1. See who can get the most places correct without using an atlas. Have a race on two or more computers to see who can get their place first. Have teams compete. They can try to guess the location as the circles are being plotted. When I tried a competition in an eighth grade class, it took a class of 28 students, each with their own computer and working in pairs, 40 minutes for 4 teams to get 5 locations. The other teams had 2 to 4 locations.
2. A GPS and earthquake stations use the same geometry to locate either your position on Earth or the position of an earthquake. You can change the simulation to an earthquake location challenge. Move the satellites the same way you moved the red dot at the lower right to locations on Earth. The dots now become earthquake observing stations and the goal is to find the location of the earthquake.
3. Further explore the theorem that three crossing circles are required to define a common point of intersection in a plane most of the time.
4. This activity is simulated in two dimensional space, but a GPS works in three dimensional space. In three dimensions it takes four intersecting spheres to determine a point in space. Have your students do some research on GPS to see if they can find out why only three satellites are needed and better understand how a GPS works in three dimensions. One of the four spheres needed to locate a position is the surface of the Earth.
- Your students can move the satellites the same way they moved the red dot at the lower right. Have your students put the satellite dots in a straight line and see what happens. What is special about this case? It is something seen often in nature and art. The fact that the satellites are in a straight line dictates that there will be two symmetric intersections and they will need information from another satellite not in line with the three. Your students need to try it to see it and they will learn more about geometry.
- Could the above case occur with GPS satellites? Your students will have to look up the geometry of GPS satellites.
- Is it possible to have two circles intersect at a common point? If so, what are the conditions? Could that happen with a system of satellites trying to locate a point on Earth? It will probably be necessary for students to draw pictures to answer these questions.
- Is there a way to locate a position on the surface of the Earth using only two circles? Hint: Can you use the Earth's surface? The Squeak activity doesn't show the actual geometry in a three dimensions so you will need to draw a different view of the GPS satellite system to answer this question.
- After exploring satellites arranged in a straight line and the preceding questions, have your students discuss what they learned and try to come up with a clarification to the theorem that three circles are generally needed to locate a point in space. Have them discuss their reasoning that leads to the clarification. Finally, have them write out a more complete statement of the theorem that includes all the exceptions.
5. Your students can take the activity apart, which is the power of Squeak. Click on the Playing button at the left for a introduction to how Squeak works. You must click Escape Browser and your resolution must be set at 1024x768 to view this properly. Go to Squeakland for tutorials and more information on using Squeak. Have your students try to take the activity apart and see how it works and then construct their own activity by modifying this one. There are many objects on the page to experiment with. They should start with the blue dot that draws the circle for the blue satellite. Open its Viewer, which can be accessed by clicking on the gray box in the upper right. Then drag out Script1 and start taking it apart by removing tiles. This script simply draws a circle, so your students can learn alot about geometry. No matter how bad a mess they make, they can always get back to the original activity by exiting Squeak and starting over, so they don't have to worry. Learners should explore, try what comes to mind, and have fun exercising their brains. You would be surprised what your students can do with Squeak! For a major challenge, have your students design a new activity where Norbert and Zot visit different places. If they look at script2 associated with the Norbert and Zot object (actually called Ted), they will see it is written in a programming language (not tiles). That isn't a great problem, because learning programming by example is common. The main information that needs to be changed is at the beginning of the script where the locations are entered (a name in single quotes followed by an x and y coordinate). Just enter new data for the new places. Even better, the world map is just another object. Delete the world map and put in another map - maybe the US or Europe or Virginia or Mars or wherever. They will need a line drawing map saved as a transparent gif so you can see through it to observe the circles trilaterate. They should be creative! When they are done go to the NASA CONNECT web site and have them submit their version of the Squeak project and we will post all that work correctly for others to learn from.
6. Have your students think about how a GPS would work in a one dimensional world. Would it still make sense to talk of being somewhere on Earth? What would Earth be? Where would the satellites be? How many satellites would you need to find your location anywhere along a line? Is it practical to have a GPS in a one dimensional world? There isn't just one correct answer to these questions, depending on how you define Earth. A fun book to read to help your students think about this problem is Flatland by Edwin Abbot.
7. It is important to foster individual aesthetic growth in your students. The sense of beauty seen in art is equally important in mathematics. Symmetry is part of beauty and it runs throughout art and mathematics. Theorems have a sense of beauty because they represent truth. Beauty gives us a sense of balance and well-being in our lives. The Squeak project was organized on the page to be functional and that often leads to a kind of beauty. But beauty is in the eye of the beholder, so challenge your students to recreate the Squeak project in a way that leaves them with a sense of balance and beauty. With the tools in Squeak, they can make their own drawings. Any of the objects can be moved and resized and their colors can often be changed. Click on any object while holding down the alt key on a PC or the command key on a Mac and a halo of handles will appear. Click on the red handle at the upper left to explore many options for changing the object. Go to Squeakland for tutorials and more information on using Squeak. When they are done go to the NASA CONNECT web site and have them submit their version of the Squeak project and we will post it to bring beauty and balance into the lives of all who choose to open their project.