The Math of GPS

 GPS sort of works in Alaska.  If you use the part that loves to give directions from point A to point B, it doesn't always work.  It will have no record of certain roads and show roads as being there that aren't.  On the other hand, the GPS can be used to identify the International Date Line.

Yes, I was in a place where they used GPS to find the International Date line so we could step across it into Russian territory and back.  A GPWS works by triangulating it's position based on distance from satellites.

The GPS measures the distance from a satellite by how long it takes for the signal to be received while keeping track of the location of one, two, or three satellites circling the earth.

One way is to use distance formulas similar to the one we use when calculating the distance between two points.  Instead of using x and y coordinates, these formulas include the z coordinates and the speed of light times the off set time of the receiver..  So basically it is d = sqrt((x-x1) + (y - y1) + (z - z1)) + c1 where c1 is the product of the speed of light times the offset of the receiver and is calculated  for each satellite in the area to create a system of equations.

 The equations in these systems are multivariate and non-linear.  One method used to solve this is Newton's method which requires one to make a guess do a bit of calculation, adjust and continue till you get the answer. Another method for finding distance is to measure the phase difference between incoming and out going continuous waves which uses basic trigonometry to find this.

The above is a short explanation, but much of the math is beyond what most of my students can do so this site has a great 27 page booklet filled with some great math activities to use in class.  It begins with having students do some real calculations.  Then it looks at the Pythagorean theorem's application to the GPS.

After going through the explanation of how GPS works, it has a variety of A variety of activities including Geocaching, using a GPS with a Topographic map, and an activity from Dr Math that uses spheres in 3D.  There are also two to three pages of links that can easily be used in the class.

This site has a nice activity to help students learn to find a position using GPS.  It is a bit simplified and uses the rate x time = distance formula but it isn't bad.  The activity is designed for students to work in groups of 3 to 4 people and is quite clear.

Both of these activities gives students the opportunity to how math is used to help them find their location.  Let me know what you think, I'd love to hear.  Have a great day.