3.+Big+Ideas+About+Taxicab+Geometry

=Why should students learn Taxicab Geometry? =

One article suggests:

"The general aim of geometry education can be described as having students learn about the environment, and use geometry in the problem solving process. Does Euclidean geometry, alone, help students understand their environment? Euclidean geometry, with its theoretical knowledge, fails to make connections and integrations in real life. For example, considering that roads are generally horizontal and vertical, a taxi driver may not always use Euclidean distance in real life. Therefore, it could be suggested that Taxicab geometry provides students with more opportunities than Euclidean geometry does in terms of making meaning out of the real world. (Tuba and Aytaç, p.109)."

Euclidean geometry does not always help students to correctly visualize a realistic environment. That is why other non-Euclidean types of geometry were created. Taxicab allows students to do math in a way that suggests a more realistic idea of what they are trying to conclude.

One way that Euclidean geometry fails in everyday life is district zones for schools in cities like New York City. Urland originally studied a math equation talking about how to find the zones for each school so that each student attends the school closest to their house. He actually improved the original answer to the problem and found the correct zones for each New York City school. Here is a figure of the original problem: Then, Urland, through a series of equations of taxicab geometry, found a way to make the zones so a student would not go to a school further from their house. Thus, each point on the line of the zones is equidistant from the other school. This is applicaple to students because they can judge if their own school zone is based off of taxicab geometry and why.

Overall in the student's learning of this subject, they will learn differently based on their grade level. Here are two ways that students will learn about this topic.

To learn more about how students construct taxicab shapes, click here.


 * Learning Taxicab Geometry for Younger Students (4th grade to 6th grade) **

Taxicab geometry takes a more realistic approach in geometry. In typical Euclidean geometry, the shortest distance between two points is a straight line. In theory, this method works perfectly. However, in real life applications, does this method always work? This is where taxicab geometry comes to play in math. The basic premise of taxicab geometry is that the shortest distance between two points is not always a straight line.

The easiest way for students to understand this concept is to relate being in a big city where you are walking down city blocks. Whenever you are in a city where the blocks resemble a grid, what would be the shortest distance? The students cannot fly a straight line to their destination. Also, students cannot walk through buildings. Thus, the students have to use taxicab geometry to figure out the shortest distance.



In the image above, say the taxi is at the coordinates of (0,2) and McDonalds is at (2,0). What would be the shortest distance driving? There are a couple of ways the taxi driver can get to McDonalds with the same distance. The driver can drive past the bank and take a right on the street where McDonalds is located and would drive four blocks total. The driver can also take an immediate right and drive two blocks and then take a left on the road with the library and McDonalds. This trip too consists of four blocks travelled in total. Thus, can there be an equation for taxicab geometry?

**Learning Taxicab Geometry for Older Students (6th grade and up)**

<span style="background-color: #ffffff; font-family: 'Times New Roman',Times,serif;">Euclidean Geometry would state that the distance from point A to point C on the following graph <span style="font-family: 'Times New Roman',Times,serif;"> <span style="background-color: #ffffff; font-family: 'Times New Roman',Times,serif;">could be solved with the following formula: <span style="font-family: 'Times New Roman',Times,serif;">

<span style="background-color: #ffffff; font-family: 'Times New Roman',Times,serif;">This formula would suggest that a diagonal line would be used to find the distance between the two points. <span style="background-color: #ffffff; font-family: 'Times New Roman',Times,serif;">The distance according to this formula would be: <span style="background-color: #ffffff; font-family: 'Times New Roman',Times,serif;">d= square root [ (3-0)^2 + (4-0)^2] <span style="background-color: #ffffff; font-family: 'Times New Roman',Times,serif;">d= square root [25] <span style="background-color: #ffffff; font-family: 'Times New Roman',Times,serif;">d= 5

<span style="background-color: #ffffff; font-family: 'Times New Roman',Times,serif;">But when we begin to think of the practicality of this formula, we find a flaw in it. This flaw is highlighted when we think of the grid lines above as streets in a city. Ideally, every square would have some sort of building in it and the lines would represent different streets. Each square would represent one block in the city. This is where Taxicab Geometry comes into play.

<span style="background-color: #ffffff; font-family: 'Times New Roman',Times,serif;">If we imagine a Taxi driver making his/her way through the city, they would want to find the distance from one point to another. If they used the Euclidean formula, that distance would suggest that they drive through buildings to get from point A to point C. This is unrealistic. Taxicab Geometry thus suggests that in order to find distance, we must only think in terms of horizontal and vertical distance. <span style="background-color: #ffffff; font-family: 'Times New Roman',Times,serif;">The formula for this is: <span style="font-family: 'Times New Roman',Times,serif;">

<span style="background-color: #ffffff; font-family: 'Times New Roman',Times,serif;">So, for the above graph, this distance would be: <span style="background-color: #ffffff; font-family: 'Times New Roman',Times,serif;">d= absolute value [3-0] + absolute value [4-0] <span style="background-color: #ffffff; font-family: 'Times New Roman',Times,serif;">d=7

<span style="background-color: #ffffff; font-family: 'Times New Roman',Times,serif;">While this is a further distance, it is a more accurate account of the distance between the two points.

<span style="background-color: #ffffff; font-family: 'Times New Roman',Times,serif;">In summary: <span style="background-color: #ffffff; font-family: 'Times New Roman',Times,serif;">Taxicab geometry takes a more realistic approach in geometry. In typical Euclidean geometry, the shortest distance between two points is a straight line. In theory, this method works perfectly. However, in real life applications, does this method always work? This is where taxicab geometry comes to play in math. The basic premise of taxicab geometry is that the shortest distance between two points is not always a straight line.