Stage 1 – Desired Results
Established Goals:
1.) Students will be able to use the definition of congruent triangles to prove congruence.
2.) Students will be able to use triangle congruence postulates to prove congruence.
3.) Students will be able to use triangle similarity postulates to prove similarity.
4.) Students will be able to use proportions to solve similar triangles.
Standards:
M11.A.2 – Understand the meanings of operations, use operations and understand how they relate
to each other.
M11.B.2 – Apply appropriate techniques, tools and formulas to determine measurements.
M11.C.1 – Analyze characteristics and properties of two- and three- dimensional geometric
shapes and demonstrate understanding on geometric relationships.
M11.A.2.1.2 – Solve problems using direct and inverse properties.
M11.A.2.1.3 – Identify and/or use proportional relationships in problem solving settings.
M11.C.1.3.1 – Identify and/or use properties of congruent or similar polygons or solids.
G.1.3.2.1 – Write, analyze, complete, or identify formal proofs (e.g., direct and/or indirect
proofs/proofs by contradiction)
Big Ideas:
1.) Properties
2.) Postulates
3.) Proportions
4.) Reasoning and Proof
Essential Questions:
1.) How can we describe relationships
between shapes and use these
relationships to better understand the
properties of shapes?
2.) How can we model real-world
situations using the properties of
similar and congruent triangles?
3.) How is congruence related to
similarity?
Enduring Understandings:
1.) Mathematical statements can be
justified through deductive and
inductive reasoning and proof.
2.) Congruent and similarity properties
can help us solve real-world
situations.
3.) Similar relationships between objects
are a form of proportional
relationships.
4.)
Stage 2 –Assessment Evidence
Performance Task:
Benchmarks
 Goal 3: Students will be able to use triangle similarity postulates to prove similarity.
 Goal 4: Students will be able to use proportions to solve similar triangles.
 Address the facet of learning Application
Task
Using a hands-on approach to similar triangles, students will be finding the heights of
four different objects in the classroom by applying the concepts that they have learned
throughout the unit.
Communication Device
 Completed graphic organizer
 Completed sheet of computations
 2 minute presentation to the class
Other Forms of Assessment:





Quizzes
o Check students’ ability to prove congruence, prove similarity, use the
postulates, and solve using proportions.
Think-Pair-Share
o Solving problems in class
Facilitated Debates
o Explanations and reasoning
Participation in everyday lessons
Journaling
o Bell work problems
Stage 3 – Learning Activities
Learning Activities:





















Math Journaling
Power Point Presentation
Graphic organizers
Think-Pair-Share
Word Chain
Word Toss
Interactive Lectures
Cornell Notes
Guided Practice
Homework
Double Entry Journal
Read With a Pencil
Facilitated Debate
Computer Lab
Exit Slips
Two-Minute Writes
Worksheet
One Word Summary
Get One, Give One
Performance Assessment
3-2-1
Similar Triangles Performance Assessment Rubric
Chart
5-Chart is complete and accurate. Zero mistakes are made.
4-Chart is complete and almost accurate. One mistake is made.
3-Chart is complete but not very accurate. Two mistakes are made.
2-Chart is complete but not accurate. Three mistakes are made.
1-Chart is complete but has more than three mistakes.
0-Chart is not complete. At least one measurement is missing.
Work
5-Computations are complete, clear, and correct.
4-Computations are complete and clear, but they have minimal errors.
3-Computations are complete and clear, but they have multiple errors.
0-Computations are either incomplete or illegible.
Accuracy
5-Heights of all four objects are correct.
4-Heights of three objects are correct.
3-Heights of two objects are correct.
2-Height of one object is correct.
Working in Class
5-Student has been working diligently in class and has been respecting others.
4-Student uses most of their time to work on the project and has been
respecting others.
3-Student has been off-task but has respected others.
0-Student has been off-task and has not been respecting others.
Presentation
5-Student presents their findings, explaining the process and problems faced
while creating it. Student shows he/she has prepared.
4-Student presents their findings, explaining the process and problems faced
while creating it. Student needs to be guided through the presentation.
3-Student presents their findings, explaining the problems faced. Student has
trouble explaining the process used.
2-Student presents their findings, but does not explain the process or problems.
0-Student does not present.
Score:_______ X 4 = _________/100
_______% A B C D F
Monday
Tuesday
Wednesday
Thursday
Day 1: Properties
of Triangles
Day 2: Similarity
Postulates
Day 3: Similarity Day 4:
Postulates
Proportions
•Math Journal
to practice
Pythagorean
Theorem
•Use a Power
Point
Presentation to
define vocabulary
•Students will
take notes using a
graphic
organizer
•Think-PairShare used for
guided practice
•Word Chain to
summarize the
new vocabulary
•Word Toss to
practice the vocab
from yesterday
•Interactive
Lecture
•Cornell Notes
•Guided practice
individually to
practice the
postulates
•Homework to
practice vocab
and postulates
•Word Toss to
practice vocab
and postulates
•Interactive
Lecture
•Double Entry
Journal will be
used for notetaking
•Think-PairShare used for
guided practice
•Read With a
Pencil to
organize a word
problem
•Facilitated
Debate will be
used to help
students reach
the correct
answer for the
word problem
•Math Journal
to practice
proportions
•Interactive
Lecture
•Double Entry
Journal for
note-taking
•Think-PairShare used for
guided practice
•Word
Problems
Worksheet
•Two-minute
write
•Homework to
practice solving
similar triangles
using
proportions
Friday
Day 5:
Proportions
•Review
Homework
•Computer Lab
to review vocab,
postulates, and
solving using
proportions
•Exit Slip
Day 6:
Congruence
Postulates
Day 7:
Congruence
Postulates
Day 8:
Reasoning and
Proof
Day 9:
Reasoning and
Proof
Day 10:
Reasoning and
Proof
•Quiz on
properties,
postulates, and
solving using
proportions
•Interactive
Lecture
•Cornell Notes
•Think-PairShare used for
guided practice
•Two-Minute
Write
•Get One, Give
One
•Congruence
Worksheet in
groups
•Facilitated
Debate to help
students check
their answers on
the worksheet
•One Word
Summary
•Math Journal
to review
postulates
•Interactive
Lecture
•Double Entry
Journal to create
a proof
•Facilitated
Debate to show
students that
congruence is a
special similarity
•Two-Minute
Write
•Homework to
practice creating
proofs
•Review
Homework
•Interactive
Lecture
•Think-PairShare used for
guided practice
•Double Entry
Journal to
create proofs
•Two-Minute
Write
•Math Journal
to practice
writing a proof
•Guided
Practice to
review
similarity and
congruence
postulates
•Homework –
finish guided
practice
Day 11
Day 12
•Introduce
Performance
Assessment
•Work on
Performance
Assessment
•Homework –
finish
performance
assessment if
needed & prepare
presentation of
findings
•Presentation of
Performance
Assessments
•3-2-1
Sensory
Register
Student Teacher Candidate: Katelyn Downey
Lesson Subject(s)/Title: Similar and Congruent Triangles
Lesson Date(s): Day 4 of calendar
Course & Grade(s): Geometry – 10th grade
STM
LTM
Focus
Attention
Connections
Organization
Recognition
Elaborations
Rehearsal
Perception
Meaning
Visualization
INSTRUCTIONAL MATERIALS:
-Guided Practice – Similar Triangles Word Problems
ESSENTIAL QUESTIONS/ SUBSIDIARY QUESTIONS:
1.) How can we describe relationships between shapes and use these relationships to
better understand the properties of shapes?
2.) How can we model real-world situations using the properties of similar and congruent
triangles?
PURPOSE:
The purpose is for students to learn how to formulate and solve proportions in order to
solve for unknowns on similar triangles.
SPECIFIC LEARNING OBJECTIVES:
 Students will set up proportions from similar triangles after being given a “perfect
model.”
 Students will solve proportions to find unknowns on similar triangles after being given
a “perfect model.”
 Students will sketch similar triangles to reflect those described in a word problem.
STANDARDS:
M11.B.2 – Apply appropriate techniques, tools and formulas to determine
measurements.
Facets of Understanding
1.
2.
3.
4.
5.
6.
Explanation
Interpretation
Application
Perspective
Empathy
Self-Knowledge
Multiple Intelligences
1.
2.
3.
4.
5.
6.
7.
8.
Linguistic [words]
Visual [pictures]
Mathematical [numbers &
reasoning]
Kinesthetic [hands-on]
Musical [music]
Interpersonal [social]
Intrapersonal [self]
Naturalist [nature]
Multiple Exposures [4 x 2]
1.
2.
3.
Dramatization
Visualization
Verbal
M11.A.2.1.3 – Identify and/or use proportional relationships in problem solving settings.
Complex Interactions
ANTICIPATORY SET:
Math Journaling – I will put two proportions on the board, and students will solve them
independently in their math journals. After, I will model solving the first problem, asking students
for input. Then, I will ask a random student to explain to me how to solve the second problem. I
will then tell students that this topic that they learned in previous math classes will be used to
determine if triangles are similar as well as solving for an unknown.
INPUT/ ACQUIRE NEW KNOWLEDGE:
Chunk 1 – Using Double Entry Journals
I will draw two triangles that appear to be the same but without any labels and ask students if the
two triangles are similar. They should be able to answer that it is unknown unless there are
labels based on the postulates that they learned the previous days. If not, I will lead them to the
correct answer by asking how they reached their answer.
I will then add labels to the triangles. (Lengths: 3,4,5 and 6,8,10). I will model how we can
determine if two triangles are similar by creating a proportion. I will remind students to be writing
the solving process as well as any notes that they feel are important in the right-hand column of
their notes.
1.
2.
Discussion
Argumentation
Bloom’s Taxonomy
1.
2.
3.
4.
5.
Knowledge [Verbatim]
Comprehension [Own Words]
Application [Problem-Solving]
Analysis [Identify components]
Synthesis [Combine
information]
6. Evaluation [Decisions]
Aspects of the Topic
1.
2.
3.
4.
5.
6.
Facts
Compare
Cause/Effect
Characteristics
Examples
Relationships
9 Effective Strategies
1.
2.
3.
4.
Similarities and Differences
Summarization and Note
Taking
Reinforcing Effort and
Providing Recognition
Homework and Practice
I will give the students a similar example. (Lengths: 5,8,12 and 10,13,17). They will use the Think-Pair-Share strategy to solve
and compare answers. I will then clear up any misconceptions.
I will then model solving for an unknown on similar triangles. (Lengths: 4,7,X and 8,14,20).
I will then give students a similar example to solve using the Think-Pair-Share strategy. (Lengths: 3,6,9 and 12,X,36). I will then
clear up any misconceptions.
Chunk 2
I will write the following word problem on the board:
A tree with a height of 4m casts a shadow 15 m long on the ground. How high is another tree that casts a shadow which is 20 m
long?
I will model solving the problem, reminding students to write the process in the right-hand column of their notes.
APPLY/ DEEPEN NEW KNOWLEDGE:
Chunk 3
I will pass out the guided practice, which is a worksheet of word problems similar to the one I solved in chunk 2. Students will
work in pairs. I will go over the answers with them, asking them to explain their answers and thought processes.
CLOSURE/ASSESSMENT:
Two-Minute Write – I will ask the students to write a summary of what they learned.
HOMEWORK: (Purpose- Practice)
2 problems of solving for unknowns
3 word problems
EVALUATION/ASSESSMENT OF STUDENTS:
 Think-Pair-Share
 Monitoring word problems
 Explanations of solving word problems
 Two-minute write
Similar Triangles: Word Problems
1. A statue, honoring Ray Hnatyshyn (1934–2002), can be found on Spadina Crescent East,
near the University Bridge in Saskatoon. Use the information below to determine the
unknown height of the statue.
2. A tree 24 feet tall casts a shadow 12 feet long. Brad is 6 feet tall. How long is Brad's
shadow? (draw a diagram and solve)
3. Triangles EFG and QRS are similar. The length of the sides of EFG are 144, 128, and 112.
The length of the smallest side of QRS is 280, what is the length of the longest side of
QRS? (draw a diagram and solve)
4. A girl 160 cm tall, stands 360 cm from a lamp post at night. Her shadow from the light is 90
cm long. How high is the lamp post?
160 cm
90 cm
360 cm
Sensory
Register
Student Teacher Candidate: Katelyn Downey
Lesson Subject(s)/Title: Similar and Congruent Triangles
Lesson Date(s): Day 7 of calendar
Course & Grade(s): Geometry – 10th grade
INSTRUCTIONAL MATERIALS:
-Congruence Worksheet
ESSENTIAL QUESTIONS/ SUBSIDIARY QUESTIONS:
1.) How can we describe relationships between shapes and use these relationships to
better understand the properties of shapes?
2.) How can we model real-world situations using the properties of similar and
congruent triangles?
PURPOSE:
The purpose is for students to practice the congruence postulates.
SPECIFIC LEARNING OBJECTIVES:
 With 90% accuracy, students will complete the congruence worksheet.
STANDARDS:
M11.C.1.3.1 – Identify and/or use properties of congruent or similar polygons or solids.
ANTICIPATORY SET:
Get One, Give One - I will ask students to write a list of the postulates that they can
remember. After a minute, I will tell the students to use the Get One, Give One strategy
to complete their lists.
I will then ask students to give me a postulate until my list is complete. I will then allow
for students to make any corrections.
APPLY/ DEEPEN NEW KNOWLEDGE:
Chunk 1
I will pass out the congruence worksheet and explain to students that they will be working in
groups of 3-4. I will remind them that participation is part of their assessment, so they should be
working diligently and cooperatively.
LTM
Focus
Attention
Connections
Organization
Recognition
Elaborations
Rehearsal
Perception
Meaning
Visualization
Facets of Understanding
7.
8.
9.
10.
11.
12.
Explanation
Interpretation
Application
Perspective
Empathy
Self-Knowledge
Multiple Intelligences
9.
10.
11.
12.
13.
14.
15.
16.
Linguistic [words]
Visual [pictures]
Mathematical [numbers &
reasoning]
Kinesthetic [hands-on]
Musical [music]
Interpersonal [social]
Intrapersonal [self]
Naturalist [nature]
Multiple Exposures [4 x 2]
4.
5.
6.
Dramatization
Visualization
Verbal
Complex Interactions
3.
4.
Students will work on their worksheets in groups. I will be walking around to monitor their
progress and answer any questions.
Chunk 2
Facilitated Debate - We will go over the answers by having a member of each group give their
answer for a question. If two groups don’t agree, I will play the devil’s advocate to create a
debate between the two groups. I will create questions to get students, not only those in the
groups that don’t agree, to give input. I will then help students reach the correct answer. We will
continue this process until all of the questions have been answered.
STM
Discussion
Argumentation
Bloom’s Taxonomy
7.
8.
9.
10.
11.
Knowledge [Verbatim]
Comprehension [Own Words]
Application [Problem-Solving]
Analysis [Identify components]
Synthesis [Combine
information]
12. Evaluation [Decisions]
Aspects of the Topic
CLOSURE/ASSESSMENT:
One Word Summary – I will ask students to write down a one word summary to
capture how they feel about proving congruence. I will call on a few random students to give me
their answer so that I can determine their thoughts about their abilities.
7.
8.
9.
10.
11.
12.
Facts
Compare
Cause/Effect
Characteristics
Examples
Relationships
HOMEWORK: None.
9 Effective Strategies
10.
11.
12.
13.
Similarities and Differences
Summarization and Note
Taking
Reinforcing Effort and
Providing Recognition
Homework and Practice
EVALUATION/ASSESSMENT OF STUDENTS:
 Get One, Give One
 Monitoring of congruence worksheet
 Facilitated Debate
 One Word Summary
Sensory
Register
Student Teacher Candidate: Katelyn Downey
Lesson Subject(s)/Title: Similar and Congruent Triangles
Lesson Date(s): Day 8 of calendar
Course & Grade(s): Geometry – 10th grade
STM
LTM
Focus
Attention
Connections
Organization
Recognition
Elaborations
Rehearsal
Perception
Meaning
Visualization
ESSENTIAL QUESTIONS/ SUBSIDIARY QUESTIONS:
1.) How can we describe relationships between shapes and use these relationships to
better understand the properties of shapes?
2.) How can we model real-world situations using the properties of similar and
congruent triangles?
3.) How is congruence related to similarity?
PURPOSE:
The purpose is for students to learn how to create a mathematical proof using the
postulates that they have learned in previous lessons. They will then have a facilitated debate
that will help them reach the understanding that congruence is a special form of similarity
(essential question #3).
SPECIFIC LEARNING OBJECTIVES:
 After being given two perfect models, students will create a mathematical proof in
pairs.
 Students will participate in a facilitated debate.
STANDARDS:
M11.A.2 – Understand the meanings of operations, use operations and understand
how they relate to each other.
G.1.3.2.1 – Write, analyze, complete, or identify formal proofs (e.g., direct and/or
indirect proofs/proofs by contradiction)
ANTICIPATORY SET:
Math Journaling – I will give students two examples of triangles. Students must state
if the triangles are congruent or not and why.
We will then discuss their answers, and I will clear up any misconceptions.
INPUT/ ACQUIRE NEW KNOWLEDGE:
Chunk 1
I will tell students to set up their notes in the Double Entry Journal format. I will then model
proving two triangles to be congruent using SAS.
I will then model another proof, but this time I will use SSS. Again, I will remind students to use
the Double Entry Journal when creating a proof.
Chunk 2
I will give students two examples to create proofs of individually. I will be monitoring their
progress and answering any questions. Once students have finished, they will check their
answers with a partner.
*These examples will prove that if two triangles are congruent, they are also similar. Some
students may have noticed this before, but this will prove it.*
Facets of Understanding
13.
14.
15.
16.
17.
18.
Explanation
Interpretation
Application
Perspective
Empathy
Self-Knowledge
Multiple Intelligences
17.
18.
19.
20.
21.
22.
23.
24.
Linguistic [words]
Visual [pictures]
Mathematical [numbers &
reasoning]
Kinesthetic [hands-on]
Musical [music]
Interpersonal [social]
Intrapersonal [self]
Naturalist [nature]
Multiple Exposures [4 x 2]
7.
8.
9.
Dramatization
Visualization
Verbal
Complex Interactions
5.
6.
Discussion
Argumentation
Bloom’s Taxonomy
13.
14.
15.
16.
17.
Knowledge [Verbatim]
Comprehension [Own Words]
Application [Problem-Solving]
Analysis [Identify components]
Synthesis [Combine
information]
18. Evaluation [Decisions]
Aspects of the Topic
13.
14.
15.
16.
17.
18.
APPLY/ DEEPEN NEW KNOWLEDGE:
Facts
Compare
Cause/Effect
Characteristics
Examples
Relationships
9 Effective Strategies
19.
20.
21.
22.
Similarities and Differences
Summarization and Note
Taking
Reinforcing Effort and
Providing Recognition
Homework and Practice
Facilitated Debate – I will ask students if they could infer anything from the two examples that they just proved. I will
play the devil’s advocate to create a better understanding. I will also lead students to the correct answer by asking questions
such as:
 Are these two examples essentially the same?
 What is the difference between the two examples?
 What can you infer from the two proofs?
CLOSURE/ASSESSMENT:
Two-Minute Write – I will ask students to write a summary about what they were able to infer from the two examples.
HOMEWORK: (Purpose- Practice)
2 mathematical proofs
EVALUATION/ASSESSMENT OF STUDENTS:
 Monitoring of creating proofs
 Facilitated Debate
 Two-minute write