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There are five Transformation manipulatives. Translation, Reflection, and Rotation are all called "Isometries" because they keep the same ("iso-") distances and angles and shape. Each can be explored independently, and then the Transformation - Composition manipulative allows two of the isometries to be performed sequentially. The other available transformation, Dilation, moves every point in the plane away from, or toward, a given Center point by some fixed scaling factor. Dilation retains shape (the images are similar to the original) but changes each length by the same scale factor, so that Dilation is not included as an isometry.

Reflection involves a "mirror line," which in this manipulative can be turned or moved freely in the workspace. When students are first exploring mirror reflections, it may be useful to give them some paper and pencil exercises to see how well they can predict by sketching what the mirror image should be for a simple polygon. It should become clear that the image of each point is located on the line perpendicular to the mirror line, at the same distance on the other side of the mirror line. That is, the mirror line is the perpendicular bisector of the segment between the point and its image. When new pieces are added to the workspace, their reflected images are also shown, so that the student can experiment with constructions, color and group pieces. The image can be moved by sliding or rotating the original objects or by moving the mirror line.

Reflection is a powerful tool for studying symmetry. By constructing an object with, say, one axis of symmetry, the object can be moved so that its reflected image coincides with the original. If the constructed object is symmetric both vertically and horizontally, then it can be rotated to see both kinds of symmetry.

By adding coordinate axes and lattice points to the workspace, it is possible to examine a reflection analytically. Students can compare coordinates resulting from reflection in the x-axis or reflection in the y-axis by entering, say, a triangle in the workspace and placing one vertex at a lattice point and checking the predicted value of the corresponding image point.

Extending the exploration to the Transformation - Composition manipulative opens more questions about the result of two given reflections. The student should be able to discover that reflection in either coordinate axis, followed by a second reflection in the other axis, moves a point with coordinates (a,b) to the image point with coordinates (-a,-b); that is, the result is the image of reflection in a point, the origin. It then becomes profitable to ask what happens with two reflections in any two perpendicular mirror-lines. What about non-perpendicular mirror-lines?

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The National Library of Virtual Manipulatives is a three-year NSF supported project to develop a library of uniquely interactive, web-based virtual manipulatives or concept tutorials, mostly in the form of Java applets, for mathematics instruction (K-8 emphasis). The project includes dissemination and extensive internal and external evaluation. For more information, please visit http://nlvm.usu.edu/en/nav/vlibrary.html.

Principal Investigators
Larry Cannon
Jim Dorward
Bob Heal
Leo Edwards

Java Applet Programming
Ethy Cannon
Joel Duffin
David Stowell
Zeke Susman
Richard Wellman
Jennifer Youngberg

Web Site Programming
Joel Duffin