Objectives (5 - 10 minutes)

1. Understand the concept of Pascal’s Triangle: Students should describe Pascal’s Triangle, its origin, and how it is constructed. They should also understand the relationship between this triangle and combinatorics.

2. Calculate the elements of Pascal’s Triangle: Students should learn how to calculate the elements of Pascal’s Triangle, being able to predict the values of the elements of a row without having to construct the entire triangle. This will be done by applying the binomial coefficient formula.

3. Know the properties and applications of Pascal’s Triangle: Students should be able to identify and explain the properties of Pascal’s Triangle, such as the sum of the elements of a row and how this sum relates to the position of the row in the triangle. In addition, they should be able to recognize situations in which the use of Pascal’s Triangle can be useful for solving problems.

Secondary Objectives:

• Develop logical and mathematical reasoning: By studying Pascal’s Triangle and its properties, students will have the opportunity to develop logical and mathematical reasoning, essential skills for mathematics and other areas of knowledge.

• Promote autonomy in studies: The flipped classroom model allows students to have greater autonomy in their studies, being able to review the content before and after class at their own pace.

Introduction (10 - 15 minutes)

1. Content review: The teacher can start the class by reviewing concepts of combinatorial analysis and binomial coefficient, which are essential for understanding Pascal’s Triangle. The teacher can use practical examples to recall how these concepts apply to everyday situations.

2. Introductory problem situations: The teacher can propose two problem situations to introduce the subject of the lesson.

   • The first situation could be: "Imagine that you have a coin and you are going to toss it three times. In how many different ways can you get heads and tails?"
   • The second situation could be: "In a classroom with 8 students, in how many different ways can the teacher choose 3 students to form a working group?"

3. Contextualizing the importance of the subject: The teacher can contextualize the importance of Pascal’s Triangle, mentioning that it is a very useful tool in solving combinatorial analysis problems and that it also has several applications in other areas of mathematics, such as geometry and statistics.

4. Introduction to the topic: The teacher can introduce the topic by presenting Pascal’s Triangle and mentioning that it was discovered by French mathematician Blaise Pascal, but that it was actually already known in China long before Pascal.

5. Curiosities and related stories: To gain the students’ attention, the teacher can tell some curiosities about Pascal’s Triangle, such as:

   • The first one is that if we add all the numbers in a row, we get a power of 2. For example, in the third row of the triangle, we have 1, 2, 1. The sum of these numbers is 4, which is equal to 2^2.

   • The second curiosity is that Pascal’s Triangle can also be used to calculate the coefficients of the development of a power of a binomial, a concept that students will learn in future classes.

Development (20 - 25 minutes)

1. Activity "Building Pascal’s Triangle" (10 - 15 minutes):

   • The teacher will divide the class into groups of up to 5 students. Each group will receive a sheet of graph paper and colored pens.

   • The teacher will explain that the objective of the activity is to build Pascal’s Triangle up to the sixth row. He will demonstrate on the board how to build the first two rows and then encourage students to continue working in their groups.

   • As the groups progress in building the triangle, the teacher will circulate around the room to guide and answer questions. He will encourage students to discover the pattern that allows them to predict the value of the elements without having to build the entire triangle.

   • When all groups are finished, the teacher will ask a representative from each group to present their triangle to the class, explaining how they arrived at the results.

   • The teacher will conclude the activity by reinforcing the pattern found by the students and explaining how it can be used to calculate the elements of Pascal’s Triangle.

2. Activity "The sum of the elements of a row of Pascal’s Triangle" (5 - 10 minutes):

   • Still in their groups, students will be given the task of calculating the sum of the elements of each row of Pascal’s Triangle that they constructed in the previous activity.

   • The teacher will encourage students to look for a pattern in the sum of the elements of each row. He will give a hint: "The sum of the elements of a row is equal to 2 to the power of the position of the row in the triangle".

   • The groups will present their findings to the class. The teacher will conclude the activity by reinforcing the property of the sum of the elements of a row of Pascal’s Triangle.

3. Activity "Solving problems with Pascal’s Triangle" (5 - 10 minutes):

   • The teacher will present to the students some problem situations that can be solved with the help of Pascal’s Triangle. For example, "In a classroom with 8 students, in how many different ways can the teacher choose 3 students to form a working group?" or "Imagine that you have a coin and you are going to toss it three times. In how many different ways can you get heads and tails?"

   • The students, in their groups, will try to solve the problems using Pascal’s Triangle. The teacher will circulate around the room to guide and answer questions.

   • At the end, each group will present their solutions to the class. The teacher will conclude the activity by reinforcing how Pascal’s Triangle can be used to solve combinatorial analysis problems.

Return (10 - 15 minutes)

1. Discussion and Sharing (5 - 7 minutes):

   • The teacher should hold a group discussion with all the students, where each group will have up to three minutes to share their solutions or conclusions from the activities carried out.

   • During this discussion, the teacher should encourage students to present their findings, difficulties, and strategies used to solve the activities.

   • In addition, this is the time for students to ask any questions that may have arisen during the activities, whether related to Pascal’s Triangle or combinatorial analysis.

   • The teacher should ensure that all students actively participate in the discussion, valuing different contributions and encouraging respect and active listening.

2. Connection to Theory (3 - 5 minutes):

   • After the discussion, the teacher should review the main theoretical points covered during the lesson and connect them to the practical activities carried out.

   • The concept of Pascal’s Triangle should be reinforced, how to calculate its elements, its properties, and its practical applications, especially in combinatorial analysis.

   • This is the time for the teacher to correct possible misconceptions that may have arisen during the discussions and to clarify concepts that are still unclear to the students.

3. Final Reflection (2 - 3 minutes):

   • To conclude the lesson, the teacher should ask students to reflect individually for one minute on the following questions:

     1. "What was the most important concept learned today?"
     2. "What questions have not yet been answered?"

   • After the minute of reflection, the teacher can ask some students to share their answers with the class.

   • The teacher should encourage students to find the answers to their unanswered questions on their own or in future classes, thus promoting students’ autonomy in their studies.

   • The teacher can also take this opportunity to ask for feedback from the students about the lesson and the teaching method used. This information will be valuable for planning and improving future lessons.

Conclusion (5 - 10 minutes)

1. Summary of key concepts: The teacher should recap the main points covered during the lesson, focusing on the key concepts of Pascal’s Triangle, such as its structure, the properties of the elements of its rows, and its applicability in problems of combinatorial analysis. The teacher should emphasize the importance of each student having understood how to calculate the elements of the triangle and how the sum of a row is a power of 2. (2 - 3 minutes)

2. Connection with practice: The teacher should reinforce how the theory presented relates to the practical activities carried out. He should explain that the construction of Pascal’s Triangle and solving problems with its help are concrete examples of how theory can be applied in practice. The teacher should emphasize that Pascal’s Triangle is a powerful tool for solving combinatorial analysis problems, an important topic in mathematics and several other areas of knowledge. (2 - 3 minutes)

3. Extra materials: The teacher can suggest materials for students to deepen their studies on Pascal’s Triangle. These materials may include math books, educational websites, explanatory videos, and learning apps. In addition, the teacher can indicate additional exercises for students to practice constructing Pascal’s Triangle and solving problems with its help. (1 - 2 minutes)

4. Importance of the subject: Finally, the teacher should summarize the importance of Pascal’s Triangle not only for mathematics but also for the students’ daily lives. He can mention that combinatorial analysis, in which Pascal’s Triangle is widely used, is used in several everyday situations, from organizing work teams to predicting outcomes in games of chance. In addition, the teacher can reinforce that the study of Pascal’s Triangle contributes to the development of students’ logical and mathematical reasoning, essential skills for their academic education and for their life. (1 - 2 minutes)