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.

2.9 
Geometry and measurement. The student applies mathematical process standards to select and use units to describe length, area, and time. The student is expected to:


2.9A 
Find the length of objects using concrete models for standard units of length.

Find
THE LENGTH OF OBJECTS USING CONCRETE MODELS FOR STANDARD UNITS OF LENGTH
Including, but not limited to:
 Length – the measurement attribute that describes a continuous distance from end to end
 Unit of length – the object or unit used to measure length
 Concrete models that represent standard units of length
 Typically used customary units of length
 Inch represented by a color tile, etc.
 Foot represented by a 12inch ruler as a single unit, etc.
 Yard represented by a yardstick as a single unit, etc.
 Typically used metric units of length
 Centimeter represented by a base10 unit cube, etc.
 Decimeter represented by a base10 long, orange Cuisenaire rod, etc.
 Meter represented by a meter stick as a single unit, etc.
 Length described to the nearest whole unit using a number and a unit
 Linear measurement – the measurement of length along a continuous line or curve
 Starting point and ending point defined
 Equal sized units of length placed end to end along the distance being measured
 Equal sized units of length iterated (repeated) with no gaps or overlaps
 Length measured using onedimensional units of length (e.g., if measuring with a color tile, measure with the edge, not the area of the color tile; if measuring with a color tile, measure with the same dimension of the color tile; etc.)
 Equal sized units of length counted to the nearest whole unit
 Last unit is not counted if the end point falls less than halfway along the unit.
 Last unit is counted if the end point falls halfway, or more than halfway, along the unit.
 Unit of length selected for efficiency
 Smaller unit of length to measure shorter objects or distances
 Larger unit of length to measure longer objects or distances
 Unit of length selected for precision
 Smaller unit of length results in a more precise measurement when measuring to the whole unit
 Larger unit of length results in a less precise measurement when measuring to the whole unit
Note(s):
 Grade Level(s):
 Grade 1 illustrated that the length of an object is the number of samesize units of length that, when laid endtoend with no gaps or overlaps, reach from one end of the object to the other.
 Grade 1 described a length to the nearest whole unit using a number and a unit.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 TxCCRS:
 I.C. Numeric Reasoning – Systems of measurement
 I.C.1. Select or use the appropriate type of method, unit, and tool for the attribute being measured.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.

2.9B 
Describe the inverse relationship between the size of the unit and the number of units needed to equal the length of an object.

Describe
THE INVERSE RELATIONSHIP BETWEEN THE SIZE OF THE UNIT AND THE NUMBER OF UNITS NEEDED TO EQUAL THE LENGTH OF AN OBJECT
Including, but not limited to:
 Length – the measurement attribute that describes a continuous distance from end to end
 Unit of length – the object or unit used to measure length
 Concrete models that represent standard units of length
 Typlically used customary units of length
 Inch represented by a color tile, etc.
 Foot represented by a 12inch ruler as a single unit, etc.
 Yard represented by a yardstick as a single unit, etc.
 Typically used metric units of length
 Centimeter represented by a base10 unit cube, etc.
 Decimeter represented by a base10 long, orange Cuisenaire rod, etc.
 Meter represented by a meter stick as a single unit, etc.
 Inverse relationship between the size of the unit and the number of units needed
 Measure the same object with different sized units of length.
 The shorter the unit of length, the more units needed
 The longer the unit of length, the fewer units needed
Note(s):
 Grade Level(s):
 Grade 1 measured the same object/distance with units of two different lengths and described how and why the measurements differ.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 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.
 I.C. Numeric Reasoning – Systems of measurement
 I.C.1. Select or use the appropriate type of method, unit, and tool for the attribute being measured.

2.9C 
Represent whole numbers as distances from any given location on a number line.

Represent
WHOLE NUMBERS AS DISTANCES FROM ANY GIVEN LOCATION ON A NUMBER LINE
Including, but not limited to:
 Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
 Characteristics of a number line
 A number line begins as a line with predetermined intervals (or tick marks) with positions/numbers labeled.
 A minimum of two positions/numbers should be labeled.
 Numbers on a number line represent the distance from zero.
 The distance between the tick marks is counted rather than the tick marks themselves.
 The placement of the labeled positions/numbers on a number line determines the scale of the number line.
 Intervals between position/numbers are proportional.
 When reasoning on a number line, the position of zero may or may not be placed.
 When working with larger numbers, a number line without the constraint of distance from zero allows the ability to “zoomin” on the relevant section of the number line.
 Number lines extend infinitely in both directions (arrows indicate the number line continues infinitely).
 Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
 Points to the left of a specified point on a horizontal number line are less than points to the right.
 Points to the right of a specified point on a horizontal number line are greater than points to the left.
 Points below a specified point on a vertical number line are less than points above.
 Points above a specified point on a vertical number line are greater than points below.
 Whole numbers represented as equally spaced lengths or distances from zero on a number line
 Relationship between a whole number represented using a strip diagram to a whole number represented on a number line
 Number lines beginning with a number other than zero
 Distance from zero to first marked increment is assumed even when not visible on the number line.
 Relationship between whole numbers as distances from zero on a number line to whole unit measurements as distances from zero on a customary ruler, yardstick, or measuring tape
 Measuring a specific length using a distance other than zero on a ruler, yardstick, or measuring tape
 Distance from zero to first marked increment not counted
 Length determined by number of whole units between starting point and ending point
Note(s):
 Grade Level(s):
 Grade 3 will represent fractions of halves, fourths, and eighths as distances from zero on a number line.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 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.

2.9D 
Determine the length of an object to the nearest marked unit using rulers, yardsticks, meter sticks, or measuring tapes.

Determine
THE LENGTH OF AN OBJECT TO THE NEAREST MARKED UNIT USING RULERS, YARDSTICKS, METER STICKS, OR MEASURING TAPES
Including, but not limited to:
 Length – the measurement attribute that describes a continuous distance from end to end
 Unit of length – the object or unit used to measure length
 Linear measurement – the measurement of length along a continuous line or curve
 Standard linear measurement tools
 Typically used customary linear measurement tools
 Ruler with inches, yardstick, measuring tape
 Typically used metric linear measurement tools
 Ruler with centimeters, meter stick, measuring tape
 Standard units of length
 Typically used customary units of length
 Typically used metric units of length
 Relationship between finding the length of objects using concrete models for standard units of length to whole unit measurements on a customary ruler, yardstick, or measuring tape
 Relationship between whole numbers as distances from zero on a number line to whole unit measurements as distances from zero on a customary ruler, yardstick, or measuring tape
 Determine length to the nearest whole unit.
 Starting point and ending point defined
 Edge of measuring tool placed along the distance being measured, aligned with the start point of the distance being measured
 Equal sized units of length counted to the nearest whole unit
 Last unit is not counted if the end point falls less than halfway along the unit.
 Last unit is counted if the end point falls halfway, or more than halfway, along the unit.
 Measuring a specific length using a starting point other than zero on a ruler, yardstick, or measuring tape
 Distance from zero to first marked increment not counted
 Length determined by number of whole units between starting point and ending point
 Selection of appropriate tool and unit of length in realworld situations
Note(s):
 Grade Level(s):
 Grade 1 used nonstandard measuring tools to measure the length of objects to reinforce the continuous nature of linear measurement.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 TxCCRS:
 I.C. Numeric Reasoning – Systems of measurement
 I.C.1. Select or use the appropriate type of method, unit, and tool for the attribute being measured.

2.9E 
Determine a solution to a problem involving length, including estimating lengths.

Determine
A SOLUTION TO A PROBLEM INVOLVING LENGTH, INCLUDING ESTIMATING LENGTHS
Including, but not limited to:
 Length – the measurement attribute that describes a continuous distance from end to end
 Mathematical and realworld problem situations
 Recognition of attributes of length embedded in mathematical and realworld problem situations (e.g., distance traveled from one place to another, length of an object, distance around the outer edges of an object, etc.)
 Onestep or multistep problems
 Measurement of one or more distances/lengths
 Multiple operations
 Addition and/or subtraction of whole unit measurements
 Solutions recorded to the nearest whole unit with a number and a unit label
 Estimation – reasoning to determine an approximate value
 Estimation prior to solving problem
 Estimation compared to actual measurement
 Benchmarks for units of length
 Finger joint (thumb works best) = approximately 1 inch
 Tip of your finger = approximately 1 centimeter
 Span of your palm = approximately 1 decimeter
 Elbow to wrist = approximately 1 foot
 Nose to fingertip of extended arm = approximately 1 yard
 Nose to fingertip of extended arm with head turned away = approximately 1 meter
 Language related to estimation
 About, a little less than, a little more than, almost, nearly, approximately, etc.
Note(s):
 Grade Level(s):
 Grade 2 introduces estimation outside of the mathematical process standards.
 Grade 3 will determine the perimeter of a polygon or a missing length when given perimeter and remaining side lengths in problems.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 TxCCRS:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.1. Use estimation to check for errors and reasonableness of solutions.
