Contextualization

Introduction

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. The word "trigonometry" comes from the Greek and literally means "triangle measure". Trigonometry is used in geometry because every shape can be broken down into smaller triangles.

In the context of trigonometry, special angles are those angles for which we can easily calculate the sine, cosine, and tangent. These special angles include 0°, 30°, 45°, 60°, and 90°. To understand trigonometry, it is crucial to understand the values of these special angles. This knowledge will not only allow you to solve more complex problems, but it will also form the basis for your understanding of the unit circle, a concept that is central to more advanced coursework in mathematics.

In this project, we will dive into the values of these special angles, learn about their applications, and engage in practical activities.

Importance and Real World Applications

Trigonometry, and more specifically, the understanding of special angles, plays a significant role in many fields. Engineers use trigonometric functions for measurements and in determining the shapes and sizes of structures. Architects use it to calculate structural load and integrity. In astronomy, it is fundamental for calculating distances between stars and planets. Even in fields such as music and sound engineering, special angles are crucial to understand sound waves.

The values of the trigonometric functions for these special angles often occur in real-life scenarios. For example, ramps inclined at 30° and mirrors angled at 45°. It is through understanding these angles and their values that we can make precise calculations and predictions about these real-life situations.

Resources

1. Khan Academy: Trigonometry - A great place to start your journey in understanding trigonometry. Provides different resources including videos, quizzes and practice exercises.
2. Wolfram MathWorld: Trigonometry - A comprehensive resource with in-depth information on different topics related to trigonometry.
3. Onlinemathlearning: Trigonometry - Offers detailed definitions and examples of trigonometric concepts.
4. Youtube: Trigonometry Course - A complete course on trigonometry covering all topics.
5. Books: Trigonometry by John Baley, and Trigonometry for Dummies by Mary Jane Sterling.
6. Documentaries: “The Story of Maths: The Language of the Universe“, “Donald in Mathmagic Land“.
7. TED Talks: "The unexpected math behind Van Gogh's "Starry Night" - Natalya St. Clair" etc.

Practical Activity

Activity Title: "Designing a Perfect Mini Golf Course with Trigonometry"

Objective of the Project

To utilize the knowledge of trigonometry and the values of special angles to design an innovative mini-golf course and to understand their applications in real-world scenarios.

Detailed Description of the Project

In this hands-on, interdisciplinary project, students will use their newly acquired trigonometric knowledge to design a mini golf course where the holes represent the special angles (0°, 30°, 45°, 60°, 90°) in trigonometry.

Each group will be responsible for creating a blueprint for the golf course, as well as the physical model of one hole. The golf hole should be designed with obstacles that are placed at special angles and the students will have to calculate the perfect shot using trigonometry.

Additionally, students will present their design to the class, explaining the theory behind the special angles used, and they will write a detailed report on the process.

This project perfectly combines mathematics with physical education and design. It challenges students not only in their comprehension of trigonometry but also their skills in design, creativity, collaboration, and presentation.

Necessary Materials

• Graph Paper
• Pencils
• Ruler
• Protractor
• Cardboard
• Art supplies for decorating
• Golf balls (or equivalent)
• Golf club (or equivalent)

Detailed Step-by-Step for Carrying Out the Activity

Step 1: Review the theory of trigonometry and special angles. What are their values? How can they be applied in real-world scenarios?

Step 2: Start brainstorming ideas for the design of the mini-golf course. Remember, each hole should represent one of the special angles and include obstacles that must be overcome using calculated shots based on that angle.

Step 3: Draw a blueprint of your mini-golf course on graph paper. Sketch the design of the hole your group will physically create. Use protractors to ensure the special angles are accurately represented.

Step 4: Build a physical model of one hole from your blueprint. Use cardboard and art supplies to make it three-dimensional, and make sure it accurately represents the special angle it's designed to illustrate.

Step 5: Test your hole by attempting to make a hole-in-one using the perfect angle. Use your theoretical knowledge to calculate the perfect shot.

Step 6: Present your hole and the trigonometric theory behind it to the class. Use your blueprint and physical model as visual aids.

Step 7: Write a report about the process of designing and building your mini-golf hole and how trigonometry was used throughout. Include the calculations for the perfect shot.

This collaborative project should take approximately 15 hours to complete over two weeks; therefore, good time management will be essential.

Deliverables

Written Document: After the practical construction and presentation, each group of students must write a document in the style of a scientific report. The report should contain the following four main sections:

1. Introduction: Provide the context of the project. Why is it relevant? Discuss the real-world application of special angles. Describe the objective of your mini-golf course project.

2. Development: Detail the theory of special angles in trigonometry. Explain how you applied this theory in designing your mini-golf course and your specific hole. Present the blueprint and the physical model of your hole and discuss the results of your tests. Indicate the methodology you used for the calculations.

3. Conclusion: Revisit the main points of your work. State what you learned from the project. Draw conclusions about the relationship between special angles and their real-world applications based on your mini-golf course design and creation.

4. Bibliography: Indicate the sources you used to work on the project, such as books, web pages, videos, etc.

Practical Model: Each group must bring their physical model of their mini-golf hole to class for presentation. Along with this, they must also present their complete course layout blueprint.

Presentation: Each group must deliver a presentation explaining their golf course, their specific hole, and how they used the theory of special angles in their designs.

This project will enhance students' understanding of special angles, their values, and their real-world applications. It also aims to build various socio-emotional skills, including teamwork, time management, problem-solving, creative thinking, and proactivity.