Lesson Name: Taxi-cab Metric A
Purpose: Introduce students to other distance functions within the context of how they experience reality.
Outline: Discuss the difference between distance and travel-distance and why sometimes travel-distance is more useful. Have students practice finding distances between points on “city” grid using travel-distance: start with a couple of examples, then ask them to do it on the computer. Have them outline their paths with colored segments and calculate the total distance. Then ask them to find a formula. Once they have found the formula explain the general concept of a “metric.”
Goals: Students will understand the basic concept of a metric. Students will be able to use the taxi-cab metric to calculate distances. Students will develop the equation for the taxi-cab metric. They will also understand he basic concept of a metric and its properties.
Implementation: Begin with a motivation. Pick a student for which there is an obstacle between you and them. Ask the student the distance between you and the student. Student will most likely respond with the actual Euclidean distance of the length of a straight line between you and the student. Then ask the student if that is the distance you would have to travel to reach the student (because of the obstacle the answer should be no). Then ask the student what distance you would have to travel in order to reach the student.
Two more examples: mapquest.com directions, and traveling around the world.
With this motivation move tot the square grid in Regular_city2.gsp. Taking the solid square to be cities, pick two point on opposite sides of one of the squares. Ask them to calculate the distance. Then ask them if this is the distance a taxi would have to travel to get from one point to the other. Using this, ask students to practice, taking two points, and finding travel distance between them. (Another full example may be given at this point for instruction.) Ask them to first pick a couple points and find any path between the two and to calculate the distance. Next, ask them to pick two points and find several paths between these two points and to calculate the distances and then compare them.
After practice the goal will be for the students to come up with an equation for the distance between any two points using travel distance. The name “taxi-cab metric” and the term metric can now be introduced. Introduction to properties of a metric can now be explained. d(a,a)=0, d(a,b)=d(b,a), d(a,b)>=0, d(a,b)+d(b,c)>=d(a,c).
See also Taxi-cab Metric B - Circle