Contextualization

Introduction

Geometric constructions are a fundamental resource in the discipline of mathematics, widely used to create graphic representations of diverse concepts and, through them, better understand and apply these concepts in problem solving. Since ancient times, geometric constructions have been used by mathematicians, engineers and artists as crucial tools in their work.

Geometric constructions are based on simple operations such as measuring a segment, constructing a line, creating a circle with a given center and radius measured from a segment, determining the point of intersection of two lines or a line and a circle, among other operations. But through these simple operations it is possible to build more complex geometric shapes, such as triangles, squares, hexagons, and even three-dimensional figures such as cubes, pyramids, etc.

In this project, we will focus on constructing mediatrices, bisectors, 90°, 60°, 45° and 30° angles and regular polygons. To carry out these constructions, we will use drawing instruments, such as a ruler and compass, as well as dynamic geometry software, which allows us to construct, animate and explore geometric figures interactively.

Contextualization

Geometric constructions have several applications in the real world. For example, they are used in engineering for the planning and construction of structures, such as bridges, buildings, tunnels, etc. In art, they are used to create works that explore shapes, patterns and symmetries. And in physics, they are used to model and analyze physical phenomena, such as the movement of a pendulum, the trajectory of a projectile, the propagation of light, among others.

Geometric constructions also play a fundamental role in mathematics education. They help develop spatial reasoning, understanding of abstract concepts and problem-solving skills. In addition, the process of making geometric constructions involves a combination of cognitive and motor skills, which makes this activity particularly rich and challenging.

Finally, the concepts and skills learned through geometric constructions are transferable to many other areas of knowledge. For example, the ability to think logically and in a structured way, so necessary for making geometric constructions, is also crucial for learning programming. And the ability to visualize and manipulate shapes in space, essential for geometry, is also very useful in disciplines such as chemistry, biology, architecture and design.

Practical Activity

Activity title: Building and Understanding Geometric Shapes

Project Objective

The objective of this project is to strengthen students' understanding of geometric constructions through the practice of creating and analyzing geometric shapes using physical materials and digital drawing software. In addition, it aims to develop communication, collaboration, time management and critical thinking skills through working on a team project.

Project Description

The students will be divided into groups of 3 to 5 people. Each group will be tasked with creating a series of geometric constructions, using both physical materials (ruler, compass, paper, etc.) and a digital drawing software of their choice (some suggestions are: GeoGebra, Desmos, Euclidea, etc.). The constructions should include mediatrices, bisectors, angles of different degrees (90°, 60°, 45° and 30°), and regular polygons. In addition, the groups must document the entire process in a written report.

The estimated time for the execution of this project, per student, is 5 to 10 hours and the deadline for the project is one month from the start of the project.

Materials needed

• Ruler
• Compass
• Paper
• Colored pencils
• Digital drawing software (GeoGebra, Desmos, Euclidea, etc.)

Step by step

1. Division of groups: Students should divide into groups of 3 to 5 people. Each group will represent a "construction team" responsible for creating a series of geometric constructions.

2. Study of constructions: Before starting the constructions, the groups must study and understand the theoretical concepts behind the constructions they will carry out. This study can be done through books, videos, websites, or any other reliable source.

3. Planning the constructions: The groups must plan their constructions, deciding which shapes they will build, which method they will use (physical or digital) and how they will divide the work among the group members.

4. Construction of the shapes: The groups will carry out the planned constructions, using the physical materials and/or the digital drawing software. During the construction, the students should record the entire process, including drawings, screenshots, notes on decisions made, etc.

5. Analysis of the constructions: After completing the constructions, the groups should analyze what they have built, seeking to understand how the theoretical concepts apply in practice.

6. Creation of the report: The groups must compile all the work done in a written report, which should include an introduction, a detailed description of the construction process (with illustrations), the analyzes performed and the bibliography used.

Project Deliverables

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

1. The constructions carried out, whether physical (photographed or scanned) or digital (files of the software used).

2. The written report, which should be organized as follows:

   • Introduction: a brief context about the theme "Geometric Constructions", its relevance and application in the real world and the objective of this project.
   • Development: a detailed explanation of the step-by-step construction process, emphasizing the theoretical understanding behind the constructions and the methodology used. This section should also present a discussion of the results obtained, with analyzes of the constructions and the processes used to create them.
   • Conclusion: a review of the main aspects of the work, reflections on what was learned and conclusions about the project. This section may also include recommendations for future similar projects.
   • Bibliography: a list of the information sources used during the project.