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

#### Unit Overview

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.

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): II. Algebraic Reasoning D2; III. Geometric Reasoning A1, A2, B1, B2, B3, C1, C3, D1, D2; IV Measurement Reasoning C3; VII. Functions B2, C2; VIII. Problem Solving and Reasoning A1, A2, A3, A4, B1, B2, C1; IX. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; X. Connections A1, B.

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.

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

Geometric and spatial reasoning are necessary to describe and analyze geometric relationships in mathematics and the real-world.

• Why are geometric and spatial reasoning necessary in the development of an understanding of geometric relationships?
• Why is it important to visualize and use diagrams to effectively communicate/illustrate geometric relationships?
• How do geometric and spatial reasoning allow for the understanding of different geometric systems as models for the world?

Transformations and compositions of transformations are used to create images.

• Why would it be necessary to create new images through transformations and compositions of transformations?
• How are transformations generated and represented algebraically?
• Why do some transformations maintain congruence and similarity and others do not?

Attributes and properties of one- and two-dimensional geometric shapes are foundational to developing geometric and measurement reasoning.

• Why is it important to compare and contrast attributes and properties of one- and two-dimensional geometric shapes?
• How does analyzing the attributes and properties of one- and two-dimensional geometric shapes develop geometric and measurement reasoning?

Application of attributes and measures of figures can be generalized to describe geometric relationships which can be used to solve problem situations.

• Why are attributes and measures of figures used to generalize geometric relationships?
• How can numeric patterns be used to formulate geometric relationships?
• Why is it important to distinguish measureable attributes?
• How do geometric relationships relate to other geometric relationships?
• Why is it essential to develop generalizations for geometric relationships?
• How are geometric relationships applied to solve problem situations?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
 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.

Numerical Understanding

• Fractions
• Perfect Squares/Square Roots

Numeric Reasoning

• Exponents
• Fractions
• Square Roots

Algebraic Reasoning

• Coordinate Plane
• Equation
• Rate of Change/Slope

Functions

• Attributes of Functions
• Linear Functions

Geometric Reasoning

• Geometric Attributes/Properties
• Geometric Relationships
• Geometric Systems
• Logical Arguments
• Postulates
• Pythagorean Theorem
• Two-Dimensional Figures
• Theorems

Measurement Reasoning

• Distance/Length
• Formulas

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Although distance and midpoint are determined in both one- and two-dimensional systems, calculations in a one-dimensional system use only the point, x, whereas calculations in a two-dimensional system use the point, (x, y).

• What are the similarities and differences between finding the length of a line segment in a one- and two-dimensional system?
• What are the similarities and differences between finding the midpoint of a line segment in a one- and two-dimensional system?

The distance and midpoint between two points on a coordinate plane can be calculated using the distance and midpoint formulas to verify geometric relationships.

• What is the relationship between the Pythagorean Theorem and the distance formula?
• What is the relationship between the distance and length?
• How can the distance formula be used to 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?
• How can the midpoint formula be used to determine the coordinates of the midpoint of a line segment?
• How is the concept of betweenness applied when finding the point a given fractional distance less than one from one end of a line segment or the midpoint of a line segment?

The slope of a line between two points on a coordinate plane can be calculated using the slope formula, and the slope can be used to determine parallelism or perpendicularity of pairs of lines.

• What relationship exists between the slopes of parallel lines?
• What relationship exists between the slopes of perpendicular lines?

An equation of a line parallel or perpendicular to a given line passing through a given point can be determined using the parallel or perpendicular slope and the given point.

• How is the slope of a line parallel to another line determined?
• How is the slope of a line perpendicular to another line determined?
• How is the equation of a line parallel or perpendicular to a given line that passes through a given point determined and represented algebraically?

Parallel lines exist in Euclidean geometry but do not exist in spherical geometry, because in spherical geometry all lines intersect at the poles.

• How would the following statement be justified as true or false for Euclidean geometry? In a plane, given a line and a point not on the line, only one line parallel to the given line can be drawn through the point.
• How would the following statement be justified as true or false for spherical geometry? In a plane, given a line and a point not on the line, only one line parallel to the given line can be drawn through the point.

Diagrams can be used to visualize and illustrate geometric relationships and aid in solving problems.

• Why are diagrams necessary for visualizing the geometric relationships found in the problem situation?
• How are diagrams used to organize information from the problem situation?
• How do diagrams aid in calculations when solving problems?
• Why is the coordinate plane used to diagram two-dimensional figures?
 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.

Algebraic Reasoning

• Patterns/Rules
• Rate of Change/Slope

Geometric Reasoning

• Congruence
• Geometric Attributes/Properties
• Scale Factors
• Similarity
• Transformations

Measurement Reasoning

• Angle Measures
• Length

Associated Mathematical Processes

• Communication
• Representations
• Relationships

Patterns can be used to formulate rules and generalizations about transformations of geometric figures represented on a coordinate system.

• How can a transformation be represented using coordinate notation?
• How can coordinate notation be used to determine the image or pre-image of a transformation?

Translations, reflections, and rotations are referred to as rigid transformations because they create an image that is congruent, exact size and shape, to the pre-image.

• What are the rigid transformations and what are the critical attributes of each?
• What is the difference between reflectional and rotational symmetry?

Dilations are referred to as non-rigid transformations because they create an image that is similar to the pre-image.

• What is a non-rigid transformation and what are its critical attributes?
• When do non-rigid transformations preserve similarity and when do they not preserve similarity?
• How is the dilation affected when the center of dilation is a point other than the origin?

Images can be created through the composition of rigid transformations, non-rigid transformations, and a mixture of both types of transformations.

• What is meant by a composition of rigid transformations?
• What is meant by a composition of non-rigid transformations?
• If the order of the sequence of transformations on the pre-image is changed, how is the final image affected?
• How can a sequence of transformations that will carry a pre-image to an image on and off the coordinate plane be identified?
• How can a composition of transformations be written as a single transformation?

#### 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 in the same plane that intersect at 90 angles whose slopes 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 Creator 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 (select CCRS from Standard Set dropdown menu)

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

TEKS# SE# TEKS Unit Level Specificity

• Bold black text in italics: Knowledge and Skills Statement (TEKS)
• Bold black text: Student Expectation (TEKS)
• Strike-through: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit(s)
• Blue text: Supporting information / Clarifications from TCMPC (Specificity)
• Blue text in italics: Unit-specific clarification
• Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS)
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:
• X. Connections
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:
• VIII. Problem Solving and Reasoning
G.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS

Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
G.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE

Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
G.1E Create and use representations to organize, record, and communicate mathematical ideas.

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
G.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and 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:
• X. Connections
G.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION

Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
G.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 an 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|x– x2| or dx|x2 – x1| and dy|y– y2| or dy|y2 – y1|
• 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
• III. Geometric Reasoning
• A1 – Identify and represent the features of plane and space figures.
• A2 – Make, test, and use conjectures about one-, two-, and three-dimensional figures and their properties.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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. Geometric Reasoning
• A1 – Identify and represent the features of plane and space figures.
• A2 – Make, test, and use conjectures about one-, two-, and three-dimensional figures and their properties.
• C1 – Make connections between geometry and algebra.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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, my2my1.
• 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, = –
• 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. Geometric Reasoning
• A1 – Identify and represent the features of plane and space figures.
• A2 – Make, test, and use conjectures about one-, two-, and three-dimensional figures and their properties.
• C1 – Make connections between geometry and algebra.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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 in the x-axis (x, y) → (x, –y)
• Reflection in the y-axis (x, y) → (–x, y)
• Reflection in the y = x line (x, y) → (y, x)
• Reflection in the y = –x line (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. Geometric Reasoning
• A1 – Identify and represent the features of plane and space figures.
• A2 – Make, test, and use conjectures about one-, two-, and three-dimensional figures and their properties.
• B1 – Identify and apply transformations to figures.
• B2 – Identify the symmetries of a plane figure.
• B3 – Use congruence transformations and dilations to investigate congruence, similarity, and asymmetries of plane figures.
• C1 – Make connections between geometry and algebra.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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 in x-axis
• Reflected in y-axis
• Reflected in y = x
• Reflected in y = –x
• Reflected in 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. Geometric Reasoning
• A1 – Identify and represent the features of plane and space figures.
• A2 – Make, test, and use conjectures about one-, two-, and three-dimensional figures and their properties.
• B1 – Identify and apply transformations to figures.
• B2 – Identify the symmetries of a plane figure.
• B3 – Use congruence transformations and dilations to investigate congruence, similarity, and asymmetries of plane figures.
• C1 – Make connections between geometry and algebra.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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. Geometric Reasoning
• A1 – Identify and represent the features of plane and space figures.
• A2 – Make, test, and use conjectures about one-, two-, and three-dimensional figures and their properties.
• B1 – Identify and apply transformations to figures.
• B2 – Identify the symmetries of a plane figure.
• B3 – Use congruence transformations and dilations to investigate congruence, similarity, and asymmetries of plane figures.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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. Geometric Reasoning
• A1 – Identify and represent the features of plane and space figures.
• A2 – Make, test, and use conjectures about one-, two-, and three-dimensional figures and their properties.
• B1 – Identify and apply transformations to figures.
• B2 – Identify the symmetries of a plane figure.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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 (point, line, plane)
• Postulates and theorems in Euclidean and spherical geometries
• Parallel lines

Note(s):