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


G.4D 
Compare geometric relationships between Euclidean and spherical geometries, including parallel lines and the sum of the angles in a triangle.

Compare
GEOMETRIC RELATIONSHIPS BETWEEN EUCLIDEAN AND SPHERICAL GEOMETRIES, INCLUDING THE SUM OF THE ANGLES IN A TRIANGLE
Including, but not limited to:
 Euclidean geometry – the study of plane and solid geometry based on definitions, undefined terms (point, line, plane) and the assumptions of mathematician Euclid
 Spherical geometry – the study of figures on the twodimensional curved surface of a sphere
 Definitions and undefined terms in Euclidean and spherical geometries
 Undefined terms
 Postulates and theorems in Euclidean and spherical geometries
 Sum of the angles in a triangle
Note(s):
 Grade Level(s)
 Geometry introduces the concept of systems of geometry, including Euclidean geometry and spherical geometry.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 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.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 CRITERIA REQUIRED FOR TRIANGLE CONGRUENCE, SPECIAL SEGMENTS OF TRIANGLES, 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)
 Criteria required for triangle congruence
 SideSideSide (SSS)
 If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.
 SideAngleSide (SAS)
 If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
 AngleSideAngle (ASA)
 If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
 AngleAngleSide (AAS)
 If two angles and a nonincluded side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
 HypotenuseLeg (HL)
 If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent.
 Special segments of triangles
 Altitude of a triangle – line segment drawn from any vertex of a triangle perpendicular to the opposite side
 Orthocenter (point of concurrency) – the intersection point of the altitudes of a triangle
 Orthocenter is not always in the interior of the triangle.
 Median of a triangle – line segment drawn from any vertex of a triangle to the midpoint of the opposite side
 Centroid (point of concurrency) – the intersection point of the medians of a triangle
 Centroid of a triangle is twice as far from a given vertex than it is from the midpoint to which the median from that vertex is drawn.
 Angle bisector – line, ray, or segment that divides an angle into two congruent angles
 Incenter (point of concurrency) – the intersection point of the angle bisectors of a triangle
 Incenter is equidistant from the three sides of the triangle (congruency of the radii of a circle).
 Incircle – a circle inscribed in a triangle, using the incenter as the center of the circle
 Perpendicular bisector of a side of a triangle – line segment that is perpendicular to a side of a triangle at the midpoint of the side
 Circumcenter (point of concurrency) – the intersection point of the perpendicular bisectors of a triangle
 Circumcenter is the center of the circle circumscribed about the triangle and is equidistant from all vertices of the triangle.
 Circumcenter of a triangle does not always exist in the interior of the triangle.
 Midsegment of a triangle – line segment drawn from the midpoints of two sides of the triangle
 Midsegment of a triangle is always parallel to the third side of the triangle whose midpoint is not included.
 Midsegment of a triangle is half as long as the third side of the triangle.
 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
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.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, CONGRUENT ANGLES, A SEGMENT BISECTOR, AN ANGLE BISECTOR, PERPENDICULAR LINES, AND THE PERPENDICULAR BISECTOR OF A LINE SEGMENT 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
 Segment bisector – point, line, ray, or segment that divides a line segment into two congruent segments
 Perpendicular bisector of a line segment – line, ray, or segment that divides a line segment into two congruent segments and forms a 90° angle at the point of intersection
 Angle bisector – line, ray, or segment that divides an angle into two congruent angles
 Perpendicular lines – lines that intersect at a 90° angle to form right angles
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, CONGRUENT ANGLES, ANGLE BISECTORS, AND PERPENDICULAR BISECTORS TO MAKE CONJECTURES ABOUT GEOMETRIC RELATIONSHIPS
Including, but not limited to:
 Geometric construction – construction 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
 Angle bisectors
 Perpendicular bisectors
 Perpendicular bisector of a segment
 Conjectures about attributes of figures related to the constructions
 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.

G.5D 
Verify the Triangle Inequality theorem using constructions and apply the theorem to solve problems.

Verify
THE TRIANGLE INEQUALITY THEOREM USING CONSTRUCTIONS
Including, but not limited to:
 Development of Triangle Inequality theorem using constructions of side lengths that form triangles and side lengths that do not form triangles
 Compass and straight edge
 Dynamic geometric software
 Patty paper
 Manipulatives
 Triangle Inequality theorem
 If a triangle has side of lengths a, b, and c, then a + b > c, a + c >b, and b + c >a.
Apply THE TRIANGLE INEQUALITY THEOREM TO SOLVE PROBLEMS Including, but not limited to:  Application of Triangle Inequality theorem to solve mathematical and realworld problems
Note(s):
 Grade Level(s)
 Previous grade levels investigated attributes of triangles.
 Grade 6 laid the foundation for the Triangle Inequality Theorem when verifying that a given set of lengths could form a triangle.
 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 verifies the Triangle Inequality Theorem using constructions.
 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.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.

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.6A 
Verify theorems about angles formed by the intersection of lines and line segments, including vertical angles, and angles formed by parallel lines cut by a transversal and prove equidistance between the endpoints of a segment and points on its perpendicular bisector and apply these relationships to solve problems.

Prove
EQUIDISTANCE BETWEEN THE ENDPOINTS OF A SEGMENT AND POINTS ON ITS PERPENDICULAR BISECTOR
Including, but not limited to:
 Methods of proof
 Coordinate geometry
 Two column proof
 Paragraph proof
 Flow chart
 Equidistant from endpoints of a segment and points on its perpendicular bisector
Apply
EQUIDISTANCE BETWEEN THE ENDPOINTS OF A SEGMENT AND POINTS ON ITS PERPENDICULAR BISECTOR TO SOLVE PROBLEMS
Including, but not limited to:
 Equidistant – equal distances
 Application of equidistant relationship between endpoints of a segment and points on its perpendicular bisector
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 proofs of geometric relationships.
 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.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.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.6B 
Prove two triangles are congruent by applying the SideAngleSide, AngleSideAngle, SideSideSide, AngleAngleSide, and HypotenuseLeg congruence conditions.

Prove TWO TRIANGLES ARE CONGRUENT BY APPLYING THE SIDEANGLESIDE, ANGLESIDEANGLE, SIDESIDESIDE, ANGLEANGLESIDE, AND HYPOTENUSELEG CONGRUENCE CONDITIONS Including, but not limited to:  Triangle congruence – triangles whose corresponding side lengths and corresponding angle measures are equal
 Congruent figures and their corresponding parts
 Corresponding parts of congruent triangles are congruent (CPCTC)
 Congruence conditions
 SideAngleSide (SAS)
 AngleSideAngle (ASA)
 SideSideSide (SSS)
 AngleAngleSide (AAS)
 HypotenuseLeg (HL)
Note(s):
 Grade Level(s)
 Previous grade levels investigated attributes of triangles.
 Geometry introduces proving triangles congruent by triangle congruence relationships.
 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.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.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.
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.

G.6C 
Apply the definition of congruence, in terms of rigid transformations, to identify congruent figures and their corresponding sides and angles.

Apply THE DEFINITION OF CONGRUENCE, IN TERMS OF RIGID TRANSFORMATIONS, TO IDENTIFY CONGRUENT FIGURES AND THEIR CORRESPONDING SIDES AND ANGLES Including, but not limited to:  Congruent figures – figures that are the same size and same shape
 Rigid transformations (isometric transformations, congruent transformations) – transformations where size and shape are preserved
 Corresponding sides and angles – sides and angles in two figures whose relative position is the same
 Congruent figures and their corresponding parts
 Corresponding parts of congruent triangles are congruent (CPCTC)
 Types of rigid transformations
 Translation
 Reflection
 Rotation
 Combinations of transformations
 Transformations on the coordinate plane
 Image points from original figure
 Original points from image figure
 Verbal description of transformation
 Application of geometric properties to find missing points
Note(s):
 Grade Level(s)
 Previous grade levels investigated attributes of geometric figures.
 Geometry introduces proving figures congruent by congruence relationships.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 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.
 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.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 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.6D 
Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems.

Verify
THEOREMS ABOUT THE RELATIONSHIPS IN TRIANGLES, INCLUDING THE SUM OF INTERIOR ANGLES, BASE ANGLES OF ISOSCELES TRIANGLES, MIDSEGMENTS, AND MEDIANS
Including, but not limited to:
 Interior angle of a polygon – angle on the inside of a polygon formed by pairs of adjacent sides
 Base angles of a triangle – the two angles that have one side in common with the base
 Concrete models and exploration activities
 Connections between models, pictures, and the symbolic formula
 Sum of interior angles
 Base angles of isosceles triangles
 Midsegments
 Medians
 Dynamic geometry software
Apply
THE RELATIONSHIPS IN TRIANGLES, INCLUDING THE SUM OF INTERIOR ANGLES, BASE ANGLES OF ISOSCELES TRIANGLES, MIDSEGMENTS, AND MEDIANS TO SOLVE PROBLEMS
Including, but not limited to:
 Determination of length and angle measurements using relationships in triangles as needed to solve realworld problem situations
 Sum of interior angles
 Bases angles of isosceles triangles
 Midsegments
 Medians
 Solutions with radical answers and rounded decimal answers
Note(s):
 Grade Level(s)
 Previous grade levels investigated attributes of triangles.
 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 proofs of conjectures about figures.
 Geometry introduces segments of a triangle.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 III. Geometric Reasoning
 A1 – Identify and represent the features of plane and space figures.
 A2 – Make, test, and use conjectures about one, two, and threedimensional figures and their properties.
 B2 – Identify the symmetries in a plane figure.
 D1 – Make and validate geometric conjectures.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

G.7 
Similarity, proof, and trigonometry. The student uses the process skills in applying similarity to solve problems. The student is expected to:


G.7A 
Apply the definition of similarity in terms of a dilation to identify similar figures and their proportional sides and the congruent corresponding angles.

Apply THE DEFINITION OF SIMILARITY IN TERMS OF A DILATION TO IDENTIFY SIMILAR FIGURES AND THEIR PROPORTIONAL SIDES AND THE CONGRUENT CORRESPONDING ANGLES Including, but not limited to:  Similar figures – shapes whose angles are congruent and side lengths are proportional (equal scale factor)
 Proportional sides – corresponding side lengths form equivalent ratios
 Corresponding angles – angles in two figures whose relative position is the same
 Scale factor
 Ratios to show dilation relationships
 Identification of similar figures
 Properties of similar triangles
 Applications to realworld situations
Note(s):
 Grade Level(s)
 Previous grade levels defined similarity, applied similarity to solve problems, and used dilations to transform 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.
 III.A.3. Recognize and apply right triangle relationships including basic trigonometry.
 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.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.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.

G.7B 
Apply the AngleAngle criterion to verify similar triangles and apply the proportionality of the corresponding sides to solve problems.

Apply THE ANGLEANGLE CRITERION TO VERIFY SIMILAR TRIANGLES Including, but not limited to:  AngleAngle criterion for triangle similarity – if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar
 Angle congruency for similar triangles
 Handson exploration
 Dynamic geometry software
Apply THE PROPORTIONALITY OF THE CORRESPONDING SIDES TO SOLVE PROBLEMS Including, but not limited to:  Formulation of equivalent ratios
 Proportional solutions for missing measures
 Applications to realworld situations
Note(s):
 Grade Level(s)
 Previous grade levels investigated similarity and proportionality.
 Previous grade levels solved problems involving similar 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 theorems of triangle similarity.
 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.A.3. Recognize and apply right triangle relationships including basic trigonometry.
 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.
 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.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.
 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.
 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.8 
Similarity, proof, and trigonometry. 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.8A 
Prove theorems about similar triangles, including the Triangle Proportionality theorem, and apply these theorems to solve problems.

Prove THEOREMS ABOUT SIMILAR TRIANGLES INCLUDING THE TRIANGLE PROPORTIONALITY THEOREM Including, but not limited to:  AngleAngle Similarity theorem for triangles
 If two angles of one triangle are congruent to two corresponding angles of another triangle, the triangles are similar.
 SideSideSide Similarity theorem
 If the measures of the corresponding sides of two triangles are proportional, the triangles are similar.
 SideAngleSide Similarity theorem
 If the measures of two sides of a triangle are proportional to the two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar.
 Triangle Proportionality theorem
 If a line intersects two sides of a triangle and is parallel to other side, then the line divides the two sides proportionally.
 Corollary of Triangle Proportionality theorem
 The line that divides the two sides of the triangle and is parallel to the other side creates a smaller triangle similar to the original
 Incorporation of a variety of methods
 Coordinate
 Transformational
 Axiomatic
 Application of a variety of formats
 Twocolumn proof
 Paragraph proof
 Flow chart
Apply THEOREMS ABOUT SIMILAR TRIANGLES TO SOLVE PROBLEMS Including, but not limited to:  Triangle Proportionality theorem
 Congruency of angles in similar triangles
 Proportionality of sides in similar triangles
 Applications to realworld situations
Note(s):
 Grade Level(s)
 Previous grade levels defined similarity and applied similarity to solve problems.
 Previous grade levels used direct variation to solve proportional problems.
 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.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.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.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.
