1. GPS THE GLOBAL POSITIONING SYSTEM
Summer Math 2002
• Who has used GPS before?!
Kathleen Smith
Anne Munroe
Central High School
Champaign, IL

2. What is GPS? PowerPoint Overview Brief Video
GPS It’s in cars, boats, planes, tractors, golf carts, cell phones, and …soon… could be implanted in your body!
What we plan to do today:
• First we will spend about 10-15minutes discussing just what GPS is, as we show this PowerPoint presentation
• Then we’ll show clips of a GPS video
• Then we’ll each make our own globe with latitude and longitude lines correctly drawn on it and guess the location of Champaign!
• Next, we’ll log on to the geocaching web site and look at nearby caches.
• Finally, we’ll get the GPS units in hand, go outside and initialize all of them, and then go to the park to find some landmarks!
What is GPS?
PowerPoint Overview
Brief Video
Longitude and Latitude Contest!
Geocaching Web Site
GPS Scavenger Hunt in the Park!

3. GPS technology has matured into a resource that goes far beyond its original design goals. These days scientists, sportsmen, farmers, soldiers, pilots, surveyors, hikers, delivery drivers, sailors, dispatchers, lumberjacks, fire-fighters, and people from many other walks of life are using GPS in ways that make their work more productive, safer, and sometimes easier. These applications fall into five broad categories: Location - determining a basic position Navigation - getting from one location to another Tracking - monitoring the movement of people and things Mapping - creating maps of the world Timing - bringing precise timing to the world

4. What is GPS?
GPS is a satellite-based radio navigation system that allows anyone anywhere on the planet to determine their 3D position with great accuracy and precision.
Some of you may know about the built in in-accuracy of the current GPS system. This intentional scrambling of the signal is intended to thwart the use of GPS by those whose intentions may be nefarious….that is the generals of enemy armies, or the intelligence gathering agencies from a foreign power. The Canadian government is working on a new type of receiver that will eliminate the intentional error. The study of accuracy and precision has it’s place in mathematics, especially when integrated with a science class, but we don’t really have time to go into that today….

5. Who maintains GPS? DOD - NAVSTAR Russia -GLONASS
EEC - Under development
•The system that we will be describing today, (and the only totally accessible system currently in operation) is the NAVigation Satellite Timing And Ranging (NAVSTAR) system US maintained by the department of defense.
•There are two other systems under development,
• From the EEC,
•And from Russia, the GLObal Navigation Satellite System with a very interesting web page that details the history of the launching of the satellites and their current status. (Only 12 of the 24 are working at the moment). Here is a graphic from their site….

7. The system is composed of three segments
Space - The orbiting satellites
There are 27 in orbit now in the NAVSTAR constellation. 24 are needed to sustain the system. 5 to 8 are visible to the receiver at any given time and they orbit the earth every 12 hours.

8. Here is a picture of a NAVSTAR satellite
Here is a picture of a NAVSTAR satellite. They have a life expectance of about 10 years….new ones are being put in to orbit annually.
They are approx. 11,000 miles above the earth.
Each one weighs about 2,000 pounds and span about 17 feet across.

9. The system is composed of three segments
Space - The orbiting satellites
Control - The monitoring stations on earth
These stations make constant accuracy checks with the satellites...

10. There are five stations set up for NAVSTAR, with the master control center located at Falcon AFB in Colorado Springs. The stations are positioned around the globe to ensure that all satellites are in touch with at least one center at all times.

11. The system is composed of three segments
Space - The orbiting satellites
Control - The monitoring stations on earth
User - The GPS units such as you will use today.
These are the human interface component. They range from vans with antennae to hand held units….

12. •There are many types of receivers on the market, this is a picture of the cheapest model available. You will notice that most of the units on the tables are of this type….All civilian units are of two types, SPS, and differential. The differential GPS units employ software that minimizes the intentional error and cost in the thousands of dollars. The SPS units are accurate to about 50 m, the differential to within 10 m and the ones the military uses to within 1 cm.
•The SPS units intended for general public use all have the same receiving system, and produce the same accuracy of reading… more expensive means more bells and whistles, but not necessarily more accuracy.

13. The GPS pre-test: When was GPS first used? Sputnik…1957 TRANSIT…1960’s
NAVSTAR…1970’s
•According to the SpatialTek web site, (address in packet.) GPS started when scientists studying the orbit of Sputnik found they could track it by its radio signal. In 1964 US submarines used a GPS system called Transit (subs had to be still). Research and development of NAVSTAR began in 1973 and the 1st four satellites went up in 1978.

14. *The word “tracking” must appear in the screen’s lower right corner!
How many satellites make up a GPS?
27 at the current time
24 to maintain a system
As was stated earlier, new satellites are being sent into space each year to replace the old. The GLONASS site has some very interesting information about the state of the Russian “constellation” as these systems are called. Only 12 of the needed 24 are currently up and running properly. They have listed all the satellites that have dropped from orbit, and those that were unsuccessfully launched….an interesting list... Here is a graphic of the NAVSTAR constellation.
Your GPS receiver must track at least 3 satellites to give you an accurate position location.
*The word “tracking” must appear in the screen’s lower right corner!

15. This graphic could be used in an advanced Algebra class where students are solving systems of conics
This next view…..

16. ….would lend itself well to a Trig class where students are studying sinusoids….

17. From Circles to Waves How would you describe the
These two graphs both represent
The NAVSTAR constellation of
satellites.
….you could put the two together and have a Unit Circle to Sine Wave lesson. This sheet is in your packet.
How would you describe the
path of a single satellite in each
graph?
How would you explain the
transition from one graph
to the other?

18. Who uses GPS? Government Industry General Public
Military
Agencies - Conservation, Weather, Rescue
Industry
Transportation
Fishing
Agriculture
General Public
Recreation
Emergency / Security Reasons
Health Related Reasons
Who uses GPS?
Of course the government is a prime user of the system. The military now issues Troops on deployment a GPS unit as standard equipment. The military also makes extensive use of GPS in spying and intelligence gathering. Aside from the military, however, the Government has found many peace time uses for GPS.
Industry is not far behind…Transportation of all sorts, sea, air and land. GPS allows you to track your UPS package…Commercial fishing and agriculture…have found many uses for the technology. And provide many applications for the classroom.
And of course the general public uses the technology for hiking, fishing, hunting, and just getting around in their up market vehicles….. Also now in order to be rescued or to track a stolen or lost car or to find a child or Alzheimer’s patient or to help a blind person navigate…

19. Each GPS satellite transmits an accurate position and time signal.
The user’s receiver measures the time delay for the signal to reach the receiver.
Measurements collected from 4 satellites are processed to solve for the 3D position, velocity and time.
How does GPS Work?
•Each satellite has it’s own binary code fingerprint (great for computer science) and time is measured using atomic clocks (article from Microgravity news Fall 99)
•The idea that it takes 4 satellites to give your latitude, longitude, altutude and velocity is a great lesson in systems…. And solving for the point of intersection of the circles generated by the satellite signal is a great days work. There is a cover article from Mathematics Teacher that takes the students through the process step by step. The article is included in the list of resources in your packet.
•You can also use this information to teach triangulation and there is a very nice description of the method on the Craig Haggart webpage inclulded in your packet.

20. So there you have it…. You now know little bit about GPS
So there you have it…. You now know little bit about GPS. And perhaps have already identified a few things you might want to do with it in your classes. Next we will give you a chance to play around with the units.
But before we go out to locate ourselves we will watch just a brief piece of a video put out by one of the big manufacturers of GPS units. We will not get into all the fancy stuff, like making routes or navigating to stored locations, because we just don’t have enough time today. You will also not get to use all the windows, but your students will love to figure out the different navigation windows for themselves. There are some very good lesson plans using the windows at the U of Texas site listed in your packet.

21. Where on Earth are we?
So, who wants to guess our current location?

22. Your Task Outside 1) Get a locked reading on your unit.
2) Chose a direction either east-west, or north-south.
3) Walk along the direction vector until the reading on your unit changes.
4) Design a method for finding the circumference of the earth based on
your observations.
When you see the little lock hooked shut on the left side of the Magellian units, or the screen changes to a fixed location on the others, you know you have a fixed reading. It would be nice if each group had at least one person with a bit of experience…
The rest is discovery, so have fun and be nice to each other…..watch traffic! Any questions?

23. What you discovered outside.
1) Did you all get the exact same reading?
2) How did you know your direction?
3) How many paces until the reading on your unit changed?
4) What was your method for finding the circumference of the earth?
5) Would the circumference be the same for N/S as E/W?
Talk about the U of I talk…show graphic….after you are fixed, the distance and direction of your movement is recorded accurately.
N/S circumference 40,008 km.
E/W here?

24. A Sample Lesson Mission Mathematics A NASA/NCTM Project
From:
Mission Mathematics
A NASA/NCTM Project
This lesson is only one of many that make up a unit on GPS technology in the NCTM Mission Mathematics book. The other units in the book address related topics like modeling the solar system, orbiting debris, elliptical orbits and advanced communication systems. This book is available at the NCTM booth.
What you need to do here, with your group at the table, is try to determine the route the ship has taken by using the GPS windows printed out for you….you have 20 minutes. We want to see an active classroom. We will answer questions and give hints if requested. Have fun..

27. How Does This Fit Into a Math Class?!
Pre Algebra: Latitude and Longitude, Integers
Algebra I: Linear Systems, Absolute Value
Geometry: Degree Measure, Circles, Spherical Geometry
Algebra II: Systems of Circles, 3D Graphing

28. How Does This Fit Into a Math Class?!
Trigonometry: Orbits as Sinusoids, Bearing,
College Algebra: Conic Sections, 3D Systems
Pre Calculus: Accuracy, Precision, Error
Calculus:
Discrete Math: Circuits, Paths, Triangulation

29. Available Materials The WEB:
Overview of GPS:
U.S. Coast Guard:
Inst. of Navigation:
GLONASS info: html
GPS sites:
"GPS for Fun":
SpatialTek:
Craig Haggart:
Trimble:
Science Junction: junction
Lesson Plans:
These sites have a wealth of information. Many are linked to each other and the students could easily do a paper based on what they find here. The overview from the U of Texas is very education oriented and many of the graphics you saw today came from there. The coast guard is thorough and clear. The GLONASS is interesting because it is Russian, GPS for fun is a newsletter geared to recreational uses, and the last two have excellent information and historical information…..new sites come up all the time. The Trimble page has a wonderful tutorrial for using the units and understanding navigation. The last two have suggestions for lesson plans and links to other sites.

30. Available Materials In Print: Mission Mathematics 9 - 12 (NCTM)
MICROGRAVITY NEWS, Fall ‘99 (NASA)
Mathematics Teacher, Nov (NCTM)
Longitude, Sobel, (Walker & CO. NY, also a PBS video)
The Librarian Who Measured the Earth
•The Mission Mathematics has the lesson we did today, along with many others. Available at the NCTM booth
• The Microgravity News is a free publication from NASA and always has articles related to mathematics…you can probably pick up a free copy of the Fall ‘99 issue at the NASA booth.
• If you need a copy of the Mathematics teacher article just let me know and I can send it to you
• The others are in print….

31. Available Materials On the Web: In Print:
Overview of GPS:
U.S. Coast Guard:
Inst. of Navigation:
GLONASS info: html
GPS sites:
"GPS for Fun":
SpatialTek:
Craig Haggart:
Trimble:
In Print:
Mission Mathematics (NCTM)
MICROGRAVITY NEWS, Fall ‘99 (NASA)
Mathematics Teacher, (NCTM)
Longitude, Sobel, (Walker & CO. NY, also a PBS video)
The Librarian Who Measured the Earth,
These sites have a wealth of information. Many are linked to each other and the students could easily do a paper based on what they find here. The overview from the U of Texas is very education oriented. The coast guard is through and clear. The GLONASS is interesting because it is Russian, GPS for fun is a newsletter geared to recreational uses, and the last two have excellent information and historical information…..new sites come up all the time. The Trimble page has a wonderful tutorrial for using the units and understanding navigation….