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:
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.

 

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

 

 

Gold Medal Problem 6.RP.A.3.C, MP3, MP6

Gold Medal Problem

By: Natasha Smith

BRONZE, SILVER, and GOLD… what place will you get?

As you can tell from the above picture, the U.S. takes home the gold when it comes to the Olympics. But have you ever wondered why they are able to take home so many medals?

In this lesson, students will discuss what effects how many medals a country receives at the Olympics. They will use data for the 2016 Olympics and calculate the percentage of medals each country received. Students will then compare and discuss the relationship between the percentage of medals each country received and its population and national wealth.

This lesson integrates Social Studies as students are looking at countries from all over the world and discussing how their different circumstances may affect their ability to receive Olympic medals. This lesson is culturally responsive as students will be critically thinking about how different countries may have advantages to winning Olympic medals and students may begin to realize that where you are from has a huge impact on your ability to medal. This lesson will also peak students interest as many of them follow the Olympics and idolize the athletes.

Extention: After comparing the data as a class, students will try and determine other factors that may have an impact on a country’s medal count. They will research and find their own data and decide whether or not they think the factor contributes to the amount of Olympic medals a country receives.

Math Standards:

CCSS.MATH.CONTENT.6.RP.A.3.C

CCSS.MATH.PRACTICE.MP3

CCSS.MATH.PRACTICE.MP6

Possible Social Studies Standard:

G.L.E.5.4.1: Analyzes multiple factors, compares two groups, generalizes, and connects past to present to formulate a thesis in a paper or presentation.

SaveSave

SaveSave

SaveSaveSaveSave

SaveSave

SaveSave

SaveSave

SaveSave

Circles and Squares 7.RP.2.a, 7.G.A.1, and 7.G.B.4 MP 4

Circles and Squares

Standard: CCSS.7.RP.2.a CCSS7.G.A.1 CCSS.7.G.B.4 Math Practice 4

By: Naomi Johnson, Rachel Van Kopp, Tracy Van Lone, Natasha Smith, and Kimberly Younger

The problem:

This problem was based on Circle-Square by Dan Meyer and had a circle and a square on a line that moved together. When one shape grew, the other would shrink by the same amount. Our job was to find out when the two shapes would be equal in area.

Connections/Solution:

We found that when the perimeter of the square is 21.1 and the circumference of the circle was 18.9, then the areas would be equal to each other. After finding the areas, we noticed that the equations for the two were very similar. The area for the square was A = 28.4234 and the area of the circles was A = 28.4234, so the square was divided by 16 and the circle was divided by 12.567, so the radius of the circle and the distance from a corner to the center of the square will be very close. Another connection that we noticed was that their perimeter and circumference will always equal 40 because that was the length of the number line it was on.


 

Extensions:

  1. Have the circle and square not touch, and looking at how that might change the perimeter and circumference. They could also extend the number line from 40 to 60 or beyond and looking at the impact that would have. For discussions, the teachers could have the students think about what would happen if, instead of the two shapes moving together, they moved apart from each other.
  2. Use a cube and sphere instead of a circle and square:

    Sphere & Cube Dimensional Relationship Exploration

    What are the dimensions of a cube and a sphere of equal volume?
    What are the dimensions of a cube and a sphere of equal surface area?
    Are there any cool relationships between the dimensions of the cube and the sphere?
    Which cube dimension most closely resembles the sphere’s radius? Why?
    Use these calculators to analyze the relationships:
    Cubehttps://www.geogebra.org/m/g6ffapbP    Spherehttps://www.geogebra.org/m/Xyc8W4Qn

(Note: The sphere radius illustrated above is 1/phi. The cube edge is 1.)

 

Cutting, Doubling, Food for All 6.RP.A.2

CCSS.MATH.CONTENT.6.RP.A.2
Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”

 

I’ve always thought pictures that showed food split in half that shouldn’t be able to be split was super interesting so I believe this would be great way to get students interested and talking at the beginning of class. It allows the students to explore what they could possibly be learning about before they are thrown into word problems. From there, it can be used to talk about ratios and fractions of food.

They will be able to find the proportion between different groups of food and the cost and find the rate for a single bit of food and apply that to making sure there is enough money to pay for a large group of people. They will also be able to compare the ratios within a given recipe. As a teacher, I would have numbers that worked well in different sizes as they begin so that the students can apply them to interesting things that they want to be working with.

Kissing Coins: The Hexagonal Unit Circle 7.G.A, 7.G.B, & HSG.CO.C

KISSING COINS 

By Rachel Van Kopp, Kimberly Younger, Tracy Van Lone, Natasha Smith, and Naomi Johnson

7 chips are arranged to kiss (touch) each other.

The chips are connected to each other and look similar to a flower.

What if the coins were different sizes?

The chips are still connected in the same pattern.

After sketching a picture including all seven chips it quickly becomes a parent that there are 6 lines of symmetry. 3 of the lines of symmetry go through the center of two circles. after connecting all of the circles’ center points it is easy to see 6 triangles and 1 hexagon.

Proof – That the radius of the center circle is equal to the six outside circles. The proof is below.

 

 

The connection between circles, triangles, and regular polygons is an important part of why the circles of all the same sizes touch. The proof shows that the center points of 3 circles will create a triangle with a 60 degree angle. If you add on an additional circle your angle will be 120 degrees. Add another circle and the angle will be 180 degrees, add another circle and the angle will be 240 degrees, and adding another circle will be 300 degrees. The proof is related to the standard CCSS.CONTENT.MATH.8.G.A.5 because the standard is discussing equilateral triangles and the cos proof we used is based off of an equilateral triangle.

Extension:

Teachers can expand on sphere packing and what the packing number would be at different dimensions, i.e. how many spheres in a 3D packed space. Teachers can expand with unequal sphere packing, which would be spheres with different radii. Or another extensions could be adding layers onto the original circles and figuring out how many circles will fit on the third or fourth layer

(Follow this link to the video.)

Click the image below to watch a GeoGebra Video about Hexagonal Circle Packing.

 


Coke vs. Sprite CCSS.Math.Content.6.RP.A.1

By: Nick Spencer, Sam Marcoe, Elizabeth Englehart, Grayson Windle

This activity is based off of Dan Meyer’s “Coke vs. Sprite” activity.  In this activity, Dan has a glass of coke, and a glass of sprite, containing equal amounts.  Dan uses a dropper to take some of the sprite and drop it into the coke, creating a coke/sprite mixture.  He then uses the dropper to take some of the coke/sprite mixture, and drops it back into the original sprite.

Dan then asks us this: Which glass contains more of its original soda?

Step 1: Our group began our investigation with a visual representation via drawings in order to solve this problem.  In the figure below, the top two circles represent our original glasses of coke and sprite, each containing 100mL of themselves.

Step 2: Next, we have the dropper extract 10mL of the sprite, and drop it into the coke.  Now we are left with a glass containing 100mL of coke and 10mL of sprite, and a glass containing 90mL of sprite.

Step 3: For the next step, our dropper takes 10mL of the coke/sprite mixture, which we will say contains 9.1mL of coke and 0.9mL of sprite, and drops this into the original sprite glass.

Step 4: Here we do some math.  After adding the coke/sprite mixture into the sprite glass, we find that the original coke glass now has 90.9mL of coke with 9.1mL of sprite, and our sprite glass has 90.9mL of sprite with 9.1mL of coke.

Conclusion: So, which glass contained more of the original soda?
Coke Glass: 90.9mL Coke & 9.1mL Sprite
Sprite Glass: 90.9mL Sprite & 9.1mL Coke

We find that the glasses actually end up containing equal amounts of their original sodas.  We also discovered that this outcome would be the same regardless of the amount of extraction from the glasses.  Had we began with taking say 20mL from the Sprite (leaving 80mL) and putting it into the Coke (which now has 100mL of Coke and 20mL of Sprite), and then taken 20mL of the Coke/Sprite mixture (lets say the dropper has 15mL of Coke and 5mL of Sprite) and then dropped this into the Sprite glass, we would simply find that each glass now contains 85mL of the original soda, and 15mL from the other soda.

CCSS: CCSS.MATH.CONTENT.6.RP.A.1
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

Coke V. Sprite CCSS.Math.Content.6.RP.A.3

In Dan Meyers Coke V. Sprite activity, found on http://mrmeyer.com/threeacts/cokevsprite/, students use both mathematical concepts and logic principles to develop an argument. The student defends their position by organizing their reasoning, then communicating a compelling argument with their peers. Students are asked to use several representations when presenting the problem, this allows students to eventually get a better understanding for how to use mathematical procedures related to ratios.

When we completed this task as a group we determined that is was necessary to give a hypothetical amount of equal proportions of the respective sodas in each glass. We then took the dropper of Sprite and mixed it with Coke. This causes a change in the ratio (or concentration) between the amount of Coke and Sprite compared to the whole. We now proceed in solving the ratio in terms of percentages. Taking the amount of Coke and dividing it by total amount of soda, we end up finding that 90.9% of the liquid is Coke and 9.1% is Sprite.

We now take our new mixture and add it to our untouched Sprite. First, we must find out how much of the sample is Coke and how much is Sprite. By using the percentages, we found of the Coke and Sprite mixture, we multiply to find the amount of each soda within the sample.  In conclusion, both glasses contain an equal amount of their respective contents. The Sprite with Coke was 90.9% Sprite and 9.1% Coke. The Coke with Sprite was practically the reciprocal of the Sprite with Coke, 90.9% Coke and 9.1% Sprite.

This activity could be covered in a 6th grade math class towards the end of their study of ratios and proportional relationships (CCSS M 6.RP 3.c.) to help the students understand percentages, and how to use percentages in a situation where they need to understand what ‘100%’ of something is that isn’t directly given as 100 units, for example. This might be quite complex for this level, however, and could also be used as a lesson for 7th grade students (CCSS M 7.RP 3.) where the students are to be expected to solve problems with multiple steps.

Ratios and Recipes! 6.RP.A

This lesson is a real world activity to be used as the last lesson of the unit. Students use a recipe to adjust the ingredients based on a new serving size. Students use their understanding of ratios and proportions to solve the equations and come up with the new serving size. This activity allows students to use what they have learned about ratios and proportions from the previous lessons and apply them to a real world problem.

CCSS.MATH.CONTENT.6.RP.A.1

Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.

CCSS.MATH.PRACTICE.MP4 Model with mathematics.

Ratios and Recipes!

Perfect Purple Classroom! 6.RP.A.3

CCSS.Math.Content.6.RP.A.3
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

CCSS.Math.Practice.MP4 Model with mathematics.

CCSS.Math.Practice.MP5 Use appropriate tools strategically.

CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of others.

Lesson objective: I can create a model that shows how two values compare to one another.

This is a real-world activity to introduce proportions and ratios. This activity allows students to see that proportions are not just something you see on math worksheets, but in everyday life. This lesson relates to the fine arts, which helps culturally diverse students learn the mathematical concept better. Fine arts such as painting has been known to express complex cultural issues and their heritage has been translated into exchangeable value. The intuitive need for students to express themselves is precisely why the arts are an ideal vehicle to develop language, deliver content, and encourage academic exploration in school in culturally responsive ways. Learning, communicating, and questioning in alignment with the arts develops a dynamic classroom environment where students are excited and engaged in this process. Therefore, integrating arts into learning results in a more engaging classroom for students of all backgrounds.

This activity relates to all students as we are talking about physically painting our classroom once students have accurately problem solved how to paint our classroom the perfect purple. In order to create the perfect purple color, we must need the perfect mixture: two cups of blue paint and three cups of red paint. After measuring our classroom, we decided we need a total of 35 cups of our perfect purple paint to coat the walls. Students will be given the opportunity to work with their peers and with manipulatives to decide how much of each color would be needed to make a total of 35 cups of perfect purple paint. The manipulative provided will be interlocking colored math cubes that the students can build the proportion of the ratio for every three cups of red there needs to be two cups of blue. Once students have had an opportunity to work through the problem using just the interlocking colored math cubes, their peers, and teacher as support, then students will present their ideas to the whole class. As the teacher guides this whole class discussion, the students and teacher will come to the conclusion of the correct amount of blue and red paint needed to create the perfect purple to paint our classroom.

Student Handout: MATH 325 perfect purple paint