Introduction

Mathematics is a universal language present in almost every aspect of our lives. And it is through mathematics that we can discover patterns, establish relationships, and better understand the world around us. In this project, we will dive into the topic of angles, specifically, we will learn about the angles formed by the hands of a clock. This is a practical way to understand how mathematical concepts manifest in everyday life.

Clocks with hands, although less common today in an increasingly digital world, are excellent examples of how mathematics is used in practice. They display the time of day using angles and circles, two fundamental mathematical concepts. By analyzing the position of a clock's hands at different times, we can learn to calculate the angle formed between them and understand the concept of an angle in a practical and meaningful way.

In the real world, understanding the concept of angles is crucial in various fields. Architects, engineers, designers, artists, pilots, navigators, and many other professionals need to understand and calculate precise angles in their daily work. Moreover, by understanding angles, we can also better understand coordinate systems, trigonometry, geometry, and by extension, the universe.

It is notable that the concept of angle is also strongly present in the field of Physics. For example, the concepts of angle and time are intertwined when we analyze the motion of celestial bodies. We could say that mathematics itself has many "hours" to tell, and this is exactly what we will try to discover in this project!

We recommend some resources to support your understanding of the key concepts:

1. Book: "Fundamentals of Elementary Mathematics: Plane Geometry - Vol. 3" (Gelson Iezzi and Carlos Murakami).
2. Video: "The use of angles in everyday life" - YouTube channel Matematica Rio with Prof. Rafael Procopio.
3. Website: "Angles" - Brasil Escola.

Activity

Activity Title: "A Day in Time: Discovering Angles in Clocks"

Project Goal

This project aims to apply the concepts learned about angles in the practical context of clocks. Additionally, it will foster important skills, such as teamwork, time management, proactivity, creative thinking, and problem-solving.

Detailed Description of the Project

Teams of students will work for two weeks to complete the project. The activity involves building an analog clock, measuring the angles formed by the hands at different times, analyzing the results obtained, performing calculations, and presenting the mathematical concepts involved through a report that connects theory and practice.

Necessary Materials

• Cardboard
• Compass
• Ruler
• Colorful markers
• Graphing software (e.g., Microsoft Excel or Google Sheets)
• Text editor software (e.g., Google Docs)

Detailed Step-by-Step Instructions

1. Clock assembly: Use the cardboard, compass, ruler, and colorful markers to build a physical analog clock. Be sure to mark the hours and minutes correctly.

2. Angle Measurements: Using the clock built, students will measure and record the angles formed by the minute and hour hands at 15-minute intervals (00:00, 00:15, 00:30, 00:45,...). This should be done considering a complete 24-hour day.

3. Angle Analysis and Calculations: Based on the measured angles, make the necessary calculations to determine the difference between the recorded angles and the theoretical angles. Remember that the hour hand moves 0.5° per minute, while the minute hand moves 6° per minute.

4. Graph Creation: Using graphing software, create a graph showing the evolution of the angles over time.

5. Report Writing: Finally, write a detailed report, in the format of a scientific paper, covering the following topics: Introduction, Development, Conclusion, and Bibliography.

Project Delivery

The final deliverables of the project are the built clock, the created graph, and the written report. In the report, students should:

• In the Introduction, contextualize the topic, explaining the relevance and real-world application of angles in clocks.
• In the Development, demonstrate the theory behind angles in clocks, explain the activity carried out in detail, indicate the methodology used, and present and discuss the results obtained.
• In the Conclusion, restate the main points of the project, mention the lessons learned, and conclude about the project.
• Finally, in the Bibliography, indicate the sources used to support the work, such as books, websites, videos, etc.