Download Critical Area 1

Common Core Geometry
Critical Area 1: Congruence with Transformations
Pacing: Weeks 2 – 10
Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Congruence with Transformations
Congruence
A. Experiment with transformations in the plane
1.
2.
3.
4.
5.
G-CO
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line distance
along a line, and distance around a circular arc.
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).
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
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.
B. Understand congruence in terms of rigid motions
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.
7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of
sides and corresponding pairs of angles are congruent.
8.
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
C. Prove geometric theorems
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; points on a perpendicular bisector of a line segment are exactly those equidistant from
the segment's endpoints.
10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a
point.
Instructional Guide for Common Core Geometry
Property of MPS
Page | 1
Rev 7.18.16
Common Core Geometry
Critical Area 1: Congruence with Transformations
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.
D. Make geometric constructions
12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic
geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the
perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Key Student Understandings
 Students will utilize precise geometric language to drive


Assessment
Formative Assessment Strategies
understanding of theorems used to prove congruence.
Students will complete constructions to visually represent geometric
theorems.
Students will perform rigid transformations on geometric figures to
prove figures congruent.
Evidence for Standards-based Grading
**Common SBG Evidence Items
Rubric
Rigid Motion (G-CO.A, SMP.3)
Are the Triangles Congruent? (G-CO.B, SMP.3)
Angle Chaser (G-CO.C, SMP.3)
Construct This! (G-CO.D, SMP.5)
Disciplinary Literacy

Disciplinary Literacy Framework
Common Misconceptions/Challenges
 Students have difficulty with transformations and should be encouraged to use tools strategically.
 Students confuse congruence with equality. It should be stressed that congruence refers to figures, while equality refers to values and distances.
 Students struggle with the use of certain geometric tools.
 When proving geometric theorems, students struggle with finding the most efficient process.
 When proving geometric theorems, students may have gaps in their reasoning and miss key steps.
 Students have difficulty seeing overlapping geometric figures, especially when the figures share a common side.
 Students may confuse the various types of transformations.
 Students may struggle to use the correct center of rotation, especially if they have simply memorized a rule (e.g., (x, y) → (y, –x) for a 90º clockwise
rotation works only if the center of rotation is the origin.)
Instructional Guide for Common Core Geometry
Property of MPS
Page | 2
Rev 7.18.16
Common Core Geometry
Critical Area 1: Congruence with Transformations
Instructional Practices
Suggested
Timeline
Language of
Geometry
(Weeks 2-3)
Suggested Learning Experiences
Precisely define, illustrate, and notate
geometric vocabulary.
Resources
**Common SBG Evidence Items
Discovering Geometry: All vocabulary and its notation found in Lessons: 1.1, 1.3, 1.4, 1.5, 1.6, 1.8,
1.9, 2.4, 2.5 and 2.6; pg. 123 (Developing Proof); pg. 132 (7); pg. 166 (17)
Additional District Created Items: Building Blocks of Geometry, Parallel Lines? , Angle Chaser**,
Constructions
(Weeks 3-7)
Construct geometric objects using:
patty paper, compass and straight
edge, and technology.
Discovering Geometry: Lesson 0.3 (constructing a regular hexagon in a circle); pg. 147 (8, 9);
Lesson 3.2; pg. 156 (5); Lesson 3.4; Lesson 3.5; pg. 183 (14); pg. 204 (14); pg. 205 (19); Lesson 4.2
(Blue Matter One Step); Lesson 4.4; Lesson 4.5
Engage NY: Construct This!**, Triangle Transformation
Illustrative Mathematics: Showing a Triangle Congruence (TE), Placing a Fire Hydrant (TE),
Locating a Warehouse (TE)
Additional District Created Items: Proving Properties of Triangles ASA, Proving Properties of
Triangles SAS, Proving Properties of Triangles SSS, Reflected Triangles
Rigid
Perform rigid transformations on
Transformations geometric figures.
(Weeks 7-10)
Prove figures are congruent.
Discovering Geometry: pg. 275 (Triangle Mid-segment Properties Investigations); pg. 373 (10); pg.
376-377 (Example A and Investigation 1); pg. 381 (7 and 8); pg. 387 (11)
Engage NY: Always a Square, Is Fred Rigid?
Illustrative Mathematics: Horizontal Stretch of the Plane (TE)
Illuminations: Congruence Theorems
Mathematics Assessment Resource Service: Transforming 2D Figures
Additional District Created Items: Are the Triangles Congruent?**, Rigid Motion**
Instructional Guide for Common Core Geometry
Property of MPS
Page | 3
Rev 7.18.16
Common Core Geometry
Critical Area 1: Congruence with Transformations
Differentiation
Literacy Connections
Geometry Teacher’s Edition - Differentiated Instruction CK-12 Foundation







Academic Vocabulary
Challenge students to “stump” each other with their transformations
so their partner cannot determine the steps to get back to the
original polygon.
Resources such as patty paper can help students perform
transformations.
Folding paper can help with reflections.
Using a small scrap of paper can help with rotations.
For rotations, connecting a vertex to the center of rotation can help
motivate the connection between rotations and circles.
Provide an already-transformed figure and ask students to describe
the transformations that carried Figure 1 onto the given figure. The
can vary in difficulty.
If students are having difficulty identifying transformations visually,
encourage them to keep an organized table of coordinates to help
them identify the types of transformations have occurred.
Vocabulary Strategies
Literacy Strategies
Additional Resources
Discovering Mathematics Online Textbook
Discovering Geometry Assessment Resources (Constructive Assessment Options)
Desmos Online Graphing Calculator
Engage NY Geometry Module 1
cK-12
learnzillion
GeoGebra
Georgia Geometry Unit 1
Instructional Guide for Common Core Geometry
Property of MPS
Page | 4
Rev 7.18.16