Lesson plan of Metric Relationships in the Right Triangle

Objectives (5-7 minutes)

1. Understand what metric relationships in right triangles are, identifying each of the relationships and how they are applied in calculating segment measurements in the triangle.

2. Apply the metric relationships in right triangles to solve practical problems, such as calculating the height of a building, the distance between two inaccessible points, among others.

3. Develop logical and mathematical reasoning skills, as well as spatial visualization capacity, which are fundamental for solving problems involving metric relationships in right triangles.

Secondary Objectives:

• Encourage students' active participation in class, promoting collaborative learning and the exchange of ideas.

• Stimulate students' critical thinking by proposing problem situations that involve the use of metric relationships in right triangles.

Introduction (10-15 minutes)

1. Previous Content Review: The teacher begins the class by reviewing basic concepts about triangles, especially right triangles, and their characteristics. They also review the Pythagorean Theorem, which is fundamental for understanding metric relationships in right triangles. This review can be done through questions to students, encouraging them to remember and explain the concepts.

2. Problem situation 1: "The building and the shadow": The teacher proposes the following situation: "Imagine that you are in an open field and you see a tall building. You want to know the height of the building, but you have no way to measure it directly. The only tool you have is a measuring stick that you can stick in the ground. How could you use this stick and the building's shadow to calculate its height?"

3. Contextualization: The teacher explains that this is a real situation, where it is often necessary to calculate distances or heights that cannot be measured directly. Metric relationships in right triangles are very useful in these cases.

4. Introduction to the topic: The teacher introduces the topic "Metric Relationships in Right Triangles", explaining that these relationships are mathematical formulas that allow calculating measurements of segments of the triangle, such as height and distance, using only the measurements of the sides of the triangle.

5. Problem situations 2 and 3: The teacher proposes two more problem situations to arouse students' interest:

• "Suppose you are on an island and want to measure the distance between two inaccessible points. You have a rope and a piece of wood that can be stuck in the ground. How could you use these materials to calculate the distance between the points?"

• "Imagine you are on a soccer field and want to calculate the distance between the two goalposts. You have a measuring tape, but you cannot use it directly. How could you solve this problem using only the measuring tape and knowledge of metric relationships in right triangles?"

Development (20-25 minutes)

1. Theory - Metric Relationships in Right Triangles (10-12 minutes): The teacher presents the theory about metric relationships in right triangles. They explain that there are two main relationships: the Pythagorean Theorem and trigonometric ratios (sine, cosine, and tangent).

• Pythagorean Theorem: The teacher explains that the Pythagorean Theorem is a fundamental relationship in a right triangle, which states that the square of the hypotenuse is equal to the sum of the squares of the legs. They demonstrate the formula and provide examples of how to use it to calculate the measure of a side in a right triangle.

• Trigonometric ratios: The teacher introduces the trigonometric ratios (sine, cosine, and tangent) and explains how they are calculated from the measurements of the sides of the triangle. They demonstrate the formula for each of them and provide examples of how to use them to calculate segment measurements in a right triangle.

2. Practice - Application Exercises (10-12 minutes): After the theoretical explanation, the teacher proposes a series of practical exercises for students to apply what they have learned. The exercises should be varied and involve calculating different segment measurements in right triangles.

• Exercise 1: The teacher presents a right triangle and asks students to calculate the measurement of one of the legs, given that the hypotenuse measures 10 and the other leg measures 6. The teacher guides students to use the Pythagorean Theorem to solve the problem.

• Exercise 2: The teacher presents another right triangle and asks students to calculate the sine of one of the angles, given that the hypotenuse measures 10 and the other leg measures 6. The teacher guides students to use the sine ratio to solve the problem.

• Exercise 3: The teacher proposes a problem situation, similar to those presented in the Introduction, and asks students to solve it using metric relationships in the right triangle. For example, "you are 20 meters away from a building and observe that the angle of elevation of the top of the building is 30 degrees. How tall is the building?"

3. Discussion - Everyday Application (5-7 minutes): The teacher ends this stage of the class by promoting a discussion on the application of metric relationships in the right triangle in everyday life. They ask students to share other situations in which they could use these relationships to solve daily life problems. They can also propose more problem situations and challenge students to solve them. For example, "how could you use the metric relationships in the right triangle to calculate the distance between the Earth and the Moon, knowing that the Earth's radius is 6,371 km and the Moon's parallax angle is 1 degree?".

Feedback (8-10 minutes)

1. Group Discussion (3-4 minutes): The teacher invites students to a group discussion about the solutions or approaches they found to solve the proposed problems. They encourage students to share ideas, doubts, and difficulties, promoting the exchange of experiences among them. This discussion is an opportunity for the teacher to assess students' understanding of metric relationships in the right triangle and for students to learn from each other.

2. Connection to Theory (2-3 minutes): The teacher then makes a connection between the group discussions and the theory presented. They highlight how the solutions found by students relate to metric relationships in the right triangle, reinforcing the applicability and importance of these concepts. They also take the opportunity to clarify any doubts that may have arisen during the discussion.

3. Individual Reflection (2-3 minutes): The teacher suggests that students individually reflect on what they learned in the lesson. They ask questions such as:

• "What was the most important concept you learned today?"
• "What questions have not yet been answered?"
• "How can you apply what you learned today in everyday situations or in other disciplines?"

Students have a minute to think and then the teacher can ask some of them to share their answers with the class. This reflection activity helps students to internalize what they have learned and to identify possible gaps in their understanding.

4. Teacher Feedback (1 minute): To end the class, the teacher provides brief feedback on the class's performance, highlighting the strengths and areas that need more attention. They also reinforce the importance of metric relationships in the right triangle and encourage students to continue practicing and exploring these concepts.

Conclusion (5-7 minutes)

1. Content Summary (2-3 minutes): The teacher summarizes the main points covered during the class, emphasizing the importance of metric relationships in the right triangle. They reinforce that these relationships allow calculating segment measurements in right triangles, such as height and distance, from the measurements of the sides of the triangle. They also review the two main topics covered: the Pythagorean Theorem and trigonometric ratios (sine, cosine, and tangent).

2. Connection Between Theory, Practice, and Applications (1-2 minutes): The teacher highlights how the class connected theory, practice, and applications. They emphasize that, after reviewing the theory, students had the opportunity to apply what they learned in solving practical exercises and real problem situations. They also reinforce the importance of understanding the theory to be able to apply it effectively and efficiently.

3. Extra Materials (1 minute): The teacher suggests some extra materials for students who wish to deepen their understanding of the subject. These materials may include online explanatory videos, interactive math websites, reference books, and additional exercises. They also encourage students to review their notes and to solve the exercises proposed during the class again.

4. Relevance of the Subject (1-2 minutes): Finally, the teacher highlights the relevance of metric relationships in the right triangle in everyday life. They mention that these relationships are frequently used to solve measurement problems and calculate distances, both in everyday situations and in various professional areas, such as engineering, architecture, physics, and geography. They end the class by encouraging students to continue exploring and applying these concepts, and emphasizing that practice is essential for deepening understanding and skill in using metric relationships in the right triangle.