Triangle Similarity G.SRT.B

johnsonkatie CCSS-Math Learning Progressions, HS Geometry, ML Ratios & Proportional Relationships


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.

 

Solving Systems of Equations HSA.REI.C

Savannah Lucero-Zey CCSS-Math Learning Progressions, HS Algebra

This learning progression will be taught in a high school algebra I class. It is about solving for systems of equations and then being able to interpret the answer that you can get out. The Common Core State Standards (CCSS) domain and cluster for this learning progression is CCSS.MATH.CONTENT.HSA.REI.C.  The main standard that the students are learning is HSA.REI.C.6.

This is based on the McDougall Litell Algebra I book. And while teaching this lesson, we assume the students have worked with general graphs and equations before. They also will have looked at systems and used the graph to find the intersection.

edTPA Learning Progression-1tb37qc

Solving for Areas 7.G.A

gilliangiles CCSS-Math Learning Progressions

This learning progression is based on the concepts of area. There are three math tasks the students will need to perform: an entry task related to a real world problem, a walk around activity, and an exit task related to the entry task. The walk around activity involves the entire class getting up and moving around the room to solve problems posted on the walls. This is one way to get the students to stretch their legs while also keeping them actively engaged in the lesson. There will be twelve problems posted on the walls around the room and the students will choose eight to solve. The idea of having the students choose which problems to do gives them the sense of freedom within their own education. The CCSS-Math that align with this learning progression are as follows:

7.G.A.1 Geometry: Draw, construct, and describe geometrical figures and describe the relationship between them.

  • The students will be finding areas and demonstrating their knowledge of the different shapes used throughout the unit.

HSG-GPE.B.7 Expressing Geometric Properties with Equations: Use coordinates to prove simple geometric theorems algebraically.

  • During lesson 6.7, there will be a couple problems where the students need to use the distance formula to algebraically solve for one side of a shape in the coordinate plane.

HSG-SRT.B.4 Similarity, Right Triangles, & Trigonometry: Prove theorems involving similarity.

During the lesson about areas, 6.7, the students will need to use the Pythagorean Theorem to solve for the height of triangles.

Learning Progression

Choose your own Adventure! Solving Systems of Linear Equations. A.REI.C

emilyivie CCSS-Math Learning Progressions, Common Core State Standards Mathematics, HS Algebra

This learning progression is an approach to teaching how to solve systems of linear equations by graphing, combination, and substitution in a student-lead learning environment.

Standards: The Common Core State Standards that will be satisfied are from the High School Algebra: Reasoning with Equations and Inequalities cluster. We will cover CCSS.MATH.CONTENT.HSA.REI.C.6 solving systems of linear equations exactly and approximately. We will also prove that given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions with standard CCSS.MATH.CONTENT.HSA.REI.C.5. In this course, students focus on mastering basic algebra 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.

 

Algebra I learning progression-1egtu8b

Learning Progression Winter 2018-1ozkjgh

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

isabelramos CCSS-Math Learning Progressions, Common Core State Standards Mathematics, Excel Geometry Curriculum, HS Algebra, HS Geometry, ML Ratios & Proportional Relationships

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

 

 

Learning Progression for Trigonometric Functions in Precalculus: CCSS.MATH.HSF.TF.A.1-4

rosemyers CCSS-Math Learning Progressions, PreCalculus

This learning progression was designed primarily for a high school Precalculus course. The three Common Core State Standards that this learning progression will be satisfying are from the cluster titled “Extend the domain of trigonometric functions using the unit circle,” these are HSF.TF.A.1, HSF.TF.A.3, and HSF.TF.A.4. In this course, students are focusing on mastering the Common Core State standards for Functions. Throughout this learning progression, students will focus on three mathematical practices which are MP5, MP7, and MP8.

One of the instructional tasks that is included in the Learning Progression is creating a unit circle.
In order to construct the unit circle students first have to form the two special right triangles. Through this activity they will from connections between the unit circle and the special right triangles which will strengthen their understanding of the concepts of the lesson.

The assessment used in this learning progression is the green sheet quiz. On this quiz students are given three angles and then they have to draw the angle and evaluate the 6 trigonometric functions for it. The tasks in the learning progression prepare students for this assessment.

Learning Progression

Extended Mathematical Practice: Learning About Numerics, Base Systems, and Modular Arithmetic Outside of the Curriculum – MP 1,3,7,& 8

torsacco CCSS-Math Learning Progressions, Improve Mathematical Thinking

Learning Progression for edTPA – Extended Mathematics

Standards:

  • CCSS.MP-1 – Make sense of problems and persevere in solving them.
  • CCSS.MP-3 – Construct viable arguments and critique the reasoning of others.
  • CCSS.MP7 – Look for and make use of structure.
  • CCSS.MP-8 – Look for and express regularity in repeated reasoning.

Task Summary: Students will be given an inquiry prompt to answer as students work through activity sheets that will have students finding values of numbers in ancient systems (e.g. Egyptian, Babylonian, Mayan), different bases, and in modular expressions. Students will also find solutions for addition and subtraction problems in different bases and with modular arithmetic. Students will also discuss their ideas, findings, and questions using mathematical thinking and reasoning. These tasks are designed to develop students’ mathematical thinking and reasoning.

Assessment Task Summary: Students will be assessed on their mathematical thinking and reasoning by their written work or mathematical discourse. Scoring of the assessment will be done via a rubric based on the standards above and learning targets for each task.

Polynomial Division Learning Progression HSA.SSE.B

Savannah Lucero-Zey CCSS-Math Learning Progressions

This learning progression is about polynomial division. The common core state standards for this learning progression:

CCSS.MATH.CONTENT.HSA.SSE.B.3

Write expressions in equivalent forms to solve problems.

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.

CCSS.MATH.CONTENT.HSA.SSE.B.3.A

Factor a quadratic expression to reveal the zeros of the function it defines.

CCSS.MATH.CONTENT.HSF.IF.C.7.C

Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

Formative Assessment

The formative assessment game used for part of this will be giving the students two cards, one of which is green and the other is red for them to choose whether they agree or not that a certain factor works for a polynomial. There will be examples up on the board and they will be given a factor. They must do the division then hold up the green card if it worked out nicely and the red if it doesn’t work.

Learning Progression Formative Assessment-3-14jpsa2

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

natashasmith2 ML Ratios & Proportional Relationships, Picture Problems, Uncategorized ,

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.

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Apartment Proportions 7.G.A.1

rachelvankopp Common Core State Standards Mathematics, Geometry Curriculum Aligned with CCSS-Math, Picture Problems , , ,

Apartment Proportions

Problem:

Congratulations! You just got your first apartment. It’s located right in the heart of downtown Ellensburg. You just realized you do not have anything to put in your new living room which is 14’ by 12.5’.

Sadly, the store you want to get your furniture from only has a few options but they come in multiple proportions. Make sure you have at least one item from each of the categories. If an item is too large or small, use ratios to change the proportions. You are only allowed to have ten of the following items in your living room!

Hint: Do some of the sizes seem odd? You should probably use ratios to change the size.

Show your work!

Below are the furniture you can select from with the measurements.

Seating                                                            Misc

Couch 8’x4’                                                    Lamp 4’x4’

Chair 2’x3’                                                     Fan 4’x2’

Bean bag square 3’x3’                                  Bookshelf 3’x2’

 

Table                                                              Entertainment

Coffee Table 16’x9’                                       TV with stand 12’x10

Side Table .5’x.5’                                           TV 10’x2’

                                                                           Gaming Consoles 4’x6’

Rug

Fuzzy Rug 15’x8’

Rug 14’x6’

 

This picture was found at Clipart-Library. This lesson will focus on the CCSS.Math.Content.7.G.A.1 which is solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

I would teach this culturally by understanding that all students come from different cultures and different cultures have different housing expectations.

Other standard:

CCSS.ELA-Literacy.W.7.4
Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience.