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LEARNING GOALS
By the end of this lesson, you should understand
- the Cartesian coordinate system is used for graphing ordered pairs and locating points.
- the horizontal x-axis intersects the vertical y-axis at the origin.
- the system is separated into 4 sections called quadrants.
- the first number in an ordered pair is known as the x-coordinate; the second number is known as the y-coordinate.
- the positive side of the x-axis is to the right of the origin and the negative side is to the left.
- the positive side of the y-axis is above the origin and the negative is below.
By the end of this lesson, you should be able to
- understand key vocabulary associated with plotting points such as ordered pair, x-coordinate y-coordinate, origin, x-axis, y-axis and quadrant.
- correctly plot points.
- name ordered pairs for given points, and correctly identify the corresponding quadrant in which the point lies.
- use number sense to determine an exact location of a point.
FUNDAMENTALS OF THE LESSON
The Cartesian coordinate system is a grid that was developed to graph ordered pairs. An ordered pair is a pair of numbers used to locate a point and is written in the form (x, y), where x is the x-coordinate and y is the y-coordinate. The coordinate system is made up of two perpendicular number lines that meet at the point (0, 0). This special point is called the origin. Each perpendicular line is an axis. The horizontal line is called the x-axis and the vertical line is called the y-axis.
The coordinate system is divided into 4 parts. These parts are called quadrants, and are labeled with numbers 1-4 in a counterclockwise direction, starting in the upper right quadrant. Usually, Roman numeral numbers I, II, III, and IV are used to label each quadrant.
An image of the coordinate system is shown below. Notice, the positive x-axis is to the right of the origin and the negative x-axis is to the left. The positive y-axis is above the origin and the negative y-axis is below.
To represent a point on the coordinate system where -5 is the x-coordinate of a point and 6 is the y-coordinate of a point, we use the ordered pair notation of (-5, 6). The -5 lets us know how far the point is to the left of the origin. The 6 lets us know how far the point is above the origin.
1. Write an ordered pair for each point on the grid.
A: _______
B: _______
C: _______
D: _______
E: _______
2. Plot the ordered pairs on the coordinate plane above. Describe the location of the point, including what quadrant the point lies in.
A. Plot M (-4,1).
B. Which quadrant is (-4,1) in?
C. Plot S ( -2,-2).
D. Which quadrant is (-2,-2) in?
E. Plot U (3,-2).
F. Which Quadrant is (3,-2) in?
TRY THESE 1
What is their location?
A family plans to take a theme park trip to different places around the United States. The family has scheduled to leave New York by plane and fly to Washington. After staying in Washington for three days, the family flies to southern California for a week. From California, the family goes to its final destination, Florida. A path21 of the trip is shown on the coordinate plane below.
3. The family leaves New York to start their trip.
A. In which quadrant does New York lie?
B. At what point does the trip begin (use ordered pair notation)?
4. Name a state that lies directly on the negative y-axis.
5. A. At what point does the trip end if each tick is a value of one unit?
B. In which quadrant does that point lie?
6. Write two different ordered pairs that represent locations in Texas.
TRY THESE 2
Navigate the map
A map of the rooms in a college hall is fit to the coordinate plane below. Sarah, a college ambassador, has given this map to each freshman student to help them navigate on their first day of classes in this particular building.
7. A. Savien is in the class that is located at (-5,-1). What class is he in?
B. Describe in words how to get from the entrance to this point.
8. A. Savien’s next class is 4 units up and 4 units to the right on the map from the class where he is now. What class is Savien going to next?
B. Find the ordered pair that states the location of that class.
9. Adrianna is at Exit 1, but has to go to the office. Give Adrianna directions on how to get to the office.
10. On the map, which classes are located in quadrant IV? Describe the coordinates of all classes in this quadrant.
FURTHER APPLICATIONS
Your campus is located at (3,-1), which is 3 blocks east and 1 block south of the center of town. To get from your apartment to the campus, you walk 4 blocks west and 2 blocks north. (Hint: It may help to draw a Cartesian coordinate system with everything labeled).
11. What ordered pair corresponds to the location of your apartment?
12. Is your apartment or your campus closer to the center of town? Explain.
Questions: Plotting Points
Tell whether the following statement is sometimes, always, or never true. Explain your reasoning.
1. The x-coordinate of points in quadrant IV are negative.
2. The x-coordinate of a point in quadrant III has the same sign as the y-coordinate of a point in quadrant II.
3. Name the ordered pair that is 4 units right and 2 units down from (-3,2).
4. In which quadrant is the point \(\left( -\dfrac{5}{2}, \dfrac{10}{3}\right)\).
5. Delanie made a map of her running trail through the woods in such a way that each intersection is a point on a coordinate plane. Delanie plans to start at point (3,2). Give the ordered pair of where she will be if she moves 5 units to the right and 7 units down on her map.
6. After Delanie moves 5 units to the right and 7 units down, what quadrant is she in?
7. Say we have a figure with the vertices A (0, 1), B (-4, 1), C (-4, -1), and D (0, -1). Draw the figure in the coordinate plane, and then find the perimeter of the figure.
8. Find the area of the figure.
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21 Image retrieved from www.magoosh.com.
