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 TITLE : Unit 02: Ordering Fractions, Decimals, and Integers SUGGESTED DURATION : 5 days

#### Unit Overview

Introduction
This unit bundles student expectations that address sets and subsets of numbers, generating equivalent forms of rational numbers, and comparing and ordering rational numbers and integers. 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. 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 determined if two given fractions were equivalent using a variety of methods. In Grade 5, students compared and ordered two decimals to the thousandths place and represented the comparison using the symbols >, <, or =. In Grade 6, Unit 01, students generated equivalent forms of fractions, decimals, and percents, including percents with fractional or decimal values such as 8.25% or .

During this Unit
Students continue their understanding of equivalency by generating and using equivalent forms of fractions, decimals, and percents to solve real-world problems, including problems involving money. The negative aspect of the number line is introduced as students explore the concept of integers and negative rational numbers. Students locate an integer or rational number on a number line and use its location to compare and order a set of numerical values, which may be presented in various forms. The number line may be used as a tool to assist in comparing or ordering a set of numbers; however, students are also expected to order a set of rational numbers arising from mathematical and real-world contexts using any strategy, such as place value, number sense, or comparisons to benchmarks. Students examine the sets and subsets of rational numbers and use a visual representation, such as a Venn diagram, to describe the relationships between the sets and subsets. This is the first time students classify a number as a natural (counting) number, whole number, integer, or rational number. Although the focus of operations in Grade 6 is with integers and positive rational numbers, students are expected to classify, compare, and order both positive and negative numerical values.

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

After this Unit
In Unit 03, students will extend previous operational experiences with adding, subtracting, multiplying, and dividing positive fractions and decimals. In Unit 04, students will apply their understanding of integers to solve problems involving integer operations. In Grade 7, students will extend previous knowledge of sets and subsets to create a visual representation to describe relationships between sets of rational numbers. Students will solve problems involving ratios, rates, and percents, including multi-step problems involving percent increase and percent decrease, and financial literacy problems. In Grade 8, students will create a visual representation to describe relationships between sets of real numbers and order a set of real numbers arising from mathematical and real-world contexts.

In Grade 6, classifying whole numbers, integers, and rational numbers using a visual representation is considered STAAR Supporting Standard 6.2A and is part of the Grade 6 Texas Response to Curriculum Focal Points (TxRCFP): Grade Level Connections. Locating, comparing, and ordering integers with a number line is identified as STAAR Supporting Standard 6.2C, while ordering rational numbers is addressed in STAAR Readiness Standard 6.2D. These standards are subsumed under the Grade 6 Focal Point: Using operations with integers and positive rational numbers to solve problems (TxRCFP). Generating equivalent forms of fractions, decimals, and percents is identified as STAAR Readiness Standard 6.4G, and part of the Grade 6 Focal Point:  Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships (TxRCFP). All of these standards are subsumed under the Grade 6 STAAR Reporting Category 1: Numerical Representations and Relationships. This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning A1, 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 (2000), students should “[connect] negative integers in appropriate ways to informal knowledge derived from everyday experiences, such as below-zero winter temperatures or lost yards on a football field. In the middle grades, students should extend these initial understandings of integers. Positive and negative integers should be seen as useful for noting relative changes or values” (p. 219 – 218). Van De Walle, Bay-Williams, Lovin, and Karp (2013) state, “Middle-school students must understand that any rational number, positive or negative, whole or not whole, can be written as a fraction and as a decimal. So, –8 can be written as the fraction or as the decimal –8 or –8.0…Fluency with equivalent representations is critical and requires much more than teaching an algorithm for moving from one representation to another” (p. 193).

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
Van de Walle, J., Bay-Williams, J., Lovin, L., & Karp, K., (2013). Teaching student-centered mathematics: Developmentally appropriate instruction for grades 6 - 8. 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? Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Rational numbers create a more sophisticated number system where new relationships exist within and between sets and subsets of numbers (counting numbers; whole numbers; integers; rational numbers).
• What representations can be used to visually demonstrate relationships between sets and subsets of numbers?
• How does organizing numbers in sets and subsets aid in understanding the relationships between numbers?
• What relationships exist between sets and subsets of numbers?
• How are the elements in counting (natural) numbers, whole numbers, integers, and rational numbers related?
• How can a number belong to the same set of numbers but not necessarily the same subset of numbers?
• What is the unique relationship between counting (natural) numbers and whole numbers?
• What relationship exists between rational numbers and the other number sets?
• Quantities are compared and ordered to determine magnitude of number and equality or inequality relations (rational numbers).
• Why is it important to identify the unit or attribute being described by numbers before comparing or ordering the numbers?
• How does understanding equivalence aid in the comparison and/or ordering of numbers?
• How can …
• place value
• numeric representations
• concrete representations
• pictorial representations
• number lines
… aid in the comparison and/or ordering of numbers?
• How can the comparison of two numbers be described and represented?
• What is the process for ordering a set of numbers?
• How are quantifying descriptors used to determine the order of a set of numbers?
• Understanding how two quantities vary together (covariation) and can be reasoned up and down in situations involving invariant (constant) relationships builds flexible numeric reasoning in order to make predictions and critical judgements about the relationship (fractions; decimals; percents).
• Fractions, decimals, and percents are modeled and described to develop an understanding of proportional relationships and these relationships are applied to represent equivalence and solve problem situations.
• How can an equivalent …
• fraction be generated when given a decimal or percent?
• decimal be generated when given a fraction or percent?
• percent be generated when given a decimal or fraction?
• What types of models and strategies can be used to generate equivalent forms of fractions, decimals, and percents?
• Why is the ability to model numbers in a variety of ways essential to solving problems in everyday life?
• How can benchmark fractions and percents be used to solve problems?
• Number and Operations
• Compare and Order
• Comparative language
• Comparison symbols
• Number
• Counting (natural) numbers
• Whole numbers
• Integers
• Rational Numbers
• Relationships and Generalizations
• Numerical
• Equivalence
• Representations
• Proportionality
• Fractions and Decimals
• Percents
• Relationships and Generalizations
• Equivalence
• Representations
• Solution Strategies
• 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 negative integers with larger absolute values are greater than negative integers with smaller absolute values.
• Some students may think that a number can only belong to one set (counting [natural] numbers, whole numbers, integers, or rational numbers) rather than understanding that some sets of numbers are nested within another set as a subset.

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 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

• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• 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.
• Order numbers – to arrange a set of numbers based on their numerical value
• Percent – a part of a whole expressed in hundredths
• Place value – the value of a digit as determined by its location in a number such as ones, tens, hundreds, one thousands, ten thousands, etc.
• Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are counting (natural) numbers
• Rational numbers – the set of numbers that can be expressed as a fraction , where a and b are integers and b ≠ 0. The set of rational numbers is denoted by the symbol Q.
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}

Related Vocabulary:

 Ascending Compare Descending Decimal Denominator Equal to Equivalent Fraction Greater than Improper fraction Interval Less than Magnitude Mixed number Number line Numerator Open number line Part Proper fraction Repeating decimal Set of numbers Subset of numbers Tick marks Venn diagram Whole
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.
6.1C

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 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
• 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:
6.2A Classify whole numbers, integers, and rational numbers using a visual representation such as a Venn diagram to describe relationships between sets of numbers.
Supporting Standard

Classify

WHOLE NUMBERS, INTEGERS, AND RATIONAL NUMBERS USING A VISUAL REPRESENTATION

Including, but not limited to:

• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• 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.
• Rational numbers – the set of numbers that can be expressed as a fraction , where a and b are integers and b ≠ 0. The set of rational numbers is denoted by the symbol Q.
• Visual representations of the relationships between sets and subsets of rational numbers

To Describe

RELATIONSHIPS BETWEEN SETS OF NUMBERS

Including, but not limited to:

• All counting (natural) numbers are a subset of whole numbers, integers, and rational numbers.
• All whole numbers are a subset of integers and rational numbers.
• All integers are a subset of rational numbers.
• All counting (natural) numbers, whole numbers, and integers are a subset of rational numbers.
• Not all rational numbers are an integer, whole number, or counting (natural) number.
• Terminating and repeating decimals are rational numbers but not integers, whole numbers, or counting (natural) numbers.

Note(s):

• Prior to Grade 6 counting (natural) numbers, whole numbers, and positive rational numbers were developed.
• Grade 6 introduces classifying whole numbers, integers, and rational numbers using a visual representation such as a Venn diagram to describe relationships between sets of numbers.
• Grade 7 will extend previous knowledge of sets and subsets using a visual representation to describe relationships between sets of rational numbers.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Grade Level Connections (reinforces previous learning and/or provides development for future learning)
• TxCCRS:
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
6.2C Locate, compare, and order integers and rational numbers using a number line.
Supporting Standard

Locate, Compare, Order

INTEGERS AND RATIONAL NUMBERS USING A NUMBER LINE

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.
• Rational numbers – the set of numbers that can be expressed as a fraction , where a and b are integers and b ≠ 0. The set of rational numbers is denoted by the symbol Q.
• Various forms of positive and negative rational numbers
• Integers
• Decimals
• Fractions
• Percents
• Relationship between equivalence of various forms of rational numbers
• All integers and rational numbers can be located as a specified point on a number line.
• Characteristics of a number line
• A number line begins as a line with predetermined intervals (or tick marks) with positions/numbers labeled.
• A minimum of two positions/numbers should be labeled.
• Numbers on a number line represent the distance from zero.
• The distance between the tick marks is counted rather than the tick marks themselves.
• The placement of the labeled positions/numbers on a number line determines the scale of the number line.
• Intervals between position/numbers are proportional.
• When reasoning on a number line, the position of zero may or may not be placed.
• When working with larger numbers, a number line without the constraint of distance from zero allows the ability to “zoom-in” on the relevant section of the number line.
• Number lines extend infinitely in both directions (arrows indicate the number line continues infinitely).
• Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
• Points to the left of a specified point on a horizontal number line are less than points to the right.
• Points to the right of a specified point on a horizontal number line are greater than points to the left.
• Points below a specified point on a vertical number line are less than points above.
• Points above a specified point on a vertical number line are greater than points below.
• Characteristics of an open number line
• An open number line begins as a line with no intervals (or tick marks) and no positions/numbers labeled.
• Numbers/positions are placed on the empty number line only as they are needed.
• When reasoning on an open number line, the position of zero is often not placed.
• When working with larger numbers, an open number line without the constraint of distance from zero allows the ability to “zoom-in” on the relevant section of the number line.
• The placement of the first two numbers on an open number line determines the scale of the number line.
• Once the scale of the number line has been established by the placement of the first two numbers, intervals between additional numbers placed are approximately proportional.
• The differences between numbers are approximated by the distance between the positions on the number line.
• Open number lines extend infinitely in both directions (arrows indicate the number line continues infinitely).
• Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
• Points to the left of a specified point on a horizontal number line are less than points to the right.
• Points to the right of a specified point on a horizontal number line are greater than points to the left.
• Points below a specified point on a vertical number line are less than points above.
• Points above a specified point on a vertical number line are greater than points below.
• Landmark (or anchor) numbers may be placed on the open number line to help locate other numbers.
• Relative magnitude of a number describes the size of a number and its relationship to another number.
• Comparison words and symbols
• Inequality words and symbols
• Greater than (>)
• Less than (<)
• Equality words and symbol
• Equal to (=)
• Quantifying descriptor in mathematical and real-world problem situations (e.g., between two given numbers, greatest/least, ascending/descending, tallest/shortest, warmest/coldest, fastest/slowest, longest/shortest, heaviest/lightest, closest/farthest, oldest/youngest, etc.)

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:
• Using operations with integers and positive rational numbers to solve problems
6.2D Order a set of rational numbers arising from mathematical and real-world contexts.

Order

A SET OF RATIONAL NUMBERS ARISING FROM MATHEMATICAL AND REAL-WORLD CONTEXTS

Including, but not limited to:

• Rational numbers – the set of numbers that can be expressed as a fraction , where a and b are integers and b ≠ 0. The set of rational numbers is denoted by the symbol Q.
• Various forms of positive and negative rational numbers
• Integers
• Decimals
• Fractions
• Percents
• Place value – the value of a digit as determined by its location in a number such as ones, tens, hundreds, one thousands, ten thousands, etc.
• Order numbers – to arrange a set of numbers based on their numerical value
• Number lines (horizontal/vertical)
• Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
• Points to the left of a specified point on a horizontal number line are less than points to the right.
• Points to the right of a specified point on a horizontal number line are greater than points to the left.
• Points below a specified point on a vertical number line are less than points above.
• Points above a specified point on a vertical number line are greater than points below.
• Quantifying descriptor in mathematical and real-world problem situations (e.g., between two given numbers, greatest/least, ascending/descending, tallest/shortest, warmest/coldest, fastest/slowest, longest/shortest, heaviest/lightest, closest/farthest, oldest/youngest, etc.)

Note(s):

• Grade 5 compared and ordered two decimals to the thousandths place and represented the comparisons using the symbols >, <, or =.
• Grade 8 will order a set of real numbers arising from mathematical and real-world contexts.
• 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.1. Compare relative magnitudes of rational and irrational numbers, and understand that numbers can be represented in different ways.
• 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.
6.4 Proportionality. The student applies mathematical process standards to develop an understanding of proportional relationships in problem situations. The student is expected to:
6.4G

Generate equivalent forms of fractions, decimals, and percents using real-world problems, including problems that involve money.

Generate

EQUIVALENT FORMS OF FRACTIONS, DECIMALS, AND PERCENTS USING REAL-WORLD PROBLEMS

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
•  Percent – a part of a whole expressed in hundredths
• Equivalent forms of positive rational numbers in real-world problem situations
• Given a fraction, generate a decimal and percent
• Given a decimal, generate a fraction and percent
• Given a percent, generate a fraction and decimal

Note(s):

• Grade 6 introduces generating equivalent forms of fractions, decimals, and percents using real-world problems, including problems that involve money.
• Grade 6 introduces ordering a set of rational numbers arising from mathematical and real-world contexts.
• Grade 7 will solve problems involving ratios, rates, and percents, including multi-step problems involving percent increase and percent decrease, and financial literacy problems.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• TxCCRS:
• 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.
• 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.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. 