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.2C 
Determine an equation of a line parallel or perpendicular to a given line that passes through a given point.

Determine
AN EQUATION OF A LINE PARALLEL TO A GIVEN LINE THAT PASSES THROUGH A GIVEN POINT
Including, but not limited to:
 Parallel lines – lines in the same plane that never intersect. Slopes of parallel lines are equal, m_{y2} = m_{y}_{1}.
 Equation of a line
 Slopeintercept form, y = mx + b where m represents slope and b represents the yintercept
 Pointslope form, y – y_{1} = m(x – x_{1}) where m represents slope and (x_{1}, y_{1}) represents the given point
 Standard form, Ax + By = C where the slope, m = –
 Determination of the equation of a line given a slope and yintercept
 Determination of the equation of a line given a graph
 Determination of the equation of a line given a slope and a point
 Determination of the equation of a line given two points
Determine
AN EQUATION OF A LINE PERPENDICULAR TO A GIVEN LINE THAT PASSES THROUGH A GIVEN POINT
Including, but not limited to:
 Perpendicular lines – lines that intersect at a 90° angle to form right angles. Slopes of perpendicular lines are opposite reciprocals, m_{y2} = – .
 Equation of a line
 Slopeintercept form, y = mx + b where m represents slope and b represents the yintercept
 Pointslope form, y – y_{1} = m(x – x_{1}) where m represents slope and (x_{1}, y_{1}) represents the given point
 Standard form, Ax + By = C where the slope, m = –
 Determination of the equation of a line given a slope and yintercept
 Determination of the equation of a line given a graph
 Determination of the equation of a line given a slope and a point
 Determination of the equation of a line given two points
Note(s):
 Grade Level(s)
 Grades 7 and 8 represented linear nonproportional situations using multiple representations, including y = mx + b, where b ≠ 0.
 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.
 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.3 
Coordinate and transformational geometry. The student uses the process skills to generate and describe rigid transformations (translation, reflection, and rotation) and nonrigid transformations (dilations that preserve similarity and reductions and enlargements that do not preserve similarity). The student is expected to:


G.3A 
Describe and perform transformations of figures in a plane using coordinate notation.

Describe, Perform
TRANSFORMATIONS OF FIGURES IN A PLANE USING COORDINATE NOTATION
Including, but not limited to:
 Transformation – one to one mapping of points in a plane such that each point in the preimage has a unique image and each point in the image has a preimage
 Rigid transformation (isometric) – a transformation that preserves the size and shape of a figure
 Translation – rigid transformation moving all points in a geometric figure the same distance and the same direction
 Representations in coordinate notation: (x, y) → (x + a, y + b), a = horizontal shift, b = vertical shift
 Reflection – rigid transformation in which each point in a geometric figure is at an equal distance on the opposite side of a given line (line of symmetry)
 Reflection across the xaxis (x, y) → (x, –y)
 Reflection across the yaxis (x, y) → (–x, y)
 Reflection across the line y = x (x, y) → (y, x)
 Reflection across the line y = –x (x, y) → (–y, –x)
 Rotation – a rigid transformation where each point on the figure is rotated about a given point
 Rotations around the origin
 Rotation of 90º counterclockwise around the origin: (x, y) → (–y, x)
 Same as a rotation of 270º clockwise around the origin: (x, y) → (–y,x)
 Rotation of 180º counterclockwise around the origin: (x, y) → (–x, –y)
 Same as a rotation of 180º clockwise around the origin: (x, y) → (–x, –y)
 Rotation of 270º counterclockwise around the origin: (x, y) → (y, –x)
 Same as a rotation of 90º clockwise around the origin: (x, y) → (y, –x)
 Rotation of 360º around the origin: (x, y) → (x, y)
 Rotation around a given point other than the origin
 Apply a translation that moves the point of rotation to the origin and translate the figure using the same translation
 Rotate the figure about the origin
 Translate the point of rotation to its original position by the opposite translation and apply the same translation to the rotated figure
 NonRigid transformation (nonisometric) – a transformation that preserves similarity and reductions and enlargements that do not preserve similarity
 Dilation – a similarity transformation in which a figure is enlarged or reduced using a scale factor and a center of dilation
 Representations in coordinate notation: (x, y) → (kx, ky), k = scale factor when the center of dilation is the origin
 Representations in coordinate notation: (x, y) → (k(x – a) + a, k(y – b) + b), k = scale factor, (a, b) is the center of dilation
Note(s):
 Grade Level(s)
 Grade 8 introduced transformations including translations, reflections, rotations, and dilations.
 Grade 8 introduced dilations with the origin as the center.
 Grade 8 generalized the properties of orientation and congruence of rotations, reflections, translations, and dilations of twodimensional shapes on a coordinate plane.
 Grade 8 differentiated between transformations that preserved congruence and those that did not.
 Grade 8 introduced using an algebraic representation to explain the effect of a given positive rational scale factor applied to twodimensional figures on a coordinate plane with the origin as the center of dilation.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 III.B. Geometric and Spatial Reasoning – Transformations and symmetry
 III.B.1. Identify transformations and symmetries of figures.
 III.B.2. Use transformations to investigate congruence, similarity, and symmetries of figures.
 III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
 III.C.1. Make connections between geometry and algebraic equations.
 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.5 
Logical argument and constructions. The student uses constructions to validate conjectures about geometric figures. The student is expected to:


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
 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.

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 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 PYTHAGOREAN THEOREM, 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
 Pythagorean Theorem and the converse of the Pythagorean Theorem
 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.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.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.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
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 III.A. Geometric and Spatial Reasoning – Figures and their properties
 III.A.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.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.
 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.
 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.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
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 III.A. Geometric and Spatial Reasoning – Figures and their properties
 III.A.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.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.
 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.
 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.10B 
Determine and describe how changes in the linear dimensions of a shape affect its perimeter, area, surface area, or volume, including proportional and nonproportional dimensional change.

Determine, Describe
HOW CHANGES IN THE LINEAR DIMENSIONS OF A SHAPE AFFECT ITS PERIMETER, AREA, SURFACE AREA, OR VOLUME, INCLUDING PROPORTIONAL DIMENSIONAL CHANGE
Including, but not limited to:
 Verbal and written description
 Dimensional change
 Perimeter and circumference
 Area and surface area
 Volume
 Proportional change
 Threedimensional proportional change – three dimensions multiplied by the same scale factor
 Comparison of the effect of proportional dimensional change
 Emphasis on connections to units
 Dimension changes in realworld problem situations
Note(s):
 Grade Level(s)
 Grade 7 and 8 modeled the effect on linear and area measurements of dilated twodimensional shapes.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 I.C. Numeric Reasoning – Systems of measurement
 I.C.1. Select or use the appropriate type of method, unit, and tool for the attribute being measured.
 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.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.1. Find the perimeter and area of twodimensional figures.
 III.D.2. Determine the surface area and volume of threedimensional figures.
 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.
 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.11 
Twodimensional and threedimensional figures. The student uses the process skills in the application of formulas to determine measures of two and threedimensional figures. The student is expected to:


G.11B 
Determine the area of composite twodimensional figures comprised of a combination of triangles, parallelograms, trapezoids, kites, regular polygons, or sectors of circles to solve problems using appropriate units of measure.

Determine THE AREA OF COMPOSITE TWODIMENSIONAL FIGURES COMPRISED OF A COMBINATION OF TRIANGLES, PARALLELOGRAMS, TRAPEZOIDS, KITES, REGULAR POLYGONS, OR SECTORS OF CIRCLES TO SOLVE PROBLEMS USING APPROPRIATE UNITS OF MEASURE Including, but not limited to:  Composites of twodimensional figures
 Triangles
 Parallelograms
 Trapezoids
 Kites
 Regular polygons
 Sectors of circles
 Applications to realworld situations
 Appropriate use of units of measure
Note(s):
 Grade Level(s)
 Previous grade levels used units, tools, and formulas to find the area of figures in problem situations.
 Previous grade levels introduced composites of twodimensional figures.
 Geometry introduces kites, regular polygons, and sectors as shapes that can make up composite figures.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
 III.D.1. Find the perimeter and area of twodimensional figures.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.3. Determine a solution.

G.11C 
Apply the formulas for the total and lateral surface area of threedimensional figures, including prisms, pyramids, cones, cylinders, spheres, and composite figures, to solve problems using appropriate units of measure.

Apply
THE FORMULAS FOR THE TOTAL AND LATERAL SURFACE AREA OF THREEDIMENSIONAL FIGURES, INCLUDING PRISMS, PYRAMIDS, CONES, CYLINDERS, SPHERES, AND COMPOSITE FIGURES TO SOLVE PROBLEMS USING APPROPRIATE UNITS OF MEASURE
Including, but not limited to:
 Connections between formulas and models, including nets
 Formulas for lateral and total surface area
 Prisms
 Lateral surface area
 S = Perimeter of the base times the height
 S = Ph
 Total surface area
 S = Perimeter of the base times the height plus 2 times the area of the base
 S = Ph + 2B
 Pyramids
 Lateral surface area
 S = times the perimeter of the base times the slant height
 S = Pℓ
 Total surface area
 S = times the perimeter of the base times the slant height plus the area of the base
 S = Pℓ + B
 Cones
 Lateral surface area
 S = π times the radius times the slant height
 S = πrℓ
 Total surface area
 S = π times the radius times the slant height plus the area of the base
 S = πrℓ +πr^{2}
 Cylinders
 Lateral surface area
 S = 2 times π times the radius times the height
 S = 2πrh
 Total surface area
 S = 2 times π times the radius times the height plus 2 times the area of the base
 S = 2πrh + 2πr^{2}
 Spheres
 Total surface area
 S = 4 times π times the radius squared
 S = 4πr^{2}
 Composite figures
 Applications to realworld situations
 Use of appropriate units of measure
Note(s):
 Grade Level(s)
 Previous grade levels used units, tools, and formulas to find area of figures in problem situations.
 Previous grade levels introduced the terminology of regular polygons.
 Grade 7 determined the surface area of pyramids and prisms using nets.
 Grade 8 connects previous knowledge of surface area and nets to the surface area formulas.
 Geometry introduces cones, pyramids, spheres, and composite figures to the surface area formulas.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
 III.D.2. Determine the surface area and volume of threedimensional figures.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.3. Determine a solution.

G.11D 
Apply the formulas for the volume of threedimensional figures, including prisms, pyramids, cones, cylinders, spheres, and composite figures, to solve problems using appropriate units of measure.

Note(s):
 Grade Level(s)
 Grades 7 and 8 introduced determining and applying formulas to solve problems involving lateral and total surface area and volume of threedimensional figures.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
 III.D.2. Determine the surface area and volume of threedimensional figures.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.3. Determine a solution.

G.12 
Circles. The student uses the process skills to understand geometric relationships and apply theorems and equations about circles. The student is expected to:


G.12B 
Apply the proportional relationship between the measure of an arc length of a circle and the circumference of the circle to solve problems.

Apply
THE PROPORTIONAL RELATIONSHIP BETWEEN THE MEASURE OF AN ARC LENGTH OF A CIRCLE AND THE CIRCUMFERENCE OF THE CIRCLE TO SOLVE PROBLEMS
Including, but not limited to:
 Arc length of a circle – a fractional distance of the circumference of a circle defined by the arc
 Measure of arc length of is denoted as
 Connecting the proportional relationship or
 Applications to realworld problem situations
 Use of appropriate units of measure
 Use of various tools
 Protractor and straightedge
 Dynamic geometric software
 Patty paper
Note(s):
 Grade Level(s)
 Previous grade levels explored characteristics of circles and proportional relationships.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 III.A. Geometric and Spatial Reasoning – Figures and their properties
 III.A.2. Form and validate conjectures about one, two, and threedimensional figures and their properties.
 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.1. Find the perimeter and area of twodimensional figures.
 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.
 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.12C 
Apply the proportional relationship between the measure of the area of a sector of a circle and the area of the circle to solve problems.

Apply
THE PROPORTIONAL RELATIONSHIP BETWEEN THE MEASURE OF THE AREA OF A SECTOR OF A CIRCLE AND THE AREA OF THE CIRCLE TO SOLVE PROBLEMS
Including, but not limited to:
 Sector of a circle – a region of the circle bounded by a central angle and its intercepted arc
 Connecting the proportional relationship
 Applications to realworld problem situations
 Use of appropriate units of measure
 Use of various tools
 Protractor and straightedge
 Dynamic geometric software
 Patty paper
Note(s):
 Grade Level(s)
 Previous grade levels explored characteristics of circles and proportional relationships.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.
 III.A. Geometric and Spatial Reasoning – Figures and their properties
 III.A.2. Form and validate conjectures about one, two, and threedimensional figures and their properties.
 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.1. Find the perimeter and area of twodimensional figures.
 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.
 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.
