4.17: Points in the Coordinate Plane
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
Previously we worked with integers. 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 axis . The vertical number line is called the 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 .
- The first number in an ordered pair identifies the coordinate . That coordinate describes the point's position in relation to the axis.
- The second number in an ordered pair identifies the coordinate . That coordinate describes the point's position in relation to the axis.
You can remember that the coordinate is listed before the coordinate in an ordered pair , because comes before in the alphabet.
Example A, B, C
Triangle has vertices , and . Graph triangle on a coordinate plane. Label the coordinates of its vertices.
Solution: Here are the steps to graphing the triangle.
- To plot vertex at (-2, -5), start at the origin. Move 2 units to the left and then 6 units down. Plot and label point .
- To plot vertex at (0, 3), start at the origin. The coordinate is zero, so do not move to the left or right. From the origin, simply move 3 units up. Plot and label point .
- To plot vertex at (6, -3), start at the origin. Move 6 units to the right and then 3 units down. Plot and label point .
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 dive boat is marked at (4, 8).
The sunken ship 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.
If each unit = 160 cubic feet, then we can multiply
Gina and Cameron will swim through 1120 cubic feet of water from the sunken ship to the boat.
Vocabulary
- Coordinate Plane
- The coordinate plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin. The coordinate plane is also called a Cartesian Plane.
- axis
- The axis is the horizontal axis in the coordinate plane, commonly representing the value of the input or independent variable.
- axis
- The -axis is the vertical number line of the Cartesian plane.
- Origin
- The origin is the point of intersection of the and axes on the Cartesian plane. The coordinates of the origin are (0, 0).
- coordinate
- The coordinate is the first term in a coordinate pair, commonly representing the value of the input or independent variable.
- coordinate
- The coordinate is the second term in a coordinate pair, commonly representing the value of the output or dependent variable.
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 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 axis means that you are moving in a negative direction. Your finger will point to a negative integer, -2, so that is the coordinate.
- Now, move your finger down from the 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 axis means that you are moving in a negative direction. Your finger will be aligned with the negative integer, -6, on the axis, so, that is the 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
This is a James Sousa video on ordered pairs and the coordinate plane.
Explore More
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 .
6.
7.
8.
Directions: Name the ordered pairs that represent the vertices of pentagon .
9.
10.
11.
12.
13.
14. On the grid below, plot point at (-6, 4).
15. On the grid below, plot point a triangle with vertices , and .
Image Attributions
Description
Learning Objectives
Here you'll learn to name and graph ordered pairs of integer coordinates in a coordinate plane.