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I. Introduction
Overview
The Unit Plan below is content from the College Preparatory Mathematics Geometry textbook, and will
be covering information on triangle similarity. The lessons breakdown shape similarity into more specific
conjectures that exist with triangles, the three covered in this lesson are SSS, AA, and SAS (side-side-
side, angle-angle, and side-angle-side respectively). I chose this for my Unit Plan because during my Fall
2012 placement at Centennial High School, I taught a lesson on similar triangles. The lesson went
relatively well using some interactive tools the textbook provides, so I felt like they would be a good
choice for the Unit Project.
I built several types of instructional strategies into the lessons, but the two that stick out the most to me
are utilizing technology to foster student understanding. The interactive geometry tools the publisher
provides allow students to visualize the extreme cases of similarity, and watch the changes happen before
their eyes. This is extremely beneficial to student learning and can foster a deeper understanding of
similarity that would normally come from repeating several examples.
The other strategy was group work. This is also built into the CPM program, and from my placement I
have seen the positive impact in can have on student discovery. This will benefit every type of student in
the class. ELL students can rely on one another to explain difficult concepts or vocabulary. Also, the
Spanish speaking Aid will be helpful for some translating. The students with IEPs may struggle with
group work, but I will be moving around ensuring they are working and staying on task. That is when it
will be helpful to have an aid in the room.
Each lesson begins with a warm-up that covers information from the previous lesson or the homework.
Then the lesson uses technology and examples to teach students, it is important to be proficient in the
interactive tools.
Objective
I expect students to gain an understanding of when to use SSS, AA, and SAS congruency. The big picture
is so students can use this information to solve real world problems which can include the height of a
person given a shadow, the height of a flagpole, etc. There are several real-world applications of similar
triangles and students can benefit from them if presented properly. I want students to gain this
understanding because it will be important to continue their understanding of geometry. Real-world
applications keep students engaged, so I think it is vital to model with mathematics. This fits in with my
personal philosophy that students must take what they learn and apply it to real life. Students often ask
why they have to learn something, and I understand why they ask. I don’t always have an answer, but in
this case, it is obvious how helpful it can be to fostering student problem solving.
Meeting the Needs of Students
Knowing how many students, which were ELL, and which were Special Education played a factor in how
I arranged my lesson plans. I have students split into groups from the start of the unit, which is a helpful
way to engage different types of students. Each student has something different to offer to each group,
and ELL students being paired can help their understanding if they have someone available who can
translate some of the more difficult terms. When students are working in groups, I plan to have the
biggest impact on individual learning. This gives time to approach each student who may be struggling
and encourage them to work with their peers to understand the concepts they do not quite get.
In addition, I attempted to change up some of the language in my lesson to fit the needs of ELL students.
Simplifying some ornate, mathematical language is useful to helping ELL students. Also, explaining
concepts in different ways may be useful to their understanding. In addition to all of this, I really tried to
keep students active in the lessons, with the hopes of holding their attention. Utilizing technology and
allowing students to work as a group to solve problems keeps students minds active and gives them all a
chance to give input and express their thoughts and ideas.
II. Lesson Plan Sequence
Day 1
Grades: 9-10
Class: Geometry
Time: 50 minutes
Number of Students: 26
I. Goals
To recognize similar triangles using SSS conjecture
To identify similar triangles, and represent the answer algebraically
II. Objectives
Students will understand that a triangle having three similar sides are called ‘Similar Triangles’
Students will understand the connection between scale factor and how it can be applied to similar
triangles
Students will determine which information is needed to satisfy similar triangles
III. Materials and Resources
Resource Page - one per student
Notes sheet – one per student to be collected at the end of class
Protractor and ruler for each student
Teacher will need a computer with access to the “Similar Triangles tool” provided by CPM
IV. Motivation
1. Warm-Up (10 minutes): Students are expected to enter the class, sit down in their seats, and begin
working on the warm-up as soon as the bell rings.
2. Project or write the following problem on the board of the classroom: “Erica believes the
triangles below might be similar. If the two shapes are similar, what must be true about their angles
and sides? Measure the angles and sides of Erica’s triangles with the tools at your table, and help her
decide if they are similar or not.”
A. Students will be using information they learned
on similar shapes in the previous lesson to determine
whether or not the shapes are similar. During the first 5
minutes, the teacher needs to go around to each group of
4 to ensure they are working and making ample
progress.
Teacher: “Okay class, great job getting started on today’s warm-up, I saw a few excellent approaches
to the problem when I was working my way around the room.” At this point, invite a group or two up
to the board to show the work they have done and explain it to the class. I expect students to
understand this concept because they were just assessed on it.
Transition: [DISCUSS OBJECTIVES] Explain to the students they are simply building off of
information they already know about similar shapes, and they will be working more specifically with
triangles for the rest of the lesson. They will use angle and length relationships to prove triangle similarity
for the next couple of problems.
V. Lesson Procedure (30 minutes)Present the class with two questions to consider, “Do we need to
measure all of the sides and all of the angles to state two triangles are similar? Do you think the triangles
have to be similar if the pairs of corresponding sides share a common ratio?”
B. Open the “Similar Triangles tool” provided by CPM. See Appendix for image (10
minutes)
C. Construct a triangle of any size and have students select a scale factor. Then, click on
‘Show: Side Lengths.’ Once the side lengths are visible, ask students to calculate the side ratios:
1.
2. After a few minutes of students computing the ratios, ask them what they notice.
They should realize that the ratios for each side are equivalent.
3. Once students recognize, select the “Modify Triangle” dot and move it around so
students can see how the side lengths change and the rations remain constant. ***Move
the cursor slowly so students can see the effect of the scale factor on each side length.
D. Write the following equation on the board for students to copy down in their Resource
Page:
1. If we have ΔABC, and ΔA’B’C’, then we can say the triangles have Side-Side-
Side similarity (SSS) if we have the following:
“This is the formula for Side-Side-Side similarity. It is a simple and useful tool in
deciding whether or not two triangles are similar. I would like you all to copy this
equality down in your notes next to two similar triangles like the picture we have here
(point to the drawing created through the CPM software).
3. Let’s me show you how to do a problem, and then you can try a few on your own. Let’s start with
the following triangles that are on your worksheet (5 minutes):
A. ΔABC has the following side lengths: AC = 5, AB = 11, BC = 7
B. ΔDEF has the following side lenths: 20, 44, 28
C. We want to answer the questions: “Are the triangles similar?” and “How do we know?”
D. “First, let’s answer are the triangles similar. We look at all of our information, and we are
only given the length of three sides, so we must try to use SSS to show they are.. What do we
need?” [Allow wait time for students to repeat the correct rations.]
E.
=
=
. We get to this by corresponding sides as a ratio, then we simplify. Remember
to work systematically by identifying which side you are looking at and its corresponding value.
Students will be given assistance on assignments by given the sides and they simply need to fill in
the ratios.
F.
=
=
. We get to this by corresponding sides as a ratio, then we simplify
G. Can someone please give me the lengths for our remaining sides?
1.
=
=
H. For your final answer with explanation, we need to write “ΔABC ~ ΔDEF by SSS”
4. Now, using what you know about side-side-side similarity, work in your groups to solve the next
two problems on your worksheet. [See Appendix for WS] (5-8 minutes)
A. Move around the room as students are working to answer questions and encourage
student learning.
1.
2.
5. At this point, I will bring all of the students back together as a class, and review the problems
above. I would ask students to share ideas they had throughout their individual and group work.
While walking through the room, identify students who are grasping the idea and have them present
their work to the class. If no students grasp the concept, then go through #2 with everyone and ask
someone to present an idea for #3. (5 minutes)
[Potential Misconception: Students may be writing their rations incorrectly, or not
matching up corresponding sides. This may become even more evident on the homework
when they are asked to prove SSS similarity and the triangles are not aligned as they have
been in the lesson. Encourage students, that when they are writing ratios to make sure to
label each corresponding side with one, two, or three dashes to show the sides they are
comparing.]
6. Some students may find it helpful to organize their thoughts before jumping into the problem.
With SSS, the following is not necessary; however, it is important to show how to organize
information for when students get to more difficult problems, and they are not told which theorem
(SSS, SAS, AA) to use. (~5 minutes)
A. When organizing your thoughts, it may be helpful to use a flow chart to simplify the
problem. For example, return to problem #3. When you were all looking at problem #3, what
facts must you know to use the SSS conjecture? [Allow wait time for students to respond they
needed the ratios of corresponding sides to be congruent].
1. Write on the board: FACTS:
2. Then, draw three bubbles since we need to satisfy three conditions to prove
similarity.
3. Finally, write the “FACTS” into the bubbles as shown below. If all three bubbles
are congruent, then this implies the triangles are congruent so we draw arrows to a
fourth bubble below which concludes similarity.
4. If a bubble does not satisfy all conditions, then we can say the triangles are not
similar like in example #3.
VI. Closure (5-10 minutes)
7. “Great job today guys, I know this lesson probably came pretty easy for most of you because it is
an extension of what we have been working on the past week with similar shapes and using their
sides and scale factor to determine whether or not they are similar. As we go on, we will be using
more of our knowledge on similar shapes to develop a few more theorems for triangles. Finally, we
will have some real world applications of these theorems, and you can all see how useful the
theorems are.
Let’s do a quick review: For two shapes to be similar,
corresponding angles must have equal measure and
corresponding sides must be proportional. However, if
you are testing for similarity between two triangles, then
it is enough to know that all three corresponding side
lengths share a common ratio. This guarantees similarity
and is referred to as the SSS Triangle Similarity
Conjecture (which can be abbreviated as “SSS
Similarity” or “SSS” for short.)
SSS: If , then the triangles are similar.
VII. Extension
If there is enough time, students’ homework asks them to explain how to set up a flowchart. If
there is enough time in class, I will ask them to move immediately to that reflection on their
homework and talk with their group members about explaining flow charts.
VII. Assessment
Assessment takes place throughout this lesson. As students are working in their groups, I will be
moving around to assess individual understanding, and ask group members to explain their
understanding to one another. I will have time while students are working to assess whether they
are linking the ideas from similar shapes to similar triangles. Next, when we are reviewing
problems 2 and 3 on the board, I can gauge the classes understanding by asking questions to
=
=
=
=
=
=
ΔABC ~ ΔDEF
several students and having them get involved. Finally, the in class worksheet is an opportunity
for me to assess students’ knowledge of today’s topic. There will be no homework for today’s
lesson, as tomorrow’s homework will cover SSS similarity. It should not take a long time to
assess each student’s work, so it will be beneficial to students to complete the WS and turn it in at
the end of class. They will be graded on completion.
IX. Standards
1. Common Core Math Standards
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 to prove
relationships in geometric figures
8. Common Core Standards for Mathematical Practice
Construct viable arguments and critique the reasoning of others – students will be working in
groups to complete some of the problems, it is extremely important they work with one another
and explain their thoughts.
Look for and make use of structure – Students are given several opportunities throughout this
lesson to look for structure to help them solve problems. The homework has been designed to
help continue the sense of structure when solving problems. This allows for students to easily
recognize the steps they need to take, and makes assessment easier on the teacher because they
are searching for a standard.
Section 3.2 – Triangle Similarity Name:_______________________
3.2 Resource Page
Conjecture Definition/Criteria Example # Flow Chart
Bubbles
SSS Side – Side - Side
AA Angle - Angle
SAS Side – Angle - Side
3.2.1 – SSS Similarity name: ____________________
Lesson Objective:
__________________________________________________________________________
Warm-Up:
Erica believes the triangles below might be similar. If the two shapes are similar, what must
be true about their angles and sides? Measure the angles and sides of Erica’s triangles with
the tools at your table, and help her decide if they are similar or not.
ΔABC ΔDEF
AB = _______ ∠ABC = ________ DE = _______ ∠DEF = ________
AC = _______ ∠ACB = ________ DF = _______ ∠DFE = ________
BC = _______ ∠CAB = ________ EF = ________ ∠FDE = ________
Part I: Similar Triangles Tool
Calculate the ratios for the triangles in the program
A
B C
D
E F
What do you notice about the ratios? How does changing the length of one of the legs
effect the scale factor? Write down your observations.
Part II: Examples
Example 1: Are the triangles below similar given the length of each side? Fill in the
blanks before and use your answers to support your answer.
=
=
=
=
=
=
Final Answer: ___________________________________________________________
Example 2: Use SSS to determine whether the triangles below are similar.
Final Answer: _______________________________________________________
5
7
11
44
28
20
Example 3: Use SSS to determine whether the triangles below are similar.
Final Answer: _______________________________________________________
Part III: Flow Charts
When organizing your thoughts, it may be helpful to use a flow chart to simplify the problem. For
example, return to problem #3. When you were all looking at problem #3, what facts must you know to
use the SSS conjecture?
FACTS:
CONCLUSION:
REMEMBER!!
For two shapes to be similar, corresponding angles must have equal
measure and corresponding sides must be proportional. However, if
you are testing for similarity between two triangles, then it is
enough to know that all three corresponding side lengths share a
common ratio. This guarantees similarity and is referred to as
the SSS Triangle Similarity Conjecture (which can be abbreviated
as “SSS Similarity” or “SSS” for short.)
SSS: If , then the triangles are similar
ΔABC ~ ΔDEF
Day 2
Grades: 9-10
Class: Geometry
Time: 50 minutes
Number of Students: 26
I. Goals
To recognize similar triangles using AA conjecture
To identify similar triangles, and represent the answer algebraically
II. Objectives
Students will understand that triangles having two similar angles are called ‘Similar Triangles’
Students will understand the connection between ‘zoom factor’ and how it can be applied to
similar triangles to determine similarity or congruency
Students will determine which information is needed to satisfy similar triangles
III. Materials and Resources
Resource Page
Notes Worksheet – one per student
Protractor and ruler for each student
Teacher will need a computer with access to the “Similar Triangles tool” provided by CPM
IV. Motivation (8-10 minutes)
Students will work on the following warm-up at the beginning of class. It will be projected onto
the wall for students to read as they enter class and take their seat. Students will have the warm-
up sitting on their desk when they enter:
“Yesterday, we discussed SSS similarity and how we can use it to show two triangles are similar.
On your warm-up, please solve question number one. Once you are finished you may move onto
question two.”
[Students have been solving similar triangles on their homework, the first problem has a scale
factor of 1, so it should be relatively easy for the students to solve. The second question asks them
to write down a guess for what they believe the definition of Angle-Angle similarity is. This
should get them to tap into prior knowledge to easily answer the first question, and use that prior
knowledge to make an educated guess for the definition of AA similarity.]
After about 5 minutes, the students will be asked to stop working and we will work on the
questions as a class: “Can I have a volunteer share their answer for the first question?”
[At this point, I expect students to say their answer to be said as follows. “The ratio of side PD to
ZY is 1, the ratio of DQ to ZX is 1, and the ratio of PQ to XY is 1 therefore the triangles are
similar with a scale factor of 1.” As students begin to answer, I will encourage them to state their
answer in the form I previously mentioned. This will be an expectation on homework and future
assessments, mathematical language is very important, especially with ELL students in class.]
[DISCUSS OBJECTIVES] Once we quickly review the first problem, students will be asked,
“Does anyone have some conjectures for the second question? Any thoughts or ideas on what
Angle-Angle similarity? What information do you think we will need? How will it be applied?”
[Do not expect students to answer with how the information will be applied, but write their
thoughts and ideas on the board to come back to later as the class works on the following
problem:
V. Lesson Procedure (35 minutes)
1. Have students begin on the first problem. Do not formally present them with the definition for
AA similarity yet, they should already understand the idea that similar shapes have congruent angles
from the previous lesson. This is giving students the opportunity to show their understanding of prior
knowledge, and apply it to solve for the missing angle.
9. After about 5 minutes, or until most groups are wrapping up, the teacher brings the class together
as a group and begins the lesson.
10. (5 minutes) Scott is looking at the set of
shapes below. He thinks that ΔEFG ∼ ΔHIJ but
he is not sure that the shapes are drawn to scale.
A. Are the corresponding angles equal?
Convince Scott that these triangles are
similar.
B. How many pairs of angles need to be
congruent to be sure that triangles are
similar?
11. Ask the students, “Can someone please tell
me what they received for the measure of angle I?
What about the measure of angle E?”
[Allow wait time for students, this is a very straightforward question, and a good opportunity to
get ELL and IEP students involved. I will focus on helping them when groups are working on the
problem in hopes they are willing to share with the class.]
A. ANGLE E = 68, ANGLE I = 25
12. (5 minutes) Now, ask students, “How many pairs of angles need to be congruent to be sure that
triangles are similar?” This is a very important question, and do not stop asking questions until the
students have fully unpacked the definition.
Teacher: “How many pairs of angles do we need to be congruent to be sure that triangles are
similar?”
Student 3: “Well, we needed all angles to be congruent in order for other shapes to be similar, so
wouldn’t it just be 3?”
Teacher: “Good answer Harold, but there is something special about triangles that tell us we do
not need all 3 sides. Who can tell me some properties of triangles?” [Write them on the board]
Student 1: “They have 3 sides”
Teacher: “True, but I am looking for something more specific. What can you tell me about their
angles?”
Student 2: “They add up to 180!”
Teacher: “Great! All triangles add up to 180 degrees, and they only have three angles. So if I
have a triangle like the ones in the example, everyone should be able to tell me how to get the
measure of the third angle. What did you do on your sheet to solve for the last angle?”
Student 1: “I subtracted 68 and 87 from 180 for the small triangle.”
Teacher: “Exactly. If we are given two angles, that means we know the length of the third. So,
what does this tell us about similarity?” [IMPORTANT TO GIVE WAIT TIME!]
Student 3: “You only need to angles to be congruent.”
Teacher: “Exactly! That is why we call it AA similarity! Take out your note sheet for the lesson,
and write in the following definition for AA similarity:”
A. AA similarity – If two corresponding angles are congruent, then we can say ΔABC and
ΔA’B’C’ are similar (ΔABC ~ ΔA’B’C’). It is also important for students to draw an example in
the final box. They can, and should, use the previous example for their drawing.
13. (~5 minutes) Flow Chart for AA Similarity: Some students may find it helpful to organize their
thoughts before jumping into the problem. It is important to show how to organize information for
when students get to more difficult problems, and they are not told which theorem (SSS, SAS, AA) to
use.
A. When organizing your thoughts, it may be helpful to use a flow chart to simplify the
problem. For example, return to problem #1. When you were all looking at problem #1, what
facts must you know to use the AA conjecture? [Allow wait time for students to respond they
needed the ratios of corresponding sides to be congruent].
1. Write on the board: FACTS:
2. Then, draw two bubbles since we need to satisfy three conditions to prove
similarity.
3. Finally, write the “FACTS” into the bubbles as shown below. If all bubbles are
congruent, then this implies the triangles are congruent so we draw arrows to a final
bubble below which concludes similarity.
4. If a bubble does not satisfy all conditions, then we can say the triangles are not
similar like in example #3.
B. Are these triangles similar? Use full sentences to explain your reasoning. Fill in the
flowchart and decide of the triangles are similar.
C. Be sure to reiterate the number of bubbles needed for each type of similarity and that if
one of the bubbles is incorrect or does not satisfy the necessary conditions, then the triangles are
not similar.
14. Next, we will consider a problem from the CPM textbook to really hammer home the idea of AA
similarity and how it differs from SSS.
15. (20 minutes) Robel’s team is using the SSS ∼ Conjecture to show that two triangles are similar.
“This is too much work,” Robel says. “When we’re using the AA ∼ Conjecture, we only need to look
at two angles. Let’s just calculate the ratios for two pairs of corresponding sides to determine that
triangles are similar.”
A. Is SS ∼ a valid similarity conjecture for triangles? That is, if two pairs of corresponding
side lengths share a common ratio, must the triangles be similar? Let’s open the dynamic
geometry software and observe.
Teacher: “For example, take a triangle with side lengths 4 cm and 5 cm. If your triangle has two
sides that share a common ratio with 4 cm and 5 cm, does your triangle have to be similar to his?”
(Here is a chance to set the lengths of two sides and then show all the possibilities of lengths that
can go between them.)
Student 1: “NO, because you can place another line in there that is 3 cm long or 4 cm long and it
will still be a triangle, just a different one.”
Teacher: “Great job! Does everyone see that? Let me restate it, if you have side lengths of 3 and
4, then there are several lengths which can fit in between 3 and 4 to create a triangle. (Begin
creating several examples in GeoGebra: 3,4,5 – 3,4,6 – 2,4,5 – etc.) Can someone give me one
more example?”
16. At this point, students are able to see a bunch of different triangles that have side lengths of 3 and
4, but are not similar. This is a great opportunity to introduce SAS similarity.
Teacher: “If I wanted to create a triangle similar to this one with side lengths 3 cm and 4 cm,
what do I need to ensure I am creating a similar triangle?
Student 1: “You need another side.”
Teacher: “That is true, but I am looking for another way to create a similar triangle. Take this for
example:”
1. Open the ‘Condition for Triangle Similarity’ software provided by CPM. Change
the scale factor to 1, and show side lengths. The software fixes two lines at 5 and 10, and
adjusts the third line depending on the angle.
Teacher: “Notice here that we have two triangles, with two fixed lengths. The software gives us
the measure of the angle in between. Can anyone guess when the triangles will be congruent?
What will the measure of the angles be?”
2. After recording a few responses, return to the software and move the angle until
the corresponding lengths are equal.
Teacher: “This is called SAS congruency, we will work more on this at the start of class
tomorrow, but I wanted to introduce it to everyone today so you can put it on your Resource
Sheet and use it for tomorrow’s class. There will be one homework problem on SAS tonight so
you can get some practice with the idea. I will only be grading for completion on that question,
not for a correct answer.”
VI. Closure
“Great work today class. I know I threw a lot of information at you, but just remember that when
you are trying to decide if two triangles are similar, use your Resource Sheet to identify which
information you have and which information you need to prove similarity. It is the same concepts we
have been working on, but just more specific to triangles. For SSS, you need to make sure all three
corresponding sides have congruent ratios. The biggest mistake I have seen is people incorrectly writing
their ratios. This is a simple mistake that you can fix by slowing down and mentally rotating the figure to
make sure the sides match up. When using AA similarity, always remember to slow down and make sure
the angles you are choosing correspond with one another. Finally, for SAS – which we have barely
covered – the important thing is to make sure the angle comes between the two sides. Repeat, “side –
angle – side” (point to a nearby triangle to show the correct order) “side – angle – side” until you are sure
you have the right order. If the order is incorrect then your whole problem will be messed up. As always,
come see me if you have any questions!”
VII. Extension
If time allows, the teacher will review two or three examples of SAS with the students. In the
examples, be sure to have students show their work properly. There must be a common scale
factor between the corresponding sides, and an equal angle. Here are some potential problems to
give students:
Organize which information is helpful for each problem, and have the students explain in their
own words why each triangle is or is not similar.
VIII. Assessment
Assessment occurs several times throughout the lesson and on the homework. As students are
working individually, I will be walking around to assess their progress compared to previous
days’ work. Also, I will be asking students and groups several questions to track their learning
and understanding. One of the best forms of assessment that occurs in this lesson is when we are
looking at SSS similarity and trying to shorten it to SS similarity. From this idea, I assessed the
students understanding of similarity and whether or not it would hold true for two sides. Then, I
asked, what conditions do we need if we want to use to similar sides – implying they needed to
unpack the definition of SAS from this incorrect way of thinking about SSS. If students could
understand this right away – or after some leading questions, I would believe they are learning at
an adequate level. The software provided by CPM is a way to reinforce student thinking, and try
out a few things they are thinking. I think this interactive software will help to assess student
understanding and foster the geometric understanding of similarity.
IX. Standards
1. Common Core Math Standards
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.3. Use the properties of similarity transformations to establish the AA criterion for two
triangles to be similar.
17. Common Core Standards for Mathematical Practice
Look for and express regularity in repeated reasoning – In this lesson, students are starting to
look for certain characteristics to determine which conjecture to use. Students need to identify
which sides and angles they need to use in order to identify if triangles are similar.
Reason abstractly and quantitatively – Students are given opportunities to look at some
abstract models in this lesson and understand them quantitatively. Students take the abstract
formula for SSS and SAS and apply it to the triangles they have to unpack similarity. The abstract
look students are given through the dynamic geometry software gives students the opportunity to
visualize a rather abstract idea. It allows bounds to be tested and helps unpack student
understanding.
3.2.2 – AA Similarity name: ____________________
Lesson Objective:
_________________________________________________________________________________
Warm-Up:
Are the triangles to the right similar?
Explain your answer using the ratios of
corresponding sides.
We know definition for Side Side Side (SSS) similarity is that two triangles are congruent when
their corresponding sides have equal ratios. (i.e. ). Using what you
know about SSS, can you think of what is needed for angle-angle
similarity (AA)? Write your guess for the definition below.
Part I: Examples
1. Scott is looking at the set of shapes below. He thinks that ΔEFG ∼ ΔHIJ but he is not sure
that the shapes are drawn to scale.
a.Are the corresponding angles equal? Convince Scott that
these triangles are similar.
b. How many pairs of angles need to be congruent to be sure
that triangles are similar?
2. a.Are these triangles similar? Use full sentences to explain your reasoning.
b. Fill in the flow chart below and decide if the triangles are similar.
FACTS:
CONCLUSION:
Part III: Unpacking SSS and AA Similarity
Is SS ∼ a valid similarity conjecture for triangles? That is, if two pairs of corresponding side lengths share a common ratio, must the triangles be similar? In this problem, you will investigate this question using Condition for Triangle Similarity a dynamic geometry tool used by your teacher.
a. Robel has a triangle with side lengths 4 cm and 5 cm. If your triangle has two sides that share a common ratio with Robel’s, does your triangle have to be similar to his? Use a dynamic geometry tool to investigate this question.
b. Kashi asks, “What if the angles between the two sides have the same measure? Would that be enough to know the triangles are similar?” Use the dynamic geometry tool to answer Kashi’s question.
c. Kashi calls this the “SAS ∼ Conjecture,” placing the “A” between the two “S”s because the angle is between the two sides. He knows it works for Robel’s triangle, but does it work on all other triangles? Test this method on a variety of triangles using the dynamic geometry tool.
1. Decide if each pair of triangles below is similar. Explain your reasoning.
a.
b.
c.
d.
2. Decide if each pair of triangles below is similar. If the triangles are similar, justify your conclusion by stating the similarity conjecture you used. If the triangles are not similar, explain how you know.
a.
b.
c.
Besides showing your reasoning, a flowchart can be used to organize your work as you determine whether or not triangles are similar.
a. Are these triangles similar? Which triangle similarity conjecture did you use?
b. What facts must you know to use the triangle similarity conjecture you chose? Julio started to list the facts in a flowchart below. Copy them on your paper and complete the third oval.
c. Once you have the needed facts in place, you can conclude that you have similar triangles. Add to your flowchart by making an oval and filling in your conclusion.
3. Ramon is examining the triangles below. He suspects they may be similar by SSS ∼.
a. Why is SSS ∼ the best conjecture to test for these triangles?
b. Set up ovals for the facts you need to know to show that the triangles are similar. Complete any necessary calculations and fill in the ovals.
c. Are the triangles similar? If so, complete your flowchart using an appropriate similarity statement. If not, explain how you know.
Extra Problem – Graded on COMPLETION!
Are the triangles to the right are similar?
Using Side - Angle –Side (SAS) conjecture. We covered this briefly at the end of class, so feel free to work with classmates to solve. If it helps, set up a flowchart. Please show all work and explain how you came to your solution.
Day 3
Grades: 9-10
Class: Geometry
Time: 50 minutes
Number of Students: 26
I. Goals
To recognize similar triangles using SAS, AA, and SSS similarity
To apply SAS, AA, and SSS to real world problems using
II. Objectives
Students will understand that SAS can only be used when the pattern goes in order as written
(side-angle-side).
Students will model real world problems and use triangle similarity to solve for missing lengths
or angles
Students will determine which information is provided can help them to decide if two given
triangles are similar
III. Materials and Resources
Each student will be given a worksheet to correspond with the lesson
Resource Page for Section 3.2
IV. Motivation
1. A warm-up will be placed on the board for the class to complete as they come in and find their
seats. They will be given 5 minutes once the bell rings to answer the following questions:
Part I: Show the triangles below are similar:
A. Which similarity conjecture will you use?
1. SAS
A. Make a flow chart showing these triangles are similar.
Part II: Write down the criteria needed to prove SAS similarity. Please include the criteria needed
for scale factor and angle measure.
2. While students are working, move around to collect homework from the last class. After about 5
minutes, ask students to bring their attention to the front of the room. After the class has quieted,
proceed with questioning to find the answers to the above questions. Students should be able to
identify they will need to use SAS conjecture and solve the ratios for corresponding sides. Then,
students need to state that angle B = angle Z. Because these three criteria are met, we can say the
triangles are similar by SAS with scale factor of 1.5. Review how to set up a flow chart, filling the
boxes with ‘FACTS’.
[remember to inform students the number of bubbles they should have will correspond with the
criteria that needs to be met for their chosen conjecture. (i.e SAS = 3, SSS = 3, AA = 2)]
Transition: [DISCUSS OBJECTIVES] “Since we were just barely introduced to SAS yesterday, let’s
continue to go through a few examples so you can really grasp the idea. Then, you will be working in
groups today to apply what you learned about SSS, AA, and SAS similarity to real world problems.”
V. Lesson Procedure (35 minutes)
1. Today’s lesson will be based on a lot of group work. First, introduce one or two more SAS
examples, showing every step along the way and creating flow charts. Here is the first example used:
2. (10 minutes) Look at the figure below and decide use SAS to prove the triangles are similar. Then
find the value of x.
A. “This problem is a little different than ones you
have seen in the past because the two triangles are
combined into one. We are going to use SAS conjecture
here to first prove the triangles are similar. Remember
when you are solving problems like this one, you must
first prove similarity. I want everyone to write on your
paper, the first step is to show the triangles are similar.”
[Give wait time for students to finish writing on their
worksheet]
B. “Now that we know we are going must first prove similarity, what information do I
need?” Draw three circles on the board to denote the three facts that must be met to prove SAS.
Student: “You need to prove the angles between corresponding sides are equal.”
Teacher: “Right! It always helps me to draw the two triangles separately, so go ahead and do that
on your paper to the left of the original figure. Can anyone tell me which angles we know are
equal even though they do not provide any measurements?”
[Students may struggle to identify the correct angle, if they are having trouble, ask a leading
question like “Does it help to split the triangles apart and label each side?”]
Student: “Angle B is the same in both triangles.”
Teacher: “Exactly! Let’s take some tissue paper and trace the two triangles. You can see that the
angle for the small triangle is the same angle for the big triangle which means it has to be equal to
itself.”
C. From here, ask students “Now that we have the angle is congruent to itself, what else do
we need to show to fill the rest of our bubbles for the flow chart?” [If students struggle, have
them pull out their Resource Sheet]. “We want to show the lengths of sides have the same scale
factor. Who can tell me how to find the scale factor in this problem?” [On the students worksheet,
there is some help as it gives the rations that need to be used. Finish proving congruency. Scale
factor = 2.5. MISCONCEPTION: Students may struggle to identify the length of AB because it is
the sum of AA’ and A’B, be ready to explain this idea to them.]
D. “Now that we know the triangles are similar and we have the scale factor, who can tell
me what I need to do to find the value of x?” [Should come easy to students, they have been
doing this throughout the entire previous unit. If it does not, remind them of some other examples
they have done when given similar shapes.]
3. (20-25 minutes) Students will be expected to review the following three problems in their groups
for the next 20 minutes. At this time, I will be moving around the room to answer any questions
students may have working on the problems.
4. Below are six triangles, none of which is drawn to scale. Among the six triangles are three pairs
of similar triangles. Identify the similar triangles, then for each pair make a flowchart justifying the
similarity.
5. Revisit the similar triangles from above.
a. Which pair of triangles are congruent? How do you know?
6. Examine the triangles below.
A. Are these triangles similar? If so, make a flowchart justifying their similarity.
B. Charles has ΔCAT ∼ ΔRUN as
the conclusion of his flowchart. Leesa
has ΔNRU ∼ ΔTCA as her conclusion.
Who is correct? Why?
C. Are ΔCAT and ΔRUN congruent? Explain how you know.
D. Find all the missing side lengths and all the angle measures of ΔCAT and ΔRUN.
VI. Closure (5-10 minutes)
Bring the class back together and review the ‘Big Picture’ or main idea. While students were
working, engage in their learning because the small groups are the teaching opportunities in this
part of the lesson. Be sure to check in with each group initially when they get started, then
monitor each group’s progress and attend to their needs. The aid will be watching over the ELL
students and helping them to work through this process. After the group is pulled back together,
cover 1 or 2 big misconceptions or issues you noticed while circulating the room. Clarify them
for the class, and then sum up by saying, “Great work on these problems, when you are working
with similar triangles, it is always important to first prove the triangles are congruent and state the
information which backs up your claim. Some of you have liked using the flowcharts to organize
your thoughts. I am no longer going to require you to use flow charts, but if they help you then I
encourage you to use them. Be sure to do the homework tonight, the problems on your quiz will
be based on the questions asked on this homework! We will have a short review at the start of
class tomorrow, but you will be expected to give the criteria needed for each similarity and
understand how to apply them to problems. Finally, be sure you can explain your work there will
be a few questions that ask you to do so! If you have any other questions, feel free to stop by my
room anytime during lunch, after school, or in the morning.”
VII. Extension
7. If time allows, I will go over an application of similarity that will be on the students quiz
tomorrow. Here is the problem:
8. Latoya was trying to take a picture of her family in front of the Big Ben clock tower in London.
However, after she snapped the photo, she realized that the top of her father’s head exactly blocked
the top of the clock tower!
While disappointed with the picture, Latoya thought she might be able to estimate the height of the
tower using her math knowledge. Since Latoya took the picture while kneeling, the camera was 2 feet
above the ground. The camera was also 12 feet from her 6-foot tall father, and he was standing about
930 feet from the base of the tower.
A. “First, we must sketch the diagram so it
is easier to read and understand.” [Redraw the
picture on the board or have the figure below
ready to display]
B. “The first step is to turn the problem into something we recognize, this can be done by
drawing a line from the camera to Big Ben clock. We then can solve using similar triangles and
similar shapes”
6’ 2'
930’ 12’
VIII. Assessment
Assessment occurs throughout this lesson. Right off the bat, students learning of the previous
lesson is assessed through the warm-up. After the warm-up, there are several informal
opportunities to assess student understanding and progress. Questioning is a very impactful way
to assess learning in this lesson. We are using the students’ knowledge from the previous lessons
and applying it all at once, so during group work is a great time to assess individual
understanding of the concepts. Finally, students are assess with their homework assignment. This
will be reviewed before the quiz tomorrow, so it will be a great opportunity for students to ask
questions on topics they may be confused about.
IX. Standards
1. Common Core Math Standards
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 to prove
relationships in geometric figures.
2. Common Core Standards for Mathematical Practice
Model with mathematics – This is the first time throughout the lesson students are presented
with the idea of modeling with mathematics. Students are given a real life example in the
extension and in the homework which gives them a look to how similarity can be used in
everyday life. An example came at the end of the lesson where students need to figure out the size
of Big Ben given the height of a man and the distance away from him a photo was taken.
Make sense of problems and persevere in solving them – Students are finally given multiple
types of triangles that have several ways of proving (or disproving) similarity. This is an
opportunity for students to really make sense of all the information they have been given over the
last 2 days and develop problem solving skills by investigating the information they have been
given. When students begin group work they are asked to identify which triangles in a random
group are similar to each other. So, students need to use all of their tools to match the similar
triangles.
3.2.3 – SAS Similarity name: ____________________
Lesson Objective:
_________________________________________________________________________________
Warm-Up:
1. Show the triangles to the right are similar:
Which similarity conjecture will you use?
Make a flow chart showing these triangles are similar.
2. Write down the criteria (all necessary information) needed to prove SAS similarity. Please
include the criteria needed for scale factor and angle measure.
Part I: Examples using SAS
1. Look at the figure below and decide use SAS to prove the
triangles are similar. Then find the value of x.
2. What is the criterion needed for SAS conjecture? Copy this information and an example
onto your Resource Page. This should complete the table!
3
x
2. Below are six triangles, none of which is drawn to scale. Among the six triangles are three pairs
of similar triangles. Identify the similar triangles, then for each pair make a flowchart justifying the
similarity.
2. Revisit the similar triangles from the previous page.
a. Which pair of triangles are congruent? How do you know?
Examine the triangles to the right. Are these triangles similar? If so, make a flowchart justifying their
similarity.
Charles has ΔCAT ∼ ΔRUN as the conclusion of his flowchart. Leesa has ΔNRU ∼ ΔTCA as her
conclusion. Who is correct? Why?
Are ΔCAT and ΔRUN congruent? Explain how you know.
Find all the missing side lengths and all the angle measures of ΔCAT and ΔRUN.
If we do not get to the following problem in class, please complete it on your own. [Hint: There is a way
to split up the figure into shapes we can work with. It will help to draw a line parallel to the ground from
the camera.]
1. Latoya was trying to take a picture of her family in front of the Big Ben clock tower in London.
However, after she snapped the photo, she realized that the top of her father’s head exactly blocked the
top of the clock tower!
While disappointed with the picture, Latoya thought she might be able to estimate the height of the tower
using her math knowledge. Since Latoya took the picture while kneeling, the camera was 2 feet above the
ground. The camera was also 12 feet from her 6-foot tall father, and he was standing about 930 feet from
the base of the tower.
a. Find the height of Big Ben!
6’ 2'
930' 12'
1. Sketch each triangle described below, if possible. If not possible, explain why it
is not possible.
a. Equilateral obtuse triangle
b. Right scalene triangle
c. Obtuse isosceles triangle
d. Acute right triangle
2. Determine which similarity conjectures (AA ∼, SSS ∼, or SAS ∼) could be
used to establish that the following pairs of triangles are similar. List as many as you
can.
3. Solve each equation to find the value of x. Leave your answers in decimal form
accurate to the thousandths place.
e. 4(x − 2) + 3(−x + 4) = −2(x − 3)
f. 2x2 + 7x − 15 = 0
g. 3x2 − 2x = −1
4. On graph paper, sketch a rectangle with side lengths of 15 units and 9 units.
Shrink the rectangle by a zoom factor of . Make a table showing the area and
perimeter of both rectangles.
5. Susan lives 20 miles northeast of Matt. Simone lives 15 miles
dues south of Susan. If Matt lives due west of Simone,
approximately how many miles does he live from Simone?
Draw a diagram and show all work
III. Assessment of Unit
Assessment
The assessment below was created to check students’ progress during the unit. The students are taken
directly from the homework, but the values are changed. Students are asked to identify if given triangles
are similar and to state whether they used SSS, SAS, or AA similarity to decide. Being able to identify
which information for each method was an important fact that recurred throughout the lesson. There were
several examples and homework problems that asked to identify which method would be best used, and a
Resource Sheet with the FACTS needed to prove it.
Another aspect students are assessed on is to solve for a missing variable given two angles, or for a
missing side given two triangles. Again, this was an idea students had a lot of practice doing, and
understanding how to apply the math skills to solve problems with algebra is important for students to
improve their mathematical knowledge. The final assessment item is in regards to a real world
application. Similar to the extension problem students were assigned as homework, they will receive a
similar picture and have to solve for the height of one of the triangles. This idea will be reviewed right
before the quiz since I expect most students to struggle. If students continue to struggle with the concept
through one example, I will give them another to make sure all of the steps are clear. The quiz will be
given for the last 30 minutes of class.
Summative assessment occurred throughout lessons as I tried to ask leading questions to students to foster
some understanding of similar triangles. Summative assessment also happened when I was walking
around the room to help individual groups. Making a specific path, and spending equal time with all
tables is an important and effective teaching method when students are working in groups.
Formative assessment occurred when homework was assigned with the final two lessons. Homework was
assigned at this time because I felt the students would have a clear understanding of how to do problems
straight from the book. The rubric for the quiz will be outlined for each question in red. The answers for
the assessment will be in blue. Total points will be at the bottom of the page.
Section 3.2.2 – Triangle Similarity
Quiz Name: __Answer Key____________ Answers in format of [Point – ‘Answer’] 1. Label the triangles below with the type of method you would choose to prove similarity [SSS, AA, or SAS]. List as many as you can for each. . SSS SAS NONE *SAS* +1 EC AA 1 – SSS 2 – SAS & AA 1- NONE, blank 0 - Incorrect/blank 1 – SAS or AA 0 - Incorrect 0 – incorrect/blank 2. Solve for y. State which method you used to prove the triangles are similar. Show all of your work.
Y = 34. SSS or SAS. 5 – correct answers, full reasoning, work shown 4 – mostly correct answers, full reasoning/work 3 – Correct answers, lacks reasoning and work 2 – Mostly correct answers, little reasoning and work 1 – Few correct answers, little or no reasoning/work
0 – No correct answers or work 3. Prove the triangles below are similar. Organize your thoughts clearly, state the method, and explain which FACTS need to be met in order for them to be similar.
Prove using SAS. Scale factor = .5 (or 2). With angle r = angle c 3 – Correct, full proof with reasoning, scale factor, corresponding walls, and congruent angles 2 – Mostly correct proof, few mistakes or incorrect scale factor
1 – Little correctness or explanation 0 – Nothing or completely incorrect
4. James thought he might be able to estimate the height of a tower using her math knowledge. The camera was 3 feet above the ground. The camera was also 8 feet from his 6-foot tall father, and he was standing about 1000 feet from the base of the tower. Sketch the diagram and locate all possible triangles. Are there any similar triangles? If there are any similar triangles, explain how you know they are similar. Correct Drawing correct labels uses SAS similarity to prove Finds the height of tower (within 5 ft) = 759 ft Labels and states scale factor, similar angle, corresponding sides 6-8 points – full proof with correct answers, labels, pictures, and explanations. Few or no mistakes 4-6 – not a full proof, or missing some information. Missing one or two correct answers with less explanation. 2-4 – not a full proof, few correct answers, incorrect picture. Misses a majority of questions with almost not explanation 0 -2 – little or no effort given. Incorrect labels, several mistakes, etc.
Section 3.2.2 – Triangle Similarity Quiz Name: ____________________ 1. Label the triangles below with the type of method you would choose to prove similarity [SSS, AA, or SAS]. List as many as you can for each.
a.
b. c. 2. Solve for y. State which method you used to prove the triangles are similar. Show all of your work.
3. Prove the triangles below are similar. Organize your thoughts clearly, state the method, and explain which FACTS need to be met in order for them to be similar.
4. James thought he might be able to estimate the height of a tower using her math knowledge. The camera was 3 feet above the ground. The camera was also 8 feet from his 6-foot tall father, and he was standing about 1000 feet from the base of the tower. Sketch the diagram and locate all possible triangles. Are there any similar triangles? If there are any similar triangles, explain how you know they are similar.
APPENDIX 1) The interactive geometry tool used in the first lesson.
2) Interactive Geometry tool used in second lesson:
