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# Ordered Pairs

## Introduction

A pair of items that have a specific significance for the order of their placements is called an ordered pair. Ordered pairs are typically used to represent a point on a coordinate plane in coordinate geometry. They can also be used to indicate relationships between items.

You’ve probably already encountered ordered pairs without recognizing them if you have used a video game or consulted a map.

Let us explore ordered pairs further, including their definition, significance, attributes, and more.

Understanding ordered pairs is generally introduced to students around the 4th grade. But don’t worry if you’re younger or older; the beauty of learning is that it has no age limit. You are in the right place whether you’re 8, 15, or even a little beyond!

## Math Domain

Ordered pairs belong to the domain of mathematics called “Coordinate Geometry.” Coordinate Geometry is like a bridge combining algebra and geometry to help us understand shapes, sizes, and positions.

## Applicable Common Core Standards

CCSS.MATH.CONTENT.5.G.A.1: Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates.

CCSS.MATH.CONTENT.6.NS.C.6.C: Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

## Definition of the Topic

An ordered pair is a set of two numbers or objects in which the order of the elements matters. It is often written in parentheses (like this: (4,5)). The first number is the x-coordinate, and the second is the y-coordinate.

## Key Concepts

Ordered Pair: An ordered pair (x, y) consists of two numbers: the x-coordinate and the y-coordinate.

Coordinate Plane: A plane spanned by two perpendicular lines called axes. The horizontal line is called the x-axis, and the vertical line is the y-axis. The point where these two lines intersect is called the origin, represented by the ordered pair (0,0).

Plotting Ordered Pairs: Ordered pairs are used to denote a point’s location on the coordinate plane. The x-coordinate denotes the horizontal position (left or right), and the y-coordinate denotes the vertical position (up or down) from the origin.

## Discussion with Illustrative Examples

Ordered pairs are composed of the x-coordinate and y-coordinate written inside the parentheses and separated by a comma. The x-coordinate is the horizontal location in the cartesian plane, called the abscissa. The y-coordinate is the vertical location in the cartesian plane, called the ordinate.

Let us imagine you’re playing a treasure hunt game. You get a hint: “Walk three steps to the right and two steps forward.” In our coordinated world, that is like saying the treasure is at the point (3,2). Here, 3 is the x-coordinate, and 2 is the y-coordinate.

When we plot this on the coordinate plane, we start at the origin (0,0), move three steps to the right along the x-axis, and then two steps upwards along the y-axis. That’s where we find our treasure!

Quadrants are quarter areas inside the coordinate plane. It can be identified by numbers and is divided into half axes. The center of the cartesian plane is called the origin and has an ordered pair of (0, 0). A cartesian plane consists of four quadrants, as shown in the figure below.

You can determine the quadrant of an ordered pair or point by checking if the coordinates are negative or positive.

### Steps to Graph Ordered Pairs

The following are the steps to follow when graphing ordered pairs:

Step 1: Start from the origin (0,0) and move horizontally x units to the right if the first coordinate is positive and to the left if it is negative.

Step 2:  From where you have stopped, move y units vertically upward if positive and downward if negative.

Step 3: Place a dot when you have stopped to represent the ordered pair.

## Examples with solutions

Example 1

Plot the point (2,4) on the coordinate plane.

Solution

Start at the origin (0,0), then move two units to the right along the x-axis and four upwards along the y-axis. Mark the point.

Example 2

Plot the ordered pair (-3,-1) on the coordinate plane.

Solution

Start at the origin (0,0), then move three units to the left along the x-axis and 1 unit downward along the y-axis. Label the point.

Example 3

What is the ordered pair for a point with five units left of the origin and three units below?

Solution

The x-coordinate is -5 (to the left of the origin), and the y-coordinate is -3 (below the y-axis). So, the ordered pair is (-5,-3).

Example 4

Identify the quadrant where each of the given ordered pair is located.

a. (1,4)

b. (-2,6)

c. (7,-4)

d. (-8,-3)

Solution

a. Since the x and y-coordinates are positive, (1,4) is in Quadrant I.

b. The ordered pair (-2,6) is in Quadrant II since the x-coordinate is negative while the y-coordinate is positive.

c. The ordered pair (7,-4) is in Quadrant IV since the x-coordinate is positive while the y-coordinate is negative.

d. Since both the x and y-coordinates are negative, (-8,-3) is in Quadrant III

## Real-life Application with Solution

Your school is planning a picnic at a park. The park is divided into a grid for different activities. The entrance is considered the origin (0,0). The playground is located four units to the right and three units up from the entrance.

In terms of an ordered pair, where is the playground located?

Solution

The playground is at the point (4,3).

## Practice Test

1. Plot these points on a coordinate plane: (3,2), (-1,4), (0,-3), (-4,-2), and (5,0).

2. What is the ordered pair for a point with two units right of the origin and three units below?

3. You’re playing a game. You move three spaces to the left and then four spaces up. What is your position as an ordered pair?

4. What ordered pair represents the origin?

5. To get to the point (2, 2), how would you move from the origin?

1.

2. The ordered pair is 2,-3.

3. You are located at -3,4.

4. The ordered pair that represents the origin is 0,0.

5. To get to the ordered pair (2,2), move two units to the right of the x-axis, then two units upward along the y-axis.

### What happens if we switch the order of the numbers in an ordered pair?

Switching the order of the numbers will denote a different point in the coordinate plane.

### Can the ordered pair be (0,0)?

Yes, that is called the origin. It is where the x-axis and the y-axis intersect.

### Do ordered pairs have negative numbers?

Absolutely! Negative numbers indicate a location left of the origin for the x-coordinate or below the origin for the y-coordinate.

### Are the coordinates always whole numbers?

No, they can be fractions or even irrational numbers.

### What is the use of ordered pairs in real life?

Ordered pairs are used in navigation, computer graphics, architecture, and more!

Remember, like in a fun game, every point in the world of ordered pairs is an adventure.

With your new knowledge, you are ready to explore the world of ordered pairs. Happy learning!

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