
Legend:  Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
 Student Expectations (TEKS) identified by TEA are in bolded, black text.
 Student Expectations (TEKS) are labeled Readiness as identified by TEA of the assessed curriculum.
 Student Expectations (TEKS) are labeled Supporting as identified by TEA of the assessed curriculum.
 Student Expectations (TEKS) are labeled Process standards as identified by TEA of the assessed curriculum.
 Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a
strikethrough.

Legend:  Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
 Unitspecific clarifications are in italicized, blue text.
 Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
 A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.

8.1 
Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:


8.1A 
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:
 Representing, applying, and analyzing proportional relationships
 Using expressions and equations to describe relationships, including the Pythagorean Theorem
 Making inferences from data
 TxCCRS:

8.1B 
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:
 Representing, applying, and analyzing proportional relationships
 Using expressions and equations to describe relationships, including the Pythagorean Theorem
 Making inferences from data
 TxCCRS:
 VIII. Problem Solving and Reasoning

8.1C 
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:
 Representing, applying, and analyzing proportional relationships
 Using expressions and equations to describe relationships, including the Pythagorean Theorem
 Making inferences from data
 TxCCRS:
 VIII. Problem Solving and Reasoning

8.1D 
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:
 Representing, applying, and analyzing proportional relationships
 Using expressions and equations to describe relationships, including the Pythagorean Theorem
 Making inferences from data
 TxCCRS:
 IX. Communication and Representation

8.1E 
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:
 Representing, applying, and analyzing proportional relationships
 Using expressions and equations to describe relationships, including the Pythagorean Theorem
 Making inferences from data
 TxCCRS:
 IX. Communication and Representation

8.1F 
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:
 Representing, applying, and analyzing proportional relationships
 Using expressions and equations to describe relationships, including the Pythagorean Theorem
 Making inferences from data
 TxCCRS:

8.1G 
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:
 Representing, applying, and analyzing proportional relationships
 Using expressions and equations to describe relationships, including the Pythagorean Theorem
 Making inferences from data
 TxCCRS:
 IX. Communication and Representation

8.4 
Proportionality. The student applies mathematical process standards to explain proportional and nonproportional relationships involving slope. The student is expected to:


8.4B 
Graph proportional relationships, interpreting the unit rate as the slope of the line that models the relationship.
Readiness Standard

Graph
PROPORTIONAL RELATIONSHIPS, INTERPRETING THE UNIT RATE AS THE SLOPE OF THE LINE THAT MODELS THE RELATIONSHIP
Including, but not limited to:
 Unit rate – a ratio between two different units where one of the terms is 1
 Slope – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or or , denoted as m in y = mx + b
 Linear proportional relationship
 Linear
 Passes through the origin (0, 0)
 Represented by y = kx
 Constant of proportionality represented as
 When b = 0 in y = mx + b, then k = the slope, m
 Graphing unit rate from various representations
 Verbal
 Numeric
 Tabular(horizontal/vertical)
 Symbolic/algebraic
 Connections between unit rate in proportional relationships to the slope of a line
Note(s):
 Grade Level(s):
 Algebra I will calculate the rate of change of a linear function represented tabularly, graphically, or algebraically in context of mathematical and realworld problems.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Representing, applying, and analyzing proportional relationships
 TxCCRS:
 I. Numeric Reasoning
 II. Algebraic Reasoning
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

8.5 
Proportionality. The student applies mathematical process standards to use proportional and nonproportional relationships to develop foundational concepts of functions. The student is expected to:


8.5A 
Represent linear proportional situations with tables, graphs, and equations in the form of y = kx.
Supporting Standard

Represent
LINEAR PROPORTIONAL SITUATIONS WITH TABLES, GRAPHS, AND EQUATIONS IN THE FORM OF y = kx
Including, but not limited to:
 Slope – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or or , denoted as m in y = mx + b
 yintercept – y coordinate of a point at which the relationship crosses the yaxis meaning the x coordinate is equal to zero, denoted as b in y = mx + b and the ordered pair (0, b)
 Linear relationship – a relationship with a constant rate of change represented by a graph that forms a straight line
 One quantity is dependent on the other
 Two quantities may be directly proportional to each other
 Can be classified as a positive or negative relationship
 Can be expressed as a pair of values that can be graphed as ordered pairs
 Graph of the ordered pairs matching the relationship will form a line
 Linear proportional problem situations
 Linear
 Passes through the origin (0, 0)
 Represented by y = kx
 Constant of proportionality represented as
 When b = 0 in y = mx + b, then k = the slope, m.
 Multiple representations of linear proportional problem situations
 Verbal
 Table (horizontal/vertical)
 Graph
 Algebraic
 Both y = kx and kx = y forms
 Association of k as multiplication by a given constant factor (including unit rate)
 Rational number coefficients and constants
 Manipulation of equations
Note(s):
 Grade Level(s):
 Grade 7 represented constant rates of change in mathematical and realworld problems given pictorial, tabular, verbal, numeric, graphical, and algebraic representations, including d = rt.
 Grade 7 converted between measurement systems, including the use of proportions and the use of unit rates.
 Algebra I will write and solve equations involving direct variation.
 Algebra I will use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions.
 Algebra I will write linear equations with two variables given a table of values, a graph, and a verbal description.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Representing, applying, and analyzing proportional relationships
 TxCCRS:
 I. Numeric Reasoning
 II. Algebraic Reasoning
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

8.5B 
Represent linear nonproportional situations with tables, graphs, and equations in the form of y = mx + b, where b ≠ 0.
Supporting Standard

Represent
LINEAR NONPROPORTIONAL SITUATIONS WITH TABLES, GRAPHS, AND EQUATIONS IN THE FORM OF y = mx + b, WHERE b ≠ 0
Including, but not limited to:
 Slope – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or or , denoted as m in y = mx + b
 yintercept – y coordinate of a point at which the relationship crosses the yaxis meaning the x coordinate is equal to zero, denoted as b in y = mx + b and the ordered pair (0, b)
 Linear relationship – a relationship with a constant rate of change represented by a graph that forms a straight line
 One quantity is dependent on the other
 Two quantities may be directly proportional to each other
 Can be classified as a positive or negative relationship
 Can be expressed as a pair of values that can be graphed as ordered pairs
 Graph of the ordered pairs matching the relationship will form a line
 Linear nonproportional problem situations
 Linear
 Does not pass through the origin (0, 0)
 Represented by y = mx + b, where b ≠ 0
 Constant slope represented as m = or m = or m =
 Multiple representations of linear nonproportional problem situations
 Verbal
 Table (horizontal/vertical)
 Graph
 Algebraic
 Both y = mx + b and mx + b = y forms
 Rational number coefficients and constants
 Manipulation of equations
Note(s):
 Grade Level(s):
 Grade 7 represented linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b.
 Algebra I will write linear equations in two variables in various forms, including y = mx + b, Ax + By = C, and y – y_{1} = m(x – x_{1}), given one point and the slope and given two points.
 Algebra I will use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions.
 Algebra I will write linear equations with two variables given a table of values, a graph, and a verbal description.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Representing, applying, and analyzing proportional relationships
 TxCCRS:
 I. Numeric Reasoning
 II. Algebraic Reasoning
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation

8.5C 
Contrast bivariate sets of data that suggest a linear relationship with bivariate sets of data that do not suggest a linear relationship from a graphical representation.
Supporting Standard

Contrast
BIVARIATE SETS OF DATA THAT SUGGEST A LINEAR RELATIONSHIP WITH BIVARIATE SETS OF DATA THAT DO NOT SUGGEST A LINEAR RELATIONSHIP FROM A GRAPHICAL REPRESENTATION
Including, but not limited to:
 Data – information that is collected about people, events, or objects
 Bivariate data – data relating two quantitative variables that can be represented by a scatterplot
 Discrete paired data – data that involves only distinct values that are finite or countable
 Scatterplot – a graphical representation used to display the relationship between discrete data pairs
 Characteristics of a scatterplot
 Title clarifies the meaning of the data represented.
 Subtitles clarify the meaning of data represented on each axis.
 Numerical data represented with labels may be whole numbers, fractions, or decimals.
 Points are not connected by a line.
 Scale of the axes may be intervals of one or more, and scale intervals are proportionally displayed.
 The scales of the axes are number lines.
 Linear relationship – a relationship with a constant rate of change represented by a graph that forms a straight line
 One quantity is dependent on the other
 Two quantities may be directly proportional to each other
 Can be classified as a positive or negative relationship
 Can be expressed as a pair of values that can be graphed as ordered pairs
 Graph of the ordered pairs matching the relationship will form a line
 Characteristics of bivariate data that suggests a linear relationship
 Linear proportional relationship
 Linear
 Passes through the origin (0, 0)
 Represented by y = kx
 Constant of proportionality represented as
 When b = 0 in y = mx + b, then k = the slope, m
 Linear nonproportional relationship
 Linear
 Does not pass through the origin (0, 0)
 Represented by y = mx + b, where b ≠ 0
 Constant slope represented as m = or m = or m =
 Characteristics of bivariate data that does not suggest a linear relationship
 Not linear
 Not represented by y = kx or y = mx + b
 No constant slope
 May or may not cross the origin (0, 0)
Note(s):
 Grade Level(s):
 Grade 8 introduces contrasting bivariate sets of data that suggest a linear relationship with bivariate sets of data that do not suggest a linear relationship from a graphical representation.
 Algebra I will calculate, using technology, the correlation coefficient between two quantitative variables and interpret this quantity as a measure of the strength of the linear association.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Representing, applying, and analyzing proportional relationships
 TxCCRS:
 II. Algebraic Reasoning
 VI. Statistical Reasoning
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation

8.5D 
Use a trend line that approximates the linear relationship between bivariate sets of data to make predictions.
Readiness Standard

Use
A TREND LINE THAT APPROXIMATES THE LINEAR RELATIONSHIP BETWEEN BIVARIATE SETS OF DATA TO MAKE PREDICTIONS
Including, but not limited to:
 Bivariate data – data relating two quantitative variables that can be represented by a scatterplot
 Characteristics of bivariate data that suggests a linear relationship
 Linear proportional relationship
 Linear
 Passes through the origin (0, 0)
 Represented by y = kx
 Constant of proportionality represented as
 When b = 0 in y = mx + b, then k = the slope, m.
 Linear nonproportional relationship
 Linear
 Does not pass through the origin (0, 0)
 Represented by y = mx + b, where b ≠ 0
 Constant slope represented as m = or m = or m =
 Graph of data suggests a constant rate of change between the independent and dependent values
 Trend line – the line that best fits the data points of a scatterplot
 A tool for making predictions by approximating the linear relationship between bivariate sets of data
 A trend line contains most of the data points and/or is situated so that the data points are evenly distributed above and below the line.
 Given or collected data
 Analysis of parts of data representation
 Title
 Labels
 Scales
 Graphed data
 Predictions of independent value when given a dependent value using a trend line that approximates the linear relationship
 Predictions of dependent value when given an independent value using a trend line that approximates the linear relationship
Note(s):
 Grade Level(s):
 Grade 8 introduces using a trend line that approximates the linear relationship between bivariate sets of data to make predictions.
 Algebra I will calculate, using technology, the correlation coefficient between two quantitative variables and interpret this quantity as a measure of the strength of the linear association.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Representing, applying, and analyzing proportional relationships
 TxCCRS:
 II. Algebraic Reasoning
 VI. Statistical Reasoning
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation

8.5I 
Write an equation in the form y = mx + b to model a linear relationship between two quantities using verbal, numerical, tabular, and graphical representations.
Readiness Standard

Write
AN EQUATION IN THE FORM y = mx + b TO MODEL A LINEAR RELATIONSHIP BETWEEN TWO QUANTITIES USING VERBAL, NUMERICAL, TABULAR, AND GRAPHICAL REPRESENTATIONS
Including, but not limited to:
 Slope – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or or , denoted as m in y = mx + b
 yintercept – y coordinate of a point at which the relationship crosses the yaxis meaning the x coordinate is equal to zero, denoted as b in y = mx + b and the ordered pair (0, b)
 Linear relationship – a relationship with a constant rate of change represented by a graph that forms a straight line
 One quantity is dependent on the other
 Two quantities may be directly proportional to each other
 Can be classified as a positive or negative relationship
 Can be expressed as a pair of values that can be graphed as ordered pairs
 Graph of the ordered pairs matching the relationship will form a line
 Linear nonproportional relationship
 Linear
 Does not pass through the origin (0, 0)
 Represented by y = mx + b, where b ≠ 0
 Constant slope represented as m = or m = or m =
 Equations in the form y = mx + b to represent relationships between two quantities
 Rational number coefficients and constants
 Various representations
 Verbal
 Numerical
 Tabular (horizontal/vertical)
 Graphical
Note(s):
 Grade Level(s):
 Grade 7 represented linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b.
 Algebra I will write linear equations in two variables given a table of values, a graph, and a verbal description.
 Algebra I will use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Representing, applying, and analyzing proportional relationships
 TxCCRS:
 II. Algebraic Reasoning
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation

8.9 
Expressions, equations, and relationships. The student applies mathematical process standards to use multiple representations to develop foundational concepts of simultaneous linear equations. The student is expected to:


8.9A 
Identify and verify the values of x and y that simultaneously satisfy two linear equations in the form y = mx + b from the intersections of the graphed equations.
Supporting Standard

Identify, Verify
THE VALUES OF x AND y THAT SIMULTANEOUSLY SATISFY TWO LINEAR EQUATIONS IN THE FORM y = mx + b FROM THE INTERSECTIONS OF THE GRAPHED EQUATIONS
Including, but not limited to:
 Slope – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or or , denoted as m in y = mx + b
 yintercept – y coordinate of a point at which the relationship crosses the yaxis meaning the x coordinate is equal to zero, denoted as b in y = mx + b and the ordered pair (0, b)
 Linear relationship – a relationship with a constant rate of change represented by a graph that forms a straight line
 Linear proportional relationship
 Linear
 Represented by y = kx or y = mx + b, where b = 0
 For y = kx and y = mx + b, k = the slope, m
 Passes through the origin (0, 0), meaning the yintercept, b, is 0
 Constant of proportionality represented as
 Constant slope represented as m = or m = or m =
 Linear nonproportional relationship
 Linear
 Represented by y = mx + b, where b ≠ 0
 Does not pass through the origin (0, 0), meaning the yintercept, b, is not 0
 Constant slope represented as m = or m = or m =
 Values of x and y that simultaneously satisfy two linear equations from a graph
 Simultaneously satisfy both linear equations means the intersection point or solution will lie on both lines
 Algebraic verification of the intersection of graphed equations as ordered pairs
 Intersection point or solution, when substituted into each equation, will result in true equations. If both equations are true, then the point of intersection simultaneously satisfies both equations.
 Intersection point and any other points on the same line result in equivalent slopes. If two lines contain the same point, then the point simultaneously satisfies both equations.
Note(s):
 Grade Level(s):
 Grade 7 determined if the given value(s) make(s) onevariable, twostep equations and inequalities true.
 Algebra I will graph systems of two linear equations in two variables on the coordinate plane and determine the solutions if they exist.
 Algebra I will solve systems of linear equations using concrete models, graphs, tables, and algebraic methods.
 Algebra I will estimate graphically the solutions to systems of two linear equations with two variables in realworld problems.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Using expressions and equations to describe relationships, including the Pythagorean Theorem
 TxCCRS:
 I. Numeric Reasoning
 II. Algebraic Reasoning
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation

8.11 
Measurement and data. The student applies mathematical process standards to use statistical procedures to describe data. The student is expected to:


8.11A 
Construct a scatterplot and describe the observed data to address questions of association such as linear, nonlinear, and no association between bivariate data.
Supporting Standard

Construct
A SCATTERPLOT
Including, but not limited to:
 Graph – a visual representation of the relationships between data collected
 Organization of data used to describe and summarize data
 Data – information that is collected about people, events, or objects
 Discrete paired data – data that involves only distinct values that are finite or countable
 Limitations
 Various forms of positive and negative rational numbers within related data pairs
 Integers
 Decimals
 Fractions
 Data representation
 Scatterplot – a graphical representation used to display the relationship between discrete data pairs
 Characteristics of a scatterplot
 Titles and subtitles
 Title represents the purpose of collected data
 Subtitles clarify the meaning of the data represented on each axis
 First quadrant of coordinate plane
 Number lines form xaxis and yaxis
 Proportional increments
 Intervals of one or more
 Break between 0 and the first marked interval indicated in one or both axes to accommodate large numbers if necessary
 Ordered pairs
 Pairs of data form each ordered pair
 Points not connected by a line
 Data pairs analyzed to find possible relationships between two sets of data
 Pairs of numbers collected to determine if a relationship exists between the two sets of data
 Relationship between each data pair is discrete although the data itself could be either continuous or discrete in nature
 Given or collected data
 Bivariate data – data relating two quantitative variables that can be represented by a scatterplot
Describe
THE OBSERVED DATA ON A SCATTERPLOT TO ADDRESS QUESTIONS OF ASSOCIATION SUCH AS LINEAR, NONLINEAR, AND NO ASSOCIATION BETWEEN BIVARIATE DATA
Including, but not limited to:
 Discrete paired data – data that involves only distinct values that are finite or countable
 Limitations
 Various forms of positive and negative rational numbers within related data pairs
 Integers
 Decimals
 Fractions
 Data representation
 Scatterplot – a graphical representation used to display the relationship between discrete data pairs
 Data pairs analyzed to find possible relationships between two sets of data
 Pairs of numbers collected to determine if a relationship exists between the two sets of data
 Relationship between each data pair is discrete although the data itself could be either continuous or discrete in nature
 Given or collected data
 Bivariate data – data relating two quantitative variables that can be represented by a scatterplot
 Association within a scatterplot
 Linear trend
 Positive trend
 Negative trend
 Nonlinear trend
 No trend or no association
Note(s):
 Grade Level(s):
 Grade 5 represented discrete paired data on a scatterplot.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Making inferences from data
 TxCCRS:
 II. Algebraic Reasoning
 VI. Statistical Reasoning
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
