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.2B 
Derive and use the distance, slope, and midpoint formulas to verify geometric relationships, including congruence of segments and parallelism or perpendicularity of pairs of lines.

Use
THE DISTANCE, SLOPE, AND MIDPOINT FORMULAS TO VERIFY GEOMETRIC RELATIONSHIPS, INCLUDING CONGRUENCE OF SEGMENTS AND PARALLELISM OR PERPENDICULARITY OF PAIRS OF LINES
Including, but not limited to:
 Distance formula: d =
 Formula for slope of a line: m =
 Formula for midpoint of a line segment on a coordinate plane:
 Congruent segments – line segments whose lengths are equal
 Midpoint of a line segment – the point halfway between the endpoints of a line segment
 Parallel lines – lines in the same plane that never intersect. Slopes of parallel lines are equal, m_{y2} = m_{y}_{1}.
 Perpendicular lines – lines that intersect at a 90° angle to form right angles. Slopes of perpendicular lines are opposite reciprocals, m_{y}_{2} = – .
 Equation of a line
 Slopeintercept form, y = mx + b
 Pointslope form, y – y_{1} = m(x – x_{1})
 Standard form, Ax + By = C
 Coordinate geometry proofs
 Identification of polygons
 Types of quadrilaterals
 Trapezium
 Kite
 Trapezoid
 Parallelogram
 Rectangle
 Rhombus
 Square
Note(s):
 Grade Level(s)
 Grade 8 introduced and applied the Pythagorean Theorem to determine the distance between two points on a coordinate plane.
 Grade 8 introduced slope as or .
 Algebra I addressed determining equations of lines using pointslope form, slope intercept form, and standard form.
 Algebra I wrote equations of lines that contain a given point and are parallel or perpendicular to a given line.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 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.
 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.5A 
Investigate patterns to make conjectures about geometric relationships, including angles formed by parallel lines cut by a transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons, and special segments and angles of circles choosing from a variety of tools.

Investigate
PATTERNS TO MAKE CONJECTURES ABOUT GEOMETRIC RELATIONSHIPS, INCLUDING DIAGONALS OF QUADRILATERALS AND INTERIOR AND EXTERIOR ANGLES OF POLYGONS CHOOSING FROM A VARIETY OF TOOLS
Including, but not limited to:
 Conjecture – statement believed to be true but not yet proven
 Investigations should include good sample design, valid conjecture, and inductive/deductive reasoning.
 Patterns include numeric and geometric properties.
 Utilization of a variety of tools in the investigations (e.g., compass and straightedge, paper folding, manipulatives, dynamic geometry software, technology)
 Diagonals of quadrilaterals
 Isosceles trapezoid: Diagonals are congruent.
 Parallelogram: Diagonals bisect each other.
 Rectangle: Diagonals are congruent and exhibit all properties of the diagonals of a parallelogram.
 Rhombus: Diagonals bisect angles, are perpendicular to each other, and exhibit all properties of the diagonals of a parallelogram.
 Square: Exhibits all properties of a parallelogram, rectangle, and rhombus.
 Interior and exterior angles of a polygon
 Interior angle of a polygon – angle on the inside of a polygon formed by pairs of adjacent sides
 Exterior angle of a polygon – angle on the outside of a polygon formed by the side of a polygon and an extension of its adjacent side
 Relationship between interior and exterior angles (one pair per vertex)
 Concave and convex polygons
 Sum of the interior angles of a polygon: sum = 180(n – 2) degrees, where n is the number of sides of the polygon
 Regular polygon – a convex polygon in which all sides are congruent (equilateral) and all angles are congruent (equiangular)
 Measure of each interior angle: ∠ measure = degrees, where n is the number of sides of the polygon
 Irregular polygon – polygon that is not equilateral or equiangular
 Sum of the exterior angles of a polygon (one angle per vertex): sum = 360°
 Regular polygon
 Measure of each exterior angle: ∠ measure = degrees, where n is the number of sides of the polygon
Note(s):
 Grade Level(s)
 Previous grade levels investigated attributes of geometric figures.
 Grade 8 used informal arguments to establish facts about the angle sum and exterior angle of triangles, the angles created when parallel lines are cut by a transversal, and the AngleAngle criterion for similarity of triangles.
 Geometry introduces analyzing patterns in geometric relationships and making conjectures about geometric relationships which may or may not be represented using algebraic expressions.
 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.
 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.
 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.
 VII.C.2. Understand attributes and relationships with inductive and deductive reasoning.

G.6 
Proof and congruence. The student uses the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as coordinate, transformational, and axiomatic and formats such as twocolumn, paragraph, and flow chart. The student is expected to:


G.6E 
Prove a quadrilateral is a parallelogram, rectangle, square, or rhombus using opposite sides, opposite angles, or diagonals and apply these relationships to solve problems.

Prove A QUADRILATERAL IS A PARALLELOGRAM, RECTANGLE, SQUARE, OR RHOMBUS USING OPPOSITE SIDES, OPPOSITE ANGLES, OR DIAGONALS Including, but not limited to:  Identification of type of quadrilateral
 Parallelogram
 Rectangle
 Square
 Rhombus
 Comparisons by coordinate proofs
 Opposite sides
 Opposite angles
 Diagonals
Apply RELATIONSHIPS IN QUADRILATERALS TO SOLVE PROBLEMS Including, but not limited to:  Mathematical problem situations involving quadrilaterals
 Realworld problem situations involving quadrilaterals
 Parallelogram
 Rectangle
 Square
 Rhombus
Note(s):
 Grade Level(s)
 Previous grade levels investigated attributes of quadrilaterals.
 Geometry introduces coordinate proofs of conjectures about figures.
 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.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VII.A.3. Determine a 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.
 VII.C.2. Understand attributes and relationships with inductive and deductive reasoning.
 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.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.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.10 
Twodimensional and threedimensional figures. The student uses the process skills to recognize characteristics and dimensional changes of two and threedimensional figures. The student is expected to:


G.10A 
Identify the shapes of twodimensional crosssections of prisms, pyramids, cylinders, cones, and spheres and identify threedimensional objects generated by rotations of twodimensional shapes.

Identify THE SHAPES OF TWODIMENSIONAL CROSSSECTIONS OF PRISMS, PYRAMIDS, CYLINDERS, CONES, AND SPHERES Including, but not limited to:  Threedimensional figures
 Prisms
 Cylinders
 Pyramids
 Cones
 Spheres
 Description of cross sectional intersections
 Representations of cross sections
 Point
 Line segment
 Region of a plane (polygon or circle)
Identify THREEDIMENSIONAL OBJECTS GENERATED BY ROTATIONS OF TWODIMENSIONAL SHAPES Including, but not limited to:  Rectangle
 Triangle
 Semicircle
 Trapezoid
 Rotation about xaxis and yaxis
Note(s):
 Grade Level(s)
 Geometry introduces solids formed by revolving two dimensional figures.
 TxCCRS
 III.A. Geometric and Spatial Reasoning – Figures and their properties
 III.A.1. Recognize characteristics and dimensional changes of two and threedimensional figures.
