Contextualization
Mathematics often seems distant from our daily lives. However, behind many phenomena of our everyday life, there are mathematical concepts acting discreetly. In this project, we will address one of these concepts, the relationship between the perimeter and area of regular polygons. This relationship is crucial for various areas of study and professions, such as architecture, engineering, and design.
Introduction
The relationship between the area and perimeter of a regular polygon is a fundamental concept in geometry. This relationship is often expressed in terms of functions and is taken as the basis for a series of mathematical theorems and principles.
Studying this relationship allows us to discover, for example, that for a fixed perimeter, the circle is the shape that maximizes the area. This has immense practical implications: it is the reason why soap bubbles are spheres and not other shapes, for example.
Importance
Understanding the relationship between area and perimeter is essential for many disciplines, including physics, architecture, construction engineering, and product design.
In the field of engineering, for example, this relationship can be used to determine the most efficient shape for a building or to calculate the amount of construction materials needed.
Activity: Designing a Theme Park
Project Objective:
This project aims to analyze the relationship between perimeter and area through the creative design of a theme park. Students will simulate the construction of a theme park in groups of 3 to 5 members, dealing with space allocation, budgets, and construction constraints, while applying mathematical concepts to optimize the design.
Detailed Project Description:
This project is interdisciplinary, encompassing Mathematics and Arts. Students must apply concepts of geometry, measurement, and calculation to design an efficient and aesthetically pleasing theme park. The design should include at least four different types of regular polygons (for example, a triangle for a water slide, a square for a restaurant, a hexagon for the Ferris wheel, etc.).
Required Materials:
- Graph paper
- Pencils and erasers
- Ruler
- Calculator
- Computer with internet access for research
- Graphic design software (optional)
Step by Step:
- Form groups of 3 to 5 students.
- Choose a theme for your group's theme park.
- Using graph paper and pencils, sketch a basic layout of the theme park according to the chosen theme. Make sure that at least four different polygons are represented (triangles, squares, pentagons, hexagons, etc.).
- Calculate the perimeter and area of each attraction represented by the polygons. Use your creativity to determine the utility of each geometric figure in the park.
- Refine the design of your park, iterating and improving the layout while keeping space efficiency and aesthetics in mind.
- Prepare a visual presentation of the theme park including the area and perimeter calculations for each attraction.
- Finally, each group should write a report detailing the design process and explaining their decisions, taking into account the guidelines for the written document.
Project Deliverables:
Students must deliver a visual plan for the theme park and a written report.
The visual plan should be clear, with each polygon labeled with its type, perimeter, and area. The calculations used to determine the perimeter and area should be clearly indicated.
The written report should follow the format of a scientific report, with sections for Introduction, Development, Conclusions, and Bibliography. The Introduction should contextualize the activity and relate it to relevant mathematical theory. The Development should detail the design process of the theme park and explain the decisions made, as well as the calculations involved. The Conclusion should synthesize the group's findings and reflect on the experience. In the Bibliography, students should cite the sources consulted during the project development.
The practical activity, along with the written report, will provide a comprehensive and applied analysis of the relationship between perimeter and area, encouraging effective collaboration, critical thinking, and creativity.
Students are expected to invest more than 12 hours to complete this project due to its complexity and the need for research and teamwork.
