Understanding By Design Unit Cover Page
Unit Title: Triangle Congruence
Grade Levels: 9th and 10th grade
Subject/Topic Areas: Congruency
Key Words: triangle, congruence, proof, rigid motion, congruence shortcuts
Designed by: Danielle Long
Time Frame: 2.5 weeks
School: Cleveland High School
Brief Summary of Unit (including curriculum context and unit goals):
This unit on congruence is designed as an application of a pervious unit on
transformations and is designed to introduce proof writing in a Geometry 1 course. Students
will learn about rigid motion transformations in terms of congruence, corresponding parts of
congruent triangles are congruent, the shortcuts for proving triangle congruence, and how to
write a formal proof. Students will be able to describe a series of rigid motions to prove triangle
congruence and they will analyze mathematical models to prove triangle congruency in two
column, paragraph, and flowchart proofs.
This unit is organized so that the performance task introduces all of the components of
the unit. The skills needed to complete the performance task will be listed in the students
“Need to know” category after reading the entry document and will be introduced daily
through inquiry investigations. Students will have multiple opportunities, in their group and
individually, to reflect on prior knowledge, develop new understandings, and apply their newly
discovered knowledge to the performance task.
Students will demonstrate acceptable evidence of understanding through two quizzes
and the culminating performance task. The performance task requires students to analyze four
ostrich race track designs and act as consults to the race track owner. Students will apply their
knowledge of triangle congruence to help them justify their evaluation of the track designs.
Students will defend their findings through both a formal proof and a series of rigid motions.
STAGE 1—IDENTIFY DESIRED RESULTS
Established Goals:
Common Core State Standards HS Geometry:
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.
G.CO.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.
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.
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.
What essential questions will be considered?
 Is there a widely accepted way to
construct viable arguments and critique
the arguments of others?
 Can we use mathematics to model real
life?
 How can rigid motion help us
understand congruence?
 How do the criteria for triangle
congruence fit with the definition of
congruence?
 How do you identify corresponding
parts of congruent triangles?
What enduring understandings are desired?
Students will understand that…
 Mathematical proofs are based on
deductive reasoning and are a sound
way to construct viable arguments.
 Rigid motion transformations move a
figure, but don’t change the angle or
side measurements.
 Corresponding parts of congruent
triangles are congruent.
What key knowledge and skills will students acquire as a result of this unit?
Students will know…
 How rigid motion relates to congruency
 Corresponding parts of congruent
triangles are congruent
 Side-side-side, angle-side-angle, sideangle-side, angle-angle-side, and
hypotenuse-leg are shortcuts for
proving two triangles are congruent.
 Angle-angle-angle and side-side-angle
do not prove congruency




Students will be able to…
Prove congruency based on rigid
motion
Find corresponding parts of congruent
triangles
Prove triangle congruence through twocolumn, paragraph, or flowchart proofs
Correctly use and identify SSS, ASA, SAS,
AAS, and HL congruence shortcuts
STAGE 2—DETERMINING ACCEPTABLE EVIDENCE
Performance Task:
Ostrich Races!
Background: In ostrich racing ostriches are either ridden like horses or they pull chariots. This type of racing
originated in Egypt, but has made its way in to America in recent years
and you can see races in states such as Arizona.
Objective: Mrs. Long loves ostrich racing and recently purchased an
ostrich race track outside of Seattle. She has recruited all of the
geometry students at Cleveland High School to consult with her on an
ostrich race track expansion. She wants to build two triangular tracks so
that she can run two races at the same time. In order for the ostrich
races to be fair, the tracks need to be exactly the same length. Four
track designers have submitted the plans below.
Your task is to determine if each track design meets Mrs. Long’s requirements. You will need to submit
a formal report with justifications supporting your findings. You must submit two types of justifications for
each track design in your report. The first type of justification must describe a series of rigid motions. The
second justification must be in the form of a formal proof.
1)
2)
H
W
A
D
O
W
Y
U
B
V
3) EH  GH , GFH  EFH
̅̅̅̅ ∥ 𝐵𝐶
̅̅̅̅ , AD  BC
𝐴𝐷
4)
B
G
F
C
C
H
Y
A
E
D
Quizzes:
Two quizzes will be given over the two and a half week unit. The first quiz will be a shuffle quiz two weeks into
the unit. The quiz will cover congruence in terms of rigid motion, CPCTC, two column, paragraph, and
flowchart proofs, and triangle congruence shortcuts. Students are given a group grade based on whether all
group members have the correct answer written down and how well each member can explain how the group
came to the answer. The second quiz will be given at the end of the unit on the same concepts covered in the
shuffle quiz. The goal of doing the shuffle quiz first is for students further their understanding of the unit
concepts through peer modeling and explanations.
WHERETO
1. Play ostrich racing YouTube video (http://www.youtube.com/watch?NR=1&v=ATZ5kvHKxE&feature=endscreen) H
2. Distribute ostrich racing performance task and have students come up with a “Know” and “Need to
know” list. W
3. Reflect on what we know about rigid transformations and the definition of congruence. E, R
4. Reintroduce corresponding parts of triangles and diagram notations and symbols. E, R
5. Apply a series of rigid transformations to find congruent triangles (interactive activity with partners) E
6. As a group analyze track designs creating multiple series of rigid transformations to prove
corresponding parts of congruent triangles are congruent. E, R
7. Reflect on “Know” and “Need to know” list to evaluate progress. E-2, R
8. Introduce triangle congruence shortcut criteria through group inquiry investigations. E
9. As a group, students compile a list of triangle congruence shortcut criteria with accompanying
diagrams. E
10. “Am I making any sense?” activity. The importance of organization. E
11. Constructing viable arguments, an introduction to writing proofs. E
12. Shuffle quiz on congruence in terms of rigid motion, CPCTC, two column, paragraph, and flowchart
proofs, and triangle congruence shortcuts. E-2
13. As a group identify given information in each track design and discuss appropriate shortcuts to use
within each proof. E
14. Reserve computer lab for students to write their ostrich racing formal report rough draft. Introduce the
structure of formal reports (table of contents and appendix). E, T
15. Peer review of formal report rough draft. Students question each other’s justifications and critique
their reasoning. E-2, R
16. Formal report is submitted and students grade themselves based on the rubric. R
17. End of unit quiz on congruence in terms of rigid motion, CPCTC, two column, paragraph, and flowchart
proofs, and triangle congruence shortcuts. E-2