Symmetry on the Rectangular Coordinate System

An Investigation


Objectives:

In this activity, students will:

  • Investigate symmetry about the x-axis, y-axis and the line y = x by using reflection, which is a form of transformation.

  • Use coordinate geometry to describe symmetric figures.

  • Make generalizations about the coordinates of symmetric points based on the line of symmetry.

Overview:

In this activity, students will construct a triangle on a grid using the Graph menu of the Geometer's Sketchpad. The students will use the Transform menu to reflect the triangle about the x-axis, y-axis and the line y=x. The students will use the dynamic feature of the Sketchpad to investigate how the coordinates of the points of the triangle change depending on the line of symmetry.

Activities:

  1. The students will work in pairs. They will be asked to follow a set of written directions to do the sketch. One student will read the worksheet while the other student is on the keyboard. They switch roles from time to time.

  2. The students will use the Sketchpad to investigate symmetry about the x-axis, y-axis and the line y=x. They will copy and paste their sketch and written conjectures on a word processing program.









Tools and Resources:

Software

The Geometer's Sketchpad
GSP Resource Center
Appleworks or Microsoft Word for word processing

Other

Activity Worksheet
Graph paper
Ruler
Paper and Pencil

Tutorial Links

  1. Online Activity Guide for GSP
  2. 101 Project Ideas for GSP
  3. Tutorial for getting started with GSP from St. Paul's Academy
Assessment:

Evaluate students in their ability to:

  1. Follow written directions.
  2. Work collaboratively.
  3. Write conjectures based on their observations.
  4. Predict coordinates of points as they are reflected about the x-axis, y-axis or the line y=x.
  5. Plot points and reflect these points about the x-axis, y-axis or the line y=x using paper and pencil.
Tips for Teachers:

The concept of symmetry is useful in graphing functions and relations. Many concepts involved in the study of functions and relations like domain, range and intercepts become easier to learn when the students are skilled in graphing. This activity is a good motivational strategy to start up a unit on graphing functions and relations. The study of symmetry about the line y=x leads to the concept of inverse functions.



Standards

Principles and Standards of School Mathematics 2000:

In grades 9-12 all students should

° use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations;
° investigate conjectures and solve problems involving two- and three-dimensional objects represented with Cartesian coordinates.
° understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices;
° use various representations to help understand the effects of simple transformations and their compositions.
NETS Technology Standard:

Prior to completion of Grade 12, students will:

3. Analyze advantages and disadvantages of widespread use and reliance on technology in the workplace and in society as a whole. (2)
8. Select and apply technology tools for research, information analysis, problem-solving, and decision-making in content learning. (4, 5)
9. Investigate and apply expert systems, intelligent agents, and simulations in real-world situations. (3, 5, 6)
10. Collaborate with peers, experts, and others to contribute to a content-related knowledge base by using technology to compile, synthesize, produce, and disseminate information, models, and other creative works. (4, 5, 6)


The Integrating Technology into Math Instruction webpages project is partially funded by a grant from The Boeing Company. Integrating Technology into Math Instruction is a project of +PLUS+ and is displayed on the Los Angeles Educational Partnership Learning Exchange. +PLUS+ is an initiative of the Los Angeles Educational Partnership.
Updated July 2000