Introduction

An angle is a measure that describes how many degrees a geometrical shape, such as a circle or a line, is tilted with respect to another. In other words, an angle is the measurement of the space between two lines that cross at a common point, also called the vertex. Angles are classified into various types depending on their magnitude, such as acute, right, obtuse, reflex and complete angles.

Angles are a fundamental concept in mathematics, and more specifically in geometry. Learning about angles provides students with important analytical skills that are applicable in a vast range of real life situations, such as architecture, engineering, sports, arts, and so on. It is also a fundamental skill for understanding other more advanced topics in mathematics.

For this project, we will explore the topic of angles in a practical and fun way, focusing on their application in our timekeeping devices: clocks. In other words, we will learn how angles show up in our everyday lives through the hands of a clock.

Contextualization

A clock, a device we use every day, works based on the idea that a complete day is a cycle of 360 degrees, which is equivalent to a complete circle. The hands of a clock form various angles depending on the time displayed. Understanding how angles are formed on a clock can help us better understand the idea of time and how it is measured.

Furthermore, understanding how to calculate the angle formed by the hands of a clock is a great example of how mathematics integrates into our daily lives. This knowledge can be useful in many real-world situations, such as coordinating daily activities, planning trips, setting work schedules, and even making it easier to read analog clocks.

Here are some resources that you can use to deepen your understanding of angles and how they relate to clocks:

1. Book: "Geometry and Algebra in High School: An approach based on the study of topics in Mathematics" by Antônio Caminha Muniz Neto and Marcelo Câmara dos Santos.
2. Website: Mundo Educação - Mathematics (https://www.mundoeducacao.com.br/matematica/angulos.htm)
3. Videos: Khan Academy (https://www.khanacademy.org/math/basic-geo/basic-geo-angle)

Practical Activity

Activity Title: The Angle of Time

Objective

To learn how to calculate the angle formed by the hour and minute hands of a clock.

Detailed Description

Students will be divided into groups of 3 to 5 members. Each group will be tasked to create their own clock using simple and widely available materials (such as cardboard, glue, hot glue sticks, colored paper, markers, etc.). In addition, students will be required to calculate the angle formed by the hour and minute hands for different times of the day.

Materials

• Cardboard
• Hot glue sticks
• Colored paper
• Markers
• Ruler
• Protractor

Step by step guide:

1. Using the cardboard, students should begin by drawing and cutting a large circle which will serve as the "face" of the clock.
2. Using markers and the ruler, students should mark the numbers of the clock where they belong (from 1 to 12).
3. Next, they should draw and cut out two clock hands from colored paper, a long one for the minutes and a short one for the hours.
4. The clock hands should be glued to the center of the clock in a way that allows them to rotate.
5. Once the clock is built, students should use the protractor to measure the angle formed by the hands at different times of the day (for example 3:00, 6:30, 12:15, etc.).
6. Each group should perform this procedure for at least 5 different times and write down the results, along with the corresponding time.
7. Document all the steps of the process, including pictures of the clock at each time and the measurements of the respective angles.

The expected duration of the project is between two and four hours per student, with a deadline of one week.

Project Deliverables

At the end of the project, each group must present:

1. The constructed clock, demonstrating the different angles for the chosen times.
2. A written report detailing the experience, divided in the following sections:

• Introduction: The group should address the relevance of studying angles and their real world applications, more specifically, in the case of clocks.

• Development: The group should explain the theory behind angles and how it applies to clocks. Next, they should detail the whole process of constructing the clock, as well as the methodology employed to calculate the angles at different times. The results obtained should be presented and discussed in detail.

• Conclusions: The group should conclude the report by revisiting the main points covered, reflecting on the lessons learned and the conclusions drawn from the project.

• Bibliography: The group should list the sources of information used to carry out the project.

Students will be encouraged to apply technical skills, such as angle calculation, as well as socio-emotional skills, such as time management, communication, problem solving, creative thinking and proactivity throughout the project.

It is worth mentioning that this project encompasses both theoretical and practical learning and fosters collaboration and teamwork among students.

Observation: In addition to evaluating the constructed clock, the written report by the group, which must be submitted together with the clock, will be assessed.