Contextualization

Theoretical Introduction

Geometry is a branch of Mathematics that focuses on the study of shapes, measurements, spaces, and their respective properties. Among its various divisions, we have Plane Geometry, which is dedicated to the study of figures bounded by straight or curved lines, all lying in the same plane. Quadrilaterals are a class of geometric figures that belong to this division.

Quadrilaterals are polygons with four sides. Depending on the properties of these sides, such as length, whether they are parallel or not, the angles they form, among others, we have different types of quadrilaterals, such as the square, rectangle, rhombus, parallelogram, and trapezoid.

Understanding the properties of quadrilaterals allows us to deduce a variety of theorems and laws that facilitate their manipulation when working with these figures. For example, we know that the sum of the interior angles of any quadrilateral always adds up to 360°, regardless of the type of quadrilateral. Furthermore, each type of quadrilateral has its own characteristics, which allow for the efficient solution of various geometric problems.

Contextualization

Understanding quadrilaterals has a central relevance not only in the field of mathematics but also in various practical applications. The study of quadrilaterals is essential to understand other concepts and techniques in geometry and trigonometry. Moreover, they have many practical applications in engineering, design, architecture, and other applied sciences.

For example, in civil construction, the blueprint of a house is usually composed of a series of quadrilaterals that delimit the spaces. Graphic designers use quadrilaterals to create shapes and layouts. In civil and mechanical engineering, quadrilaterals are used to calculate stresses and strains in structures. In short, quadrilaterals are a fundamental tool in many disciplines dealing with space, shape, and measurement.

Research Materials

Below are suggestions for materials for further study on the topic:

1. Book: IEZZI, Gelson; et al. Fundamentals of Elementary Mathematics - Vol. 1 - Sets and Functions. São Paulo: Atual. This book offers a good introduction to the fundamental concepts of mathematics, including plane geometry and quadrilaterals.
2. Website: Just Mathematics - On this website, you will find an entire section dedicated to plane geometry, including quadrilaterals.
3. Video: Professor Ferretto - Quadrilaterals - This video provides a detailed explanation of the properties of quadrilaterals, with solved examples.

Practical Activity

Activity Title: "Architecting with Quadrilaterals"

Project Objective

The activity "Architecting with Quadrilaterals" aims to practically explore the characteristics and properties of quadrilaterals in the creation of a house project using only these geometric figures. Through this activity, students will be able to apply their theoretical knowledge to solve a concrete problem, developing teamwork, communication, critical and creative thinking skills, as well as technical skills associated with mathematics and geometry.

Project Description

The project will be carried out in groups of 3 to 5 students. Each group will be tasked with producing a simplified architectural project of a house using only quadrilateral figures. The dimensions of the house and the area of the spaces must be calculated and presented, explaining the importance of using quadrilaterals in its construction.

Required Materials:

1. Graph paper
2. Ruler
3. Pencil and eraser
4. Calculator
5. Books and online resources for research

Step-by-step for the activity

1. Study the concepts and properties of quadrilaterals, considering their peculiarities and practical applications.
2. As a group, discuss ideas for the house project. The house should include different types of rooms, such as living room, bedrooms, kitchen, and bathrooms, and all of them should be represented by different types of quadrilaterals.
3. Draw the house plan on graph paper using only quadrilateral figures. Ensure that all lines are straight and all measurements are correct. Use an appropriate scale to represent the area of the house.
4. Calculate the area of each room in the house and the total area of the house. Remember that the area of a quadrilateral can vary according to the type of quadrilateral.
5. Prepare a detailed report of the project, including the house drawing, the measurements of each room, the total area of the house, and an explanation of the choice of different types of quadrilaterals for each room.

Project Deliverables

At the end of the activity, each group must deliver:

1. The drawing of the house plan, properly dimensioned and calculated.

2. A written report on the project. The report should include the following topics:

   • Introduction: In this section, the student should briefly explain the relevance of studying quadrilaterals and how they were used in the house project. Additionally, they should clarify the objective of the activity.
   • Development: In this part, the student should describe the theory about quadrilaterals and their properties. Then, they should explain the project methodology: how the house was designed, which types of quadrilaterals were chosen and why. Each step of the activity should be properly explained and justified. Furthermore, the calculations of the areas of the rooms and the total area of the house should be presented in this topic.
   • Conclusions: Here, the student should summarize the main points of the work, mention the main learnings, and discuss and interpret the results obtained. Finally, they should reflect on the importance of geometry and particularly of quadrilaterals in the real world.
   • Bibliography: Indicate all sources consulted for the project development.

The idea of the project is for students to integrate theory and practice, observing concretely how the knowledge of mathematics, especially plane geometry, can be applied in daily activities, such as in planning a house.

Duration: The project should take between two to four hours per student to complete and will have a one-week deadline.