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


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

Apply
MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE
Including, but not limited to:
 Mathematical problem situations within and between disciplines
 Everyday life
 Society
 Workplace
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxCCRS:
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.1. Interpret results of the mathematical problem in terms of the original realworld situation.
 IX.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
 IX.A.2. Connect mathematics to the study of other disciplines.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.
 IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
 IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.

G.1B 
Use a problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process and the reasonableness of the solution.

Use
A PROBLEMSOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEMSOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:
 Problemsolving model
 Analyze given information
 Formulate a plan or strategy
 Determine a solution
 Justify the solution
 Evaluate the problemsolving process and the reasonableness of the solution
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxCCRS:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.1. Use estimation to check for errors and reasonableness of solutions.
 V.A. Statistical Reasoning – Design a study
 V.A.1. Formulate a statistical question, plan an investigation, and collect data.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VII.A.2. Formulate a plan or strategy.
 VII.A.3. Determine a solution.
 VII.A.4. Justify the solution.
 VII.A.5. Evaluate the problemsolving process.
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.2. Evaluate the problemsolving process.

G.1C 
Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select
TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS
Including, but not limited to:
 Appropriate selection of tool(s) and techniques to apply in order to solve problems
 Tools
 Real objects
 Manipulatives
 Paper and pencil
 Technology
 Techniques
 Mental math
 Estimation
 Number sense
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxCCRS:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.1. Use estimation to check for errors and reasonableness of solutions.
 V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
 V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.

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

Communicate
MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE
Including, but not limited to:
 Mathematical ideas, reasoning, and their implications
 Multiple representations, as appropriate
 Symbols
 Diagrams
 Graphs
 Language
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxCCRS:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
 IX.B. Connections – Connections of mathematics to nature, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld situations.

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

Create, Use
REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:
 Representations of mathematical ideas
 Organize
 Record
 Communicate
 Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
 Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxCCRS:
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.

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

Analyze
MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:
 Mathematical relationships
 Connect and communicate mathematical ideas
 Conjectures and generalizations from sets of examples and nonexamples, patterns, etc.
 Current knowledge to new learning
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxCCRS:
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
 VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
 VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
 IX.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
 IX.A.2. Connect mathematics to the study of other disciplines.

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

Display, Explain, Justify
MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION
Including, but not limited to:
 Mathematical ideas and arguments
 Validation of conclusions
 Displays to make work visible to others
 Diagrams, visual aids, written work, etc.
 Explanations and justifications
 Precise mathematical language in written or oral communication
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxCCRS:
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.4. Justify the solution.
 VII.B. Problem Solving and Reasoning – Proportional reasoning
 VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
 VII.C. Problem Solving and Reasoning – Logical reasoning
 VII.C.1. Develop and evaluate convincing arguments.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
 VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII. C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.

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


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

Determine
THE COORDINATES OF A POINT IN A TWODIMENSIONAL COORDINATE SYSTEM THAT IS A GIVEN FRACTIONAL DISTANCE LESS THAN ONE FROM ONE END OF A LINE SEGMENT TO THE OTHER, INCLUDING FINDING THE MIDPOINT
Including, but not limited to:
 Twodimensional coordinate system – two axes at right angles to each other, forming a xyplane, consisting of points (x, y)
 A point in a twodimensional coordinate system that is a given fractional distance less than one from one end of a line segment to the other
 Distance between coordinates for (x, y): d_{x} = x_{1} – x_{2} or d_{x} = x_{2} – x_{1} and d_{y} = y_{1} – y_{2} or d_{y} = y_{2} – y_{1}
 For any k (where 0 < k < 1), the coordinates for a point on a segment that is the fraction k of the distance from the first endpoint (x_{1}, y_{1}) to the second endpoint (x_{2}, y_{2}) can be found using the expression (x_{1} + k(x_{2} – x_{1}), y_{1} + k(y_{2} – y_{1})).
 Formula for the midpoint of a line segment:
Note(s):
 Grade Level(s)
 Prior grade levels addressed points and distance on a number line.
 Prior grade levels addressed points and lines on a coordinate plane.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 I.A. Numeric Reasoning – Number representations and operations
 I.A.2. Perform computations with rational and irrational numbers.
 III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
 III.C.1. Make connections between geometry and algebraic equations.
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.1. Analyze given information.
 VII.B. Problem Solving and Reasoning – Proportional reasoning
 VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
 IX.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.

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

Derive
THE DISTANCE, SLOPE, AND MIDPOINT FORMULAS
Including, but not limited to:
 Distance formula: d =
 Connection between Pythagorean Theorem and distance formula
 Formula for slope of a line: m =
 Formula for midpoint of a line segment on a coordinate plane:
 Relationship between distance, slope, and midpoint formulas
 Involve the use of the horizontal and vertical distances from one point to another
 Distance formula applies these distances using the Pythagorean Theorem to find the length of the slanted segment (hypotenuse of the right triangle) that connects the two points.
 Slope formula is the ratio of these two distances.
 Midpoint formula uses the sum of these 2 distances to their respective coordinates to find the coordinate of the midpoint.
Use
THE DISTANCE, SLOPE, AND MIDPOINT FORMULAS TO VERIFY GEOMETRIC RELATIONSHIPS, INCLUDING CONGRUENCE OF SEGMENTS AND PARALLELISM OR PERPENDICULARITY OF PAIRS OF LINES
Including, but not limited to:
 Distance formula: d =
 Formula for slope of a line: m =
 Formula for midpoint of a line segment on a coordinate plane:
 Congruent segments – line segments whose lengths are equal
 Midpoint of a line segment – the point halfway between the endpoints of a line segment
 Parallel lines – lines in the same plane that never intersect. Slopes of parallel lines are equal, m_{y2} = m_{y}_{1}.
 Perpendicular lines – lines that intersect at a 90° angle to form right angles. Slopes of perpendicular lines are opposite reciprocals, m_{y}_{2} = – .
 Equation of a line
 Slopeintercept form, y = mx + b
 Pointslope form, y – y_{1} = m(x – x_{1})
 Standard form, Ax + By = C
Note(s):
 Grade Level(s)
 Grade 8 introduced and applied the Pythagorean Theorem to determine the distance between two points on a coordinate plane.
 Grade 8 introduced slope as or .
 Algebra I addressed determining equations of lines using pointslope form, slope intercept form, and standard form.
 Algebra I wrote equations of lines that contain a given point and are parallel or perpendicular to a given line.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS
 III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
 III.C.1. Make connections between geometry and algebraic equations.
 VII.B. Problem Solving and Reasoning – Proportional reasoning
 VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
 IX.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.

G.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
 Transformations that do not preserve similarity
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.3B 
Determine the image or preimage of a given twodimensional figure under a composition of rigid transformations, a composition of nonrigid transformations, and a composition of both, including dilations where the center can be any point in the plane.

Determine
THE IMAGE OR PREIMAGE OF A GIVEN TWODIMENSIONAL FIGURE UNDER A COMPOSITION OF RIGID TRANSFORMATIONS, A COMPOSITION OF NONRIGID TRANSFORMATIONS, AND A COMPOSITION OF BOTH, INCLUDING DILATIONS WHERE THE CENTER CAN BE ANY POINT IN THE PLANE
Including, but not limited to:
 Preimage – original figure prior to a transformation
 Image – figure after a transformation
 Rigid transformation (isometric) – a transformation that preserves the size and shape of a figure
 NonRigid transformation (nonisometric) – a transformation that preserves similarity and reductions and enlargements that do not preserve similarity
 Composition of transformations – combination of two or more transformations
 Composition of rigid transformations
 Reflection
 Reflected across the xaxis
 Reflected across the yaxis
 Reflected across the line y = x
 Reflected across the line y = –x
 Reflected across horizontal or vertical lines
 Translation
 Rotation
 Rotated around the origin
 Rotated around a point other than the origin
 Composition of nonrigid transformations
 Dilation
 Center of dilation can be any point in the plane
 Multiple compositions of both rigid and nonrigid transformations
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.
 Geometry graphs and describes a composition of transformations.
 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.

G.3C 
Identify the sequence of transformations that will carry a given preimage onto an image on and off the coordinate plane.

Identify THE SEQUENCE OF TRANSFORMATIONS THAT WILL CARRY A GIVEN PREIMAGE ONTO AN IMAGE ON AND OFF THE COORDINATE PLANE Including, but not limited to:  Preimage – original figure prior to a transformation
 Image – figure after a transformation
 Transformations on the coordinate plane
 Transformation of preimage onto an image
 Reflection
 The line of reflection is the perpendicular bisector of a segment that joins a point on the preimage to the corresponding point on the image.
 Translation
 A composite of reflections through an even number of parallel lines.
 Rotation
 A composite of reflections through an even number of intersecting lines.
 Composition of transformations
 A composite of transformations is commutative, meaning the order of the sequence of transformations does not matter.
 Dilation
 Reductions and enlargements that do not preserve similarity
 Transformations not on a coordinate plane
 Transformations in tessellations
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.

G.3D 
Identify and distinguish between reflectional and rotational symmetry in a plane figure.

Identify
REFLECTIONAL AND ROTATIONAL SYMMETRY IN A PLANE FIGURE
Including, but not limited to:
 Reflectional symmetry – symmetry in which one half of the image is a mirror image of the other over a line of reflection
 Horizontal reflectional (line) symmetry – reflectional symmetry about a horizontal line of reflection
 Vertical reflectional (line) symmetry – reflectional symmetry about a vertical line of reflection
 Characteristics of reflection
 Line of symmetry – line dividing an image into two congruent parts that are mirror images of each other
 Symmetric points – two points in a plane such that the line segment joining the points is bisected by a point or center
 Rotational symmetry – symmetry that occurs if a figure can be rotated less than 360° around a central point and still look the same as the original. The number of times a figure fits into itself in one complete rotation is called the order of rotational symmetry.
 Point symmetry – 180^{o} rotation around a point
 Characteristics of rotation
 Point of rotation – point around which a figure is rotated (center of rotation)
 Degree of rotation – number of degrees a figure is rotated about the point (center) of rotation
Distinguish BETWEEN REFLECTIONAL AND ROTATIONAL SYMMETRY IN A PLANE FIGURE Including, but not limited to:  Reflectional and rotational symmetry in polygons
 Reflectional and rotational symmetry in circles
 Comparison of characteristics of reflectional and rotational symmetry
Note(s):
 Grade Level(s)
 Grade 8 introduced symmetry and transformations including reflections and rotations.
 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.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
 III.C.1. Make connections between geometry and algebraic equations.

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 PARALLEL LINES
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
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.
