Using Radio Navigation and Aerodynamics and other aviation examples to teach Mathematics

This draft of a proposed middle school academic program has been liberally plagiarized from the celestial navigation curriculum of the Island School in Eleuthera. (Find and replace used)

Program Overview Draft– for discussion- Dave Ahl Draft

This draft was written using the Brunswick Middle School as a “special place” example. Clearly, The Island School in Eleuthera is a VERY “special place” in the Bahamas where the night sky is perfect for learning celestial navigation. While many middle school kids would love to travel to learn. Travel is beyond the resources of many middle school parents. This proposed academic mathematics course has been designed to capitalize on the same Island School academic concept but translate it so it can work in “ordinary” places. No international travel required. There are thousand of airports and this course could be taught anywhere.

 The Brunswick School Mathematics program seeks to fully utilize the resources inherent to special location adjacent to Westchester Airport . We aim to produce capable, creative problem solvers who can understand the world through the lens of mathematics. Specifically, our mathematics program is guided by four enduring understandings we strive to instill in each of our students:

• Mathematics is the study of how to think;
• Mathematics is a skill that needs to be practiced;
• Everyone is capable of learning and mastering mathematics;
• Mathematics explains the universe in which we live.

We hope to show students that mathematics is a creative, thoughtful enterprise that leads directly to understanding our world. In accordance with these aims and The Brunswick School’s commitment to fostering an experience that is truly place-based, our mathematics program teaches the theory and practice and history of radio and celestial navigation. 

The Brunswick School Mathematics course in radio navigation seeks to develop an appreciation for the power of mathematics to analyze the world in which we live, as well as nurture a sense of wonder about aviation.  Our program focuses on challenging students to solve the classic problems from the history of science, mathematics and navigation: What is the circumference of the Earth? How do we find the longitude? What does the altitude of a celestial body at my meridian indicate about my latitude? What is the angular speed of the moon’s revolution around the Earth? What is the arc-measure between any two points on the surface of a sphere? Where am I?

How did the Wright brothers do wind tunnel research? What is their contribution to propeller research that is being implemented today with wind turbines?

What specific changes did Lindbergh make to the Spirit of St. Louis to increase time aloft? How did he calculate weight and balance for his take off roll to clear and obstacle? Why did he decide on a single engine? What the burn rate in gallons bounds per hour take off weight versus landing weight?

These questions, and others like them, are explored in detail during the xx week Brunswick School term.

Radio navigation combined with celestial navigation connects the most interesting problems in modern geometry and trigonometry to the practice of determining one’s location on the surface of the Earth. In addition, radio navigation links the study of mathematics to the artistic, scientific and philosophical musings about the flight of birds that are ancient as human history.  In this way, the course becomes a multidisciplinary synergy of mathematics with the other partitions of the school. We are pleased to offer our students this exciting opportunity that is fully integrated with our special place next to HPN.

Radio Navigation

Course Description

Radio / celestial navigation is an applied trigonometry course that teaches students the theory and practice of navigation by starlight and radio wave. Until the advent of GPS, radio navigation combined with celestial was the bedrock of all navigation science because it is failsafe, elegant and remarkably accurate.  A skilled navigator, in optimal conditions, working only with a sextant and a watch, can determine the location of his or her ship to within .2 minutes of arc—a position error of only 400 yards. The scientific theory at the foundation of radio navigation teaches students about the motion of the Earth, the seasons, the wandering daylight problem, the stars, the planets and the celestial sphere.  Within this rich conceptual framework, students develop spherical geometry and spherical trigonometry in order to model the surface of the Earth and its luminous container, the night sky. Brunswick students learn the practice of celestial navigation although the Brunswick campus due to airport light pollution does not allow for uninterrupted views of the Northern, Southern, and Western horizons. Students may have to find another night location to develop skill using a sextant to find the altitude of a range of celestial bodies. However The Brunswick campus is an excellent location for radio navigation practice because of line of sight access to VHF navigation aids; VOR as well as the ILS including the outer, middle and inner markers. Because much of the radio navigation spectrum overlaps with cell phone spectrum many experiments can be done using existing student cell phones.

Does cell phone useage aboard an airplane during final approach interfere with the ILS glide slope?

For students more interested nautical navigation the course will teach use of a nautical almanac to find ephemeris data, to reduce sights with and without the use of H.O. tables and to plot the position of a vessel at sea on a chart.  In particular, the practical skills students acquire include:

• Charting a dead reckon track
• The Use, Maintenance and Correction of a Mariner’s Sextant
• Shooting the noon sun
• Shooting a dusk star (or planet)
• Shooting a twilight star (or planet)
• Taking a bearing with a compass
• Correcting a compass for magnetic deviation
• Plotting a position fix on a chart

The mathematical component of the course is driven by world problems that develop the geometry and trigonometry required for celestial navigation.  Problem sets are preceded by readings that equip students with the conceptual framework to create the mathematics necessary to answer a particular problem. Through this process students develop their mathematical modeling and problem solving skills.  The aim is to graduate students who are confident, mature problem solvers.

The course is an applied geometry and trigonometry course.  While many ideas from the plane geometry of Euclid are useful when modeling navigation problems, such as finding the distance to the horizon, students are also introduced to the geometry of curved surfaces.  The familiar properties of lines and shapes inherent to figures incident with a plane do not hold for figures lying on a sphere.  In particular, students study non-Euclidean geometry including great circles, loxodromes, spherical triangles and tangent planes.  The specific list of mathematics topics covered during the course includes:

I. Coordinate systems:

              a. Definition of a coordinate system;

              b. Coordinates on a plane;

              c. Coordinates on the surface of a sphere;

II. Spherical Geometry:

                 a. The elliptical parallel postulate;

                 b. Great circles, small circles and defining distance;

                 c. Arc measure, arc length, central angles, latitude angles;

III. Tangents and parallel lines:

              a.  Definitions of parallel lines;

              b.  Definitions of tangent lines;

              c.  Relations between tangents to a circle and parallel lines;

IV. Introduction to trigonometry:

              a. Special right triangles;

              b. Trigonometric ratios defined on a right triangle;

              c.  Trigonometric ratios defined on the unit circle;

              d. Solving triangles;

              e. Inverse trigonometry;

V.  Constructions:

              a. Introduction to the history, theory of compass and straightedge constructions.

              b. Constructing line segments and circles;

              c. Bisecting angles and segments;

              d. Constructing perpendiculars, parallels, similar triangles;

              e. Constructing the trigonometric ratios;

VI. Applied trigonometry:

              a. Law of sines;

              b. Law of cosines;

              c. Problems relating angular diameter, parallax angle and distance;

VII. Spherical Trigonometry:

                 a. The angle-arc relations of spherical triangles;

                 b. The Navigational Triangle;

                 c. The spherical law of sines;

                 d. The spherical law of cosines;

Other mathematics topics may include vector algebra, systems of linear and non-linear equations, the geometry of map projections and a comparison of orthodromes to loxodromes.  The problem sets sample a broad spectrum of assumed mathematical knowledge; this wide range allows students to be challenged at their individual level of mathematics. 

Physics – (Not sure of the relationship between Physics department and Math department at Middle school level?)

A study of the electromagnet spectrum and wave theory. Special emphasis on spectrum from 88Mhz (FM radio) to VHF (Broad cast TV and aviation) to 3Ghz

(Cell phones, other hand held wireless devices (GPS) and cell phone towers and base stations and Wifi base stations wireless internet

Aerodynamics- can be expanded

Meteorology- can be expanded

Aviation physiology – the impact of altitude on physical performance (sports/training overlap)

If these areas are included this course would provide much of the basis for passing the FAA Private Pilot written examination.

In addition to hours spent in the classroom and during study hall, the course includes frequent opportunities for students to take their mathematics education in their own hands, into the world.  To better understand the movements of the celestial sphere, students often appeal to the night sky to illustrate the spherical coordinate systems that describe the positions of objects in the sky. 

Field trips include excursions to the control tower, flight simulators, and radio navigation transmitters. airplane cockpits.

To investigate the distance to the horizon, afternoons with sextants to develop skill shooting stars and treks to chart the latitude and longitude and altitude of local landmarks. Through connecting mathematical theory to the practice of answering interesting, available questions, the course cultivates the students’ ability to solve problems that require creativity, patience and persistent effort.