Introduction to the Rectangular Coordinate System and Ordered Pairs

The rectangular coordinate system is formed by the intersection of two axes at right angles.
The horizontal axis is identified as the x-axis.
The vertical axis is identified as the y-axis.
The point where the x-axis and y-axis intersect is called the origin.
The intersection of these two axes divides the system into four distinct regions known as quadrants.
The quadrants are numbered I, II, III, and IV in a counterclockwise direction starting from the upper right region:
- Quadrant I: Upper right area.
- Quadrant II: Upper left area.
- Quadrant III: Lower left area.
- Quadrant IV: Lower right area.

Every point in the rectangular coordinate system is represented as a dot and corresponds to an ordered pair of real numbers.
The notation for an ordered pair is $(x, y)$ .
In this pairing, the order of the numbers is essential, which is why it is strictly referred to as an "ordered pair."
The x-coordinate (the first number in the pair) denotes the horizontal distance and direction from the origin.
The y-coordinate (the second number in the pair) denotes the vertical distance and direction from the origin.

The mathematical signs of the $x$ and $y$ coordinates vary depending on which quadrant the point is located in:
- Quadrant I: Both coordinates are positive. To plot a point here, move right (positive $x$ ) and up (positive $y$ ). The signs are $(+, +)$ .
- Quadrant II: The $x$ coordinate is negative and the $y$ coordinate is positive. To plot a point here, move left (negative $x$ ) and up (positive $y$ ). The signs are $(-, +)$ .
- Quadrant III: Both coordinates are negative. To plot a point here, move left (negative $x$ ) and down (negative $y$ ). The signs are $(-, -)$ .
- Quadrant IV: The $x$ coordinate is positive and the $y$ coordinate is negative. To plot a point here, move right (positive $x$ ) and down (negative $y$ ). The signs are $(+, -)$ .

Step 1: Always begin the plotting process at the origin $(0, 0)$ .
Step 2: Move the horizontal distance specified by the $x$ coordinate. Move right if the value is positive and left if the value is negative.
Step 3: From that new horizontal position, move the vertical distance specified by the $y$ coordinate. Move up if the value is positive and down if the value is negative.
Step 4: Once the distances have been traversed, place a dot to represent the point.

Scenario: Plotting Examples
- Participants were asked to plot points in the rectangular coordinate system and state which quadrant or axis each point is located on.
- Point A: $(2, 6)$ . The student plotted this and correctly identified it as being in Quadrant I (Q1) because they moved right and up.
- Point B: $(-4, 2)$ . The student plotted this as being in Quadrant II (Q2) because it involves moving left 4 units and up 2 units.
- Point C: $(0, -5)$ . This point is located on an axis rather than in a quadrant. Because the $x$ value is zero, the point stays on the vertical axis (y-axis).
- Point D: $(-\frac{5}{2}, -\frac{7}{2})$ . These are fractional coordinates requiring movement in the negative horizontal and negative vertical directions. This point is located in Quadrant III.
- Point E: $(4, 0)$ . This point is located on the horizontal axis (x-axis) because the vertical movement ( $y$ coordinate) is zero.
Classroom Management and Software Interaction:
- Future graphing examples will incorporate exercises on the Pearson platform.
- During the plotting activity, volunteers were encouraged to use different colors and to label their points (e.g., labeling the point "A" and writing "Q1" next to it).