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 TITLE : Unit 06: Fractions SUGGESTED DURATION : 18 days

#### Unit Overview

Introduction
This unit bundles student expectations that address decomposing, comparing, adding, and subtracting fractions, developing the relationship between fractions and decimals, and generating equivalent fractions. 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.

Prior to this Unit
In Grade 3, students composed and decomposed a fraction with a numerator greater than zero and less than or equal to b as a sum of parts . They compared two fractions having the same numerator or denominator in problems by reasoning about their sizes and justifying the conclusion using symbols, words, objects, and pictorial models. Grade 3 students explained that two fractions are equivalent if and only if they are both represented by the same point on the number line or represent the same portion of a same size whole for an area model. Students also represented fractions between zero and one (specifically those with denominators 2, 3, 4, 6, and 8) with strip diagrams and number lines, and determined the corresponding fraction given specified points on a number line. In Grade 4 Unit 01, students represented, compared, and ordered decimals, including tenths and hundredths, using concrete and visual models (e.g., number lines, decimal disks, decimal grids, base-10 blocks), and money. Students also determined the corresponding decimal to the tenths or hundredths place of a specified point on a number line.

During this Unit
Students relate their understanding of decimal numbers to fractions that name tenths and hundredths, and represent both types of numbers as distances from zero on a number line. Along with representing fractions (including those that represent values greater than one) as sums of unit fractions, students decompose fractions into sums of fractions with the same denominator using concrete and pictorial models and record the results with symbolic representations. These expectations support understanding as students represent and solve addition and subtraction of fractions with equal denominators using objects and pictorial models that build to the number line and properties of operations. Students use benchmark fractions of 0, , and 1, referring to the same whole, to evaluate the reasonableness of sums and differences of fractions that may or may not have equal denominators. Using a variety of methods to determine equivalence of two fractions underlies students’ abilities to compare two fractions with different numerators and different denominators and represent those comparisons using symbols.

Other considerations: Reference the Mathematics COVID-19 Implementation Tool Grade 4

After this Unit
In Grade 5, students will represent the value of the digit in decimals through the thousandths using expanded notation and numerals. They will also extend their knowledge of fraction operations as they represent and solve addition and subtraction of fractions with unequal denominators referring to the same whole using objects and pictorial models and properties of operations. Students will be expected to add and subtract positive rational numbers fluently. In Grade 6, students will use equivalent fractions, decimals, and percents to show equal parts of the same whole, and will locate, compare, and order integers and rational numbers (positive and negative) using a number line. Students will also extend representations for division to include fraction notation such as represents the same number as a ÷ b where b ≠ 0.

Research
According to the recommendations of the National Mathematics Advisory Panel (2008), “The most important foundational skill not presently developed appears to be proficiency with fractions (including decimals, percent, and negative fractions). The teaching of fractions must be acknowledged as critically important and improved before an increase in student achievement in algebra can be expected” (p. 18). The National Council of Teachers of Mathematics (2010) states that, “Flexible reasoning about the unit is a key component of rational number sense, and it is something that develops over time and through many experiences. The challenge for teachers is to build on students’ intuitive understanding of rational numbers to help them identify and reason about the unit” (p. 74).

National Council of Teachers of Mathematics. (2010). Developing essential understanding of rational numbers grades 3 – 5. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the national mathematics advisory panel. Washington, DC: U.S. Department of Education.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9
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?
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.
• Fractions are an extension of the number system used to communicate and reason about the infinite values that exist between whole values.
• When might fractions be used in real life?
• Why is it important to be able to identify or work with fractional parts of a whole?
• What relationships exist between …
• whole numbers and fractions?
• proper fractions, improper fractions, and mixed numbers?
• decimals and fractions?
• Why is it important to determine the unit or whole when working with fractions?
• How can a set of objects be described as a whole?
• How does changing the size of the whole affect the size or amount of a fractional partition?
• How are names of fractional parts determined?
• How is the …
• denominator
• numerator
… of a fraction determined?
• Why can the denominator not be represented as zero?
• How can the numerator and the denominator of a fraction be described as a multiplicative relationship?
• What are some ways a fraction can be represented?
• Why is it important to be able to recognize and create a variety of representations for a quantity?
• How could representing a fraction …
• using words
• using concrete models
• using numerals and symbols
• in simplified form
• as a distance from zero on a number line
… improve understanding and communicating about the value of a fraction and the equivalence of the representations?
• How can composing or decomposing a …
• whole as a sum of unit fractions
• fraction as a sum of parts
… aid in understanding the value of a fraction and solving problems?
• How does the ability to work with fractions aid in solving problems?
• Estimation strategies can be used to mentally approximate solutions and determine reasonableness of solutions.
• How are benchmark fractions used to estimate solutions to problems involving fractions?
• What are the similarities and differences between estimating solutions involving fractions and solutions involving whole numbers or decimals?
• When might an estimated answer be preferable to an exact answer?
• How can an estimation aid in determining the reasonableness of an actual solution?
• Recognizing and understanding operational relationships in a variety of problem situations leads to efficient, accurate, and flexible representations and solution strategies (addition and subtraction of fractions with equal denominators).
• 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 can representing a problem situation using objects and pictorial models that build to the number line and properties of operations aid in problem solving?
• How does understanding …
• relationships within and between operations
• properties of operations
… aid in determining an efficient strategy or representation to investigate problem situations?
• What strategies can be used to determine …
• the sum
• the difference
• any unknown
… in an addition or subtraction situation involving whole numbers and fractions with like denominators?
• When using addition to solve a problem situation, why can the order of the addends be changed?
• When using subtraction to solve a problem situation, why can the order of the minuend and subtrahend not be changed?
• Number and Operations
• Composition and Decomposition of Numbers
• Estimation
• Benchmark fractions
• Number
• Counting (natural) numbers
• Whole numbers
• Fractions
• Decimals
• Number Representations
• Operations
• Subtraction
• Properties of Operations
• Relationships and Generalizations
• Numerical
• Operational
• Equivalence
• 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.

 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?
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.
• Fractions are an extension of the number system used to communicate and reason about the infinite values that exist between whole values.
• When might fractions be used in real life?
• Why is it important to be able to identify or work with fractional parts of a whole?
• What relationships exist between …
• whole numbers and fractions?
• proper fractions, improper fractions, and mixed numbers?
• decimals and fractions?
• Why is it important to determine the unit or whole when working with fractions?
• How can a set of objects be described as a whole?
• How does changing the size of the whole affect the size or amount of a fractional partition?
• How are names of fractional parts determined?
• How is the …
• denominator
• numerator
… of a fraction determined?
• Why can the denominator not be represented as zero?
• How can the numerator and the denominator of a fraction be described as a multiplicative relationship?
• What are some ways a fraction can be represented?
• Why is it important to be able to recognize and create a variety of representations for a quantity?
• How could representing a fraction …
• using words
• using concrete models
• using numerals and symbols
• in simplified form
… improve understanding and communicating about the value of a fraction and the equivalence of the representations?
• What strategies can be used to …
• generate equivalent fractions?
• determine if two fractions are equivalent?
• compare fractions having different numerators and different denominators?
• What relationship exists between the number of partitions in a whole and the size of the partition?
• Why can equal partitions of identical wholes look different?
• Why can a fraction vary in representation but the value of the fraction stay the same?
• Number
• Compare and Order
• Comparative language
• Comparison symbols
• Number
• Counting (natural) numbers
• Whole numbers
• Fractions
• Decimals
• Number Representations
• Relationships
• Numerical
• Equivalence
• 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 view whole numbers and fractions or decimals numbers as separate and unrelated systems of numbers rather than understanding that fraction and decimal numbers are a natural extension of whole numbers.
• Students may mistakenly think that represents a greater value than because in the numerators, 3 is greater than 2, and in the denominators, 5 is greater than 3.
• Students may think that when adding fractions with equal denominators that they must add the numerators, and then add the denominators.

Underdeveloped Concepts:

• Some students may be proficient at understanding a basic part-whole model for fractions and decimals but need to transition to a more complex understanding of multiple representations, particularly number line models.

#### Unit Vocabulary

• Associative property of addition – if three or more addends are added, they can be grouped in any order, and the sum will remain the same; a + b + c = (a + b) + c = a + (b + c)
• Commutative property of addition – if the order of the addends are changed, the sum will remain the same; a + b = c; therefore, b + a = c
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Decimal number – a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
• Denominator – the part of a fraction written below the fraction bar that tells the total number of equal parts in a whole or set
• Equivalent fractions – fractions that have the same value
• Fraction – a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division.
• Improper fraction – a number in the form where a and b are whole numbers and a > b where b is not equal to zero
• Least common denominator (LCD) – the least common multiple of the denominators of two or more fractions
• Least common multiple (LCM) – the smallest multiple that two or more numbers have in common
• Least common numerator – the least common multiple of the numerators of two or more fractions
• Mixed number – a number that is composed of a whole number and a fraction
• Numerator – the part of a fraction written above the fraction bar that tells the number of fractional parts specified or being considered
• Proper fraction – a number in the form where a and b are whole numbers and a < b where b is not equal to zero
• Strip diagram – a linear model used to illustrate number relationships
• Unit fraction – a fraction in the form representing the quantity formed by one part of a whole that has been partitioned into b equal parts where b is a non-zero whole number
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}

Related Vocabulary:

 Area model Benchmark fractions Difference Equal to (=) Greater than (>) Less than (<) Number line Simplify fractions Sum Tick marks Unit fraction
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 4 Mathematics TEKS

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

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

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
TEKS# SE# TEKS SPECIFICITY
4.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
4.1A 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:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• 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.
4.1B 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:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• 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.
4.1C 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:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• 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.
4.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Process Standard

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• 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.
4.1E 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:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• 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.
4.1F 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:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• 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.
4.1G 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:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• 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.
4.2 Number and operations. The student applies mathematical process standards to represent, compare, and order whole numbers and decimals and understand relationships related to place value. The student is expected to:
4.2G Relate decimals to fractions that name tenths and hundredths.

Relate

DECIMALS TO FRACTIONS THAT NAME TENTHS AND HUNDREDTHS

Including, but not limited to:

• Whole numbers (0 – 1,000,000,000)
• 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}
• Decimals (less than or greater than one to the tenths and hundredths)
• Decimal number – a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
• Fractions (proper, improper, and mixed numbers)
• Fraction – a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division.
• Proper fraction – a number in the form where a and b are whole numbers and a < b where b is not equal to zero
• Improper fraction – a number in the form where a and b are whole numbers and a > b where b is not equal to zero
• Mixed number – a number that is composed of a whole number and a fraction
• Various concrete and visual models
• Number line (horizontal/vertical)
• Number line representing values less than one
• Number line representing values greater than one
• Number line representing values between tick marks
• Area model (tenths and hundredths grids)
• Decimals and fractions of the same whole
• Decimals and fractions less than one
• Decimals and fractions greater than one
• Decimal disks
• Decimals and fractions of the same whole
• Decimals and fractions less than one
• Decimals and fractions greater than one
• Base-10 blocks
• Decimals and fractions to same whole
• Decimals and fractions less than one
• Decimals and fractions greater than one
• Money
• Decimal and fraction relationships of a dollar
• Fraction language

Note(s):

• Grade 4 introduces relating decimals to fractions that name tenths and hundredths.
• Grade 6 will use equivalent fractions, decimals, and percents to show equal parts of the same whole.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding decimals and addition and subtraction of decimals
• 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.
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.2. Interpret the relationships between the different representations of numbers.
4.3 Number and operations. The student applies mathematical process standards to represent and generate fractions to solve problems. The student is expected to:
4.3A Represent a fraction a/b as a sum of fractions 1/b, where a and b are whole numbers and b > 0, including when a > b.
Supporting Standard

Represent

A FRACTION  AS A SUM OF FRACTIONS , WHERE a AND b ARE WHOLE NUMBERS AND b > 0, INCLUDING WHEN a > b

Including, but not limited to:

• Fractions (proper, improper, or mixed numbers with equal denominators)
• Fraction – a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division.
• Proper fraction – a number in the form where a and b are whole numbers and a < b where b is not equal to zero
• Improper fraction – a number in the form where a and b are whole numbers and a > b where b is not equal to zero
• Mixed number – a number that is composed of a whole number and a fraction
• Unit fraction – a fraction in the form representing the quantity formed by one part of a whole that has been partitioned into b equal parts where b is a non-zero whole number
• Relationship between the whole and the part
• Numerator – the part of a fraction written above the fraction bar that tells the number of fractional parts specified or being considered
• Denominator – the part of a fraction written below the fraction bar that tells the total number of equal parts in a whole or set
• Represent an amount less than, equal to, or greater than 1 using a sum of unit fractions
• Multiple Representations
• Concrete models of whole objects
• Linear model
• Fraction bars, customary ruler, linking cube trains, folded paper strips, etc.
• Area models
• Fraction circles or squares, pattern blocks, etc.
• Concrete models of a set of objects
• Pattern blocks, color tiles, counters, etc.
• Pictorial models
• Fraction strips, fraction bar models, number lines, etc.

Note(s):

• Grade 3 composed and decomposed a fraction  with a numerator greater than zero and less than or equal to b as a sum of parts .
• Grade 6 will extend representations for division to include fraction notation such as  represents the same number as a ÷ b where b ≠ 0.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.2. Interpret the relationships between the different representations of numbers.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
4.3B Decompose a fraction in more than one way into a sum of fractions with the same denominator using concrete and pictorial models and recording results with symbolic representations.
Supporting Standard

Decompose

A FRACTION IN MORE THAN ONE WAY INTO A SUM OF FRACTIONS WITH THE SAME DENOMINATOR USING CONCRETE AND PICTORIAL MODELS AND RECORDING RESULTS WITH SYMBOLIC REPRESENTATIONS

Including, but not limited to:

• Fractions (proper, improper, or mixed numbers with equal denominators)
• Fraction – a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division.
• Proper fraction – a number in the form where a and b are whole numbers and a < b where b is not equal to zero
• Improper fraction – a number in the form where a and b are whole numbers and a > b where b is not equal to zero
• Mixed number – a number that is composed of a whole number and a fraction
• Relationship between the whole and the part
• Numerator – the part of a fraction written above the fraction bar that tells the number of fractional parts specified or being considered
• Denominator – the part of a fraction written below the fraction bar that tells the total number of equal parts in a whole or set
• Decompose fractions into smaller fractional parts represented by a sum of unit fractions or multiples of unit fractions with the same denominator
• Concrete models of whole objects
• Linear models
• Fraction bars, customary ruler, linking cube trains, folded paper strips, etc.
• Area models
• Fraction circles or squares, pattern blocks, etc.
• Concrete models of a set of objects
• Pattern blocks, color tiles, counters, etc.
• Pictorial models
• Fraction strips, bar models, number lines, etc.

Note(s):

• Grade 3 composed and decomposed a fraction  with a numerator greater than zero and less than or equal to b as a sum of parts .
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.2. Interpret the relationships between the different representations of numbers.
• 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.
4.3C Determine if two given fractions are equivalent using a variety of methods.
Supporting Standard

Determine

IF TWO GIVEN FRACTIONS ARE EQUIVALENT USING A VARIETY OF METHODS

Including, but not limited to:

• Fractions (proper, improper, or mixed numbers with equal or unequal denominators)
• Fraction – a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division.
• Proper fraction – a number in the form where a and b are whole numbers and a < b where b is not equal to zero
• Improper fraction – a number in the form where a and b are whole numbers and a > b where b is not equal to zero
• Mixed number – a number that is composed of a whole number and a fraction
• Relationship between the whole and the part
• Numerator – the part of a fraction written above the fraction bar that tells the number of fractional parts specified or being considered
• Denominator – the part of a fraction written below the fraction bar that tells the total number of equal parts in a whole or set
• Equivalent fractions – fractions that have the same value
• Comparisons of fractions are only valid when referring to the same size whole.
• Variety of methods to determine if two fractions are equivalent
• Equivalency using a number line
• Equivalency using an area model
• Equivalency using a strip diagram
• Strip diagram – a linear model used to illustrate number relationships
• Equivalency using a numeric approach
• Multiply and/or divide the numerator and denominator by the same non-zero whole number
• Simplify each fraction
• Equivalency using numeric reasoning
• Relationship between numerators and denominators within fractions being compared

Note(s):

• Grade 3 explained that two fractions are equivalent if and only if they are both represented by the same point on the number line or represent the same portion of a same size whole for an area model.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.2. Interpret the relationships between the different representations of numbers.
• 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.
4.3D Compare two fractions with different numerators and different denominators and represent the comparison using the symbols >, =, or <.

Compare

TWO FRACTIONS WITH DIFFERENT NUMERATORS AND DIFFERENT DENOMINATORS

Including, but not limited to:

• Fractions (proper, improper, or mixed with equal or unequal denominators)
• Fraction – a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division.
• Proper fraction – a number in the form where a and b are whole numbers and a < b where b is not equal to zero
• Improper fraction – a number in the form where a and b are whole numbers and a > b where b is not equal to zero
• Mixed number – a number that is composed of a whole number and a fraction
• Relationship between the whole and the part
• Numerator – the part of a fraction written above the fraction bar that tells the number of fractional parts specified or being considered
• Denominator – the part of a fraction written below the fraction bar that tells the total number of equal parts in a whole or set
• Benchmarks
• Comparisons of fractions are only valid when referring to the same size whole.
• Equivalent fractions to determine common denominator or common numerator prior to comparing fractions
• Common denominators
• Common denominators standardize the size of the pieces; therefore, compare the number of pieces (numerator).
• Larger numerator → more equal-size fractional pieces → larger fraction
• Smaller numerator → fewer equal-size fractional pieces → smaller fraction
• Least common multiple (LCM) – the smallest multiple that two or more numbers have in common
• Least common denominator (LCD) – the least common multiple of the denominators of two or more fractions
• Common numerators
• Common numerators standardize the number of pieces; therefore, compare the size of each piece (denominator).
• Larger denominator → smaller fractional piece → smaller fraction
• Smaller denominator → larger fractional piece → larger fraction
• Least common multiple (LCM) – the smallest multiple that two or more numbers have in common
• Least common numerator – the least common multiple of the numerators of two or more fractions
• Compare improper fractions and mixed numbers
• Concrete or pictorial models
• Comparisons of fractions are only valid when referring to the same size whole.

Represent

THE COMPARISON OF TWO FRACTIONS WITH DIFFERENT NUMERATORS AND DIFFERENT DENOMINATORS USING THE SYMBOLS >, =, OR <

Including, but not limited to:

• Whole numbers (0 – 1,000,000,000)
• 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}
• Fractions (proper, improper, or mixed numbers with equal or unequal denominators)
• Fraction – a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division.
• Proper fraction – a number in the form where a and b are whole numbers and a < b where b is not equal to zero
• Improper fraction – a number in the form where a and b are whole numbers and a > b where b is not equal to zero
• Mixed number – a number that is composed of a whole number and a fraction
• Relationship between the whole and the part
• Numerator – the part of a fraction written above the fraction bar that tells the number of fractional parts specified or being considered
• Denominator – the part of a fraction written below the fraction bar that tells the total number of equal parts in a whole or set
• Comparative language and symbols
• Inequality words and comparison symbols
• Greater than (>)
• Less than (<)
• Equality words and symbol
• Equal to (=)

Note(s):

• Grade 3 compared two fractions having the same numerator or denominator in problems by reasoning about their sizes and justifying the conclusion using symbols, words, objects, and pictorial models.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Building foundations for addition and subtraction of fractions
• 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.
4.3E Represent and solve addition and subtraction of fractions with equal denominators using objects and pictorial models that build to the number line and properties of operations.

Represent, Solve

ADDITION AND SUBTRACTION OF FRACTIONS WITH EQUAL DENOMINATORS USING OBJECTS AND PICTORIAL MODELS THAT BUILD TO THE NUMBER LINE AND PROPERTIES OF OPERATIONS

Including, but not limited to:

• Fractions (proper, improper, or mixed numbers with equal denominators)
• Fraction – a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division.
• Proper fraction – a number in the form where a and b are whole numbers and a < b where b is not equal to zero
• Improper fraction – a number in the form where a and b are whole numbers and a > b where b is not equal to zero
• Mixed number – a number that is composed of a whole number and a fraction
• Sums of fractions limited to equal denominators
• Subtraction
• Differences of fractions limited to equal denominators
• Fractional relationships
• Relationship between the whole and the part
• Numerator – the part of a fraction written above the fraction bar that tells the number of fractional parts specified or being considered
• Denominator – the part of a fraction written below the fraction bar that tells the total number of equal parts in a whole or set
• Referring to the same whole
• Fractions are relationships, and the size or the amount of the whole matters
• Common whole is needed when adding or subtracting fractions
• Equivalent fractions to simplify solutions
• Concrete objects and pictorial models for addition of fractions with equal denominators that build to the number line
• Pattern blocks and other shapes (circles, squares, rectangles, etc.)
• Fraction strips and other strip models
• Relationships between concrete objects and pictorial models for addition of fractions with equal denominators, number lines, and properties of operations
• Properties of operations
• Commutative property of addition – if the order of the addends are changed, the sum will remain the same
• a + b = c; therefore, b + a = c
• Associative property of addition – if three or more addends are added, they can be grouped in any order, and the sum will remain the same
• a + b + c = (a + b) + c = a + (b + c)
• Pattern blocks and other shapes (circles, squares, rectangles, etc.)
• Fraction strips and other strip models
• Concrete objects and pictorial models for subtraction of fractions with equal denominators that build to the number line
• Pattern blocks and other shapes (circles, squares, rectangles, etc.)
• Fraction strips and other strip models
• Recognition of addition and subtraction in mathematical and real-world problem situations

Note(s):

• Grade 4 introduces representing and solving addition and subtraction of fractions with equal denominators using objects and pictorial models that build to the number line and properties of operations.
• Grade 5 will represent and solve addition and subtraction of fractions with unequal denominators referring to the same whole using objects and pictorial models and properties of operations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
4.3F Evaluate the reasonableness of sums and differences of fractions using benchmark fractions 0, 1/4, 1/2, 3/4, and 1, referring to the same whole.
Supporting Standard

Evaluate

THE REASONABLENESS OF SUMS AND DIFFERENCES OF FRACTIONS USING BENCHMARK FRACTIONS 0, AND 1, REFERRING TO THE SAME WHOLE

Including, but not limited to:

• Fractions (proper, improper, or mixed numbers)
• Fraction – a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division.
• Proper fraction – a number in the form where a and b are whole numbers and a < b where b is not equal to zero
• Improper fraction – a number in the form where a and b are whole numbers and a > b where b is not equal to zero
• Mixed number – a number that is composed of a whole number and a fraction
• Fractional relationships
• Relationship between the whole and the part
• Numerator – the part of a fraction written above the fraction bar that tells the number of fractional parts specified or being considered
• Denominator – the part of a fraction written below the fraction bar that tells the total number of equal parts in a whole or set
• Referring to the same whole
• Fractions are relationships and the size or the amount of the whole matters
• Common whole is needed when adding or subtracting fractions
• Estimate and evaluate the reasonableness of sums and differences using fraction benchmarks
• Mathematical and real-world problem situations
• With and without models

Note(s):

• Grade 4 evaluates the reasonableness of sums and differences of fractions using benchmark fractions 0, and 1, referring to the same whole.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
4.3G Represent fractions and decimals to the tenths or hundredths as distances from zero on a number line.
Supporting Standard

Represent

FRACTIONS AND DECIMALS TO THE TENTHS OR HUNDREDTHS AS DISTANCES FROM ZERO ON A NUMBER LINE

Including, but not limited to:

• Fractions (proper, improper, and mixed numbers)
• Fraction – a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division.
• Proper fraction – a number in the form where a and b are whole numbers and a < b where b is not equal to zero
• Improper fraction – a number in the form where a and b are whole numbers and a > b where b is not equal to zero
• Mixed number – a number that is composed of a whole number and a fraction
• Decimals (less than or greater than one to the tenths and hundredths)
• Decimal number – a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
• 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.
• Fractions or decimals to the tenths or hundredths as distances from zero on a number line
• Relationship between a fraction represented using a strip diagram to a fraction represented on a number line and the relationship between a decimal represented using a strip diagram to a decimal represented on a number line
• Strip diagram – a linear model used to illustrate number relationships
• Fractions or decimals as distances from zero on a number line greater than 1
• Point on a number line read as the number of whole units from zero and the fractional or decimal amount of the next whole unit
• Number line beginning with a number other than zero
• Distance from zero to first marked increment is assumed even when not visible on the number line.
• Relationship between fractions as distances from zero on a number line to fractional measurements as distances from zero on a customary ruler, yardstick, or measuring tape
• Measuring a specific length using a starting point other than zero on a customary ruler, yardstick, or measuring tape
• Distance from zero to first marked increment not counted
• Length determined by number of whole units and the fractional amount of the next whole unit
• Relationship between fractions and decimals as distances from zero on a number line to fractional and decimal measurements as distances from zero on a metric ruler, meter stick, or measuring tape
• Measuring a specific length using a starting point other than zero on a metric ruler, meter stick, or measuring tape
• Distance from zero to first marked increment not counted
• Length determined by number of whole units and the fractional amount of the next whole unit

Note(s):

• Grade 3 represented fractions greater than zero and less than or equal to one with denominators of 2, 3, 4, 6, and 8 using concrete objects and pictorial models, including strip diagrams and number lines.
• Grade 3 determined the corresponding fraction greater than zero and less than or equal to one with denominators of 2, 3, 4, 6, and 8 given a specified point on a number line.
• Grade 5 will represent the value of the digit in decimals through the thousandths using expanded notation and numerals.
• Grade 6 will identify a number, its opposite, and its absolute value.
• Grade 6 will locate, compare, and order integers and rational numbers using a number line.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding decimals and addition and subtraction of decimals
• 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.
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.2. Interpret the relationships between the different representations of numbers.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.