High School: Geometry
2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
In today's world, it seems more and more people are trying to transform themselves in some way. Whether they want to transform their inner selves through self-help books and meditation or their outer selves with exercise and more cosmetic surgery than Joan Collins can handle, we shouldn't forget that shapes sometimes need a change as well. These changes are called transformations.
No, not Transformers. They're cool and all, but they're not the same.
A transformation means somehow altering a shape on the coordinate plane. Whether we move, flip, stretch, shrink, or turn the shape, we're performing a transformation. Unfortunately, an Extreme Makeover doesn't count when it comes to shapes.
The three main types of transformations—translations, reflections, and rotations—are called "rigid" because they preserve the distance and angles of their shapes.
Though it means something much different in your students' foreign language classes, translation means moving a shape in one direction. They can think of it as "sliding" that shape. No turning, no flipping, no complex movements. Just a straight slide. The shape stays exactly the same, but in a new place.
Just as it sounds, rotation is a circular movement. The shape moves around a central point. It's like turning a dial in one direction or the other, causing the object to stay where it is, but spin clockwise or counterclockwise. After 360°, though, you end up right where you started.
Students should think of reflection as looking at a shape in a mirror. Everything about the shape is the same, except one is the mirror image of the other. If imagining a mirror is strange or difficult, your right and left hand are reflections of each other, too.
Students should know that when any of these transformations are done in the coordinate plane, they can be described as functions that take points in the plane as inputs and give other points as outputs. The best way to see this is through coordinates of a simple shape, like a triangle. Moving up one unit means we add 1 to every y coordinate. Our input is y and our output is y + 1. Easy peasy.
We've been talking about transformations that move shapes without altering the shape itself—only the coordinates on which it sits. In other words, these transformations "preserve distance and angle," but there are some transformations that change them. This happens when multiplication is used, as opposed to addition or subtraction.
When you multiply, the distances between lines of the shape on the coordinate plane are lengthened. Multiply the x coordinates by 2, and the shape grows twice as wide; multiply the y coordinates by 2, and the shape grows twice as tall. This is called a "stretch" and tell your students to be careful with them. Otherwise, we may end up with gigantic shapes that'll want to take over the world.
On the other hand, if you multiply the coordinates by a number less than 1, the transformation is called compression. If compression were made into a Hollywood movie, it'd be called, "Honey, I Shrunk the Shapes."
Students should know how to move a given figure around the coordinate plane according to the different types of transformations. They should also represent a translation as a function using coordinates and understand the difference between transformations that preserve distance and angle and those that do not.
In no time, they'll be transforming themselves into the best geometry students you could hope for! Awww.