Download Unit 4 - Schoolwires.net

Unit 4
Grade 9 and 10
Similar Triangles
Regular and Honors Geometry
Lesson Outline
BIG PICTURE/LESSON ABSTRACT
The study of triangles includes proving many properties that students may already
be familiar with. The angle sum of a triangle being 1800 and the relationship
between an exterior angle the sum of the remote interior angles are familiar and
connected ideas that can be proved. Discovering and applying combinations of sides
and angles that are sufficient conditions for similarity or congruence of two
triangles (for similarity: AA, SSS, SAS, and for congruence: SSS, SAS, ASA, AAS,
HL) provides experience in making conjectures. The results of these relationships
can be used to reason further about additional properties of triangles, isosceles
triangles, and many quadrilaterals. Proofs related to triangles can again take many
different forms including coordinate proofs.
In addition to congruence relationships, similarity is an important area of study in
triangles. In fact, it is reasonable to begin with the properties of similarity and
then move to congruence properties as a special case of similarity. The properties
of congruence and similarity should be used to solve problem situations.
Focus Question:
What are the similarities and differences between similar and congruent triangles?
Common Core Essential State Standards
Domain: Congruence(G-CO)
Similarity, Right Triangles and Trigonometry(G-SRT)
Clusters: EXPERIMENT with transformations in the plane.
UNDERSTANAD similarity in terms of similarity transformations.
PROVE theorems involving similarity.
Standards:
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-SRT.2 Given two figures, USE the definition of similarity in terms of similarity
transformations to DECIDE if they are similar; EXPLAIN using similarity
transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles, and the proportionality of all corresponding pairs of
sides.
G-SRT.5 USE congruence and similarity criteria for triangles to solve problems and
1
to PROVE relationships in geometric figures.
G-SRT.1 VERIFY experimentally the properties of dilations GIVEN by a center and
a scale factor:
a. A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.
G-SRT.3 USE the properties of similarity transformations to ESTABLISH the AA
criterion for two triangles to be similar.
G-SRT.4 PROVE theorems about triangles. Theorems include: a line parallel to one
side of a triangle divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity.
Gaps from 8th Grade Common Core (after 2012-13, students will come to high school
with the following):
8.G.4 UNDERSTAND that a two-dimensional figure is similar to another if the
second can be obtained from the first by a sequence of rotations, reflections,
translations, and dilations; given two similar two-dimensional figures, DECSRIBE a
sequence that EXHIBITS the similarity between them.
8.G.5 USE informal arguments to ESTABLISH interior and exterior angles
CREATED when parallel lines are cut by a transversal, and the angle-angle criterion
for similarity of triangles. For example, arrange three copies of the same triangle so
that the sum of the three angles appears to form a line, and give an argument in
terms of transversals why this is so.
Standards for Mathematical Practices:
2) Reason abstractly and quantitatively.
3) Construct viable arguments and critique the reasoning of others.
5) Use appropriate tools strategically.
6) Look for and make use of structure.
Intellectual Processes:
Representation: Use representations of triangles to model and interpret physical,
social, and mathematical phenomenas.
Reasoning and Proof: Develop and evaluate mathematical arguments and proofs
related to similar and congruent triangles.
Problem Solving: Build new mathematical knowledge of triangles through problem
solving.
Key Concepts/Vocabulary:
Similarity, composition, rigid motion, dilation, center, ratio, angle measure, side
length/segment length, proportional, corresponding sides, corresponding angles,
2
proof, Pythagorean Theorem, parallel, intersect, congruence, triangle similarity,
distance, image, preimage, vertex angle, Triangle Congruency (SSS, SAS, AAS, ASA,
*HL) Triangle Similarity (SSS, SAS, AA)
Day Title/Topic
Learning Goal/Objectives
Expectations
1
Who is Thales?
 Discover Thales theorems
G-SRT.2
 Investigate similar triangles
G-SRT.4
 Understand the Basic Proportionality 8.G.4
Theorem and its Converse.
8.G.5
What is
 Investigate the properties of similar
Similarity?
triangles
2
It’s all just
 Investigate similar figures,
G-SRT.1
similar to me.
corresponding sides, and scale
factors
 Investigate the measures of angles in
similar figures
3
Transformation
 Recall properties of transformation.
G-SRT.2
to
 Investigate dilation using
G-SRT.5
Dilation
4
measurement
G-SRT.4
 Use inductive reasoning to formulate
G-SRT.1
Showing
reasonable conjectures and use
G-CO.2
Triangle
deductive reasoning to justify
8.G.4,8.G.5
Congruence
formally or informally
5
Showing
 Identify and use AA, SAS and
G-SRT.3
Triangle
SSS similarity to solve a variety
Similarity
of problems.
 Discover indirect measurement
6
How high? How
 Solve problems involving similar
G-SRT.2
far?
triangles using measurement data
G-SRT.5
 Solve problems involving similar
Using What
triangles from given situations.
You have
Learned
7
 Assessment
3
Unit 4: Day 1: Similar Triangles: Who is Thales? What is Similarity
Minds On: 40 Min
Learning Goals
Materials
 Thales story
Action: 30 Min
Students will apply proportionality in
 Worksheet
Consolidate/
the context of parallel lines theorems.
 Ruler
Connection: 10 Min
Prior Knolwedge: special angles
 Glue
formed by parallel lines, ratios,
 Calculator
Total = 80 Min
proportions
 Computer ( Geogebra
or TI -83 or Nspire)
Anticipated Challenges:
Students not reading on grade level.
The sum of the triangles not congruent
to 180 degrees due to human error.
The student measurement verses a
computer answer.
Assessment Opportunities
Minds On…
Individual → Pretest
Review the
Groups of 4 → Read: Thales Story
cooperative
Students will divide the reading assignment in
learning skills.
their groups, take notes, and discuss their
findings with their group members.
Encourage
each groups to
Whole Class → Discussion
Discuss the reading as a whole group and add share, then
next group to
to their individual notes as needed. If the
add what is
students need help Facilitate a discussion by
new or unique
asking leading questions such as:
and so on until
all groups have
 What theorems did Thales discover that
shared.
we have discussed?
 How did he use ratios? Proportions?
Assess
Action!
Groups of 4→ Guided Investigation
initiative
Students will complete the activity.
learning skill,
measuring
using ruler,
and
protractor.
Consolidate/
Connection
Whole Class → Guided Discussion
Consider the results of the investigation.
Share different solutions.
Ask students to write a summary of what they
learned during the investigations.
Assess
student
understanding.
4

If needed say: “In order for two
triangles to be considered similar, all
three _____ ______ (corresponding
angles) must be congruent and all three
pairs of ________
_______(corresponding sides) must be
______proportional.”
Extension/PREP/Hwk
Students will complete pages 11 and 12 for homework. This will also help
prepare students for the SAT and ACT exams.
Accommodations/Special Needs: 1) Have students draw pictures by the
Greeks to give them a better understanding about right triangles. *2)
Given a triangle with an internal segment parallel to a side, ask students to
give and justify three true proportions for the figure.
* This can also be used for an opener of the next lesson or part of the
closure if time permits.
Teacher Reflection on Lesson: I really enjoyed presenting the history of Thales story
at the beginning of this unit. Students realized that he is the father of Geometry. This
motivated the students during the lesson. It was the foundation for the reminder of the
unit. The students read about how he used indirect measurement to find the height of a
pyramid.
Looking Ahead: Ratio will become the scale factor in the world of similar figures, and
proportions will be heavily utilized and manipulated in working with similar figures.
Aspects that Worked.





Discussion of Thales was great.
Communicating ideas.
Sketching Thales theorem.
Presenting the ratio and
proportional at the beginning of the
lesson.
Students relied on prior knowledge
from Algebra I in solving algebraic
proportion.
Things to change for next lesson.
Reserve the computer lab for all my classes.
I would also use Geogebra to complete the
investigation.
5
Unit 4: Day 2: Similar Triangles: It’s all just similar to me.
Minds On: 20 Min
Learning Goals: Investigate
Materials
Student
Action: 25 Min
properties of similar triangles,
handout
Consolidate/Connection:
corresponding angles are equal and
scissors
20 Min
corresponding sides are proportional
Protractors
Total =
using concrete materials.
rulers(cm)
Prior Knowledge
colored
pencils
Measuring angles with a protractor
Calculator
and measuring lengths with a ruler.
Frayer
Anticipated Challenges:
graphic
Fear of fractions and the numerical
organizer
values.
Assessment
Opportunities
Minds On…
Whole Class → Guided Discussion
Conduct bell ringer.
Whole Class → Guided Discussion
Students will begin the activity by
cutting out the triangles and then
grouping the triangles.
Whole Class → Guided Discussion
The students will share with the
class how they grouped the
triangles.
Next, ask students what similar
triangles are: same shape,
different size.
Lastly, tell the students to group
the similar triangles together.
Whole Class → Guided
Instructions
Guide the students through labeling
the triangles in the following way:
1a, 1b, 1c, 2a, 2b, 2c, 3a,3b,3c with
Assess how the
different groups
grouped the
triangles.
Assess that the
students are
labeling
correctly.
6
the number being the similar groups:
1- acute triangles, 2-right triangles,
3-obtuse triangles, and the letter
being the size: a – smallest, bmiddle size, c- largest.
Have students take the groups of
similar triangles and match them
with the corresponding angles. So
that they can see the corresponding
angles of similar triangles are
congruent.
Help students to label the
corresponding angles in groups of
similar triangles. Students can use
different colored pencils to mark
the corresponding angles or they
can mark the angles using arcs with
one slash, two slashes, or three
slashes.
Action!
Groups of 4→ Guided Investigation
Students will continue with the
activity in their groups.
Answer question 2.
Next, tell them to determine the
measure of all other angles without
measuring the angles. Then label
the triangles appropriately and
complete the chart.
Now they will use a protractor to
measure the angles of 1c, 2, and 3c.
Using the discovery they made
about angles in similar triangles,
they will find all the other angles
without measuring them.
Lastly, students will discover what a
scale factor is.
Whole Class → Guided Discussion
Assess that
students are
labeling the
triangles with
the appropriate
measurement.
7
Discuss the concept of
corresponding sides with the
students. Have them label the
corresponding sides of each set of
triangles. They can use different
colored pencils to mark the
corresponding side or they can mark
them using slashes.
Consolidate
Connection
Individual → Practice
Students will complete a Frayer
model for similar triangles based on
their learning.
Optional: Discuss briefly the
differences and similarities between
similar shapes and congruent shapes.
Assess students
understanding.
Extension/PREP/Hwk: Option 1)Write a summary of today’s
lesson. Option 2) Find the missing information for pairs of similar
triangles.
Accommodations/Special Needs:
 This lesson incorporates different techniques typically
utilized for diverse learners (hands on manipulates and
interactive online manipulatives).
 Another option is for students to work in pairs.
Teacher Reflection on Lesson: This lesson was a reinforcement lab to the
previous activity. The students manipulated the triangles to visualize the parallel
lines proportionality from the previous lessons. This was a great way to explore
similar triangles. The students were able to think logically, using inductive reasoning
to formulate reasonable conjectures.
Aspects that Worked.
 The hands on manipulative gave
students an opportunity to
visualize that angles are
congruent and sides are
proportional.
 Use precise mathematical
language and use symbolic
notation.
Things to change for next lesson.
I pondered eliminating this activity
from my honors class, due to the high
number of sophomores enrolled in
Honors Geometry, I left it in as a
challenge. This activity is a
reinforcement lesson.
8
This lab also served as a way for
students to work cooperatively
and independently to explore
similar triangles.
Unit 4: Day 3and 4: Similar Triangles: Transformation and
Showing Triangle Congruence.
Minds On: 30 to 50 Min
Learning Goals
Materials
Worksheet,
 Students will identify and
Action: 60-90 Min
protractor,
compare the three congruent
Consolidate/Connection: 20

Min
Total = 1 to 2 days





Minds On…
transformations.
Apply the three congruence
transformation to coordinates
of the vertices of figures.
Identify and apply dilations.
Students will verify congruent
and similar figures.
Students will investigate, and
justify the conclusion for
triangle congruence (SSS, SAS,
ASA, and AAS)
You can use short cuts to
determine if triangles are
congruent.
ruler,
straws,
construction
paper
Graphic
Organizer
Prior Knowledge
Unit 1 transformation, isometric and
knowledge of rigid motion.
Anticipated Challenges:
 New students may not have
the prior knowledge of
transformation as needed.
 Some of the measurements
will vary due to the length of
the straws.
 How to use the straws to
measure the angles.
Assessment
Opportunities
Assess students
Groups of 4→ Guided
understanding.
Investigation
9
Action!
Students will complete Activity
One
Individual → Practice
1) Discuss the ideas of
transformations that occurred. 2)
Which of the six the
transformations were congruent or
similar.
Individual → Practice
Students will write a summary of
the activity.
Groups of 4→ Guided
Investigation
Students will complete Activity
2-6.
Assess students
understanding and
justifications for
their reasoning.
Assess students’
ability to use
inductive,
deductive, and
analytical methods.
Whole Class → Guided Discussion
Consider the results of the
investigation. Facilitate a
discussion about proving triangles
congruent by SSS. This is a short
cut. You can prove triangles are
congruent if the three sides of the
triangles are congruent. The
students will also verify this by
measuring the three angles.
Assess students
are making the
correct notation
for congruent sides
and angles.
Whole Class → Guided Discussion
Students should be discussing
triangles are congruent by SAS.
Facilitate a discussion about
proving triangles congruent by
SAS. This is a short cut. You can
prove triangles are congruent if
the two sides of the triangles are
Assess students
understanding and
the short cut of
proving triangles
are congruent by
using two sides and
the angle between
the two sides.
Assess students
understanding and
short cut of
proving triangles
are congruent by
using three sides
of a triangle.
10
congruent and the angle between
those two sides is also congruent.
The students will also verify this
by measuring the remaining
corresponding parts.
Discuss why some student’s third
length varied from 8 to 9.5.
Whole Class → Guided Discussion
Students should be discussing
triangles are congruent by ASA.
Facilitate a discussion about
proving triangles congruent by
ASA. This is a short cut. You can
prove triangles are congruent if
the two angles of the triangles are
congruent and the side between
those two angles is also congruent.
The students will also verify this
by measuring the remaining
corresponding parts.
Whole Class → Guided Discussion
Students should be discussing
triangles are congruent by AAS.
This is a short cut. You can prove
triangles are congruent if the two
angles of the triangles are
congruent and the side not
between those two angles is also
congruent. The students will also
verify this by measuring the
remaining corresponding parts.
Assess if students
are correctly
placing the straws
on top of the
protractor and
then making
markings to
construct their
angles.
Assess students
understanding and
the short cut of
proving triangles
are congruent by
using two sides and
the angle between
the two sides
Assess students
understanding and
the short cut of
proving triangles
are congruent by
using two angles
and the side not
between the two
angles.
Whole Class → Guided Discussion
Students should be discussing
11
triangles are not congruent when
they have three congruent angles.
The triangles are similar but not
congruent.
Consolidate
Connection
Assess students
understanding of
AAA and SSA are
not short cuts in
proving triangles
are congruent.
Whole Class → Guided Discussion
Students will do a group summary
for activities 2 -6.
Whole Class → Guided Discussion
Consider the results of the
investigation. Share results, and
ask students to write any concerns
of their findings in the
investigations of the activities 2-6.
Which of the following short cuts
work and which did not and explain
(SSS, SAS, ASA, AAS, AAA, and
SSA)
Extension/PREP/Hwk: Briefly discuss why SSA and AAA do
not work for triangle congruence.
Assigned as needed due to conflict with honor roll celebrations.
Students had an opportunity to debrief before the start of the
following day about the activities.
Accommodations/Special Needs:
This lesson incorporates different techniques typically utilized for
diverse learners (hands on manipulates and interactive online
manipulatives.
Another option is for students to complete the graphic organizer on
triangle congruence in pairs.
Teacher Reflection on Lesson: I love the flow of the information. This activity
allowed students to develop a step-by step plan for which they have prior
experience. The students were able to grasp the concepts and proceeded with ease
through the rest of the activities. The students did understand that you could use
the short cuts to prove triangles congruent instead of verifying the measurements
of the angles and the segment lengths for both triangles every time. My regular
students grasp this concept and enjoyed using the short cut.
Looking Ahead: This activity will help students to find which shortcuts may be used to
12
prove triangles congruent.
Aspects that Worked.
The students were required to write individual
summary statements for their first activity.
Collaborative group summaries for activities 2
through 6 were require. This gave me an
opportunity to assess the students
understanding and grade them using a rubric
model.
This also gave me an opportunity to talk about
technology measurements verses human
measurements. I gave the students this
example: If I have new carpet installed in my
house and the carpenter measured the
perimeter incorrectly, it would cost the
company and myself more money.
Things to change or modify for
next lesson.
Due to bench mark exams, I was not
able to do this lab using Geogebra. I
was only able to use the computer lab
with one Regular group of students.
The computer lab would have given the
students an opportunity for the
measurement to work every time
verses human measurements.
Unit 4: Day 5 and 6: Similar Triangles: Showing Triangles Similarity
and How High? How far? Using What you have Learned.
Minds On: 10 Min
Learning Goals
Materials
Discovery
Action: 60 Min
 Students will be able to
Activity
Consolidate/Connection: 15
identify and use AA, SAS,
Sheet
Min
and SSS.
Graphic
Total = 85 Min
 Similarity to solve a variety
Organizer
of problems including real
world applications.
Prior knowledge
 Proving triangles congruence,
 Using short cuts to prove
triangles congruent.
 Thales indirect measurement
of the pyramid.
Anticipated Challenges:
In writing geometric statements,
students tend to write the word
“because” instead of using the
symbol notation for therefore.
Assessment
Opportunities
Minds On…
Individual → Complete the
Bellringer
13
Action!
Students will complete the bell
ringer. Facilitate the students
answers before beginning the
Activity of Proving Triangles are
Similar.
Pairs → Guided Investigation :
Pairs work through the Discovery
Activity. Encourage students to
show their work and present their
solution in an organized manner.
Whole Class → Guided
Investigation
Students will discuss the short
cuts of proving triangles similar by
AA, SSS and SAS. Facilitate the
discussion as needed.
Whole Class → Guided Discussion
Demonstrate to the class how to
justify their reasoning. For
example, write the statements
using the correct notation.
Given /A = /D, /ACD =/FCD,
vertical angles are congruent( :.)
ΔACD = ΔFCD.
Whole Class → Guided
Discussion
Facilitate the discussion how tall is
the wall activity and referring back
to Thales indirect measurement of
the pyramid.
Assess students
understanding of
the short cuts of
proving triangle
similarity.
Assess students
understanding of
indirect
measurement.
Assess students
understanding and
the students are
drawing the
triangles and
labeling the
information
correctly.
Assess students
are correctly using
their notations and
marking their
vertical angles and
reflexive
segments.
14
Groups of 4→ Guided
Investigation
Facilitate by asking the students
to sketch the drawing using two
right triangles.
Consolidate
Connection
Whole Class → Guided Discussion
Have students to put their
sketches and answers on the
boards. Facilitate the discussion
of the results.
Compare and contrast similarity
and congruence. What makes two
figures similar?
Assess students
are drawing the
two right triangles
and labeling the
segments with the
correct lengths.
Assess to make
sure students are
setting up the
proportions
correctly.
Extension/PREP/Hwk
Students will complete the following: When are two triangles
similar? Give examples of situations in which similar triangles occur.
Compare and contrast SSS similarity and SSS congruence.
Review for assessment.
Use the Similar Postulate/Theorems worksheets for students to
practice using notation and justify why the triangles are similar.
Accommodations/Special Needs:
This lesson incorporates different techniques typically
utilized for diverse learners (hands on manipulates and
interactive online manipulatives.
Students will complete the graphic organizer for proving
triangles similar.
Teacher Reflection on Lesson: After this lesson my students were able to
answer the focus question: What are the similarities and differences between
similar and congruent triangles? As I review this lesson I think the format of the
groups played an essential role in this unit. The groups of four were selected by
the students and my self. The students were ask to write down one person they did
want to work with and two people they preferred not to work with. I sorted the
groups based on their request, and work ethics.
Looking Ahead: A classic proof of the Pythagorean Theorem and the use of
the geometric means, the similar triangles created when the altitude to the
hypothesis is drawn. The study of indirect measurement will continue to be
15
used in our right triangle unit. Similarity is also key to theorems in circle
geometry.
Aspects that Worked.
Things to change for next
 The lab worked very well. Once
lesson.
the students had completed the
activities, I returned their
pretest, and individually they
completed a review.
Changing the desk to diagonal rows
facing the door was a great idea for the
indirect measurement activity. I did not
want the students to think of my
classroom as a traditional room or
depend on their classmates or myself
for completion activity. My rationale
was the importance of the indirect
measurement. The concepts of problem
solving will become more and more
evident when we start our trigonometry
unit. Students will be faced with similar
activities of problem solving and I
wanted them to be prepared to work
outside of their groups individually.
Unit 4: Day 7: Similar Triangles: Assessment
The pretest is used as an informal assessment. It provided me with
the following rationales: what to teach, in what order, to provide
appropriate activities to meet the needs of all students, and to
include concrete and technology activities filled with continuous
assessment opportunities:
 I start with the history of Thales and his contributions to
geometry, which laid the foundation for them to discover the
relevant relationship. This activity also promoted
mathematical thinking on part of the students.
 Next, to use the Proportionality Theorem for the following
purposes: 1) Give students prior knowledge of parallel lines
and the algebraic portion would be separated from the
indirect measurement. It would give students the opportunity
to practice with solving proportions before having to solve
16



proportions and set up proportions based on applications.
Review of Transformation would also be used as a prior
knowledge and allow the students to see geometry at work as
a cohesive subject.
Congruence and Similarity of Triangle will give the students
an opportunity to make comparison.
Direct measurement
The activities during the lessons served as informal assessments, and
I was able to make adjustments quickly. The informal assessments
served as a check of how well the students were grasping the
concepts. The activities were also used informally to assess the
mathematical communication that occurred between students.
The Post Assessment for this unit consists of 20 questions. The
questions format included five True/False, ten multiple choice, and 5
open ended questions. The pre assessment and the post assessment
was a common exam that all Geometry teachers at my school used.
We will meet next week to discuss this assessment.
Based on the post assessment data my students did learn the
material and the instructional goals were met. (Please see graph
below pg 18.)
As I review the post assessment,
 I should continue to work on indirect measurement, a few of
the students set up the proportion correctly. However, they
just made simple mathematical mistakes.
 A few students did not mark their triangles with the correct
notations to prove the angles were congruent, which sides were
congruent and therefore did not choose the correct answer.
To address this issue I will have students to use colored
pencils to mark the drawings based on given information.
17
# of Correct Answers
Pretest Vs. Protest
30
25
20
PRETEST
15
POSTTEST
10
5
0
GGGGGGCO.2 SRT.1 SRT.2 SRT.3 SRT.4 SRT.5
Goal
18