6.1 
Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:


6.1A 
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:
 Using operations with integers and positive rational numbers to solve problems
 Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
 Using expressions and equations to represent relationships in a variety of contexts
 Understanding data representation
 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.

6.1B 
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:
 Using operations with integers and positive rational numbers to solve problems
 Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
 Using expressions and equations to represent relationships in a variety of contexts
 Understanding data representation
 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.

6.1C 
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 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
 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:
 Using operations with integers and positive rational numbers to solve problems
 Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
 Using expressions and equations to represent relationships in a variety of contexts
 Understanding data representation
 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.

6.1D 
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, 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:
 Using operations with integers and positive rational numbers to solve problems
 Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
 Using expressions and equations to represent relationships in a variety of contexts
 Understanding data representation
 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.

6.1E 
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:
 Using operations with integers and positive rational numbers to solve problems
 Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
 Using expressions and equations to represent relationships in a variety of contexts
 Understanding data representation
 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.

6.1F 
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:
 Using operations with integers and positive rational numbers to solve problems
 Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
 Using expressions and equations to represent relationships in a variety of contexts
 Understanding data representation
 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.

6.1G 
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:
 Using operations with integers and positive rational numbers to solve problems
 Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
 Using expressions and equations to represent relationships in a variety of contexts
 Understanding data representation
 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.

6.4 
Proportionality. The student applies mathematical process standards to develop an understanding of proportional relationships in problem situations. The student is expected to:


6.4A 
Compare two rules verbally, numerically, graphically, and symbolically in the form of y = ax or y = x + a in order to differentiate between additive and multiplicative relationships.
Supporting Standard

Compare
TWO RULES VERBALLY, NUMERICALLY, GRAPHICALLY, AND SYMBOLICALLY IN THE FORM OF y = ax OR y = x + a IN ORDER TO DIFFERENTIATE BETWEEN ADDITIVE AND MULTIPLICATIVE RELATIONSHIPS
Including, but not limited to:
 Rational numbers – the set of numbers that can be expressed as a fraction , where a and b are integers and b ≠ 0. The set of rational numbers is denoted by the symbol Q.
 Various forms of positive and negative rational numbers
 Integers
 Decimals
 Limited to positive decimal values
 Fractions
 Limited to positive fractional values
 Additive relationship – when a constant nonzero value is added to an input value to determine the output value (y = x + a)
 Multiplicative relationship – when a constant nonzero value is multiplied by an input value to determine the output value (y = ax)
 Independent variable – the variable in an equation or rule which represents the input value
 Dependent variable – the variable in an equation or rule which represents the output value
 Various representations of relationships
 Verbally
 Numerically
 Graphically
 Symbolically
 Relationships between multiple representations of additive and multiplicative relationships
Note(s):
 Grade Level(s):
 Grade 5 generated a numerical pattern when given a rule in the form y = ax or y = x + a and graph.
 Grade 5 recognized the difference between additive and multiplicative numerical patterns given in a table or graph.
 Grade 7 will represent constant rates of change in mathematical and realworld problems given pictorial, tabular, verbal, numeric, graphical, and algebraic representations, including d = rt.
 Grade 7 will determine the constant of proportionality () within mathematical and realworld problems.
 Grade 7 will represent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b.
 Grade 8 will represent linear proportional situations with tables, graphs, and equations in the form of y = kx.
 Grade 8 will represent linear nonproportional situations with tables, graphs, and equations in the form of y = mx + b, where b ≠ 0.
 Grade 8 will distinguish between proportional and nonproportional situations using tables, graphs, and equations in the form y = kx, or y = mx + b, where b ≠ 0.
 Grade 8 will identify examples of proportional and nonproportional functions that arise from mathematical and realworld problems.
 Grade 8 will write an equation in the form y = mx + b to model a linear relationship between two quantities using verbal, numerical, tabular, and graphical representations
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
 TxCCRS:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 VI.A. Functions – Recognition and representation of functions
 VI.A.2. Recognize and distinguish between different types of functions.

6.6 
Expressions, equations, and relationships. The student applies mathematical process standards to use multiple representations to describe algebraic relationships. The student is expected to:


6.6A 
Identify independent and dependent quantities from tables and graphs.
Supporting Standard

Identify
INDEPENDENT AND DEPENDENT QUANTITIES FROM TABLES AND GRAPHS
Including, but not limited to:
 Independent quantities are represented by the x coordinates or the input.
 Dependent quantities are represented by the y coordinates or the output.
 Identification of independent and dependent quantities
 Tables (horizontal/vertical)
 Graphs
Note(s):
 Grade Level(s):
 Grade 5 graphed in the first quadrant of the coordinate plane ordered pairs of numbers arising from mathematical and realworld problems, including those generated by number patterns or found in an inputoutput table.
 Grade 6 introduces identifying independent and dependent quantities from tables and graphs.
 Grade 7 will represent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Using expressions and equations to represent relationships in a variety of context
 Understanding data representation
 TxCCRS:
 VI.B. Functions – Analysis of functions
 VI.B.1. Understand and analyze features of functions.

6.6B 
Write an equation that represents the relationship between independent and dependent quantities from a table.
Supporting Standard

Write
AN EQUATION THAT REPRESENTS THE RELATIONSHIP BETWEEN INDEPENDENT AND DEPENDENT QUANTITIES FROM A TABLE
Including, but not limited to:
 Rational numbers – the set of numbers that can be expressed as a fraction , where a and b are integers and b ≠ 0. The set of rational numbers is denoted by the symbol Q.
 Independent quantities are represented by the x coordinates or the input.
 Dependent quantities are represented by the y coordinates or the output.
 Equations from a table of data
 In the form y = kx
 Coefficient – a number that is multiplied by a variable(s)
 Integers
 Products of integers limited to an integer multiplied by an integer
 Decimals
 Limited to positive decimal values
 Fractions
 Limited to positive fractional values
 In the form y = x + b
 Constant – a fixed value that does not appear with a variable(s)
 Integers
 Decimals
 Limited to positive decimal values
 Fractions
 Limited to positive fractional values
 Equations from a table of related data pairs, where the yvalue (output) is dependent upon the xvalue (input)
Note(s)
 Grade Level(s):
 Grade 5 graphed in the first quadrant of the coordinate plane ordered pairs of numbers arising from mathematical and realworld problems, including those generated by number patterns or found in an inputoutput table.
 Grade 6 introduces writing an equation that represents the relationship between independent and dependent quantities from a table.
 Grade 7 will represent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Using expressions and equations to represent relationships in a variety of contexts
 TxCCRS:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.

6.6C 
Represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b.
Readiness Standard

Represent
A GIVEN SITUATION USING VERBAL DESCRIPTIONS, TABLES, GRAPHS, AND EQUATIONS IN THE FORM y = kx OR y = x + b
Including, but not limited to:
 Independent quantities are represented by the x coordinates or the input.
 Dependent quantities are represented by the y coordinates or the output.
 Various representations to describe algebraic relationships
 Verbal descriptions
 Tables
 Graphs
 Equations
 In the form y = kx, where k is the nonzero scale factor (constant of proportionality), from multiplicative problem situations
 Coefficient – a number that is multiplied by a variable(s)
 Integers
 Products of integers limited to an integer multiplied by an integer
 Decimals
 Limited to positive decimal values
 Fractions
 Limited to positive fractional values
 In the form y = x + b, where b is the constant nonzero addend, from additive problem situations
 Constant – a fixed value that does not appear with a variable(s)
 Integers
 Decimals
 Limited to positive decimal values
 Fractions
 Limited to positive fractional values
Note(s):
 Grade Level(s):
 Grade 6 introduces representing a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b.
 Grade 7 will represent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b.
 Grade 8 will represent linear proportional situations with tables, graphs, and equations in the form of y = kx.
 Grade 8 will represent linear nonproportional situations with tables, graphs, and equations in the form of y = mx + b, where b ≠ 0.
 Grade 8 will write an equation in the form y = mx + b to model a linear relationship between two quantities using verbal, numerical, tabular, and graphical representations.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Using expressions and equations to represent relationships in a variety of contexts
 TxCCRS:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
 VI.C. Functions – Model realworld situations with functions
 VI.C.2. Develop a function to model a situation.
 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.

6.11 
Measurement and data. The student applies mathematical process standards to use coordinate geometry to identify locations on a plane. The student is expected to:
Readiness Standard


6.11A 
Graph points in all four quadrants using ordered pairs of rational numbers.
Readiness Standard

Graph
POINTS IN ALL FOUR QUADRANTS USING ORDERED PAIRS OF RATIONAL NUMBERS
Including, but not limited to:
 Rational numbers – the set of numbers that can be expressed as a fraction , where a and b are integers and b ≠ 0. The set of rational numbers is denoted by the symbol Q.
 Various forms of positive and negative rational numbers as ordered pairs
 Integers
 Decimals
 Fractions
 Coordinate plane (coordinate grid) – a twodimensional plane on which to plot points, lines, and curves
 Axes – the vertical and horizontal lines that act as a reference when plotting points on a coordinate plane
 Intersecting lines – lines that meet or cross at a point
 Origin – the starting point in locating points on a coordinate plane
 Quadrants – any of the four areas created by dividing a plane with an xaxis and yaxis
 Attributes of the coordinate plane
 Two number lines intersect perpendicularly to form the axes which are used to locate points on the plane
 The horizontal number line is called the xaxis.
 The vertical number line is called the yaxis.
 The xaxis and the yaxis cross at 0 on both number lines and that intersection is called the origin.
 The ordered pair of numbers corresponding to the origin is (0, 0).
 Four quadrants are formed by the intersection of the xand yaxes and are labeled counterclockwise with Roman numerals beginning with Quadrant I that includes the positive x and yvalues.
 Quadrant I (+, +)
 Quadrant II (–, +)
 Quadrant III (–, –)
 Quadrant IV (+, –)
 Iterated units are labeled and shown on both axes to show scale.
 Intervals may or may not be increments of one.
 Intervals may or may not include decimal or fractional amounts.
 Relationship between ordered pairs and attributes of the coordinate plane
 A pair of ordered numbers names the location of a point on a coordinate plane.
 Ordered pairs of numbers are indicated within parentheses and separated by a comma. (x, y)
 The first number in the ordered pair represents the parallel movement on the xaxis, left or right starting at the origin.
 The second number in the ordered pair represents the parallel movement on the yaxis, up or down starting at the origin.
 Process for graphing ordered pairs of numbers on the coordinate plane
 To locate the x coordinate, begin at the origin and move to the right or left along the xaxis the appropriate number of units according to the x coordinate in the ordered pair.
 To locate the y coordinate, begin at the origin and move up or down along the yaxis the appropriate number of units according to the y coordinate in the ordered pair.
 The point of intersection of both the parallel movements on the xaxis and the yaxis is the location of the ordered pair.
 Plot or select a point on a coordinate plane to satisfy a situation
Note(s):
 Grade Level(s):
 Grade 5 described the key attributes of the coordinate plane, including perpendicular number lines (axes) where the intersection (origin) of the two lines coincides with zero on each number line and the given point (0, 0), the x coordinate, the first number in the ordered pair, indicates movement parallel to the xaxis starting at the origin, the y coordinate, the second number, indicates movement parallel to the yaxis starting at the origin.
 Grade 5 described the process for graphing ordered pairs of numbers in the first quadrant of the coordinate plane.
 Grade 5 graphed in the first quadrant of the coordinate plane ordered pairs of numbers arising from mathematical and realworld problems, including those generated by number patterns or found in an inputoutput table.
 Grade 8 will use an algebraic representation to explain the effect of a given positive rational scale factor applied to twodimensional figures on a coordinate plane with the origin as the center of dilation.
 Grade 8 will explain the effect of translations, reflections over the x or yaxis, and rotations limited to 90°, 180°, 270°, and 360° as applied to twodimensional shapes on a coordinate plane using an algebraic representation.
 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)
