Objectives (5 - 7 minutes)

1. Understand the concept of trigonometric functions and how they can be applied to solve real-world problems.
2. Develop the ability to graph trigonometric functions on the coordinate plane.
3. Develop the ability to interpret these graphs, identifying features such as amplitude, period, and midline, among others.

Secondary objectives:

• Encourage critical thinking and problem solving through hands-on activities.
• Develop the ability to work with precision and attention to detail when graphing complex functions.
• Encourage active student participation, promoting discussion and collaboration in the classroom.

Introduction (10 - 12 minutes)

Review of Necessary Concepts:

1. The teacher begins the lesson with a brief review of the concepts of function, angle, sine, and cosine. These concepts are essential for understanding the topic of trigonometric functions and graphing. (3 - 4 minutes)

2. Next, the teacher gives a quick summary of how to graph a simple function on the coordinate plane, reinforcing the importance of the x- and y-axes, and how to interpret the information presented on the graph. (2 - 3 minutes)

Contextualization:

3. The teacher presents two everyday situations where trigonometric functions are widely used:

   • In engineering, in calculations of trajectories, periodic movements, and sound waves.
   • In physics, in problems of oscillation, circular motion, and electromagnetic waves. (2 - 3 minutes)

4. To motivate the importance of the subject, the teacher can mention how GPS uses trigonometric functions to calculate the precise location of a device, or how music is composed based on trigonometric functions that represent sound waves. (2 - 3 minutes)

Gaining Attention:

5. The teacher introduces the topic of the lesson with two pieces of trivia:
   • The first piece of trivia is about the origin of the term "trigonometry", which comes from the Greek "trigonon" (triangle) and "metron" (measure), in other words, the measurement of triangles, which is one of the fundamental applications of trigonometric functions.
   • The second piece of trivia is about the fact that trigonometric functions are periodic, which means that their graphs repeat themselves after a certain interval. This characteristic is widely used in several areas of knowledge. (2 - 3 minutes)

Development (20 - 25 minutes)

Activity 1: Building a Carousel (10 - 12 minutes)

1. The teacher divides the class into groups of 3 to 4 students and provides each group with a large sheet of paper, colored pens, a ruler, and a compass.

2. The teacher explains the activity: each group must draw a carousel on their sheet of paper, with the seats arranged in a circle. The carousel must have at least 5 seats.

3. Next, the teacher gives each group an angle, which will be used to determine the initial position of the seats on the carousel. For example, if the angle given is 60 degrees, the first seat will be drawn 60 degrees to the right of the vertical axis.

4. Now, the teacher asks each group to determine the positions of the other seats, by adding or subtracting the initial angle. For example, if the initial angle is 60 degrees, the second seat will be drawn 120 degrees to the right of the vertical axis, the third 180 degrees, and so on.

5. After all groups have drawn their carousels, the teacher asks one student from each group to stand on the "carousel" and turn slowly, while another student records the position of the "carousel" every 30 seconds.

6. At the end of the activity, the teacher asks each group to plot a graph on the coordinate plane, with time on the x-axis and position on the y-axis. The students should notice that the graph is a complete cycle of a trigonometric function, with a period of 2π.

Activity 2: Modeling a Wave (10 - 12 minutes)

1. The teacher continues the hands-on activity, now asking the groups to graph a sine wave. He provides each group with a point on the x-axis and asks the students to determine the value of the sine function at that point.

2. Then, the teacher asks each group to choose a point on the y-axis and determine the value of x where the sine function reaches that value. The students should realize that there are infinite values of x for each value of y, showing that the sine function is not invertible.

3. The teacher reinforces that the sine function is periodic, and asks each group to determine the period of the function and the amplitude value. They should verify that the amplitude is half the difference between the maximum and minimum values of the function, and the period is the distance between two points with the same y-value.

4. Finally, the teacher asks each group to draw the complete graph of the sine function on the coordinate plane, using the initial point and the amplitude and period determined. Students should be able to identify features of the graph, such as the axis of symmetry, the maximum and minimum points, and the points of inflection.

Recap (10 - 12 minutes)

Review of Activities (5 - 6 minutes)

1. The teacher asks each group to share their solutions and conclusions with the class. Each group should explain how they built the carousel and the graph of the sine wave, and which characteristics of the trigonometric function they were able to identify.

2. During the presentations, the teacher should ask questions that stimulate reflection and deepen the students' understanding. For example:

   • "How did you determine the position of the seats on the carousel? What was the role of the given angle?"
   • "How did you determine the position of the seats at other times? Why did you add or subtract the initial angle?"
   • "In the sine wave graph, how did you determine the value of the sine function for a given point on the x-axis?"
   • "How did you determine the value of x for a given point on the y-axis? Why are there infinite values of x for each value of y?"
   • "How did you determine the amplitude and period of the sine function? How do these characteristics affect the graph?"
   • "What other features of the sine function graph were you able to identify? Why are these features important?"

3. After all the presentations, the teacher summarizes the main conclusions, reinforcing the concepts and skills worked on. He also clarifies any possible doubts that may have arisen during the presentations.

Connection with Theory (2 - 3 minutes)

1. The teacher takes this opportunity to make the connection between the hands-on activities and the theory presented in the Introduction. He emphasizes how building the carousel and the sine wave graph illustrates the concepts of trigonometric function, period, amplitude, axis of symmetry, among others.

2. The teacher also explains how trigonometric functions are widely used in several areas of knowledge, such as engineering, physics, and music, to model periodic phenomena and waves. He reinforces the importance of knowing how to interpret and work with graphs of trigonometric functions to solve practical problems in these areas.

3. To consolidate the connection between theory and practice, the teacher can propose a short review exercise, where students must apply what they have learned to solve a practical problem. For example, they can be challenged to use trigonometric functions to model the movement of a pendulum, the sound of a musical note, or the trajectory of a rocket.

Final Reflection (3 - 4 minutes)

1. The teacher encourages students to reflect on what they have learned in the lesson, asking questions such as:

   • "What was the most important concept you learned today?"
   • "What questions still haven't been answered? What else would you like to learn about trigonometric functions and graphs?"
   • "How can you apply what you learned today to everyday situations or other disciplines?"

2. The students have a minute to think about these questions, and then they are invited to share their answers with the class. The teacher should encourage everyone's participation and value all contributions, even if they are different from the expected answer.

3. Finally, the teacher thanks the students for their participation and reinforces the importance of continuous study and practice for learning mathematics. He ends the lesson by reminding the students of the next steps: reviewing the lesson content, doing the assigned exercises, and preparing for the next lesson.

Conclusion (5 - 7 minutes)

1. Summary and Recapitulation (2 - 3 minutes)

   • The teacher begins the Conclusion by recalling the main points discussed during the lesson. This includes the concept of trigonometric function, graphing on the coordinate plane, and interpreting these graphs.
   • He recapitulates the hands-on activities carried out, highlighting the main insights gained through them, such as identifying features of the graphs of trigonometric functions and understanding how these functions can model periodic phenomena and waves.

2. Connection between Theory, Practice, and Applications (1 - 2 minutes)

   • The teacher highlights how the lesson connected the theory, practice, and applications of trigonometric functions. He reinforces how the theoretical understanding of these functions allowed the students to carry out the hands-on activities and how the application of these functions to real-world problems demonstrated their relevance and usefulness.
   • He also mentions again the applications of trigonometric functions in several areas, such as engineering, physics, and music, and how knowledge of these functions can help students better understand these areas and solve practical problems related to them.

3. Supplementary Materials (1 - 2 minutes)

   • The teacher suggests some additional study materials for students who want to deepen their understanding of the topic. These may include textbooks, math websites, educational videos, and math learning apps.
   • He also recommends that students practice plotting graphs of trigonometric functions and interpreting these graphs more, as this is a fundamental skill for solving problems involving these functions.

4. Importance of the Topic and Closing (1 minute)

   • Finally, the teacher reiterates the importance of trigonometric functions, both in mathematics and in various areas of knowledge and everyday life. He encourages students to continue studying and practicing this topic, promising to be available to clarify any doubts that may arise.
   • The teacher thanks the class for their participation and ends the lesson, leaving the students with anticipation for the next meeting.