Lesson Plan #12 - Parametric Arrows
By Timothy J. Cody
Physics/Calculus Lesson

The bow and arrow have a rich history among the Native American and its use as a tool for hunting and a weapon of defense or war. Many lessons can be created to include the use of the bow and arrow into a variety of curriculums. Among some of the curriculums that could utilize the bow and arrow in various units could be: Art, Social Studies, Language Arts, Physics and Mathematics.

Objectives:
For the students to connect with the use of the bow and arrow for hunting among the Native Americans and apply it to properties of Physics and parametric equations to be used in Calculus. Included in the lessons would be several of the National Council of Teaching Mathematics Standards, which would be applied with the lessons.

Time required:
Depending on the depth of each curriculum's lesson, the one involving the team effort of Physics and Calculus could be 2-3 weeks.

Materials required:
Computers with Internet access, Ti graphing calculators, Vernier products CBL (calculator based laboratories) and photo gates attachments, bow and arrows, targets, measuring tapes

Procedures:
Students will first be asked to keep a journal of their work to include in a report on the total experiment and all related work. Students will need to connect with an on-line sight in the Internet to download programs for photo gates with CBL units and the Tt calculators. Students will then be ready to set up CBL and photo gates on a course with a target approximately 50 or 100 meters away. Photo gates will be placed on the ground to record various intervals during the flight path of the arrow from which data points will generate a parametric equation. This equation will be used to further explore concepts of vectors in Physics and derivatives in Calculus. Needed will be preliminary calculations for the height of targets and release points of the arrows. This will be done with the tape measures as well as the distances between the targets and release points. From this also will come data collected to map out the path of the arrow on grid paper with a rectangular coordinate system. In addition to the data collection and calculation work, there will also be presentations by Native American community resources giving talks on the construction of the bow and arrows in traditional fashion as well as a bit of historical background on the cultural uses of these implements.

Mathematical Standards:

From the National Standards for Mathematics, the following would be covered in this lesson: Standard 1-Mathematics as Problem Solving with investigation application, recognition, formulation of real-world situations of this experiment; Standard 2-Mathematics as Communication with reflection, journalizing, formulating, expressing, extending, and appreciation of the process involved; Standard 3-Mathematics as Reasoning with testing, conjecturing, constructing, and formulating assertions of this exploration. Standard 4-Mathematical Connections with the relation of math to the flight and properties involved in shooting a bow and arrow.
Standard 5-Algebra with both calculation work and equations for Physics
and Calculus topics; Standard 6-Functions with real-world situations involving graphs tables and parametric changes.

Assessment:
For each particular discipline, a variety of methods of assessments could be used from the proper use of technology and the ability to use and set up equipment and materials to the correctness of grammar in journals. A whole host of things could be evaluated from this project and its different concepts available to be taught.