4.1 
Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:


4.1A 
Apply mathematics to problems arising in everyday life, society, and the workplace.
Process Standard

Apply
MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE
Including, but not limited to:
 Mathematical problem situations within and between disciplines
 Everyday life
 Society
 Workplace
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
 Measuring angles
 Understanding decimals and addition and subtraction of decimals
 Building foundations for addition and subtraction of fractions
 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.

4.1B 
Use a problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process and the reasonableness of the solution.
Process Standard

Use
A PROBLEMSOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEMSOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:
 Problemsolving model
 Analyze given information
 Formulate a plan or strategy
 Determine a solution
 Justify the solution
 Evaluate the problemsolving process and the reasonableness of the solution
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
 Measuring angles
 Understanding decimals and addition and subtraction of decimals
 Building foundations for addition and subtraction of fractions
 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.

4.1C 
Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.
Process Standard

Select
TOOLS, INCLUDING 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 fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
 Measuring angles
 Understanding decimals and addition and subtraction of decimals
 Building foundations for addition and subtraction of fractions
 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.

4.1D 
Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Process Standard

Communicate
MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, AND LANGUAGE AS APPROPRIATE
Including, but not limited to:
 Mathematical ideas, reasoning, and their implications
 Multiple representations, as appropriate
 Symbols
 Diagrams
 Language
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
 Measuring angles
 Understanding decimals and addition and subtraction of decimals
 Building foundations for addition and subtraction of fractions
 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.

4.1E 
Create and use representations to organize, record, and communicate mathematical ideas.
Process Standard

Create, Use
REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:
 Representations of mathematical ideas
 Organize
 Record
 Communicate
 Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
 Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
 Measuring angles
 Understanding decimals and addition and subtraction of decimals
 Building foundations for addition and subtraction of fractions
 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.

4.1F 
Analyze mathematical relationships to connect and communicate mathematical ideas.
Process Standard

Analyze
MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:
 Mathematical relationships
 Connect and communicate mathematical ideas
 Conjectures and generalizations from sets of examples and nonexamples, patterns, etc.
 Current knowledge to new learning
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
 Measuring angles
 Understanding decimals and addition and subtraction of decimals
 Building foundations for addition and subtraction of fractions
 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.

4.1G 
Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Process Standard

Display, Explain, Justify
MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION
Including, but not limited to:
 Mathematical ideas and arguments
 Validation of conclusions
 Displays to make work visible to others
 Diagrams, visual aids, written work, etc.
 Explanations and justifications
 Precise mathematical language in written or oral communication
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
 Measuring angles
 Understanding decimals and addition and subtraction of decimals
 Building foundations for addition and subtraction of fractions
 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.

4.7 
Geometry and measurement. The student applies mathematical process standards to solve problems involving angles less than or equal to 180 degrees. The student is expected to:


4.7A 
Illustrate the measure of an angle as the part of a circle whose center is at the vertex of the angle that is "cut out" by the rays of the angle. Angle measures are limited to whole numbers.

Illustrate
THE MEASURE OF AN ANGLE AS THE PART OF A CIRCLE WHOSE CENTER IS AT THE VERTEX OF THE ANGLE THAT IS "CUT OUT" BY THE RAYS OF THE ANGLE. ANGLE MEASURES ARE LIMITED TO WHOLE NUMBERS.
Including, but not limited to:
 Ray – part of a line that begins at one endpoint and continues without end in one direction
 Degree – the measure of an angle where each degree represents of a circle
 Unit measure labels as “degrees” or with symbol for degrees (°)
 Angle – two rays with a common endpoint (the vertex)
 Various angle types/names
 Right angle, 90°, used as a benchmark to identify and nae angels
 Acute angle – an angle that measures less than 90°
 Right angle – an angle (formed by perpendicular lines) that measures exactly 90°
 Notation is given as a box in the angle corner to represent a 90° angle.
 Obtuse angle – an angle that measures greater than 90° but less than 180°
 Straight angle – an angle that measures 180° (a straight line)
 Angle measures limited to whole numbers, 0° to 180°
 Center of the circle – the point equidistant from all points on the circle
 Circle
 A figure formed by a closed curve with all points equal distance from the center
 No straight sides
 No vertices
 No parallel or perpendicular sides
 A circle measures 360° for one full rotation around the center of the circle.
 Representation of an angle measure as a “turn” around the center point of a circle “cut out” by the rays of the angle where the vertex of the angle is aligned to the center of the circle.
Note(s):
 Grade Level(s):
 Grade 4 introduces illustrating the measure of an angle as the part of a circle whose center is at the vertex of the angle that is "cut out" by the rays of the angle. Angle measures are limited to whole numbers.
 Foundational for work with central angles in Geometry and radian measures in Precalculus.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:

4.7B 
Illustrate degrees as the units used to measure an angle, where 1/360 of any circle is one degree and an angle that "cuts" n/360 out of any circle whose center is at the angle's vertex has a measure of n degrees. Angle measures are limited to whole numbers.

Illustrate
DEGREES AS THE UNITS USED TO MEASURE AN ANGLE, WHERE OF ANY CIRCLE IS ONE DEGREE AND AN ANGLE THAT "CUTS" OUT OF ANY CIRCLE WHOSE CENTER IS AT THE ANGLE'S VERTEX HAS A MEASURE OF n DEGREES. ANGLE MEASURES ARE LIMITED TO WHOLE NUMBERS
Including, but not limited to:
 Degree – the measure of an angle where each degree represents of a circle
 Unit measure labels as “degrees” or with symbol for degrees (°)
 Angle measures limited to whole numbers, 0° to 360°
 Angle – two rays with a common endpoint (the vertex)
 Center of the circle – the point equidistant from all points on the circle
 Circle
 A figure formed by a closed curve with all points equal distance from the center
 No straight sides
 No vertices
 No parallel or perpendicular sides
 A circle measures 360° for one full rotation around the center of the circle.
 Representations of the “cuts” out of a circle as degrees of angle measures
Note(s):
 Grade Level(s):
 Grade 4 introduces illustrating degrees as the units used to measure an angle, where of any circle is one degree and an angle that "cuts" out of any circle whose center is at the angle's vertex has a measure of n degrees. Angle measures are limited to whole numbers.
 Leads to the development of radian measures in Precalculus.
 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.

4.7C 
Determine the approximate measures of angles in degrees to the nearest whole number using a protractor.
Readiness Standard

Determine
THE APPROXIMATE MEASURES OF ANGLES IN DEGREES TO THE NEAREST WHOLE NUMBER USING A PROTRACTOR
Including, but not limited to:
 Degree – the measure of an angle where each degree represents of a circle
 Unit measure labels as “degrees” or with symbol for degrees (°)
 Angle measures limited to whole numbers, 0° to 180°
 Various angle types/names
 Right angle, 90°, used as a benchmark to identify and name angles
 Acute angle – an angle that measures less than 90°
 Right angle – an angle (formed by perpendicular lines) that measures exactly 90°
 Notation is given as a box in the angle corner to represent a 90° angle.
 Obtuse angle – an angle that measures greater than 90° but less than 180°
 Straight angle – an angle that measures 180° (a straight line)
 Protractor – a tool used to determine the measure of an angle
 Two sets of measures from 0° to 180° going in opposite directions
 Relationships between a protractor and a circle
 One protractor is a semicircle, 180°
 Two protractors make a complete circle, 360°
 Measurement or “m” notation indicates the measure of the angle in degrees (e.g., m∠1 = 50°)
 Measure angles with a ray aligned at zero degrees.
 When aligning the ray to zero degrees on the right side of the protractor, read the angle measurement using the inner set of measures from right to left.
 When aligning the ray to zero degrees on the left side of the protractor, read the angle measurement using the outer set of measures from left to right.
 Measure angles whose rays may lie between numerically marked intervals.
 Relate to reading unmarked whole number intervals on a number line.
 Measure angles where a ray of the angle does not lie on zero degrees.
 Read measure of both rays using either the inner or the outer set of measures, then subtract smaller measure from larger measure to determine angle measure.
 Measure angles within twodimensional figures.
 Treat the sides of the figure that form the angle as rays.
 Use a right angle, 90°, as a benchmark to determine angle classifications (acute, obtuse, and right) to determine reasonableness of angle measures.
Note(s):
 Grade Level(s):
 Grade 4 introduces determining the approximate measures of angles in degrees to the nearest whole number using a protractor.
 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.

4.7D 
Draw an angle with a given measure.
Supporting Standard

Draw
AN ANGLE WITH A GIVEN MEASURE
Including, but not limited to:
 Degree – the measure of an angle where each degree represents of a circle
 Unit measure labels as “degrees” or with symbol for degrees (°)
 Angle measures limited to whole numbers, 0° to 180°
 Angle – two rays with a common endpoint (the vertex)
 Various angle types/names
 Right angle, 90°, used as a benchmark to identify and name angles
 Acute angle – an angle that measures less than 90°
 Right angle – an angle (formed by perpendicular lines) that measures exactly 90°
 Notation is given as a box in the angle corner to represent a 90° angle.
 Obtuse angle – an angle that measures greater than 90° but less than 180°
 Straight angle – an angle that measures 180° (a straight line)
 Protractor – a tool used to determine the measure of an angle
 Use a protractor to draw an angle of a given measure
 Use the straight edge of the protractor to draw a ray.
 Place the vertex of the protractor on the endpoint of the ray.
 Align the vertex and the 0° mark on the protractor to the ray.
 Use the scale beginning with 0 and mark the given angle measure.
 Use the straightedge of the protractor to draw a ray from the vertex to the angle mark.
Note(s):
 Grade Level(s):
 Grade 4 introduces drawing an angle with a given measure.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:

4.7E 
Determine the measure of an unknown angle formed by two nonoverlapping adjacent angles given one or both angle measures.
Supporting Standard

Determine
THE MEASURE OF AN UNKNOWN ANGLE FORMED BY TWO NONOVERLAPPING ADJACENT ANGLES GIVEN ONE OR BOTH ANGLE MEASURES
Including, but not limited to:
 Degree – the measure of an angle where each degree represents of a circle
 Unit measure labels as “degrees” or with symbol for degrees (°)
 Angle measures limited to whole numbers, 0° to 180°
 Angle – two rays with a common endpoint (the vertex)
 Adjacent angles – two nonoverlapping angles that share a common vertex and exactly one ray
 Complementary angles – two angles whose degree measures have a sum of 90°
 Supplementary angles – two angles whose degree measures have a sum of 180°
 Congruent angles – angles whose angle measurements are equal
 Angle congruency marks – angle marks indicating angles of the same measure
 Decompose and compose angle measures
 Angle measures up to 360°
 The angle measure of the whole is the sum of the angle measure of the parts
 Given the measure of one angle, and the whole, find the measure of the other angle.
 Given the measure of two angles, find the measure of the whole angle.
 Given the measure of the whole angle divided equally, find the measure of the equal sized angles
 Multiple steps to find a missing measure
 Adjacent angles within twodimensional figures
 Angles in context without graphics
Note(s):
 Grade Level(s):
 Grade 4 introduces determining the measure of an unknown angle formed by two nonoverlapping adjacent angles given one or both angle measures.
 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.2. Perform computations with rational and irrational numbers.
 III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
 III.C.1. Make connections between geometry and algebraic equations.
 III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
 III.D.3. Determine indirect measurements of geometric figures using a variety of methods.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.3. Determine a solution.
 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.
