2.1 
Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:


2.1A 
Apply mathematics to problems arising in everyday life, society, and the workplace.

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 proficiency in the use of place value within the base10 numeration system
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 Measuring length
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 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.

2.1B 
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.

Use
A PROBLEMSOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEMSOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:
 Problemsolving model
 Analyze given information
 Formulate a plan or strategy
 Determine a solution
 Justify the solution
 Evaluate the problemsolving process and the reasonableness of the solution
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Developing proficiency in the use of place value within the base10 numeration system
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 Measuring length
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 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.

2.1C 
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.

Select
TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING 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
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Developing proficiency in the use of place value within the base10 numeration system
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 Measuring length
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 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.

2.1D 
Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

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 proficiency in the use of place value within the base10 numeration system
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 Measuring length
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 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.

2.1E 
Create and use representations to organize, record, and communicate mathematical ideas.

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 proficiency in the use of place value within the base10 numeration system
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 Measuring length
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 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.

2.1F 
Analyze mathematical relationships to connect and communicate mathematical ideas.

Analyze
MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:
 Mathematical relationships
 Connect and communicate mathematical ideas
 Conjectures and generalizations from sets of examples and nonexamples, patterns, etc.
 Current knowledge to new learning
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Developing proficiency in the use of place value within the base10 numeration system
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 Measuring length
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 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.

2.1G 
Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

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 proficiency in the use of place value within the base10 numeration system
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 Measuring length
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 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.

2.3 
Number and operations. The student applies mathematical process standards to recognize and represent fractional units and communicates how they are used to name parts of a whole. The student is expected to:


2.3A 
Partition objects into equal parts and name the parts, including halves, fourths, and eighths, using words.

Partition
OBJECTS INTO EQUAL PARTS, INCLUDING HALVES, FOURTHS, AND EIGHTHS
Name
THE EQUAL PARTS OF PARTITIONED OBJECTS, INCLUDING HALVES, FOURTHS, AND EIGHTHS, USING WORDS
Including, but not limited to:
 Fraction – a number that can be used to name part of an object or part of a set of objects
 Equal parts – fractional parts that are the same size in area and may or may not be the same shape
 Partition – separation or division of an object into parts
 Determination of the whole
 One object or shape defined as the whole
 Multiple connected shapes or objects defined as the whole
 A set of separate objects defined as the whole
 Whole divided into two, four, or eight equal parts
 Appropriate oral and written mathematical language to name equal parts, including halves, fourths, and eighths
 Fractions written in word form do not include hyphens.
 Two equal parts
 One half, two halves or one whole
 Four equal parts
 One fourth, two fourths, three fourths, four fourths or one whole
 One quarter, two quarters, three quarters, four quarters or one whole
 Eight equal parts
 One eighth, two eighths, three eighths, four eighths, five eighths, six eighths, seven eighths, eight eighths or one whole
 Hyphen is used only when the fraction written in word form modifies another word
 Each fractional part of an object is the same size in area and the same shape.
 Orientation of the partitioned parts does not affect the size in area or shape of the parts.
 A whole may be partitioned different ways.
 Equal parts of a whole that are the same size in area may not be the same shape.
 Direct comparison of parts to justify equal size in area
 Equal parts of nonidentical wholes may have the same fractional name but may not have equal size in area or the same shape.
 Concrete models of whole objects
 Linear models
 Cuisenaire rods, fraction bars, linking cube trains, folded paper strips, etc.
 Area models
 Fraction circles or squares, pattern blocks, geoboards, etc.
 Concrete models of a set of objects
 Pattern blocks, color tiles, counters, realworld objects, etc.
 Relationship and distinction between ordinal numbers and the number of parts named in a fraction
Note(s):
 Grade Level(s):
 Grade 1 partitioned twodimensional figures into two and four fair shares or equal parts and described the parts using words.
 Grade 2 is not expected to identify the relationship between equivalent fractions (e.g., twofourths is the same as onehalf, etc.).
 Grade 3 will introduce fraction symbols to represent pictorial models of fractions and fractional parts of a set of objects.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 TxCCRS:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.2. Interpret the relationships between the different representations of numbers.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.

2.3B 
Explain that the more fractional parts used to make a whole, the smaller the part; and the fewer the fractional parts, the larger the part.

Explain
THE MORE FRACTIONAL PARTS USED TO MAKE A WHOLE, THE SMALLER THE PART; AND THE FEWER THE FRACTIONAL PARTS, THE LARGER THE PART
Including, but not limited to:
 Fraction – a number that can be used to name part of an object or part of a set of objects
 Inverse relationship between the size of the fractional part and the number of equal parts in the whole when given the same size whole
 The greater the number of parts, the smaller the size of the parts
 The smaller the number of parts, the greater the size of the parts
 Whole divided into two, four, or eight equal parts
 Concrete models of whole objects
 Linear models
 Cuisenaire rods, fraction bars, linking cube trains, folded paper strips, etc.
 Area models
 Fraction circles or squares, pattern blocks, geoboards, etc.
 Realworld problem situations
Note(s):
 Grade Level(s):
 Grade 1 partitioned twodimensional figures into two and four fair shares or equal parts and described the parts using words.
 Grade 3 will introduce fraction symbols to represent pictorial models of fractions and fractional parts of a set of objects.
 Grade 3 will explain that the unit fraction represents the quantity formed by one part of a whole that has been partitioned into b equal parts where b is a nonzero whole number.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.1. Compare relative magnitudes of rational and irrational numbers, and understand that numbers can be represented in different ways.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
 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.

2.3C 
Use concrete models to count fractional parts beyond one whole using words and recognize how many parts it takes to equal one whole.

Use
CONCRETE MODELS TO COUNT FRACTIONAL PARTS BEYOND ONE WHOLE USING WORDS
Recognize
HOW MANY PARTS IT TAKES TO EQUAL ONE WHOLE
Including, but not limited to:
 Fraction – a number that can be used to name part of an object or part of a set of objects
 Relationship between counting whole numbers and counting fractional parts of a whole
 Hierarchical inclusion – concept of nested numbers, meaning each prior number in the counting sequence is included in the set as the set increases (e.g., 18 is 17 increased by 1; 18 decreased by 1 is 17; etc.)
 Appropriate oral and written mathematical language
 Determination of the whole
 One object or shape defined as the whole
 Multiple connected shapes or objects defined as the whole
 A set of separate objects defined as the whole
 Wholes divided into two, four, or eight equal parts
 Recognition of the number of parts that equal one whole
 Two halves equal one whole; four fourths equal one whole; eight eighths equal one whole
 Recognition of the number of parts being considered
 Number of parts being considered within one whole
 Number of parts being considered beyond one whole
 Concrete models of the whole
 Linear models
 Cuisenaire rods, fraction bars, linking cube trains, folded paper strips, etc.
 Area models
 Fraction circles or squares, pattern blocks, geoboards, etc.
 Set models
 Pattern blocks, color tiles, counters, realworld objects, etc.
 Count fractional parts up to one whole using concrete models
 Correct sequence of fractional names
 Two equal parts
 One half, two halves or one whole
 Four equal parts
 One fourth, two fourths, three fourths, four fourths or one whole
 One quarter, two quarters, three quarters, four quarters or one whole
 Eight equal parts
 One eighth, two eighths, three eighths, four eighths, five eighths, six eighths, seven eighths, eight eighths or one whole
 Count fractional parts beyond one whole using concrete models
 Correct sequence of fractional names
 Two equal parts
 One half, two halves, three halves, four halves, five halves, etc.
 One half, one whole, one and one half, two wholes, two and one half, etc.
 Four equal parts
 One fourth, two fourths, three fourths, four fourths, five fourths, six fourths, seven fourths, eight fourths, nine fourths, etc.
 One fourth, two fourths, three fourths, one whole, one and one fourth, one and two fourths, one and three fourths, two wholes, two and one fourth, etc.
 One quarter, two quarters, three quarters, four quarters, five quarters, six quarters, seven quarters, eight quarters, nine quarters, etc.
 One quarter, two quarters, three quarters, one whole, one and one quarter, one and two quarters, one and three quarters, two wholes, two and one quarter, etc.
 Eight equal parts
 One eighth, two eighths, three eighths, four eighths, five eighths, six eighths, seven eighths, eight eighths, nine eighths, ten eighths, eleven eighths, twelve eighths, thirteen eighths, fourteen eighths, fifteen eighths, sixteen eighths, seventeen eighths, etc.
 One eighth, two eighths, three eighths, four eighths, five eighths, six eighths, seven eighths, one whole, one and one eighth, one and two eighths, one and three eighths, one and four eighths, one and five eighths, one and six eighths, one and seven eighths, two wholes, two and one eighth, etc.
Note(s):
 Grade Level(s):
 Grade 3 will represent fractions greater than zero and less than or equal to one with denominators of 2, 3, 4, 6, and 8 using concrete objects and pictorial models, including strip diagrams and number lines.
 Grade 3 will introduce fraction symbols to represent pictorial models of fractions and fractional parts of a set of objects.
 Grade 3 will solve problems involving partitioning an object or a set of objects among two or more recipients using pictorial representations of fractions with denominators of 2, 3, 4, 6, and 8.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 TxCCRS:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.2. Interpret the relationships between the different representations of numbers.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
