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 – Realworld problem solving
 VII.D.1. Interpret results of the mathematical problem in terms of the original realworld 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, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld 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 problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process and the reasonableness of the solution.
Process Standard

Use
A PROBLEMSOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEMSOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:
 Problemsolving model
 Analyze given information
 Formulate a plan or strategy
 Determine a solution
 Justify the solution
 Evaluate the problemsolving 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 problemsolving process.
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.2. Evaluate the problemsolving 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, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld 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 nonexamples, 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.
Readiness Standard

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 base10 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
 Base10 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 Level(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 nonzero 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.

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 Level(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 nonzero whole number
 Simplify each fraction
 Equivalency using numeric reasoning
 Relationship between numerators and denominators within fractions being compared
Note(s):
 Grade Level(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 <.
Readiness Standard

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 equalsize fractional pieces → larger fraction
 Smaller numerator → fewer equalsize 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.
 Shaded portions of models may or may not be adjacent.
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
Note(s):
 Grade Level(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.
Readiness Standard

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
 Addition
 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 realworld problem situations
Note(s):
 Grade Level(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 realworld problem situations
 With and without models
Note(s):
 Grade Level(s):
 Grade 4 evaluates the reasonableness of sums and differences of fractions using benchmark fractions 0, and 1, referring to the same whole.
 Grade 5 will add and subtract positive rational numbers fluently.
 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 base10 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 “zoomin” 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 Level(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.
