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4.17: Points in the Coordinate Plane

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Have you ever wanted to find a sunken ship? Take a look at this dilemma.

Gina and Cameron are very excited because they are going to go on a dive to see a sunken ship. The dive is quite shallow which is unusual because most dives are found at depths that are too deep for two junior divers. However, this one is at 40 feet, so the two divers can go to see it.

They have the following map to chart their course. Cameron wants to figure out exactly how far the boat will be from the sunken ship. Each square represents 160 cubic feet of water.

First, he makes a note of the coordinates. Then he can use the map to calculate the distance.

We use coordinate grids like this one all the time. Use the information in this lesson to help Cameron figure out the coordinates of his boat and the sunken ship. Then you will be able to estimate the distance between them.

Guidance

In working with integers in previous Concepts, we used both horizontal (left-to-right) and vertical (up-and-down) number lines. Imagine putting a horizontal and a vertical number line together. In doing this, you could create a coordinate plane .

In a coordinate plane like the one shown, the horizontal number line is called the x- axis . The vertical number line is called the y- axis . The point at which both of these axes meet is called the origin .

We can use coordinate planes to represent points, two-dimensional geometric figures, or even real-world locations. If you think about a map, you will realize that you see a coordinate plane on a map. Then you use coordinates to find different locations. Let’s look at how we can use a coordinate plane.

How do we name points on a coordinate plane?

Each point on a coordinate plane can be named by an ordered pair of numbers, in the form (x, y) .

  • The first number in an ordered pair identifies the x- coordinate . That coordinate describes the point's position in relation to the x- axis.
  • The second number in an ordered pair identifies the y- coordinate . That coordinate describes the point's position in relation to the y- axis.

You can remember that the x- coordinate is listed before the y- coordinate in an ordered pair (x, y) , because x comes before y in the alphabet.

Write these terms and their definitions in your notebook.

Identifying the coordinates of a point is similar to locating a point on a number line. The main difference is that you need to find the point that corresponds to both of the given coordinates.

Name the ordered pair that represents the location of point Z below.

Here are the steps to naming the coordinates.

  • To start, place your finger at the origin.
  • Then move your finger to the right along the x- axis until your finger is lined up under point Z . You will need to move 4 units to the right to do that. Moving to the right along a number line means you are moving in a positive direction. So, the x- coordinate is a positive integer 4.
  • Now, move your finger up from the x- axis until your finger reaches point Z . You will need to move 5 units up to do that. Moving up along the y- axis means you are moving in a positive direction. So, the y- coordinate is a positive integer 5.

The arrows below show how you should have moved your finger to determine the coordinates.

To name the ordered pair, write the x- coordinate first and the y- coordinate second. Separate the coordinates with a comma and put parentheses around them, like this (4, 5).

So, the ordered pair (4, 5) names the location of point Z .

Now that you know how to name points using an ordered pair, it is time learn how to graph them from an ordered pair.

Graphing points on a coordinate plane is similar to naming them. Given an ordered pair, you can move your finger left or right along the x- axis and then up or down parallel to the y- axis until you find the location named by the ordered pair. Then you can plot a point at that location.

There are a few key points to remember.

  • If the x- coordinate is positive, move to the right of the origin. If the x- coordinate is negative, move to the left.
  • If the y- coordinate is positive, move up parallel to the y- axis. If the y- coordinate is negative, move down.

Here is another one.

Plot the ordered pair (-5, 3) as a point on the coordinate plane.

Here are the steps:

  • The x- coordinate is a negative integer, -5, so move your finger 5 units to the left along the x- axis. Your finger should be pointing to the integer -5 on the x- axis.
  • The y- coordinate is a positive integer, 3, so move your finger 3 units up from the x- axis.

Plot a point at that location. That point represents the ordered pair (-5, 3).

Sometimes, the points you plot on a coordinate grid will form the vertices of a geometric figure, such as a triangle. Try this one on your own.

Example A, B, C

Triangle ABC has vertices A (-2, -5), \ B(0, 3) , and C(6, -3) . Graph triangle ABC on a coordinate plane. Label the coordinates of its vertices.

Solution: Here are the steps to graphing the triangle.

  • To plot vertex A at (-2, -5), start at the origin. Move 2 units to the left and then 6 units down. Plot and label point A .
  • To plot vertex B at (0, 3), start at the origin. The x- coordinate is zero, so do not move to the left or right. From the origin, simply move 3 units up. Plot and label point B .
  • To plot vertex C at (6, -3), start at the origin. Move 6 units to the right and then 3 units down. Plot and label point C .

Connect the vertices with line segments to show the sides of the triangle, as shown.

Here is the original problem once again. Use this information to help Cameron with the coordinates and the distance.

Gina and Cameron are very excited because they are going to go on a dive to see a sunken ship. The dive is quite shallow which is unusual because most dives are found at depths that are too deep for two junior divers. However, this one is at 40 feet, so the two divers can go to see it.

They have the following map to chart their course. Cameron wants to figure out exactly how far the boat will be from the sunken ship. Each square represents 160 cubic feet of water.

First, he makes a note of the coordinates. Then he can use the map to calculate the distance.

First, here are the coordinates of each item on the map.

The sunken ship is marked at (4, 8).

The dive boat is marked at (-3, 7).

Notice the arrows. Once they get to the sunken ship, Gina and Cameron will swim up 1 unit and over 6 units.

1 + 6 = 7

If each unit = 160 sq. feet, then we can multiply 160 \times 7

Gina and Cameron will swim through 1120 cubic feet of water from the boat to the sunken ship.

Vocabulary

Here are the vocabulary words in this Concept.

Coordinate Plane
a plane with four quadrants where locations are marked according to coordinates.
x - axis
the horizontal line on the coordinate plane
y - axis
the vertical line on the coordinate plane
Origin
the point where the x- axis and the y- axis meet
x- coordinate
the first coordinate in an ordered pair.
y- coordinate
the second coordinate in an ordered pair.

Guided Practice

Here is one for you to try on your own.

This coordinate grid shows locations in Jimmy's city. Name the ordered pair that represents the location of the city park.

Answer

Here are the steps to figuring out the coordinates of the city park.

  • Place your finger at the origin.
  • Next, move your finger to the right along the x- axis until your finger is lined up above the point representing the city park. You will need to move 2 units to the left to do that. Moving to the left along the x- axis means that you are moving in a negative direction. Your finger will point to a negative integer, -2, so that is the x- coordinate.
  • Now, move your finger down from the x- axis until your finger reaches the point for the city park. You will need to move 6 units down to do that. Moving down parallel to the y- axis means that you are moving in a negative direction. Your finger will be aligned with the negative integer, -6, on the y- axis, so, that is the y- coordinate.

The arrows below show how you should have moved your finger to find the coordinates.

So, the ordered pair (-2, -6) names the location of the city park.

Video Review

Here is a video for review.

- This is a James Sousa video on ordered pairs and the coordinate plane.

Practice

Directions : Use what you have learned to complete this practice section.

1. Name the ordered pair that represents each of these points on the coordinate plane.

Directions: Below is a map of an amusement park. Name the ordered pair that represents the location of each of these rides.

2. roller coaster

3. Ferris wheel

4. carousel

5. log flume

Directions: Name the ordered pairs that represent the vertices of triangle FGH .

6. F

7. G

8. H

Directions: Name the ordered pairs that represent the vertices of pentagon ABCDE .

9. A

10. B

11. C

12. D

13. E

14. On the grid below, plot point V at (-6, 4).

15. On the grid below, plot point a triangle with vertices R (4, -1), \ S (4, -4) , and T (-3, -4) .

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Difficulty Level:

Basic

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Date Created:

Dec 21, 2012

Last Modified:

Mar 17, 2014
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