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THEY'RE EVERYWHERE!
THEY'RE EVERYWHERE!
Grades 7-12

Envision a world without geometric shapes--no houses, no buildings, no roads, no airplanes, no television, no computers. This lesson is designed to give students an appreciation of the polygons and polyhedrons around them that make their world one of order and strength. After reviewing the definitions and attributes of polygons, students will view the video segments to see how these polygons are found in every building. Polygons make up the repeating structure and support of these buildings. Students will view segments that include an architect who uses geometry to identify her limitations in designing buildings; a structural engineer who uses geometry as supports for the massive structures he designs; and an inventor who uses geometry to generate structures that change from one shape to another. Students will build polygons and polyhedrons, and then construct airplanes out of polygons. Students will then view a "movie" on the invasion of polygons, which will be topic of their final project, "You Can't Get Away From Them." This investigation can extend over two class periods.
The Eddie Files #103: Geometry: Invasion of the Polygons

Students will be able to:
1. review the vocabulary of polygons.
2. build two and three dimensional geometric shapes.
3. build flexible and inflexible shapes.
4. construct a paper airplane to determine how shape can control distance and accuracy.
5. create expanding and contracting shapes.
6. investigate the use of polygons and polyhedrons in architecture and geometric inventions.

Texas Assessment of Academic Skills (TAAS), Exit Level
Math Objectives:
#3: Demonstrate an understanding of geometric properties and relationships.
#11: Determine solution strategies and solve problems.
#12: Express or solve problems using mathematical representation.

NCTM Standards for Grades 9-12
Standard 1: Mathematics as Problem Solving
Standard 3: Mathematics as Reasoning
Standard 7: Geometry from a Synthetic Perspective
Standard 14: Mathematical Structure
Per class:
Per group of 4 students:
Per student:
Day One
When students enter the room, ask them to build a three sided figure from three gumdrops (modeling clay may be substituted for gumdrops) and three toothpicks. The gumdrops will be the vertices of the shapes and the toothpicks will be the sides. Ask students to name the shape they have built. (a triangle) On the overhead or chalkboard, make a list of all the kinds of triangles they know, categorizing by sides and by angles. (scalene, isosceles, equilateral, obtuse, acute, right) Ask students to work in groups of four, with each of the students building one of these shapes with their toothpicks and gumdrops. Tell students that they have just built the simplest polygon. Ask students for a definition of polygon. (a closed shape made of segments)
Ask students what the next polygon with four sides is called. (a quadrilateral) On the overhead, generate a list of all the special quadrilaterals. (parallelogram, rectangle, rhombus, square, trapezoid, kite) Call out the description of one of the quadrilaterals and ask each group to build the quadrilateral described with toothpicks and gumdrops. Have the students name the quadrilateral. For example, tell the students to build the quadrilateral that has the following:
four congruent sides and four congruent angles (a square)
four congruent angles (a rectangle)
only one pair of parallel sides (a trapezoid)
four congruent sides (a rhombus)
opposite sides parallel (a parallelogram)
two pair of adjacent sides congruent, but opposites
not (a kite)
Next have the students generate a five sided polygon (a pentagon), a six sided polygon (a hexagon), and an eight sided polygon (an octagon). Have students note that these are all two dimensional figures (flat).
Now have each group try to solve the polygon puzzle included as part of this lesson. Each side of the triangle marked with a picture, word, or definition must be matched with its corresponding picture, word, or definition. The puzzle will be complete when all the pieces form a large triangle.

Tell the students they will watch a segment of video where two construction workers find polygons in the three dimensional buildings that they are helping to construct. The student's responsibility will be to find polygons in the buildings. Tell students they will be asked to design a building without polygons. Students will also be asked to find polyhedrons in the architect's designs and actual buildings. Ask the students to watch what the worker on the left is doing during this segment.

BEGIN The Eddie Files video after Eddie says, "So I hit the streets, looking for polygons." PAUSE when the male worker on the left stands up and says, "...basically very common in buildings, but there..." when the camera shows the construction site. Ask students to find polygons in the structure. RESUME the video. PAUSE again when the male worker admits, "That is a parallelogram," and the camera again shows the skyscraper being built. Ask students what predominant shape is seen. (a parallelogram or rectangle) Ask students what kinds of buildings could be built without polygons. Could a building made totally of circles and curves be possible? Have students try to draw the shape they visualize. Ask for volunteers to share their drawings. Ask the class for the limitations of these buildings. For example, there would be no corners. How would that affect the walls and floors of the buildings? Would skyscrapers be possible?
RESUME video until the female worker throws a green pepper into the male worker's lunch pail. Ask students what she is doing. (stealing his lunch) PAUSE after they have finished talking, on the frame of the cylindrical appearing building. Ask students if the building is really formed from circles or cylinders. (The building is really made of rectangles hinged together to form the appearance of a cylinder.)
RESUME the video briefly. PAUSE the video on the building with the blue triangular roof with two chimneys. Ask students to find the polygons that form the roof and the entire building. The shapes of the sides are polygons, which are two dimensional. However, the actual building is a three dimensional shape, which is called a polyhedron. A polyhedron is a three dimensional shape made of polygons.
Have students build polyhedrons with their toothpick polygons. Have them pick four polygons they constructed earlier and connect them with more toothpicks. Choose one and ask the students what polygons were used to construct the shape. Now ask each group of four students to build a polyhedron with at least one hexagon, one pentagon, two squares, and two triangles. Note that the more vertices connected, the more sturdy the shape. Have each group write a description of their polyhedron. Have the groups exchange the descriptions. Each group will now try to build another group's polyhedron. Ask the groups to check for accuracy with the original group when they are finished. Ask students if every group built the same polyhedron. (no) Ask students how many possible polyhedrons there are in the world. (too many to count)
Inform the students that they are now going to see an architect who makes her living designing polyhedrons for specific purposes. Tell students they have specific responsibilities during the viewing. They are to remember how many possible choices the architect has to solve the dilemma of how the train station should look; to identify the limitations of her building design; and to determine why she decided on a rectangular prism of glass.
RESUME the video. PAUSE after she says, "...letting them know where the front door was." Ask students what limitations she had for the building design. (only 40 feet wide and 50 feet long-a small space to get 100,000 people through) RESUME the video. PAUSE after she says, "What if the whole building was just one big triangle?" What polygons are used to make this triangular polyhedron tower? (primarily triangles and some rectangles) Ask students if they think this tower fits in with the rest of the neighborhood and why. RESUME the video. PAUSE after, "You can see it from very far away." Ask students why she picked the glass rectangular prism. (same shape as the surrounding buildings, appears to be the same height, glass tower can be seen from very far away) RESUME video. PAUSE on the view of New York City at night.
Have students place a toothpick and gumdrop triangle made with three toothpicks, one gumdrop square made with four toothpicks, and one gumdrop pentagon made with five toothpicks on their desks. Ask students which structure is the sturdiest. (triangle) Ask why the triangle is so sturdy. (cannot bend it inwards) Explain to the students that triangles are the only shapes that exist only in the convex form. Tell students that convex polygons are polygons which have no vertices which would cause a line going through one of the segments of the polygon to contain a point in the interior of the polygon. Any line drawn through the triangle segments would not enter the triangle. Now have students push one vertex of the square inwards. It is possible. The shape is no longer a square. It is a nonconvex quadrilateral. Have students trace along the sides of the quadrilateral. Have them notice that their pencil will go through the interior of the shape. Repeat the process for the pentagon. Again it is possible to turn the pentagon into a nonconvex pentagon. Ask students to write a conclusion on what makes the sturdiest polygon.
Remind students that flexible shapes are not necessarily sturdy and strong. These two attributes are critical when building structures for human habitation and use. The next segment of video introduces students to a structural engineer. Tell the students their responsibility will be to define what a structural engineer does. The students will also build some simple shapes which add strength and rigidity to flexible materials.
RESUME video. PAUSE after he says, "...space frame that is all made out of triangles." Ask students what the triangles do for the ceiling. (make it rigid) Ask what is the main function of a structural engineer in building. (to make buildings sturdy) RESUME video. PAUSE after he says, "...give the material strength to hold the weight of my hand." Have students roll a piece of paper and test the strength as the engineer did. RESUME video. STOP after he says, "This is why it is so important to understand the shape of a building."

Cut a piece of string approximately two meters long. Divide the students into groups of no more than ten. Place a trash can in the center of each group and have each member of the group stand exactly one string length from the trash can. What shape will the students form? (circle) Have each student, one at a time, try to throw a piece of standard notebook paper without any folds or creases into the trash can from where the student is standing. Ask students to describe what happened. (paper hard to control) Have students take the same piece of paper and wad it into a tight ball. Repeat the process with students throwing the paper into the trash can. Ask students to describe what happened. (The more rigid ball was easily thrown into the trash can.) Have students go back to their desks to fold a paper airplane from a piece of notebook paper. First have them fold the paper in half the long way. Take two corners (the unfolded ones) and fold them back to form two right triangles on either side of the airplane. A right trapezoid should be formed.

Now fold one obtuse triangle on each side, bringing B down to segment DF and creasing through A and F to form the longest side of the obtuse triangle.

Now fold one more triangle on each side by folding segment AF onto segment DF.

Have students look at the airplane from the rear. Ask them if they notice the same fold the structural engineer used to show how a triangular fold kept the shape from falling down. Have students go back to their circle. Again one at a time (for safety's sake), have students throw the airplane into the trash can. Students should discuss how much more movement is made possible by building with polygons instead of a wadded up piece of paper. Ask students to note how sturdy their shape is as they throw it. Finally, ask them to pick up their plane and flatten it. Have students note the number of triangles they actually folded. This is an example of a shape that was molded into another form, but could be put back to the original.

Pre-Viewing Activity Day 2
Briefly review the activities of the previous day. Then have each student make a toothpick and gumdrop hexagon and push in one vertex to make it nonconvex. Without breaking it, have students try to collapse the entire hexagon into as small a shape as possible. Have students try to fold it back out into a hexagon. Tell students that by having a flexible shape, they are able to "morph" one shape into another and back again.

Focus for Viewing
Tell students that they are about to view a video segment about a geometric inventor. This inventor takes what the class did on a small scale and does it on a large one. The students' responsibility is to find the polygons that morph into other polygons to form beautiful polyhedrons. Students should also determine what the word "morph" means. Students should listen to find the one mistake he makes when describing certain polygons.

Viewing Activities
Begin The Eddie Files video where it was stopped on the previous day. Allow the students to watch the segment uninterrupted. Pause the video after he says, "That's the first step of inventing.'' Ask the students if they would like to see the inventor segment again. If so, rewind to the segment where his first invention is shown. Resume the video and ask students to tell where he made the mistake. (He calls triangles and pentagons three sided shapes.) Pause the video at the end of the inventor sequence when he says, "That's the first step of inventing." To prepare for the viewing segment summarizing the lesson, fast forward to the black and white section of the video which shows the "4" on the screen and stop the video here to first conduct the following activity.

Post-Viewing Activity
Tell students that they are going to invent a continuous band of paper without a beginning or an end that will go around the room. The students will only be allowed one sheet of notebook paper and scissors for the actual activity. Tell students that they will "morph" this piece of paper into this band, and then will "morph" it back to a piece of notebook paper. Have students work in groups of four. Give them some time to experiment with cutting the paper. If no one comes up with a solution, ask the students to follow these directions. Have the students fold their notebook paper in half to form a rectangle 8 1/2 by 5 1/2 inches. Have the students cut through both layers of paper according to the following diagram. Cut along the dotted lines.

Now have students cut along the crease, being certain to leave the outside segments on both ends uncut.

Now have students unfold their shape. Ask them how they would have to cut to make the paper band go around the room, or at least around a large desk.
Remind students that by now they should have a better appreciation of how useful polygons and polyhedrons are in our world. Students will now watch a spoof of a movie called "The Invasion of the Polygons." Student responsibility is to decide how many quadrilaterals the man under investigation at the end of the spoof could have seen. Students will also decide what is meant by, "You can run but you cannot hide!" as it relates
to polygons.
Begin the video at the "4." Have students watch the entire segment. Stop after the words, "Coming to a theater near you." Ask students what quadrilaterals the actor could have seen. (rectangle, parallelogram, rhombus, trapezoid, kite, square) Ask students the purpose of this segment. (It illustrates that polygons are everywhere. Unless we are out in nature, we are surrounded by them.)
Within the community, take students to a construction site to observe and document polygons and polyhedrons in building structures. Have students interview an architect or structural engineer.
Have students make a collage of polygons and polyhedrons they find in their house, car, church, grocery store, doctor's office, shopping mall, or neighborhood. Students will take photographs or sketch pictures of the polygons they find. Students will document the polygons and polyhedrons they display in the pictures. Students will then take a picture of a "polygonless" situation. The title for this activity will be "You Can't Get Away From Them."
Have students use hinges and other materials to design and build polygon inventions that "morph".
Have students visit an art gallery to see how artists use geometric shapes in their creations.
Literature: Have students read Flatland, A Romance in Many Dimensions by Edwin Abbott.
Art: Have students study the tessellation work by M.C. Escher to discover how he used polygons as the basis of his patterns.
Architecture: Have students study the ancient buildings of the Greeks, Mayans, Moors, and Romans to see the use of polygons in their structure. Have students study medieval architecture in Europe versus the architecture of the Renaissance to note structural differences.
Science: Have students investigate how scientists use polygons as the basis of a chemical substance, such as a hexagon for a Benzene ring.
Careers/Industry: Have students investigate the use of polygons and polyhedrons in the planning of automobiles and airplanes. Have students investigate the different ways to cut precious stones in
geometric cuts.

1995-1996 National Teacher Training Institute / Austin

Master Teacher: Linda Shaub

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