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.

4.3 
Number and operations. The student applies mathematical process standards to represent and generate fractions to solve problems. The student is expected to:


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.

4.4 
Number and operations. The student applies mathematical process standards to develop and use strategies and methods for whole number computations and decimal sums and differences in order to solve problems with efficiency and accuracy. The student is expected to:


4.4A 
Add and subtract whole numbers and decimals to the hundredths place using the standard algorithm.
Readiness Standard

Add, Subtract
WHOLE NUMBERS AND DECIMALS TO THE HUNDREDTHS PLACE USING THE STANDARD ALGORITHM
Including, but not limited to:
 Whole numbers
 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}
 Addition and subtraction of whole numbers
 Connection between place value and the standard algorithm
 Standard algorithm
 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
 Addition and subtraction of decimals
 Relate addition and subtraction of decimals to the hundredths place using concrete objects and pictorial models to the standard algorithm for adding and subtracting decimals.
 Trailing zeros – a sequence of zeros in the decimal part of a number that follow the last nonzero digit, and whether recorded or deleted, does not change the value of the number
Note(s):
 Grade Level(s):
 Grade 3 solved with fluency onestep and twostep problems involving addition and subtraction within 1,000 using strategies based on place value, properties of operations, and the relationship between addition and subtraction.
 Grade 4 extends adding and subtracting of whole numbers from 1,000 to 1,000,000 and introduces adding and subtracting decimals, including tenths and hundredths.
 Grade 5 will estimate to determine solutions to mathematical and realworld problems involving addition, subtraction, multiplication, or division.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
 Understanding decimals and addition and subtraction of decimals
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.2. Perform computations with rational and irrational numbers.

4.4H 
Solve with fluency one and twostep problems involving multiplication and division, including interpreting remainders.
Readiness Standard

Solve
WITH FLUENCY ONE AND TWOSTEP PROBLEMS INVOLVING MULTIPLICATION AND DIVISION, INCLUDING INTERPRETING REMAINDERS
Including, but not limited to:
 Whole numbers
 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}
 Fluency – efficient application of procedures with accuracy
 Standard algorithms for the four operations
 Automatic recall of basic facts
 Multiplication
 Product – the total when two or more factors are multiplied
 Factor – a number multiplied by another number to find a product
 Products of twodigit factors by twodigit factors and up to fourdigit factors by onedigit factors
 Division
 Quotient – the size or measure of each group or the number of groups when the dividend is divided by the divisor
 Dividend – the number that is being divided
 Divisor – the number the dividend is being divided by
 Quotients up to fourdigit dividends by onedigit divisors
 Quotients may include remainders
 Remainder dependent upon the mathematical or realworld situation
 Various ways to record remainder
 Ignore the remainder
 Add one to the quotient
 Remainder is the answer
 Remainder recorded as a fraction
 One and twostep problem situations
 Onestep problems
 Recognition of multiplication and division in mathematical and realworld problem situations
 Twostep problems
 Twostep problems must have onestep in the problem that involves multiplication and/or divison; however, the other step in the problem can involve addition and/or subtraction.
 Recognition of multiplication and division in mathematical and realworld problem situations
 Equation(s) to reflect solution process
Note(s):
 Grade Level(s):
 Grade 4 introduces solving with fluency one and twostep problems involving multiplication and division, including interpreting remainders.
 Grade 5 will multiply with fluency a threedigit number by a twodigit number using the standard algorithm.
 Grade 5 will solve with proficiency for quotients of up to a fourdigit dividend by a twodigit divisor using strategies and the standard algorithm.
 Various mathematical process standards will be applied to this student expectation as appropriate
 TxRCFP:
 Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.2. Perform computations with rational and irrational numbers.

4.5 
Algebraic reasoning. The student applies mathematical process standards to develop concepts of expressions and equations. The student is expected to:


4.5A 
Represent multistep problems involving the four operations with whole numbers using strip diagrams and equations with a letter standing for the unknown quantity.
Readiness Standard

Represent
MULTISTEP PROBLEMS INVOLVING THE FOUR OPERATIONS WITH WHOLE NUMBERS USING STRIP DIAGRAMS AND EQUATIONS WITH A LETTER STANDING FOR THE UNKNOWN QUANTITY
Including, but not limited to:
 Whole numbers
 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}
 Addition
 Subtraction
 Differences of whole numbers
 Multiplication
 Product – the total when two or more factors are multiplied
 Factor – a number multiplied by another number to find a product
 Products of whole numbers up to twodigit factors by twodigit factors and up to fourdigit factors by onedigit factors
 Division
 Quotient – the size or measure of each group or the number of groups when the dividend is divided by the divisor
 Dividend – the number that is being divided
 Divisor – the number the dividend is being divided by
 Quotients of whole numbers up to fourdigit dividends by onedigit divisors
 Quotients may include remainders
 Representations of an unknown quantity in an equation
 Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
 Any single letter to represent the unknown quantity (e.g., 24 – 8 = y, etc.)
 Equal sign at beginning or end and unknown in any position
 Multistep problem situations involving the four operations in a variety of problem structures
 Recognition of addition, subtraction, multiplication, and/or division in mathematical and realworld problem situations
 Representation of problem situations with strip diagrams and equations with a letter standing for the unknown
 Strip diagram – a linear model used to illustrate number relationships
 Relationship between quantities represented and problem situation
Note(s):
 Grade Level(s):
 Grade 3 represented one and twostep problems involving addition and subtraction of whole numbers to 1,000 using pictorial models, number lines, and equations.
 Grade 3 represented and solved one and twostep multiplication and division problems within 100 using arrays, strip diagrams, and equations.
 Grade 3 determined the unknown whole number in a multiplication or division equation relating three whole numbers when the unknown is either a missing factor or product.
 Grade 5 will represent and solve multistep problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
 TxCCRS:
 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.B. Communication and Representation – Interpretation of mathematical work
 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.

4.5B 
Represent problems using an inputoutput table and numerical expressions to generate a number pattern that follows a given rule representing the relationship of the values in the resulting sequence and their position in the sequence.
Readiness Standard

Represent
PROBLEMS USING AN INPUTOUTPUT TABLE AND NUMERICAL EXPRESSIONS TO GENERATE A NUMBER PATTERN THAT FOLLOWS A GIVEN RULE REPRESENTING THE RELATIONSHIP OF THE VALUES IN THE RESULTING SEQUENCE AND THEIR POSITION IN THE SEQUENCE
Including, but not limited to:
 Whole numbers
 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}
 Addition
 Subtraction
 Differences of whole numbers
 Multiplication
 Product – the total when two or more factors are multiplied
 Factor – a number multiplied by another number to find a product
 Products of whole numbers up to twodigit factors by twodigit factors and up to fourdigit factors by onedigit factors
 Division
 Quotient – the size or measure of each group or the number of groups when the dividend is divided by the divisor
 Dividend – the number that is being divided
 Divisor – the number the dividend is being divided by
 Quotients of whole numbers up to fourdigit dividends by onedigit divisors
 Data sets of whole numbers
 Sets may or may not begin with 1
 Sets may or may not be listed in sequential order
 Sequence – a list of numbers or a collection of objects in a specific order that follows a particular pattern or rule
 Expression – a mathematical phrase, with no equal sign or comparison symbol, that may contain a number(s), an unknown(s), and/or an operator(s)
 Rule – an expression describing the relationship between the input and output values in a pattern or sequence
 Various representations of problem situations
 Inputoutput table – a table which represents how the application of a rule on a value, input, results in a different value, output
 Relationship between inputoutput tables and number patterns
 When the input is the position in the sequence, then the output is the value in the sequence.
 When the input is the value in the sequence, then the output is the position in the sequence.
 Relationship between values in a number pattern
 Additive numerical pattern – a pattern that occurs when a constant nonzero value is added to an input value to determine the output value
 Multiplicative numerical pattern – a pattern that occurs when a constant nonzero value is multiplied by an input value to determine the output value
 Relationship between numerical expressions and rules to create inputoutput tables representing the relationship between each position in the sequence and the value in the sequence
Note(s):
 Grade Level(s):
 Grade 3 represented realworld relationships using number pairs in a table and verbal descriptions.
 Grade 5 will generate a numerical pattern when given a rule in the form y = ax or y = x + a and graph.
 Grade 5 will recognize the difference between additive and multiplicative numerical patterns given in a table or graph.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
 TxCCRS:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 V.B. Statistical Reasoning – Describe data
 V.B.4. Describe patterns and departure from patterns in the study of data.
 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.

4.5D 
Solve problems related to perimeter and area of rectangles where dimensions are whole numbers.
Readiness Standard

Solve
PROBLEMS RELATED TO PERIMETER AND AREA OF RECTANGLES WHERE DIMENSIONS ARE WHOLE NUMBERS
Including, but not limited to:
 Rectangle
 4 sides
 4 vertices
 Opposite sides congruent
 2 pairs of parallel sides
 4 pairs of perpendicular sides
 4 right angles
 Square (a special type of rectangle)
 4 sides
 4 vertices
 All sides congruent
 2 pairs of parallel sides
 4 pairs of perpendicular sides
 4 right angles
 Perimeter – a linear measurement of the distance around the outer edge of a figure
 Perimeter is a onedimensional linear measure.
 Whole number side lengths
 Recognition of perimeter embedded in mathematical and realworld problem situations
 Formulas for perimeter from STAAR Grade 4 Mathematics Reference Materials
 Square
 P = 4s, where s represents the side length of the square
 Rectangle
 P = l + w + l + w or P = 2l + 2w, where l represents the length of the rectangle and w represents the width of the rectangle
 Determine perimeter when given side lengths with or without models
 Determine perimeter by measuring to determine side lengths
 Ruler, STAAR Grade 4 Mathematics Reference Materials ruler, yardstick, meter stick, measuring tape, etc.
 Determine missing side length when given perimeter and remaining side length
 Perimeter of composite figures
 Area – the measurement attribute that describes the number of square units a figure or region covers
 Area is a twodimensional square unit measure.
 Whole number side lengths
 Recognition of area embedded in mathematical and realworld problem situations
 Formulas for area from STAAR Grade 4 Mathematics Reference Materials
 Square
 A = s × s, where s represents the side length of the square
 Rectangle
 A = l × w, where l represents the length of the rectangle and w represents the width of the rectangle
 Determine area when given side lengths with and without models
 Determine area by measuring to determine side lengths
 Ruler, STAAR Grade 4 Mathematics Reference Materials ruler, yardstick, meter stick, measuring tape, etc.
 Determine missing side length when given area and remaining side length
 Area of composite figures
 Multiple ways to decompose a composite figure to determine perimeter and/or area
Note(s):
 Grade Level(s):
 Grade 4 introduces solving problems related to perimeter and area of rectangles where dimensions are whole numbers.
 Grade 5 will represent and solve problems related to perimeter and/or area and related to volume.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.2. Perform computations with rational and irrational numbers.
 III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
 III.D.1. Find the perimeter and area of twodimensional figures.

4.8 
Geometry and measurement. The student applies mathematical process standards to select appropriate customary and metric units, strategies, and tools to solve problems involving measurement. The student is expected to:


4.8C 
Solve problems that deal with measurements of length, intervals of time, liquid volumes, mass, and money using addition, subtraction, multiplication, or division as appropriate.
Readiness Standard

Solve
PROBLEMS THAT DEAL WITH MEASUREMENTS OF LENGTH, INTERVALS OF TIME, LIQUID VOLUMES, MASS, AND MONEY USING ADDITION, SUBTRACTION, MULTIPLICATION, OR DIVISION AS APPROPRIATE
Including, but not limited to:
 Whole numbers (0 – 1,000,000,000)
 Products of twodigit factors by twodigit factors and up to fourdigit factors by onedigit factors
 Quotients up to fourdigit dividends by onedigit divisors
 Decimals (greater than one and less than one)
 Addition and subtraction of money amounts up to hundredths
 Conversions limited to multiples of halves (e.g., 0.5, 1.5, 4.5, etc.)
 Determined using reasoning that half of any value is that value divided by 2
 Fractions (proper, improper, and mixed numbers)
 Addition and subtraction of fractions with like denominators
 Conversions limited to multiples of halves (e.g., etc.)
 Determined using reasoning that half of any value is that value divided by 2
 Typically used customary and metric units
 Customary
 Length: miles, yards, feet, inches
 Volume (liquid volume) and capacity: gallons, quarts, pints, cups, fluid ounces
 Weight: tons, pounds, ounces
 Metric
 Length: kilometer, meter, centimeters, millimeters
 Volume (liquid volume) and capacity: kiloliter, liter, milliliter
 Mass: kilogram, gram, milligram
 Based on prefixes attached to base unit
 Base units include meter for length, liter for volume and capacity, and gram for weight and mass.
 Kilo: one thousand base units
 Deci: onetenth of a base unit
 Centi: onehundredth of a base unit
 Milli: onethousandth of a base unit
 Typically used measurement tools
 Customary
 Length: rulers, yardsticks, measuring tapes
 Volume (liquid volume) and capacity: measuring cups, measuring containers or jars
 Metric
 Length: rulers, meter sticks, measuring tapes
 Volume (liquid volume) and capacity: beakers, graduated cylinders, eye droppers, measuring containers or jars
 Mass: pan balances, triple beam balances
 Problem situations that deal with measurements of length
 Addition, subtraction, multiplication, and/or division of measurements of length with or without conversion
 May or may not include using measuring tools to determine length
 Problem situations that deal with intervals of time (clocks: hours, minutes, seconds)
 Addition and subtraction of time intervals in minutes
 Such as a 1 hour and 45minute event minus a 20minute event equals 1 hour 25 minutes
 Time intervals given
 Pictorial models and tools
 Measurement conversion tables
 Analog clock with gears, digital clock, stop watch, number line, etc.
 Time conversions
 1 hour = 60 minutes; 1 minute = 60 seconds
 Fractional values of time limited to multiples of halves
 Elapsed time
 Finding the end time
 Finding the start time
 Finding the duration
 Problem situations that deal with intervals of time (calendar: years, months, weeks, days)
 Time conversions
 1 year = 12 months; 1 year = 52 weeks; 1 week = 7 days; 1 day = 24 hours
 Fractional values of time limited to multiples of halves
 Problem situations that deal with measurements of volume (liquid volume) and capacity
 Addition, subtraction, multiplication, and/or division of measurements of volume (liquid volume) and capacity with or without conversion
 May or may not include using measuring tools to determine volume (liquid volume) and capacity
 Problem situations that deal with measurements of mass
 Addition, subtraction, multiplication, and/or division of measurements of mass with or without conversion
 May or may not include using measuring tools to determine mass
 Problem situations that deal with money
 Addition and subtraction may include whole number or decimal amounts
 Multiplication and division limited to amounts expressed as cents or dollars with no decimal values
 Comparison of money amounts
 Making change
 Range of dollar amounts
Note(s):
 Grade Level(s):
 Grade 3 determined solutions to problems involving addition and subtraction of time intervals in minutes using pictorial models or tools such as a 15minute event plus a 30minute event equals 45 minutes.
 Grade 4 introduces solving problems that deal with measurements of length, intervals of time, liquid volumes, mass, and money using addition, subtraction, multiplication, or division as appropriate.
 Grade 5 will solve problems by calculating conversions within a measurement system, customary or metric.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.2. Perform computations with rational and irrational numbers.
 III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
 III.D.3. Determine indirect measurements of geometric figures using a variety of methods.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.3. Determine a solution.

4.9 
Data analysis. The student applies mathematical process standards to solve problems by collecting, organizing, displaying, and interpreting data. The student is expected to:


4.9A 
Represent data on a frequency table, dot plot, or stemandleaf plot marked with whole numbers and fractions.
Readiness Standard

Represent
DATA ON A FREQUENCY TABLE, DOT PLOT, OR STEMANDLEAF PLOT MARKED WITH WHOLE NUMBERS AND FRACTIONS
Including, but not limited to:
 Graph – a visual representation of the relationships between data collected
 Organization of data used to interpret data, draw conclusions, and make comparisons
 Data – information that is collected about people, events, or objects
 Categorical data – data that represents the attributes of a group of people, events, or objects
 May include numbers or ranges of numbers
 Numerical data – data that represents values or observations that can be measured and placed in ascending or descending order
 Can be counted or measured.
 Limitations
 Whole numbers
 Fractions (proper, improper, and mixed numbers)
 Data representations
 Frequency table – a table to organize data that lists categories and the frequency (number of times) that each category occurs
 Characteristics of a frequency table
 Titles and column headers
 Title represents the purpose of collected data
 Column headers clarify the meaning of the data represented in the table
 Representation of categorical or numerical data
 Table format
 Each category label listed in a row of the table
 Tally marks used to record the frequency of each category
 Numbers used to represent the count of tally marks in each category
 Every piece of data represented using a onetoone correspondence
 Value of the data in each category
 Determined by the count of tally marks in that category
 Represents the frequency for that category
 Dot plot – a graphical representation to organize small sets of data that uses dots (or Xs) and an axis to show the frequency (number of times) that each category or number occurs
 Characteristics of a dot plot
 Titles, subtitles, and labels
 Title represents the purpose of collected data
 Subtitle clarifies the meaning of categories or number line
 Labels identify each category or numerical increment below the line
 Representation of categorical or numerical data
 Dots (or Xs)
 Placed in a horizontal or vertical linear arrangement
 Vertical graph beginning at the bottom and progressing up above the line
 Horizontal graph beginning at the left and progressing to the right of the line
 Spaced approximately equal distances apart within each category
 Axis
 Categorical data represented by a line segment labeled with categories
 Numerical data represented by a number line labeled with proportional increments
 Every piece of data represented using a onetoone or scaled correspondence, as indicated by the key
 Dots (or Xs) generally represent one count
 May represent multiple counts if indicated with a key
 Value of the data in each category
 Determined by the number of dots (or Xs) or total value of dots (or Xs), as indicated by the key if given
 Represents the frequency for that category
 Density of the dots relates to the frequency distribution of the data
 Stemandleaf plot – a graphical representation used to analyze and compare groups or clusters of numerical data by separating the digits in numerical values based on place value. The left digit(s) of the data form the stems and the remaining digit(s) or fraction form the leaves that correspond with each stem, as designated by a key.
 Characteristics of a stemandleaf plot
 Titles and column headers
 Title represents the purpose of collected data
 Column headers indicate stems and leaves
 Representation of numerical data
 Vertical line, such as in a Tchart, separates stems from their corresponding leaves
 Stems listed to the left of the vertical line with their corresponding leaves listed in a row to the right of the vertical line
 Determination of place value(s) that represents stems versus place value(s) that represents leaves is dependent upon how to best display the distribution of the entire data set and then indicated by a key
 Left digit(s) of the data forms the stems and remaining digit(s) or fraction forms the leaves that correspond with each stem, as indicated by the key
 Every piece of data represented using a onetoone correspondence, including repeated values
 Stem represents one or more pieces of data in the set
 Leaf represents only one piece of data in the set
 Leaves provide frequency counts for the range of numbers included in that row of the stemandleaf plot
 Density of the leaves relates to the frequency distribution of the data
 Connection between graphs representing the same data
 Dot plot to stemandleaf plot
 Stemandleaf plot to dot plot
 Same data represented using a frequency table, dot plot, or stemandleaf plot
Note(s):
 Grade Level(s):
 Grade 1 represented data with picture and bartype graphs.
 Grade 2 represented data with pictographs and bar graphs with intervals of one.
 Grade 3 summarized a data set with multiple categories using a frequency table, dot plot, pictograph, or bar graph with scaled intervals.
 Grade 4 introduces representing data on a stemandleaf plot.
 Grade 5 will represent categorical data with bar graphs or frequency tables and numerical data, including data sets of measurements in fractions or decimals, with dot plots or stemandleaf plots.
 Grade 6 will represent numeric data graphically, including dot plots, stemandleaf plots, histograms, and box plots.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
 Understanding decimals and addition and subtraction of decimals
 Building foundations for addition and subtraction of fractions
 TxCCRS:
 V.B. Statistical Reasoning – Describe data
 V.B.2. Construct appropriate visual representations of data.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.

4.9B 
Solve one and twostep problems using data in whole number, decimal, and fraction form in a frequency table, dot plot, or stemandleaf plot.
Supporting Standard

Solve
ONE AND TWOSTEP PROBLEMS USING DATA IN WHOLE NUMBER, DECIMAL, AND FRACTION FORM IN A FREQUENCY TABLE, DOT PLOT, OR STEMANDLEAF PLOT
Including, but not limited to:
 Graph – a visual representation of the relationships between data collected
 Organization of data used to interpret data, draw conclusions, and make comparisons
 Data – information that is collected about people, events, or objects
 Categorical data – data that represents the attributes of a group of people, events, or objects
 Numerical data – data that represents values or observations that can be measured and placed in ascending or descending order
 Limitations
 One or twostep problems
 Addition
 Sums of whole numbers
 Sums of decimals up to the hundredths
 Sums of fractions limited to equal denominators
 Subtraction
 Differences of whole numbers
 Differences of decimals with values limited to the hundredths
 Differences of fractions limited to equal denominators
 Multiplication
 Products of whole numbers up to twodigit factors by twodigit factors and up to fourdigit factors by onedigit factors
 Division
 Quotients of whole numbers up to fourdigit dividends by onedigit divisors
 Data Representations
 Frequency table – a table to organize data that lists categories and the frequency (number of times) that each category occurs
 Dot plot – a graphical representation to organize small sets of data that uses dots (or Xs) and an axis to show the frequency (number of times) that each category or number occurs
 Stemandleaf plot – a graphical representation used to analyze and compare groups or clusters of numerical data by separating the digits in numerical values based on place value. The left digit(s) of the data form the stems and the remaining digit(s) or fraction form the leaves that correspond with each stem, as designated by a key.
 Solve problems using data represented in frequency tables, dot plots, or stemandleaf plots
Note(s):
 Grade Level(s):
 Grade 3 solved one and twostep problems using categorical data represented with a frequency table, dot plot, pictograph, or bar graph with scaled intervals.
 Grade 5 will solve one and twostep problems using data from a frequency table, dot plot, bar graph, stemandleaf plot, or scatterplot.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
 Understanding decimals and addition and subtraction of decimals
 Building foundations for addition and subtraction of fractions
 TxCCRS:
 V.B. Statistical Reasoning – Describe data
 V.B.3. Compute and describe the study data with measures of center and basic notions of spread.
