Cartesian Coordinate System

Abstract

This lesson is designed to familiarize students to the Cartesian Coordinate System and its many uses in the world of mathematics. The Cartesian coordinate system was developed by the mathematician Descartes during an illness. As he lay in bed sick, he saw a fly buzzing around on the ceiling, which was made of square tiles. As he watched he realized that he could describe the position of the fly by the ceiling tile he was on. After this experience he developed the coordinate plane to make it easier to describe the position of objects.

This lesson is best if the students work in small groups of two or three.

Objectives

Upon completion of this lesson, students will:

  • have been introduced to the Cartesian coordinate plane
  • be able to plot points on the plane
  • be able to read coordinates for a point from a graph
  • be able to give the ratio of rise over run for slope

Standards Addressed

Grade 3

  • Geometry

    • The student demonstrates understanding of position and direction.

Grade 4

  • Geometry

    • The student demonstrates understanding of position and direction.

Grade 5

  • Geometry

    • The student demonstrates understanding of position and direction.

Grade 6

  • Geometry

    • The student demonstrates understanding of position and direction.

Grade 7

  • Geometry

    • The student demonstrates understanding of position and direction.

Grade 8

  • Geometry

    • The student demonstrates understanding of position and direction.

Grade 9

  • Geometry

    • The student demonstrates understanding of position and direction when solving problems (including real-world situations).

Grade 10

  • Geometry

    • The student demonstrates understanding of position and direction when solving problems (including real-world situations).

Fifth Grade

  • Geometry

    • Graph points on the coordinate plane to solve real-world and mathematical problems.

Grades 6-8

  • Algebra

    • Represent and analyze mathematical situations and structures using algebraic symbols

Grade 7

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 5: The learner will demonstrate an understanding of linear relations and fundamental algebraic concepts.

8th Grade

  • Data Analysis and Probability

    • The student will demonstrate through the mathematical processes an understanding of the relationships between two variables within one population or sample.

  • Geometry

    • The student will demonstrate through the mathematical processes an understanding of the Pythagorean theorem; the use of ordered pairs, equations, intercepts, and intersections to locate points and lines in a coordinate plane; and the effect of a dilation
    • The student will demonstrate through the mathematical processes an understanding of the Pythagorean theorem; the use of ordered pairs, equations, intercepts, and intersections to locate points and lines in a coordinate plane; and the effect of a dilation in a coordinate plane.

Elementary Algebra

  • Elementary Algebra

    • Standard EA-4: The student will demonstrate through the mathematical processes an understanding of the procedures for writing and solving linear equations and inequalities.
    • Standard EA-5: The student will demonstrate through the mathematical processes an understanding of the graphs and characteristics of linear equations and inequalities.
    • Standard EA-6: The student will demonstrate through the mathematical processes an understanding of quadratic relationships and functions.

Intermediate Algebra

  • Algebra

    • The student will demonstrate through the mathematical processes an understanding of functions, systems of equations, and systems of linear inequalities.
    • The student will demonstrate through the mathematical processes an understanding of algebraic expressions and nonlinear functions.

7th Grade

  • Probability and Statistics

    • 7.17 The student, given a problem situation, will collect, analyze, display, and interpret data, using a variety of graphical methods, including frequency distributions; line plots; histograms; stem-and-leaf plots; box-and-whisker plots; and scattergrams.

4th Grade

  • Geometry

    • 4.18 The student will identify the ordered pair for a point and locate the point for an ordered pair in the first quadrant of a coordinate plane.

  • Geomety

    • 4.18

8th Grade

  • Patterns, Functions, and Algebra

    • 8.16 The student will graph a linear equation in two variables, in the coordinate plane, using a table of ordered pairs.
    • 8.16 The student will graph a linear equation in two variables, in the coordinate plane, using a

Textbooks Aligned

Grade Six

  • Data About Us

    • Investigation Four: Coordinate Graphs

6th

  • Module 7 - Wonders of the World

    • Section 5: Temperature, Integers, and Coordinate Graphs
      • Reason for Alignment: This lesson contains discussion and teacher directions included to guide students through the concepts. The level is appropriate for this stage of a student skill. The discussion in the lesson should be useful for working with students in understanding the four quadrants and examples.

7th

  • Module 1 - Search and Rescue

    • Section 2: Integers and Coordinates
      • Reason for Alignment: The Cartesian Coordinate System is much the same as the Graphing lesson, with the same uses and applications though has more of an algebra feel to it.

Student Prerequisites

  • Arithmetic: Students must be able to:
    • perform integer and fractional arithmetic
  • Algebraic: Students must be able to:
    • work with very simple linear algebraic expressions
  • Technological: Students must be able to:
    • perform basic mouse manipulations such as point, click and drag
    • use a browser for experimenting with the activities

Teacher Preparation

Key Terms

coordinate plane

A plane with a point selected as an origin, some length selected as a unit of distance, and two perpendicular lines that intersect at the origin, with positive and negative direction selected on each line. Traditionally, the lines are called x (drawn from left to right, with positive direction to the right of the origin) and y (drawn from bottom to top, with positive direction upward of the origin). Coordinates of a point are determined by the distance of this point from the lines, and the signs of the coordinates are determined by whether the point is in the positive or in the negative direction from the origin

coordinates

A unique ordered pair of numbers that identifies a point on the coordinate plane. The first number in the ordered pair identifies the position with regard to the x-axis while the second number identifies the position on the y-axis

function

A function f of a variable x is a rule that assigns to each number x in the function's domain a single number f(x). The word "single" in this definition is very important

graph

A visual representation of data that displays the relationship among variables, usually cast along x and y axes.

negative numbers

Numbers less than zero. In graphing, numbers to the left of zero. Negative numbers are represented by placing a minus sign (-) in front of the number

Lesson Outline

  1. Focus and Review

    Remind students what has been learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson:

    • Choose a student in the class, ask another student to describe that person's location in the classroom. For example 3rd row 4th seat back. Use this as an application of the coordinate system

  2. Objectives

    Let the students know what it is they will be doing and learning today. Say something like this:

    • Today, class, we are going to learn about cartesian coordinate system.
    • We are going to use the computers to learn cartesian coordinate system, but please do not turn your computers on until I ask you to. I want to show you a little about this activity first.

  3. Teacher Input

  4. Guided Practice

    • Have students practice their skills with the General Coordinates Game.
    • For further practice or an alternative game, have the students play the Maze Game.
    • To show students that the coordinate plane is useful in more than just describing the location of objects lead a discussion on reading points off a graph. This will show the students that they can read graphs and find the equations of lines using their knowledge of the coordinate plane.

  5. Independent Practice

  6. Closure

    • You may wish to bring the class back together for a discussion of the findings. Once the students have been allowed to share what they found, summarize the results of the lesson.

Alternate Outline

This lesson can be rearranged in several ways.

  • Omit one or the other of the computer activities to reduce the amount of time spent.
  • Add a discussion about fractional movement on the coordinate plane.
  • For students who aren't ready to handle negative numbers yet, replace the Coordinates activity with the positive numbers only alternate versions:

Suggested Follow-Up

After these discussions and activities, students will be have learned to plot points on the coordinate plane and to read the coordinates off of a graph. The next lesson Function and Graphs will introduce students to the graphical representation of functions.