4.8: The Coordinate Plane
Introduction
The Sunken Ship
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 sunken ship 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.
What You Will Learn
By the end of this lesson, you will be able to demonstrate the following skills:
- Name ordered pairs of integer coordinates representing points in a coordinate plane.
- Graph ordered pairs of integer coordinates as points in a coordinate plane.
- Find lengths, widths and areas of rectangles on a coordinate plane.
- Make scatter plots of paired real-world data to recognize patterns and make predictions.
Teaching Time
I. Name Ordered Pairs of Integer Coordinates Representing Points in a Coordinate Plane
In working with integers in previous lessons, 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 would create a coordinate plane.
In a coordinate plane like the one shown, the horizontal number line is called the \begin{align*}x-\end{align*}axis. The vertical number line is called the \begin{align*}y-\end{align*}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 look at a map, you will realize that you often see a coordinate plane (or lines of longitude and latitude) on a map. You use the 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 \begin{align*}(x, y)\end{align*}.
- The first number in an ordered pair identifies the \begin{align*}x-\end{align*}coordinate. That coordinate describes the point's position in relation to the \begin{align*}x-\end{align*}axis.
- The second number in an ordered pair identifies the \begin{align*}y-\end{align*}coordinate. That coordinate describes the point's position in relation to the \begin{align*}y-\end{align*}axis.
You can remember that the \begin{align*}x-\end{align*}coordinate is listed before the \begin{align*}y-\end{align*}coordinate in an ordered pair \begin{align*}(x, y)\end{align*}, because \begin{align*}x\end{align*} comes before \begin{align*}y\end{align*} 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.
Example
Name the ordered pair that represents the location of point \begin{align*}Z\end{align*} 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 \begin{align*}x-\end{align*}axis until your finger is lined up under point \begin{align*}Z\end{align*}. 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 \begin{align*}x-\end{align*}coordinate is a positive integer 4.
- Now, move your finger up from the \begin{align*}x-\end{align*}axis until your finger reaches point \begin{align*}Z\end{align*}. You will need to move 5 units up to do that. Moving up along the \begin{align*}y-\end{align*}axis means you are moving in a positive direction. So, the \begin{align*}y-\end{align*}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 \begin{align*}x-\end{align*}coordinate first and the \begin{align*}y-\end{align*}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 \begin{align*}Z\end{align*}.
Example
This coordinate grid shows locations in Jimmy's city. Name the ordered pair that represents the location of the city park.
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 left along the \begin{align*}x-\end{align*}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 \begin{align*}x-\end{align*}axis means that you are moving in a negative direction. Your finger will point to a negative integer, -2, so that is the \begin{align*}x-\end{align*}coordinate.
- Now, move your finger down from the \begin{align*}x-\end{align*}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 \begin{align*}y-\end{align*}axis means that you are moving in a negative direction. Your finger will be aligned with the negative integer, -6, on the \begin{align*}y-\end{align*}axis, so, that is the \begin{align*}y-\end{align*}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.
Now that you know how to name points using an ordered pair, it is time learn how to graph them from an ordered pair.
II. Graph Ordered Pairs of Integer Coordinates as Points in a Coordinate Plane
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 \begin{align*}x-\end{align*}axis and then up or down parallel to the \begin{align*}y-\end{align*}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 \begin{align*}x-\end{align*}coordinate is positive, move to the right of the origin. If the \begin{align*}x-\end{align*}coordinate is negative, move to the left.
- If the \begin{align*}y-\end{align*}coordinate is positive, move up parallel to the \begin{align*}y-\end{align*}axis. If the \begin{align*}y-\end{align*}coordinate is negative, move down.
Example
Plot the ordered pair (-5, 3) as a point on the coordinate plane.
Here are the steps:
- The \begin{align*}x-\end{align*}coordinate is a negative integer, -5, so move your finger 5 units to the left along the \begin{align*}x-\end{align*}axis. Your finger should be pointing to the integer -5 on the \begin{align*}x-\end{align*}axis.
- The \begin{align*}y-\end{align*}coordinate is a positive integer, 3, so move your finger 3 units up from the \begin{align*}x-\end{align*}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.
Example
Triangle \begin{align*}ABC\end{align*} has vertices \begin{align*}A (-2, -5), \ B(0, 3)\end{align*}, and \begin{align*}C(6, -3)\end{align*}. Graph triangle \begin{align*}ABC\end{align*} on a coordinate plane. Label the coordinates of its vertices.
Here are the steps to graphing the triangle.
- To plot vertex \begin{align*}A\end{align*} at (-2, -5), start at the origin. Move 2 units to the left and then 6 units down. Plot and label point \begin{align*}A\end{align*}.
- To plot vertex \begin{align*}B\end{align*} at (0, 3), start at the origin. The \begin{align*}x-\end{align*}coordinate is zero, so do not move to the left or right. From the origin, simply move 3 units up. Plot and label point \begin{align*}B\end{align*}.
- To plot vertex \begin{align*}C\end{align*} at (6, -3), start at the origin. Move 6 units to the right and then 3 units down. Plot and label point \begin{align*}C\end{align*}.
Connect the vertices with line segments to show the sides of the triangle, as shown.
There are other figures that can be graphed on the coordinate plane as well. When you graph a rectangle, you can also look at length, width and area of the rectangle. Let’s look at how this works using the coordinate plane.
III. Find Lengths, Widths and Areas of Rectangles on a Coordinate Plane
Rectangles and squares are four-sided polygons with four right angles. In a rectangle, opposite sides are equal in length. A square is a special type of rectangle that has all four sides equal in length.
Sometimes, a rectangle will be graphed on the coordinate plane. If two sides of the rectangle are horizontal and the other two sides are vertical, we can determine the length and width of the rectangle by counting units along the sides. Once we know the length and width of a rectangle, we can use those measures to find the rectangle's area, \begin{align*}A\end{align*}, using this formula:
\begin{align*}A=lw\end{align*}, where \begin{align*}l =\end{align*} length and \begin{align*}w =\end{align*} width.
The area of a figure measures the number of square units that fit inside the figure.
Example
Rectangle \begin{align*}WXYZ\end{align*}. is graphed on this coordinate plane.
Find the length and width of the rectangle, and then find the area of the rectangle.
It does not matter which side we call the length and which side we call the width, but we usually use the term “length” to describe the longer of the two sides of the rectangle.
Here are the steps:
- Side \begin{align*}XY\end{align*} is one of the longer sides of the rectangle. Count the units from point \begin{align*}X\end{align*} to point \begin{align*}Y\end{align*}. There are 7 units, so the length of the rectangle is 7 units.
- Side \begin{align*}XW\end{align*} is one of the shorter sides of the rectangle. Count the units from point \begin{align*}X\end{align*} to point \begin{align*}W\end{align*}. There are 3 units, so the width of the rectangle is 3 units.
Use the formula for the area of a rectangle. Substitute 7 for the length, \begin{align*}l\end{align*}, and substitute 3 for the width, \begin{align*}w\end{align*}. Then calculate the area.
\begin{align*}A=lw = 7 \times 3 = 21\end{align*}
So, the area of the rectangle is 21 square units.
Remember, the area of the rectangle measures the number of square units that fit inside the rectangle. So, to check this answer, we can count the squares inside rectangle \begin{align*}WXYZ\end{align*}. There are 21 squares inside the rectangle, so its area is 21 square units. That is the same answer we found by using the formula, so that answer must be correct.
We can also find the area of a square graphed on the coordinate plane using this formula.
\begin{align*}A=s^2\end{align*}, where \begin{align*}s =\end{align*} side length.
Example
Points \begin{align*}J, \ K,\end{align*} and \begin{align*}L\end{align*}, shown below, are three vertices of a square.
a. Plot the fourth vertex of the square on the grid. Name it vertex \begin{align*}M\end{align*} and identify its coordinates.
b. Find the area of square \begin{align*}JKLM\end{align*}.
Points \begin{align*}J\end{align*} and \begin{align*}K\end{align*} will form one side of the square, and every side of the square will have the same length. We can count the units between those two points to determine the length of each side of the square.
There are 4 units between points \begin{align*}J\end{align*} and \begin{align*}K\end{align*}, so all the sides of the square will be 4 units long.
That means that the distance between the fourth vertex, vertex \begin{align*}M\end{align*}, and vertex \begin{align*}L\end{align*} will be 4 units also. From point \begin{align*}L\end{align*}, we can count 4 units to the left to find that point \begin{align*}M\end{align*} should be plotted at. Plot that vertex and connect the sides. The completed square will look like this.
We already know that the length of one side of the square is 4 units. So, we can substitute 4 for \begin{align*}s\end{align*}, the side length, in the formula and find the area of a square.
\begin{align*}A=s^2=4^2=4 \times 4 = 16\end{align*}
So, the area of the square is 16 square units.
To check this answer, we can count the squares inside square \begin{align*}JKLM\end{align*}. There are 16 squares inside the square, so its area must be 16 square units. That is the same answer we found when we used the formula.
Next, we can look at how to use the coordinate grid to organize data.
IV. Make Scatter Plots of Paired Real-World Data to Recognize Patterns and Make Predictions
A scatter plot is a way of representing real-world data in a visual way. A scatter plot is a graph showing many data points. It allows us to determine if there us a relationship between pairs of data. A scatter plot can show a positive relationship, a negative relationship, or no relationship, as shown below.
Here is how we can determine the relationships between the data on a scatter plot.
- If the points on the scatter plot seem to form a line that slants up from left to right, it shows a positive relationship.
- If the points on the scatter plot seem to form a line that slants down from left to right, it shows a negative relationship.
- If the points on the scatter plot are scattered randomly, it shows no relationship.
For the first two plots shown above, you could draw a straight line through the points called a line of best fit. That line would not go through every point, but it would show the general trend in the data.
Example
A teacher wants to know if there is a relationship between the amount of time her students spent working on a social studies report and the grade each student received. She surveyed 10 students and recorded the data below.
Student | Number of Hours Worked | Grade |
---|---|---|
Ahmad | 5 | 90 |
Becky | 3 | 80 |
Darrell | 3.5 | 80 |
Emma | 1 | 60 |
Guillermo | 4.5 | 90 |
Helene | 1 | 70 |
Kiet | 3 | 75 |
Nykeisha | 4 | 85 |
Ollie | 2 | 70 |
Zivia | 2.5 | 75 |
Make a scatter plot to show the data in the table.
First, make a scatter plot for the data. The teacher wants to know if the number of hours worked is related to the grade a student earned, so use the horizontal axis of the scatter plot to show the number of hours worked. In the table, the numbers of hours worked range from 1 to 5, so give that axis a scale of 0 to 6, increasing by intervals of 1.
Use the vertical axis to show the grades students received. The grades in the table range from 60 to 90. Showing every grade from 0 to 100 would make the plot very large, so include a break in the axis between 0 and 60. Use intervals of 10 for the rest of scale.
Plot a point for each pair of data in the table. For example, Ahmad worked for 5 hours and earned a grade of 90. So, plot a point at (5, 90) on the scatter plot. Do this until you have plotted all 10 data points and the scatter plot looks like this.
The scatter plot above represents the data that is in the table.
Now, let's interpret the scatter plot and use it to make a prediction.
Example
Interpret and make a prediction based on the scatter plot you created in Example 7.
a. Determine if there is a relationship between the number of hours worked and the grade received. If so, describe the relationship.
b. Suppose an eleventh student spent 6 hours working on her report. Based on the scatter plot, predict the grade you would expect her to receive.
Let's look back at the scatter plot in Example 7.
Draw a line of best fit for this scatter plot. Remember, a line of best fit will not go through every single point on the plot, however, it will fit the general trend of the data. Here is how a line of best fit might look for this scatter plot.
The line of best fit slants up from left to right. So, this scatter plot shows a positive relationship between the data. That means that, in general, the longer a student spent working on the project, the higher the student's grade.
Now, let's predict what a student who spent 6 hours working on the project would probably receive as a grade.
Find 6 hours on the horizontal axis. Move your finger up. You can see that the line of best fit predicts that if a student works for 6 hours, she will receive a grade of about 100.
So, that would be a good prediction of the grade the eleventh student would receive.
Real-Life Example Completed
The Sunken Ship
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 sunken ship 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.
\begin{align*}1 + 6 = 7\end{align*}
If each unit = 160 sq. feet, then we can multiply \begin{align*}160 \times 7\end{align*}
Gina and Cameron will swim through 1120 cubic feet of water from the boat to the sunken ship.
Vocabulary
- Coordinate Plane
- a plane with four quadrants where locations are marked according to coordinates.
- \begin{align*}x -\end{align*}axis
- the horizontal line on the coordinate plane
- \begin{align*}y -\end{align*}axis
- the vertical line on the coordinate plane
- Origin
- the point where the \begin{align*}x-\end{align*} axis and the \begin{align*}y-\end{align*} axis meet
- \begin{align*}x-\end{align*}coordinate
- the first coordinate in an ordered pair.
- \begin{align*}y-\end{align*}coordinate
- the second coordinate in an ordered pair.
- Rectangle
- a figure with four right angles and opposite sides parallel and congruent
- Square
- a figure with four right angles and four equal sides that are also parallel.
- Area
- the measure of the space inside a figure
- Scatter Plot
- a data chart that uses points to show the relationship between data and events.
Technology Integration
Khan Academy, The Coordinate Plane
James Sousa, Plotting Points on The Coordinate Plane
James Sousa, Identify a Quandrant of a Point on the Coordinate Plane
James Sousa, Example of Plotting Points on the Coordinate Plane
James Sousa, Example of Identifying the Coordinates of Points on the Coordinate Plane
Other Videos:
This is a Brightstorm video on plotting points and naming quadrants.
http://www.mathplayground.com/mv_plotting_points_naming_quadrants.html
Time to Practice
Directions: Name the ordered pairs.
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 \begin{align*}FGH\end{align*}.
6. \begin{align*}F\end{align*}
7. \begin{align*}G\end{align*}
8. \begin{align*}H\end{align*}
Directions: Name the ordered pairs that represent the vertices of pentagon \begin{align*}ABCDE\end{align*}.
9. \begin{align*}A\end{align*}
10. \begin{align*}B\end{align*}
11. \begin{align*}C\end{align*}
12. \begin{align*}D\end{align*}
13. \begin{align*}E\end{align*}
14. On the grid below, plot point \begin{align*}V\end{align*} at (-6, 4).
15. On the grid below, plot point a triangle with vertices \begin{align*}R (4, -1), \ S (4, -4)\end{align*}, and \begin{align*}T (-3, -4)\end{align*}.
Directions: Square \begin{align*}UVWX\end{align*} is graphed on the coordinate plane below.
16. Find the length of one side of this square.
17. Find the area of this square.
Directions: Points \begin{align*}K, \ L,\end{align*} and \begin{align*}M\end{align*}, shown below, are three vertices of a rectangle.
18. Plot the fourth vertex of the rectangle on the grid. Name it vertex \begin{align*}N\end{align*} and identify its coordinates. Then connect the vertices to form rectangle \begin{align*}KLMN\end{align*}.
19. Find the length and width of rectangle \begin{align*}KLMN\end{align*}.
20. Find the area of rectangle \begin{align*}KLMN\end{align*}.
Directions: This scatter plot shows the relationship between the last digit of ten students' phone numbers and their vocabulary quiz scores.
21. Does this scatter plot show a positive relationship, a negative relationship, or no relationship?
22. Serena wants to know if there is a relationship between a person's age and the number of DVDs they purchased in one year. She surveyed a group of people and recorded the data in the table below.
Person | Age | Number of DVDs Purchased |
---|---|---|
Person 1 | 18 | 14 |
Person 2 | 19 | 13 |
Person 3 | 20 | 13 |
Person 4 | 20 | 12 |
Person 5 | 21 | 11 |
Person 6 | 22 | 12 |
Person 7 | 22 | 11 |
Person 8 | 23 | 10 |
Person 9 | 24 | 9 |
Person 10 | 25 | 9 |
a. Use the grid below to make a scatter plot for the data in the table. Draw a line of best fit for the scatter plot.
b. Does the scatter plot you created show a positive relationship, a negative relationship, or no relationship?
c. If the trend in the scatter plot continues, predict the number of DVDs you would expect a 27-year-old person to buy in one year.
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