### Summary

Students learn that math is important in navigation and engineering. Ancient land and sea navigators started with the most basic of navigation equations (speed x time = distance). Today, navigational satellites use equations that take into account the relative effects of space and time. However, even these high-tech wonders designed by engineers cannot be created without pure and simple math concepts — basic geometry and trigonometry — that have been used for thousands of years. In this lesson, these basic concepts are discussed and illustrated in the associated activities.

### Engineering Connection

### Educational Standards

Each TeachEngineering lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the *Achievement Standard Network (ASN)*, a project of *JES & Co. *(www.jesandco.org).

In the ASN, standards are hierarchically structured: first by source; *e.g.*, by state; within source by type; *e.g.*, science or mathematics; within type by subtype, then by grade, *etc*.

Click on the standard groupings to explore this hierarchy as it applies to this document.

- Colorado: Math
- Colorado: Science
- b. Describe methods and equipment used to explore the solar system and beyond (Grade 8) [2009] ...show

- Common Core State Standards for Mathematics: Math
- 6. Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. (Grade 7) [2010] ...show
- 7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. (Grade 8) [2010] ...show
- 7. Explain and use the relationship between the sine and cosine of complementary angles. (Grades 9 - 12) [2010] ...show
- 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.<sup>★</sup> (Grades 9 - 12) [2010] ...show

- International Technology and Engineering Educators Association: Technology

### Learning Objectives

- Describe how navigation and engineering are based on mathematics
- Describe how the Pythagorean Theorem solves real-world problems.
- Explain the difference between accuracy and precision

### Introduction/Motivation

### Lesson Background and Concepts for Teachers

*triangle*is always 180°.

*Pythagorean theorem*was an idea discovered by Pythagoras, a Greek mathematician who lived from 569-500 B.C. It is said that he discovered the special property of right-angled triangles while looking at the tiles of an Egyptian Palace. Pythagoras said, "In a right-angled triangle, the area of the square on the hypotenuse equals the sum of the squares on the other two sides."

*radian*is the angle made when the radius of a circle represents an arc on its perimeter.

**1 radian = 57.30 degrees**

Trigonometry

*right*triangle, which has one angle equal to 90º. By definition, the 90° angle is made by two lines that are perpendicular to each other (like the corner of a square), and the third side of the triangle is made by a sloping line connecting the two perpendiculars. This sloping line is called the

*hypotenuse*, and the name comes from the Greek words

*hypo*(meaning

*under*) and

*teinein*(meaning to

*stretch*). Essentially, hypotenuse means

*to stretch under the 90º angle*. It is easiest to show this visually.

*SOH CAH TOA*can effectively help students remember which sides go with which functions (Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, etc.). Mnemonics may help students memorize the relations: "Some Old Hag Caught A Hippie Tripping On Art" or "Some Oaf Happily Cut A Hole Through Our Apartment."

### Vocabulary/Definitions

geometry: |
The mathematical relationships of points, lines, angles, surfaces and solids. |

trigonometry: |
The mathematical relationships between the sides and the angles of triangles. |

### Associated Activities

- Stay in Shape - Students learn how distances on the surface of the Earth are arcs, and not straight lines. They also learn how to calculate the arc length. The come to understand why knowing about geometric shapes such as triangles and circles is fundamental to understanding navigation.
- Trig River - Students act as engineers and use trigonometry to learn basic surveying. They also learn how to determine the width of a "river" without actually crossing it!

### Lesson Closure

### Assessment

Pre-Lesson Assessment

*Discussion Question:*Solicit, integrate and summarize student responses.

- How important is math in navigation? (Answer: It depends on the goal of the traveler. If you have unlimited time and your destination is visible from miles away, you may not need math. But, if you want to get somewhere as fast as possible or the destination is not visible until you are practically on top of it, then math is essential.)

Post-Introduction Assessment

*Voting:*Ask true/false questions and have students vote by holding thumbs up for true and thumbs down for false. Tally the votes, and write the totals on the board. Give the right answer.

- Thumbs Up: if you think it is good to use math when navigating.
- Thumbs Down: if you think navigation can be close enough without math.
- Thumbs Up: if you think math is important for engineers to know how to use.

Lesson Summary Assessment

*Student-Generated Questions:*Solicit, integrate and summarize student responses.

- Have students come up with one question of their own to ask the rest of the class. Be prepared to provide help to some students form questions. Have students take turns asking their questions to the class for as long as time permits.

### Lesson Extension Activities

### Additional Multimedia Support

### References

TrigRatios. April 22, 1998. University of Tennessee at Chattanooga. October 16, 2003. http://www.staff.vu.edu.au/mcaonline/units/trig/trigraddegrees.html

### Contributors

Jeff White, Penny Axelrad, Janet Yowell, Malinda Schaefer Zarske

### Copyright

© 2004 by Regents of the University of Colorado.

### Supporting Program

Integrated Teaching and Learning Program, College of Engineering, University of Colorado Boulder

### Acknowledgements

Last modified: March 4, 2015