Help students to broaden their outlook on geometry in their everyday environment. Explore the properties of parallel lines and parallelograms. Discover ways to prove the existence of parallel lines and discuss the concept of parallel lines. Tie in the concept of parallel lines with science, history, and business.
ITV series
Landscape of Geometry: Lines That Don't Cross (TV Ontario)
Students will be able to define and use the following in practical applications:
(per group):
Design a worksheet tha has pictures or designs and ask the class what they see in common with the different pictures( they are all examples of parallel design and the knowledge of parallel lines has aided in the construction and invention of these everyday objects.. Put responses on board or an easel board. (Sample responses: all hold something; all hang; all are household items; all stack*; save space *; are organized neatly*; all have right angles to something.* The asterisked responses allow the teacher to point If necessary, discuss the concept of parallellism.
Divide the class into groups of four with individually assigned responsibilities. One member picks up the following materials: 6 pieces of patty paper, a pencil, and a ruler. Assign one member to draw, one to compare, one to record results. Have the group draw two lines using the ruler. Draw a third line that intersects the other two (the transversal). Trace this exact figure on a second piece of patty paper. Label the angles from 1 to 8 on both sheets. Have the third member of the group twist and turn one sheet over the other to compare the measure of the angles. Record the results. (Use a protractor if so desired.) Next take two other sheets of patty paper and draw parallel lines using both edges of the ruler. Draw a third line (not perpendicular) intersecting the parallel lines. Label all angles from 1 to 8. Again have the group trace the exact diagram on a second piece of patty paper and label the angles exactly the same. Proceed as before and have the group record the relationships they discover about the eight angles. (As an option they can use a protractor.) At this point, refer to worksheet # 2 to complete the activity.
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 their homework will include a journal entry discussing at least four uses of parallel lines or parallelograms that were discussed in the video, paying strict attention to uses in history, science and business. Include examples in your discussion.
Inform them that they will have a project due on the use of the Rose Compass so they should pay attention to this part of the video.
Begin the video and pause the tape when David (the narrator) is riding on a railroad car and crashes. He says, "You should be able to rely on things that stay the same distance apart." Ask what the definition of parallel lines is. Resume the video.
Pause when David says, "What does "same distance apart" mean if you look at them by eye?" Ask, "What does same distance apart mean?" Use the overhead and draw several transversals between two parallel lines. Resume the video.
Pause when David takes out a book and the screen begins to show a line and a point. Tell the students this is very important for Euclidian Geometry of a flat surface. (Parallel line postulate) Tell them to copy down the definition.
Stop and rewind the tape until they get this definition. (You could, at this time, ask what other possibility could exist? Explain about geometry of the globe and many lines that are parallel going through one point. If you have a globe, demonstrate and explain that the shortest distance between two points is the distance of a great circle. Use string to measure the distance from San Francisco to Japan flying over the Pacific and going over the Arctic Circle.) Resume the video.
Pause when the sun shows and rays of the sun come out side by side. Ask about parallel beams of light. They will discuss flashlight, headlights of a car and maybe lasers. (You could bring in a newspaper or magazine article (i.e. Dec. 1996 Readers Digest article on "Collision of Asteroids and Earth" by Dana Desonie.) Resume the video and pause when David walks with a sign post and says, "East pole not remembered since Winnie the Pooh." Ask, "What is the measure of the angle between the north and east pole?" Resume the video.
Pause when David says, "North line East line... these are corresponding angles." Ask the class what other angles could be considered corresponding angles? What letter does it form? Resume the video.
Pause when David appears on a ship. Tell the class to pay strict attention as they will have to use this information to complete their treasure map and compass rose. Resume the video and pause when the scene changes from the ship. Stop and rewind. Play the ship sequence again in order to emphasize the image of the sails for future discussion. Resume the video.
Pause when David says, "Great lines are parallel but I'm still not going anywhere. Back to the book." Ask what he should do. See if the class can explain how the tracks should come together. Hopefully they'll see that you need to measure the distance between lines twice to see that the distance stays the same; thus you have a parallelogram. Resume the video.
Pause after David discusses warp and weft of cloth and says, "They started to build in panels. Show your display of cloth. Demonstrate stretch on the bias. Ask what "panels" could mean. Ask them to pay attention to the two different sails and come up with the difference between them. Resume the video.
Freeze the frame when it shows the ship America. Point out how the sails are attached totally on top and bottom of the panel. Resume the video.
Pause when David appears in a woman's dress and talks about the design of clothes. Explain that this is supposed to be Queen Victoria and then ask if anyone plans to go into the fashion industry. Explain about seams in design (bring in a pattern and explain the arrows). Resume the video.
Pause when you see the railroad and David says, "Any pair of corresponding angles work." Ask for an explanation. Resume the video and pause when David says, "Geometers assume that on a flat plane the lines will never meet. But they can't prove it." Ask, "What do we call a statement without proof?" (postulate or axiom) Stop the video.
Write the journal entry discussing at least four ways that parallel lines are used in real-life situations noting especially connections to history, science and business.
Make a treasure map using a rose compass and parallel rule.
Form groups with three or four students. Have them build a parallel rule. (Cut from overhead transparencies parallel strips one inch wide by eleven-and-a-half long. Punch holes in the ends and secure four together with paper fasteners forming a collapsible rhombus; or instead of overhead transparencies, pre-cut plexiglass with holes punched in the ends could be used to form the collapsible rhombus. Hand out worksheet #3 to practice using the parallel rule. Next, attach the "rose compass" to the lower right hand of a piece of drawing paper. Use scotch tape to fasten the rose with the origin at the corner of the paper and the north lined up with the right edge and the west lined up with the bottom. Have the group draw a treasure map. Put in trees, rocks and waterways or the school. (I also used small butterfly stickers and had the students draw a design around them. I had the students number these identical objects and pick one of them to be their treasure.) Now using your parallel rule give the following precise directions for finding your treasure: may not use the numbered objects as reference points; may say go 6 inches at 120 degrees; use at least 3 directional changes; must tell starting point. Put the location in a sealed envelope and attach it to your treasure map. Have groups exchange maps and see if they can follow another groups' directions to find the hidden treasure.
Give a prize for the best map and directions.
Arrange a visit to see the Tall Ships when they are in the region.
Have a sailor visit the classroom and demonstrate how the parallel rule is used in navigation.
Have a seamstress or tailor demonstrate how a pattern is designed relative to the weave of the material used.
Math and Social Studies: Research and report on the significance of sail design in commerce.
Math and Art: Bring a variety of art works demonstrating perspective in design to class.
Math and Business: Research and report on the geometry of the globe and its effect on the airline industry.
WORKSHEET #2
Go to the overhead and summarize results from the groups. Ask which are vertical angles? Which are equal? Which are supplementary? (Put one patty paper over the other to form a straight line with the supplementary angles to demonstrate.) Show the group
One letter F (4 and 8) and ask them to find the others. Define corresponding angles and ask the relationship. Show one letter Z (3 and 6) and ask them to find the other. Define alternate interior angles.
Show one letter C (4 and 6) and ask them to find the other pair. Define the relationship of alternate interior angles on the same side of the transversal. Ask them to turn the patty paper 90 degrees and again find the relationships. Ask what relationships exist between the angles if the lines are parallel (either equal or supplementary).
Use two more pieces of patty paper and draw a second set of parallel lines intersecting the first pair. (Make sure the second set of lines are a different distance apart -- could use a pencil boxes' edges and make sure they are not at right angles.) Again trace the exact same figure on the second piece of patty paper. Compare angles and sides by moving one patty paper over the other. Again summarize results of the parallelogram on the overhead.