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:
 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.

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:
 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.

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:
 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.

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:
 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.

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:
 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.

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:
 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.

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:
 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.

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:
 VI.C. Functions – Model realworld situations with functions
 VI.C.2. Develop a function to model a situation.
 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.
 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.
 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.

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:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
 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.
 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.

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:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
 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.

8.5E 
Solve problems involving direct variation.
Supporting Standard

Solve
PROBLEMS INVOLVING DIRECT VARIATION
Including, but not limited to:
 Direct variation – a linear relationship between two variables, x (independent)and y (dependent), that always has a constant unchanged ratio, k, and can be represented by y = kx
 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.
 Recognition that when a problem situation has two variable quantities with a constant ratio, then the variable quantities have a relationship reflecting direct variation.
 Ratio, k, represents constant of variation or constant of proportionality
 Direct variation can also be phrased as direct proportion or directly proportional.
 Problem situations (e.g., circumference, conversions, unit rates, similarity, percents, etc.) involving prediction and comparison with justification
 Various solution methods for solving problems involving direct variation
 Table (horizontal/vertical)
 Graph
 Algebraic
Note(s):
 Grade Level(s):
 Grade 7 determined the constant of proportionality () within mathematical and realworld problems.
 Algebra I will write and solve equations involving direct variation.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Representing, applying, and analyzing proportional relationships.
 TxCCRS:
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 VI.B. Functions – Analysis of functions
 VI.B.2. Algebraically construct and analyze new functions.
 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.
 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.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.

8.5F 
Distinguish between proportional and nonproportional situations using tables, graphs, and equations in the form y = kx or y = mx + b, where b ≠ 0.
Supporting Standard

Distinguish
BETWEEN PROPORTIONAL AND NONPROPORTIONAL SITUATIONS USING TABLES, GRAPHS, AND EQUATIONS IN THE FORM y = kx OR 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 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 =
 Various representations
 Table (horizontal/vertical)
 Graph
 Equation
Note(s):
 Grade Level(s):
 Grade 7 represented linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = kx.
 Algebra I will write linear equations in 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:
 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.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.

8.5G 
Identify functions using sets of ordered pairs, tables, mappings, and graphs.
Readiness Standard

Identify
FUNCTIONS USING SETS OF ORDERED PAIRS, TABLES, MAPPINGS, AND GRAPHS
Including, but not limited to:
 Relation – a set of ordered pairs (x, y) where the x is associated with a specific y
 Function – relation in which each element of the input (x) is paired with exactly one element of the output (y)
 Distinguish between relations and functions
 All functions are relations but not all relations are functions.
 Various representations
 Sets of ordered pairs
 Tables (horizontal/vertical)
 Mappings – the process of pairing input and output in a function. Mappings are usually demonstrated by a diagram consisting of two lists, usually in ovals, with arrows associating items from the first list to items in the second list.
 Visual representation of a relation or a pairing of inputs with outputs
 Arrows connect inputs to corresponding outputs
 Can be used to quickly determine if a relation is a function
 Graphs
 Vertical line test can be used to determine if a relation is a function when graphing
Note(s):
 Grade Level(s):
 Grade 8 introduces identifying functions using sets of ordered pairs, tables, mappings, and graphs.
 Algebra I will introduce function notation.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Representing, applying, and analyzing proportional relationships
 TxCCRS:
 VI.A. Functions – Recognition and representation of functions
 VI.A.1. Recognize if a relation is a function.

8.5H 
Identify examples of proportional and nonproportional functions that arise from mathematical and realworld problems.
Supporting Standard

Identify
EXAMPLES OF PROPORTIONAL AND NONPROPORTIONAL FUNCTIONS THAT ARISE FROM MATHEMATICAL AND REALWORLD PROBLEMS
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
 Function – relation in which each element of the input (x) is paired with exactly one element of the output (y)
 Linear proportional function
 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 function
 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 =
 Various representations
 Verbal
 Table (horizontal/vertical)
 Graph
 Equation
 Generalizations about functions and linear proportional and linear nonproportional relationships in mathematical and realworld problem situations
 All linear proportional and linear nonproportional relationships are functions.
 Not all functions are linear proportional or linear nonproportional functions.
 Not all linear relationships are functions.
Note(s):
 Grade Level(s):
 Grade 8 introduces examples of proportional and nonproportional functions that arise from mathematical and realworld problems.
 Algebra I will introduce function notation.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Representing, applying, and analyzing 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.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.

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:
 II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
 II.C.1. Describe and interpret solution sets of equalities and inequalities.
 II.D. Algebraic Reasoning – Representing relationships
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.3. Determine a solution.
 VII.A.4. Justify the solution.
 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.

8.12 
Personal financial literacy. The student applies mathematical process standards to develop an economic way of thinking and problem solving useful in one's life as a knowledgeable consumer and investor. The student is expected to:


8.12C 
Explain how small amounts of money invested regularly, including money saved for college and retirement, grow over time.
Supporting Standard

Explain
HOW SMALL AMOUNTS OF MONEY INVESTED REGULARLY, INCLUDING MONEY SAVED FOR COLLEGE AND RETIREMENT, GROW OVER TIME
Including, but not limited to:
 Principal of an investment – the original amount invested
 Various types of investments
 Savings account – a bank or credit union account in which the money deposited earns interest so there will be more money in the future than originally deposited
 Traditional savings account – money put into a savings account much like paying a monthly expense such as a light bill or phone bill
 Taxable investment account – many companies will create an investment portfolio with the specific purpose of saving and building a strong portfolio to be used to pay for college
 Annuity – deductible and nondeductible contributions may be made, taxes may be waived if used for higher education; sold by financial institutions
 U.S. savings bond – money saved for a specific length of time and guaranteed by the federal government
 529 account – educational savings account managed by the state, and is usually taxdeferred
 Retirement savings – optional savings plans or accounts to which the employer can make direct deposits of an amount deducted from the employee's pay at the request of the employee
 401(k) – a set amount of money, or percentage of pay, that is set aside from an employee’s pay check by their employer, before the employee’s wages are taxed. The employer may or may not contribute as well to the employee’s 401(k) fund depending on employer’s policy. The money is taxed when it is withdrawn at retirement age. In addition, if withdrawn prior to retirement age an additional penalty tax is assessed.
 403(b) – a set amount of money, or percentage of pay, that is set aside from an employee’s pay check by their employer, before the employee’s wages are taxed. The money is taxed when it is withdrawn at retirement age. In addition, if withdrawn prior to retirement age an additional penalty tax is assessed.
 Similar to a 401(k); however, 403(b) plans are offered by nonprofit organizations
 Individual retirement account (IRA) – a set amount of money, or percentage of pay, that is invested by an individual with a bank, mutual fund, or brokerage.
 Social Security – a percentage of an employee's pay required by law that the employer withholds from the employee's pay for social security savings which is deposited into the federal retirement system; payment toward that employee's eventual retirement; the employer also is required to pay a matching amount for the employee into the federal retirement system.
 Generalizations of investing money regularly, including money for college and retirement
 Small amounts of money invested regularly build a larger principal amount to earn more interest
 A small amount of money invested for a longer period of time has the potential to earn as much interest as one large lump sum investment.
 Investing small amounts of money regularly may be more manageable for most people and demonstrates longterm financial planning and responsibility.
Note(s):
 Grade Level(s):
 Grade 7 analyzed and compared monetary incentives, including sales, rebates, and coupons.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 TxCCRS:
 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.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.

8.12G 
Estimate the cost of a twoyear and fouryear college education, including family contribution, and devise a periodic savings plan for accumulating the money needed to contribute to the total cost of attendance for at least the first year of college.
Supporting Standard

Estimate
THE COST OF A TWOYEAR AND FOURYEAR COLLEGE EDUCATION, INCLUDING FAMILY CONTRIBUTION
Including, but not limited to:
 Various considerations for each college
 School related costs
 Tuition (in state or out of state)
 Fees
 Room and board
 Books
 Cost of living in location (various costs of living depending on the city and state of college)
 Inflation – the general increase in prices and decrease in the purchasing value of money
 When planning ahead of time for college savings, the increase in all expenses based on inflation must be considered (e.g., tuition, room and board, etc.)
 Family contribution
Devise
A PERIODIC SAVINGS PLAN FOR ACCUMULATING THE MONEY NEEDED TO CONTRIBUTE TO THE TOTAL COST OF ATTENDANCE FOR AT LEAST THE FIRST YEAR OF COLLEGE
Including, but not limited to:
 Periodic savings plan
 Accumulating money to contribute to a savings plan
 Savings account – a bank or credit union account in which the money deposited earns interest so there will be more money in the future than originally deposited
 529 account – educational savings account managed by the state, and is usually taxdeferred
 Family contribution
 Plan for saving for college
 Estimate the total cost of attendance for each year at the college
 Determine what, if any, family contributions will be received
 Determine if a savings account was established to pay for college
 Calculate the cost of attending college and subtract the amount saved or contributed to determine yearly or monthly payments toward a college savings plan
 Creation of a budget to include a savings plan to cover the cost of college
Note(s):
 Grade Level(s):
 Grade 6 explained various methods to pay for college, including through savings, grants, scholarships, student loans, and workstudy.
 Grade 7 calculated and compared simple and compound interest earnings.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 TxCCRS:
 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.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.
