High School: Geometry

SIMILARITY WITH TRIANGLES

The class I will be teaching will be a 9th grade high school

Geometry class. The textbook used will be Geometry:

Integration, Application, Connection by Glencoe and McGraw-

Hill Companies published in 2001. This learning progression

will follow the student through meeting the Common Core

State Standards about proving theorems involving similarities

with triangles. These Common Core State Standards will be

covered in chapter 7 of the textbook, sections 3, 4, and 5. The

Standards for this learning progression will be the cluster

involving proving theorems involving similarity, which are

CCSS.Math.Content.HSG.SRT.B.4 and

CCSS.Math.Content.HSG.SRT.B.5. The math practices that are

going to be used throughout the learning progression are as

follows: MP3, MP4 and MP5.

CCSS.Math.Content.HSG.SRT.B.4:

Prove theorems about triangles. Theorems

include: a line parallel to one side of a

triangle divides the other two proportionally,

and conversely.

CCSS.Math.Content.HSG.SRT.B.5:

Use congruence and similarity criteria for

triangles to solve problems and to prove

relationships in geometric figures.

COMMON CORE STATE STANDARDS

MATHEMATICAL PRACTICES

CCSS.Math.Practice.MP3:

Construct viable arguments and critique the

reasoning of others.

CCSS.Math.Practice.MP4:

Model with mathematics.

CCSS.Math.Practice.MP5:

Use appropriate tools strategically.

Lesson 7-3 Identifying Similar Triangles

For this lesson I will start out by putting a picture under the

document camera of the pyramids in Egypt and explain how

Greek mathematician Thales used geometry for the first

time to solve for the height of the Great Pyramids. I will

write the hinge question on the white board on the side for

the students to consider throughout the lesson. The

students should be able to answer the hinge question by the

end of the lesson. The CCSS covered in this lesson is the first

part of the math standard HSG.SRT.B.5 about solving

problems for triangles with similarity and congruence.

I will follow the introduction with notes for the class in

which I will go through some example of the concepts under

the document camera for the student to copy down. I will

go through three similarities and examples for each to show

the students how they can use the similarity to solve

problems. The similarities are angle-angle (AA), side-sideside

(SSS), and side-angle-side (SAS). While going through

examples, I will be very student involved and ask students

questions as often as possible to have them solving the

problems with me. Once we have finished that, I will ask the

student if they have any questions on anything we have

done so far.

Next, we will do a hand on activity. The students will need a

ruler and protractor. Since the students will be using the

tools and drawing the triangles they will be using MP4 and

MP5. The students will draw a triangle and measure all the

sides of the triangle. They will then draw another triangle

with a scale factor of ½ of 2. They measure the angles of the

triangles to compare. They will answer the questions: Are

these triangles similar? Why? Which triangle similarity is

this? Answer: Since the sides are proportionate the triangles

are similar and this is the SSS similarity. The students

checked their answer by measuring the angles which could

be AA similarity. The students will be assigned homework in

which some problems will be basic problems directly using

the formulas while some of the problems will be real world

problems where the students will have to apply the material

and make connections to solve the problem. Therefore,

being able to answer the hinge question.

Hinge Question:

How can you use similar triangles to

solve problems?

Common Core State Standard:

CCSS.Math.Content.HSG.SRT.B.5:

Use congruence and similarity criteria

for triangles to solve problems and to

prove relationships in geometric

figures.

Angle-Angle (AA) Similarity:

If two angles of one triangle are

congruent to two angles of another

triangle, then the triangles are similar.

Side-Side-Side (SSS) Similarity:

If the measures of the corresponding

sides of two triangles are proportional,

then the triangles are similar.

Side-Angle-Side (SAS) Similarity:

If the measures of two sides of a

triangle are proportional to the

measures of two corresponding sides of

another triangle and the included

angles are congruent, then the triangles

are similar.

CCSS.Math.Practice.MP4:

Model with mathematics.

CCSS.Math.Practice.MP5:

Use appropriate tools strategically.

Lesson 7-4 Parallel Lines and Proportional Parts

I will start this activity with a warm-up related to the last

activity. I will give the students two triangles with two sides

labeled and the angle between the sides labeled as well. The

students will have to show these are similar triangles using

SAS similarity. This will be used as a way to review the

material from the day before.

I will then move onto the new material. The students will

have to prove the two theorems, 7-4 and 7-5. Similar to the

last lesson, I will walk the students through the proofs for

the theorems under the document camera. I will ask

questions to get the students involved in the proofs and

have them assisting me to solve the proof. Once the proofs

are done, I will make sure the students understand the

material and see if anybody has any questions.

I will use the rest of the class to give the students an activity.

Prior to class, I will print out an assortment of triangles with

lines through them, some parallel and some not. The

students will use rulers and protractors to make

measurements based on the theorems to determine if the

line is parallel or not. The students will put their name on

the back and tape it to the board. Once everybody has done

one problem and taped it to the front, then the students will

look at other students’ answers and discuss each other’s

answers and critique their answers and give reasons, which

is using the practice MP3.

Again, the students will be given practice problems for

homework in which some of them will be simple while

others are more challenging and will cause the students to

need to make connections and apply the concepts.

Hinge Question:

Are these lines parallel?

Common Core State Standard:

CCSS.Math.Content.HSG.SRT.B.4:

Prove theorems about triangles.

Theorems include: a line parallel to one

side of a triangle divides the other two

proportionally, and conversely.

Theorem 7-4:

If a line is parallel to one side of a

triangle and intersects the other two

sides in two distinct points, then it

separates these sides into segments of

proportional lengths.

Theorem 7-5:

If a line intersects two sides of a triangle

and separates the sides into

corresponding segments of

proportional lengths, then the line is

parallel to the third side.

CCSS.Math.Practice.MP3:

Construct viable arguments and critique

the reasoning of others.

Lesson 7-5

This lesson, I will start with a warm-up from the first lesson. I

will give the students two triangles with different sides and

angles labeled as it relates to each similarity. The warm up

will have three problems each about one of the following

similarities: AA, SSS, SAS. I will then have a follow up

question to solve for the other side using proportions. The

CCSS covered by this lesson is the second part of

HSG.SRT.B.5 about proving relationships in triangles from

similarity.

I will use the warm up to move into the new concepts. In

this lesson, the students will learn about four theorems

which come from triangle similarity. Like the other lessons, I

will use the document camera to guide the students through

examples for each theorem. We will go through the

theorems 7-7, 7-8, 7-9, and 7-10 as listed on the right. I will

go over one example for each theorem and when I am

finished I will see if the students have any questions about

any of the material covered.

We will then move to an activity where the students will

have a chance to model the concepts in a problem. The

students will use a ruler to make a diagram for the problem

45 on page 376 of the book. Two similar triangular jogging

paths are laid out in a park with one path inside the other.

The dimensions of the inner path are 300, 350, and 550

meters. The shortest side of the outer path is 600 meters.

Will a jogger on the inner path run half as far as the one on

the outer path? Explain. The students will be allowed to

work with their neighbor on this problem but they are not

allowed to move around the room. This give the students

the chance to use math practices MP4 and MP5 by using a

ruler to move the problem. The students will turn this in at

the end of class.

For the students’ homework this time they will be given a

take home quiz. There will be three matching problems to

start for the similarities AA, SSS, and SAS with three pairs of *triangles and different sides or angles labeled. The next section will have theorems 7-4 and 7-5
and the students will need to show an example for each like we did in class. The final section
will have two triangles and the students will need to explain two of the theorems from today’s
lesson, that is theorems 7-7,7-8,7-9, and 7-10. This should be an easy take home quiz if the
students took notes. They will not be allowed to use the same examples as in class so they will
be showing they understand the notes.
Hinge Question:
What can similar triangles tell us about
other characteristics of the triangle?
Common Core State Standard:
CCSS.Math.Content.HSG.SRT.B.5:
Use congruence and similarity criteria
for triangles to solve problems and to
prove relationships in geometric figures.
Theorem 7-7
If two triangles are similar, then the
perimeters are proportional to the
measures of corresponding sides.
Theorem 7-8
If two triangles are similar, then the
measures of corresponding altitude are
proportional to the measures of the
corresponding sides.
Theorem 7-9
If two triangles are similar, then the
measures of corresponding angle
bisectors of the triangles are
proportional to the measures of the
corresponding sides.
Theorem 7-10
If two triangles are similar, then the
measures of corresponding medians are
proportional to the measures of the
corresponding sides.
CCSS.Math.Practice.MP4:
Model with mathematics.
CCSS.Math.Practice.MP5:
Use appropriate tools strategically.*