Contextualization

The study of angles is a crucial part of mathematics, especially when it comes to geometry. Since antiquity, humans have realized the importance of angles in constructions, navigation, and eventually in sciences like physics. We will see that even our daily lives are guided by angles, as we look at something as common to us as an analog clock.

Analog clocks are a physical representation of the passage of time. They are made up of three hands - seconds, minutes, and hours - that move around a circle. Despite their apparent simplicity, there is a complex geometry behind these movements. For example, the angles between the hour and minute hands can vary widely depending on the time of day.

The angles on a clock are a wonderful example of how abstract mathematical concepts can be incorporated into everyday objects. With each passing minute, the angles between the hour and minute hands change, creating a fascinating pattern of shapes and relationships. This shows how relevant the study of angles is, as it reveals the connection between this seemingly abstract concept and the world around us.

Introduction

To start this project, it is essential that you have a solid understanding of the concept of angles. An angle is defined as the amount of rotation needed to bring a line (or radius) from its original position to a final position, always taking a fixed direction as a reference. In our case, the angle will be measured between the hands of an analog clock.

Angles are measured in degrees or radians. In a clock, we use the concept of degrees, where a full rotation (360 degrees) represents the daily cycle of a 12-hour clock. This means that each hour on a clock represents an angle of 30 degrees and each minute represents an angle of 6 degrees.

It is important to emphasize that, although we are talking about an analog clock, the concepts and principles used here apply to many other areas of science and engineering. In other words, angles play a central role in understanding how the world works.

For further theoretical study, we recommend the following references as a basis:

1. Flat Geometry - Angles
2. Introduction to Angles

Remember: theory and practice go hand in hand. So, besides studying, let's get our hands dirty and apply these theories!

Practical Activity

Activity Title: Mathematical Clock

Project Objective:

The main objective of this project is to allow students to understand and apply the concepts of angles, using an everyday object, the analog clock, to calculate the angle formed by the hour and minute hands at different times.

Project Description:

Students will be divided into groups of 3 to 5 members. Each group will use the theory of angles and an analog clock to calculate the angle formed by the hour and minute hands for different times of the day.

The project requires students to apply theoretical concepts of angles in practice, using their observation skills, problem-solving abilities, teamwork, and critical thinking.

Required Materials:

1. An analog clock with movable hands, visibly marked for hours, minutes, and, if possible, seconds.
2. Pencils and paper for notes.
3. Calculators for necessary calculations.
4. Books or online resources for reference.

Step by Step:

1. Each group should study and discuss the concept of angles as recommended in the introduction section.

2. Next, each group should analyze the structure of the clock and identify how the hands move and form angles.

3. Set up the clock to display different times. For example, 3:00, 4:30, 6:15, 8:45, etc.

4. Students should calculate the angle formed by the hour and minute hands for each set hour.

5. Repeat this process for at least 10 different times.

6. All observations, calculations, and conclusions should be noted and discussed within the group.

7. Finally, each group should prepare a detailed project report as described in the introduction section.

Project Deliverables:

After completing the practical part, each group must produce a written document, in the form of a report, containing:

1. Introduction: The group must contextualize the theme, explaining its relevance and real-world application, and the project's objective.

2. Development: The group must explain the theory of angles, describe the activity carried out in detail, indicate the methodology used, and finally present and discuss the results obtained.

3. Conclusion: The group must conclude the work by summarizing its main points, explaining the learnings obtained, and drawing conclusions about the project.

4. Bibliography: The group must indicate all sources used to work on the project, such as books, web pages, videos, etc.

This report must be submitted to the teacher within one month of the project's start. The document will be evaluated not only for the understanding and application of the theory of angles but also for the group's ability to collaborate, manage time, and effectively solve problems.