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 TITLE : Unit 03: Operations with Positive Fractions and Decimals SUGGESTED DURATION : 10 days

#### Unit Overview

Introduction
This unit bundles student expectations that address operations with positive rational numbers, specifically focusing on the relationships between multiplication and division of positive rational numbers. 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 Unit 01, students extended their mathematical foundations of equivalency within rational numbers to include percents. Students applied their understandings of percents to solve real-world problems. Methods for solving real-world problem situations involving percents, such as the use of proportions or scale factors between ratios, were not included in Unit 01. In Grade 5, students added and subtracted positive rational numbers fluently, as well as solved for products and quotients of decimals to the hundredths, multiplied a whole number and a fraction, and divided whole numbers by unit fractions and unit fractions by whole numbers.

During this Unit
Students expand their understanding of the representations for division to include fraction notation such as , which is equivalent to a ÷ b where b ≠ 0. Students recognize that dividing by a rational number and multiplying by its reciprocal result in equivalent values as well as determine whether a quantity is increased or decreased when multiplied by a fraction greater than or less than one. Exposure to solving mathematical and real-world situations assists students in generalizing operations with positive fractions and decimals, which builds fluency and reasonableness of solutions. All of these standards are encompassed within the additional expectation in this unit, which is for students to multiply and divide positive rational numbers fluently.

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

After this Unit
In Unit 04, students will begin solving problems involving all operations of integers. In Units 05 – 11, students will continue to apply their knowledge of operations with positive rational numbers. In Grade 7, students will apply and extend previous understandings of operations to solve problems using addition, subtraction, multiplication, and division of rational numbers with the expectation of being able to add, subtract, multiply, and divide positive and negative rational numbers fluently.

In Grade 6, extending representations for division is identified as STAAR Supporting Standard 6.2E and part of the Grade 6 STAAR Reporting Category Numerical Representations and Relationships. Recognizing that dividing by a rational number and multiplying by its reciprocal result in equivalent values, and determining whether a quantity is increased or decreased when multiplied by a fraction are identified as STAAR Supporting Standards 6.3A and 6.3B. Multiplying and dividing positive rational numbers is identified as STAAR Readiness Standard 6.3E. These standards are within the Grade 6 STAAR Reporting Category 2: Computations and Algebraic Relationships. All of the standards in this unit are subsumed under 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 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 the National Council of Teachers of Mathematics (NCTM), Principles and Standards for School Mathematics (2000), “In the middle grades, students should continue to refine their understandings of addition, subtraction, multiplication, and division as they use these operations with fractions, decimals, percents, and integers. Teachers need to be attentive to conceptual obstacles that many students encounter as they make the transition from operations with whole numbers” (p. 218). In addition, NCTM (1997) states, “solutions to many problems in measurement, geometry, algebra, probability, and statistics require knowledge of, and facility with, rational numbers” (p. 2). The expectations for operational fluency within Grade 6 are foundational to expectations of operational fluency within Grade 7. "Developing fluency requires a balance and connection between conceptual understanding and computational proficiency. On the one hand, computational methods that are over-practiced without understanding are often forgotten or remembered incorrectly...On the other hand, understanding without fluency can inhibit the problem-solving process"(NCTM, 2000 pg. 35).

National Council of Teachers of Mathematics. (1997). Curriculum and evaluation standards for school mathematics. Reston, VA:  National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency. (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

 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 develops the understanding of equivalence and allows for working flexibly with numbers in order to communicate and reason about the value of the number (positive rational numbers).
• Which symbols are used to represent division?
• How is fraction notation used to represent division?
• What relationships exist between the quantities represented by a fraction in the form and the quotient a ÷ b?
• Estimation strategies can be used to mentally approximate solutions and determine reasonableness of solutions (positive rational numbers).
• What strategies can be used to estimate solutions to problems?
• When might an estimated answer be preferable to an exact answer?
• How can an estimation aid in determining the reasonableness of an actual solution?
• When might one estimation strategy be more beneficial than another?
• Recognizing and understanding operational relationships in a variety of problem situations leads to efficient, accurate, and flexible representations and solution strategies (positive rational numbers).
• 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
• relationships between fractions and decimals
… aid in determining an efficient strategy or representation to investigate and solve problem situations?
• Why is it important to understand when and how to use standard algorithms?
• Why is it important to be able to perform operations with positive rational numbers fluently?
• What relationships exist between multiplication and division?
• How can the reciprocal of a positive rational number be determined?
• Operational understandings lead to generalizations that aid in determining the reasonableness of solutions (positive rational numbers).
• When dividing by a non-zero rational number and multiplying by its reciprocal, why are the results equivalent values?
• When multiplying two non-zero positive rational numbers greater than one, why is the product always greater than each of the factors?
• When multiplying two non-zero positive rational numbers where one of the factors is greater than one, why is the product always greater than the smallest factor?
• When multiplying two non-zero positive rational numbers where both factors are less than one, why is the product always less than the smallest factor?
• When dividing two non-zero positive rational numbers where the dividend is less than the divisor, why is the quotient always greater than zero and less than one?
• When dividing two non-zero positive rational numbers where the dividend is greater than the divisor, why is the quotient always greater than one?
• Number and Operations
• Number
• Whole numbers
• Rational numbers
• Reciprocals
• Operations
• Multiplication
• Division
• Relationships and Generalizations
• Numerical
• Operational
• Equivalence
• Solution Strategies and Algorithms
• 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.

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that they will need to find a common denominator when multiplying or dividing rational numbers.
• Some students may multiply two mixed numbers by multiplying the whole number parts and then the fractional parts.
• Some students may incorrectly place the decimal point in the final product or quotient of a problem without regard to place value relationships.
• Some students may attempt to perform computations with percents without converting them to equivalent decimals or fractions for multiplying or dividing.

Underdeveloped Concepts:

• Some students may think that a percent may not exceed 100%.
• Some students may think that a percent may not be less than 1%.
• Some students may not realize which operation is easier to use when converting between number forms.
• Some students may confuse decimal place values when converting decimals to fractions.
• Some students may have difficulty recognizing the part and the whole in problem situations.
• Some students may believe every fraction relates to a different rational number instead of realizing equivalent fractions relate to the same relative amount.
• Some students may divide a decimal by 100 by moving the decimal two places to the left when trying to convert it to a percent rather than multiplying by 100 and moving the decimal two places to the right.
• Some students may try to convert a fraction to a decimal by placing the denominator in the dividend rather than the numerator.
• Some students may think the value of 43% of 35 is the same value of 43% of 45 because the percents are the same rather than considering that the wholes of 35 and 45 are different, so 43% of each quantity will be different.
• Some students may think that a fraction can be converted to a decimal by simply writing the numerator and denominator as digits after a decimal (e.g., is equivalent to 0.78).

#### Unit Vocabulary

• Fluency– efficient application of procedures with accuracy
• Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are counting (natural) numbers
• Reciprocal – a quantity that is used to multiply by a given quantity which results in the product of one

Related Vocabulary:

 Decimal Denominator Dividend Division notation Divisor Equivalent Factor Fraction Fraction notation Greater than Improper fraction Less than Mixed number Numerator Product Proper fraction Quotient
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 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
• 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:
6.2E Extend representations for division to include fraction notation such as a/b represents the same number as a ÷ b where b ≠ 0.
Supporting Standard

Extend

REPRESENTATIONS FOR DIVISION TO INCLUDE FRACTION NOTATION SUCH AS REPRESENTS THE SAME NUMBER AS a ÷ b WHERE b ≠ 0

Including, but not limited to:

• Division notation
• Numeric: a ÷ b
• a represents the dividend.
• b represents the divisor, where b ≠0.
• Fraction notation
• is the algebraic notation for any rational number represented as a fraction.
• a represents the numerator of the fraction. A numerator denotes the number of equal parts from the whole or set.
• b represents the denominator of the fraction. A denominator denotes the total number of equal parts in a whole or set.
• Relationship between fraction notation and division
• is the same as a divided by b, where b ≠0.
• Algebraic: a ÷ b =  = = a
• The numerator of a fraction is the same as the dividend in a division problem.
• The denominator of a fraction is the same as the divisor in a division problem.
• Relationship between fraction notation and decimal notation
• Dividing a numerator by a denominator yields the decimal notation of the fraction.

Note(s):

• Grade 4 represented a fraction as a sum of fractions , where a and b are whole numbers and b > 0, including when a > b.
• 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
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:
6.3A Recognize that dividing by a rational number and multiplying by its reciprocal result in equivalent values.
Supporting Standard

Recognize

THAT DIVIDING BY A RATIONAL NUMBER AND MULTIPLYING BY ITS RECIPROCAL RESULT IN EQUIVALENT VALUES

Including, but not limited to:

• Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are counting (natural) numbers
• Various forms of positive rational numbers
• Counting (natural) numbers
• Decimals
• Fractions
• Percents converted to equivalent decimals or fractions for multiplying or dividing
• Reciprocal – a quantity that is used to multiply by a given quantity which results in the product of one
• Relationship between multiplication and division
• Dividing a number a by a given number b is equivalent to multiplying a by the reciprocal of b.
• Algebraic: a ÷ b = = = a
• Relationships between equivalent positive rational number representations

Note(s):

• Grade 3 determined a quotient using the relationship between multiplication and division.
• 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:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
6.3B Determine, with and without computation, whether a quantity is increased or decreased when multiplied by a fraction, including values greater than or less than one.
Supporting Standard

Determine

WITH AND WITHOUT COMPUTATION, WHETHER A QUANTITY IS INCREASED OR DECREASED WHEN MULTIPLIED BY A FRACTION, INCLUDING VALUES GREATER THAN OR LESS THAN ONE

Including, but not limited to:

• Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are counting (natural) numbers
• Positive fractions less than one
• Written as , where a is less than b and where a > 0 and b > 0
• Located between 0 and 1 on a number line, 0 < < 1
• Fractions greater than one
• Written as , where a is greater than b and where a > 0 and b > 0
• Located to the right of 1 on a number line, > 1
• Product of a given positive rational number and a positive fraction less than one
• Various forms of given positive rational numbers
• Counting (natural) numbers
• Decimals
• Fractions
• Percents converted to equivalent decimals or fractions for multiplying or dividing
• A positive rational number is decreased when multiplied by a positive fraction less than one.
• Product of a given positive rational number and a fraction greater than one
• Various forms of positive rational numbers
• Counting (natural) numbers
• Decimals
• Fractions
• Percents converted to equivalent decimals or fractions for multiplying or dividing
• A positive rational number is increased when multiplied by a positive fraction greater than one.
• Generalizations of fraction computations
• A positive rational number is increased when multiplied by a fraction greater than one.
• A positive rational number is decreased when multiplied by a fraction less than one.

Note(s):

• Grade 6 introduces determining, with and without computation, whether a quantity is increased or decreased when multiplied by a fraction, including values greater than or less than one.
• Grade 7 will apply and extend previous understandings of operations to solve problems using addition, subtraction, multiplication, and division of rational numbers.
• 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:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• 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.
6.3E Multiply and divide positive rational numbers fluently.

Multiply, Divide

POSITIVE RATIONAL NUMBERS FLUENTLY

Including, but not limited to:

• Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are counting (natural) numbers
• Various forms of positive rational numbers
• Counting (natural) numbers
• Decimals
• Fractions
• Percents converted to equivalent decimals or fractions for multiplying or dividing fluently
• Fluency – efficient application of procedures with accuracy
• Relationship between dividing by a fraction and multiplying by its reciprocal
• Reciprocal – a quantity that is used to multiply by a given quantity which results in the product of one

Note(s):

• Grade 5 represented multiplication of decimals with products to the hundredths using objects and pictorial models, including area models.
• Grade 5 solved for products of decimals to the hundredths, including situations involving money, using strategies based on place-value understandings, properties of operations, and the relationships to the multiplication of whole numbers.
• Grade 5 represented quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using objects and pictorial models, including area models.
• Grade 5 solved for quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using strategies and algorithms, including the standard algorithm.
• Grade 5 represented and solved multiplication of a whole number and a fraction that refers to the same whole using objects and pictorial models, including area models.
• Grade 5 represented division of a unit fraction by a whole number and the division of a whole number by a unit fraction such as ÷ 7 and 7 ÷  using objects and pictorial models, including area models.
• Grade 5 divided whole numbers by unit fractions and unit fractions by whole numbers.
• Grade 6 introduces multiplying and dividing positive rational numbers fluently.
• 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:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.