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 TITLE : Unit 05: All Operations SUGGESTED DURATION : 14 days

#### Unit Overview

Introduction
This unit bundles student expectations that address input-output tables, sequences, expenses, and solving one-, two-, or multistep problems using all four operations. 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 Unit 02, students used addition or subtraction with whole numbers and decimals to tenths and hundredths to solve problems. In Units 03 – 04, students examined multiplication and division separately using strip diagrams and equations to represent problem situations. Students solved one- or two-step problems involving multiplication and division, which included interpreting remainders. In Grade 3, students represented real-world relationships using number pairs in a table and verbal descriptions.

During this Unit
Students apply previously learned concepts to solve one-, two-, or multi-step problems involving addition and subtraction of whole numbers and decimals to the hundredths place, multiplication of whole numbers up to two-digit factors and up to four-digit factors by one-digit factors, and division of whole numbers up to four-digit dividends by one-digit divisors with remainders in appropriate contexts. Students examine financial literacy situations that involve calculating a profit and learn to distinguish between fixed and variable expenses. Representations of these real-life situations that continue to be utilized include strip diagrams and equations with a letter standing for the unknown quantity. This unit further requires students to represent problems using an input-output table and numerical expressions to generate a number pattern that follows a given rule. These identified rules incorporate an algebraic understanding of the relationship of the values in the resulting sequence and their position in the sequence.

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

After this Unit
In Units 06 – 08, students will apply all four operations to problem situations involving measurement situations as well as situations that require analyzing and interpreting data. In Grade 5, students will work with addition and subtraction of decimal numbers to the thousandths place. 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. Grade 5 students will also generate a numerical pattern when given a rule in the form y = ax or y = x + a and graph. They will also be expected to recognize the difference between additive and multiplicative numerical patterns given in a table or graph. Concepts of financial literacy will be extended as students define income tax, payroll tax, sales tax, and property tax.

Research
According to the National Mathematics Advisory Panel (2008), “to prepare students for Algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency, and problem-solving skills” (p. 19). Chapin and Johnson (2000), acknowledge that “when students explore patterns and generalize relationships among numbers, they are developing informal understanding of one of the most important topics in high school and college algebra – functions. A function is a relationship in which two sets are linked by a rule that pairs each element of the first set with exactly one element of the second set. We use functions every day without realizing it ” (p. 205).

Chapin, S & Johnson, A. (2000). Math matters: Understanding the math you teach. Sausalito, CA: Math Solutions Publications.
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)
• Recognizing and understanding operational relationships in a variety of problem situations leads to efficient, accurate, and flexible representations and solution strategies (addition and subtraction of whole numbers and decimals; multiplication and 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 …
• an equation(s) with a letter standing for the unknown
• strip diagrams or other pictorial models
• input-output tables and numerical expressions
… 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?
• words, number patterns, tables, rules, and expressions 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 sum
• the difference
• any unknown
… in an addition or subtraction situation involving whole numbers and decimals?
• What strategies can be used to determine …
• the product
• an unknown factor
… in a multiplication situation?
• 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?
• When adding or subtracting decimal numbers, why is it important to align the place values?
• What relationships exist between …
• multiplication and division?
• subtraction and division?
• quotients and remainders?
• operations with whole numbers and operations with decimals?
• input-output tables, number patterns, numerical expressions, rules, and sequences?
• When using addition to solve a problem situation, why can the order of the addends be changed?
• When using subtraction to solve a problem situation, why can the order of the minuend and subtrahend not be changed?
• When using multiplication to solve a problem situation, why can the order of the factors be changed?
• 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 (addition and subtraction of whole numbers and decimals; multiplication and division of whole numbers).
• When adding two non-zero whole numbers and/or positive decimals, why is the sum always greater than each of the addends?
• When subtracting two non-zero whole numbers and/or positive decimals with the minuend larger than the subtrahend, why is the difference always less than the minuend?
• When multiplying two counting numbers greater than one, why is the product always greater than each of the factors?
• 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
• Number
• Counting (natural) numbers
• Whole numbers
• Decimals
• Operations
• Subtraction
• Division
• Multiplication
• Problem Types
• Properties of Operations
• Relationships and Generalizations
• Operational
• Equivalence
• Solution Strategies and Algorithms
• Algebraic Reasoning
• Patterns and Relationships
• Input-output tables
• 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.

 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? Financial and economic knowledge leads to informed and rational decisions allowing for effective management of financial resources when planning for a lifetime of financial security. Why is financial stability important in everyday life? What economic and financial knowledge is critical for planning for a lifetime of financial security? How can mapping one’s financial future lead to significant short and long-term benefits? How can current financial and economic factors in everyday life impact daily decisions and future opportunities?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Recognizing and understanding operational relationships in a variety of problem situations leads to efficient, accurate, and flexible representations and solution strategies (addition and subtraction of whole numbers and decimals; multiplication and 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 …
• 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
• income, expenses, costs, and profit
… aid in determining an efficient strategy or representation to investigate problem situations?
• What strategies can be used to determine …
• the sum
• the difference
• any unknown
… in an addition or subtraction situation involving whole numbers and decimals?
• What strategies can be used to determine …
• the product
• an unknown factor
… in a multiplication situation?
• 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?
• When adding or subtracting decimal numbers, why is it important to align the place values?
• 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?
• operations with whole numbers and operations with decimals?
• operations and calculating profit?
• When using addition to solve a problem situation, why can the order of the addends be changed?
• When using subtraction to solve a problem situation, why can the order of the minuend and subtrahend not be changed?
• When using multiplication to solve a problem situation, why can the order of the factors be changed?
• 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 (addition and subtraction of whole numbers and decimals; multiplication and division of whole numbers).
• When adding two non-zero whole numbers and/or positive decimals, why is the sum always greater than each of the addends?
• When subtracting two non-zero whole numbers and/or positive decimals with the minuend greater than the subtrahend, why is the difference always less than the minuend?
• When multiplying two counting numbers greater than one, why is the product always greater than each of the factors?
• 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?
• Understanding expenses and profit aids in making informed financial management decisions, which promotes a more secured financial future.
• What are some examples of …
• fixed expenses?
• variable expenses?
• How can an expense be fixed for one person but variable for another person?
• How is profit calculated?
• How is profit affected by …
• expenses (costs)?
• income (revenue)?
• Number and Operations
• Number
• Counting (natural) numbers
• Whole numbers
• Decimals
• Operations
• Subtraction
• Division
• Multiplication
• Problem Types
• Properties of Operations
• Relationships and Generalizations
• Operational
• Equivalence
• Solution Strategies and Algorithms
• Algebraic Reasoning
• Representations
• Concrete models
• Pictorial models
• Expressions
• Equations
• Personal Financial Literacy
• Expenses
• Fixed and variable
• Profit
• 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 misunderstand the distinction between fixed and variable expenses.
• Some students may attempt to solve multistep problems by using only a one-step process.
• Some students may apply a rule for an additive numerical pattern to a multiplicative pattern.
• Some students may misinterpret values in an input-output table by comparing input values to other input values, or by comparing output values to other output values, rather than interpreting the relationship between input values and corresponding output values.

Underdeveloped Concepts:

• Some students may think that decimal numbers should be lined up vertically according to the maximum number of digits in order to use the standard algorithm rather than realizing that they must be lined up according to place values.
• Some students may lack fluency with some of the basic multiplication and 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 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 have learned tricks or shortcuts to remember multiplication facts rather than learning ways to reason about and justify the procedure.
• Some students may be able to perform a symbolic procedure for multiplication or division with limited understanding of the multiplication or division concepts involved.
• Some students may be emergent thinkers when making connections among strip diagrams, equations, and problem situations.

#### Unit Vocabulary

• Additive numerical pattern – a pattern that occurs when a constant non-zero value is added to an input value to determine the output value
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Decimal number – a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
• 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
• Expense – payment for goods and services
• 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)
• Factor – a number multiplied by another number to find a product
• Fixed expenses – expenses that are consistent from month to month
• Fluency – efficient application of procedures with accuracy
• Income – money earned or received
• Input-output table – a table which represents how the application of a rule on a value, input, results in a different value, output
• Multiplicative numerical pattern – a pattern that occurs when a constant non-zero value is multiplied by an input value to determine the output value
• Product – the total when two or more factors are multiplied
• Profit – money that is made in a business after all the costs and expenses are paid
• Quotient – the size or measure of each group or the number of groups when the dividend is divided by the divisor
• Rule – an expression describing the relationship between the input and output values in a pattern or sequence
• Sequence – a list of numbers or a collection of objects in a specific order that follows a particular pattern or rule
• Strip diagram – a linear model used to illustrate number relationships
• Trailing zeros – a sequence of zeros in the decimal part of a number that follow the last non-zero digit, and whether recorded or deleted, does not change the value of the number
• Variable expenses – expenses that vary in cost from month to month
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}

Related Vocabulary:

 Difference Input Output Place value Position Remainder Rule/process Sequential Standard algorithm Sum Unknown Value
Unit Assessment Items System Resources Other Resources

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

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

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

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

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

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

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

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

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

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

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
TEKS# SE# TEKS SPECIFICITY
4.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
4.1A Apply mathematics to problems arising in everyday life, society, and the workplace.
Process Standard

Apply

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

Including, but not limited to:

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

Note(s):

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

Use

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

Including, but not limited to:

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

Note(s):

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

Select

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

Including, but not limited to:

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

Note(s):

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

Communicate

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

Including, but not limited to:

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

Note(s):

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

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

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

Note(s):

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

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

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

Note(s):

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

Display, Explain, Justify

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

Including, but not limited to:

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

Note(s):

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

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 base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
• 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 non-zero digit, and whether recorded or deleted, does not change the value of the number
• Standard algorithm

Note(s):

• Grade 3 solved with fluency one-step and two-step 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 real-world 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 two-step problems involving multiplication and division, including interpreting remainders.

Solve

WITH FLUENCY ONE- AND TWO-STEP 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 two-digit factors by two-digit factors and up to four-digit factors by one-digit 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 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 multiplication and division in mathematical and real-world problem situations
• Two-step problems
• Two-step problems must have one-step 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 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 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}
• Sums of whole numbers
• 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 two-digit factors by two-digit factors and up to four-digit factors by one-digit 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 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 addition, subtraction, multiplication, and/or 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.
4.5B Represent problems using an input-output 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.

Represent

PROBLEMS USING AN INPUT-OUTPUT 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}
• Sums of whole numbers
• 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 two-digit factors by two-digit factors and up to four-digit factors by one-digit 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 four-digit dividends by one-digit 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
• Input-output table – a table which represents how the application of a rule on a value, input, results in a different value, output
• Relationship between input-output 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 non-zero value is added to an input value to determine the output value
• Multiplicative numerical pattern – a pattern that occurs when a constant non-zero value is multiplied by an input value to determine the output value
• Relationship between numerical expressions and rules to create input-output tables representing the relationship between each position in the sequence and the value in the sequence

Note(s):

• Grade 3 represented real-world 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.
4.10 Personal financial literacy. The student applies mathematical process standards to manage one's financial resources effectively for lifetime financial security. The student is expected to:
4.10A Distinguish between fixed and variable expenses.
Supporting Standard

Distinguish

BETWEEN FIXED AND VARIABLE EXPENSES

Including, but not limited to:

• Expense – payment for goods and services
• Fixed expenses – expenses that are consistent from month to month
• Allows for greater planning in spending
• Often associated with necessary spending
• Often reflects needs
• Sometimes reflects wants
• Variable expenses – expenses that vary in cost from month to month
• Allows for greater personal control in spending
• Often associated with discretionary spending
• Often reflects wants
• Sometimes reflects needs
• Relationship between fixed and variable expenses
• Some expenses do not change from month to month and some expenses do change each month
• Some expenses that may be fixed for you may be variable for others depending on the situation

Note(s):

• Grade 3 explained the connection between human capital/labor and income.
• Grade 5 will define income tax, payroll tax, sales tax, and property tax.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Financial Literacy
4.10B Calculate profit in a given situation.
Supporting Standard

Calculate

PROFIT IN A GIVEN SITUATION

Including, but not limited to:

• Whole numbers
• Decimals (less than or greater than one to the tenths and hundredths)
• Sums of whole numbers
• Sums of decimals up to the hundredths
• Subtraction
• Differences of whole numbers
• Differences of decimals with values limited to the hundredths
• Multiplication
• Products of whole numbers up to two-digit factors by two-digit factors and up to four-digit factors by one-digit factors
• Division
• Quotients of whole numbers up to four-digit dividends by one-digit divisors
• Income – money earned or received
• Income in a business also called revenue
• Expense – payment for goods and services
• Expenses in a business also called costs
• Profit – money that is made in a business after all the costs and expenses are paid
• Profit is calculated by subtracting expenses (costs) from income (revenue).
• Income – expenses = profit
• Revenue – costs = profit
• Determining profit from a single source for income and/or expenses
• Determining profit from multiple sources for incomes and/or expenses
• Relationship between income, expenses, and profit
• When income is greater than expenses there is a profit.
• When income is less than expenses, there is no profit or the costs exceed the income.

Note(s):

• Grade 3 described the relationship between the availability or scarcity of resources and how that impacts cost.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Financial Literacy
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• 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.2. Connect mathematics to the study of other disciplines. 