Hello, Guest!
 TITLE : Unit 04: Division of Whole Numbers SUGGESTED DURATION : 11 days

#### Unit Overview

Introduction
This unit bundles student expectations that address division of whole numbers and using estimation to justify reasonable solutions. According to the Texas Education Agency, mathematical process standards including application, a problem-solving model, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this Unit
In Grade 3, students solved one-step and two-step problems involving division within 100 using various strategies (e.g., objects; pictorial models, including arrays, area models, and equal groups; properties of operations; or recall of facts) and were also able to represent those division situations using strip diagrams and equations. Students also used rounding to the nearest 10 or 100 and compatible numbers to estimate solutions in addition and subtraction situations. In Grade 4, Units 02 – 03, students used rounding to the nearest 10, 100, or 1,000 as well as compatible numbers to make reasonable estimates in addition and subtraction problems. They also used various models and representations (e.g., arrays, area models, properties of operations, strip diagrams, equations) to solve multiplication problems.

During this Unit
Students represent quotients of up to four-digit whole number dividends by one-digit whole number divisors using arrays, area models, and equations. As students experience various division strategies and algorithms, including the standard algorithm, they begin to develop fluency of the operation of division. Students are expected to solve fluently one- and two-step problems involving division, including the interpretation of remainders. This unit also includes representing division problem situations using strip diagrams and equations with a letter standing for the unknown quantity. Students continue to use estimation, rounding and compatible numbers, to find estimates and justify reasonableness for solutions to problems involving division.

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

After this Unit
In Units 05 – 08, students will continue to apply division skills as they use estimation and knowledge of other operations to make connections across the curriculum. In Grade 5, students will solve with proficiency for quotients of up to a four-digit dividend by a two-digit divisor using strategies and the standard algorithm. They will represent and solve multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity. Students will also extend their application of strategies for rounding numbers to decimals to the tenths or hundredths place, and will estimate to determine solutions to mathematical and real-world problems involving addition, subtraction, multiplication, or division.

In Grade 4, solving with fluency one- and two-step problems involving division is identified as STAAR Readiness Standard 4.4H, and representing multi-step problems involving the four operations with whole numbers using strip diagrams and equations with a letter standing for the unknown quantity is addressed in STAAR Readiness Standard 4.5A. STAAR Supporting Standard 4.4E addresses representing the quotient of up to a four-digit whole number divided by a one-digit whole number using arrays, area models, or equations. Using strategies and algorithms, including the standard algorithm, to divide up to a four-digit dividend by a one-digit divisor is identified as STAAR Supporting Standard 4.4F, while rounding to the nearest 10, 100, or 1,000 or using compatible numbers to estimate solutions involving whole numbers is STAAR Supporting Standard 4.4G. These standards are all included in Grade 4 STAAR Reporting Category 2: Computations and Algebraic Relationships, and are further subsumed within the Grade 4 Texas Response to Curriculum Focal Points (TxRCFP): Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems. This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning A2, B1; II. Algebraic Reasoning D1 D2; V. Statistical Reasoning A1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
According to the National Council of Teachers of Mathematics (2011), “An understanding of algorithms and the use of properties of whole number multiplication and division to justify them provides a basis for developing fluency in rational number computations and algebraic manipulations” (p. 51). These researchers further state that, “In many situations, problem solvers need to understand how the divisor, the quotient, and the remainder can be used in the context. Even when they have calculated the quotient and remainder, they need to interpret each of them in the context of the situation and understand how to use each to find an answer” (p. 24). The National Mathematics Advisory Panel (2008) concludes from their findings that “Conceptual understanding of mathematical operations, fluent execution of procedures, and fast access to number combinations together support effective and efficient problem solving” (p. 26).

National Council of Teachers of Mathematics. (2011). Developing essential understanding of multiplication and division grades 3 – 5. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the national mathematics advisory panel. Washington, DC: U.S. Department of Education.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9
Texas Education Agency. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from https://www.texasgateway.org/resource/txrcfp-texas-response-curriculum-focal-points-k-8-mathematics-revised-2013

 Understanding and generalizing operational relationships leads to more sophisticated representations and solution strategies in order to investigate or solve problem situations in everyday life. What relationships exist within and between mathematical operations? How does generalizing operational relationships lead to developing more flexible, efficient representations and/or solution strategies? Why is understanding the problem solving process an essential part of learning and working mathematically? Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Estimation strategies can be used to mentally approximate solutions and determine reasonableness of solutions (division of whole numbers).
• What strategies can be used to estimate solutions to problems?
• What are the similarities and differences between rounding numbers and using compatible numbers to estimate a solution?
• When might an estimated answer be preferable to an exact answer?
• How can an estimation aid in determining the reasonableness of an actual solution?
• When might one estimation strategy be more beneficial than another?
• Recognizing and understanding operational relationships in a variety of problem situations leads to efficient, accurate, and flexible representations and solution strategies (division of whole numbers).
• How does the context of a problem situation affect the representation, operation(s), and/or solution strategy that could be used to solve the problem?
• How can representing a problem situation using …
• arrays
• area models
• an equation(s) with a letter standing for the unknown
• strip diagrams or other pictorial models
… aid in problem solving?
• What patterns and relationships can be found within and between the words, pictorial models, and equations used to represent a problem situation?
• How does understanding …
• relationships within and between operations
• properties of operations
• place value
… aid in determining an efficient strategy or representation to investigate problem situations?
• What strategies can be used to determine …
• the quotient
• an unknown
… in a division situation?
• Why is it important to understand when and how to use standard algorithms?
• Why is it important to be able to perform operations with whole numbers fluently?
• What relationships exist between …
• multiplication and division?
• subtraction and division?
• quotients and remainders?
• When using division to solve a problem situation, why can the order of the dividend and divisor not be changed?
• Operational understandings lead to generalizations that aid in determining the reasonableness of solutions (division of whole numbers).
• When dividing two counting numbers greater than one with the dividend greater than the divisor, why is the quotient always …
• less than the dividend?
• greater than one?
• Number and Operations
• Estimation
• Rounding
• Compatible numbers
• Number
• Counting (natural) numbers
• Whole numbers
• Operations
• Division
• Problem Types
• Properties of Operations
• Relationships and Generalizations
• Operational
• Equivalence
• Solution Strategies and Algorithms
• Algebraic Reasoning
• Representations
• Concrete models
• Pictorial models
• Expressions
• Equations
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that arrays and area models only represent multiplication problems rather than recognizing that one of the dimensions of an array or area model represents the quotient of a division problem.
• Some students may misrepresent a problem situation such as “8 pages of an album with 56 baseball cards arranged equally on the pages results in 7 cards on each page” with the equation “8 ÷ 56 = 7”.
• Some students may think that a division problem situation must contain key words such as “how many are in each group?”
• Some students may attempt to interpret division situations based on the presence of one number that is larger than the other number.
• Some students may think that a division equation written as the traditional algorithm is read from left-to-right.
• Some students may believe that using the standard algorithm to solve division problems is not related to the partial quotients found when using arrays, area models, equations, or properties of mathematics.
• Some students may think that the remainder in a division problem is always part of the answer in a problem situation rather than a quantity that must be interpreted within a certain context.

Underdeveloped Concepts:

• Some students may lack fluency with some of the basic division facts.
• Although some students may recognize the relationship between multiplication and division when using basic facts, they do not apply this knowledge beyond the basic facts.
• Some students may be able to perform a symbolic procedure for division with limited understanding of the division concepts or problem types involved (e.g., 12 ÷ 3 = 4 could represent 12 separated into 3 groups with 4 in each group or 12 separated into groups of 3 creating 4 groups).
• Some students may have limited or no experience with strip diagrams and their relationship to equations that represent problem situations.

#### Unit Vocabulary

• Compatible numbers – numbers that are slightly adjusted to create groups of numbers that are easy to compute mentally
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Dividend – the number that is being divided
• Divisor – the number the dividend is being divided by
• Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• Estimation – reasoning to determine an approximate value
• Fluency – efficient application of procedures with accuracy
• Quotient – the size or measure of each group or the number of groups when the dividend is divided by the divisor
• Rounding – a type of estimation with specific rules for determining the closest value
• Strip diagram – a linear model used to illustrate number relationships
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}

Related Vocabulary:

 About Approximately Area model Around Array Close Consecutive Estimate Halfway Little less Little more Magnitude Multiple Number line Open number line Partial quotients Remainder Standard algorithm
Unit Assessment Items System Resources Other Resources

Show this message:

Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Center if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards

Texas Education Agency – Texas Response to Curriculum Focal Points for K-8 Mathematics Revised 2013

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Grade 4 Mathematics TEKS

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

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

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
• A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
TEKS# SE# TEKS SPECIFICITY
4.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
4.1A Apply mathematics to problems arising in everyday life, society, and the workplace.
Process Standard

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE

Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world situation.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
• IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.
4.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.
Process Standard

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION

Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.A. Statistical Reasoning – Design a study
• V.A.1. Formulate a statistical question, plan an investigation, and collect data.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.2. Formulate a plan or strategy.
• VII.A.3. Determine a solution.
• VII.A.4. Justify the solution.
• VII.A.5. Evaluate the problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving process.
4.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.
Process Standard

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS

Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
• V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
4.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Process Standard

Communicate

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

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
4.1E Create and use representations to organize, record, and communicate mathematical ideas.
Process Standard

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
4.1F Analyze mathematical relationships to connect and communicate mathematical ideas.
Process Standard

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
4.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Process Standard

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION

Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.4. Justify the solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
4.4 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 four-digit whole number divided by a one-digit whole number using arrays, area models, or equations.
Supporting Standard

Represent

THE QUOTIENT OF UP TO A FOUR-DIGIT WHOLE NUMBER DIVIDED BY A ONE-DIGIT 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 four-digit dividends by one-digit divisors
• Quotients may include remainders
• Relationships between multiplication and division to help in solution process
• a ÷ b = c, so b × c = a
• Recognition of division in mathematical and real-world 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 3 solved one-step and two-step 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 four-digit dividend by a two-digit 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 four-digit dividend by a one-digit divisor.
Supporting Standard

Use

STRATEGIES AND ALGORITHMS, INCLUDING THE STANDARD ALGORITHM, TO DIVIDE UP TO A FOUR-DIGIT DIVIDEND BY A ONE-DIGIT 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 four-digit dividends by one-digit divisors
• Quotients may include remainders
• Recognition of division in mathematical and real-world problem situations
• Automatic recall of basic facts
• Relationships between multiplication and division to help in solution process
• a ÷ b = c, so b × c = a
• 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 4 introduces using strategies and algorithms, including the standard algorithm, to divide up to a four-digit dividend by a one-digit divisor.
• Grade 5 will solve with proficiency for quotients of up to a four-digit dividend by a two-digit 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 four-digit dividends by one-digit divisors
• Recognition of operations in mathematical and real-world problem situations
• Multi-step problems
• 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 (pre-determined 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 two-digit numbers are being considered in the problem; round to the nearest 100 if only three-digit numbers are being considered in the problem; round to the nearest 1,000 if only four-digit numbers are being considered; round to the nearest 10 if both two-digit and three-digit numbers are being considered in the problem; round to the nearest 100 if both three-digit and four-digit numbers are being considered; etc.).
• Vocabulary indicating estimation in mathematical and real-world 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 two-digit numbers are being considered in the problem; round to the nearest 100 if only three-digit numbers are being considered in the problem; round to the nearest 1,000 if only four-digit numbers are being considered; round to the nearest 10 if both two-digit and three-digit numbers are being considered in the problem; round to the nearest 100 if both three-digit and four-digit numbers are being considered; etc.).
• Vocabulary indicating estimation in mathematical and real-world 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 four-digit dividends by one-digit divisors
• Recognition of operations in mathematical and real-world problem situations
• Multi-step problems
• 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 real-world 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 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 two-step problems involving multiplication and division, including interpreting remainders.

Solve

WITH FLUENCY ONE- AND TWO-STEP 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 four-digit dividends by one-digit divisors
• Quotients may include remainders
• Remainder dependent upon the mathematical or real-world situation
• Various ways to record remainder
• Ignore the remainder
• Add one to the quotient
• Remainder recorded as a fraction
• One- and two-step problem situations
• One-step problems
• Recognition of division in mathematical and real-world problem situations
• Two-step problems
• Two-step problems must have one-step 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 real-world problem situations
• Equation(s) to reflect solution process

Note(s):

• Grade 4 introduces solving with fluency one- and two-step problems involving multiplication and division, including interpreting remainders.
• Grade 5 will multiply with fluency a three-digit number by a two-digit number using the standard algorithm.
• Grade 5 will solve with proficiency for quotients of up to a four-digit dividend by a two-digit 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 multi-step problems involving the four operations with whole numbers using strip diagrams and equations with a letter standing for the unknown quantity.

Represent

MULTI-STEP 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 four-digit dividends by one-digit 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
• Multi-step problem situations involving the four operations in a variety of problem structures
• Recognition of division in mathematical and real-world 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 3 represented one- and two-step 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 two-step 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 multi-step 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.