FALLS CHURCH, Virginia—On paper, plane geometry can seem pretty flat and uninteresting. But when eighth-graders have a chance to use geometric concepts to plot shots on a pool table or sink a putt in miniature golf, they see that the subject is not so dull after all.

George Mason Middle School teacher Kathy Hawthorne begins her Geometry in Motion unit by having students use tracing paper and a geometry construction tool on the computer to explore the concepts of reflection, translation, and rotation. Hawthorne guides students to discover the algebraic rules for reflecting points over different lines or rotating points about a center point. She likes the software because "it allows students to explore complex geometrical problems without getting hung up on the constructions by hand. Once they create a sketch, students can test their conjectures for various shapes and sizes by clicking and dragging shapes." No erasing required. 

Once they understand these basic functions, students are ready to put their learning into action. Hawthorne uses an overhead projector to show them a calculator program that animates a shape by moving it around the coordinate plane. She explains the code necessary to write such a program, then asks students to write their own program that will move a shape on the coordinate plane with a minimum of two transformations.

"Some students already have lots of experience programming," she says, so she asks them to challenge themselves by including either a size transformation or a glide reflection. Sometimes she just tells advanced students, "Impress me!" And they do. "Many of these students are more expert at programming than I am. They really like the idea of showing me something new," she says. One student, for instance, created a program that showed a snowboarder riding down a mountain slope and rotating in mid-air. "It's incredible what kids can come up with when they're interested," Hawthorne says.

On the due date, Hawthorne links students' calculators to her computer, which projects onto an overhead screen that the whole class can see. Graphing software enables her to download a hard copy of the code, which students can then take home. She has them write comments alongside the code, explaining the process of their program step-by-step and further cementing their understanding.

The applications get even more creative when Hawthorne introduces students to a master of tessellation, the late artist M.C. Escher. She hands out art paper and challenges students to create their own unique tessellation designs. "The results," she says, "are fabulous."

Finally, Hawthorne sets the stage for students to apply what they've learned about geometry to problem-solving situations. That's where billiards comes in. Reminding them what they've learned about the law of reflection—which states that the angle of incidence will equal the angle of reflection—she ushers them to the school computer lab for an online game of pool. "I give them five minutes to use their knowledge of reflection to perfect a banked shot," she says. Once they have mastered this skill, she has them switch to another Web site that features online miniature golf. "They calculate the path of the ball to make as many hole-in-one shots as possible."

All the pieces come together in the last activity of Geometry in Motion. Hawthorne challenges students to arrange a laser pointer and mirrors so that the beam of laser light will reflect off all three mirrors and then strike a target. "Students work in groups and start with a scale drawing of the classroom. Then they go to the computer lab and create a scale drawing of the room using the geometry construction tool," she explains. Back in the classroom, groups set up their mirrors and target, then demonstrate to their classmates whether their plan results in a "hit" or a "miss."