To introduce hyperbolas, have students find real-world objects which can be described through hyperbolas. Any cone can be cross-sectioned into hyperbolas, so students should look for cones. Examples include traffic cones, ice cream cones, conic paper cups and party hats. Ask more advanced students to measure the hyperbola and to approximate its shape through an equation. Encourage less advanced students to bring in or photograph cones or simply describe the shape of its hyperbolas to the class.
The history of hyperbolas overlaps with the history of the other two conic sections, the ellipse and the parabola. Conic sections were first explored by the ancient Greeks and are among the oldest branches of mathematics. In modern use, hyperbolas are used in optics, engineering and even air travel. Have students research the Greek mathematician who is said to have first described hyperbolas and compare his mathematic description of hyperbolas to modern equations.
The equation of a hyperbola is found in every geometry textbook and should be familiar to your students as they learn about conic sections. Present your students with a series of hyperbola equations and have them race to see who can graph the hyperbolas the quickest. Have students work individually or in groups. For another challenge, present students with a set of already graphed hyperbolas and have them attempt to approximate the hyperbolas' equations. Crown the winning student(s) the Hyperbola Hero.
Hyperbolas describe the path a satellite or probe will take as it passes an orbital body at escape velocity or higher. For an advanced project, instruct your students to research the escape velocity for earth and other bodies in the solar system. They should then calculate the potential orbital trajectories for spacecraft flying to and from earth. Have them research and analyze how the angle of approach and entry speed alters the hyperbolas. For extra credit, have students review the orbital trajectory equations for historical space flights, such as the Galileo probe, which flew by Jupiter, or the Voyager probe, which flew by both Jupiter and Saturn.