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4.1: The Coordinate Plane

Difficulty Level: At Grade Created by: CK-12
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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
\begin{align*}x-\end{align*} and \begin{align*}y-\end{align*}axes
origin
quadrant
ordered pair
\begin{align*}x\end{align*} and \begin{align*}y\end{align*} coordinates
positive \begin{align*}x\end{align*}, negative \begin{align*}x\end{align*}
positive \begin{align*}y\end{align*}, negative \begin{align*}y\end{align*}
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 \begin{align*}xy-\end{align*}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, \begin{align*}(2, 7)\end{align*} is not the same point as \begin{align*}(7,2)\end{align*}.
  • If a coordinate of a point is \begin{align*}0\end{align*}, 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 \begin{align*}(-,+)\end{align*} resides in quadrant II. Points with coordinates having signs \begin{align*}(-,-)\end{align*} 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 \begin{align*}x-\end{align*}coordinates and \begin{align*}y-\end{align*}coordinates.
  • Resize a graph by rescaling the axes. In general, the \begin{align*}x\end{align*} and \begin{align*}y-\end{align*}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, \begin{align*}x = 0, 1, -1, 2,\end{align*} 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.

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