Context

The concept presented here is the Theorem of Thales, one of the first theorems studied in the Mathematics discipline that has a direct application in the real world. Questions like 'How tall is the building?' or 'How far away is that object?' can be answered using this theorem. In fact, the Theorem of Thales is one of the main foundations of Geometry, being essential to be understood for the study of all other areas within Mathematics.

The Theorem of Thales is a geometric theorem related to triangles and their respective angles. It was proposed by Thales of Miletus, a philosopher, mathematician, and astronomer from Ancient Greece. Thales was one of the first to apply mathematical methods to practical questions, such as measuring the heights of pyramids and the distance of ships from the coast.

This theorem is a great tool for solving problems involving proportionality relationships, similarity of figures, and length measurements. With the Theorem of Thales, for example, architects can calculate the best way to take advantage of natural light in a room, saving energy.

Mathematics is present in our daily lives in countless ways, and we do not always realize it. From going to the supermarket, using technology, or decorating the house, Mathematics is present. Understanding the Theorem of Thales and how to apply it effectively means having a powerful tool to solve complex problems in simple ways.

Practical Activity

Activity Title: Practicing the Theorem of Thales

Project Objective

The main objective of this activity is to empower students to understand, apply, and solve practical problems using the Theorem of Thales. Additionally, it aims to develop skills in working as a team, communicating effectively, and managing time.

Detailed Project Description

In this project, each group of 3 to 5 students will have to apply the Theorem of Thales to solve a practical problem: finding the height of an unknown object, such as a tree, a lamppost, or a building, using only a ruler, a rod, and the Theorem of Thales. In addition, students must document the entire process in a report.

Required Materials

1. Ruler
2. Approximately 1-meter rod
3. Pencil and paper for notes
4. Camera to document the process (optional, but recommended)

Detailed Step-by-Step for Activity Execution

1. Choose the object whose height you want to measure, it can be a tree, a lamppost, or a building.
2. Use the rod and the ruler to, following the Theorem of Thales, measure the height of the chosen object. Establish the proportion between the length of the rod's shadow (to be measured with the ruler) and the height of the rod, in relation to the length of the shadow of the unknown object.
3. Repeat the measurement at least 3 times to confirm the accuracy of the results.
4. Document the entire process with photos and/or videos.
5. Discuss as a group and interpret the results obtained.

The groups have one week to complete the activity.

Project Deliverables

1. Written Report: each group must produce a report detailing the entire process. The report should include:

• Introduction: The student must contextualize the theme, its relevance and application in the real world as well as the objective of this project.
• Development: Detailed description of the entire activity, from the choice of the object to the obtaining of the final results. It should explain how the Theorem of Thales was applied in practice and the methodology used to obtain the measurements. Presentation and discussion of the results should also be addressed in this section.
• Conclusion: Conclusions about the project, summarizing its main points, explaining the learnings obtained, and discussing the effectiveness of the Theorem of Thales as a tool to solve day-to-day problems.
• Bibliography: Indication of the sources consulted during the project.

2. Oral Presentation: Each group will have the opportunity to present the project to the class, emphasizing the methodology used and the results obtained. This allows the development of communication skills and the exposure of ideas.