1.1 
Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:


1.1A 
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 an understanding of place value
 Solving problems involving addition and subtraction
 Analyzing attributes of twodimensional shapes and threedimensional solids
 Developing the understanding of length
 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.

1.1B 
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 an understanding of place value
 Solving problems involving addition and subtraction
 Analyzing attributes of twodimensional shapes and threedimensional solids
 Developing the understanding of length
 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.

1.1C 
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 an understanding of place value
 Solving problems involving addition and subtraction
 Analyzing attributes of twodimensional shapes and threedimensional solids
 Developing the understanding of length
 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.

1.1D 
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 an understanding of place value
 Solving problems involving addition and subtraction
 Analyzing attributes of twodimensional shapes and threedimensional solids
 Developing the understanding of length
 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.

1.1E 
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 an understanding of place value
 Solving problems involving addition and subtraction
 Analyzing attributes of twodimensional shapes and threedimensional solids
 Developing the understanding of length
 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.

1.1F 
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 an understanding of place value
 Solving problems involving addition and subtraction
 Analyzing attributes of twodimensional shapes and threedimensional solids
 Developing the understanding of length
 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.

1.1G 
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 an understanding of place value
 Solving problems involving addition and subtraction
 Analyzing attributes of twodimensional shapes and threedimensional solids
 Developing the understanding of length
 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.

1.8 
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:


1.8A 
Collect, sort, and organize data in up to three categories using models/representations such as tally marks or Tcharts.

Collect, Sort, Organize
DATA IN UP TO THREE CATEGORIES USING MODELS/REPRESENTATIONS
Including, but not limited to:
 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 represent numbers or ranges of numbers
 Limitations
 Up to three categories
 Data values limited to whole numbers up to 20
 Data collected in the form of responses to a question
 Survey – to ask a group of people a question in order to collect information about their opinions or answers
 Common characteristics in a collection of objects
 Data sorted in a variety of ways
 Data organized and represented in a variety of ways
 Data organized using Tcharts, sorting mats, etc.
 Data represented by realworld objects, pictures, drawings, or tally marks
 One unit of data represented by each object, picture, drawing, or tally mark
Note(s):
 Grade Level(s):
 Kindergarten collected, sorted, and organized data into two or three categories.
 Grade 2 will organize a collection of data with up to four categories using pictographs and bar graphs with intervals of one or more.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing an understanding of place value
 TxCCRS:
 V.B. Statistical Reasoning – Design a study
 V.B.2. Construct appropriate visual representations 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.

1.8B 
Use data to create picture and bartype graphs.

Use
DATA
To Create
PICTURE AND BARTYPE GRAPHS
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
 Data collected in the form of responses to a question with up to three categories
 Survey – to ask a group of people a question in order to collect information about their opinions or answers
 Common characteristics in a collection of objects sorted into up to three categories
 Limitations
 Up to three categories
 Data values limited to whole numbers up to 20
 Data representations
 Picture graph – a graphical representation to organize data that uses pictures or symbols evenly spaced or placed in individual cells, where each picture or symbol represents one unit of data, to show the frequency (number of times) that each category occurs
 Characteristics of picture graphs
 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 or placed in individual cells within each category
 Different picture or symbol used to represent each category
 Every piece of data represented using a onetoone correspondence
 One unit of data represented by each picture or symbol
 Value of the data represented by the pictures
 Determined by the total number of pictures or symbols in that category
 Represents the frequency of each category
 Bartype graph – a graphical representation to organize data that uses bars divided into individual cells, where each cell represents one unit of data, to show the frequency (number of times) that each category occurs
 Characteristics of bartype graphs
 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
 Bars composed of individual cells
 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
 Bars divided into equalsized cells with no gaps between cells
 Bars equal in width
 Independent bars that do not touch
 May use a different color to represent each category
 Does not include a scaled axis
 Every piece of data represented using a onetoone correspondence
 One unit of data represented by each shaded cell
 Value of the data represented by the bar
 Determined by the total number of cells in the bar for each category
 Represents the frequency for that category
 Connection between graphs representing the same data
 Picture graph to bartype graph
 Bartype graph to picture graph
 Same data represented using a picture graph and a bartype graph
Note(s):
 Grade Level(s):
 Kindergarten used data to create realobject and picture graphs.
 Grade 2 will 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.
 Grade 2 will organize a collection of data with up to four categories using pictographs and bar graphs with intervals of one or more.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing an understanding of place value
 TxCCRS:
 V.B. Statistical Reasoning – Design a study
 V.B.2. Construct appropriate visual representations 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.

1.8C 
Draw conclusions and generate and answer questions using information from picture and bartype graphs.

Draw
CONCLUSIONS USING INFORMATION FROM PICTURE AND BARTYPE GRAPHS
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 three categories
 Data values limited to whole numbers up to 20
 Data values limited to addition or subtraction within 10
 Data representations
 Picture graph – a graphical representation to organize data that uses pictures or symbols evenly spaced or placed in individual cells, where each picture or symbol represents one unit of data, to show the frequency (number of times) that each category occurs
 One unit of data represented by each picture or symbol
 Bartype graph – a graphical representation to organize data that uses bars divided into individual cells, where each cell represents one unit of data, to show the frequency (number of times) that each category occurs
 One unit of data represented by each shaded cell
 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 without numbers
 Comparative language used with numbers
 Changes in orientation do not affect data values
Generate, Answer
QUESTIONS USING INFORMATION FROM PICTURE AND BARTYPE GRAPHS
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 three categories
 Data values limited to whole numbers up to 20
 Operations limited to onestep addition or subtraction within 10
 Data representations
 Picture graph – a graphical representation to organize data that uses pictures or symbols evenly spaced or placed in individual cells, where each picture or symbol represents one unit of data, to show the frequency (number of times) that each category occurs
 One unit of data represented by each picture or symbol
 Bartype graph – a graphical representation to organize data that uses bars divided into individual cells, where each cell represents one unit of data, to show the frequency (number of times) that each category occurs
 One unit of data represented by each shaded cell
 Generate and answer questions using data in in picture graphs and bartype graphs
 Description of data
 Comparison using data values
 Operations using data values
Note(s):
 Grade Level(s):
 Kindergarten drew conclusions from realobject and picture graphs.
 Grade 2 will draw conclusions and make predictions from information in a graph.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing an understanding of place value
 TxCCRS:
 V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
 V.C.3. Make predictions using summary statistics.
 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.
