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.

4.4 
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:


4.4E 
Represent the quotient of up to a fourdigit whole number divided by a onedigit whole number using arrays, area models, or equations.
Supporting Standard

Represent
THE QUOTIENT OF UP TO A FOURDIGIT WHOLE NUMBER DIVIDED BY A ONEDIGIT WHOLE NUMBER USING ARRAYS, AREA MODELS, OR EQUATIONS
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}
 Division of whole numbers
 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
 Relationships between multiplication and division to help in solution process
 Recognition of division in mathematical and realworld problem situations
 Representations of quotients
 Arrays
 Arrangement of a set of objects in rows and columns
 Area models
 Arrangement of squares/rectangles in a grid format
 Connect the factors as the length and width, and the product as the area
 Equations
 Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
 Dividend ÷ divisor = quotient
 Quotient = dividend ÷ divisor
 Division is not commutative even though multiplication is commutative.
Note(s):
 Grade Level(s):
 Grade 3 solved onestep and twostep problems involving multiplication and division within 100 using strategies based on objects; pictorial models, including arrays, area models, and equal groups; properties of operations; or recall of facts.
 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.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.

4.4F 
Use strategies and algorithms, including the standard algorithm, to divide up to a fourdigit dividend by a onedigit divisor.
Supporting Standard

Use
STRATEGIES AND ALGORITHMS, INCLUDING THE STANDARD ALGORITHM, TO DIVIDE UP TO A FOURDIGIT DIVIDEND BY A ONEDIGIT DIVISOR
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}
 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
 Recognition of division in mathematical and realworld problem situations
 Automatic recall of basic facts
 Relationships between multiplication and division to help in solution process
 Division structures
 Partitive division
 Total amount known
 Number of groups known
 Size or measure of each group unknown
 Quotative division (also known as Measurement division)
 Total amount known
 Size or measure of each group known
 Number of groups unknown
 Relationship between division and multiples of 10
 When the value of the dividend increases by a multiple of 10 and the value of the divisor remains the same, then the value of the quotient is multiplied by the same multiple of 10.
 Strategies and algorithms for division
 Decomposing division problem situations into partial quotients (using numbers that are compatible with the divisor)
 Standard algorithm using the distributive method
 Record steps that relate to the algorithm used including distributing the value in the quotient according to place value.
 Standard algorithm
 Equation(s) to reflect solution process
Note(s):
 Grade Level(s):
 Grade 4 introduces using strategies and algorithms, including the standard algorithm, to divide up to a fourdigit dividend by a onedigit divisor.
 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.

4.4G 
Round to the nearest 10, 100, or 1,000 or use compatible numbers to estimate solutions involving whole numbers.
Supporting Standard

Round
TO THE NEAREST 10, 100, OR 1,000 TO ESTIMATE SOLUTIONS INVOLVING WHOLE NUMBERS
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}
 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
 Recognition of operations in mathematical and realworld problem situations
 Estimation – reasoning to determine an approximate value
 Rounding – a type of estimation with specific rules for determining the closest value
 To the nearest 10; 100; or 1,000
 Number lines
 Proportionally scaled number lines (predetermined intervals)
 Open number line (no marked intervals)
 Relative magnitude of a number describes the size of a number and its relationship to another number.
 Rounding to the nearest 10 on a number line
 Determine the two consecutive multiples of 10 that the number being rounded falls between.
 Begin with the value of the original tens place within the number and then identify the next highest value in the tens place.
 Determine the halfway point between the consecutive multiples of 10.
 Locate the position of the number being rounded on the number line.
 Determine if the number being rounded is before, past, or on the halfway point between the consecutive multiples of 10 on the number line.
 If the number being rounded is before the halfway point on the number line, round to the value of the original tens place.
 If the number being rounded is past the halfway point on the number line, round to the value of the next highest tens place.
 If the number being rounded is on the halfway point on the number line, round to the value of the next highest tens place.
 Rounding to the nearest 100 on a number line
 Determine the two consecutive multiples of 100 that the number being rounded falls between.
 Begin with the value of the original hundreds place within the number and then identify the next highest value in the hundreds place.
 Determine the halfway point between the consecutive multiples of 100.
 Locate the position of the number being rounded on the number line.
 Determine if the number being rounded is before, past, or on the halfway point between the consecutive multiples of 100 on the number line.
 If the number being rounded is before the halfway point on the number line, round to the value of the original hundreds place.
 If the number being rounded is past the halfway point on the number line, round to the value of the next highest hundreds place.
 If the number being rounded is on the halfway point on the number line, round to the value of the next highest hundreds place.
 Rounding to the nearest 1,000 on a number line
 Determine the two consecutive multiples of 1,000 that the number being rounded falls between.
 Begin with the value of the original thousands place within the number and then identify the next highest value in the thousands place.
 Determine the halfway point between the consecutive multiples of 1,000.
 Locate the position of the number being rounded on the number line.
 Determine if the number being rounded is before, past, or on the halfway point between the consecutive multiples of 1,000 on the number line.
 If the number being rounded is before the halfway point on the number line, round to the value of the original thousands place.
 If the number being rounded is past the halfway point on the number line, round to the value of the next highest thousands place.
 If the number being rounded is on the halfway point on the number line, round to the value of the next highest thousands place.
 Round a given number to the closest multiple of 10; 100; or 1,000 on a number line.
 Round a given number to the higher multiple of 10; 100; or 1,000 if it falls exactly halfway between the multiples on a number line.
 Round numbers to a common place then compute.
 If not designated, find the greatest common place value of all numbers in the problem to determine the place value to which you are rounding (e.g., round to the nearest 10 if only twodigit numbers are being considered in the problem; round to the nearest 100 if only threedigit numbers are being considered in the problem; round to the nearest 1,000 if only fourdigit numbers are being considered; round to the nearest 10 if both twodigit and threedigit numbers are being considered in the problem; round to the nearest 100 if both threedigit and fourdigit numbers are being considered; etc.).
 Vocabulary indicating estimation in mathematical and realworld problem situations (e.g., about, approximately, estimate, etc.)
 Vocabulary descriptors of the effects of the adjustment on the estimation compared to the actual solution (e.g., about, close, little more/little less, around, approximately, estimated, etc.)
 Variation of the estimate from the actual solution is dependent upon the magnitude of the adjustment(s) of the actual numbers.
 Rounding numerically based on place value
 Find the place to which you are rounding.
Look at the digit of the next lowest place value, the digit to the right of which you are rounding. If the digit in that place is less than 5, then the digit in the rounding place remains the same. If the digit in that place is greater than or equal to 5, then the digit in the rounding place increases by 1. The digit(s) to the right of the place of which you are rounding is replaced with “0”.
 Round numbers to a common place then compute.
 If not designated, find the greatest common place value of all numbers in the problem to determine the place value to which you are rounding (e.g., round to the nearest 10 if only twodigit numbers are being considered in the problem; round to the nearest 100 if only threedigit numbers are being considered in the problem; round to the nearest 1,000 if only fourdigit numbers are being considered; round to the nearest 10 if both twodigit and threedigit numbers are being considered in the problem; round to the nearest 100 if both threedigit and fourdigit numbers are being considered; etc.).
 Vocabulary indicating estimation in mathematical and realworld problem situations (e.g., about, approximately, estimate, etc.)
 Vocabulary descriptors of the effects of the adjustment on the estimation compared to the actual solution (e.g., about, close, little more/little less, around, approximately, estimated, etc.)Variation of the estimate from the actual solution is dependent upon the magnitude of the adjustment(s) of the actual numbers.
 Variation of the estimate from the actual solution is dependent upon the magnitude of the adjustment(s) of the actual numbers.
Use
COMPATIBLE NUMBERS TO ESTIMATE SOLUTIONS INVOLVING WHOLE NUMBERS
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}
 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
 Recognition of operations in mathematical and realworld problem situations
 Estimation – reasoning to determine an approximate value
 Compatible numbers – numbers that are slightly adjusted to create groups of numbers that are easy to compute mentally
 Determine compatible numbers then compute.
 Vocabulary indicating estimation in mathematical and realworld problem situations (e.g., about, approximately, estimate, etc.)
 Vocabulary descriptors of the effects of the adjustment on the estimation compared to the actual solution (e.g., about, close, little more/little less, around, approximately, estimated, etc.)
 Variation of the estimate from the actual solution is dependent upon the magnitude of the adjustment(s) of the actual numbers.
Note(s):
 Grade Level(s):
 Grade 3 rounded to the nearest 10 or 100 or use compatible numbers to estimate solutions to addition and subtraction problems.
 Grade 5 will round decimals to tenths or hundredths.
 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.B. Numeric Reasoning – Number sense and number concepts
 I.B.1. Use estimation to check for errors and reasonableness of solutions.

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 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
 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 division in mathematical and realworld problem situations
 Twostep problems
 Twostep problems must have onestep in the problem that involves divison; however, the other step in the problem can involve addition and/or subtraction.
 Recognition of the four operations 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 DIVISION 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}
 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 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.
