Download Unit 2 - Middletown Public Schools

Middletown Public Schools
Mathematics Unit Planning Organizer
Grade/Course Geometry
20 instructional days + 4 days for reteaching/enrichment
Duration
Subject
Unit 2
Math
Congruence, Proof and Constructions
Big Idea(s)
Students will apply ideas about congruence and similarity to describe and analyze two-dimensional figures and to solve problems.
What does it mean for two figures to be congruent?
Essential
Question(s) How is rigid motion used to prove congruence?
How is coordinate geometry used to prove congruence?
Mathematical Practices
Practices in bold are to be emphasized in the unit.
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.
Domain and Standards Overview
Understand congruence in terms of rigid motions.
Prove geometric theorems.
Priority and Supporting Common Core State Standards
Bold Standards are Priority
CC.9-12.G.CO.7 Use the definition of congruence in terms of rigid
Grade/Course, Unit Title
2014
1
Explanations and Examples
A rigid motion is a transformation of points in space consisting of a
Date Created/Revised: November 18,
motions to show that two triangles are congruent if and only if
corresponding pairs of sides and corresponding pairs of angles are
congruent.
sequence of one or more translations, reflections, and/or rotations. Rigid
motions are assumed to preserve distances and angle measures.
Congruence of triangles
Two triangles are said to be congruent if one can be exactly
superimposed on the other by a rigid motion, and the congruence
theorems specify the conditions under which this can occur.
CC.9-12.G.CO.6 Use geometric descriptions of rigid motions to A rigid motion is a transformation of points in space consisting of a
transform figures and to predict the effect of a given rigid motion on a sequence of one or more translations, reflections, and/or rotations. Rigid
given figure; given two figures, use the definition of congruence in terms motions are assumed to preserve distances and angle measures.
of rigid motions to decide if they are congruent.
Students may use geometric software to explore the effects of rigid
motion on a figure(s).
CC.9-12.G.CO.8 Explain how the criteria for triangle congruence
(ASA, SAS, and SSS) follow from the definition of congruence in
terms of rigid motions
Students may use geometric simulations (computer software or graphing
calculator) to explore theorems about lines and angles.
CC.9-12.G.CO.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
Grade/Course, Unit Title
2014
2
Date Created/Revised: November 18,
segment's endpoints.
Students may use geometric software to make geometric constructions.
CC.9-12.G.CO.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
Examples:
● Construct a triangle given the lengths of two sides and the measure
of the angle between the two sides.
● Construct the circumcenter of a given triangle.
Bloom’s Taxonomy Levels
Depth of Knowledge Levels
Concepts
What Students Need to Know
● definition of congruence
Skills
What Students Need to Be Able to Do
● USE
3
● two triangles are congruent if and only if
corresponding pairs of sides and
corresponding pairs of angles are
congruent
● SHOW
2
● criteria for triangle congruence
● EXPLAIN
2
o ASA
o SAS
o SSS
Grade/Course, Unit Title
2014
3
Date Created/Revised: November 18,
Learning Progressions
The standards below represent prior knowledge and enrichment opportunities for standards in this unit.
Standard
Prerequisite Skills
Accelerate Learning
CC.9-12.G.CO.7 Use the definition of
congruence in terms of rigid motions to
show that two triangles are congruent if and Understand congruence and similarity using
only if corresponding pairs of sides and physical models, transparencies, or geometry
software. 8.G.1-5
corresponding pairs of angles are
congruent.
CC.9-12.G.CO.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.
CC.9-12.G.CO.8 Explain how the criteria
for triangle congruence (ASA, SAS, and
SSS) follow from the definition of
congruence in terms of rigid motions
CC.9-12.G.CO.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
Grade/Course, Unit Title
2014
Understand congruence and similarity using
physical models, transparencies, or geometry
software. 8.G.1-5
Understand congruence and similarity using
physical models, transparencies, or geometry
software. 8.G.1-5
Understand congruence and similarity using
physical models, transparencies, or geometry
software. 8.G.1-5
4
Date Created/Revised: November 18,
endpoints.
CC.9-12.G.CO.12 Make formal geometric
constructions with a variety of tools and
methods (compass and straightedge, string,
reflective devices, paper folding, dynamic
Understand and apply the Pythagorean
geometric software, etc.). Copying a segment;
Theorem. 8.G.6-8
Draw,
construct,
and describe geometrical
copying an angle; bisecting a segment;
figures and describe the relationships between
bisecting an angle; constructing perpendicular
them. 7.G.1-3
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
Unit Assessments
Performance Task
Common Formative Assessment
Grade/Course, Unit Title
2014
CC.9-12.G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
5
Date Created/Revised: November 18,