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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.

## Date Created:

Feb 22, 2012

Aug 22, 2014
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