High School: Geometry
6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
The words "rigid" and "motion" sound like complete opposites, don't they? Well, not in geometry. A rigid motion is a motion—or transformation, in geometric lingo—that preserves distance and lengths; in other words, every segment of the image is congruent to its preimage.
Students should already be comfortable with the ideas of translations, reflections, and rotations. They have to use rigid motions to not only transform figures and predict the effect it has on a specific figure, but also to determine when two figures are congruent.
Congruence between two geometric objects can be defined as a rigid motion on the coordinate plane that maps one object onto another. Students should be able to recite this this definition verbatim. Or, more importantly, they should actually understand it.
The key here is to differentiate between congruence and carrying one object onto another. While two reflected objects can't always be carried onto one another, they can still be congruent because all that separates them is reflection—a rigid motion. Make sense?
Once students fully understand how these topics fit together, they will be able to use transformations to not only find and prove properties of geometry, but to create patterns, including tessellations.
If your students are giving you a hard time with why these things are important, remind them that if they ever want to dabble in design, whether it's fashion, interior, or video game, they have to identify and create patterns. At the root of patterns is, you guessed it, geometry. That'll get them thinking about transformations and congruence the next time they watch Project Runway or play Call of Duty.