Module Title: Launching a Line

Standards:

HSF.TF.A.2 - Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
HSF.TF.B.5 - Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
HSF.TF.C.8 - Prove the Pythagorean identity sin2 (θ) + cos2 (θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

PBL Goal:
Students will build a zipline model, perform various calculations, collect trial data, and reflect upon the data and prototyping process.

Structure:

Each lesson in the "Launching a Line" module follows the 5E lesson framework to guide students through a comprehensive learning experience. The structure includes:

Engage: Capture students' interest with real-world scenarios, like zipline design, and introduce key concepts through interactive discussions or videos.
Explore: Allow students to investigate mathematical principles, such as the unit circle or angles of depression, through hands-on activities or virtual simulations.
Explain: Guide students to articulate their understanding of trigonometric identities and their application to zipline design.
Elaborate: Encourage deeper thinking by applying their knowledge to complex problems, such as calculating platform height and analyzing the safety of their zipline designs.
Evaluate: Assess students' comprehension with formative assessments, data analysis from their prototype testing, and reflective discussions on their design process.

Overview:
In the "Launching a Line" module, students embark on a project-based journey where trigonometry meets engineering design in the creation of a zipline in the Sangre de Cristo Mountain range. The module aligns with key standards such as HSF.TF.A.2, HSF.TF.B.5, and HSF.TF.C.8, guiding students through the application of trigonometric functions, the unit circle, and Pythagorean identities.

Lesson 1:
Lesson 1 lays the groundwork with an exploration of the unit circle and SOHCAHTOA, predominantly as a refresher from skills they should have acquired during Geometry. This lesson serves as an introduction to how trigonometry can be applied in real-world engineering scenarios like determining the height of a platform or the angle of a zipline.

Lesson 2:
In Lesson 2, students delve into the concepts of angles of depression and elevation, which are essential for designing the slope of the zipline. Using trigonometric identities, students calculate the ideal angle for both safety and performance. They will also explore how these angles directly impact the rider’s speed and experience. At the end of this lesson, students will be invited to start working on their lab sheets. Within the course, this is their third experience crafting an experiment with two treatment groups, one independent variable, one dependent variable, and a procedure. The will be invited to start building the line to take sample calculations.

Lesson 3:
Lesson 3 focuses on prototyping the zipline model, where students apply the Pythagorean identities, SOHCAHTOA, and their understanding of angles to ensure the design is a true right triangle. By using the unit circle and calculating average speed, students finalize their designs and test their models through data collection, reinforcing the mathematical concepts they've learned. By the end of this lesson, they have completed the first part of the project.

Place-Based Significance:
The Sangre de Cristo Mountain range was selected for its geographical significance, stretching from New Mexico, into Colorado. This area provides a dramatic backdrop for the zipline project, offering students the chance to work with real-world geographical data while emphasizing the importance of eco-tourism and environmental conservation. Students not only engage with the mathematical aspects of zipline design but also connect their learning to the natural world, understanding how these mountains, visible from Santa Fe, offer both recreational and educational opportunities. Through this project, students learn to balance the thrill of adventure with sustainable design and environmental stewardship.