Lesson Plan | Active Learning | Trigonometry in the Right Triangle

Assumptions: This Active Lesson Plan assumes: a 100-minute class, prior student study with both the Book and the start of Project development, and that only one activity (among the three suggested) will be chosen to be conducted during the class, as each activity is designed to take up a significant portion of the available time.

Objectives

Duration: (5 - 10 minutes)

The objectives stage is essential for directing the focus of students and the teacher on what will be achieved by the end of the lesson. This section establishes the learning goals that students should reach, ensuring a clear and shared understanding of what is expected. By detailing specifically what skills will be developed, students can better prepare and engage in the proposed activities during the lesson.

Main Objectives:

1. Develop students' ability to understand the fundamental trigonometric relationships (sine, cosine, and tangent) and their practical applications in right triangles.

2. Enable students to solve problems involving the calculation of angles and sides in right triangles, using the trigonometric functions sine, cosine, and tangent.

Side Objectives:

1. Encourage logical reasoning and the application of mathematical concepts in practical situations.

Introduction

Duration: (15 - 20 minutes)

The introduction serves to engage students with the content they studied previously, using problem situations that stimulate the practical application of trigonometric concepts. Furthermore, the contextualization shows the relevance of the topic in daily life, increasing students' interest and facilitating the connection between theory and practice. These approaches help establish an active and meaningful learning environment.

Problem-Based Situations

1. Given a 10-meter-high pole, a student wishes to calculate the distance the shadow of this pole casts on the ground at noon on a sunny day. The student knows that the sun's angle at noon is 45 degrees. How would they use trigonometry to solve this problem?

2. Imagine an airplane flying in a straight line at an altitude of 5000 feet. The pilot spots a landmark on the ground that forms a 30-degree angle with the horizon. Using trigonometry, what is the horizontal distance between the airplane and the landmark?

Contextualization

Trigonometry in the right triangle is not just a mathematical tool but a concept applicable in various everyday situations. Since ancient times, navigators and builders have used these principles to navigate and construct structures. Today, trigonometry is essential in fields such as engineering, architecture, physics, and even in computer games to create three-dimensional environments. This applicability makes learning trigonometry practical and relevant for students, showing how mathematics is present and useful in the real world.

Development

Duration: (65 - 75 minutes)

The Development stage is designed to allow students to practically and contextually apply the concepts of trigonometry in the right triangle that they have previously studied. Through playful and challenging activities, students will work in groups to solve real and fictional problems, developing collaboration skills, critical thinking, and the application of mathematical knowledge in diverse situations. This approach not only reinforces learning but also demonstrates the relevance and utility of trigonometry in the real world.

Activity Suggestions

It is recommended to carry out only one of the suggested activities

Activity 1 - Mathematical Explorers Challenge

> Duration: (60 - 70 minutes)

- Objective: Apply the concepts of trigonometry to solve practical measurement and estimation problems in a geographical exploration context.

- Description: Students are divided into groups of up to 5 people, with each group representing a team of explorers who must use trigonometry to map an unknown area. The map is a smaller-scale replica of an outdoor area near the school. The challenge consists of determining the height of a tower and the distance to two points on the ground, using only a protractor and a measuring tape.

- Instructions:

• Each group receives a map and the measuring material.

• Students must first identify on the map the location of the tower and the two reference points.

• Using the protractor, they should measure the angle of inclination of the line of sight from the points to the top of the tower.

• With the measuring tape, measure the distance between the observation point and the tower, creating a right triangle on the map.

• Apply trigonometry to discover the height of the tower and the distance to the points on the ground.

• Present the findings and the method used to the rest of the class.

Activity 2 - The Mystery of the Missing Telescope

> Duration: (60 - 70 minutes)

- Objective: Develop problem-solving and teamwork skills by applying trigonometry in an investigation and deduction context.

- Description: In this scenario, students act as mathematical detectives. They receive a set of clues that include witness reports about angles of observation, distances, and heights, all related to a telescope that was used to observe a special event. The challenge is to recreate the crime scene (the exact location where the telescope was) using trigonometry to resolve the imprecise information from the witnesses.

- Instructions:

• Groups receive an envelope with the case clues.

• Study the clues and decide which information is relevant to apply trigonometry.

• Solve the angles and distances that form right triangles in the described scenarios.

• Use trigonometric functions to determine the approximate location of the telescope.

• Develop a final report explaining the reasoning and solutions found.

• Present the solution to the class, justifying each step of the process.

Activity 3 - Building the Amusement Park

> Duration: (60 - 70 minutes)

- Objective: Apply trigonometric knowledge in a practical engineering project, developing calculation and presentation skills.

- Description: Students, organized into groups, are hired as engineers to design part of an amusement park that includes a roller coaster. They need to calculate various angles and distances to ensure the safety and fun of the visitors. They use trigonometry to calculate the height and length of the roller coaster segments to meet safety requirements and provide an exciting experience.

- Instructions:

• Each group receives a set of design requirements that include specific heights and angles for the roller coaster.

• Students must use trigonometry to calculate the necessary dimensions, ensuring that the roller coaster is safe and fun.

• After the calculations, they must draw the design on paper, showing the measurements and angles calculated.

• Present the design to the class, explaining how trigonometry was applied and justifying the design choices.

Feedback

Duration: (15 - 20 minutes)

This feedback stage is crucial for consolidating learning, allowing students to articulate and reflect on what they have learned and how they applied the concepts of trigonometry. The group discussion helps to identify gaps in understanding and promote a deeper comprehension of the topics covered. Additionally, by hearing from their peers, students can acquire new perspectives and problem-solving strategies, enriching the learning process.

Group Discussion

After completing the activities, organize a large discussion circle with all the students to share discoveries and challenges faced. Start the discussion with a brief introduction about the importance of reflecting on what was learned and how this applies in everyday life. Encourage students to discuss not only the results obtained but also the problem-solving process. Use questions such as 'What surprised you most during the activity?' and 'How would you apply these concepts in other situations?' to guide the conversation.

Key Questions

1. What strategies did you find most effective when applying trigonometry to solve the proposed problems?

2. Was there any stage of the process that you found particularly challenging? How did you overcome it?

3. How can you use what you learned today in practical situations outside the classroom?

Conclusion

Duration: (10 - 15 minutes)

The conclusion serves to synthesize the main points addressed during the lesson, reinforcing students' learning and ensuring that the connection between theory and practice has been clearly established. Additionally, this stage provides a moment for reflection on the importance and applicability of trigonometric concepts in everyday life, highlighting the value of what was learned and potential extensions for knowledge beyond the classroom.

Summary

Throughout this lesson, students were able to explore and apply the fundamental concepts of trigonometry in the right triangle, focusing on the calculations of sine, cosine, and tangent. Using practical activities such as the 'Mathematical Explorers Challenge', the 'Mystery of the Missing Telescope', and the 'Building the Amusement Park', they solved real and fictional problems, solidifying theoretical understanding in applied contexts.

Theory Connection

Today's lesson highlighted the importance of connecting theory with practice. Through group activities, students not only reviewed and applied the theoretical concepts previously studied but also visualized and experienced the practical utility of trigonometry in everyday situations and in simulated engineering projects. This approach reinforces understanding of the content and its relevance outside the school environment.

Closing

Trigonometry in the right triangle is a powerful mathematical tool, essential for various fields such as engineering, architecture, physics, and even in everyday activities. Understanding and being able to apply these concepts not only enriches students' mathematical knowledge but also prepares them to face real and everyday challenges where mathematics is needed.