Through the use of video and a hands-on activity students will observe how certain geometric shapes can be used to make some structures rigid and others flexible. The activity involves construction of a 30-60 right triangle, folding while following directions, and identifying polygons.
ITV Series
The Eddie Files: Invasion of the Polygons (Fase Productions)
Overhead materials:
(each group)
(each student)
Vocabulary
Hold up a commercially designed transformer toy and demonstrate the transformation several times. Discuss what geometric shapes appear to facilitate the transformation. (parallelograms). Demonstrate the flexible cube. Discuss what could be done to make the cube rigid. (Insert one diagonal bar to make triangles.)
Then hold a piece of paper and demonstrate that it has no support strength. Discuss (without conclusions) what could be done to the paper in order to make it strong enough to support the weight of your hand. (rolling, folding)
Finally, place the paper on a table and discuss (without conclusions) what could be done in order to make the paper pop into the air (folding).
The focus for viewing is a specific responsibility or task(s) students are responsible for during or after watching the video to focus and engage students' viewing attention. Tell the students that they will be watching a video of a middle grades mathematics class and a subsequent homework assignment involving the use of polygons in the work place. Tell the students they will be expected to identify polygons and how they are used but more importantly, they should look for specific examples where a certain shape and/or design for connection causes a structure to be either flexible or rigid.
Begin the video with Miss Tolliver holding a structure. and pause when she says, "What name can we give to this structure?" Ask the students the same question. Ask the students to explain the difference between a polygon and a polyhedron. Resume the video.
Pause when she flattens the figure and says, "polygon...polyhedron." Ask the students whether the flattened figure is what they would call a polygon and test it against the definition. (A polygon is a plane, closed figure with straight line segments for sides.) Develop the idea of concave vs. convex polygons. Then ask the students to define a polyhedron. (A polyhedron is a closed structure with polygons for sides.) Discuss whether a polyhedron could be concave. Ask the students if the structure was flexible or rigid. Discuss what caused the flexibility. Was it shape or construction? Introduce the next segment by telling the students they will follow a student as he completes a homework assignment to find out where he sees polygons and polyhedra and what they are used for.
Fast forward the video and resume playing the the tape at the New York buildings (after the comedy skit), pause when the architect, Frances Halsband, says, "letting them know where the front door was." Discuss the problems involved in designing an entrance with dimensions of 40x50 feet through which 1000,000 people are expected to pass. Ask what they would consider for solutions to the problems. Discuss what special problems occur when you are converting an existing building rather than starting from the bottom up. Resume the video.
Pause the tape when there is a night scene. Discuss how she proceeded to solve the problem. Ask: What basic shapes were used? Why did she use so much glass? Would the use of so much glass affect the shapes to be used? Resume the video.
Pause when Eddie says, "Israel Seinuk is a structural engineer." Ask what shapes he said were used in the space frame and why they were used.
Resume the video and pause after he shows the triangle and says, "See, now it doesn't fall anymore." Recap on the previewing activity involving support strength of a piece of paper and the suggestions the students had offered. Then tell the students about how architecture changed when they wanted to build skyscrapers. We are used to seeing the rectangles and triangles which are the building blocks of today's tall structures but years ago such buildings were difficult to erect because they had to construct extremely thick walls to hold up the roof. The network of beams and girders which is the first stage in construction today is called "skeleton construction." Resume the video.
Pause when the inventor, Chuck Hoberman, jumps off the side. Briefly discuss what structures we have seen so far. Ask what purpose such figures might have and discuss whether an invention has to have a purpose in order to exist. Resume the video and pause when he says, "The folds themselves make it strong." Ask what (were) the dominant shapes (triangles). Resume the video and then stop when he says, "That's the first step in inventing." After a brief discussion about the beauty of design in the structures, ask what shapes made them flexible and/or rigid. Wrap-up with an overview connecting the pre-viewing activities with scenes from the video. Also, lead the students to the conclusion that parallelograms enhance flexibility and triangles make a figure rigid.
Construct a polyPOPagon: Design by Phil Winter, TIES magazine, November-December, 1990.
The basic element for this activity is a rectangle cut out of oak tag with dimensions such that a diagonal would divide the rectangle into two 30-60 degree right triangles. This part could be adapted to suit levels of difficulty. EASY: give each student a rectangle with the right dimensions and have them draw the diagonal and measure angles to discover that they are 30-60 degree right triangles.
AVERAGE: Give each student an 8x11 piece of oak tag and challenge them to cut out the required rectangle. This could involve much brainstorming as to the variety of possible methods.
DIFFICULT: Give each group of three students one piece of oak tag and challenge them to cut three different sized, but similar, rectangles with the required dimensions. They should discuss procedures only within the group.
Once every student has an acceptable rectangle to work with, follow the step by step directions. Using the overhead projector, you should work along with the students to create a demonstration model. With each line segment drawn, stop to identify any polygons which develop in the pattern. Expect some disruption of order when the students test the "popability" of their design. When the class returns to order, discuss what allowed the figure to collapse (the parallelograms), what caused the popability (the folding section and the elastic) and what, if anything, the requirement of a 30-60 right triangle had to do with it (the hexagon was a rigid figure).
PolyPOPagon re-visited: The class activity involved a polyPOPagon with the hexagon as the basic polygon. Select three different polygons and try to design a similar structure for each. Describe in detail why or why not the design was possible.
Analyze construction of wooden children's safety gates, flexible wooden wall-racks, and industrial scissors-jacks. Identify ways in which the use of polygons to create flexibility is similar. Unfortunately, the flexibility element also causes accidents to happen. Design a simple lock device for each of these which would allow for a variety of "openings" without collapsing.
Research the construction of keystone stone arches. Find at least one in the region and build a model of it using real stones or clay-formed stones. The entire model should be no larger than the top of your classroom desk.
Research and write a report on structures which have failed because of geometric design. The report should be at least three pages long (typewritten), refer to at least three failed structures and provide details about the geometric design causing the failure.
Design pop-up simple shelters and furniture which could be stored flat for convenience and portability but which could be used by the Red Cross in catastrophic situations or to shelter the homeless.
group work
Math and Art: Design a pop-open birthday card. Then, prepare a cost effective plan for packaging and marketing two thousand units.
Math and Art: Design a pop-up calendar of activities for a local organization (Rotary, Library Association, Football Boosters, etc.). Then contact the group to try to publish the calendar as a joint effort.
Math and Technology: Design an original but simple transformer type toy or design as seen in the video.