Education Technology

Sailing Through the Fog: Solving Systems of Linear Equations Graphically

Activity Overview

Tracking equipment is used to help ship navigators traveling in dense fog determine if they are in danger of colliding with other ships that may be in the area. For this activity, we are going to assume two ships are traveling along intersecting paths. We will develop linear equations that represent the path of each ship and then find the intersection point graphically.

Before the Activity

If you are using the TI-Navigator System with the TI-73 you will need the attached .pdf file.

  • Students should be familiar with using the TI-Navigator System, particularly with regards to contributing points, lines, and equations in the Activity Center.

  • Students should have a thorough understanding of the coordinate plane.

  • Teacher may want to review finding the slope and developing a linear equation given the slope and a point.

  • The teacher should change the window of the coordinate plane of the activity center in TI-Navigator such that each unit represents ten miles.
  • During the Activity

    1. Coordinate plane is to be considered a map, with each unit representing ten miles. Students are to consider movement in the positive vertical direction as moving north, movement in the positive horizontal direction as moving east, and so on.
    2. Using the TI-Navigator system, the teacher should place two points anywhere on the coordinate plane in the activity center. These points should be sent to the students' calculators.
    3. The teacher then should indicate how the ship moves each hour (i.e., in one hour, one ship moves 20 miles north and 30 miles east while the other ship moves 30 miles south and 35 miles west). The teacher should indicate that this movement represents the slope of the line that each ship will follow.
    4. Without performing any calculations, students should contribute the point where they believe the paths of the ships will intersect.
    5. The class should be divided into two groups. One group will find the linear function that represents the path of one ship, while the other group will find the linear function that represents the path of the other ship. All students should contribute their equations/lines to the activity center. 6. A quick poll should be used to determine if the majority of the class believes the ships will collide or not. The point of intersection then can be found by finding where the two paths intersect. Finally, it can then be determined if the ships will collide or not.
    7. Other examples should be used, including one in which the ships following parallel paths.

    After the Activity

    This activity could be extended so that students are asked to find how far apart the two ships are at certain intervals using the distance formula.