Contextualization

Area and perimeter are fundamental concepts in geometry and are often used together in real-world math problems. Both are measurements that describe something about the shape of a figure, but they measure different things. The perimeter of a figure is the distance around its boundary, while the area of a figure is the number of square units needed to completely fill the figure.

To explore the relationship between area and perimeter, we can consider two geometric figures with the same perimeter. For example, a square and a rectangle. Both can have the same perimeter (the sum of all sides), but the areas will be different. This is because areas are determined by multiplying the length by the width for both shapes, and in the case of the square, the length and width are equal, while in the rectangle they are different.

Now, let's consider the reverse case, where we have figures with the same area but different perimeters. This is illustrated by considering a square and a rectangle with the same area. For a given area, the figure with the smallest perimeter is always the square. This shows that the area and perimeter of a figure are not necessarily related in a simple way.

The importance of understanding these measurements goes beyond the classroom and shows us their application in the real world. For example, when calculating the cost of fencing a field (perimeter) or the amount of paint needed to paint a wall (area), these concepts help make the correct calculations. Additionally, designers use these measurements when creating layouts for interior spaces, and scientists use them to understand how things move and grow.

In the real world, the dichotomy between area and perimeter is also present in conservation science and environmental issues. An example of this is the discussion on habitat fragmentation: a large uniform habitat may have the same area as several smaller fragments, but the border area (equivalent to the concept of perimeter) between the habitat and its surroundings is highly different, therefore having different effects on fauna and flora (Bender, Tischendorf & Fahrig, 2003).

Some references are:

1. BENDER, D. J.; TISCHENDORF, L.; FAHRIG, L. Using patch isolation metrics to predict animal movement in binary landscapes. Landscape Ecology, v. 18, n. 1, p. 17–39, 2003.
2. Book: Matemática sem mistério - Cesar Borges and Antonio Marques de França - Editora Ática, 2005.
3. Mundo Educação Website - Access here
4. Brasil Escola Website - Access here

Practical Activity

Activity Title: "Exploring areas and perimeters with geometric manipulatives"

Project Objective

The project aims to stimulate the understanding of the concept of area and perimeter through a playful and collaborative approach. Students will be challenged to build different geometric shapes using math manipulatives to compare their areas and perimeters.

Detailed Project Description

In this project, students will create and compare different geometric shapes, such as squares, rectangles, and triangles, to deeply understand the concept of area and perimeter and the complex relationship between them.

This project will allow students to investigate how figures with equal perimeters can have different areas and vice versa. Students will use math manipulatives to build their geometric shapes and record their findings.

Additionally, students will relate their mathematical knowledge to real-world situations, enhancing their problem-solving skills and critical thinking.

The project will be carried out in groups of 3 to 5 people and is expected to last approximately 15 hours, divided between group discussion stages, practical experimentation, recording of discoveries, and writing the final report.

Required Materials

1. Geometric manipulatives (e.g., plastic building blocks of different shapes)
2. Graph paper
3. Ruler
4. Pencils and pens of different colors
5. Notebook for notes

Detailed Step-by-Step for Activity Execution

1. Initial Discussion: Students will start a group discussion to review the concepts of area and perimeter and discuss their ideas about the relationships between them.

2. Experiment Planning: Each group will plan the shapes they will create, considering the criteria of having figures with equal perimeters but different areas, and figures with equal areas but different perimeters.

3. Construction and Measurement: Students will build their figures using geometric manipulatives and graph paper, taking the necessary measurements to calculate the areas and perimeters.

4. Recording Discoveries: Each group will record their findings and observations in a notebook. These findings include the drawings of the shapes, measurements, area and perimeter calculations, and comparisons between the shapes.

5. Relation to the Real World: Students will be challenged to think about real-world situations where the concepts of area and perimeter apply, recording these ideas in their notebook.

6. Writing the Final Report: Groups will prepare a report that includes the introduction (with the project context and objective), the development (with the theory, methodology used, and results found), and the conclusion (with the main points, learnings, and conclusions about the project). Additionally, the report should include the bibliography used.

Project Deliverables

1. Notebook: Should contain all observations, discoveries, measurements, and calculations made during the project.
2. Final Report: Should be written clearly and precisely, explaining the theory, methodology, results, and conclusions of the project.
3. Group Presentations: Each group will present their findings to the class, including a discussion on how the findings relate to the real world.