Common Core Geometry Unit 1 Starting Points
Unit 1: Transformations, Similarity, and Congruence
Part 1: Transformations and the Coordinate Plane
Essential Questions:
o How is visualization essential to the study of geometry?
o How does the concept of rigid motion connect to the concept of congruence?
o How does geometry explain or describe the structure of our world?
o How do geometric constructions enhance understanding of the geometric properties
of objects?
o How can reasoning be used to establish or refute conjectures?
o What are the characteristics of a valid argument?
o What is the role of deductive or inductive reasoning in validating a conjecture?
o What facts need to be verified in order to establish that two figures are congruent?
o How can the properties of geometric figures be verified using a coordinate plane?
o What is the connection between parallel and perpendicular lines in coordinate
geometry as it relates to systems of equations studied in Algebra?
o What is the connection between the distance formula and the Pythagorean theorem?
o How can coordinate geometry be used to prove simple theorem algebraically?
Curriculum Standards:
Use coordinates to prove simple geometric theorems algebraically.
G.GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. For example,
prove or disprove that a figure defined by four given points in the coordinate plane is a
rectangle)
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.
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.
Use coordinates to prove simple geometric theorems algebraically.
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).
This document represents one sample starting points for the unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.
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.
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.
Approximate Length of Unit Part 1 (Transformations in the Coordinate Plane): 18 days
Standard(s)
G.GPE.B.4
G.GPE.B.7
G.GPE.B.5
G.CO.C.12
Days
8
Notes
Big Ideas:
Identify polygons represented on the coordinate plane and
verify their properties.
Calculate distances, areas, and perimeters on the
coordinate plane using Pythagorean theorem and/or the
distance formula. Pre-assess for student knowledge of the
Pythagorean theorem and distance formula.
Investigate parallel and perpendicular slopes on the
coordinate plane and use them to find equations of lines.
Construct parallel and perpendicular lines.
Resources:
 Lesson: Perimeter and Area
 Lesson: Proving Slope Criteria for Parallel and
Perpendicular Lines
 Task: Pooltastic!
 Task: Safe Crossings
 Sketchpad Tutorial: Proving/Disproving a Rectangle
 Sketchpad Tutorial: Proving Slope Criteria for
Parallel/Perpendicular Lines
 Sketchpad Tutorial: Constructing
Parallel/Perpendicular Lines
 Patty Paper Tutorial: Constructing
Parallel/Perpendicular Lines
 Compass Tutorial: Constructing Parallel/Perpendicular
Lines
 Illustrative Math Task: Is This a Rectangle?
This document represents one sample starting points for the unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.
 Illustrative Math Task: Unit Squares and Triangles
 Illustrative Math Task: A Midpoint Miracle
PARCC Assessment Limits/Clarification:
 These standards are part of major content clusters
assessed on PARCC and are fluency standards, with
the exception of G.CO.C.12.
 G.CO.C.12 is part of supporting content cluster
assessed on PARCC.
G.CO.A.2
G.CO.A.4
6
Big Ideas:
Transform figures in the coordinate plane using multiple
tools.
Describe translations, reflections, rotations, and dilations.
Compare those that preserve distance with those that do
not. Pre-assess for student knowledge of transformations.
Resources:
 Lesson: Experiment with Transformations
 Lesson: Defining Transformations that Create
Congruent Figures
 Lesson Seed: Defining Similarity
 Sketchpad Tutorial: Transformations in the Coordinate
Plane
 Patty Paper Tutorial: Transformations that Preserve
Distance
G.CO.C.9
4
PARCC Assessment Limits/Clarification:
 These standards are part of supporting content
standards assessed on PARCC.
Big Ideas:
Use multiple methods to prove theorems involving angles.
Include methods involving both transformations as well as
the coordinate plane.
Resources:
 Lesson: Constant Rate Exploration
 Task: Meet Me at the Corner
 Task: Soccer Angles
 Sketchpad Tutorial: Vertical Angles Proof
 Sketchpad Tutorial: AIA/Corresponding Angles Proof
 Patty Paper Tutorial: Vertical Angles Proof
 Patty Paper Tutorial: AIA/Corresponding Angles
Proof
 Proof Block Tutorial: Vertical Angles Proof
This document represents one sample starting points for the unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.
 Proof Resource: Vertical Angles Proof Samples
 Proof Resource: AIA/Corresponding Angles Proof
Samples
PARCC Assessment Limits/Clarification:
 This standard is part of major content cluster
assessed on PARCC.
This document represents one sample starting points for the unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.
Part 2: Similarity
Essential Questions:
o How can transformations be used to explain similarity?
o How is similarity of geometric figures applied and verified?
o What is the relationship between transformations that produce congruent figures and
transformations that produce similar figures?
Curriculum Standards:
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.
Understand similarity in terms of similarity transformations.
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.
Approximate Length of Unit Part 2(Similarity): 8 days
Standard(s)
G.SRT.A.1
Days
3
Notes
Big Ideas:
Verify the properties of dilation through experimentation.
Resources:
 Lesson: Verifying Dilation
 Illustrative Math Task: Dilating a Line
PARCC Assessment Limits/Clarification:
 This standard is part of major content cluster
assessed on PARCC.
G.SRT.A.2
5
Big Ideas:
Develop the definition of similarity and use it to decide if two
figures are similar.
Explain similarity transformations and relate the transformation
to the proportionality of similar figures.
Resources:
 Lesson: Similar Figures
 Sketchpad Tutorial: Similar Figures
 Illustrative Math Task: Are They Similar?
This document represents one sample starting points for the unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.
PARCC Assessment Limits/Clarification:
 This standard is part of major content cluster
assessed on PARCC.
Part 3: Congruence
Essential Questions:
o Which transformations preserve distance and angle?
o How can a figure be transformed to carry on to itself?
o What tools can be used to perform given transformations?
o What are the properties of various parallelograms and how can they be verified using
rigid motions?
o Can transformations be applied in any sequence in order to carry a shape onto itself?
o How do congruence and similarity relate? How are they similar? How are they
different?
Curriculum Standards:
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, focus 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.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: copying a segment; copying an angle.
This document represents one sample starting points for the unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.
Total Approximate Length of Unit Part 3 (Congruence): 12 days
Standard(s)
G.CO.A.3
G.CO.C.11
Days
4-6
Notes
Big Ideas:
Describe transformations that would carry a given figure onto
itself.
Use transformations to verify properties of parallelograms, such as
the opposite sides of a rectangle are congruent. This will be
revisited in Unit 2.
Resources:
 Lesson: Carrying a Figure Onto Itself
 Lesson: Investigating Properties of Parallelograms
 Sketchpad Tutorial: Parallelogram Theorems
 Patty Paper Tutorial: Angles in a Parallelogram
 Patty Paper Tutorial: Diagonals of Parallelograms
 Proof Resource: Sample Parallelogram Proofs
PARCC Assessment Limits/Clarification:
 G.CO.A.3 is part of supporting content cluster assessed
on PARCC.
 G.CO.C.11 is part of major content cluster assessed on
PARCC.
G.CO.B.6
G.CO.A.5
G.CO.C.12
6-8
Big Ideas:
Use descriptions of rigid motions to transform figures or predict
transformations on given figures.
Define congruence in terms of rigid motion and use the definition
to determine if two figures are congruent.
Transform figures using given transformations using multiple
tools.
Describe a sequence of transformations that would carry a given
figure onto another.
Copy a segment and an angle using construction tools.
Resources:
 Lesson: Congruent Figures
 Task: Transformation Sensation
 Task: First to Finish
 Sketchpad Resource: Find the Transformation
This document represents one sample starting points for the unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.
 Patty Paper Tutorial: Copying a Segment or Angle
 Compass Tutorial: Copying a Segment or Angle
 Illustrative Math Task: Building a Tile Pattern by
Reflecting Hexagons
 Illustrative Math Task: Reflected Triangles
PARCC Assessment Limits/Clarification:
 G.CO.B.6 is part of major content cluster assessed on
PARCC.
 G.CO.A.5 and G.CO.C.12 are part of supporting content
clusters assessed on PARCC.
 G.CO.C.12 is a fluency standard.
Culminating Task: Illustrative Math Task: Building a Tile Pattern by Reflecting Octagons
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.
This document represents one sample starting points for the unit. It is not all-inclusive and is only one
planning tool. Please refer to the wiki for more information and resources.