Download Section 8.5 - Renee Beach: Teaching Portfolio

Section 8.1
Class: Geometry
Grade Level: 10th grade
Unit: Similarity
Teacher: Ms. Renee Beach
Common Core State Standards (CCSS)
Similarity, Right Triangles, and Trigonometry
 Define trigonometric ratios and solve problems involving right triangles
o 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. (G-SRT.6.)
Iowa Core Curriculum-Subject Area Standard(s)
Geometry
 Understands and applies properties and relationships of geometric figures.
21st Century Skill(s)
 Communication and collaboration
Essential Question
What are the requirements for two polygons to be similar? How can you use similar
polygons to solve real-life problems?
Objectives
Students will be able to:
 Solve the ratio of two numbers
 Use proportions to solve real-life problems
Anticipatory Set
(10 minutes) Complete the pre-assessment labeled Section 8.1 Anticipatory Set. Then go
over the answers in class. Before going over the answers, students will write everything
they know about similarity between polygons, what they want to know about similar
polygons, and how they might find out about similar polygons on their KWHL chart
(Inquiry-based learning).
Teaching: Activities
(10 minutes) Hand out definitions sheet to students. Have them write down the
definitions of ratio, proportion, means, and extremes. It may be necessary to talk them
through the difference between ratio and proportion and means and extremes.
(7 minutes) Go over two examples of ratios. Walk through the first one and let them try
the second on their own.
1.
Answer:
2.
Answer:
(20 minutes) Have students pair up into groups of 2. Give each group of two a piece of
string approximately 12 inches long and a ruler.
1. They will measure the circumference of the base of each other’s thumb, the
circumference of each other’s wrist, and the circumference of each other’s neck
by wrapping the string till it connects and measuring that out on the ruler. Record
the results in a table.
2. Once the pair has completed this by both partners, they will each find the ratio of
their wrist measurement to their thumb measurement. Then similarly the ratio of
the neck to wrist.
3. They should then compare the two ratios.
4. Then compare your ratios to others in the class.
5. Would it matter if the data was recorded all in inches or all in centimeters?
(5 minutes) Go over this ratio problem with the class. The measure of the angles in ∆JKL
are in the extended ratio of 1:2:3. Find the measures of the angles.
We then know x+2x+3x=180  6x=180  x=30. Therefore the angle measures are 30,
60, and 90.
(5 minutes) Have students write down the cross product property and the reciprocal
property in their notes.
(8 minutes) Have students solve the proportions. Walk through the first one and let them
try the second one on their own.
1.
2.
Answer:
Answer:






(10 minutes) On the projector, show the picture below for students to see. Below you are
shown Bev Dolittle’s painting Music in the Wind. Her actual painting is 12 inches high.
How wide is it?
Thus we can set up the ratio such that we have
, where x=width of the
painting, 12=height of the painting, 4.375=width of the photo, and 1.25=height of the
photo. By multiplicaiton you then get
to have x=42 inches wide.
. By simplification you can solve
Given the same photo as above, we estimate the flute to be about 1(7/8) inches long. How
long might the flute be in the actual painting?
Answer:
 f=18 inches long
Closure
(10 minutes) They will get this time to work on the homework for tomorrow and ask
questions as needed. This will also give me time to talk to them about any homework
they haven’t turned in. They will need to write a bumper sticker about what they learned
today to hand in before leaving class.
Independent Practice
Problems from the end of the section.
Assessment
(5 minutes) Put this equation on the board. This is like a question that they will have to
solve on a test. They can solve this on a piece of paper of their own.
Materials
Paper and Pencil
String
Ruler
Calculator
Duration
Total time: 90 minutes
Pre-Assessment
The pre-assessment activity that I used in this lesson was a review of previous material
that they should have learned a while ago. By doing this I am able to tell their previous
knowledge of the information.
Section 8.2
Class: Geometry
Grade Level: 10th grade
Unit: Similarity
Teacher: Ms. Renee Beach
Common Core State Standards (CCSS)
Similarity, Right Triangles, and Trigonometry
 Understand similarity in terms of similarity transformations
o 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.2.)
Iowa Core Curriculum-Subject Area Standard(s)
Algebra
 Understand and apply proportionality
21st Century Skill(s)
 Communication and Collaboration
 Life and Career Skills
Essential Question
What are the requirements for two polygons to be similar? How can you use similar
polygons to solve real-life problems?
Objectives
Students will be able to:
 Use properties of proportions
 Use proportions to solve real-life problems
Anticipatory Set
(10 minutes) Have students get homework out. This is a time for them to ask any
questions about the homework that they might not have understood or if one of the
questions was difficult to understand. Once complete with that, collect the homework
from students.
Teaching: Activities
(10 minutes) Have students get out their definitions sheets and refer to the additional
properties of proportions. They will use these properties to determine whether the
following statements are true.
1. If
is true.
, then
. Answer:


. Therefore statement

2. If
, then
. Answer:
Therefore the statement is false.
(5 minutes) In the diagram


.
. Find the length of BD.


x=8. Therefore BD=8
(5 minutes) Now on their definitions sheet, have them write down the definition of
geometric mean. Go through the example to find the geometric mean of 8 and 18. They
can either find it by
or
.
(10 minutes) Let the students try this example on their own or in pairs of 2 or 3:
International stndard paper sizes are commonly used all over the world. The various sizes
all have the same width-to-lenth ratios. Two sizes of paper are shown, called 4A and A3.
The distance labeled x is the geometric mean of 210mm and 420mm. Find the value of x.
Answer:


Therefore the geometric mean of 210 and 420 is
diagram is about 297 mm.


, or about 297. So x in the
.
(25 minutes) Here is a real-life example for the students to work through:
A scale model of the Titanic is 107.5 inches long and 11.25 inches wide. The Titanic
itself was 882.75 feet long. How wide was it?
Answer: First they should set up a proprtion using a variable to represent the width of the
Titanic.
about 92.4 feet wide.


. Therefore the Titanic was
Some students may be wondering why you didn’t have to convert the units to all be the
same. Well when writing a proportion with unlike units, the numerators should have the
same units and the denominators would have the same units.
Now ask the students if there are other ways you could write the proportion. If you wrote
the proporiton in different ways would you have to convert the units to all be the same?
Don’t solve the other ways but write them down on the board for them to see.
(10 minutes) Have students break up into small groups. They should try to compile a list
of other ways you can use scale models to figure out the actual size of a shape. Once they
have a decent sized list every student in the class should come to the board and write
down at least one idea that they came up with. Discuss these ideas as a class and what
type of career they could go into if they wanted to work with ideas like this.
Closure
(10 minutes) They will get this time to work on the homework for tomorrow and ask
questions as needed. This will also give me time to talk to them about any homework
they haven’t turned in. Students must write a newspaper headline to turn in before
leaving the classroom.
Independent Practice
Problems from the back of the book.
Assessment
(5 minutes) This will be like a question they will have on a test.
In 1997, the ratio of the population of South Carolina to the population of Wyoming was
47:6. The population of South Carolina was about 3,760,000. Find the population of
Wyoming.
Materials
Paper
Pencil
Computer
Calculator
Duration
Total time: 90 minutes
Pre-Assessment
NA
Section 8.3
Class: Geometry
Grade Level: 10th grade
Unit: Similarity
Teacher: Ms. Renee Beach
Common Core State Standards (CCSS)
Similarity, Right Triangles, and Trigonometry
 Understand similarity in terms of similarity transformations
o Given two figures, use the definition of similarity in terms of similarity
transformations to decide fi 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.2.)
Iowa Core Curriculum-Subject Area Standard(s)
Geometry and Measurement
 Understand and apply similarity, with connections to proportion
21st Century Skill(s)
 Critical Thinking and Problem Solving
 Flexibility and Adaptability
Essential Question
What are the requirements for two polygons to be similar? How can you use similar
polygons to solve real-life problems?
Objectives
Students will be able to:
 Identify similar polygons
 Use similar polygons to solve real-life problems
Anticipatory Set
(5 minutes) On a piece of paper, students will write down what they think it means for
two polygons to be similar and what the difference between similar and congruent are (if
there is one). Then show them the definition of similar polygons and by either a thumbs
up, to the side, or down, students will let me know if they got that one right, somewhat
right, or not at all. Then explain that for two polygons to be similar they must have
congruent angles and proportional sides but for two polygons to be congruent, they must
have congruent angles and congruent sides (questioning strategies).
Teaching: Activities
(5 minutes) Have students get out their definitions paper and write down the definition of
similar polygons and scale factor.
(45 minutes) This time will be used to go through the power point up until the last slide
(15 minutes) This will be used to explain the problem that they will develop and present
to the class and also this will be the time they use to work on their problem.
Closure
(15 minutes) This time will be used for students to present their problems that they came
up with. They have to write one thing on a piece of paper that they found interesting
about someone else’s problem.
Independent Practice
Problems from the back of the book.
Assessment
(5 minutes) This will be like a question they will have on a test.
The similarity statement for the following figures is □WXYZ ~ □PQRS. Find the
statement of proportionality and the scale factor.
Materials
Paper
Pencil
Graph paper (not necessary but they can chose to use it)
Duration
Total time: 90 minutes
Pre-Assessment
The pre-assessment is them trying to come up with their definition of what they think it
means for two polygons to be similar and if there is a difference between similar and
congruent.
Section 8.4
Class: Geometry
Grade Level: 10th grade
Unit: Similarity
Teacher: Ms. Renee Beach
Common Core State Standards (CCSS)
Similarity, Right Triangles, and Trigonometry
 Understand similarity in terms of similarity transformations
o Using the properties of similarity transformations to establish the AA
criterion for two triangles to be similar. (G-SRT.3.)
Iowa Core Curriculum-Subject Area Standard(s)
Geometry
 Understands and applies properties and relationships of geometric figures.
Reasoning and Proof
 Develops and evaluates mathematical arguments and proofs.
21st Century Skill(s)
 Life and Career Skills
 Learning and Innovation Skills
Essential Question
What are the requirements for two polygons to be similar? How can you use similar
polygons to solve real-life problems?
Objectives
Students will be able to:
 Identify similar triangles
 Use similar triangles in real-life problems
Anticipatory Set
(5 minutes) Show the following triangles on the board:
∆BTW~∆ETC. The students should use their previous knowledge to write the statement
of proportionality, the measure of angle TEC, and find ET and BE.
Answer: statement of proportionality is
, and ET is found by

, the measure of angle TEC is


. BE is found by
Teaching: Activities
(5 minutes) Have students get out their definitions sheet and write down the definition of
postulate 25, the Angle-Angle (AA) Similarity Postulate.
(10 minutes) Draw the figure on a graph that is projected on the board.
Use properties of similar triangles to explain why any two points on a line can be used to
calculate the slope. Find the slope of the line using both pairs of points shown.
Answer: By the AA similarity postulate ∆BEC~∆AFD, so the ratios of corresponding
sides are the same. In particular,
. By a property of proportions,
. The
slope of a line is the ratio of the change in y to the corresponding change in x. The ratios
represent the slopes of BC and AD, respectively. Because the two slopes are
equal, any two points on a line can be used to calculate the slope. You can verify this
with specific values from the diagram.
and
.
(10 minutes) Walk through this problem as a class about using similar triangles in real
life.
Low-level aerial photos can be taken using a remote-controlled camera suspended from a
blimp. You want to take an aerial photo that covers a ground distance g of 50 meters. Use
the proportions
to estimate the altitude h that the blimp should fly at to take the
photo. In the proportion, use f=8 cm and n=3 cm. These two variables are determined by
the type of camera being used.
Show this immage if it will help them visualize the problem better
Answer:



. Therefore the blimp should fly
at an altitude of about 133 meters to take a photo that covers a ground distance of 50
meters.
(10 minutes) Find the length of QS using scale factors for the following figures
Answer: Find the scale factor of ∆NQP to ∆TQR.
. Now since
the ratio of the lengths of the altitutdes is equal to the scale factor, you can write the
following equation:
. Substitute 6 for QM and solve for QS to show that QS=4.
(35 minutes) Hand out the learning contract. This time will be used to fill out the learning
contract, get their choices approved, and then to work on the problems. They should work
on these individually. This will be due at the next class period. (Self-paced strategy)
Closure
(10 minutes) Discuss with the students that in the previous lesson over 8.3, they learned
that the perimeters of similar polygons are in the same ratio as the lengths of the
corresponding side. This concept can be generalized as follows. If two polygons are
similar, then the ratio of any two corresponding lengths (such as altitudes, medians, angle
bisector segments, and diagonals) is equal to the scale factor of the similar polygons.
This should help them understand that the scale factor can be used for so much more than
just the ratio of side lengths of corresponding sides. By discussing this, they should be
able to connect lessons to each other and better understand what is being asked of them.
Students will fill out a 3-2-1 card on what they learned today, what they found
interesting, and what they still have questions on.
Independent Practice
Finish their problems from the learning contract that is due at the next class period.
Assessment
(5 minutes) This will be like a question they will have on a test.
Prove the two triangles are similar. What postulate makes them similar? Write the
similarity statement for these triangles.
Materials
Paper
Pencil
Calculator
Duration
Total time: 90 minutes
Pre-Assessment
None
Section 8.5
Class: Geometry
Grade Level: 10th grade
Unit: Similarity
Teacher: Ms. Renee Beach
Common Core State Standards (CCSS)
Similarity, Right Triangles, and Trigonometry
 Prove theorems involving similarity
o Use congruence and similarity criteria for triangles to solve problems and
to prove relationships in geometric figures. (G-SRT.5.)
Iowa Core Curriculum-Subject Area Standard(s)
Geometry
 Understands and applies properties and relationships of geometric figures.
Reasoning and Proof
 Develops and evaluates mathematical arguments and proofs.
21st Century Skill(s)
 Information, media and technology skills
 Life and career skills
Essential Question
What are the requirements for two polygons to be similar? How can you use similar
polygons to solve real-life problems?
Objectives
Students will be able to:
 Use similarity theorems to prove two triangles are similar
 Use similar triangles to solve real-life problems
Anticipatory Set
(5 minutes) Show this video to the class to get them ready to prove similar triangles.
http://www.youtube.com/watch?v=iIISJmO0VGw
Teaching: Activities
(5 minutes) Students will get out their theorems sheet and write down theorem 8.2 on
side-side-side similarity theorem and theorem 8.3 on side-angle-side similarity theorem.
(20 minutes) Go through the power point with the class.
(15 minutes) Hand out worksheets labeled 8.5 Group 1, 8.5 Group 2, and 8.5 Group 3.
The students getting 8.5 Group 1 are the above target students who will work by
themselves to find the solution to the problem. The students who get the 8.5 Group 2
worksheet are the on target students who will also work on the problem by themselves.
Then the students who get the 8.5 Group 3 are the below target students who should also
work individually. Once they complete the problem, they can confer with the other
students in their group to see if they got similar results (tiered assignment).
(10 minutes) Each group will pick one student to come to the board and explain to the
rest of the class how they got their answer.
(25 minutes) Students should split up into groups of 3 or 4, they can decide their own
groups. Hand out the Tic-Tac-Toe chart to each group. As a group they will pick one of
the options listed and complete the option as best they can. They should be ready to
perform their choice at the next class (choices and web 2.0 mind tool).
Closure
(5 minutes) Each group must show me what they have done so far for their choice and
ask any questions they might have. Students must write on a post-it note what their
choice was and how far they have gotten in it.
Independent Practice
Finish the Tic-Tac-Toe assignment before the next class.
Assessment
(5 minutes) This will be like a question they will have on a test.
Prove the two triangles are similar. What postulate makes them similar? Write the
similarity statement for these triangles.
Materials
Paper
Pencil
Calculator
Computer
Duration
Total time: 90 minutes
Pre-Assessment
None
Section 8.6
Class: Geometry
Grade Level: 10th grade
Unit: Similarity
Teacher: Ms. Renee Beach
Common Core State Standards (CCSS)
Similarity, Right Triangles, and Trigonometry
 Understand similarity in terms of similarity transformations
o 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.2.)
Iowa Core Curriculum-Subject Area Standard(s)
Geometry
 Understand an apply similarity, with connections to proportion
21st Century Skill(s)
 Learning and Innovation Skills
 Life and Career Skills
Essential Question
What are the requirements for two polygons to be similar? How can you use similar
polygons to solve real-life problems?
Objectives
Students will be able to:
 Use proportionality theorems to calculate segment lengths
 Use proportionality theorems to solve real-life problems
Anticipatory Set
(20 minutes) Students will share or perform their choice from their tic-tac-toe chart.
Teaching: Activities
(5 minutes) Have students get out their theorems sheet and write down theorem 8.4, 8.5,
8.6, and 8.7.
(40 minutes) Split the class into 4 groups. Each group will go to a different part of the
room to solve a problem. One teacher will be in charge of stations 1 and 2, the other
teacher will be in charge of stations 3 and 4. Students should spend no more then 10
minutes at each station.
Station 1: In the diagram AB|| ED, BD=8, DC=4, and AE=12. Use the triangle
proportionality theorem to find the length of EC.
Answer:



.
Stantion 2: Given the diagram, use theorem 8.5 to determine whether MN||GH.
Answer: Begin b finding and simplifying the ratios of the two sides devided by MN.
and
Station 3: In the diagram,
8.6 to find the length of TU.
. Because
, MN is not parallel to GH.
, and PQ=9, QR=15, and ST=11. Use theorem
Answer: Because corresponding angles are congruent the lines are parallel and you can
use Theorem 8.6.

therefore the length of TU is 55/3.
Station 4: In the diagram,
of DC.



. Use the given side lengths to find the length
Answer: Since AD is an angle bisector of CAB, you can apply thoerem 8.7. Let x=DC.
Then BD=14-x.




. Therefore the length of DC is 8.75 units.
(multiple intelligences and co-teaching strategy)
(10 minutes) Display the following figure on the board:

In the diagram KL||MN. Find the values of the variables.
Answer: To find he value of x, you can set up a proportion.


since KL||MN, ∆JKL~∆JMN and
.
.



. Then

Closure
(10 minutes) They will get this time to work on the homework for tomorrow and ask
questions as needed. This will also give me time to talk to them about any homework
they haven’t turned in. Be sure to remind them that they will have a test in the next class.
Students must give two examples of when they would use proportionality theorems in
real-life.
Independent Practice
Problems from the end of the section.
Assessment
(5 minutes) This will be like a question they will have on a test.
Find the value of the variable.
Materials
Paper
Pencil
Calculator
Duration
Total time: 90 minutes
Pre-Assessment
None
Section 8.7
Class: Geometry
Grade Level: 10th grade
Unit: Similarity
Teacher: Ms. Renee Beach
Essential Question
What are the requirements for two polygons to be similar? How can you use similar
polygons to solve real-life problems?
Objectives
Students will be able to:
 Identify dilations
 Use properties of dilations to create a real-life perspective drawing
Teaching: Activities
On this day they would learn about dilation and how it can be used in real life. They
would be able to identify the type of dilation and how much of a reduction or
enlargement took place.
Independent Practice
Problems from the back of the book
Assessment
This will be like a question they will have on a test.
Identify the dilation and find its scale factor when the blue triangle is mapped onto the
red triangle.
Materials
Paper
Pencil
Calculator
Duration
Total time: 90 minutes
Pre-Assessment
None
Test Day
Class: Geometry
Grade Level: 10th grade
Unit: Similarity
Teacher: Ms. Renee Beach
Common Core State Standards (CCSS)
Similarity, Right Triangles, and Trigonometry
 Understand similarity in terms of similarity transformations
o 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.2.)
 Understand similarity in terms of similarity transformations
o Using the properties of similarity transformations to establish the AA
criterion for two triangles to be similar. (G-SRT.3.)
 Prove theorems involving similarity
o Use congruence and similarity criteria for triangles to solve problems and
to prove relationships in geometric figures. (G-SRT.5.)
 Define trigonometric ratios and solve problems involving right triangles
o 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. (G-SRT.6.)
Iowa Core Curriculum-Subject Area Standard(s)
Geometry
 Understands and applies properties and relationships of geometric figures.
Algebra
 Understand and apply proportionality
Geometry and Measurement
 Understand and apply similarity, with connections to proportion
Geometry
 Understands and applies properties and relationships of geometric figures.
Reasoning and Proof
 Develops and evaluates mathematical arguments and proofs.
Geometry
 Understand an apply similarity, with connections to proportion
21st Century Skill(s)
 Life and Career Skills
Essential Question
What are the requirements for two polygons to be similar? How can you use similar
polygons to solve real-life problems?
Objectives
Students will be able to:
 Solve problems related to chapter 8
 Relate the topics of chapter 8 to real life problems
Anticipatory Set
(40 minutes) Hand out the sheet titled Chapter 8 Test Review. Give the students about 25
minutes to work on this by themselves or in small groups. Then go through the answers
and if they don’t understand something then further explain it.
Teaching: Activities
(50 minutes) Now hand out Chapter 8 test to the students. They can use their calculator
on the test but nothing else. If they complete the test early they can work on missing or
late homework for this class or homework for another class.
Closure
The closure is just making sure that all the students turn in their test. If everyone finishes
before class is over, give them a preview of what they will be learning in the next section
which is special right triangles.
Independent Practice
No homework.
Assessment
The assessment is the chapter test.
Materials
Paper
Pencil
Calculator
Duration
Total time: 90 minutes
Pre-Assessment
None