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 Instructional Focus DocumentGeometry
 TITLE : Unit 02: Coordinate Geometry and Transformations SUGGESTED DURATION : 15 days

#### Unit Overview

Introduction
This unit bundles student expectations that address geometric explorations of distance, midpoint, slope, and parallel and perpendicular lines in a two-dimensional coordinate system, including determining equations of lines. The student expectations also address rigid and non-rigid transformations both on and off the coordinate plane making connections between algebra and geometry. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this Unit
In Grade 8, students transformed geometric figures through translations, reflections, rotations around the origin, and dilations with the origin as the center of dilation. Students described transformations using algebraic representations. Students also used the Pythagorean Theorem to determine distance between two points in a coordinate plane. In Grade 8 and Algebra I, students studied the graphs and characteristics of linear functions, including slope of a line. In Algebra I, students wrote the equation of a line that contains a given point and is parallel or perpendicular to a given line.

During this Unit
Students investigate the undefined terms point, line, and plane in a two-dimensional coordinate system in Euclidean and spherical geometries. Using coordinate points, students derive the distance formula and apply the distance formula to determine lengths and congruence of line segments and fractional distances less than one from one end of a line segment to the other. Coordinate points are also used to derive and apply the midpoint formula and slope formula. Slope is applied to define and investigate parallel and perpendicular lines, including comparison of parallel lines in Euclidean and spherical geometry. Students algebraically determine the equation of a line when given a point on the line and a line parallel or perpendicular to the line. In addition, students build upon their knowledge of coordinate geometry to analyze the critical attributes of transformations, including translations, reflections, rotations with points of rotation other than the origin, and dilations where the center of dilation can be any point on the coordinate plane. Students examine patterns to generalize rigid transformations (translations, reflections, and rotations) in the coordinate plane. Students also explore non-rigid transformations or dilations in the coordinate plane using scale factors. They compare and contrast dilations to other geometric transformations and examine relationships in terms of similarity. Students perform rigid transformations, non-rigid transformations, and composite transformations using coordinate notation. Students identify the sequence of transformations performed for a given pre-image or image on or off a coordinate plane. Reflection symmetry and rotational symmetry in plane figures are identified and differentiated.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool HS Geometry

After this Unit
In Units 03 – 08, conjectures, postulates, and theorems will be used to investigate geometric relationships in parallel lines and transversals, polygons, and circles. Deductive reasoning will continue to be used to verify conjectures and theorems, and connections will continue to be made between algebra and geometry. The concepts in this unit will also be applied in subsequent mathematics courses.

This unit is supporting the development of Texas College Career Readiness Standards (TxCCRS): I. Numeric Reasoning A2, B1; II. Algebraic Reasoning D1, D2; III. Geometric and Spatial Reasoning B1, B2, C1; V. Statistical Reasoning A1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, C2, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
According to the National Council of Teachers of Mathematics (NCTM) Focus in High School Mathematics: Reasoning and Sense Making (2009), students should develop multiple geometric approaches to solving problems, including the use of coordinate geometry and transformations. (NCTM) According to NCTM, coordinate geometry helps students apply algebraic concepts to analyze geometric concepts, and vice versa. NCTM also affirms that transformations provide another useful approach to understanding geometric relationships.

National Council of Teachers of Mathematics. (2009). Focus in high school mathematics: Reasoning and sense making. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9

 Geometric, spatial, and measurement reasoning are foundational to visualizing, analyzing, and applying relationships within and between scale, shapes, quantities, and spatial relations in everyday life. Why is developing geometric, spatial, and measurement reasoning essential? How does geometric, spatial, and measurement reasoning affect how one sees and works in the world? Logical reasoning can be used to make sense of claims, determine their validity, and construct and communicate arguments. Why is developing logical reasoning in mathematics important and how does this reasoning influence decision making in everyday life? What elements of logical reasoning influence the truth of a statement? How is logical reasoning used to uncover truths and/or make sense of, construct, and determine the validity of arguments and claims?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding and working with variance and invariance within geometry builds flexible algebraic and geometric reasoning and deepens understanding of intrinsic properties of geometric relationships.
• What invariant (unchanging) and variant (changing) relationships exist between …
• Euclidean and spherical geometries?
• length and distance?
• Accurate representations, models, or diagrams within a geometric system allows for visualizing, illustrating, and analyzing geometric relationships to aid in making and validating conjectures about those geometric relationships and is central to geometric thinking.
• What types of problem situations represent geometric relationships on the coordinate plane?
• How can representations and appropriate geometric language be used to effectively communicate and illustrate geometric relationships on the coordinate plane?
• What tools and processes can be used to …
• represent fractional distances between two points on a coordinate plane
• represent midpoint of two points on a coordinate plane
• represent equation of a line parallel or perpendicular to a given line that passes through a given point
… in Euclidean geometry?
• How are undefined terms used to develop the concepts of distance (measure), congruence, and betweenness?
• What conjectures can be made and validated about two-dimensional distance relationships in line segments, the coordinate plane, and segment addition?
• Attributes and quantifiable measures of geometric figures can be generalized to describe, determine, and represent algebraic and geometric relationships and be applied to solve problem situations.
• How can understanding geometric relationships on the coordinate plane be applied when solving problem situations?
• How can measurable attributes related to …
• distance
• length
• slope
… be distinguished and described in order to generalize geometric relationships?
• What processes can be used to determine the …
• coordinates of the point that is a given fractional distance less than one from one end of a line segment?
• coordinates of the midpoint of a line segment?
• equation of a line parallel or perpendicular to a given line that passes through a given point?
• How are different processes used to determine whether lines are parallel or perpendicular?
• Coordinate and Transformational Geometry
• Geometric Relationships
• Congruence
• Equidistance
• Equivalence
• Parallelism
• Perpendicularity
• Measure relationships
• Geometric Representations
• One-dimensional
• Two-dimensional figures
• One- and Two-Dimensional Coordinate Systems
• Distance
• Slope
• Midpoint
• Parallel and perpendicular lines
• Equations of lines
• Patterns, Operations, and Properties
• Logical Arguments and Constructions
• Geometric Relationships
• Parallelism
• Geometric Representations
• Lines
• Geometries
• Euclidean
• Spherical
• Patterns, Operations, and Properties
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 Geometric, spatial, and measurement reasoning are foundational to visualizing, analyzing, and applying relationships within and between scale, shapes, quantities, and spatial relations in everyday life. Why is developing geometric, spatial, and measurement reasoning essential? How does geometric, spatial, and measurement reasoning affect how one sees and works in the world?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding and working with variance and invariance within geometry builds flexible algebraic and geometric reasoning and deepens understanding of intrinsic properties of geometric relationships.
• What invariant (unchanging) and variant (changing) relationships exist between rigid and non-rigid transformations?
• Accurate representations, models, or diagrams within a geometric system allows for visualizing, illustrating, and analyzing geometric relationships to aid in making and validating conjectures about those geometric relationships and is central to geometric thinking.
• What types of problem situations represent transformations of geometric figures?
• How can representations and appropriate geometric language be used to effectively communicate and illustrate geometric relationships in transformations?
• What tools and processes can be used to …
• generate and represent rigid and non-rigid transformations
• represent composition of transformations
• represent the sequence of transformations
• identify and distinguish between reflectional and rotational symmetry
… in Euclidean geometry?
• How can models be used to make and validate conjectures about geometric relationships in transformations that occur on and off the coordinate plane?
• How can a composite figure be transformed to create a unique design?
• Why do some transformations preserve similarity while other transformations do not?
• Attributes and quantifiable measures of geometric figures can be generalized to describe, determine, and represent algebraic and geometric relationships and be applied to solve problem situations.
• How can understanding transformations of geometric figures be applied when solving problem situations?
• How can proportionality be used to describe the relationship between an image and pre-image of a transformed figure?
• What processes can be used to determine the …
• coordinates of an image resulting from transformations?
• scale factors of dilations?
• coordinate notation to describe transformations?
• composition of transformations?
• sequence of transformations?
• Coordinate and Transformational Geometry
• Geometric Relationships
• Congruence
• Similarity
• Proportionality
• Corresponding sides and angles
• Measure relationships
• Geometric Representations
• Two-dimensional figures
• Patterns, Operations, and Properties
• Transformations
• Rigid transformations
• Non-rigid transformations
• Composition of transformations
• Sequence of transformations
• Reflectional and rotational symmetry
• Associated Mathematical Processes
• Communication
• Representations
• Relationships
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may confuse the x- and y-coordinates, or their order, in the formulas for slope, distance, and midpoint such as using x-values in the numerator of the slope rather than as the denominator.
• Some students may confuse the order of operations in the formulas for slope, distance, and midpoint such as students squaring the numbers before subtracting the coordinates in the distance formula rather than first subtracting the coordinates then squaring the result.

Underdeveloped Concepts:

• Some students may think that the minus sign in the slope and distance formulas are representative of the negative of a coordinate rather than understanding it is part of the formula. (e.g., For x2x1 given x1 = –3 and x2 = –4, then –4–(–3) → –4 + 3).
• Although most students may easily recognize the different transformations from diagrams, these students may be unable to define these transformations by their critical attributes. (e.g., Students may be able to recognize a rotation, but they may not be able to identify the angle measure, direction, or center of the rotation.).

#### Unit Vocabulary

• Composition of transformations – combination of two or more transformations
• Congruent segments – line segments whose lengths are equal
• Degree of rotation – number of degrees a figure is rotated about the point (center) of rotation
• Dilation – a similarity transformation in which a figure is enlarged or reduced using a scale factor and a center of dilation
• Euclidean geometry – the study of plane and solid geometry based on definitions, undefined terms (point, line, plane) and the assumptions of mathematician Euclid
• Horizontal reflectional (line) symmetry – reflectional symmetry about a horizontal line of reflection
• Image – figure after a transformation
• Line of symmetry – line dividing an image into two congruent parts that are mirror images of each other
• Midpoint of a line segment – the point halfway between the endpoints of a line segment
• Non-rigid transformation (non-isometric) – a transformation that preserves similarity and reductions and enlargements that do not preserve similarity
• Parallel lines – lines in the same plane that never intersect. Slopes of parallel lines are equal, my2 = my1.
• Perpendicular lines – lines that intersect at a 90° angle to form right angles. Slopes of perpendicular lines are opposite reciprocals, • Point of rotation – point around which a figure is rotated (center of rotation)
• Point symmetry – 180o rotation around a point
• Pre-image – original figure prior to a transformation
• 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)
• Reflectional symmetry – symmetry in which one half of the image is a mirror image of the other over a line of reflection
• Rigid transformation (isometric) – a transformation that preserves the size and shape of a figure
• Rotation – a rigid transformation where each point on the figure is rotated about a given point
• 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.
• Spherical geometry – the study of figures on the two-dimensional curved surface of a sphere
• Symmetric points – two points in a plane such that the line segment joining the points is bisected by a point or center
• Transformation – one to one mapping of points in a plane such that each point in the pre-image has a unique image and each point in the image has a pre-image
• Translation – rigid transformation moving all points in a geometric figure the same distance and the same direction
• Two-dimensional coordinate system – two axes at right angles to each other, forming a xy-plane, consisting of points (x, y)
• Vertical reflectional (line) symmetry – reflectional symmetry about a vertical line of reflection

Related Vocabulary:

 Center of dilation Congruence Coordinates Definitions Distance Distance formula Line segment Midpoint Midpoint formula Opposite reciprocals Point-slope form, y – y1 = m(x – x1) Postulates Scale factor Similarity Slope Slope formula Slope-intercept form, y = mx + b Standard form, Ax + By = C Theorems Undefined terms
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Center if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Geometry Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

• Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
• Student Expectations (TEKS) identified by TEA are in bolded, black text.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
• A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
TEKS# SE# TEKS SPECIFICITY
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 – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world 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, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world 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 problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION

Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving 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 problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving 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, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world 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 non-examples, 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 two-dimensional 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 two-dimensional coordinate systems, including finding the midpoint.

Determine

THE COORDINATES OF A POINT IN A TWO-DIMENSIONAL 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:

• Two-dimensional coordinate system – two axes at right angles to each other, forming a xy-plane, consisting of points (x, y)
• A point in a two-dimensional 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): dx = |x1x2| or dx = |x2x1| and dy = |y1y2| or dy = |y2y1|
• 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 (x1, y1) to the second endpoint (x2, y2) can be found using the expression (x1 + k(|x2x1|), y1 + k(|y2y1|)).
• Formula for the midpoint of a line segment: Note(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, my2my1.
• Perpendicular lines – lines that intersect at a 90° angle to form right angles. Slopes of perpendicular lines are opposite reciprocals, my2 = – .
• Equation of a line
• Slope-intercept form, y = mx + b
• Point-slope form, yy1 = m(xx1)
• Standard form, Ax + By = C

Note(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 point-slope 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, my2 = my1.
• Equation of a line
• Slope-intercept form, y = mx + b where m represents slope and b represents the y-intercept
• Point-slope form, yy1 = m(xx1) where m represents slope and (x1, y1) represents the given point
• Standard form, Ax + By = C where the slope, m = – • Determination of the equation of a line given a slope and y-intercept
• 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, my2 = – .
• Equation of a line
• Slope-intercept form, y = mx + b where m represents slope and b represents the y-intercept
• Point-slope form, yy1 = m(xx1) where m represents slope and (x1, y1) represents the given point
• Standard form, Ax + By = C where the slope, m = – • Determination of the equation of a line given a slope and y-intercept
• 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):

• Grades 7 and 8 represented linear non-proportional situations using multiple representations, including y = mx + b, where b ≠ 0.
• Algebra I addressed determining equations of lines using point-slope 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 non-rigid 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 pre-image has a unique image and each point in the image has a pre-image
• 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 x-axis (x, y) → (x, –y)
• Reflection across the y-axis (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
• Non-Rigid transformation (non-isometric) – 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(xa) + a, k(yb) + b), k = scale factor, (a, b) is the center of dilation
• Transformations that do not preserve similarity

Note(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 two-dimensional 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 two-dimensional 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 pre-image of a given two-dimensional figure under a composition of rigid transformations, a composition of non-rigid transformations, and a composition of both, including dilations where the center can be any point in the plane.

Determine

THE IMAGE OR PRE-IMAGE OF A GIVEN TWO-DIMENSIONAL FIGURE UNDER A COMPOSITION OF RIGID TRANSFORMATIONS, A COMPOSITION OF NON-RIGID TRANSFORMATIONS, AND A COMPOSITION OF BOTH, INCLUDING DILATIONS WHERE THE CENTER CAN BE ANY POINT IN THE PLANE

Including, but not limited to:

• Pre-image – 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
• Non-Rigid transformation (non-isometric) – 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 x-axis
• Reflected across the y-axis
• 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 non-rigid transformations
• Dilation
• Center of dilation can be any point in the plane
• Multiple compositions of both rigid and non-rigid transformations

Note(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 two-dimensional 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 two-dimensional 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 pre-image onto an image on and off the coordinate plane.

Identify

THE SEQUENCE OF TRANSFORMATIONS THAT WILL CARRY A GIVEN PRE-IMAGE ONTO AN IMAGE ON AND OFF THE COORDINATE PLANE

Including, but not limited to:

• Pre-image – original figure prior to a transformation
• Image – figure after a transformation
• Transformations on the coordinate plane
• Transformation of pre-image onto an image
• Reflection
• The line of reflection is the perpendicular bisector of a segment that joins a point on the pre-image 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 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 two-dimensional 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 two-dimensional 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 – 180o 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 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 two-dimensional curved surface of a sphere
• Definitions and undefined terms in Euclidean and spherical geometries
• Undefined terms • Postulates and theorems in Euclidean and spherical geometries
• Parallel lines

Note(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. 