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# 11.6: The Coordinate Plane

Difficulty Level: At Grade Created by: CK-12

## Introduction

The Map

Kevin and his pen pal Charlotte are both creating maps of their neighborhoods to show each other what it looks like where they live. Kevin has decided to name the most important things on his map. He has decided to include his house, his school, the skate park and the library. Since Kevin lives close to each of these things, he is sure that he can draw them on a map.

Kevin has decided to use a coordinate grid to show each location. He wants to send Charlotte a key that will match each location with its accurate coordinates.

Here is the grid that Kevin starts off with.

Given this map, which coordinates should Kevin use to name each location?

Pay close attention and you will learn how to write coordinates to name locations.

What You Will Learn

By the end of this lesson you will be able to complete the following:

• Graph ordered pairs of integer coordinates as points in all four quadrants.
• Graph geometric figures given coordinates of vertices.
• Locate places on maps using integer coordinates.
• Describe paths between given points as integer translations.

Teaching Time

I. Graph Ordered Pairs of Integer Coordinates as Points in all Four Quadrants

Way back in Chapter 2, you learned how to graph points on a coordinate grid. This coordinate grid only had one quadrant or section to it. This was necessary at the time because you didn’t know about integers yet. Here is a picture of the coordinate grid with only one quadrant.

Now let’s think back to that section of Chapter 2 and review some of the vocabulary associated with coordinate grids and graphing points.

Now if we are going to plot a point on the coordinate grid pictured above, we will have an \begin{align*}x\end{align*} coordinate and a \begin{align*}y\end{align*} coordinate. We go across the \begin{align*}x\end{align*} axis to the \begin{align*}x\end{align*} value and then up to the \begin{align*}y\end{align*} value and that is where we plot the point.

Let’s practice.

Example

Plot (3, 5) on the coordinate grid then label it point \begin{align*}A\end{align*}

Now we have point (3, 5) graphed on the coordinate grid.

But this isn’t the only coordinate grid! Now that you know about integers, we can see all four quadrants of the coordinate grid. While in the past we only graphed points in one quadrant, there are actually FOUR quadrants to the coordinate grid. Let’s take a look.

Here you can see all four quadrants of the coordinate grid. If you look at each axis, you will see that there are positive and negative values on each axis. The \begin{align*}x\end{align*} axis has positive values to the right of the origin, and negative values to the left of the origin. The \begin{align*}y\end{align*} axis has positive values above the origin and negative values below the origin. We can plot points in all four quadrants.

How can we graph points in all four quadrants?

We can work on this in the same way that we did when we had only one quadrant. We use ordered pairs. There will be an \begin{align*}x\end{align*} value and a \begin{align*}y\end{align*} value in the ordered pair. The \begin{align*}x\end{align*} value can be positive or negative and the \begin{align*}y\end{align*} value can be positive or negative. We start at the origin, move to the \begin{align*}x\end{align*} value and then to the \begin{align*}y\end{align*} value. Then we can graph the point.

Let’s look at an example.

Example

Graph the point (-4, 3) and name it point \begin{align*}P\end{align*}.

Here we started at the origin. Worked our way to the left to negative four on the \begin{align*}x\end{align*} axis then worked our way up to positive three on the \begin{align*}y\end{align*} axis. That is where we graphed point \begin{align*}P\end{align*}.

What about if one of the values is a 0?

You can graph one of these points in the same way. You just use the 0 as the value for \begin{align*}x\end{align*} or \begin{align*}y\end{align*} based on the ordered pair. Here is an example.

Example

Graph (0, -2) and name it point \begin{align*}B\end{align*}.

Because the first value is 0, the \begin{align*}x\end{align*} coordinate doesn’t move left or right of the origin. Then we move down because the \begin{align*}y\end{align*} value is negative two. That is where we have graphed point \begin{align*}B\end{align*}.

Practice Identifying each ordered pair on the Coordinate grid.

Take a few minutes to check your work with a friend.

II. Graph Geometric Figures Given Coordinates of Vertices

Now that you have learned how to graph points in all four quadrants, you can look at graphing lines and figures on the coordinate grid. If you have the coordinates of each vertex of a figure, you can easily graph it on the coordinate grid. Remember that the coordinates are the ordered pairs of each point. The coordinates let you know where to graph each point.

Let’s look at an example.

Example

Graph a figure with the coordinates \begin{align*}A(-4,3) \ B(2,3) \ C(2, -1) \ D(-4,-1)\end{align*}. When finished, name the figure that has been drawn on the grid.

To start with, we plot each point on the coordinate grid and then we connect the lines. This will give us a geometric figure.

Now that we have it graphed, you can use what you have already learned about geometric figures to name the figure.

This is a rectangle.

You can graph any geometric figure on the coordinate grid as long as you have been given the coordinates.

First, plot each point.

Then, connect the points to form a figure.

Finally, use what you have learned to name the figure drawn.

Let’s practice one more.

Example

Graph and name the following figure with these coordinates \begin{align*}D(1,3) \ E(5,3) \ F(7,-1) \ G(1,-1)\end{align*}

Here we have graphed a four sided figure with one pair of parallel sides. This is a trapezoid.

III. Locate Places on Maps Using Integer Coordinates

When we graphed geometric figures, we used integer coordinates to find the location of each point. Then we graphed each point according to its location. Maps also use integer coordinates to identify different locations. If you look at a map, you will see some numbers and sometimes letters around the border of the map. This can assist you in figuring out the location of cities or even different locations.

Some maps use integers to identify different locations. Let’s look at a map that does this. Here we have used a coordinate grid to identify where different places are in a town. Let’s look at this map.

We can say that Kara’s house is blue, Mark’s house is pink and Chase’s house is green. Each house has coordinates. We can say that the center of each house marks its coordinates on the map.

Kara’s house is at (-3, 1)

Mark’s house is at (3, -2)

Chase’s house is at (3, 4)

Local maps use letters and numbers to identify locations. World maps use degrees written in latitude and longitude. Let’s learn about this real life use of coordinates.

Longitude is the measure of lines horizontally on a map.

Latitude is the measure of lines vertically on a map.

We can measure longitude and latitude using degrees. These degrees are written as ordered pairs.

Here you can see degrees of latitude as horizontal measures. The degrees of longitude are the vertical measures.

We can identify different locations on a map if we have the coordinates of the location. Notice that the degrees of latitude are written first, those are the horizontal degrees, and the degrees of longitude are written second. Those are the vertical degrees.

Use the map of the United States, pictured below, for the following example.

Example

Which state is at \begin{align*}45^\circ, 70^\circ\end{align*}?

To answer this question, we start with the horizontal degrees, the latitude. That says \begin{align*}45^\circ\end{align*}. We start at 45 and then move to 70 degrees.

You can see that we are at the state of Maine.

Maine is our answer.

As long as you have values on a map, you can use coordinates to identify any location.

www.EnchantedLearning.com

Practice working in degrees. Identify the states according to their locations in latitude and longitude.

1. \begin{align*}30^\circ, 83^\circ\end{align*}
2. \begin{align*}42^\circ, 100^\circ\end{align*}
3. \begin{align*}30^\circ, 100^\circ\end{align*}

Check your answers with a friend. Did you identify each state correctly?

IV. Describe Paths Between Given Points As Integer Translations

A translation is when a figure or a point is moved on a coordinate grid. It is when you slide a figure or a point on a grid. We can use integers to assist us in indentifying different translations. Let’s look at an example.

Here, we started with point \begin{align*}A\end{align*}. Then point \begin{align*}A\end{align*} was moved on the coordinate grid. It has been translated to a new location. When we slide or translate a point or figure, the new point has a little symbol next to it. Here is how we write a translation.

\begin{align*}A\end{align*} to \begin{align*}A'\end{align*}

We can use integers to show the path of the translation.

How many units did \begin{align*}A\end{align*} move on the \begin{align*}x\end{align*} axis?

If you count, you can see that it moved +6 units.

How many units did \begin{align*}A\end{align*} move on the \begin{align*}y\end{align*} axis?

If you count, you can see that it moved -2 units (remember that negative 'y' means down).

We write the translation as (6, -2).

Try one of these on your own. Write the path of \begin{align*}B\end{align*} as an integer translation.

Take a few minutes to check your work with a partner.

## Real Life Example Completed

The Map

Here is the original problem once again. Reread the problem and then use what you have learned to write the coordinates to match Kevin’s map.

Kevin and his pen pal Charlotte are both creating maps of their neighborhoods to show each other what it looks like where they live. Kevin has decided to name the most important things on his map. He has decided to include his house, his school, the skate park and the library. Since Kevin lives close to each of these things, he is sure that he can draw them on a map.

Kevin has decided to use a coordinate grid to show each location. He wants to send Charlotte a key that will match each location with its accurate coordinates.

Here is the grid that Kevin starts off with.

Given this map, which coordinates should Kevin use to name each location?

Now that you have finished this lesson, let’s work on writing coordinates to match Kevin’s map.

First, let’s start with his home. His house is located at (4, 5).

His school is located close to his home at (4, 2)

The library is located at (-1, 3).

Finally, the skate park is the farthest away from his home at (0, -3).

Kevin is ready to send his map and coordinates to Charlotte. He can’t wait to see her map.

## Vocabulary

Here are the vocabulary words that are found in this lesson.

the four sections of a coordinate grid
Origin
the place where the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} axis’ meet at (0, 0)
Ordered Pair
the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values used to locate points on a coordinate grid \begin{align*}(x,y)\end{align*}
\begin{align*}x\end{align*} axis
the horizontal axis on the coordinate grid
\begin{align*}y\end{align*} axis
the vertical axis on the coordinate grid
Coordinates
the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values of an ordered pair
Longitude
vertical measure of degrees on a map
Latitude
horizontal measure of degrees on a map

## Technology Integration

Other Videos:

1. http://www.mathplayground.com/mv_plotting_points_naming_quadrants.html – This is a Brightstorm video on plotting points and naming quadrants.

Resources

1. http://www.EnchantedLearning.com – maps and coordinates
2. http://www.wikipedia.com – world map with longitude and latitude
3. http://www.travelcity.com – information on cities and locations

## Time to Practice

Directions: Identify the coordinates of each of the points plotted on the coordinate grid.

1. \begin{align*}A\end{align*}

2. \begin{align*}B\end{align*}

3. \begin{align*}C\end{align*}

4. \begin{align*}D\end{align*}

5. \begin{align*}E\end{align*}

6. \begin{align*}F\end{align*}

7. \begin{align*}G\end{align*}

8. \begin{align*}H\end{align*}

9. \begin{align*}I\end{align*}

10. \begin{align*}J\end{align*}

Directions: Graph each figure using the vertices. Then name the graphed figure.

11. \begin{align*}A(-2, 2)\!\\ B(2, 2)\!\\ C(2, -2)\!\\ D(-2, -2)\end{align*}

12. \begin{align*}D(-4, 3)\!\\ E(-1, 1)\!\\ F (-4, 1)\end{align*}

Directions: Use integers to identify each translation.

13. \begin{align*}A\end{align*} to \begin{align*}A'\end{align*}

14. \begin{align*}B\end{align*} to \begin{align*}B'\end{align*}

15. \begin{align*}C\end{align*} to \begin{align*}C'\end{align*}

16. \begin{align*}D\end{align*} to \begin{align*}D'\end{align*}

17. \begin{align*}E\end{align*} to \begin{align*}E'\end{align*}

18. \begin{align*}F\end{align*} to \begin{align*}F'\end{align*}

19. \begin{align*}G\end{align*} to \begin{align*}G'\end{align*}

20. \begin{align*}H\end{align*} to \begin{align*}H'\end{align*}

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