Contextualization

Mathematics is everywhere around us, not only when we are doing calculations, but also when we are evaluating shapes and sizes. Today, we will explore two fundamental concepts of mathematics: perimeter and area. The perimeter is the measure of the outline of a plane figure, while the area is the measure of the surface of that figure. By understanding these two concepts and their relationships and interactions, we can apply this knowledge in different real-life situations.

The perimeter and area are even more amazing tools when applied to regular figures. Regular polygons are those that have all sides and angles equal, such as the square, the equilateral triangle, and the regular hexagon. When we manipulate the measure of the sides of these figures, we can observe interesting behaviors of the perimeter and area.

Now you must be wondering, where do we use the perimeter and area in real life? Well, think about situations such as building a garden, where you need to calculate the amount of fence to delimit it (perimeter) and the amount of soil or grass that you will need (area). Or when we think about building a house or a building, where the amount of material depends on both the perimeter and the area of the rooms. Even in art and design, these concepts are deeply relevant. Therefore, understanding how the perimeter and area relate is essential for various activities.

Hands-on Activity: "Exploring the World of Plane Figures"

Project Aim

The aim of this project is to explore the relationship between the perimeter and area of regular polygons and to understand how varying the length of the side affects these quantities. The students should work in groups of 3 to 5 people and the project should be completed in one week, with a workload of 2 to 4 hours per student.

Project Description

The students will choose three types of regular polygons (for example, a square, a hexagon, and a decagon). For each polygon, they will vary the length of the side, calculate the perimeter and area for each length, and plot the results on a graph.

Materials Required

• Graph paper for creating the polygons and drawing the graphs.
• Pencil, eraser, ruler, and compass.
• Calculator to assist in the calculations.
• Optional: graphing software.

Step by Step

1. Choice of Polygons: Each group must choose three regular polygons for analysis.

2. Drawing the Polygons: Using a ruler and compass, each group should draw on graph paper the chosen polygons with different side lengths.

3. Calculating Perimeter and Area: The students should then calculate the perimeter and area of each polygon drawn, using the correct formulas for each type of polygon.

4. Graphs of Perimeter vs. Side Length and Area vs. Side Length: With the data collected, the students should create two graphs for each polygon: one showing how the perimeter changes with different side lengths and another showing how the area changes with different side lengths.

5. Analysis of the Graphs: Based on the graphs, the students should discuss and analyze the relationships between perimeter, area, and side length.

Written Document

After concluding the practical part of the project, each group must prepare a written report with four sections: Introduction, Development, Conclusions, and Bibliography.

• Introduction: The students must explain the purpose of the project, its objectives, and the importance of understanding the relationship between perimeter and area.

• Development: In this section, the students should detail the procedures followed during the activity, including the choice of polygons, the drawing process, calculation, and plotting of the graphs. They should also explain the theory behind the perimeter and area calculations. In addition, it is necessary to present and discuss the graphs and results obtained.

• Conclusion: The students should revisit the main points of the project, highlight what they learned, and the conclusions they drew about how the perimeter and area vary with the length of the side.

• Bibliography: The students should list all the sources of information they used during the project, including books, websites, and videos.

The final delivery of the project will consist of the presentation of the drawings and graphs created during the practical activity, as well as the written report.