Podcasting Parabolas

Objectives:
*Students should be familiar with solving a system of equations in three variables.
1. Students will be able to identify the basic parts of a parabola: vertex, focus and directrix.
2. Students will discover through their own findings the relationship between three points of a parabola and construct the equation of specific parabola defined.
3. Students will be able to identify and digitally capture real-life representations of parabolas
4. Students will organize the information collected to create and publish a podcast that summarizes their individual discoveries about the history of parabolas, everyday use (beyond architectural structures), the construction of a specific parabola's equation, given three points and the relationship between these points and the shape of the graph.

Introduction to the Activity:
Formal Definition of the Parabola will be explored with a model cone that has been intersected with a plane that is parallel to the side of a cone.

Exploration Activity:
Parabolic Trajectory will be modeled and discussed with a tennis ball. Students will assist in marking the first and second point of impact, as well as the height of the tennis ball that is bounced in front of a pre-measured wall (A digital camera can be used to photograph the actual parabolic path of the ball)

The process for finding the equation of the parabola will be discussed as outlined below:
The general equation of a parabola is y = ax^2 + bx + c
You have three pairs of points that are (x,y) ordered pairs. Substitute the x and y values of each point into the equation for a parabola. You will get three LINEAR equations in three unknowns, the three constants.
You can then solve this system of three equations for the values
of A, B, and C, and you'll have the equation of the parabola that
intersects your 3 points.

Students will then proceed to collaborate on the correct mathematical calculation of the parabolic trajectory of the tennis ball. (Multiple "bounces" can be performed and calculated to explore the relationship between the three points and the shape of the parabola)

Class discussion to follow.

Rearch/Data Collection
Students will then work with a partner and use the internet to research the history of the parabola, everyday uses/occurrences of parabolas (beyond structural entities), and find/create supporting graphics to be used in their Parabola Podcast. Students will also be assigned to capture their own images of parabolas they see in everyday life. [17th Century - Galileo's discovery to benefit canoneers; McDonald's Logo; Satellite Dish receptors (focus); Nature - Rainbows, Mountains, Quarter-Moon; Cave entrances, splash of water, stream of water from hose or geyser; Moire Fabric; Subaru Telescope in Hawaii; Flashlights; Headlights; Amphi-Theaters; Fireworks; ball trajectories (soccer, football, basketball, etc..) Gymnast's trajectory of a back-flip; back-bend - to name a few acceptable findings, but by no means all]

Organization/Planning
After gathering necessary components, partners will organize the data and compose a script to be used in their podcast. (Students will be required to submit a written commentary of their podcast for teacher's approval prior to publishing/sharing contents) Groups can choose to self-edit their script or have other groups help with editing requirements.

Integration of Multiple Technologies
Creating the podcast - students will record their commentary (to include an intro jingle with correct/appropriate ducking); Students will then import supporting pictures/graphics that correspond to the scripted recording. (*A minimum of two links should be included in the publication) Again groups may self edit or ask other groups/teacher to critique prior to sharing. After final edits have been made and an ending has been set - students will export files to flash drives - which will then be uploaded and published by the school IT personnel.
(Students will be required to create an enhanced podcast - but may choose to make a video podcast if desired (and capable).

Assessment
Podcasts will be graded with a student made rubric designed by the class members(and approved by the teacher).
Informal Student Reflections of the activity.

OPTIONAL ENRICHMENT ACTIVITY: After uploading student made pictures - the locations of the pictures can be mapped and included in the podcasts to share the specific locations of known parabolas in our community.