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 TITLE : Unit 04: Operations with Integers SUGGESTED DURATION : 10 days

#### Unit Overview

Introduction
This unit bundles student expectations that address identifying a number, its opposite, and its absolute value and representing and modeling integer operations fluently, including standardized algorithms. According to the Texas Education Agency, mathematical process standards including application, a problem-solving model, 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. The introduction to the grade level standards state, “While the use of all types of technology is important, the emphasis on algebra readiness skills necessitates the implementation of graphing technology."

Prior to this Unit
In Grade 4, students represented fractions and decimals to the tenths or hundredths as distances from zero on a number line. In Grade 6 Unit 02, students continued their understanding of equivalency by ordering whole numbers, positive and negative rational numbers, and integers using both models and real-world contexts.

During this Unit
Students examine number relationships involving identifying a number, its opposite, and absolute value. Previous work with number lines transitions to the understanding that absolute value can be represented on a number line as the distance a number is from zero. This builds to the relationship that since distance is always a positive value or zero, then absolute value is always a positive value or zero. Although students have been introduced to the concept of integers, this is the first time students are exposed to operations with negative counting (natural) numbers, which is a subset of integers. The development of integer operations with concrete and pictorial models is foundational to student understanding of operations with integers. Forgoing the use of concrete and pictorial models as a development of integer operations could be detrimental to future success with computations involving negative quantities, such as negative fractions and decimals. The use of concrete and pictorial models for integer operations is intended to be a bridge between the abstract concept of operations with integers and their standardized algorithms. It is expected that once the concept of integer operations has been sufficiently developed and connected to the standardized algorithms, students should add, subtract, multiply, and divide integers fluently.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool Grade 6

After this Unit
In Units 06 – 07, students will apply their understanding of integer operations to simplify expressions and solve one-variable, one-step equations and inequalities, which may include negative values and solutions. In Grade 7, students will extend integer operations to fractions and decimals as they are expected to add, subtract, multiply, and divide rational numbers fluently.

In Grade 6, identifying a number, its opposite, and its absolute value is identified as Supporting Standard 6.2B and is part of the Grade 6 STAAR Reporting Category 1: Numerical Representations and Relationships. Representing integer operations with concrete models and connecting the actions with the models to standardized algorithms is identified as Supporting Standard 6.3C; whereas, adding, subtracting, multiplying, and dividing integers is identified as Readiness Standard 6.3D. Both of these are subsumed under the Grade 6 STAAR Reporting Category 2: Computations and Algebraic Relationships. All of these standards are included in the Grade 6 Texas Response to Curriculum Focal Points (TxRCFP): Using operations with integers and positive rational numbers to solve problems. This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning A1, A2, B1; II. Algebraic Reasoning 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 Van de Walle, Bay-Williams, Lovin, and Karp (2014) state, “Calculating with integers can become a lesson in memorization when students are rushed to follow rules such as ‘two like signs become a positive’ and ‘two unlike signs become a negative’… Through discussions that explicitly focus on the mathematical concepts over time, the connections between manipulatives and related concepts are developed” (p. 24). According to the National Council of Teachers of Mathematics (2014), “in higher grades, fractions, and integers become more prominent. An understanding of numbers allows computational procedures to be learned and recalled with ease. Students should be able to perform computations in different ways. They should use mental methods and estimations in addition to doing paper-and-pencil calculations. Having computational fluency allows students to make good decisions about the use of calculators. Regardless of the method used to compute, students should be able to explain their method, understand that many methods exist, and see the usefulness of methods that are efficient, accurate, and general” (p. 3).

National Council of Teachers of Mathematics. (2000). Executive summary: Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from https://www.texasgateway.org/resource/txrcfp-texas-response-curriculum-focal-points-k-8-mathematics-revised-2013
Van de Walle, J., Bay-Williams, J., Lovin, L., & Karp, K., (2014). Teaching student-centered mathematics: Developmentally appropriate instruction for grades 6 - 8. (2nd ed., Vol. 3). Boston, MA: Pearson.

 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 recognize and represent numbers in various forms allows for working flexibly with numbers in order to communicate and reason about the value of the number (rational numbers).
• How can real world descriptors aid in distinguishing between a positive number and a negative number?
• Why is the number line the most efficient tool to model absolute value of a number?
• What relationships exist between …
• a number and its opposite?
• the absolute value of a number and the absolute value of its opposite?
• When a positive number decreases but is still greater than zero, why does the absolute value of the positive number decrease?
• When a positive number increases, why does the absolute value of the positive number increase?
• When a negative number decreases, why does the absolute value of the negative number increase?
• When a negative number increases but is still less than zero, why does the absolute value of the negative number decrease?
• Why is the absolute value of zero, zero?
• Number and Operations
• Number
• Rational
• Absolute value
• Relationships and Generalizations
• Numerical
• Operational
• Equivalence
• Representations
• 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.

 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)
• Recognizing and understanding operational relationships in a variety of problem situations leads to efficient, accurate, and flexible representations and solution strategies (integers).
• How does the context of a problem situation affect the representation, operation(s), and/or solution strategy that could be used to solve the problem?
• How does understanding …
• relationships within and between operations
• properties of operations
… aid in determining an efficient strategy or representation to investigate and solve problem situations?
• How can representing integer operations with …
• concrete (two-color counters, etc.)
• pictorial models (e.g., number lines, etc.)
… aid in solving problems involving …
• subtraction?
• multiplication?
• division?
• How does connecting the actions of the models to standardized algorithms develop understanding and fluency of integer operations?
• Why is it important to understand when and how to use standard algorithms?
• Why is it important to be able to perform operations with integers fluently?
• When subtracting a negative integer in an expression, why is the expression usually rewritten as addition?
• Operational understandings lead to generalizations that aid in determining the reasonableness of solutions (integers).
• When adding two integers, if a pair of addends have the same sign, why does the sum have the sign of both addends?
• When adding two integers, if a pair of addends have opposite signs, why does the sum have the sign of the addend with the greatest absolute value?
• When multiplying or dividing two or more integers with no negative signs or an even number of negative signs, why is the sign of the product or quotient always positive?
• When multiplying or dividing two or more integers with one negative sign or an odd number of negative signs, why is the sign of the product or quotient always negative?
• Number and Operations
• Number
• Whole numbers
• Integers
• Operations
• Subtraction
• Multiplication
• Division
• Relationships and Generalizations
• Numerical
• Operational
• Equivalence
• Representations
• Solution Strategies and Algorithms
• 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 the absolute value is the opposite of a number rather than the distance of the number away from zero (e.g. A student may think that the absolute value for 5 is –5, but –5 is actually the opposite.)
• Some students may forget to attach the sign of the integers to the sum or difference when adding or subtracting integers.
• Some students may have difficulty rewriting subtraction problems involving integers as the addition of an opposite.
• Some students may think that subtracting a negative integer from a negative integer always results in a difference of a negative integer.
• Some students may think that multiplying a negative integer by a negative integer results in a product that is negative.
• Some students may thing that dividing a negative integer by a negative integer results in a quotient that is negative.

#### Unit Vocabulary

• Absolute value – the distance of a value from zero on a number line
• Fluency– efficient application of procedures with accuracy
• Integers – the set of counting (natural numbers), their opposites, and zero {–n, …, –3, –2, –1, 0, 1, 2, 3, ..., n}. The set of integers is denoted by the symbol Z.

Related Vocabulary:

 Addend Ascend Compare Credit Debit Deposit Descend Difference Dividend Divisor Factor Gain Like signs Loss Number line Negative Opposite Positive Product Profit Quotient Sum Unlike signs Withdrawal
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 – Texas Response to Curriculum Focal Points for K-8 Mathematics Revised 2013

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 Grade 6 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.
• A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
TEKS# SE# TEKS SPECIFICITY
6.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
6.1A 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.
• TxRCFP:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• 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.
6.1B 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.
• TxRCFP:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• 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.
6.1C 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.
• TxRCFP:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• 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.
6.1D

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, AND LANGUAGE AS APPROPRIATE

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• 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.
6.1E 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.
• TxRCFP:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• 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.
6.1F 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.
• TxRCFP:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• 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.
6.1G 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.
• TxRCFP:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• 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.
6.2 Number and operations. The student applies mathematical process standards to represent and use rational numbers in a variety of forms. The student is expected to:
6.2B Identify a number, its opposite, and its absolute value.
Supporting Standard

Identify

A NUMBER, ITS OPPOSITE, AND ITS ABSOLUTE VALUE

Including, but not limited to:

• Numbers
• Positive numbers are to the right of zero on a horizontal number line and above zero on a vertical number line.
• Represented with a (+) symbol or no symbol at all
• Negative numbers are to the left of zero on a horizontal number line and below zero on a vertical number line.
• Represented with a (–) symbol
• Zero is neither positive nor negative.
• Quantities from mathematical and real-world problem situations are represented with positive and negative numbers.
• Relationships between a number and its opposite
• All numbers have an opposite and are represented with positive and negative values.
• The term “opposite” refers to the additive inverse of a number, meaning the number that when added to a given number results in zero.
• Opposite numbers are equidistant from zero on a number line.
• The opposite of the opposite of a number is the number itself.
• Relationships between a number and its absolute value
• Absolute value – the distance of a value from zero on a number line
• Notation for absolute value is |x|, where x is any number
• Distance is always a positive value or zero.
• The distance of a number from zero is the same as the distance of its opposite from zero.
• As a positive number decreases, the absolute value of the positive number decreases.
• As a positive number increases, the absolute value of the positive number increases.
• As a negative number decreases, the absolute value of the negative number increases.
• As a negative number increases, the absolute value of the negative number decreases.
• The absolute value of zero is zero.
• Relationship between a number, its opposite, and its absolute value
• The absolute value of a number and its opposite are equidistant from zero.
• The absolute value of a number and the absolute value of its opposite are equivalent.

Note(s):

• Grade 4 represented fractions and decimals to the tenths or hundredths as distances from zero on a number line.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Use operations with integers and positive rational numbers to solve problems
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.1. Compare relative magnitudes of rational and irrational numbers, and understand that numbers can be represented in different ways.
6.3 Number and operations. The student applies mathematical process standards to represent addition, subtraction, multiplication, and division while solving problems and justifying solutions. The student is expected to:
6.3C Represent integer operations with concrete models and connect the actions with the models to standardized algorithms.
Supporting Standard

Represent

INTEGER OPERATIONS WITH CONCRETE MODELS

Including, but not limited to:

• Integers – the set of counting (natural numbers), their opposites, and zero {–n, …, –3, –2, –1, 0, 1, 2, 3, ..., n}. The set of integers is denoted by the symbol Z.
• Operations
• Add, subtract, multiply, and/or divide
• Verbal actions expressed symbolically and vice versa
• Concrete models and pictorial representations
• Number lines (horizontal/vertical)
• Two-color counters

Connect

THE ACTIONS OF INTEGER OPERATIONS WITH THE CONCRETE MODELS TO STANDARDIZED ALGORITHMS

Including, but not limited to:

• Integer operations include using the additive inverse for subtraction by adding the opposite of the integer following the subtraction.
• Various representations of multiplication
• Various representations of division
• Connections between the actions of models for integer operations to standardized algorithms for integer operations
• Standardized algorithms of operations
• If a pair of addends has the same sign, then the sum will have the sign of both addends.
• If a pair of addends has opposite signs, then the sum will have the sign of the addend with the greatest absolute value.
• Subtraction
• A subtraction problem may be rewritten as an addition problem by adding the opposite of the integer following the subtraction symbol and then applying the rules for addition.
• Multiplication
• If a pair of factors has the same sign, then the product is positive.
• If a pair of factors has opposite signs, then the product is negative.
• Division
• If the dividend and divisor have the same sign, then the quotient is positive.
• If the dividend and divisor have opposite signs, then the quotient is negative.

Note(s):

• Grade 6 introduces representing integer operations with concrete models and connecting the actions with the models to standardized algorithms.
• Grade 7 will add, subtract, multiply, and divide rational numbers fluently.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using operations with integers and positive rational numbers to solve problems
• TxCCRS:
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
6.3D Add, subtract, multiply, and divide integers fluently.

INTEGERS FLUENTLY

Including, but not limited to:

• Integers – the set of counting (natural numbers), their opposites, and zero {–n, …, –3, –2, –1, 0, 1, 2, 3, ..., n}. The set of integers is denoted by the symbol Z.
• Fluency – efficient application of procedures with accuracy
• Various representations of multiplication
• Various representations of division
• Generalizations of integer operations
• If a pair of addends has the same sign, then the sum will have the sign of both addends.
• If a pair of addends has opposite signs, then the sum will have the sign of the addend with the greatest absolute value.
• A subtraction problem may be rewritten as an addition problem by adding the opposite of the integer following the subtraction symbol, and then applying the rules for addition.
• Multiplication and division
• If two rational numbers have the same sign, then the product or quotient is positive.
• If two rational numbers have opposite signs, then the product or quotient is negative.
• When multiplying or dividing two or more rational numbers, the product or quotient is positive if there are no negative signs or there is an even number of negative signs.
• When multiplying or dividing two or more rational numbers, the product or quotient is negative if there is one negative sign or there is an odd number of negative signs.

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