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 TO SOLVE PROBLEMS
Including, but not limited to:
 Appropriate selection of tool(s) to apply in order to solve problems
 Tools
 Real objects
 Manipulatives
 Paper and pencil
 Technology
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.

4.6 
Geometry and measurement. The student applies mathematical process standards to analyze geometric attributes in order to develop generalizations about their properties. The student is expected to:


4.6A 
Identify points, lines, line segments, rays, angles, and perpendicular and parallel lines.
Supporting Standard

Identify
POINTS, LINES, LINE SEGMENTS, RAYS, ANGLES, AND PERPENDICULAR AND PARALLEL LINES
Including, but not limited to:
 Point – a specific location in space
 Has no dimension and is usually represented by a small dot
 Line – a set of points that form a straight path that goes in opposite directions without ending
 Line labels
 Lines named according to two points on a line
 Lines named by one lower case cursive letter
 Parallel lines – lines that lie in the same plane, never intersect, and are always the same distance apart
 Various orientations including vertical, horizontal, diagonal, and parallel lines of even, uneven, or offset lengths
 Notation may be given using chevrons or arrows to represent parallel lines.
 If more than one set of parallel lines are present, the number of chevrons or arrows distinguishes the sets of parallel lines.
 Intersecting lines – lines that meet or cross at a point
 Various orientations including vertical, horizontal, diagonal, and intersecting lines of even, uneven, or offset lengths
 Perpendicular lines – lines that intersect at right angles to each other to form square corners
 Various orientations including vertical, horizontal, diagonal, and perpendicular lines of even, uneven, or offset lengths
 Notation is given as a box in the angle corner to represent a 90° angle.
 Lines in pictorial models and polygons
 Extending lines beyond pictorial models
 Line segment – part of a line between two points on the line, called endpoints of the segment
 Ray – part of a line that begins at one endpoint and continues without end in one direction
 Relationships between line segments, rays, and lines
 A line segment is part of a ray and part of a line
 A ray is part of a line
 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)
 Angle labels for a single angle
 Angle with one letter, the letter of the vertex
 Angle label with three letters, where the middle letter is the vertex of the angle
 Angle label with a number or letter designated within the angle
 Angle symbol with one letter, the letter of the vertex
 Angle symbol with three letters, where the middle letter is the vertex of the angle
 Angle symbol with a number or letter designated within the angle
 Angle labels for adjacent angles
 Adjacent angles – two nonoverlapping angles that share a common vertex and exactly one ray
 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)
 Angles in pictorial models and polygons
Note(s):
 Grade Level(s):
 Grade 3 used attributes to recognize rhombuses, parallelograms, trapezoids, rectangles, and squares as examples of quadrilaterals and drew examples of quadrilaterals that do not belong to any of these subcategories.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Measuring angles
 Grade Level Connections (reinforces previous learning and/or provides development for future learning)
 TxCCRS:
 III.A. Geometric and Spatial Reasoning – Figures and their properties
 III.A.1. Recognize characteristics and dimensional changes of two and threedimensional figures.

4.6B 
Identify and draw one or more lines of symmetry, if they exist, for a twodimensional figure.
Supporting Standard

Identify, Draw
ONE OR MORE LINES OF SYMMETRY, IF THEY EXIST, FOR A TWODIMENSIONAL FIGURE
Including, but not limited to:
 Line of symmetry – line dividing an image into two congruent parts that are mirror images of each other
 Twodimensional figure – a figure with two basic units of measure, usually length and width
 Twodimensional figures and realworld figures
 Shapes with more than one line of symmetry
 Shapes with no lines of symmetry
 Shapes on which lines of symmetry have not been drawn
 Across a vertical line, across a horizontal line, or across a diagonal line of symmetry
 A line of reflection exists for a figure if for every point on one side of the line of reflection, there is a corresponding point the same distance from the line.
Note(s):
 Grade Level(s):
 Grade 3 used attributes to recognize rhombuses, parallelograms, trapezoids, rectangles, and squares as examples of quadrilaterals and drew examples of quadrilaterals that do not belong to any of these subcategories.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Grade Level Connections (reinforces previous learning and/or provides development for future learning)
 TxCCRS:
 III.A. Geometric and Spatial Reasoning – Figures and their properties
 III.A.1. Recognize characteristics and dimensional changes of two and threedimensional figures.
 III.B. Geometric and Spatial Reasoning – Transformations and symmetry
 III.B.1. Identify transformations and symmetries of figures.

4.6C 
Apply knowledge of right angles to identify acute, right, and obtuse triangles.
Supporting Standard

Apply
KNOWLEDGE OF RIGHT ANGLES TO IDENTIFY ACUTE, RIGHT, AND OBTUSE TRIANGLES
Including, but not limited to:
 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°
 Triangle – a polygon with three sides and three vertices
 Triangles are named based on their largest angle.
 Acute triangle – a triangle in which each of the three angles is acute (less than 90°)
 Right triangle – a triangle with one right angle (exactly 90°) and two acute angles
 Obtuse triangle – a triangle that has one obtuse angle (greater than 90°) and two acute angles
Note(s):
 Grade Level(s):
 Grade 3 used attributes to recognize rhombuses, parallelograms, trapezoids, rectangles, and squares as examples of quadrilaterals and drew examples of quadrilaterals that do not belong to any of these subcategories.
 Grade 4 introduces formal and symbolic geometric language for lines, line segments, rays, and angles.
 Grade 5 will classify twodimensional figures in a hierarchy of sets and subsets using graphic organizers based on their attributes and properties.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 TxCCRS:
 III.A. Geometric and Spatial Reasoning – Figures and their properties
 III.A.1. Recognize characteristics and dimensional changes of two and threedimensional figures.

4.6D 
Classify twodimensional figures based on the presence or absence of parallel or perpendicular lines or the presence or absence of angles of a specified size.
Readiness Standard

Classify
TWODIMENSIONAL FIGURES BASED ON THE PRESENCE OR ABSENCE OF PARALLEL OR PERPENDICULAR LINES OR THE PRESENCE OR ABSENCE OF ANGLES OF A SPECIFIED SIZE
Including, but not limited to:
 Twodimensional figure – a figure with two basic units of measure, usually length and width
 Regular figure – a polygon with all sides and angles congruent
 Irregular figure – a polygon with sides and/or angles that are not all congruent
 Classify – applying an attribute to categorize a sorted group
 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°
 Line – a set of points that form a straight path that goes in opposite directions without ending
 Parallel lines – lines that lie in the same plane, never intersect, and are always the same distance apart
 means is parallel to .
 Notation may be given using chevrons or arrows to represent parallel lines.
 If more than one set of parallel lines are present, the number of chevrons or arrows distinguishes the sets of parallel lines.
 Perpendicular lines – lines that intersect at right angles to each other to form square corners
 means is perpendicular to .
 Notation is given as a box in the angle corner to represent a 90° angle.
 Sides of twodimensional figures are composed of line segments, the part of a line between two points on the line
 Congruent – of equal measure, having exactly the same size and same shape
 Angle congruency marks – angle marks indicating angles of the same measure
 m∠A ≅ m∠C means ∠A is congruent to ∠C.
 Side congruency marks – side marks indicating side lengths of the same measure
 Equilateral – all side lengths of a polygon are congruent in measure
 Equiangular – all angles in a polygon are congruent in measure
 Types of twodimensional figures
 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
 Polygon – a closed figure with at least 3 sides, where all sides are straight (no curves)
 Types of polygons
 Triangle
 3 sides
 3 vertices
 No parallel sides
 Types of triangles
 Triangles are named based on their largest angle.
 Scalene triangle
 3 sides
 3 vertices
 No congruent sides
 No parallel sides
 Up to one possible pair of perpendicular sides
 Right triangle with two sides that are perpendicular to form a right angle and three different side lengths
 No congruent angles
 Right triangle with one 90° angle and two other angles each of different measures
 Obtuse triangle with one angle greater than 90° and two other angles each of different measures
 Acute triangle with all angles less than 90° and all angles of different measures
 Isosceles triangle
 3 sides
 3 vertices
 At least 2 congruent sides
 No parallel sides
 Up to one possible pair of perpendicular sides
 Right triangle with two sides that are perpendicular to form a right angle and are each of the same length
 At least 2 congruent angles
 Right triangle with one 90° angle and two other angles each of the same measure
 Obtuse triangle with two angles of the same measure and one angle greater than 90°
 Acute triangle with all angles measuring less than 90° and at least two of the angles of the same measure
 Equilateral triangle/Equiangular triangle (a special type of isosceles triangle)
 3 sides
 3 vertices
 All sides congruent
 No parallel or perpendicular sides
 All angles congruent
 Acute triangle with all angles measuring 60°
 Quadrilateral
 4 sides
 4 vertices
 Types of quadrilaterals
 Trapezoid
 4 sides
 4 vertices
 Exactly one pair of parallel sides
 Types of trapezoids
 Isosceles trapezoid
 4 sides
 4 vertices
 Exactly one pair of parallel sides
 At least 2 congruent sides, where 2 of the sides are opposite each other
 Right trapezoid
 4 sides
 4 vertices
 Exactly one pair of parallel sides
 2 pairs of perpendicular sides
 2 right angles
 Parallelogram
 4 sides
 4 vertices
 Opposite sides congruent
 2 pairs of parallel sides
 Opposite angles congruent
 Types of parallelograms
 Rectangle
 4 sides
 4 vertices
 Opposite sides congruent
 2 pairs of parallel sides
 4 pairs of perpendicular sides
 4 right angles
 Rhombus
 4 sides
 4 vertices
 All sides congruent
 2 pairs of parallel sides
 Opposite angles congruent
 Square (a special type of rectangle and a special type of rhombus)
 4 sides
 4 vertices
 All sides congruent
 2 pairs of parallel sides
 4 pairs of perpendicular sides
 4 right angles
 Pentagon
 5 sides
 5 vertices
 Possible parallel and/or perpendicular sides
 Possible acute, obtuse, and/or right angles
 Hexagon
 6 sides
 6 vertices
 Possible parallel and/or perpendicular sides
 Possible acute, obtuse, and/or right angles
 7gon (heptagon)
 7 sides
 7 vertices
 Possible parallel and/or perpendicular sides
 Possible acute, obtuse, and/or right angles
 Octagon
 8 sides
 8 vertices
 Possible parallel and/or perpendicular sides
 Possible acute, obtuse, and/or right angles
 9gon (nonagon)
 9 sides
 9 vertices
 Possible parallel and/or perpendicular sides
 Possible acute, obtuse, and/or right angles
 Decagon
 10 sides
 10 vertices
 Possible parallel and/or perpendicular sides
 Possible acute, obtuse, and/or right angles
 11gon (hendecagon)
 11 sides
 11 vertices
 Possible parallel and/or perpendicular sides
 Possible acute, obtuse, and/or right angles
 12gon (dodecagon)
 12 sides
 12 vertices
 Possible parallel and/or perpendicular sides
 Possible acute, obtuse, and/or right angles
 Classification of twodimensional figures based on attributes of sides and angles
Note(s):
 Grade Level(s):
 Grade 3 classified and sorted two and threedimensional figures, including cones, cylinders, spheres, triangular and rectangular prisms, and cubes, based on attributes using formal geometric language.
 Grade 5 will classify twodimensional figures in a hierarchy of sets and subsets using graphic organizers based on their attributes and properties.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 TxCCRS:
 III.A. Geometric and Spatial Reasoning – Figures and their properties
 III.A.1. Recognize characteristics and dimensional changes of two and threedimensional figures.
