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 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 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.

2.10 
Data analysis. The student applies mathematical process standards to organize data to make it useful for interpreting information and solving problems. The student is expected to:


2.10A 
Explain that the length of a bar in a bar graph or the number of pictures in a pictograph represents the number of data points for a given category.

Explain
THAT THE LENGTH OF A BAR IN A BAR GRAPH OR THE NUMBER OF PICTURES IN A PICTOGRAPH REPRESENTS THE NUMBER OF DATA POINTS FOR A GIVEN CATEGORY
Including, but not limited to:
 Graph – a visual representation of the relationships between data collected
 Organization of data used to interpret data, draw conclusions, and make comparisons
 Limitations
 Up to four categories
 Intervals limited to 1, 2, 5, or 10
 Data representations
 Bar graph – a graphical representation to organize data that uses solid bars that do not touch each other and a scaled axis to show the frequency (number of times) that each category occurs
 Length of the bar in a bar graph represents the number of data points for a given category
 Length of the bar represents the distance from zero on the axis scale
 Axis represented as a number line with scaled intervals of one or more units proportionally displayed
 Value of the data represented by the bar is determined by reading the number on the scaled axis associated with the length of the bar.
 Pictograph – a graphical representation to organize data that uses a picture or symbol, where each picture or symbol may represent one or more than one unit of data, to show the frequency (number of times) that each category occurs
 Number of pictures or symbols in pictograph represents the number of data points for a given category
 Key identifies the value of each picture or symbol
 Value of each picture or symbol may be one or more
 Partial symbols represent the fractional value of the whole picture or symbol
 Value of the data represented by pictures is determined by the total value of pictures and partialpictures or symbols, as indicated by the key.
Note(s):
 Grade Level(s):
 Grade 1 represented data to create picture and bartype graphs.
 Grade 2 introduces bar graphs and pictographs.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing proficiency in the use of place value within the base10 numeration system
 TxCCRS:
 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.

2.10B 
Organize a collection of data with up to four categories using pictographs and bar graphs with intervals of one or more.

Organize
A COLLECTION OF DATA WITH UP TO FOUR CATEGORIES USING PICTOGRAPHS AND BAR GRAPHS WITH INTERVALS OF ONE OR MORE
Including, but not limited to:
 Graph – a visual representation of the relationships between data collected
 Organization of data used to interpret data, draw conclusions, and make comparisons
 Data – information that is collected about people, events, or objects
 Categorical data – data that represents the attributes of a group of people, events, or objects
 May include numbers or ranges of numbers
 Limitations
 Up to four categories
 Intervals limited to 1, 2, 5, or 10
 Data representations
 Pictograph – a graphical representation to organize data that uses a picture or symbol, where each picture or symbol may represent one or more than one unit of data, to show the frequency (number of times) that each category occurs
 Characteristics of a pictograph
 Titles, subtitles, and labels
 Title represents the purpose of collected data
 Subtitle clarifies the meaning of categories
 Labels identify each category below the line
 Representation of categorical data
 Pictures or symbols
 Placed in a horizontal or vertical linear arrangement
 Vertical graph beginning at the bottom and progressing up above the line
 Horizontal graph beginning at the left and progressing to the right of the line
 Spaced approximately equal distances apart within each category
 One picture or symbol used to represent all categories
 Partial picture or symbol represents the fractional value of the whole picture or symbol
 Key
 Identifies the value of each picture or symbol
 Every piece of data represented using a onetoone or scaled correspondence, as indicated by the key
 Value of the data in each category
 Determined by the total value of pictures and partialpictures or symbols, as indicated by the key
 Represents the frequency for that category
 Bar graph – a graphical representation to organize data that uses solid bars that do not touch each other and a scaled axis to show the frequency (number of times) that each category occurs
 Characteristics of a bar graph
 Titles, subtitles, and labels
 Title represents the purpose of collected data
 Subtitles clarify the meaning of data represented on each axis
 Labels identify each category
 Representation of categorical data
 Bars
 Placed in a horizontal or vertical linear arrangement to represent data
 Solid bars that are equal in width
 Independent bars that do not touch
 Length of the bar represents the distance from zero on the axis scale
 Axis
 Represented as a number line
 Scale intervals proportionally displayed
 Intervals of one or more units
 Every piece of data represented using a onetoone or scaled correspondence as indicated by the intervals on the axis
 Value of the data represented by the bar
 Determined by reading the number on the scaled axis associated with the length of the bar
 Represents the frequency for that category
 Connection between graphs representing the same data
 Picture graph to pictograph
 Bartype graph to bar graph
 Pictograph to bar graph
 Bar graph to pictograph
 Same data represented using a pictograph and a bar graph
Note(s):
 Grade Level(s):
 Grade 1 used data to create picture and bartype graphs.
 Graph 3 will summarize a data set with multiple categories using a frequency table, dot plot, pictograph, or bar graph with scaled intervals.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing proficiency in the use of place value within the base10 numeration system
 TxCCRS:
 V.B. Statistical Reasoning – Describe data
 V.B.2. Construct appropriate visual representations of data.

2.10C 
Write and solve onestep word problems involving addition or subtraction using data represented within pictographs and bar graphs with intervals of one.

Write, Solve
ONESTEP WORD PROBLEMS INVOLVING ADDITION OR SUBTRACTION USING DATA REPRESENTED WITHIN PICTOGRAPHS AND BAR GRAPHS WITH INTERVALS OF ONE
Including, but not limited to:
 Graph – a visual representation of the relationships between data collected
 Organization of data used to interpret data, draw conclusions, and make comparisons
 Data – information that is collected about people, events, or objects
 Categorical data – data that represents the attributes of a group of people, events, or objects
 Limitations
 Up to four categories
 Intervals limited to 1
 Operations limited to onestep addition or subtraction
 Data representations
 Pictograph – a graphical representation to organize data that uses a picture or symbol, where each picture or symbol may represent one or more than one unit of data, to show the frequency (number of times) that each category occurs
 Bar graph – a graphical representation to organize data that uses solid bars that do not touch each other and a scaled axis to show the frequency (number of times) that each category occurs
 Write and solve mathematical and realworld problems using data represented within pictographs and bar graphs.
Note(s):
 Grade Level(s):
 Grade 3 will solve one and twostep problems using categorical data represented with a frequency table, dot plot, pictograph, or bar graph with scaled intervals.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.2. Perform computations with rational and irrational numbers.
 V.B. Statistical Reasoning – Describe data
 V.B.3. Compute and describe the study data with measures of center and basic notions of spread.

2.10D 
Draw conclusions and make predictions from information in a graph.

Draw
CONCLUSIONS FROM INFORMATION IN A GRAPH
Including, but not limited to:
 Graph – a visual representation of the relationships between data collected
 Organization of data used to interpret data, draw conclusions, and make comparisons
 Data – information that is collected about people, events, or objects
 Categorical data – data that represents the attributes of a group of people, events, or objects
 Factual data – actual quantities represented in a graph used to interpret data, draw conclusions, and make comparisons
 Limitations
 Up to four categories
 Intervals limited to 1, 2, 5, or 10
 Data representations
 Pictograph – a graphical representation to organize data that uses a picture or symbol, where each picture or symbol may represent one or more than one unit of data, to show the frequency (number of times) that each category occurs
 Bar graph – a graphical representation to organize data that uses solid bars that do not touch each other and a scaled axis to show the frequency (number of times) that each category occurs
 Description of data represented
 Identification of title and category labels
 Explanation of what the graph represents
 Conclusions related to the question that led to the data collection
 Numerical conclusions in the data
 Quantities represented by the data
 Number in each category
 Number in a category(s) may be zero
 Combined total(s)
 Comparisons of data represented
 Comparative language used with numbers
 Comparative language used without numbers
 Changes in orientation do not affect data values
Make
PREDICTIONS FROM INFORMATION IN A GRAPH
Including, but not limited to:
 Graph – a visual representation of the relationships between data collected
 Organization of data used to interpret data, draw conclusions, and make comparisons
 Data – information that is collected about people, events, or objects
 Categorical data – data that represents the attributes of a group of people, events, or objects
 Inferential data – existing data used to make predictions about future data
 Limitations
 Up to four categories
 Intervals limited to 1, 2, 5, or 10
 Data representations
 Pictograph – a graphical representation to organize data that uses a picture or symbol, where each picture or symbol may represent one or more than one unit of data, to show the frequency (number of times) that each category occurs
 Bar graph – a graphical representation to organize data that uses solid bars that do not touch each other and a scaled axis to show the frequency (number of times) that each category occurs
 Make predictions based on patterns in the data collected
 Make predictions based on comparisons of quantities in the data collected
 Make predictions about future actions based on the purpose of the data collection
Note(s):
 Grade Level(s):
 Grade 1 drew conclusions and generated and answered questions using information from picture and bartype graphs.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 TxCCRS:
 V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
 V.C.3. Make predictions using summary statistics.
