 4x2, 2mn
• Binomial – two term expression; e.g., 4 – 2y, 3a + 1, 5x2 – 2x, mnpq
• Trinomial – three term expression; e.g., x2 + 2x + 1, a2 – 2ab – 8b2
• 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.
• Second degree polynomial – polynomial whose highest degree term contains one variable with a power of two, or two variables each having a power of one
• Ex: 3x2 + x – 5; The highest degree term is 3x2 and the one variable has a power of two.
• Ex: x2xy + 12y2; All three terms are degree two. The single variable terms, x2 and y2 both have degree two. For the xy term, both have a power of one which adds to a degree of two for the term.
• Simplifying polynomials by multiplication using concrete models
• Algebra tiles
• Simplifying polynomials by multiplication algebraically
• Box method
• Distributive method
• Clear grouping symbols using the distributive property.
• Combine like terms.
• Place terms in order
• Alphabetical order
• Decreasing degree order
• Applications of multiplication of polynomials in mathematical problem situations
• Applications of multiple operations of polynomials in mathematical problem situations

Note(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:
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• II.B. Algebraic Reasoning – Manipulating expressions
• II.B.1. Recognize and use algebraic properties, concepts, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
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 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

Including, but not limited to:

• Degree one polynomial divided by another degree one polynomial
• Degree two polynomial divided by a degree one polynomial
• Degree two polynomial divided by another degree two polynomial
• Degree of divisor not to exceed degree of dividend
• 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):

• 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:
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• II.B. Algebraic Reasoning – Manipulating expressions
• II.B.1. Recognize and use algebraic properties, concepts, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
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 AND DEGREE TWO 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
• Second degree polynomial – polynomial whose highest degree term contains one variable with a power of two, or two variables each having a power of one
• Factorization of the greatest common factor (GCF)
• Operations on polynomials
• Multiplication

Note(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:
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• II.B. Algebraic Reasoning – Manipulating expressions
• II.B.1. Recognize and use algebraic properties, concepts, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
A.10E Factor, if possible, trinomials with real factors in the form ax2 + bx + c, including perfect square trinomials of degree two.

Factor

TRINOMIALS WITH REAL FACTORS IN THE FORM ax2 + bx + c, INCLUDING PERFECT SQUARE TRINOMIALS OF DEGREE TWO, IF POSSIBLE

Including, but not limited to:

• Trinomial – three term expression
• Factorization of trinomials
• Form ax2 + bx + c
• a and b – coefficients of the variables
• c – the constant term
• Terms in descending order alphabetically and by degree
• First check for a greatest common factor (GCF).
• Leading coefficient, a, equal to 1
• Ex: x2 – 2x – 63; x2 + 5x + 25; p2 + 13pq + 40q2
• Leading coefficient, a, real number other than 1
• Ex: 3x2 – 24x + 36; 2x2 – 9x – 5; 15a2 + 11ab + 2b2
• Perfect square trinomial – first term a perfect square, third term a perfect square, middle term double the product of the square roots of the first and last terms
• Ex: 4x2 – 12x + 9; • Identification of factorable trinomials and non-factorable (prime) trinomials
• Factorization of factorable trinomials
• Factor tables
• Box method
• Leading coefficient other than 1
• Box method
• Grouping
• Multiplication/division method (Bottoms Up)

Note(s):

• Algebra I introduces factorization of polynomials of degree one and degree two.
• Algebra II will extend factorization to polynomials of degree three and degree four, including factoring by grouping.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II.B. Algebraic Reasoning – Manipulating expressions
• II.B.1. Recognize and use algebraic properties, concepts, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
A.10F Decide if a binomial can be written as the difference of two squares and, if possible, use the structure of a difference of two squares to rewrite the binomial.
Supporting Standard

Decide

IF A BINOMIAL CAN BE WRITTEN AS THE DIFFERENCE OF TWO SQUARES

Including, but not limited to:

• Binomial – two term expression
• Binomial whose terms are the difference of two squares
• Both terms are perfect squares.
• The two terms have opposite signs.
• Difference of squares written as a2b2
• Ex: x2 – 9; 81x2 – 121y2; 25a2 – 9b2; Use

THE STRUCTURE OF A DIFFERENCE OF TWO SQUARES TO REWRITE THE BINOMIAL, IF POSSIBLE

Including, but not limited to:

• First check for a greatest common factor (GCF).
• Difference of squares written as a2b2
• Factorization of binomial that is the difference of two squares
• Factors into two binomials
• (square root of first term + square root of second term)(square root of first term – square root of second term)
• a2b2 = (a + b)(ab)
• Identification of factorable trinomials and non-factorable (prime) binomials
• Factorization of factorable trinomials

Note(s):

• Algebra I introduces factorization of polynomials of degree one and degree two.
• Algebra II will extend factorization to polynomials of degree three and degree four, including sum and difference of two cubes and factoring by grouping.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II.B. Algebraic Reasoning – Manipulating expressions
• II.B.1. Recognize and use algebraic properties, concepts, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
A.11 Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to rewrite algebraic expressions into equivalent forms. The student is expected to:
A.11B Simplify numeric and algebraic expressions using the laws of exponents, including integral and rational exponents.

Simplify

NUMERIC AND ALGEBRAIC EXPRESSIONS USING THE LAWS OF EXPONENTS, INCLUDING INTEGRAL AND RATIONAL EXPONENTS

Including, but not limited to:

• Algebraic expression – a generalization that is a combination of variables, numbers (constants and coefficients), and operators
• 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 = • Zero exponent: a0 = 1
• Rational exponent: • Simplification of expressions using laws (properties) of exponents
• Numeric expressions, including scientific notation
• Algebraic expressions
• Variables can appear as either the base or the exponent, but in either case must be rational numbers.
• Applications of algebraic expressions involving exponents

Note(s):

• Prior grade levels simplified numeric expressions involving whole number exponents.
• Grade 8 introduced scientific notation.
• Algebra I introduces exponential functions.
• Algebra I applies laws (properties) of exponents to simplify numeric and algebraic expressions.
• Algebra II will introduce equations involving rational exponents.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• II.B. Algebraic Reasoning – Manipulating expressions
• II.B.1. Recognize and use algebraic properties, concepts, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions). 