Geometry (& GT)
Unit 1: Transformations, Similarity, and Congruence
Part 1: Transformations and the Coordinate Plane
Students are expected to use coordinates to prove simple geometric theorems algebraically. Building on their work with the Pythagorean theorem in 8th grade to find distances, students use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines. For example, students will prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle.
Students will also experiment with transformations in the plane. In Grade 8, students had experience with transformations: translations, reflections, rotations, and dilations. Comparisons of transformations will provide the foundation for understanding similarity and congruence. Additionally, students will prove theorems, using a variety of formats, and solve problems about triangles, quadrilaterals, and other polygons. Students will be exposed to multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in twocolumn format, and using diagrams without words. They also apply reasoning to complete geometric constructions and explain why they work. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning.
Part 2: Similarity
Students apply their earliest experience with dilations and proportional reasoning to build a formal understanding of similarity. The only transformations that do not absolutely preserve congruence are dilations. Focus will be on the power of dilations to effect size change in geometric figures. Dilations preserve angle measure while enlarging or reducing lengths proportionally. Students will be able to determine if two figures are similar, essentially a dilation of each other, by comparing their angles for congruence and their sides for proportionality. Students will discover that the other transformations (reflections, rotations, shifts, translations) produce a similarity in a 1:1 ratio and thus are a special subset of similarity transformations, namely transformations that produce congruent figures. (A metaphor for this would be that of a square being a specifically defined rectangle in that way that a congruence is a specifically defined similarity).
Part 3: Congruence
Students will build on their prior experience with rigid motions, including translations, reflections and rotations, in order to develop notions about what it means for two objects to be congruent. Rigid motions are at the foundation of the definition of congruence, and students should recognize that they preserve distance and angle. Students will continue to develop their understanding that congruence is a special case of similarity with a 1:1 ratio. Students should experiment with rigid motions in the coordinate plane in order to carry given polygons onto themselves, in order to verify major properties of parallelograms, and in order to determine the congruence of given polygons. Students should be encouraged to use a variety of tools, such as protractors, compasses, MIRA boards, patty paper, graph paper, and geometry software in order to transform given figures to prove congruence. In addition, students should use a variety of methods for proof, both formal and informal. Students will apply their understanding of congruence to copy segments and angles using a variety of tools and construction methods.
Students are expected to use coordinates to prove simple geometric theorems algebraically. Building on their work with the Pythagorean theorem in 8th grade to find distances, students use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines. For example, students will prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle.
Students will also experiment with transformations in the plane. In Grade 8, students had experience with transformations: translations, reflections, rotations, and dilations. Comparisons of transformations will provide the foundation for understanding similarity and congruence. Additionally, students will prove theorems, using a variety of formats, and solve problems about triangles, quadrilaterals, and other polygons. Students will be exposed to multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in twocolumn format, and using diagrams without words. They also apply reasoning to complete geometric constructions and explain why they work. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning.
Part 2: Similarity
Students apply their earliest experience with dilations and proportional reasoning to build a formal understanding of similarity. The only transformations that do not absolutely preserve congruence are dilations. Focus will be on the power of dilations to effect size change in geometric figures. Dilations preserve angle measure while enlarging or reducing lengths proportionally. Students will be able to determine if two figures are similar, essentially a dilation of each other, by comparing their angles for congruence and their sides for proportionality. Students will discover that the other transformations (reflections, rotations, shifts, translations) produce a similarity in a 1:1 ratio and thus are a special subset of similarity transformations, namely transformations that produce congruent figures. (A metaphor for this would be that of a square being a specifically defined rectangle in that way that a congruence is a specifically defined similarity).
Part 3: Congruence
Students will build on their prior experience with rigid motions, including translations, reflections and rotations, in order to develop notions about what it means for two objects to be congruent. Rigid motions are at the foundation of the definition of congruence, and students should recognize that they preserve distance and angle. Students will continue to develop their understanding that congruence is a special case of similarity with a 1:1 ratio. Students should experiment with rigid motions in the coordinate plane in order to carry given polygons onto themselves, in order to verify major properties of parallelograms, and in order to determine the congruence of given polygons. Students should be encouraged to use a variety of tools, such as protractors, compasses, MIRA boards, patty paper, graph paper, and geometry software in order to transform given figures to prove congruence. In addition, students should use a variety of methods for proof, both formal and informal. Students will apply their understanding of congruence to copy segments and angles using a variety of tools and construction methods.
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What will my child learn?
PART I: TRANSFORMATIONS, SIMILARITY, AND CONGRUENCE
Students will:
Use coordinates to prove simple geometric theorems algebraically.
G.GPE.B.4
Use coordinates to prove simple geometric theorems algebraically.
G.GPE.B.7
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
G.GPE.B.5
Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
Make geometric constructions.
G.CO.C.12
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Constructions include: constructing perpendicular lines, constructing a line parallel to a given line through a point not on the line.
Experiment with transformations in the plane.
G.CO.A.2
Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
Experiment with transformations in the plane.
G.CO.A.4
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
Prove geometric theorems.
G.CO.C.9
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent.
PART II: SIMILARITY
Understand similarity in terms of similarity transformations.
G.SRT.A.1
Verify experimentally the properties of dilations given by a center and a scale factor.
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
G.SRT.A.2
Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
PART III: CONGRUENCE
Experiment with transformations in the plane.
G.CO.A.3
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
Prove geometric theorems.
G.CO.C.11
Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. (Note: This standard will be revisited in Unit 2. In this unit, the focus is on verifying relationships in the coordinate plane.)
Understand congruence in terms of rigid motions.
G.CO.B.6
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
Experiment with transformations in the plane.
G.CO.A.5
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Make geometric constructions.
G.CO.D.12
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Constructions include: copying a segment; copying an angle.
PART I: TRANSFORMATIONS, SIMILARITY, AND CONGRUENCE
Students will:
Use coordinates to prove simple geometric theorems algebraically.
G.GPE.B.4
Use coordinates to prove simple geometric theorems algebraically.
 Background Info. & Guided Practice
 Check for Understanding: Geometry Problems on the Coordinate Plane
 Review/Rewind: Challenge Problem: Points of Two Circles
 Enrichment Tasks: Midpoint Miracle
G.GPE.B.7
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
 Background Info.: Distance Formula*
 Check for Understanding: Coordinate Plane Word Problems with Polygons
 Review/Rewind: Coordinate Plane Word Problem
 Enrichment Tasks: Squares on a Coordinate Grid  Triangle Perimeters
G.GPE.B.5
Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
 Background Info.: Equations of Parallel and Perpendicular Lines*
 Check for Understanding: Write the Equation of Perpendicular Lines
 Review/Rewind: Parallel Lines from Equation  Perpendicular Lines from Equation
 Enrichment Tasks: Parallel Lines in the Coordinate Plane  When are two lines perpendicular?
Make geometric constructions.
G.CO.C.12
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Constructions include: constructing perpendicular lines, constructing a line parallel to a given line through a point not on the line.
 Background Info.: Constructing Perpendicular Lines Using a Compass and Straightedge*
 Check for Understanding: Constructing a Perpendicular Line
 Review/Rewind: Geometric Constructions: Perpendicular Bisector
 Enrichment Tasks: Origami Regular Octagon  Bisecting an Angle
Experiment with transformations in the plane.
G.CO.A.2
Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
 Background Info.: Possible Transformations Examples*
 Check for Understanding: Qualitatively Defining Rigid Transformations  Quantitatively Defining Rigid Transformations
 Review/Rewind: Intro to Geometric Transformations
 Enrichment Tasks: Fixed Points of Rigid Motions  Horizontal Stretch of the Plane
Experiment with transformations in the plane.
G.CO.A.4
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
 Background Info. & Guided Practice: Exploring Rigid Transformations*  Possible Transformation Examples*
 Check for Understanding: Qualitatively Defining Rigid Transformations
 Review/Rewind: Identifying Type of Transformation
 Enrichment Tasks: Identifying Translations  Identifying Rotations
Prove geometric theorems.
G.CO.C.9
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent.
 Background Info.: Vertical Angles Are Congruent Proof*  Figuring Out Angles Between Transversal and Parallel Lines*
 Check for Understanding: Line and Angle Proofs
 Review/Rewind: Angle Relationships Example
 Enrichment Tasks: Congruent Angles Made by Parallel Lines and a Transverse  Points Equidistant From Two Points in the Plane
PART II: SIMILARITY
Understand similarity in terms of similarity transformations.
G.SRT.A.1
Verify experimentally the properties of dilations given by a center and a scale factor.
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
 Background Info.
 Check for Understanding: Dilations
 Review/Rewind: Performing Dilations
 Enrichment Tasks: Dilating a Line
G.SRT.A.2
Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
 Background Info.
 Check for Understanding: Defining Similarity through AnglePreserving Transformations
 Review/Rewind: Similar Shapes and Transformations
 Enrichment Tasks: Congruent and Similar Triangles  Similar Triangles
PART III: CONGRUENCE
Experiment with transformations in the plane.
G.CO.A.3
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
 Background Info.
 Check for Understanding: Symmetry of TwoDimensional Shapes
 Review/Rewind: Congruent Shapes and Transformations
 Enrichment Tasks: Symmetries of a Quadrilateral II  Symmetries of Rectangles
Prove geometric theorems.
G.CO.C.11
Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. (Note: This standard will be revisited in Unit 2. In this unit, the focus is on verifying relationships in the coordinate plane.)
 Background Info.
 Review/Rewind: Proof: Opposite Sides of a Parallelogram
 Enrichment Tasks: Midpoints of the Sides of a Parallelogram  Parallelograms and Translations
Understand congruence in terms of rigid motions.
G.CO.B.6
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
 Check for Understanding: Congruent Triangles 1  Congruent Triangles 2
 Review/Rewind: Corresponding Angles in Congruent Triangles
 Enrichment Tasks: Building a Tile Pattern by Reflecting Hexagons  Building a Tile Pattern by Reflecting Octagons
Experiment with transformations in the plane.
G.CO.A.5
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
 Check for Understanding: Performing Transformations on the Coordinate Plane  Transforming Polygons
 Review/Rewind: Performing Translations  Performing Rotations  Performing Reflections
 Enrichment Tasks: Showing a Triangle Congruence : A Particular Case  Showing a Triangle Congruence: General Case
Make geometric constructions.
G.CO.D.12
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Constructions include: copying a segment; copying an angle.
 Check for Understanding: Construct an Angle Bisector
 Review/Rewind: Geometric Constructions: Angle Bisector
 Enrichment Tasks: Origami Equilateral Triangle  Construction of Perpendicular Bisector
What are some signs of student mastery?
Transformations and the Coordinate Plane

Tools & Technology
GeoGebra is a dynamic math software tool that allows users to explore geometry, algebra, tables, graphing, statistics and other areas of math in one easytouse package. It can be downloaded for FREE on a computer or tablet device. Math Open Reference  a free interactive tool that allows students to explore the area and perimeter of rectangles on the coordinate plane. National Library of Virtual Manipulatives  a free interactive tool that allows students to explore the dilations of figures on coordinate planes. 
More 4 U
Have you ever heard your child talk about using "Patty Paper" in Geometry class? See how Patty Paper is used in the study of geometry:
Here's a tutorial on using a Compass and Straightedge to: Properties and Definitions of Transformations 