How Far Between Cities?     
by Suzanne Alejandre

Teacher Lesson Plan

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NCTM Standards:
      Understand processes of measurement
      Solve problems involving scale factors
      Make decisions about units and scales involving measurement

      Specify locations and describe spatial relationships using coordinate geometry
      Use geometric modeling to solve problems
      Create deductive arguments concerning the Pythagorean relationship

      Students will plot three points to form a triangle.
      Students will find the hypotenuse of a triangle using the Pythagorean Theorem.
      Students will use a given scale to calculate distance.

Part I: Tutorial

This section is more of a "tutorial" than a lesson. The idea is that after following a set of directions, students will see the right triangles on the map. Once they realize that the line segment drawn between two city points can be thought of as the hypotenuse of a right triangle, they can more easily apply the Pythagorean Theorem.

If you feel your students already have a good understanding of this concept, have them skip (or possibly skim) Part I and go to Part II.

Open the Java Applet

Note: It will open in a separate window. If you are displaying the page for students, arrange your browser windows so that the applet and the directions can be easily viewed. If students are working individually they should be encouraged to do this.

Direct students to follow the steps individually on their computers or as a class have one student follow the steps as the class observes:

  1. Select Clear Points (a good habit in case a point had been marked earlier)

  2. Click on the dot on the map indicating the location of Rialto. Use one of the 4 arrows to fine tune the location of the point. The readout in the upper right corner should be x: 196 y: 180

  3. Select Mark Point [Result of this action: nothing visible yet]

  4. Click on the dot on the map indicating the location of Loma Linda. Use one of the 4 arrows to fine tune the location of the point. The readout in the upper-right corner should be x: 402 y: 57

  5. Select Mark Point [Result of this action: a line will appear]

  6. Mark this point x: 196 y: 57. Use one of the 4 arrows to fine tune the location of the point.

  7. Select Mark Point [Result of this action: a second line will appear]

  8. Click on Rialto again. Use one of the 4 arrows to fine tune the location of the point. The readout in the upper right corner should be x: 196 y: 180

  9. Select Mark Point [Result of this action: a third line will appear]

  10. Click in upper-left area of the map. (For example: anywhere northwest of Interstate 15.)

  11. Observe the triangle that has been drawn on your map.

  12. This is a right triangle. The line segment connecting the locations of the two cities forms the hypotenuse of the right triangle.

  13. What are the coordinates (x, y) of the endpoints of the line segment that is the hypotenuse of the right triangle?

NOTE: Reinforce the importance of the straight line segments. Remind students that the side of the triangle parallel to the x axis and the side of the triangle parallel to the y axis should be straight! In order to accomplish this, attention needs to be given to the x= and y= readout in the upper right corner of the window.

Practice: Ask students to "construct" two other triangles using the following pairs of cities:

Fontana and San Bernardino
Muscoy and Loma Linda


Ask students to generalize this method and write an explanation of the steps they would take to make a right triangle with any two cities as the end points of the hypotenuse.

Part II:

Students will apply what they have know about right triangles and finding the hypotenuse to calculating the distance between two cities on a map.

Go to Part II


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