Mathematics, Grades 9-12
1998-1999 Master Teacher Michael Powell, High School of Science and Technology, Springfield, MA
Overview
In
this lesson the students will explore the concept of similarity. They will
learn about such topics as ratio of corresponding sides, internal ratios and
scaling factors. These explorations will be assisted by two activities. The
first is to watch a video that explains the concept of similarity and gives
several examples to illustrate this concept. The second is to use the plan
of action they develop as the guide for measuring indirectly the height of
their school building.
ITV
series
Project
Mathematics; "Similarity" produced by CALTECH
Learning
objectives
Students will be able to:
identify corresponding parts of similar polygons; calculate the scaling factoring
if the relative sizes of corresponding sides is known; calculate the length
of unknown sides if the internal ratio is known; calculate the length of an
unknown side if the scaling factor is known.
Materials
per three or four students:
meter sticks (or similar
measuring devices)
calculators
1
piece of chalk (to mark off meter lengths on pavement)
Pre Viewing Activities
Have
the students divide an 8.5 x 11” sheet of paper into 3 columns. Label the
columns "What I already knew", "What I just learned",
"Questions I still have."
Focus For Viewing
The
focus for viewing is a specific responsibility or task that the student is
responsible for during or after watching the video designed to focus or engage
the attention of the student. The first focusing task is to identify certain
ideas presented in the video and to categorize them as either new learning
or old knowledge.
Address the class and say, "You are about to watch a video that describes a concept called 'Similarity'. There will be much on this tape that you already know and much that is new to you. As you watch this video I would like you to identify those ideas that are new to you and to list them in the column on your paper that is labeled 'New Ideas'. If you see something that you already know list that idea in the column labeled 'Old Knowledge'. If you have a question that is not answered by the video, write the question down in the third column which is labeled 'Questions I Still Have'. If there are things you would like to write but missed the exact words in the video, feel free to ask me to pause and rewind. I will be happy to do that for you."
Viewing Activity
Begin the video at the "Review of Prerequisites.” Pause immediately after the announcer says, “and volumes of solids”. Ask the students, " How many of you know what a ratio is? Is it just another name for a fraction?” Many of the students will know already that a ratio is generally expressed as a fraction and will respond to the above question positively. Now ask the students, "Did you write that down in the column labeled ‘What I already knew’? If you didn't know that a ratio was just another name for a fraction, then be sure to write that in the column labeled ‘What I just Learned’ Refocus their attention by reminding them that they will be seeing many things that are new to them and they should be ready to write these things down. Ask the students to be ready to identify two uses for scale models.
Resume the video. Pause the tape immediately after the actress flees from the gigantic praying mantis. Ask the students, "What were the two uses for scale models?" (hobby, toys, designing cars and bridges) Ask them, "How did you like the pictures of the bridge falling?" (It was real. It was the Tacoma Narrows Bridge in Washington state during the 1940's) Ask the students, "How would making a scale model have helped the engineers build that bridge better?" (by allowing wind tunnel tests that would have shown the susceptibility to harmonic waves caused by the wind) Ask the students, "What about those manhole monsters. Can insects really grow that big?" Refocus their attention by asking them to identify two ancient Greeks who first discussed similar shapes. Say, "In the coming segment you will see a few uses of a scaling factor.”
Resume the video. Pause the tape immediately after the announcers says, "...we also use similarity when we make test a model in a wind tunnel. ...." Ask the students, "What were the uses of similarity that you saw in this last segment?” Respond to the students’ answers and any other comments. Ask them, "Did you already know that a map was a scale model of a city?" Ask them, "What column did you write that item in?” Refocus by asking the students to be ready to identify the "center of scaling" that will be introduced in the next segment.
Resume the video. Pause the tape immediately after the announcer says, "...scaling changes all the line segments by the same factor." Ask the students, "Who can tell me what is meant by the scaling factor?" Wait for student responses and clarify as needed. Since this is the main point of the entire lesson, spend as much time as necessary to be sure that the concepts of corresponding parts, scaling factors and ratio of sides are all understood by the students. If necessary, put additional examples on the board or overhead and ask the students to identify these concepts. Say to the students, " I would like you to take a few minutes, maybe five minutes to write a few sentences about the ideas you've seen in this video so far." Wait five minutes and ask students to read their entries, discussing in whole class format, providing clarification, emphasis, or correction as needed. Refocus the students by asking them to be ready to say how Thales measured the height of the pyramid.
Resume the video. Pause immediately after the announcer says, "...the angles do indeed form a straight angle. That's all there is to it.” Ask the students, "How did Thales measure the height of the pyramid?" Wait for student responses. Wait until the complete explanation is given, including measuring the length of the shadows at the same time. Ask the students, "Suppose we wanted to measure the height of this building. How could we do it using only calculators, poles and meter sticks?" When the class has formulated a plan of action (make sure it gets written down step by step), say to the class, "After we finish watching the video, we are going to go outside to actually measure the height of this building.” Refocus their attention by asking them whether the ideas of similarity that they have just seen for the triangles can be applied to other polygons or shapes. Ask also "What is the effect the scaling factor has on the perimeter and area of polygons." You may have to quickly review the meaning of perimeter and area.
Resume the video. Pause the video immediately after the announcer says, "...and the same is true for a pyramid, a sphere or any three dimensional creature.” Say, "You have just seen the effect that scaling has on perimeter, area and volume. What was that effect?” Wait for student responses and clarify as needed. Refocus their attention by asking, "Remember the manhole monsters? What would happen if an insect really did grow that big?"
Resume the video. Stop the video after the announcer says, "...oversized household pests.”
Post Viewing Activity
Ask
the students if anyone has questions or comments about the video. Some of
the class will have entered questions onto their 8.5 x 11” paper. They should
have the opportunity to ask questions and have them answered.
The class has developed a plan for measuring the height of the building based on the method Thales used to measure the height of the pyramid.
Take the class outside and break them up into groups of three or four students per group. Have them measure the length of the building's shadow as well as their own height and the length of their shadow. This is best done as far before or after noon as possible to get the longest shadows.
Then by calculating the proportion,
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they can calculate the height of their school building.
Action Plan
Students
could contact a local architectural firm and ask to see blueprints of buildings
and find out what scaling factor the architect used for the drawings.
Students could purchase scale model kits of cars, airplanes, ships, etc. and use the scaling factor identified on the instructions to determine the size of the actual car, plane or ship.
Extensions
Social Studies: Students could make a scale map of their town showing roads, rivers, etc, drawn to scale.