G.1 
Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:


G.1A 
Apply mathematics to problems arising in everyday life, society, and the workplace.

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.
 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.

G.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.

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.
 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.

G.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.

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
Including, but not limited to:
 Appropriate selection of tool(s) and techniques to apply in order to solve problems
 Tools
 Real objects
 Manipulatives
 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.
 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.

G.1D 
Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

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.
 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.

G.1E 
Create and use representations to organize, record, and communicate mathematical ideas.

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.
 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.

G.1F 
Analyze mathematical relationships to connect and communicate mathematical ideas.

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.
 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.

G.1G 
Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

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.
 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.

G.2 
Coordinate and transformational geometry. The student uses the process skills to understand the connections between algebra and geometry and uses the one and twodimensional coordinate systems to verify geometric conjectures. The student is expected to:


G.2A 
Determine the coordinates of a point that is a given fractional distance less than one from one end of a line segment to the other in one and twodimensional coordinate systems, including finding the midpoint.

Determine
THE COORDINATES OF A POINT IN A ONEDIMENSIONAL COORDINATE SYSTEM THAT IS A GIVEN FRACTIONAL DISTANCE LESS THAN ONE FROM ONE END OF A LINE SEGMENT TO THE OTHER, INCLUDING FINDING THE MIDPOINT
Including, but not limited to:
 A point in a onedimensional coordinate system that is a given fractional distance less than one from one end of a line segment to the other
 Onedimensional coordinate system – numbers used as locations of points on a number line
 Line segment – part of a line between two points on the line, called endpoints of the segment
 Distance formula: d = x_{1} – x_{2} or dx_{2}– x_{1}
 Midpoint of a line segment – the point halfway between the endpoints of a line segment
 For any k (where 0 < k < 1), the location on a number line that is the fraction k of the distance from any point, x_{1} to any other point, x_{2} and can be found using the expression x_{1} + k(x_{2} – x_{1}).
Note(s):
 Grade Level(s)
 Prior grade levels addressed points and distance on a number line.
 Prior grade levels addressed points and lines on a coordinate plane.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 I.A. Numeric Reasoning – Number representations and operations
 I.A.2. Perform computations with rational and irrational numbers.
 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.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 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.
 IX.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.

G.4 
Logical argument and constructions. The student uses the process skills with deductive reasoning to understand geometric relationships. The student is expected to:


G.4A 
Distinguish between undefined terms, definitions, postulates, conjectures, and theorems.

Distinguish
BETWEEN UNDEFINED TERMS, DEFINITIONS, POSTULATES, AND CONJECTURES
Including, but not limited to:
 Undefined terms – terms not formally defined and used to define other terms/concepts in a mathematical system
 Undefined terms in Euclidean geometry
 Definitions – words or terms used to describe new terms/concepts
 Postulates (axioms) – statements accepted as true without requiring proof
 Conjecture – statement believed to be true but not yet proven
 Representation and notation for undefined and defined terms
 Connections between undefined terms and defined terms
 Connections between definitions, postulates, and conjectures
 Conjectures are suspected of being true but can be disproved with a counterexample or proven with a logical argument at which point a conjecture becomes a theorem.
 Definitions and postulates can be used to support a logical argument that a conjecture is true.
Note(s):
 Grade Level(s)
 Geometry introduces the vocabulary of undefined terms, definitions, postulates, conjectures, and theorems.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
 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.

G.4B 
Identify and determine the validity of the converse, inverse, and contrapositive of a conditional statement and recognize the connection between a biconditional statement and a true conditional statement with a true converse.

Identify, Determine
THE VALIDITY OF THE CONVERSE, INVERSE, AND CONTRAPOSITIVE OF A CONDITIONAL STATEMENT
Including, but not limited to:
 Conditional statement – statement composed of a hypothesis (if) and a conclusion (then)
 Converse of a conditional statement – statement in which the hypothesis and conclusion of the original conditional statement are interchanged
 Inverse of a conditional statement – statement in which the hypothesis and conclusion of the original conditional statement are negated
 Contrapositive of a conditional statement – statement in which the hypothesis and conclusion of the original conditional statement are interchanged and negated
 Validity – determination of statement as either true or false
 Connections between conditional, converse, inverse, and contrapositive statements
 Conditional statements in a variety of forms
 If p then q
 q if p
 p implies q
 Symbolic representations
 p → q for if p then q
 ~p → ~q for if not p then not q
 Logical equivalence of a statement and its contrapositive
 Mathematical and nonmathematical conditional statements
 Examples and counterexamples
Recognize THE CONNECTION BETWEEN A BICONDITIONAL STATEMENT AND A TRUE CONDITIONAL STATEMENT WITH A TRUE CONVERSE Including, but not limited to:  Connection between a conditional statement and a biconditional statement
 Biconditional statement – statement for which both the conditional statement and its converse are true
 Truth value of a biconditional or related statement is determined by analysis of its hypothesis and conclusion (truth table, Venn diagram)
 A test for determining if a definition is a good definition is to test its converse, meaning a good definition is a biconditional statement.
Note(s):
 Grade Level(s)
 Geometry introduces conditional statements, including converse, inverse, contrapositive, and biconditional statements.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 III.A. Geometric and Spatial Reasoning – Figures and their properties
 III.A.2. Form and validate conjectures about one, two, and threedimensional figures and their properties.
 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.
 VII.C.2. Understand attributes and relationships with inductive and deductive reasoning.
 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.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.

G.4C 
Verify that a conjecture is false using a counterexample.

Verify A CONJECTURE IS FALSE USING A COUNTEREXAMPLE Including, but not limited to:  Counterexample – example used to disprove a statement
 Mathematical and nonmathematical examples and counterexamples
Note(s):
 Grade Level(s)
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 III.A. Geometric and Spatial Reasoning – Figures and their properties
 III.A.2. Form and validate conjectures about one, two, and threedimensional figures and their properties.
 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.
 VII.C.2. Understand attributes and relationships with inductive and deductive reasoning.
 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.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.

G.5 
Logical argument and constructions. The student uses constructions to validate conjectures about geometric figures. The student is expected to:


G.5B 
Construct congruent segments, congruent angles, a segment bisector, an angle bisector, perpendicular lines, the perpendicular bisector of a line segment, and a line parallel to a given line through a point not on a line using a compass and a straightedge.

Construct
CONGRUENT SEGMENTS AND CONGRUENT ANGLES USING A COMPASS AND A STRAIGHTEDGE
Including, but not limited to:
 Geometric construction – construction of accurate representations of lengths, angles, and geometric figures using only a straight edge and compass
 Congruent segments – line segments whose lengths are equal
 Congruent angles – angles whose angle measurements are equal
Note(s):
 Grade Level(s)
 Previous grade levels investigated attributes of geometric figures.
 Geometry introduces constructions.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 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.

G.5C 
Use the constructions of congruent segments, congruent angles, angle bisectors, and perpendicular bisectors to make conjectures about geometric relationships.

Use
THE CONSTRUCTIONS OF CONGRUENT SEGMENTS AND CONGRUENT ANGLES TO MAKE CONJECTURES ABOUT GEOMETRIC RELATIONSHIPS
Including, but not limited to:
 Geometric construction – creation of accurate representations of lengths, angles, and geometric figures using only a straight edge and compass
 Use of various tools
 Compass and straightedge
 Dynamic geometric software
 Patty paper
 Constructions
 Congruent segments
 Congruent angles
 Conjectures about attributes of figures related to the constructions
 Number line and segment addition
 Angle measure and angle addition
Note(s):
 Grade Level(s)
 Previous grade levels investigated attributes of geometric figures.
 Geometry introduces the use of constructions to make conjectures about geometric relationships.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 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.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.
