6.1 
Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:


6.1A 
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:
 Using operations with integers and positive rational numbers to solve problems
 Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
 Using expressions and equations to represent relationships in a variety of contexts
 Understanding data representation
 TxCCRS:
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.1. Interpret results of the mathematical problem in terms of the original realworld situation.
 IX.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
 IX.A.2. Connect mathematics to the study of other disciplines.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.
 IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
 IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.

6.1B 
Use a problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process and the reasonableness of the solution.
Process Standard

Use
A PROBLEMSOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEMSOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:
 Problemsolving model
 Analyze given information
 Formulate a plan or strategy
 Determine a solution
 Justify the solution
 Evaluate the problemsolving process and the reasonableness of the solution
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Using operations with integers and positive rational numbers to solve problems
 Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
 Using expressions and equations to represent relationships in a variety of contexts
 Understanding data representation
 TxCCRS:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.1. Use estimation to check for errors and reasonableness of solutions.
 V.A. Statistical Reasoning – Design a study
 V.A.1. Formulate a statistical question, plan an investigation, and collect data.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VII.A.2. Formulate a plan or strategy.
 VII.A.3. Determine a solution.
 VII.A.4. Justify the solution.
 VII.A.5. Evaluate the problemsolving process.
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.2. Evaluate the problemsolving process.

6.1C 
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 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
 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:
 Using operations with integers and positive rational numbers to solve problems
 Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
 Using expressions and equations to represent relationships in a variety of contexts
 Understanding data representation
 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.

6.1D 
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, AND LANGUAGE AS APPROPRIATE
Including, but not limited to:
 Mathematical ideas, reasoning, and their implications
 Multiple representations, as appropriate
 Symbols
 Diagrams
 Language
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Using operations with integers and positive rational numbers to solve problems
 Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
 Using expressions and equations to represent relationships in a variety of contexts
 Understanding data representation
 TxCCRS:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.

6.1E 
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:
 Using operations with integers and positive rational numbers to solve problems
 Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
 Using expressions and equations to represent relationships in a variety of contexts
 Understanding data representation
 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.

6.1F 
Analyze mathematical relationships to connect and communicate mathematical ideas.
Process Standard

Analyze
MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:
 Mathematical relationships
 Connect and communicate mathematical ideas
 Conjectures and generalizations from sets of examples and nonexamples, patterns, etc.
 Current knowledge to new learning
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Using operations with integers and positive rational numbers to solve problems
 Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
 Using expressions and equations to represent relationships in a variety of contexts
 Understanding data representation
 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.

6.1G 
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:
 Using operations with integers and positive rational numbers to solve problems
 Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
 Using expressions and equations to represent relationships in a variety of contexts
 Understanding data representation
 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.

6.2 
Number and operations. The student applies mathematical process standards to represent and use rational numbers in a variety of forms. The student is expected to:


6.2E 
Extend representations for division to include fraction notation such as a/b represents the same number as a ÷ b where b ≠ 0.
Supporting Standard

Note(s):
 Grade Level(s)
 Grade 4 represented a fraction as a sum of fractions , where a and b are whole numbers and b > 0, including when a > b.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Using operations with integers and positive rational numbers to solve problems

6.3 
Number and operations. The student applies mathematical process standards to represent addition, subtraction, multiplication, and division while solving problems and justifying solutions. The student is expected to:


6.3A 
Recognize that dividing by a rational number and multiplying by its reciprocal result in equivalent values.
Supporting Standard

Recognize
THAT DIVIDING BY A RATIONAL NUMBER AND MULTIPLYING BY ITS RECIPROCAL RESULT IN EQUIVALENT VALUES
Including, but not limited to:
 Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are counting (natural) numbers
 Various forms of positive rational numbers
 Counting (natural) numbers
 Decimals
 Fractions
 Percents converted to equivalent decimals or fractions for multiplying or dividing
 Reciprocal – a quantity that is used to multiply by a given quantity which results in the product of one
 Relationship between multiplication and division
 Dividing a number a by a given number b is equivalent to multiplying a by the reciprocal of b.
 Algebraic: a ÷ b = = = a •
 Relationships between equivalent positive rational number representations
Note(s):
 Grade Level(s):
 Grade 3 determined a quotient using the relationship between multiplication and division.
 Grade 7 will add, subtract, multiply, and divide rational numbers fluently.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Using operations with integers and positive rational numbers to solve problems
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.2. Perform computations with rational and irrational numbers.

6.3B 
Determine, with and without computation, whether a quantity is increased or decreased when multiplied by a fraction, including values greater than or less than one.
Supporting Standard

Note(s):
 Grade Level(s):
 Grade 6 introduces determining, with and without computation, whether a quantity is increased or decreased when multiplied by a fraction, including values greater than or less than one.
 Grade 7 will apply and extend previous understandings of operations to solve problems using addition, subtraction, multiplication, and division of rational numbers.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Using operations with integers and positive rational numbers to solve problems
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.2. Perform computations with rational and irrational numbers.
 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.

6.3E 
Multiply and divide positive rational numbers fluently.
Readiness Standard

Multiply, Divide
POSITIVE RATIONAL NUMBERS FLUENTLY
Including, but not limited to:
 Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are counting (natural) numbers
 Various forms of positive rational numbers
 Counting (natural) numbers
 Decimals
 Fractions
 Percents converted to equivalent decimals or fractions for multiplying or dividing fluently
 Fluency – efficient application of procedures with accuracy
 Relationship between dividing by a fraction and multiplying by its reciprocal
 Reciprocal – a quantity that is used to multiply by a given quantity which results in the product of one
Note(s):
 Grade Level(s):
 Grade 5 represented multiplication of decimals with products to the hundredths using objects and pictorial models, including area models.
 Grade 5 solved for products of decimals to the hundredths, including situations involving money, using strategies based on placevalue understandings, properties of operations, and the relationships to the multiplication of whole numbers.
 Grade 5 represented quotients of decimals to the hundredths, up to fourdigit dividends and twodigit whole number divisors, using objects and pictorial models, including area models.
 Grade 5 solved for quotients of decimals to the hundredths, up to fourdigit dividends and twodigit whole number divisors, using strategies and algorithms, including the standard algorithm.
 Grade 5 represented and solved multiplication of a whole number and a fraction that refers to the same whole using objects and pictorial models, including area models.
 Grade 5 represented division of a unit fraction by a whole number and the division of a whole number by a unit fraction such as ÷ 7 and 7 ÷ using objects and pictorial models, including area models.
 Grade 5 divided whole numbers by unit fractions and unit fractions by whole numbers.
 Grade 6 introduces multiplying and dividing positive rational numbers fluently.
 Grade 7 will add, subtract, multiply, and divide rational numbers fluently.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Using operations with integers and positive rational numbers to solve problems
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.2. Perform computations with rational and irrational numbers.
