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7. The Coordinate Plane
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Pre-Algebra /
Chapter 1
7. 

The Coordinate Plane

If one is an aspiring athlete like Dominika, the coordinate plane can be used to improve performance in sports. The coordinate plane is a two-dimensional grid created by the intersection of a vertical and a horizontal number line. It is divided into four areas known as quadrants. Points on this plane are represented by ordered pairs, where the first number indicates the distance along the horizontal axis and the second number indicates the distance along the vertical axis. These ordered pairs can mark the locations of homes, friends' homes, and sports facilities. The concept of reflection can also be explored, which involves flipping a point across an axis to produce its mirror image. This concept can help identify advantageous positions on a sports field or court. The coordinate plane is not merely a mathematical abstraction; it is a practical tool that can be applied in various real-life situations such as sports strategy, navigation, and design.
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Lesson Settings & Tools
13 Theory slides
13 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
Image Credits expand_more
The Coordinate Plane
Slide of 13
A number line helps visualize one-dimensional quantities like temperature and time. However, there are situations where two dimensions are needed, such as when latitude and longitude when locating places on a map. The coordinate plane is used in these situations! This lesson will teach how to locate points on a coordinate plane and explore some of its uses.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

The Battleship Game Board

Get ready for an adventure! Discover the location of enemy ships hidden on the game board by using specific points. Each point on the board is represented by the combination of a letter and a number. Place the boats by dragging them from the stern, or the back, and rotate them by clicking on the bow, or the front.
The Battleship Game Board
External credits: Freepik
Consider the following questions.
  • How does using the letter and number combination to locate enemy ships in the game relate to finding a location on a map in real life?
  • What are other situations where two numbers, two letters, or a combination of both help to indicate a location?
Discussion

Coordinate System

A coordinate system is a reference framework used to describe the positions of objects like points, lines, and surfaces in a space. A fixed point called the origin is used as a reference in the coordinate system. The most common types of coordinate systems are one-, two-, or three-dimensional.

Coordinate Plane

A coordinate plane is a two-dimensional coordinate system. It is a grid that results from intersecting a vertical number line with a horizontal number line at their zero points. The horizontal number line is usually named the axis and the vertical number line is usually the axis.

A figure showing that a coordinate plane has two perpendicular axes: the horizontal x-axis and the vertical y-axis. They intersect at the origin.
The positive numbers on the axis are above zero and the negative ones are below zero. The origin is where the lines intersect, which is the point
Discussion

Coordinate

A coordinate is the position of an object in a coordinate system relative to the corresponding axis. Coordinates are often seen together with other coordinates, which can describe the position of an object in a coordinate system with one, two, three, or even more dimensions.

In Two Dimensions

In a two-dimensional coordinate system, points are usually expressed as a coordinate pair — also called an ordered pair — denoted by The first coordinate states the position along the axis and the second coordinate states the position along the axis.

Point on the 2D coordinate plane
Pop Quiz

Random Points on a Coordinate System

Identify the coordinates of the given point. Write the coordinates as an ordered pair where represents the coordinate and the coordinate.

Random point on a coordinate plane.
Example

The Path From Home to the Basketball Court

Dominika plays basketball with her friends every afternoon. She warms up by running and picks up two friends, Jordan and Emily, on the way to the court. Dominika uses ordered pairs to represent the positions of her and her friends' houses.

Position
Dominika's House
Jordan's House
Emily's House
a Dominika wants to display this information on a coordinate plane. Which graph correctly displays the positions of the three houses?
Four coordinate planes
b The basketball court is at the point , and Dominika's whole path from her house to the court and back can be represented by joining points D, J, E, C, and D. Which graph shows Dominika's whole path?
Four coordinate planes

Hint
a An ordered pair can be graphed on a coordinate plane by moving horizontally from the origin the number of units specified in its coordinate and then vertically the number of units specified in its coordinate.
b Plot the coordinates of the basketball court. Connect the points using line segments in the given order.

Solution
a The positions of the girls' houses are given as ordered pairs. Recall that an ordered pair can be graphed on a coordinate plane by moving horizontally from the origin the number of units specified in its coordinate and vertically the number of units specified in its coordinate.
How to graph an ordered pair.
With this information in mind, consider the position of Dominika's house.
The coordinate is and the coordinate is Move units to the right of the origin and then unit up to graph Dominika's house on the coordinate plane.
Dominika's house position on a coordinate plane.
Now, consider the coordinates of Jordan's and Emily's houses.
On the coordinate plane, move units horizontally to the right of the origin and then units up to graph the position of Jordan's house. To plot Emily's house, start at the origin again. Move unit to the left and units up to graph Emily's house.
Girl's house positions.
Notice that this corresponds to graph III.
b Begin by considering the coordinates of the position of the basketball court.
Add this point to the graph from the previous part. Move unit horizontally to the left of the origin and then unit up.
Point F added to the graph of Girl's house positions.
The points can now be joined according to the given order D, J, E, F, and D. Draw a line starting from point D to point J. Next, from point J, go to point E, and so on until point C is connected to point D.
A square drawn from the points representing the field and the girl's houses positions.
This graph matches the one given in option I.
Example

Dominika's Running Routine

Dominika's house is represented by the point on a coordinate plane. Every morning, she runs to a place five blocks away from her house, then returns home using the same route.
Dominika's location on a coordinate plane.
If Dominika picks a different spot to visit every day, which graph displays all the places she could go? Consider that every unit on the coordinate plane represents one block and that Dominika stays on the straight roads.
Join all the potential places Dominika could go. What shape do these points form?

Hint

Moving one unit on the axis or axis on the coordinate plane is counted as moving one block. Since Dominika can only stay on the straight roads, her position can only move up, down, left, and right on the graph.

Solution

Moving one unit on the axis or axis on the coordinate plane is the same as Dominika moving one block. Dominika only runs on the straight roads, so her progress on the graph can only move in four directions — up, down, left, and right.

Allowed movements

Keeping this in mind, one route she can take is moving five blocks directly to the right. Move units to the right starting her house to show this possible destination. Remember that Dominika's house is located at not the origin!

Showing the point (7,1) on a coordinate plane as a possible resting point.

Another spot Dominika might go is one block up and four blocks to the right of her starting point. It does not matter if she moves up on her first, second, third, fourth, or final move. Her destination will always be the same if she moves one block up out of the five total moves.

Showing (6,1) as a possible resting point on a coordinate plane.

Follow the same reasoning to find the other places that Dominika can go. Plot each as ordered pairs on the coordinate plane.

All the ordered pairs graphed on a coordinate plane.

Now that all the points are plotted on the graph, connect them with lines to see the overall shape.

All ordered pairs connected with lines to form a square.

The shape is a rotated square.

Showing Our Work

 Finding All the Places Dominika Can Go

The places Dominika can go can be determined by identifying all the possible routes she can take. Remember, she can only go five blocks away from her home. Moving left or down indicates a negative direction on the corresponding axis, represented by negative numbers.

Route from Home Calculations Destination
blocks to the right
blocks to the right and block up
blocks to the right and blocks up
blocks to the right and blocks up
block to the right and blocks up
blocks up
block to the left and blocks up
blocks to the left and blocks up
blocks to the left and blocks up
blocks to the left and block up
blocks to the left
blocks to the left and block down
blocks to the left and blocks down
blocks to the left and blocks down
block to the left and blocks down
blocks down
block to the right and blocks down
blocks to the right and blocks down
blocks to the right and blocks down
blocks to the right and block down
Discussion

Quadrants

In a coordinate plane, the intersection of the axis and the axis produces four regions called quadrants. The quadrants are numbered counterclockwise from the top right quadrant as Quadrant I to Quadrant IV in the bottom right.

Quadrants labeled on a coordinate plane.

The signs of the coordinates of a point can be determined based on which quadrant the point lies in.

Quadrants labeled on a coordinate plane.
Conversely, the quadrant of a point can be found by looking at the signs of its coordinates.
Points on an axis do not belong to any quadrant.
Pop Quiz

Finding the Quadrant in Which a Point Lies

A coordinate system has four quadrants that provide information about the signs of the and coordinates of a point. In the following applet, identify in which quadrant the given point lies.

Example

Dominika is drafting a logo for her basketball team to wear on their jerseys. She wants the logo to have a cool geometric shape, so she sets some points that she thinks will look good. Help her design the logo by matching each point with its corresponding quadrant on the coordinate plane.

Hint

Look at the signs of the coordinates of each point to identify their quadrants.

Solution
The quadrant in which a point lies can be determined by looking at the signs of its coordinates. For instance, a point in Quadrant I has a positive coordinate and a positive coordinate. Consider the given points and see if any of them fit this description.
The point is located in Quadrant I because both of its coordinates are positive. In contrast, a point in Quadrant II has a negative coordinate and a positive coordinate. Look at the remaining points to see which one fits this description.
The point is in Quadrant II because its coordinate is negative but its coordinate is positive. Follow a similar reasoning to find the corresponding quadrants of the remaining points.
Point Signs of Coordinates Quadrant
Quadrant III
Quadrant IV
Quadrant II
Quadrant I
Now that the quadrant of each point has been identified, plot the points on a coordinate plane. Move horizontally from the origin the number of units indicated by the coordinate and vertically the number of units specified by the coordinate of each point.
Points plotted on a coordinate plane.
External credits: Image by Freepik
The logo is a hit, and the team gains more and more fans as a result!
Discussion

Reflecting a Point Across an Axis on the Coordinate Plane

Reflecting a point means making a mirror image of that point by flipping it across a certain line or axis. The following rules show how to find reflected points across a specific axis.

Point Axis of Reflection Procedure Reflected Point
Change the coordinate to its opposite.
Change the coordinate to its opposite.
and Change the and coordinates to their opposites.

Next, the process for reflecting points across the axis, the axis, or both, on a coordinate plane will be shown. Consider the points and

Point Axis of Reflection Reflected Point
 ?
 ?
and  ?
There are two steps to follow to reflect a point.
1
Find the Reflected Point
expand_more
Keep the same coordinate of a point and change the coordinate to its opposite to reflect it across the axis. With this information, point can be reflected across the axis by changing its coordinate to its opposite,
In contrast, change the coordinate of a point to its opposite and keep its coordinate to reflect it across the axis. Point can be reflected across the axis by changing the coordinate to its opposite,
Finally, change the coordinate and the coordinate of a point to their opposites to reflect it across both the axis and axis. Therefore, reflect point across both axes by changing its coordinate from to and its coordinate from to


2
Plot the Point and Its Reflection on the Coordinate Plane
expand_more

Once each initial point and its reflection across a given axis are found, plot both points on the coordinate plane.

Each point and its reflection plotted on a coordinate plane.
Example

Improving Basketball Skills With Math

Dominika's team is only one game away from winning the basketball tournament! Dominika is a shooting guard and wants to use her math skills to improve her game. She starts by figuring out the point on the court from where she usually scores using a coordinate plane. This point is

Basketball court on a coordinate plane.
a Dominika knows that if she mirrors that point across the horizontal central line of the court, she can score from there, too. Find this mirror point.
b When Dominika is defending, she needs to be quick on her feet to get the rebounds. To do that, she needs to know the point that is the mirror image of across the vertical central line of the court. What is this point?

Hint
a Reflect the point across the axis. The coordinate remains the same and the coordinate is changed to its opposite.
b Reflect the point across the axis. The coordinate remains the same and the coordinate is changed to its opposite.

Solution
a Reflect the point across the axis to find the mirror point of Keep the coordinate the same and change the coordinate to its opposite. The coordinate is The opposite of the coordinate is
Graph this point on the coordinate plane. Start at the origin and move units to the left and units up.
The mirror shooting point on a coordinate plane.

Dominika is interested in the mirror point of because it is a great shooting position. This point is as far away from the basket as the original point but is oriented in a different direction. This can give her an advantage over her opponents and help her score more points during game!

b Reflect the point across the axis to find the defense point. Keep the coordinate and change the coordinate to its opposite. In this case, the opposite of is
Add this point to the graph.
The defending point graphed on a coordinate plane.

Now that Dominika understands her attack and defense positions better, she can perform much better on the court. Her team is doing great and they keep scoring point after point until before they know it, they win first place in the tournament. They are the champions!

Closure

Introducing the Three-Dimensional Coordinate System

This lesson showed how to locate points with respect to the and axis on a coordinate plane. A third axis, usually called the axis, can be used to provide more information about the position of objects. Think of it like this.

Description
axis Tells if something is in front of or behind the origin.
axis Tells if something is to the left or right of the origin
axis Tells if something is above or below the origin.

These three axes create a coordinate system that is similar to the coordinate plane but with the addition of an extra dimension. This additional dimension means that points are represented as ordered triples instead of as ordered pairs, as in the coordinate plane.

3D coordinate system



Level 1
Level 2
Level 3
The Coordinate Plane
Exercises
The point is in the second quadrant. Without graphing, use this information to match each point with its corresponding quadrant.

Let's figure out which reflections were used on the point with coordinates ( a, b) to get to the given points. This could help us determine the quadrant to which each option belongs.

Point (a,- b)

Let's look at point ( a, - b). Point&Reflected Point ( a, b)&( a, - b) Notice how these two points share the same x-coordinate but have opposite y-coordinates. Remember, when we keep the x-coordinate the same and change the y-coordinate to its opposite, we are reflecting the point across the x-axis. Point&Reflection Acrossx-axis ( a, b)&( a, - b) If the starting point is in Quadrant II, its reflection across the x-axis ( a, - b) is in the quadrant below, which is Quadrant III.

Point (- a,b)

Now let's find the quadrant of point ( - a, b). Point&Reflected Point ( a, b)&( - a, b) These two points share the same y-coordinate but have opposite x-coordinates. Remember that if we change the x-coordinate to its opposite while keeping the y-coordinate the same, the point will be reflected across the y-axis. Point&Reflection Acrossy-axis ( a, b)&( - a, b) Since the initial point is in Quadrant II, its reflection across the y-axis ( - a, b) is in Quadrant I.

Point (- a,- b)

Finally, let's look at the point ( - a, - b). Point&Reflected Point ( a, b)&( - a, - b) Here, we can observe that the reflected point has both coordinates opposite to the initial coordinates. This implies that point ( a, b) is reflected across both the x- and y-axes. Point&Reflection Acrossx- andy-axes ( a, b)&( - a, b) Let's apply one reflection at a time. When reflecting the point across the x-axis, the point falls in Quadrant III. After that, when reflecting the new point across the y-axis, the point lands in Quadrant IV. This means that the point ( -a, -b) belongs to Quadrant IV.

Conclusion

Now that we have determined the quadrant to which each point belongs, let's organize our information.

Ordered Pair Quadrant
(a,- b) Quadrant III
(- a, b) Quadrant I
(- a,- b) Quadrant IV

Magdalena is sketching a rectangle on a coordinate plane and has already drawn three of its corners.

Three vertices of a rectangle on a coordinate plane.
 What are the coordinates of the missing corner?
What is the area of the rectangle?

Let's look at the given diagram.

The point (-4, -2) in Quadrant III is the reflection of the point (-4, 2) in Quadrant II over the x-axis. Similarly, let's reflect the point (2,2) in Quadrant I over the x-axis. We can do this by changing its y-coordinate to its opposite. Point&Reflection Acrossx-axis ( 2, 2)&( 2, -2) Therefore, the missing corner has coordinates ( 2, -2).


We multiply the length and width of a rectangle to find its area. Area of the Rectangle Length*Width Let's count the number of units horizontally from point (-4, -2) to (2, -2) to find the length of the rectangle. Next, let's count the units vertically from point (-4, -2) to (-4, 2) to find its width.

The length of the rectangle is 6 and its width is 4. Let's multiply these two values to find the area of the rectangle. Area of the Rectangle 6* 4=24 The area of the rectangle is 24 square units.

The Coordinate Plane

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