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Mathematics Governor’s Institute 2003

 

(Download as Microsoft Word documents: Measuring Mountains, Worksheet)

 

Title of Project:  Measuring Mountains

 

Team Members:  Thomas Danias, Sharon Hlusko, Theresa Philbin, Karen Tucker

 

Grade Level and/or Course:  Grades 9-12, Trigonometry, Advanced Geometry

 

Concepts Used:Trigonometric functions,  Right triangles, Angle Measurement, Linear measurement

 

PA Standards Addressed: 

Trigonometry 2.10.11B  Identify, create and solve practical problems involving right triangles using the trigonometric functions and the Pythagorean Theorem.

            Geometry 2.9.11I  Model situations geometrically to formulate and solve problems.

            Problem Solving 2.5.11A  Select and use appropriate mathematical concepts and techniques from different areas of mathematics and apply them to solving non-routine and multi-step problems.

 

NCTM Standards Addressed:

            Geometry, Problem Solving

 

Introduction /Applications:  We would like to know the height of a structure that cannot be measured directly…even the height of a mountain!

 

Question:  What is the height of the school (any building)?

 

Model:  Solve a right triangle using trigonometry.

 

Resources and Materials:  Each group of students will need:  a tape measure, a protractor, string (dark-colored)—approximately 12 inches, a large metal washer (or similar weight), worksheets.

 

Procedures and Activities: 

1.      Using a protractor, measure an angle of elevation: 

-         attach string to center hole in bottom of protractor

-         attach weight to bottom of string

-         hold the protractor with the degree measures beneath the straightedge (in other words, upside down)

-         point the protractor to the top of an object in the room…aligning the straight-edge of the protractor with your line of sight to the top of the object

-         allowing the string to hang vertically, record the degree measure at which the string crosses the protractor…this will be the measure of the angle of elevation (or its complement) depending on the way the protractor is held and the way in which it is read.  NOTE:  Be sure that students record an appropriate angle measure for the angle of elevation.  This is a good opportunity to remind students of using estimation to check the reasonableness of their answer.

2.      Using the model, determine the height of a measurable object, such as the height of a ceiling, flagpole, doorway, etc.:

-         measure the linear distance you are standing from the base of an object

-         determine the angle of elevation to the top of the object, using the method just described

-         sketch a right triangle to model the situation, label the appropriate side and angle

-         solve for the height using right triangle trigonometry

-         compare the result to the actual height of the object…note the difference, the actual height is higher than the calculated height

-         determine the reason for this discrepancy…students should come to realize that the height they indirectly measured was not measured from ground level, but from eye level.  Therefore, the actual height would be the calculated height plus the height the protractor is from the floor.  (Be cautious about unit conversions so that all measurements are in the same unit of measure.)

-         Repeat this process with another measurable object(s) to verify this formula

3.      Once students have mastered this method, they are now ready to measure the height of the school!

 

Rubric:

            Drawing a two-dimensional model

            Appropriate labels with correct units of measure

            Correct measurements, both distance and angle measures

            Use of appropriate trigonometric functions

            Correct numerical answer with appropriate label

 

Accommodations:

ESL:  Appropriate definition of terms/materials.  Students may need instruction about the units of measure. In particular, the English units may not be familiar to them.  ESL students may work better in groups with other ESL students who have a better understanding of the English language.

Special Education:  A structured worksheet with the model pictured may be appropriate for these students.

 

Enrichment:  Make a scale drawing of the front of your house using these techniques to determine the proper dimensions of relevant features of the house and use an appropriate scale.

 

 

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