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An Explorer lesson that uses triangles and trigonometry to measure height.

Author: Pat Jungkeit and Joanne Caniglia

Grade Level: 9-12

Subject: Geometry

Objective: To use similar triangles and trigonometric ratios to compute the approximate height of the school smokestack.

- To use outdoor measurement to collect approximate data.
- To use computer technology students to create experimental heights and use statistical analysis to predict the actual height of the stack.
- To connect all phases of the project by creating a complete report on the computer.
- To communicate the results of their project to the class.

Materials:

A tall structure near the school, 50' or 100' tape measure, mirrors, clinometers (these can be "Homemade" with cardboard, straws, and string) computers, software having spreadsheet, word processing, and graph/chart capabilities.

Directions:

Students should work cooperatively with a partner for the project.

Students should be encouraged to treat the project as a bid for a corporate contract. It is important that they measure carefully and calculate accurately. However, it is equally important that they communicate their findings in a convincing and appealing manner.

In the Classroom:

- Explore the connection between light reflection in a mirror and similar triangles. Discuss how the triangles might be used to calculate heights.
- Explore the use of shadows to create similar triangles that can also be used to determine approximate heights.
- Explore angles of elevation, and use them with the tangent function to calculate heights.
- Have each team estimate the height before beginning the experiments.
- Challenge students to discover a new method for determining the height of the stack.
- Before going outside, give the students careful instructions about the expectations and goals of the experiment.

Outside

Mirror Experiment:(minimum of 4 locations)Place mirror flat on the ground. Measure the distance from the stack to the mirror. Slowly move back from the mirror until the top of the stack is visible in the mirror. Measure the distance from the mirror to the person who has sighted the stack. Measure the height of the person (to eye level). Record all data.

Angle of Elevation:(minimum of 4 locations)Use the altimeter to sight the top of the stack and record the angle. Measure the distance from the stack to the person doing the sighting. Record the height of the person (to eye level).

Shadow Experiment:(minimum of 2 data sets)Align your body so the top of your shadow matches the top of the shadow of the stack. Measure the length of your shadow and the length of the stack's shadow. Record your height. Each partner should do this.

The Computer Laboratory

- Each pair of students will create a spreadsheet that clearly displays their data and incorporates algebraic formulas to calculate the experimental heights for each of the experiments.
- Using the experimental heights, students will find the measures of central tendency, create charts and or graphs, and use statistical reasoning to decide on the best estimate of the height. Outlines should be discussed.
- A formal report will be created on a word processor to connect all the phases of the project. It should include: a cover page; an explanation of each experiment, with drawings, labels and formulas, spreadsheets; charts or graphs, statistical data, and a conclusion containing the estimated height of the smokestack.

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