Objectives (5 - 7 minutes)

1. Introduce the concept of midpoint in analytical geometry, explaining that it is the point that divides a line segment into two equal parts.

2. Present the formula to find the midpoint of a line segment, which consists of calculating the arithmetic mean of the coordinates of the points that delimit the segment.

3. Provide students with a practical understanding of the concept of midpoint, through examples and exercises involving the application of the formula.

Secondary Objectives:

• Encourage critical thinking and problem-solving through exercises applying the concept of midpoint.

• Develop the ability to work with Cartesian coordinates, an essential tool in analytical geometry.

• Foster active student participation through group discussions and constructive feedback.

The teacher should start the lesson by reviewing basic concepts of analytical geometry, such as the Cartesian coordinate system and the representation of points on the plane. This will help prepare students for the introduction of the new concept of midpoint.

Introduction (10 - 15 minutes)

1. The teacher begins the lesson with an Introduction activity, involving the use of a string and two clips. He explains that the clips represent points A and B, and the string is the line segment AB. The challenge is to find the exact point on the string that divides AB into two equal parts. Students are encouraged to work in groups and discuss possible strategies to solve the problem. After some time, the teacher reveals that the solution is simply folding the string in half, thus finding the midpoint. (5 minutes)

2. The teacher then contextualizes the importance of the concept of midpoint in analytical geometry, explaining that it is widely used in various practical applications, such as locating a point in the middle of a line, determining the center of mass of an object, and even in computer graphics algorithms. (3 minutes)

3. Next, the teacher presents two problem situations involving the concept of midpoint. The first situation involves determining the midpoint of a bridge over a river, to ensure that the bridge is built equidistant from the banks. The second situation involves determining the midpoint of a soccer field, to ensure that the goal is placed exactly in the middle of the field. The teacher asks students to think of possible strategies to solve these problems. (5 minutes)

4. Finally, the teacher formally introduces the topic of the lesson, explaining that they will learn how to calculate the midpoint of a line segment in analytical geometry. He emphasizes that although the formula may seem complex, it is actually quite simple and straightforward. (2 minutes)

Development (20 - 25 minutes)

Activity 1: "Finding the Midpoint in Practice" (10 - 12 minutes)

1. The teacher divides the class into groups of up to five students and hands out sheets of paper with a Cartesian plane drawn on them.

2. Next, the teacher hands out cards with the coordinates of two points (for example, A(2,3) and B(6,7)) and instructs the students to draw the line segment connecting these two points on the Cartesian plane. (2 minutes)

3. The students, in their groups, must calculate the midpoint of the line segment using the formula presented in the Introduction. The teacher circulates around the room, assisting the groups as needed. (5 minutes)

4. After calculating the midpoint, the students must mark the point on the Cartesian plane and verify if the line segment was indeed divided into two equal parts. (3 minutes)

5. The teacher then asks each group to present their result to the class. He reinforces the importance of understanding the concept of midpoint and how it is calculated, rather than just "plug and chug" into the formula. (2 minutes)

Activity 2: "Midpoint and Positioning" (10 - 13 minutes)

1. The teacher proposes a new challenge to the groups: they must now determine the midpoint of a line segment connecting two coordinates that were not provided, but are located at specific points on the Cartesian plane that the teacher previously drew.

2. The teacher draws a small scenario on the board, such as a maze game, where students must find the midpoint of a short path. (3 minutes)

3. The students, in their groups, must identify the coordinates of the initial and final points of the path, and then calculate the midpoint. They should then position a "character" (a drawing or figure on a separate piece of paper) at the midpoint of the path to verify if they calculated correctly. (5 minutes)

4. After all groups have presented their solutions, the teacher leads a brief discussion on how determining the midpoint is a useful tool in the real world, especially in areas like engineering and architecture. (2 minutes)

5. To conclude, the teacher highlights that although the formula for calculating the midpoint may seem abstract, it is actually a way to formalize a process that we intuitively use in many everyday situations. (2 minutes)

Return (8 - 10 minutes)

1. The teacher begins the Return stage with a brief review of the main points covered in the lesson. He reinforces the definition of midpoint, the formula to calculate it, and the importance of this concept in analytical geometry and various practical applications. (2 - 3 minutes)

2. Next, the teacher suggests that students reflect individually on what they learned in the lesson. He asks the following questions:

   1. What was the most important concept learned today?
   2. What questions have not been answered yet?
   3. How can you apply what you learned today in everyday situations? (2 - 3 minutes)

3. After students have had time to think about their answers, the teacher asks them to share their reflections aloud, if they feel comfortable doing so. The teacher encourages students to be honest and open, and not to worry about having the "right" answers. The goal is to promote critical reflection and independent thinking. (2 - 3 minutes)

4. The teacher then provides constructive feedback and answers any questions students may have. He also shares his own perspective on the students' answers, highlighting strengths and identifying possible areas for improvement. (1 minute)

5. Finally, the teacher concludes the lesson by reinforcing the importance of the concept of midpoint and encouraging students to continue practicing and applying what they have learned. He also announces the topic of the next lesson, which will be about the concept of distance between two points in analytical geometry. (1 minute)

Conclusion (5 - 7 minutes)

1. The teacher begins the Conclusion of the lesson by reviewing the key points of the topic discussed. He highlights the definition of midpoint, the formula to calculate it, and the importance of this concept in analytical geometry. (2 minutes)

2. Next, the teacher connects theory to practice, recalling the activities carried out during the lesson. He emphasizes how the midpoint formula was applied to solve practical problems, such as determining the midpoint of a bridge over a river or a soccer field. (2 minutes)

3. The teacher then suggests additional materials for independent study, such as explanatory videos online, additional exercises in textbooks and math websites, and math learning apps. He encourages students to explore these resources to deepen their understanding of the concept of midpoint. (1 minute)

4. Finally, the teacher makes the connection of the topic to the real world. He highlights how analytical geometry, and more specifically the concept of midpoint, is present in various everyday situations and in various careers, such as engineering, architecture, and computer science. The teacher concludes the lesson by reinforcing the relevance of the subject studied and motivating students to continue exploring mathematics and its applications. (2 minutes)