Contextualization
Theoretical Introduction
A triangle is a geometric figure composed of three sides and three angles. It is among the first geometric concepts we learn and it is highlighted in various areas of mathematics. One of the most important and fundamental aspects of the triangle is its area, which is calculated by half the product of the base by the height (A = (b * h) / 2).
In many situations, however, the area of a triangle can be calculated without knowing its height. There are alternative forms for the calculation that use, for example, Heron's Theorem, which can be used to find the area of a triangle when we know the length of all its sides. Another method used is Brahmagupta's formula, used when we have a cyclic quadrilateral.
In addition to learning how to calculate the area of a triangle, it is important to understand the relationship between triangles and other geometric figures, such as quadrilaterals. Did you know that every quadrilateral can be divided into two triangles? This is one of the tricks we use to calculate the area of a quadrilateral.
Importance and Applications
The area of a triangle is a measurement that indicates the space bounded by its sides, and this information is used in several real-life applications. For example, a surveyor may need to calculate the area of a triangular piece of land, or an architect may need to calculate the area of a triangular roof when designing a house.
Moreover, the ability to calculate the area of a triangle is essential for understanding many other mathematical concepts, such as calculating the area of complex polygons, understanding the Pythagorean theorem, and solving problems involving trigonometry.
Practical Activity
Activity Title: Get Triangled
Project Goal
The main objective of this project is to apply the theoretical concepts about the area of the triangle in practical and tangible situations. By the end of this project, the students will be able to calculate the area of a triangle in different ways, as well as the area of a quadrilateral by dividing it into triangles. Students will also develop teamwork, time management, communication, and critical thinking skills.
Detailed Project Description
Students will be divided into groups of 3 to 5 participants each. Each group will create a "Mathematical Park", where each attraction will be a different type of triangle or quadrilateral.
Each group will be tasked with:
- Designing the triangles and quadrilaterals, both on paper and on the online GeoGebra tool.
- Calculating the area of each drawn figure.
- Creating a detailed report with the descriptions of the "attractions" and the calculations performed.
The area of the triangles must be calculated using at least three different methods: the traditional method (base x height / 2), Heron's Theorem, and calculating from a quadrilateral divided into two triangles (if applicable).
Required Materials
- Graph paper (or blank paper)
- Pencils and erasers
- Ruler or set square
- Access to the GeoGebra software
- Internet access for research and report writing
Detailed Step-by-Step
- Divide the class into groups of 3 to 5 students.
- Each group should draw at least three different types of triangles and two quadrilaterals on paper and in the GeoGebra software.
- After drawing the figures, the students should calculate the area of each one using the appropriate methods.
- Each group should create a detailed report explaining each of its "attractions", the type of triangle or quadrilateral, how the area was calculated, and the results obtained.
- The report should follow the structure: Introduction, Development, Conclusion, and Bibliography used.
Introduction:
The introduction should provide the context of the project, indicating the importance of studying the area of triangles and quadrilaterals, the relevance of such content in everyday life, and the objective of the proposed activity.
Development:
This should be the most elaborate topic of the report. Here, students need to explain the theoretical concepts necessary to calculate the area of triangles in various possible ways. It is in this part that the step-by-step explanation of how the areas of the geometric figures were calculated should be presented, as well as the methodology used in the activity (materials and tools) and the presentation and discussion of the results obtained.
Conclusion:
Review the concepts learned, what was achieved with the project, and the main difficulties faced and overcome. The conclusion should be a space for students to reflect on their learning during the project.
Bibliography:
List all materials consulted for the project (books, websites, videos, etc.) following ABNT standards.
The project should be completed in one month, with an average dedication of five to ten hours per student.
