Contextualization

We need to talk about a fundamental topic in mathematics: the Metric Relations in the Right Triangle. Before we get to the subject of our project, we need to review some important definitions. Do you remember what a right triangle is? Correct, it is a type of triangle that has a 90-degree angle, also called a right angle. And what are metric relations? They are equations that express the triangle's measurements in terms of other measurements, which can be heights, angles, or even the area.

In mathematics, the metric relations of a right triangle are expressions that relate the measurements of the sides of this triangle. The three most well-known are: the Pythagorean theorem, the theorem of the altitude relative to the hypotenuse, and the acute angle theorem. Now, you must be wondering: why should I learn about this?

The metric relations in the right triangle are widely used in mathematics and in many other disciplines. It is thanks to them that we can understand concepts such as trigonometry and analytical geometry, so important for physics, engineering, architecture, among other areas.

For example, when an architect designs a ramp for access, he needs to ensure that it is not so steep as to hinder the passage of users. For this, he uses trigonometry, which is based on the metric relations of right triangles.

We can find several other examples in our daily lives: when we position a ladder so that we can climb safely, when we calculate the shortest route between two points, and even when we play a basketball game, where we need to calculate the best angle to throw the ball and make that impressive basket.

To delve deeper into the subject, we recommend reading the book Fundamentals of Elementary Mathematics Vol. 1 - Sets and Functions which provides a didactic and complete approach to metric relations in right triangles. In addition, the websites Just Mathematics and World Education are also great sources of information with practical examples and solved exercises.

Now that we know the importance of the topic, let's move on to our challenge!

Practical Activity

Activity Title: "Building and Understanding Right Triangles"

Project Objective

Through the construction of models of right triangles and solving related problems, students will be able to explore and understand the metric relations of the right triangle, including the Pythagorean theorem, and the similarity of triangles.

Detailed Project Description

Students will be divided into groups of 3 to 5 and will work together to:

1. Build models of right triangles;
2. Develop and solve problems involving the metric relations of the right triangle;
3. Write a report detailing the development process, the problems solved, and the conclusions reached.

This project should be completed in one week, with an estimated two to four hours of work per student.

Required Materials

• Ruler
• Compass
• Cardboard or cardstock
• Pencil
• Scissors

Step-by-Step for Activity Execution

1. Construction of Models: Each group should build at least three models of right triangles, choosing the sizes of the sides according to the metric relations of the right triangle. The choice of side sizes should be justified in the report;

2. Problem Development: Next, each group should develop at least five problems involving the metric relations of the right triangle and solve them using the models. The problems should involve calculations of areas, side lengths, and angles. The problems and their solutions should be described in the report;

3. Report: After completing the practical activities, students should prepare a report detailing the development process, the problems solved, the methodology used, the discussion of the results obtained, and the conclusions. The report should include:

   • Introduction: With the contextualization of the theme, its relevance and application, in addition to the project's objective;
   • Development: With the explanation of the activity in detail, including the description of the construction of the models, the methodology used, the problems developed, their solutions, and the discussion of the results;
   • Conclusion: Reiterating the main points of the work, the learnings obtained, the conclusions drawn from the project, and how the activity helped students understand the metric relations of the right triangle.
   • Bibliography: With all the references used during the project development.

This project will bring students not only the understanding of the metric relations in the right triangle but also the opportunity to develop socio-emotional skills such as time management, communication, problem-solving, creative thinking, and proactivity.

By working in teams, students learn to listen to others' ideas, to negotiate solutions, and to make decisions together, essential skills for the 21st century.