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# 4.1: The Coordinate Plane

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

At the end of this lesson, students will be able to:

• Identify coordinates of points.
• Plot points in a coordinate plane.
• Graph a function given a table.
• Graph a function given a rule.

## Vocabulary

Terms introduced in this lesson:

coordinate plane
$x-$ and $y-$axes
origin
ordered pair
$x$ and $y$ coordinates
positive $x$, negative $x$
positive $y$, negative $y$
relation
domain
range
graph of a function
continuous function
discrete function
independent variable
dependent variable
linear relationship
discrete problem
slope
intercept

## Teaching Strategies and Tips

Introduction: Motivate $xy-$coordinates with examples from daily life that employ rectangular coordinate systems.

• Examples: a city map, the game of Battleship, a chessboard, spreadsheets, assigned seating at a theater.
• Discuss how to find a particular location in each example: a seat in a theater can be found by row number and then by seat number.
• Point out that the examples are lattices, differing from the Cartesian coordinate system in that they are discrete.

Use Examples 1-3 to demonstrate finding coordinates of points on a graph and Examples 4 and 5 to plot points given their coordinates. Allow the class to make observations such as:

• The coordinates of a point cannot be interchanged since the first coordinate specifies going left/right and the second coordinate, up/down. For example, $(2, 7)$ is not the same point as $(7,2)$.
• If a coordinate of a point is $0$, then the point resides on an axis.
• Quadrants can be distinguished by the signs of the coordinates contained in them. For example, a point having coordinates with the signs $(-,+)$ resides in quadrant II. Points with coordinates having signs $(-,-)$ belong to quadrant III.
• In Example 4, it is necessary to display four quadrants so that all points will be visible. The set of points in Example 5 have only positive coordinates; it is convenient therefore to display only the first quadrant. As an informal rule, axes do not need to be extended farther than the largest and smallest $x-$coordinates and $y-$coordinates.
• Resize a graph by rescaling the axes. In general, the $x$ and $y-$axes can be scaled differently. Axis tick marks do not need to be unit increments.

General graphing tips:

• In applied problems, the independent and dependent variables should be distinguished early. Ask:

What quantity is depending on the other?

• In setting up the axes, a suitable scale must be chosen. Ask:

Will the important features of the graph be visible?

Will it be necessary to use different increments along the two axes?

• Constructing tables is a valuable tool. See Examples 6 & 7. Allow students to use the simple inputs, $x = 0, 1, -1, 2,$ in their tables when appropriate.

The second method in Example 7 will be returned to in greater detail in a subsequent chapter.

## Error Troubleshooting

General Tip: To determine the graph of a linear relationship, no more than two points are needed. Students can be encouraged to plot at least three to ensure no arithmetical errors were made.

Feb 22, 2012

Aug 22, 2014