Download Taliesin West - PolygonMath

Kirkman, Anne
ED550
Lesson Plan, Geometry
Model Polygon Area Unit, Ch. 11
Lesson Plan Topic: Finding Areas and Angles Measures of
Polygons
Polygonal Angle and Area Measures
Mathematics Content Objectives:
Understand how to find interior and exterior angle measures of polygons. Determine the areas of any n-gon.
Compare and contrast perimeters and areas of similar figures. Apply functional problem-solving skills to
real-life situations.
Mathematical Process Skills:
Students will draw on prior knowledge of polygons, n-gons, convex polygons, regular polygons, similar
polygons, and trigonometric ratios. They will then use this information to evaluate polygons and determine
what the areas and angle measures are. Applications investigated include stained glass window creation,
landscape architecture hardscape, and hexagonal basaltic columns formed by rapidly cooling lava.
Mass Curriculum Framework: G.CO . 11.1, Congruence, part 3
Instructional Outcomes:
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Restate prior knowledge about geometric identities for sine, cosine, and tangent functions such as
angle measure, side lengths, and ratios of sides through classroom discussion.
Formulate the relationship between area of equilateral triangles and area of regular polygons.
Collaborate with other students in small groups to develop areas of polygons using knowledge of
bases, angle, and leg variables.
Continue with the same critical thinking skills to develop other strategies for finding areas and angle
measures.
Identify the ratios of similar figures.
Briefly introduce geometric technology methodology.
Complete an in-class assignment to create a Venn diagram as a study strategy for assessment review.
Summarize and reflect on what has been learned.
Learning Strategies
Elaborations of prior knowledge; selective attention for listening and performing calculations; note-taking;
cooperating.
Materials
Text, Geometry (2004 edition) by Larson, Boswell, Stiff; graph paper, protractor, rulers, colored pencils,
calculators, computers with geometry software such as Geometers Sketchpad.
Procedures
Introduction
(5 minutes)
Teacher displays pictures and graphics of the Giant Causeway in Ireland. After pointing out that the "rocks"
are actually hexagonal columns are formed from lava that has rapidly cooled. These columns range in height
from two to fifty two feet, but their crosswise area is much smaller. The teacher then introduces topics of
area and angle measures of polygons. Teacher states that they have learned about sine and cosine functions
in a previous lesson and asks if students can name real-life examples of using these functions to find lengths.
What is the difference between sine and cosine functions?
Presentation 1: Finding Interior and Exterior Angle Measures of Polygons, Ch.11.1
(10 minutes)
Teacher uses a chart depicting specific polygons and asks students how their names relate to number of
sides. The teacher explains that the sum of the measure of the interior angles of a polygon also depends on
the number of sides. Recall that the sum of the measure of interior angles of a quadrilateral can be found by
dividing the quadrilateral into two triangles. Teacher then has students create and fill out a chart to
investigate the sum of polygon angle measures as described on p. 661.
From this chart, students should be able to derive the sum of the interior angles of a convex n-gon:
(n-2)180 degrees
Also, they can correlate the measure of each interior angle:
n  2  180
180
At this point, students can construct a pentagon and draw all side segments as lines. Have them shade one
exterior angle at each vertex. Following instructions listed at the top of p. 663, cut out the exterior angles
and arrange them in a circle. From this, they should see:
The sum of the measures of the exterior angles of a convex polygon is 360.
The measure of each exterior angle of a regular n-gon is:
360
n
They can study the examples #1-5 on pp. 662-664 and complete the guided practice on p. 665.
Presentation 2: Finding Area Measures of Regular Polygons, Ch. 11.2, p. 669
(15 minutes)
Recall, the area of any triangle is:
1
bh
2
1
s  s2
4
Discuss the height of a triangle and relate to the "height" of a triangle formed in a polygon with central angle.
This height is called the apothem denoted "a".
Now: the area of an equilateral triangle can be found with: A 
The area of any regular n-gon with side length s, perimeter P, and apothem a is:
1
A  aP
2
Students can study examples #1-4, p. 669-672 and complete the guided practice on p. 672.
Presentation 3: Comparing Perimeter and Areas of Similar Figures, Ch. 11.3, p. 677
(10 minutes)
Recall similar triangles have the same ratio of corresponding side lengths. Now, the areas of similar
a2
polygons is:
where a is the area of polygon 1 and b is the area of polygon 2. A great realb2
life example is noted as the Chicago Board of Trade octagonal trading pit.
Students can study examples #1-3, p. 677, and complete the guided practice on p. 679.
Expansion: Area Relationships in Similar Figures
(10 minutes)
An extension to help develop these concepts would be the Similar Figures activity on p. 676.
Students can work in small groups.
They will draw polygons on graph paper with rulers. Then, they determine the areas of their figure after
dividing it up into rectangles.
Students make a chart, compare and discuss finding with group mates.
Together, they draw conclusions and discuss their conjectures.
Summarize & Reflect
(10 minutes)
As a wrap-up, students will continue working in groups and create a Venn diagram that they can use as a
study guide for the concepts learned. This can be added to as they progress through chapter 11. Finally, the
teacher asks one person from each group to stand up and summarize what was learned.
Images:
Basaltic Columns: The Giant's Causeway, Ireland
Amazingly, you can make columnar joints in your kitchen! Just mix 50:50 corn starch and water and put it
into a coffee cup. Dry the mixture with a bright light above it --- this might take up to a week. When the
mixture is dry, carefully break it apart and you will see that the interior is broken up into small “starch
columns” a few mm wide [picture] [picture] [picture]. In this experiment, water replaces the heat in the lava,
and the columns are 1000X smaller. Why?
We have learned how to control this “kitchen experiment” very precisely so that we can study the process
in detail. We have also studied real lava columns in Washington state, Scotland and Ireland. We can
measure how the columns evolve by using X rays to see the cracks inside the starch [movie]. We have
explained why the size of the columns in lava is 1000X that of starch. The answer lies in a certain ratio of
the speed of the cracks, the size of the columns and the diffusion of the water or heat.
Controlled starch experiments make very regular looking columns, just like those found in real
lava [picture]. The size of the columns depends on the speed that the cracks advance, and the
rate at which the water can move through the starch. The analogous properties of lava are known
from measurements
Amazingly, you can make columnar joints in your kitchen! Just mix 50:50 corn starch and water and put it
into a coffee cup. Dry
Stained Glass Window creation: relates to problem: #39-41, p. 666, ch.11.1
Foucault Pendulum, Houston Museum of Natural Sciences, Houston, Texas.
Stop Sign
Hobby-Eberly Telescope in Fort Davis, TX
Taliesin West, Scottsdale, AZ - Triangular Pool and walkway
Taliesin West
Imagine the opportunity to experience the environment which gave birth to some of the world’s greatest
architectural wonders, walking where the great Frank Lloyd Wright walked, feeling the organic connection between
earth and element that inspired such remarkable creations as the Guggenheim Museum in New York City and
countless private residences throughout the United States, including much of the design work for Fallingwater,
Wright’s residential masterpiece near Pittsburgh, Pennsylvania. For those visiting Scottsdale, a trip to Taliesin West
is just such an opportunity.
Wright’s Winter Home.
Taliesin West was designed by Wright in the 1930s as a winter home away from the cold, harsh climate of his
native Wisconsin. Built by artists and apprentices studying under Wright, Taliesin West serves as home to the
Frank Lloyd Wright School of Architecture and is a National Historic Landmark. Operated today by the Frank Lloyd
Wright Foundation, Taliesin West is a breathtaking example of the Wright form, blending nature with structure and
provides the public with the opportunity to witness first hand how the preeminent American architect created some
of his stunning work.
Things to See.
Taliesin West is a collection of buildings, ranging from the Wright family living quarters to
the facility’s office to Wright’s original outdoor entertainment venue, the Cabaret
Theater. Even the drive approach up to the facility is an architectural wonder that should
not be missed. Some of the highlights include:
The Garden Room – Considered the most stunning of all the structures at Taliesin
West, the Garden Room was built by Wright himself and continually remodeled
throughout his life. The room is large and spacious, open to a spectacular garden on the east and magnificent
Arizona panorama at the south. A fireplace at the far end of the room reflects the warm flow of the sunset and is
made from natural stone found on the property. The opposite end of the room leads to a hidden alcove that was
the private living quarters for the Wright family.
Sun Cottage – The first living area for the Wrights, this early building on the campus was actually a collection of
tiny sleeping alcoves surrounding an open space with a fireplace at one end. After the architect’s death in 1959,
Mrs. Wright continued to remodel the space to include additional sleeping quarters for friends and family.
Office – The first building you approach when making your way up the scenic entrance drive, the Office is typical
of Wright’s desert style, constructed of natural stone and sporting a see-through roof held up by thick beams of
strong native wood and industrial steel. Preserved in much the same manner as it was when Wright greeted
visitors each winter season, the Office is representative of the architectural style that is found throughout the
campus.
Tours of Taliesin West.
There are several guided tours available at Taliesin West, including the dramatic Night Lights on the Desert tour.
As Taliesin West is set high above the valley, sunset provides an entirely new perspective as the stone structures
take on an artistic quality and the shimmering lights from the city below dance dreamily across the pavilions and
patios that surround the buildings.
Specific tours of the different structures, including the Apprentice Court and the Garden Room living quarters are
also available. The most popular tour at Taliesin West is the 90 minute Insights tour, and allows visitors to not only
tour the rooms, bit sit in furniture designed by Wright himself and take in the atmosphere of visiting the home and
its famous residents.
Special Events.
The dramatic setting of Taliesin West makes it the perfect venue for special events and gatherings of up to 300
people. The many patios and courtyards are exceptionally beautiful in the evening hours and several meeting
rooms can accommodate small daytime groups as well. If you are planning a reception or intimate gathering in the
near future, consider Taliesin West for a spectacular backdrop to your special occasion.
Getting to Taliesin West.
Taliesin West is open daily from 8:30 a.m. to 5:30 p.m. It is closed Thanksgiving, Christmas, New Years Day and
Easter Sunday. Located at the intersection of Cactus Road and Frank Lloyd Wright Boulevard in northeast
Scottsdale, Taliesin West is just east of Highway 101 and northwest of Highway 87.
12621 North Frank Lloyd Wright
Scottsdale, AZ 85250
480-860-2700
More Information.
Frank Lloyd Wright School of Architecture
Frank Lloyd Wright Foundation
Chicago Board of Trade trading pit
Fort Jefferson, Key West, Fl
Archimedes
colin percival, 1999, pi
hyperlink to create a venn diagram