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, TO SOLVE PROBLEMS 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.

8.3 
Proportionality. The student applies mathematical process standards to use proportional relationships to describe dilations. The student is expected to:


8.3C 
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.
Readiness Standard

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
Including, but not limited to:
 Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are counting (natural) numbers
 Various forms of positive rational numbers
 Counting (natural) numbers
 Decimals
 Fractions
 Percents
 Scale factor – the common multiplicative ratio between pairs of related data which may be represented as a unit rate
 Dilation – a transformation in which an image is usually enlarged or reduced, depending on the scale factor, in such a way that the angles of the original figure and the dilated figure are congruent and the sides of the original figure and the dilated figure are proportional creating similar figures
 Enlargement (scale factor >1)
 Reduction (0 < scale factor < 1)
 Congruent (scale factor = 1)
 Coordinate plane (all four quadrants)
 Center of dilation – a coordinate point that serves as the focal point for generating a dilation
 The ratio of the distance from the center of dilation to any point on the image compared to the distance from the center of dilation to the corresponding point on the preimage will result in the scale factor, k.
 Lines drawn through each point on the preimage and its corresponding image point will intersect at the center of dilation.
 Origin as center of dilation
 Algebraic representation to describe effects of dilations
 (x, y) → (kx, ky), where k is the scale factor used to dilate a figure about the origin
 Various representations of dilations
 Verbal
 Graphical
 Tabular
 Algebraic
Note(s):
 Grade Level(s):
 Grade 8 introduces using 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.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Representing, applying, and analyzing proportional relationships
 TxCCRS:
 III.B. Geometric and Spatial Reasoning – Transformations and symmetry
 III.B.1. Identify transformations and symmetries of figures.
 III.B.2. Use transformations to investigate congruence, similarity, and symmetries of figures.
 III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
 III.C.1. Make connections between geometry and algebraic equations.
 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.

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.

8.4C 
Use data from a table or graph to determine the rate of change or slope and yintercept in mathematical and realworld problems.
Readiness Standard

Use
DATA FROM A TABLE OR GRAPH TO DETERMINE THE RATE OF CHANGE OR SLOPE AND yINTERCEPT IN 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
 Determining rate of change or slope from various representations
 Table (horizontal/vertical)
 Graph
 Connections between unit rate, rate of change, and slope in mathematical and realworld problems
 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)
 Determining yintercept from various representations
 Table (horizontal/vertical)
 Graph
 Connections between the “starting point” (the output value when the input value is 0) and yintercept in mathematical and realworld problem situations
 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 =
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.
 Algebra I will graph linear functions on the coordinate plane and identify key features, including xintercept, yintercept, zeros, and slope, in 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.B. Functions – Analysis of functions
 VI.B.1. Understand and analyze features of functions.
 VI.C. Functions – Model realworld situations with functions
 VI.C.2. Develop a function to model a situation.
 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:


8.5D 
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:
 V.B. Statistical Reasoning – Describe data
 V.B.4. Describe patterns and departure from patterns in the study of data.
 VI.C. Functions – Model realworld situations with functions
 VI.C.1. Apply known functions to model realworld situations.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 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.

8.5G 
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.

8.5I 
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.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.

8.7 
Expressions, equations, and relationships. The student applies mathematical process standards to use geometry to solve problems. The student is expected to:


8.7C 
Use the Pythagorean Theorem and its converse to solve problems.
Readiness Standard

Use
THE PYTHAGOREAN THEOREM AND ITS CONVERSE TO SOLVE PROBLEMS
Including, but not limited to:
 Right triangle – a triangle with one right angle (exactly 90°) and two acute angles
 Legs of a right triangle – the two shortest sides of a right triangle
 Hypotenuse – the longest side of a right triangle, the side opposite the right angle
 Pythagorean Theorem
 Verbal
 The sum of the squares of the legs of a right triangle equals the square of the hypotenuse.
 Formula
 a^{2} + b^{2} = c^{2}, where a and b represent the legs of a right triangle and c represents the hypotenuse
 When solving for a, b, or c both the positive and negative numerical values should be considered, but since the applications are measurements the negative values do not apply.
 Converse of Pythagorean Theorem
 Verbal
 If the sum of the squares of the two shortest sides of a triangle equals the square of the third side, then the triangle is a right triangle.
 Formula
 a^{2} + b^{2} = c^{2}, where a and b represent the legs of a right triangle and c represents the hypotenuse
Note(s):
 Grade Level(s):
 Grade 8 introduces using the Pythagorean Theorem and its converse to solve 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.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 III.A. Geometric and Spatial Reasoning – Figures and their properties
 III.A.3. Recognize and apply right triangle relationships including basic trigonometry.
 III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
 III.D.3. Determine indirect measurements of geometric figures using a variety of methods.

8.8 
Expressions, equations, and relationships. The student applies mathematical process standards to use onevariable equations or inequalities in problem situations. The student is expected to:


8.8C 
Model and solve onevariable equations with variables on both sides of the equal sign that represent mathematical and realworld problems using rational number coefficients and constants.
Readiness Standard

Model, Solve
ONEVARIABLE EQUATIONS WITH VARIABLES ON BOTH SIDES OF THE EQUAL SIGN THAT REPRESENT MATHEMATICAL AND REALWORLD PROBLEMS USING RATIONAL NUMBER COEFFICIENTS AND CONSTANTS
Including, but not limited to:
 Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
 Variable – a letter or symbol that represents a number
 One variable on both sides of the equation
 Coefficient – a number that is multiplied by a variable(s)
 Integers
 Decimals (positive or negative)
 Fractions (positive or negative)
 Constant – a fixed value that does not appear with a variable(s)
 Integers
 Decimals (positive or negative)
 Fractions (positive or negative)
 Characteristics of equations
 Equates two expressions
 Equality of the variable
 One solution
 Equality words and symbol
 Relationship of order of operations within an equation
 Order of operations – the rules of which calculations are performed first when simplifying an expression
 Parentheses/brackets: simplify expressions inside parentheses or brackets in order from left to right
 Exponents: rewrite in standard numerical form and simplify from left to right
 Multiplication/division: simplify expressions involving multiplication and/or division in order from left to right
 Addition/subtraction: simplify expressions involving addition and/or subtraction in order from left to right
 Models to solve onevariable equations with variables on both sides of the equal sign (concrete, pictorial, algebraic)
 Solutions to onevariable equations with variables on both sides of the equal sign from mathematical and realworld problem situations
 Possible solutions
 One real solution
 No solution
 Infinite solutions (all real solutions)
Note(s):
 Grade Level(s):
 Grade 7 modeled and solved onevariable, twostep equations and inequalities.
 Algebra I will solve linear equations in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides.
 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.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
 II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations 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.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.1. Interpret results of the mathematical problem in terms of the original realworld situation.
 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.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 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.
 IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.

8.10 
Twodimensional shapes. The student applies mathematical process standards to develop transformational geometry concepts. The student is expected to:


8.10C 
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.
Readiness Standard

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
Including, but not limited to:
 Prime notation of image points
 Coordinate plane (all four quadrants)
 Single transformations
 Effects of transformations as algebraic representations
 Translation – a transformation frequently described as a slide of a figure
 Algebraic notation
 Translation h units horizontally
 Translation k units vertically
 Translation h units horizontally and k units vertically
 Reflection – a transformation frequently described as a flip or a mirror image of the original figure
 Algebraic notation
 Reflection across a vertical axis (yaxis)
 Reflection across a horizontal axis (xaxis)
 Rotation – a transformation frequently described as a turn of a figure around a designated point
 Origin as center of rotation
 Algebraic notation
 Rotation of 90° counterclockwise around the origin: (x, y) → (–y, x)
 Same as a rotation of 270° clockwise around the origin: (x, y) → (–y, x)
 Rotation of 180º counterclockwise around the origin: (x, y) → (–x, –y)
 Same as a rotation of 180º clockwise around the origin: (x, y) → (–x, –y)
 Rotation of 270º counterclockwise around the origin: (x, y) → (y, –x)
 Same as a rotation of 90º clockwise around the origin: (x, y) → (y, –x)
 Rotation of 360° counterclockwise or clockwise around the origin: (x, y) → (x, y)
 Determine the transformation performed from a graphed set of figures.
 Graph a transformation based on a given rule.
Note(s):
 Grade Level(s):
 Grade 8 introduces explaining 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.
 Geometry introduces rotations and dilations that may or may not be about the origin.
 Geometry introduces composite transformations.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Grade Level Connections (reinforces previous learning and/or provides development for future learning)
 TxCCRS:
 III.B. Geometric and Spatial Reasoning – Transformations and symmetry
 III.B.1. Identify transformations and symmetries of figures.
 III.B.2. Use transformations to investigate congruence, similarity, and symmetries of figures.
 III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
 III.C.1. Make connections between geometry and algebraic equations.

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:


8.12D 
Calculate and compare simple interest and compound interest earnings.
Readiness Standard

Calculate, Compare
SIMPLE INTEREST AND COMPOUND INTEREST EARNINGS
Including, but not limited to:
 Principal of an investment – the original amount invested
 Simple interest for an investment – interest paid on the original principal in an account, disregarding any previously earned interest
 Compound interest for an investment – interest that is calculated on the latest balance, including all compounded interest that has been added to the original principal investment
 Formulas for interest from STAAR Grade 8 Mathematics Reference Materials
 Simple interest
 I = Prt, where I represents the interest, P represents the principal amount deposited, r represents the interest rate in decimal form, and t represents the number of years the principal amount is deposited
 Compound interest
 A = P(1+ r)^{t}, where A represents the total accumulated amount including the principal and earned compounded interest, P represents the principal amount deposited, r represents the interest rate in decimal form, and t represents the number of years the principal amount is deposited
 Comparing simple and compound interest earnings
Note(s):
 Grade Level(s):
 Grade 7 calculated and compared simple interest and compound interest earnings.
 Grade 8 solves realworld problems comparing how interest rate and loan length affect the cost of credit.
 Algebra I will refer to 1 + r in the compound interest formula, A = P(1 + r)^{t}, as the factor and will be given the variable b.
 Mathematical Models with Applications will introduce analyzing compound interest for multiple compounding periods within a year.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.1. Compare relative magnitudes of rational and irrational numbers, and understand that numbers can be represented in different ways.
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 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.
