Context

Mathematics is a discipline that, through one of its branches, Geometry, allows us to analyze and understand the shapes that surround us. Geometry is essential for various areas of knowledge, from Art, Architecture, Engineering, to even Biology and Chemistry. Among the various shapes we can study, the hexagon is one of them. A polygon with six sides, present in various natural and artificial structures.

Hexagons are one of the most efficient shapes in nature. For example, bees create hexagonal cells in their hives. The reason? The regular hexagon allows bees to use less wax to build more cells, enhancing honey storage. Engineers, in turn, use the hexagon in structures due to its resistance and balanced distribution of forces.

Introduction

The concept of area is one of the fundamental in mathematics and is widely applied in many areas of science and industry. In this project, we will explore the theory and practice of calculating the area of a hexagon, especially the regular hexagon. A regular hexagon is a polygon with six equal sides, and, unlike most polygons, it has a simple formula to calculate its area.

When calculating the area of a regular hexagon, we apply our knowledge of trigonometry and geometry. This polygon can be divided into 6 equilateral triangles, which facilitates the calculation of the area. This skill is crucial in solving many practical problems, including, for example, the design of soap bubbles, packaging, mosaics, and even in the creation of board and digital games.

Each stage of this project will challenge you to explore the concept of the area of a regular hexagon under different circumstances. Thus, you can develop your mathematical skills while working as a team to solve problems and challenges presented.

Practical Activity

Activity Title: "Exploring Areas: The Power of the Hexagon"

Project Objective

This project aims to develop technical skills in calculating the area of a regular hexagon and promote essential socio-emotional skills such as teamwork, time management, and communication. Additionally, it aims to enhance students' understanding of the practical application of mathematical concepts in the real world.

Detailed Project Description

Students will be divided into groups of 3 to 5 people. Each group will be tasked with assembling and calculating the area of a regular hexagon using materials such as popsicle sticks and string. In addition, the groups will also be challenged to create a simulation in a program like GeoGebra or similar, corroborating their practical results.

Required Materials

• Popsicle Sticks.
• String.
• Ruler.
• GeoGebra or similar software.
• Pens, pencils, paper, and eraser.

Step-by-Step Guide for the Activity

1. Assembly of the hexagon: Students will use popsicle sticks to build a regular hexagon, fixing the vertices with string. Students must measure and record the length of the hexagon's side.

2. Manual calculation of the area: With the side measurement, students should calculate the area of the regular hexagon using the formula Area = (3 * √3 / 2) * side².

3. GeoGebra Simulation: Students will replicate the assembled hexagon in GeoGebra or similar software, using the side measurements obtained earlier. With the program, they will calculate the area of the hexagon and check if it matches the manual calculations.

4. Group Discussion: After the practical stage, students will gather to discuss the similarities and differences found in manual area calculations and those done through the software. They should reflect on the importance of calculating the area of hexagons and how they are applied in everyday life.

Project Deliverables

Groups should prepare a final report containing:

• Introduction: Providing context on the hexagon, its relevance and application in the real world, as well as the objective of this project.
• Development: A detailed explanation of the hexagon assembly procedure, manual area calculation, GeoGebra simulation, group discussion, and the differences (if any) between manual area calculations and software calculations.
• Conclusion: Reflections on what they learned during the project, the importance of calculating the area of the hexagon in everyday life, as well as the skills they developed throughout the project.
• Bibliography: All sources of information that were used to assist in the project.

Students should strive to ensure that the report is cohesive, coherent, and free of grammar errors. They will be evaluated on both teamwork and their ability to calculate the area of the regular hexagon and effectively communicate the results.