Coordinates in 11-Plus Mathematics

Understanding Coordinates

Coordinates are a fundamental concept in mathematics, particularly in geometry. They allow us to identify specific points on a grid using a pair of numerical values. In this section, we will explore how to read and plot coordinates in all four quadrants of the Cartesian plane.

The Cartesian Plane

The Cartesian plane is divided into four quadrants:

• Quadrant I: Both x and y coordinates are positive (e.g., (3, 2)).
• Quadrant II: x coordinates are negative, and y coordinates are positive (e.g., (-3, 2)).
• Quadrant III: Both x and y coordinates are negative (e.g., (-3, -2)).
• Quadrant IV: x coordinates are positive, and y coordinates are negative (e.g., (3, -2)).

Reading Coordinates

Coordinates are written in the form (x, y), where:

• x: Represents the horizontal position (left or right).
• y: Represents the vertical position (up or down).

To read coordinates, locate the x value on the horizontal axis and the y value on the vertical axis. The point where these two values intersect is the location of the coordinate on the grid.

Plotting Coordinates

To plot a coordinate on a grid:

1. Start at the origin (0, 0).
2. Move along the x-axis to the x-coordinate.
3. From that point, move vertically to the y-coordinate.
4. Mark the point where you land.

Identifying Positions and Shapes

Coordinates can be used to identify positions and shapes on a grid. For example, a rectangle can be defined by its corner coordinates:

Example: Plotting a Rectangle

Consider a rectangle with corners at (1, 1), (1, 4), (5, 1), and (5, 4).

Steps:

1. Plot the points (1, 1) and (1, 4) on the grid.
2. Plot the points (5, 1) and (5, 4).
3. Connect the points to form the rectangle.

Finding Midpoints

The midpoint of a line segment can be found using the formula:

Midpoint M = ((x1 + x2)/2, (y1 + y2)/2)

This formula averages the x-coordinates and the y-coordinates of the endpoints.

Example: Finding the Midpoint

Find the midpoint between the points (2, 3) and (4, 7).

Solution:

• Using the formula: M = ((2 + 4)/2, (3 + 7)/2)
• M = (6/2, 10/2) = (3, 5)

Problem-Solving with Coordinates

Coordinates can also be used to solve various mathematical problems. For instance, you might be asked to determine the distance between two points or to identify whether a point lies within a certain shape.

Understanding how to read, plot, and manipulate coordinates is essential for success in the 11-plus mathematics exam. Practice these skills regularly to enhance your proficiency.