Contextualization

The study of Mathematics provides tools for understanding patterns, orders, and relationships present in our daily lives. Thus, the Congruence of angles and Proportionality are fundamental concepts of this discipline that allow us to understand, for example, the similarity and difference between figures, the enlargement or reduction of figures without altering their properties, and even the functioning of natural phenomena and architectural constructions.

To exemplify, when we talk about congruent angles, we refer to angles that have the same measure. This can be observed in different situations in our daily lives, such as the arrangement of the hands on a clock, the angle of a staircase, the geometry of a building, and even in the formation of shadows depending on the position of the sun.

Proportionality, on the other hand, allows us to establish a relationship of equivalence between different quantities. This concept is widely applied in various areas, from cooking, when we need to adjust the quantities of ingredients in a recipe, to civil construction, when we need to design the dimensions of a building while maintaining the same proportion.

Introduction

In mathematics, congruence is a relationship that exists between two geometric figures that have the same shape and size, although their position or orientation may be different. In this sense, two angles are congruent if, and only if, their measures are equal. Therefore, we frequently refer to this property when studying Euclidean geometry.

Proportionality, on the other hand, is not limited to geometry alone but extends to various other areas of knowledge. In the context of geometry, proportionality can be identified when the ratio between the measures of two corresponding sides of two similar figures is constant. That is, when we enlarge or reduce a figure, if the proportion is maintained, the angles remain congruent, and the proportionality relationship between the corresponding sides is preserved.

These concepts are fundamental tools for the development of logical and analytical thinking. Furthermore, their practical application in the real world is vast, ranging from calculating distances and areas to interpreting and solving everyday problems.

Students can delve deeper into the concepts of Congruence and Proportionality in the following resources:

1. SOS Professor: Congruence of angles
2. Mundo Educação: Proportionality
3. Nova Escola: Activities on congruence of angles
4. YouTube: Matemática Rio with Prof. Rafael Procopio

Practical Activity

Activity Title: "Miniature Architects: Congruence and Proportionality in Action"

Project Objective:

The objective of this activity is to provide students with the opportunity to apply the concepts of congruence of angles and proportionality in creating a floor plan and a scaled-down model of a building or a house. Additionally, it is expected that they also develop socio-emotional skills while working in groups, such as time management, communication, problem-solving, and creative thinking.

Detailed Project Description:

This project will be carried out by students divided into groups of 3 to 5 members. Each group will be responsible for choosing an existing building or house and creating, based on their observations and real measurements, a detailed floor plan and a scaled-down model of that construction, respecting the principles of congruence of angles and proportionality.

The activity will be divided into four main stages:

1. Research and data collection
2. Elaboration of the floor plan and the design of the scaled-down model
3. Construction of the scaled-down model
4. Analysis, discussion, and preparation of the final report

Required Materials:

• Graph paper for the elaboration of the floor plan
• Ruler, protractor, and compass
• Materials for the construction of the scaled-down model (can be cardboard, popsicle sticks, clay, etc.)
• Measuring equipment (tape measure)
• Writing materials (pencils, pens, colored pencils)
• Computer with internet access (for research and report preparation)

Detailed Step-by-Step for Carrying Out the Activity:

1. Research and Data Collection: Each group must choose a building or house to study. They should note the characteristics of the building, take photos if possible, and take measurements of the main dimensions (height, width, and length), angles, and proportions. If necessary, they can use images and plans available on the internet to assist in this stage.

2. Elaboration of the Floor Plan and the Design of the Scaled-Down Model: With the collected data, students should elaborate the floor plan of the chosen building or house on graph paper, maintaining the real proportions. Then, they should project how the scaled-down model will be, establishing the new dimensions and proportions in relation to the original construction.

3. Construction of the Scaled-Down Model: Using the chosen materials, students should build the scaled-down model of the building or house, following the previously elaborated design. They should observe if the angles remained congruent and if the proportions between the measurements were respected.

4. Analysis, Discussion, and Preparation of the Final Report: After completing the scaled-down model, students should analyze and discuss the construction process and how the concepts of congruence and proportionality were applied. They should highlight the difficulties encountered, the proposed solutions, and what they learned from the experience. Finally, they should prepare a written report containing: Introduction (contextualizing the theme, relevance, application, and objective of the project), Development (theory, activity carried out, methodology, and results), Conclusion (learning and final conclusions), and Bibliography.

Project Deliverables:

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

1. The floor plan of the chosen building or house and the design of the scaled-down model.

2. The constructed scaled-down model.

3. Photos documenting the construction process and the final model.

4. A written report containing:

   • Introduction: where they explain the reason for choosing the building or house, the relevance of studying congruence and proportionality, and the project's objective.
   • Development: where they detail the involved theory, the methodology used, the activity carried out, and the results obtained.
   • Conclusion: where they reflect on the difficulties encountered, the learning obtained, and the final conclusions of the project.
   • Bibliography: where they indicate all sources of information used.