Download Proportions and Similarity

Geometry Unit Plan
Mr. Ives
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Rocinante High School
Key Concepts:
Proportions
Proportions and
Similar Figures
Solving Proportions, Solving Similar
Figures, Fractals, Pythagorean
Theorem, and Geometric Mean
An equation in the
form of 𝑎/b = c/d
stating that two ratios
are equal.
Similarity
Figures that have the
same shape, but not
necessarily the same
size.
Fractals
Fractals are figures
that are self-similar,
with infinite detail.
Geometric Mean
The measure of an
altitude in a right
triangle is the
geometric mean of the
segments of the
hypotenuse.
The Pythagorean Theorem
The square of the two
legs of a right triangle
is equal to the square
of the hypotenuse.
+ Overview:
This unit is an
exploration of the rules
governing figures that
have the same shape
but not the same size.
It includes both
Fractals, nature's most
beautiful patterns, and
the most important and
famous theorem in
Geometry, the
Pythagorean Theorem.
Goals and Objectives:
After this Proportion Unit is completed the students will be able to:
1. Define Proportionality and Similarity.
2. Calculate missing parts of Proportional and Similar Figures, including
constructed response proportion problems.
3. Understand the use of Fractals in nature and technology.
4. Apply the Pythagorean Theorem in real world situations.
5. Find the Geometric Mean of two figures.
more on
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Proportions and Similar Figures Unit
Mr. Ives
Mnemonic:
"This is to This as That is
to That, Cross Multiply
and Divide!"
Proportions have the
same ratio when using
the Mean / Extremes
Formula.
Common Core Standards Addressed:
Proportions and Similarity: Understand similarity in terms of similarity transformations and
prove theorems involving similarity.
CCSS.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.
CCSS.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.
CCSS.MATH.CONTENT.HSG.SRT.A.3
Use the properties of similarity transformations to establish
the AA criterion for two triangles to be similar.
Prove theorems involving similarity
CCSS.MATH.CONTENT.HSG.SRT.A.3
Use the properties of similarity transformations to establish
the AA criterion for two triangles to be similar.
Prove theorems involving similarity
Importance:
Similar figures are used to represent many real-world situations and involving a scale factor
for the corresponding parts. The word, "model" itself is often taken to mean a scaled similar
version of a larger object. Some examples include photography that uses similar triangles to
find focal lengths, map making which uses proportions, and, of course, the Pythagorean
theorem which is used in the construction of every modern building, skyscraper, and bridge.
New to the math world, are fractals, which are the blueprint for literally everything in nature
from tree growth, nervous systems and circulatory systems, to weather patterns and crystal
growth.
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Proportions and Similar Figures Unit
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Mr. Ives
Lesson Plan Overview
Two Week Timeline
Week 1 Day 1: Introduction to Proportions
Focus Activity: Predict: How would you go about measuring something too tall or difficult to climb?
The students will write the focus activity in their data folder on the top of the page of Cornell Notes they will take for the day. At
the bottom of the notes will be the reflection on learning.
Teaching Strategies and Activities:
Slideshow: Proportions and their applications for use in the real world including Da Vinci's drawings of the human body, triangle
in construction and roofing, and model airplanes. I'm including some amazing models that have recently been made at a
nanoscale.
Essential Questions:
"What is a proportion?"
"What kinds of situations are modeled by proportions?"
"How can I recognize a proportion problem?" (Problem Recognition Skills)
"Is there a repeated pattern or mathematical rule for quickly solving any proportion?" (Common Core Math skill 6: Look for and express
regularity in repeated calculations, recognize short cuts.)
(continued)
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Proportions and Similar Figures Unit
Mr. Ives
Presentation: Solving proportions, estimating proportions, scale factor, and recognizing
proportional problems by number of data points.
Cornell Notes: The mnemonic, "This is to This as That is to That, Cross Multiply and
Divide," needs to be introduced in this first lesson. I have the students use their bodies
kinesthetically to re-enforce this with multiple intelligences. This is a better name for the
Means / Extremes Formula, since it more easily recognizable and easily memorized,
although I still use the correct vocabulary so that the students will know the technical
Unit Vocabulary:
Midsegment
Proportion
Ratios
Means /
extremes
name.
Agile Mind: Dilations: Proving Triangles Similar: The software uses dilations to
introduce similar figures. This is a good way to do that since dilations are, by definition
similar and it links to reflections and rotations that were learned in an earlier unit.
Guided and Independent Practice: The students will definitely need to remediate the
skills included in this lesson. It is important to model a variety of problems using proportions,
from recipes, to distance, time and rate, to money problems, to percent problems since there are
Crossmultiplication
Parallel lines
Corresponding
angles
MANY kinds of proportional relationships that can be tested in standardized testing. Other
Week 1 Day 2:
resources included will be practice problems in the basic skill.
Measures of Central Tendency
Geometric mean
Applications
Right triangle
Reflection on Learning: The students reflect in a one or two sentence paragraph on
Altitude
what, if anything, they learned today. They may reflect on any learning, even learning of Focus Activity: Name one thing you
behavior, real world applications, just the basic facts of the lesson, or even a personal
know about gambling!
reflection of their own journey.
The students will write the focus activity
Hypotenuse
Similar
I like to use the classic "Tree and Shadow" problem. The problem, which of Cornell
Notes they will take for the
triangles
in their data folder on the top of the page
Best Practices:
is time-honored, uses the shadow of a person or another object and the given height of
day. At the bottom of the notes will be
that object with the shadow of a tree. The students calculate the height of the tree using
the reflection on learning. The focus
similar triangles and proportionality.
Special Right
Triangles
activities
at the beginning of the week
lead up to the Probability Simulation on
Wednesday.
Teaching Strategies and Activities:
Practice and Vocabulary Activity:
Probability Concentration: Students
pick pairs of hidden cards for a match.
When a match is found the student must
correctly name the given probability or
(continued)
correct definition to get a point. If
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successful, the student gets to pick again.
Turns rotate after a student misses.
Proportions and Similar Figures Unit
"Similar
objects are the
same shape,
but different
sizes!"
Mr. Ives
Week 1 Day 2: Similar Triangles Field Experiment
Focus Activity: What do you know about Christopher Columbus? Since Christopher
Columbus used a sextant and a compass to cross the Atlantic, his story makes a nice segue into
the use of a sextant for remotely measuring distances. During the story, cross connection to
Social Studies is easy by talking about why Columbus wanted to cross in the first place.
Similar Triangles Field Experiment / Simulation:
Teaching Strategies and Activities:
Today I will run a real world field experiment to remotely measure the height of the school's flagpole. (This will also work
with any tall, inaccessible object on school grounds.)
Materials needed:
Ruler
Pencil
One 8.5 by 11 inch standard piece of copy paper
One 10 inch long piece of string
One penny
Two pieces of masking tape.
One 30 ft. or greater measuring tape. (Teacher only)
One piece of chalk (Teacher only)
1. Have the student mark the bottom edge of the paper in one half inch increments, numbering from right to left instead of the
traditional left to right. (Watch this closely, since it is hard to break a habit pattern!)
2. From the top right corner of the paper, tape the end of the string to the corner.
3. Now tape the penny to the other end of the string.
4. This makes a homemade sextant that can be used for sighting objects. Make sure that any students (who have not shot a
gun before) know how to aim down the top edge of the sextant.
5. Assign each student a partner who will read off the measurement from the bottom edge of the paper.
6. Given weather conditions, check for needed glasses, sunglasses, and coats.
7. Out at the flagpole, designate a line exactly 30 feet away from the bottom of the flagpole by using a measuring tape and a
piece of chalk to mark the spot.
8. Ask the students to line up on the line all the same distance away from the flagpole. They will most likely naturally make a
straight line, so now is a good time to talk about error and reasons why they might all get differing answers. It will be obvious
to a few that they are further away than the students closer to the mark. That should lead to a discussion of the fact that they
all should be at a radius from the flagpole to make sure they are the same distance away. Other error indicators are the height
of the student and whether they are higher or lower elevations.
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Proportions and Similar Figures Unit
Mr. Ives
Now have one of the student pairs sight on the top of the flagpole, while the other reads off their number on the scale. Then
have the other one of the pair do this. Once everyone has a number, return to the classroom.
9. Back in the classroom, the data can be used to set up a similar triangles problem. I like to have the students estimate the
height before beginning actual calculation.
Leading questions include:
"In what situations would you need to measure something from a
remote distance?"
"How could you use proportions in your own business?"
"Surveyors use proportions, what are they measuring and why?"
Essential Questions:
"What is a sextant?"
"How can I remotely measure something that is too high to
climb?"
"How do modern surveying tools work?"
"Why is it necessary to survey land before building?"
Cornell Notes: Definitions of Proportions and Similarity will be
included, as well as examples of both.
Guided and Independent Practice: A little independent
practice is always good idea. The students usually pick
these concepts quickly and easily so time is not an issue.
Other resources included will be practice problems in the
basic skill.
Reflection on Learning: The students reflect in a one or
two sentence paragraph on what, if anything, they
learned today. They may reflect on any learning, even
learning of behavior, real world applications, just the
basic facts of the lesson, or even a personal reflection of
their own journey.
Best Practices: Even though the idea of error analysis was introduced outside, it really has impact when the students do the
actual calculations in the classroom and find that they got slightly differing answers from each other. Now is a good time to ask
them why they think their answer was different from their partner's answer or another pair's answer.
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Probability and Statistics Unit
Resources/Materials:
Agile Mind Software and Resources
Technology / Science / Mathematics Slideshows
Mr. Ives
Proportions can model
any object that is
larger or smaller! It
can make cities small
or atoms big!
Education YouTube videos
Glencoe Algebra II textbook and resources
Manipulatives and Puzzles
Laptop Computers
Pencil and Paper
Data Folder and Formula Sheet
Proportions Manipulatives: string, pennies, chalk, tape, etc.
Week 1 Day 3
Similar Triangles Applications Day
Focus Activity: Concentration!
Differentiated Learning: Visual / Spatial: A fun activity to re-enforce definitions, triangle classification, angle pairs,
conversion units, and symbols such as Pi, is to make out a set of thirty six 3X5 cardboard cards into eighteen pairs with magnets
attached to both sides. I include a pair of wild cards that will match anything. Arrange these cards in a random order with the
blank side facing out and have the students select pairs one at a time. If the student matches a pair, they must correctly identify
the given information to win the pair. If they win, they get to go again. I repeat this until all pairs are gone from the board.
Each student gets one bonus point for every pair they collect!
Teaching Strategies: Constructed Response Problems.
This day is designed to make sure that students can handle word problems with similar figures. Again, problem recognition is
very important. Students should look for problems that have three pieces of information and are missing the third one.
Essential Questions:
"How do similar triangles model situations in the real world?"
"Is there a repeated pattern that can make solving similar triangle problems easier?"
"How do I recognize a similar triangle problem?"
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Proportions and Similar Figures Unit
Probability can be
used to predict fire
danger!
Mr. Ives
Assessment: Formative:
I usually work through the class one-on-one and see if I can head off problems involving
similar triangles. I also look for difficulty in reading, understanding, and setting up the
constructed response questions. I have a mnemonic device for setting up constructed
response problems:
A.
Analyze. What kind of problem is it? Do you have a formula? What do you know
about these problems?
D.
Draw a Picture. Try to draw a picture, a graph, a table, or a graphic organizer of
this problem.
D.
Design the Problem. Set the problem up or fill in the formula.
S.
Solve. Solve the problem or evaluate the formula.
Assessment: Summative
The students will evaluate their work on the basis of the rubric in their data folders.
Reflection on Learning: The students reflect in a one or two sentence paragraph on what, if
anything, they learned today. They may reflect on any learning, even learning of behavior,
real world applications, just the basic facts of the lesson, or even a personal reflection of their
own journey.
Week 1 Day 4:
Fractals: Writing in Mathematics
On this day I will introduce the mathematics of Fractals, the most beautiful patterns in
nature and, until the advent of computers, virtually unknown and inaccessible. Fractal math
can be seen in the patterns of the body's nervous and circulatory systems. It is in the clouds
and the rivers. Modern programmers use fractals to generate complex computer graphics.
The lead off is a fractal video made by a student, followed by a Mandelbrot dive to 10232
magnification. The students can see that the pattern is self-similar and the iterations are
infinite.
I like to include a slideshow of videogame in-screen shots from such games as Skyrim and
Oblivion. They clearly show the amazing and life-like detail that fractals can generate
without prior mapping.
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Proportions and Similar Figures Unit
Mr. Ives
Best Practices:
This will be a writing in mathematics lesson, so the students will email me reports. I use the
school email since it will automatically tag the student's name to the assignment.
The email is to be written in essay form, with an introduction, a body, and a conclusion. The
essay must have at least three pictures of fractals and must answer the following essential
questions:
1. What is a fractal?
2. When and how were they discovered?
3. Where are fractals found in nature and why are they important to living creatures?
4. How are fractals used in technology and computer programming?
Formative Assessment Criteria for Success:
(What does success on this lesson’s outcomes look like?)
The student will write a report containing an introduction, a body, and a conclusion on the
discovery, uses, and importance of fractals in technology and nature. This essay will include at
least three pictures and will have correct spelling and punctuation as well as vocabulary.
The student will complete the assignment with a rubric of 7 or higher.
All notes and diagrams will be complete in their data folder.
Week 1 Day 5:
Kinesthetic Activity: Math Battle Chess!
At the end of a week filled with new concepts, it's good to take a break and do some kinesthetic
activity. For this unit, the students will stage a simplified chess game on a giant room-sized
chessboard using their bodies as pieces. I usually mark off a five by nine space grid with
masking ape on the floor. Since making the entire board would use too much tape, just mark
the actual places the students will stand. Each mark should be two feet from any other mark in
any direction.
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Proportions and Similar Figures Unit
Mr. Ives
Rules:
1. Pawns may move only straight ahead one space at a time but
can only battle and capture diagonally. In a battle, the first player
to correctly identify the flash card answer, vocabulary word, or
symbol is the winner. There is no penalty for multiple incorrect
guesses
2. Kings / Queens (Depending on the student who is currently
in that capacity) can move one space in any direction and can
capture in any direction.
3. Ninth ranked pawn becomes a king or queen.
4. First team to capture the king / queen wins.
Essential Questions:
1. How do pawns move and capture?
2. How do kings / queens move and capture?
3. How do I win the game?
Best practice:
The teacher should take into account the skill levels of the two battlers and make sure to choose a flash card that is possible for
both players to solve.
Assessment: Formative:
The students self-evaluate during the activity by forming multiple guesses. If
the students are both having trouble, I will give hints or say "higher," or
"lower."
Assessment: Summative
The winning team is the team with the most wins at the end of the period.
Personal Reflection on Lesson:
I thought students might not like the face-to-face social interaction of a game
like this, but they absolutely love it. It is often remarked as the most fun game
they have ever played. This is a much better way to review dry conversion facts
and definitions than a written quiz.
Week 2 Day 1:
"The Battle of Pythagoras":
It's time now for a drama and reading-based lesson. To introduce this concept I use a simulated "battle" fought around a large
triangle on the floor between the evil "Geometricians," and the good "Pythagoreans." Two student volunteers play the parts of
Pythagoras and his Commander in the army as they try to hold off the attack of the Geometricians at the north and south
passes. After a devastating battles near the "Aay" river and the "Bee' Mountains, Pythagoras uses his newly discovered
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Proportions and Similar Figures Unit
Theorem to compute the distance across the "Cee" Desert.
Guided / Independent Practice:
The students will now solve Pythagorean problems independently. Included in this
lesson is a set of scrambled sentences using all the vocabulary words in the unit.
The students will unscramble the sentences by rewriting them with the correct
italicized word. For example:
"The robbers got away with a theorem amount of money."
Mr. Ives
Modifications:
Extended time for completion of
assignments or tests.
Additional time for reading
assignments.
Time for repeated drill and review.
"The longest side of a right triangle is called the tremendous."
Small groups.
Best Practice - Flexibility
Data folders with study sheets,
vocabulary sheets, and other
supplemental aids.
It is good to give the students both kinds of Pythagorean problems: the problems
involving finding the hypotenuse given the two legs and the problems involving
using algebra to find a missing leg.
Reflection on Learning: The students reflect in a one or two sentence paragraph on
what they learned today.
Visual demonstrations that include
the presentation of material in
small steps.
Multisensory materials and
activities (written, spoken,
demonstration, kinesthetic.)
Mnemonic devices and aids.
Word processor with spell checker
and calculator.
Assisted note-taking and notetaking guidelines.
Credit for participation, effort and
attendance as well as completed
work.
One-to-one teacher contact daily.
Best Practices: I like to dress up in a Roman toga for a little extra fun and to set up
the atmosphere!
Assistance with organization and
planning as well as organized
classroom procedures and
materials.
Peer / pair tutoring and
cooperative learning.
Emphasis on success.
Seating to reduce distractions.
Clearly defined limits and
classroom procedures.
Cooling off periods and concrete,
positive re-enforcement.
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Proportions and Similar Figures Unit
Mr. Ives
Week 2 Day 2:
Geometric Mean
Finding Geometric Mean is very similar to solving other proportions and the students
will be well versed in that process by now. The only difference is, of course, that solving
radical equations is involved in many of the problems. I take a little time to teach a minilesson on radicals and square roots, with emphasis on rational and irrational answers and
approaches to writing them and rounding them. Our class standard is the one
hundredths place value since that is often the standard in college.
Personal Reflection:
I want to really hit this hard, since it is Common Core and easily lends itself to
constructed response.
Since it involves exponents, I will have to teach solving radical equations to the class. Most of the students will not have had
Algebra II and will not be strong in this skill, even though they should have had it in Algebra I.
I also will continue my drive for problem
recognition and tie this lesson to the other lesson
on proportions by stating that the students
should watch for problems with three given
information points and one missing one.
Activities/Tasks: (What learning experiences
will students engage in?)
Focus activity: journaling and goal setting.
Video of appropriate science and technology
(time permitting)
Presentation of geometric mean problems.
Guided practice of geometric mean problems.
Independent practice of geometric mean
problems.
Formative Assessment Criteria for Success: (What does success on this lesson’s outcomes look like?)
The students will correctly solve geometric mean problems using the means – extremes formula and solving radical equations.
The student will complete the assignment with a rubric of 7 or higher.
All notes and diagrams will be complete in their data folder.
Week 2 Day 3
Reflection
on Learning:
The students reflect in a one or two sentence paragraph on
Formal Summative
Assessment:
what
they learned
A twenty
questiontoday.
test covering the entire unit with at least two
vocabulary questions and two constructed response questions.
My thanks to Charles Shultz and the Charles Shultz Museum for
permission to use the "Charlie Brown" cartoon.
© 2014 Jonathan Ives
Rocinante High School
All Rights Reserved
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