Introducing Cartesian System

“Learning is the relative change in behaviour of an individual due to experience or training.” -Baron

No learning is complete unless it finds some application in the life of an individual at some point of time or the other.  When there is a power cut suddenly in our house, it is not difficult for us to find the place where we can find the candle, matchbox etc. Similarly identifying a particular child becomes easy if we know the school, class and the section in which he/she studies. Mathematics plays a major role in developing this kind of reference-based learning and it is the concept of Cartesian System that helps us develop this kind of thinking. Here is a way in which this concept can be introduced in the classroom.

Creating the situation
The teacher takes four post cards to the class and shows each one of them to the students.

Teacher: Which of these four would reach to Rohit?
Students: The fourth one.
Teacher: Why?
Students: Since the address is complete.
Teacher: Yes because it is easy to locate Rohit out of the so many Rohits, only if we know the exact address.

Leading to the concept- arousing curiosity
In Geometry the surface of the black board is an example of a plane. We find infinite points in this plane. Out of these infinite points, how do we locate the desired point? It is possible only if we know its address. Then what would be the address of a point? The answer for this question lies in the concept of Cartesian System [1].

Development of the concept
With the help of two perpendicular lines, the entire plane is divided into four parts and each part is called as a QUADRANT.

These two mutually perpendicular lines are called the reference lines. The horizontal reference line is called the X-axis and the vertical reference line is called the Y-axis. Their point of intersection is called the origin denoted by O.

In a Cartesian system, the quadrants are always marked in the anti-clockwise direction. This kind of grouping separates the infinite set of points into four groups.
Points in each quadrant have their own identity or address. This identity of a point is represented by an ordered pair.

What is an ordered pair?
A pair of numbers separated by comma and enclosed in a bracket say (2, 3) constitutes an ordered pair. Every ordered pair has two numbers in it called the coordinates. The first one is called the x-coordinate or abscissa and the second one is called the y-coordinate or the ordinate. It is conventional to write the x-coordinate first and then the y-coordinate.

How to plot a point in the Cartesian system?
Consider the point (2,3). In order to identify this point, we always start from the origin. Here the x-coordinate is 2. So move 2 steps along x-axis in the positive direction.
Then the y-coordinate is 3. From 2 on the x-axis move three steps upwards along the line parallel to y-axis. That gives us the position of the point (2,3).

Once the child becomes familiar with the plotting of a point in the Cartesian plane, realization that the point (2,3) and (3,2) are not the same is quite important.
Note: (a,b) = (b,a) only if a = b.

Extended Learning
Any point on the x-axis will have its y-coordinate as 0.
Any point on the y-axis will have its x-coordinate as 0.

Historical Note
This Cartesian System used for describing the position of a point in a plane is named in the honour of the great French Mathematician Rene Descartes of the 17th century.

Special Features
This branch of mathematics integrates the application of algebraic principles to solve the geometry problems and vice versa. It is very much used in fields such as navigation, architecture etc.,

Footnotes:
1. This is only one answer, there are other systems such as Polar Coordinates to give the location of a point in space

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