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:

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:
 VIII. Problem Solving and Reasoning

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:
 VIII. Problem Solving and Reasoning

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:
 IX. Communication and Representation

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:
 IX. Communication and Representation

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:

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:
 IX. Communication and Representation

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, 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
 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.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
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
 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.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.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 PROOF OF THE PYTHAGOREAN THEOREM
Including, but not limited to:
 Concrete models and exploration activities
 Connections between models, pictures, and the symbolic formula
 Proof of the Pythagorean Theorem
 Dynamic geometry software
Apply
THE RELATIONSHIPS IN TRIANGLES, INCLUDING THE PYTHAGOREAN THEOREM, 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
 Pythagorean Theorem and the converse of the Pythagorean Theorem
 Solutions with radical answers and rounded decimal answers
Note(s):
 Grade Level(s)
 Previous grade levels investigated attributes of triangles.
 Grade 8 introduced and applied the Pythagorean Theorem and the converse of the Pythagorean Theorem to solve problems.
 Grade 8 used models and diagrams to explain the Pythagorean Theorem.
 Geometry proves the Pythagorean Theorem and uses the Pythagorean Theorem and the converse of the Pythagorean Theorem to solve problems.
 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. Geometric Reasoning
 B1 – Identify and apply transformations to figures
 B3 – Use congruence transformations and dilations to investigate congruence, similarity, and symmetries of plane figures.
 D1 – Make and validate geometric conjectures.
 IV. Measurement Reasoning
 C3 – Determine indirect measurements of figures using scale drawings, similar figures, Pythagorean Theorem, and basic trigonometry.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

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.8B 
Identify and apply the relationships that exist when an altitude is drawn to the hypotenuse of a right triangle, including the geometric mean, to solve problems.

Identify, Apply
THE RELATIONSHIPS THAT EXIST WHEN AN ALTITUDE IS DRAWN TO THE HYPOTENUSE OF A RIGHT TRIANGLE, INCLUDING THE GEOMETRIC MEAN, TO SOLVE PROBLEMS
Including, but not limited to:
 Altitude of a triangle – line segment drawn from any vertex of a triangle perpendicular to the opposite side
 Hypotenuse of a right triangle – the longest side of a right triangle, the side opposite the right angle
 In a right triangle, the measure of the altitude from the vertex of the right angle to the hypotenuse is the geometric mean between the measures of the two segments formed where the altitude intersects the hypotenuse.
 If an altitude is drawn to the hypotenuse of a right triangle, then the length of either leg is the geometric mean of the length of the hypotenuse and the length of the segment of the hypotenuse adjacent to that leg.
 Geometric mean – a positive number x such that , so x^{2} = ab and x =
 Connection to similar triangles
 Concrete models and exploration activities
 Dynamic geometric software
 Realworld problem situations
Note(s):
 Grade Level(s)
 Geometry introduces the geometric mean.
 Previous grade levels solved problems involving similar figures.
 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.
 B1 – Identify and apply transformations to figures.
 B3 – Use congruence transformations and dilations to investigate congruence, similarity, and asymmetries of plane figures.
 D1 – Make and validate geometric conjectures.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

G.9 
Similarity, proof, and trigonometry. The student uses the process skills to understand and apply relationships in right triangles. The student is expected to:


G.9A 
Determine the lengths of sides and measures of angles in a right triangle by applying the trigonometric ratios sine, cosine, and tangent to solve problems.

Determine
THE LENGTHS OF SIDES AND MEASURES OF ANGLES IN A RIGHT TRIANGLE BY APPLYING THE TRIGONOMETRIC RATIOS SINE, COSINE, AND TANGENT TO SOLVE PROBLEMS
Including, but not limited to:
 Trigonometric ratios – a ratio of the measures of two sides of a right triangle based on their position in relation to an acute angle in the right triangle
 Sine
 Cosine
 Tangent
 Right triangles
 Side lengths
 Angle measures
 Applications to realworld situations
Note(s):
 Grade Level(s)
 Middle School introduced ratios and unit rates when developing proportionality.
 Grade 8 uses the Pythagorean Theorem and its converse to solve problems.
 Geometry introduces trigonometric ratios.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 III. Geometric Reasoning
 A3 – Recognize and apply right triangle relationships including basic trigonometry.
 C1 – Make connections between geometry and algebra.
 C3 – Make connections between geometry and measurement.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

G.9B 
Apply the relationships in special right triangles 30°60°90° and 45°45°90° and the Pythagorean theorem, including Pythagorean triples, to solve problems.

Note(s):
 Grade Level(s)
 Grade 8 used the Pythagorean Theorem and its converse to solve problems.
 Geometry introduces special right triangles.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 III. Geometric Reasoning
 A3 – Recognize and apply right triangle relationships including basic trigonometry.
 C3 – Make connections between geometry and measurement.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections
