Download Grade Level: Unit:

Math 1:
Connections to Previous Learning:
Unit 2: Geometry
Approximate Time Frame: 4-6 weeks
In Grade 8, students are introduced to the concepts of congruence and similarity through the use of physical models and dynamic geometry software. They are
introduced to four types of transformations: rotations, reflections, translations, and dilations. These transformations are then used to establish whether or not
two-dimensional figures are congruent to one another and whether or not two-dimensional figures are similar to one another. Students are also introduced to
the notion of proof through informal arguments that establish theorems such as the sum of the angles of a triangle is 180° and the Pythagorean Theorem.
Focus of this Unit:
Since concepts such as rotation, reflection, and translation were treated in the Grade 8 standards mostly in the context of hands-on activities, and with an
emphasis on geometric intuition, one focus of this unit is to develop more precise definitions for these transformations. Precision and proof are critical elements
of this unit. Students' understanding of what constitutes proof will be developed as students are introduced to the logical structure behind hypotheses and
conclusions. Geometric objects will be more precisely defined using the basic building blocks of geometry, point, line, and distance. One very specific outcome
of the unit is to develop a precise definition of congruence in terms of rigid motions, to use this definition to develop criteria for determining when two triangles
are congruent, and finally, to develop triangle congruence theorems which follow from the definition of congruence in terms of rigid motions. In addition,
attributes of geometric objects observed at an informal level in earlier grades will now be looked at more precisely through proof, and in many instances, these
proofs will make use of triangle congruence and its consequences. Along with physical models, dynamic geometry environments will provide students with tools
for investigating, experimenting with, conjecturing about, and modeling geometric phenomena.
Connections to Subsequent Learning:
Rigid motions will be combined with the non-rigid motion of dilation to develop a definition of similarity in terms of these motions. The definitions and
properties of geometric objects considered in this unit will reappear in future units that deal with other geometric concepts from an analytic perspective
(coordinate geometry) and with modeling. Understanding of the criteria for triangle congruence will help students, not just during their investigations of the
properties of lines, triangles, and quadrilaterals in this unit, but also in their future investigations of circles and trigonometric ratios and in many geometric
modeling tasks. Proof concepts will be spiraled into subsequent units and courses as it will be used later on in both geometric and non-geometric settings. The
problem solving techniques that will be used to solve problems from this unit will help students deal with similar types of problems that they will see in later
units, particularly those units which involve modeling.
10/9/2013 10:09:22 AM
Major Standards
Adapted from UbD framework
Supporting Standards
Page 1
Additional Standards
Math 1:
Unit 2: Geometry
Desired Outcomes
Standard(s):
Understand congruence in terms of rigid motions.
 G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effects 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.
 G.CO.7 Use 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.
 G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motion.
Prove geometric theorems.
 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 endpoints.
 G.CO.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.
 G.CO.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.
Experiment with transformations in the plane.
 G.CO.1 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.
 G.CO.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).
 G.CO.3 Given a rectangle parallelogram trapezoid, or regular polygon, describe the rotations and reflections that carry onto itself.
 G.CO.4 Develop definitions of rotations, reflections, and translations, in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
 G.CO.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 transformation that will carry a given figure onto another.
Transfers: Students will be able use their geometry skills of rigid motion to create proofs of theorems.
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Ex. Using rigid motion to take the vertices of a triangle and transform them in to a straight angle, to prove the sum of the angles of a triangle is 180o.
10/9/2013 10:09:22 AM
Major Standards
Adapted from UbD framework
Supporting Standards
Page 2
Additional Standards
Math 1:
Unit 2: Geometry
WIDA Standard: (English Language Learners)
English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.
English language learners benefit from:
 explicit vocabulary instruction with regard to geometric tools and terminology.
 explicit instruction with regard to theorems and their meanings.
Understandings: Students will understand…
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Two geometric figures are congruent if there is a sequence of rigid motions (rotations, reflections, or translations) that carries one onto the other.
A proof consists of a hypothesis and conclusion connected with a series of logical steps.
The basic building blocks of geometric objects are formed from the undefined notions of point, line, distance along a line, and distance around a circular arc.
Two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles of the triangles are congruent.
It is possible to prove two triangles congruent without proving corresponding pairs of sides and corresponding pairs of angles of the triangle are congruent if
certain subsets of these 6 congruence relationships are known to be true (e.g. SSS, SAS, ASA, but not SSA).
Different observed relationships between lines, between angles, between triangles, and between parallelograms are provable using basic geometric building
blocks and previously proven relationships between these building blocks and between other geometric objects.
The geometric relationships that come from proving triangles congruent may be used to prove relationships between geometric objects.
Essential Questions:
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In terms of rigid motions, when are two geometric figures congruent?
What are the undefined building blocks of geometry and how are they used?
What are possible conditions that are necessary to prove two triangles congruent?
What are the roles of hypothesis and conclusion in a proof?
What criteria are necessary in proving a theorem?
10/9/2013 10:09:22 AM
Major Standards
Adapted from UbD framework
Supporting Standards
Page 3
Additional Standards
Math 1:
Unit 2: Geometry
Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.)
*1. Make sense of problems and persevere in solving them. Students recognize the hypothesis and conclusion in a proof statement and generate the
*2.
*3.
4.
5.
*6.
*7.
8.
requisite proof using givens, along with known facts, definitions, postulates, and theorems.
Reason abstractly and quantitatively. Students use figures and information, including auxiliary lines, pertaining to a specific geometric object as an aid in
reasoning about that geometric object in general.
Construct viable arguments and critique the reasoning of others. Students create and present proofs, and critique the proofs and deductive reasoning of
others.
Model with mathematics.
Use appropriate tools strategically. Students use physical models, drawings, and dynamic geometry software to form conjectures about geometric objects
and to reason from information about the geometric object provided by these tools.
Attend to precision. Students recognize that incorrect initial attempts at definitions, conjectures, and theorems may be corrected through a process of
refinement.
Look for and make use of structure. Students use the structure of geometric objects to gain insights into, make conjectures about, and create proofs
pertaining to these objects.
Look for and express regularity in repeated reasoning.
Prerequisite Skills/Concepts:
Advanced Skills/Concepts:
Students should already be able to:
Some students may be ready to:
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Understand what constitutes a rigid motion.
Have an informal understanding of what it means for two
geometric figures to be congruent.
Given an object, perform specific reflections, rotations, and
translations to that object.
Understand geometric objects such a point, line, angle, triangle,
parallelogram, circle, and so on.
Use informal methods of proof.
10/9/2013 10:09:22 AM
Major Standards
Recognize how counterexamples can be used to refute conjectures.
Know the definitions of, make conjectures about, and create proofs pertaining to
quadrilaterals that are not parallelograms, such as kites or cyclic quadrilaterals.
Adapted from UbD framework
Supporting Standards
Page 4
Additional Standards
Math 1:
Unit 2: Geometry
Knowledge: Students will know…
Skills: Students will be able to …
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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.
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10/9/2013 10:09:22 AM
Major Standards
Develop definitions of rotations, reflections, and translations, in terms of angles,
circles, perpendicular lines, parallel lines, and line segments.
Represent transformations in the plane.
Predict the effects of a given rigid motion on a given figure.
Use the definition of congruence in terms of rigid motions to decide if 2 given
figures are congruent.
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 and those that do not.
Given two geometric objects, recognize if and describe how one can be
transformed to the other through a sequence of rigid motions.
Describe the necessary transformations to carry a figure onto itself.
Draw a figure that represents specific given information.
Construct a hypothesis and proof from a given conjecture.
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from
the definition of congruence in terms of rigid motions.
Prove theorems about lines, angles, triangles and parallelograms.
Adapted from UbD framework
Supporting Standards
Page 5
Additional Standards
Math 1:
Unit 2: Geometry
Academic Vocabulary:
Critical Terms:
Supplemental Terms:
Transformation
Rigid motion
Congruence
Hypothesis
Conclusion
Proof
Postulate
Theorem
Corresponding
Rotation
Reflection
Translation
Dilation
Point
Line
Angle
Assessment
Pre-Assessments
Formative Assessments
Pre-Assessment Four Corners
Unit 2 Pre-Assessment
Vocabulary Activity
Vocabulary Crossword Puzzle
G.CO.7-8 Formative Assessment
G.CO.11 Spot the Parallelogram
Vocabulary Worksheet
G.CO.11 Prove It!
G.CO.9 Build Your Own City
Quilting with Transformations (use
with dynamic geometry software)
Summative Assessments
Self-Assessments
Transformations Summative
Assessment G.CO.1-7
Summative Assessment G.CO.9-11
Post-Assessment Four Corners
Vocabulary Graphic Organizer
Sample Lesson Sequence
1.
2.
3.
4.
Transformations G.CO.1-7 (Look at www.utah as a reference)
Theorems about Lines and Angles G.CO.9
Congruence and Theorems about Triangles G.CO.10, G.CO.8
Theorems about Parallelograms G.CO.11 (Model Lesson)
10/9/2013 10:09:22 AM
Major Standards
Adapted from UbD framework
Supporting Standards
Page 6
Additional Standards