Contextualization

When studying mathematics and its various disciplines, we often ask ourselves: "Why should I learn this? Where will I use this knowledge in my life?" These are valid questions that deserve clear answers. In the case of studying areas, specifically the area of polygons, and in this project, the hexagon, we can see the practical value of this knowledge in various fields of study and work; architecture, design, physics, engineering, and even in biology and chemistry, where hexagonal shapes regularly appear.

For example, the molecular structure of graphene, an extremely strong and lightweight material, follows a hexagonal pattern. In the environment, the hexagonal shape is often found in the honeycomb of bees. The efficiency of this shape in terms of material economy and structural resistance is proof of its importance in the real world. In terms of design and architecture, the hexagon is often used for its appearance and also for its unique packing and accommodation properties.

Introduction

Mathematics is the language of the universe, and geometry is one of its most vital disciplines. Geometry allows us to understand shapes, measure distances, areas, volumes, and many other aspects of the world around us. Among the numerous figures that geometry studies, one stands out for its omnipresence in the natural world and the beauty of its symmetry: the hexagon.

In this sense, the hexagon is a figure with six sides. In a regular hexagon, all sides and angles are equal. Calculating the area of a hexagon is a bit more complex than calculating the area of a square or rectangle, as more dimensions need to be considered.

To achieve a proper understanding of how to calculate the area of a hexagon, it is important to have a good grasp of the properties of basic geometry and polygons in general, as well as familiarity with area calculation principles. The key concept for calculating the area of a hexagon is the concept of decomposing a complex figure into simpler figures - such as triangles - whose area we can easily calculate.

Practical Activity: "Unraveling the Hexagon"

Project Objective

The objective of this project is to provide an understanding of the area of a hexagon, applying geometry concepts and area calculation in a practical and collaborative way. Students will work in groups of 3 to 5, managing time, solving problems, and effectively communicating throughout the project development.

Detailed Project Description

This project will involve creating a three-dimensional (3D) model of a hexagon and subsequently determining its area. The idea is that students can break down the hexagon into six equilateral triangles, calculate the area of these triangles, and sum these areas to obtain the total area of the hexagon. The project should also emphasize the relevance and application of the acquired knowledge in the real world.

Required Materials

1. Cardboard or cardstock
2. Ruler
3. Compass
4. Pencil
5. Scissors
6. Glue
7. Calculator

Detailed Step-by-Step Guide for Activity Execution

1. Each group should start by drawing a regular hexagon on the cardboard or cardstock. Use the compass to ensure all sides and angles are equal.

2. Next, students should break down the hexagon into six equilateral triangles.

3. Each group should calculate the area of one of the equilateral triangles using the formula: A = (L² √3) / 4, where L is the length of the triangle's side.

4. After that, students should multiply the area of one of the triangles by six to obtain the total area of the hexagon.

5. Once the measurements and calculations are done, the groups should model the hexagon in 3D using the dissected triangles, gluing the edges to form the geometric figure.

6. With the model in hand, the groups should present to others the creation process, explaining how the area calculation was performed, and discuss the practical applications of this knowledge in the real world.

Written Document

After completing the practical project, each group should prepare a report that thoroughly describes the hexagon creation process and the calculations performed.

1. Introduction: Students should contextualize the theme, its relevance and application in the real world, as well as the objective of this project.

2. Development: Students should explain the theory behind calculating the area of a hexagon, detail the activity, indicate the methodology used, and finally present and discuss the results obtained.

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

4. Bibliography: Students should indicate the sources they relied on to work on the project, such as books, web pages, videos, etc.

Note that the practical activity and the written document complement each other. While in practice groups apply studied concepts and perform calculations, the document allows for the consolidation of learning, as well as the development of writing and argumentation skills.