# Slopes of Parallel and Perpendicular Lines HSG.GPE.B.5

This learning progression is based on the students being able find the slopes of lines and determine if two lines are parallel or perpendicular based on their slopes. It a compilation of three math tasks that will help students understand how to find the slope of a line, classify lines as parallel or perpendicular, and writing equations in slope intercept and point slope forms.

Students will use the slope formula to find the slope a line that is graphed as well as find the slope of a line when two points for a line are given.

• MATH.CONTENT.HSG.GPE.B.5– Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

Students will be using the slopes of two different lines to determine if they are parallel or perpendicular. They will recognize that parallel lines have the same slope and slopes for perpendicular lines are opposite reciprocals.

• MATH.CONTENT.HSG.CO.C.9– Prove theorems about lines and angles.

Students will learn to write equations in point slope form and slope intercept form. They will use the information from these equations, as well as knowledge from the previous tasks to graph lines.

• MATH.CONTENT.HSS.ID.C.7– Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

Formative assessment in the form of observation and discussion will be the primary procedure to support student learning. I will observe how they are finding solutions and ask them questions about their understanding of the learning targets.  At the end of each task students will be given a handout, which will also show how well they are able to meet the learning targets.

Slopes of Parallel and Perpendicular Lines Learning Progression

# Proportion and Similarity HSG.SRT.A.2

This learning progression is provided to support student’s understanding of ratios and proportionality in relation to similar figures. There is a series of three consecutive math tasks that will be completed by students to help guide them through using ratios and proportions to prove figures are similar.

Students will use their knowledge of algebra to express proper ratios and simplify them correctly. They will also be using the cross product of ratios to solve the proportions.

• MATH.CONTENT.HSA.SSE.B.3-Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Students will identify congruent angles and corresponding sides of a polygon so they can find the similarity ratio. Once they do this, they will be able to recognize the sides and angles that are proportional and determine if polygons are similar.

• MATH.CONTENT.HSG.SRT.A.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.

Students will use triangle similarity theorems to show how congruent angles and corresponding sides that are proportional determine if triangles are similar.

• MATH.CONTENT.HSG.SRT.A.3-Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Formative assessment in the form of observation and discussion will be the primary procedure to support student learning. I will observe how they are finding solutions and ask them questions about their understanding of the learning targets.  At the end of each task students will be given a handout, which will also show how well they are able to meet the learning targets.

Proportion and Similarity Learning Progression

# Trigonometry Ratios G.SRT

This learning progression will take place at a 9th grade Algebra 1 classroom. The class is consisted of 32 students where the students have an individual sitting arrangements. This allows the students to work on their own and makes it easy to work with a partner as well. The book use for this class is “Geometry: Integration, Applications, and Connections” by Glencoe and McGraw-Hill. The lesson will cover chapter 8: Applying Right Triangles and Trigonometry but will focus on using sine, cosine, and tangent ratios in order to solve/find the sides and angles of a right triangle.

CCSS.Math:

1.) CCSS.Math.G-SRT.6

Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

2.) CCSS.Math.G-SRT 7

Explain and use the relationship between the sine and cosine of complementary angles.

3.) CCSS.Math.G-SRT.8

Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Learning Progression Trigonometry Ratios-1fej0wt

# Triangle Similarity G.SRT.B

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:
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
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:
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.

# Fun in the sun with converting Percents, Decimals, and Fractions CCSS.MATH.CONTENT.4.NF.C.5

My learning progression is on converting percents, decimals, and fractions. It will lead to comparing ratios and using fractions and percents for ratio lengths. I will cover how to convert from percent to decimal to fraction and vise versa. I will start off with a hinge question then go in with my lesson. I plan to have the students in groups of four and they will be handed a worksheet of conversion.

learning progression percents edtpa-12sg8wn

# Yeah He Caught It, But How Far Was It? 8.G.B.7

Problem: For this lesson, students will be doing an extreme version of utilizing the Pythagorean Theorem. The number in the picture, 400ft, is the distance from home plate to dead center field at Miller Park in Milwaukee, Wisconsin. The goal of the lesson is for students to find the distance of the ball when it is caught by Keon Broxton. Almost all of the information is present in the picture without much background knowledge, but they will also be told that he was estimated to be about 4ft off of dead center when he caught the ball.

The beginning of the lesson will have students discussing what information they have and what information they will need to complete the assignment. The groups will realize that they need to use the Pythagorean Theorem to solve the problem. The groups will come together for a class discussion and asked what information they still need.

After that, students will do their own research to find the missing components. These components are things like where in the stadium the ball was caught, where the ball was when it was caught, etc. The students will use this information to solve the problem, and present to the class how they solved the problem and how they found the missing information.

Connected Math Standards:

CCSS.MATH.CONTENT.8.G.B.7
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

CCSS.MATH.CONTENT.8.G.B.8
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Extension: If students are grasping the content quickly, have them estimate the angle at which the ball entered Keon Broxton’s glove. This can be done using basic trigonometry, which is referenced in the standard CCSS.MATH.HSG.SRT.D.10. While this is technically beyond their grade level, they have a lot of the background knowledge needed to solve the problem so long as they have the height of the ball at the apex of its flight.

Cultural Relevance: This lesson is culturally relevant because baseball is widely played around the world, especially in Hispanic cultures like Mexico, Puerto Rico, Costa Rica, and especially in the Dominican Republic. Plus many students idolize baseball players in a way that they will like the opportunity to do extra research with certain players in them.

Standard from Another Content Area:
WA.EALR.5.4.1
Uses sources within the body of the work to support positions in a paper or presentation.

This standard will be used when students are finding extra information for the assignment.

# How to Prove Two Triangles are Similar G.CO

High School: Geometry I

Proving Geometric Theorems

By Emily Ivie

This learning progression was designed primarily for students who are in a traditional classroom
setting and have similar mathematical abilities. This class consists of 28 students, mainly
sophomores, who are studying geometry as part of their graduation requirement. The geometry
class has been designed to cover topics at a pace that meets the state requirements of content
topics in high school geometry. Since the class is set up on a semester system, the majority of the
students in this class have been a part of this class since the start of the school year and have had
the same exposure and background knowledge about geometry. Many students enjoy talking and
taking part in discussions during class, that is why I am going to teach this learning progression

Standards:
The Common Core State Standards that will be satisfied are from the High School Geometry:
Congruence: Prove Geometric Theorems cluster. In the Congruence domain we will cover
CCSS.MATH.CONTENT.HSG.CO.C.9. Prove theorems about lines and angles and we will also
prove theorems about triangles and parallelograms with standards
CCSS.MATH.CONTENT.HSG.CO.C.10 and CCSS.MATH.CONTENT.HSG.CO.C.11. In this
course, students focus on mastering basic geometry knowledge that is required by the state, while
integrating in common core standards and mathematical practices. In this learning progression
the students will use four mathematical practices including: MP4, MP5, and MP7.
The curriculum these students are learning is based off the McDougal Littell: Geometry by
Larson, Boswell, and Stiff. This textbook is a resource used to design lectures, find worksheets,
and create practice problems through. The start to each class day will begin with a warm up
problem, followed with an introduction to a new topic with notes, classroom activity, and then
discussion. The notes are put up on the projector at the beginning of class so that students who
finish their warm up activity early can start writing the notes.

Accommodations:
Throughout the learning progression accommodations are made for students with IEPs and ELL
students. For the students who need a longer amount of time to write down their notes, they are
able to get the notes from me a day early so that they do not fall behind during lecture and are
able to participate in the discussion during note. Another accommodation made is that I let my 2
ELL students sit next to each other because they feel most comfortable working in partners this
way. In addition to these accommodations, the learning progression designed has activities that
are accessible to all learners. They do not require internet or any other tools beside from the
classroom whiteboards, pen, and paper.

Central Focus:
The central focus of this learning progression is for students to understand how specific theorems
about lines, triangles, and parallelograms were proved and how you can apply them. Students
will also have an understanding as to why it is important to know these theorems when applying
them to everyday life situations.

Lesson 1: Lines
We will start with an introduction to lines: parallel, transversal, and perpendicular. Since many of
the students have already had an introduction to this topic, it will be much easier to engage in a
class in student lead discussion when I ask them, “What does it mean for two lines to be parallel/
transversal/perpendicular?” This discussion of defining certain types of lines will begin to build
their mathematical thinking and conceptual understanding that they will use again during their
partner tasks. During the entry task students will demonstrate MP 7 looking for a way to identify
structure. After reviewing the entry task and understanding these definitions, I will give students
different geometric pictures and we will play a game of “I Spy”, students will be given colored
pencils and required to make a key. We will go around the class and identify specific types of
lines and angles. Once each student has identified and color marked one type of each line and
angle, I will show the students how to prove theorems about line angles. My example I will show
in class will be proving how vertical angle are congruent. I will ask students specific questions to
guide their learning such as: “What do we know about the lines and angles in this diagram? Are
their any linear pairs? What about supplementary?” These questions will formatively assess my
students knowledge about how well they conceptually understand how to identify lines and
angles. When students answer these questions, I will be looking for them to make the connection
between the next steps such as, “since we have angles that are linear pairs, we can use the linear
pair postulate.” This assessment will show students understanding of
CCSS.MATH.CONTENT.HSG.CO.C.9. proving theorems about lines and angles. After students
have worked through the vertical angle theorem, I will ask them to prove that alternate interior
angles are congruent. They will turn in their proof as part of their summative assessment. Once
looking at their proof, I will give feedback based on their reasoning and mathematical thinking.

Lesson 2: Triangles
be expanding our proof knowledge of triangles building off of the prior lesson where students
learned about proving lines and angles and we will try to prove properties about triangles. I will
work through one property about triangles and hold a discussion. After this, we will break up into
groups and I will give each group one theorem about triangles to solve. Then once enough time
has passed, each group will go up to the front board and give a presentation about the theorem
they proved. This lesson aligns with the following standard
CCSS.MATH.CONTENT.HSG.CO.C.10 proving theorems about triangles. Finally once groups
have put their proof up on the whiteboard, I will ask questions to assess their understanding such
as “What does this theorem tell us?”, “How can we apply this postulate to our problem?”, and
“Where do you start when proving this theorem?” These questions are important to ask students
to make sure that they are using tools (such as theorems) appropriately MP5. These
presentations will be a formative and summative assessment to make sure that students can
properly use the new information we just learned as well as explain their answers using old
vocabulary.

Lesson 3: Parallelograms
We will be expanding students understanding of proving line angles and triangles by introducing
parallelogram theorems. Ideally this lesson should be a fun activity that helps students with their
understanding of parallelogram proofs. Students will begin the lesson with a warm up in which
we will cover material learned in the previous day. Students will find a partner and share their
proof completed from the homework the night before. After we finish the entry task I plan to go
over the learning outcomes for the day, which is, students will be able to use their learned
understanding of parallelogram proofs. Shortly after that we will have a class discussion about
what a parallelogram is and I will encourage the use of specific vocabulary words like length,
adjacent, and angle. During our discussion I will hand out 4 parallelogram figures made from
construction paper to each student. We will go through each theorem about parallelograms:
labeling, folding, and drawing on each figure to show understanding for each theorem. Then I
will have the students take the time to try about write up their proofs of these theorems. While
the students working on folding there diagrams I will be walking around the classroom asking
questions about the theorems and how they proved the theorem. During this activity students will
achieve their learning target of CCSS.MATH.CONTENT.HSG.CO.C.11 and MP 4: “Model with
Mathematics” because students will use their parallelogram cuts outs to model their proofs.

Problem: The dimensions of the area known as the “key” in high school basketball are above.  The rectangular area is 12 feet by 19 feet.  The area connected to the top of this rectangle is a half circle and has a radius of 6 feet.  Find the combined area of the two shaded regions.

The CCSS aligned with this problem is:

CCSS.MATH.CONTENT.HSG.MG.A.1
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).*

I selected this problem because basketball season is beginning and many students connect with the sport.  However, many students who do follow basketball aren’t aware of the dimensions of the court, so I felt that it would be an interesting topic.  This task requires students to use their knowledge of areas of rectangles and circles, as well as their mathematical reasoning abilities to solve for the overall area.  If I was teaching this lesson I might leave the dimensions off of the picture and have the students measure the dimensions in the school gym as partners, then solve for the area using their own measurements.  This makes the activity more involved and engaging.

# Pizza Price MG.A.3

CCSS.Math.Content.HSG.MG.A.3
Apply geometric methods to solve design problems

For this picture, I would create a problem related to area of a circle and ratios. The students would be given the diameter of three different sized pizzas then the price of each of those pizza and the students have to determine what deal is the best, that is what has the lowest price per area of pizza ratio. At Domino’s Pizza they have three carry out deals, a Medium for \$5.99, and Large for \$7.99, and an Extra Large for \$9.99. The Medium pizza has a 12 inch diameter, the Large has a 14 inch diameter, and the Extra Large has a 16 inch diameter. This problem relates to the CCSS because it is about using geometric areas and ratios to solve a problem. This picture and problem will get the students interested because I am sure all students like pizza and would love to find the best deal for when they want to buy pizza themselves. Since high school students do not have a lot of money, finding the best deal will definitely draw their attention. The students will work with the concept of ratios and how they can relate to areas.

# Laws of Sine & Cosine (Shadows) SRT.10&11

Problem: Mike is looking at a building that is 300 feet away from him. If the angle of elevation between his position and the building is 25 degrees, how tall is the building?
HSG-SRT.D.10 Prove the Laws of Sines and Cosines and use them to solve problems.
HSG-SRT.D.11 Understand and apply the Law of Sines and the Law of Cosines in right and non-right triangles.
The purpose of this picture is to guide the lesson of the day. By using the shadow of a building students can relate the usage of the Law of Sines and Cosines to real life scenarios. This will provide a better understanding of the properties of the laws and provide a visual representation.