Objectives (5 - 7 minutes)

1. Understanding the properties of polynomials: The teacher should ensure that students have a solid understanding of the properties of polynomials, including the number of terms, degree, dominant term, among others. This can be achieved through a quick review of the topic at the beginning of the class.

2. Identifying the properties of polynomials through practical exercises: Students should be able to identify the properties of polynomials through practical exercises. This can be done through a series of problems that students must solve, where they will have to apply their knowledge of the properties of polynomials.

3. Problem-solving skills: In addition to identifying the properties of polynomials, students should be able to solve problems related to this topic. They should be able to use the properties of polynomials to simplify them, add and subtract polynomials, and multiply polynomials.

   Secondary objectives:

   • Logical reasoning: Through complex problem solving, students will also be developing their logical reasoning.

   • Critical thinking improvement: Working with polynomials and their properties, students will be stimulated to think critically about the topic in order to solve the problems presented.

Introduction (10 - 15 minutes)

1. Review of previous concepts: The teacher should begin the class by doing a brief review of previous concepts that are essential for understanding the properties of polynomials. This can include defining terms such as monomials, binomials, and trinomials, as well as the rule for adding and multiplying monomials. The teacher can do this through a quick interactive quiz or math games.

2. Problem situations: The teacher should then present students with two problem situations that involve solving polynomials, but that will not be solved immediately. These problem situations can be, for example, simplifying a complex polynomial or adding two polynomials with different degrees. This will serve to arouse students' interest in the topic and to show the applicability of the subject.

3. Contextualization: The teacher should then contextualize the importance of polynomials in the real world. It can be mentioned that they are widely used in engineering, physics, computer science, and economics to model real-world situations. For example, Ohm's law in electricity can be expressed as a polynomial.

4. Topic introduction: To introduce the topic in an interesting and engaging way, the teacher can share two curiosities about polynomials. The first can be the history of polynomials, mentioning that they have been studied since ancient times and that solving certain polynomials led to the development of new branches of mathematics, such as number theory. The second curiosity can be about the existence of a million-dollar prize offered by the Clay Mathematics Institute for solving a set of seven unsolved problems, one of which is about solving polynomials.

Development (20 - 25 minutes)

1. Polynomial Domino Game: The teacher should provide each group of students with domino cards with different polynomial expressions. Each domino will have a polynomial expression on one side and the expression's result on the other. The students must then play the game trying to match the expressions that are equal. This game will help students to practice adding and subtracting polynomials, as well as to identify equivalent expressions. (10 - 12 minutes)

   • Preparation: The teacher should prepare the domino cards in advance, ensuring that there is a variety of polynomial expressions with different degrees and numbers of terms.
   • Rules: Students should play the dominoes in the traditional way, but instead of matching numbers, they should match polynomial expressions that are equal.
   • Monitoring: The teacher should circulate around the room, monitoring the game and providing help and feedback as needed.

2. Group Problem Solving Activity: The teacher should provide the student groups with a series of problems that involve simplifying, adding, and subtracting polynomials. Students should work together to solve the problems, applying the properties of polynomials that they have learned. The teacher should circulate around the room, offering support as needed and encouraging discussion among the members of the group. (10 - 12 minutes)

   • Preparation: The teacher should prepare a series of problems that are challenging but feasible for students.
   • Rules: The students should work in groups to solve the problems, but each member of the group should contribute to the solution.
   • Monitoring: The teacher should circulate around the room, monitoring the work of the groups, providing help, and feedback as needed.

3. Discussion on the Applicability of Polynomials: To end the Development stage, the teacher should conduct a short discussion with the class about the applicability of polynomials in the real world. The teacher should ask students to think about and share examples of how polynomials are used in different areas, such as engineering, physics, computer science, and economics. This will help to reinforce the relevance of the topic and the connection between theory and practice. (5 - 7 minutes)

   • Preparation: The teacher should prepare some questions to start the discussion and, if possible, have some examples ready to share.
   • Rules: The teacher should establish rules for the discussion, such as giving each student a turn to speak and respecting the opinions of others.
   • Monitoring: The teacher should monitor the discussion, ensuring that all students have the opportunity to participate and that the discussion remains focused on the topic.

Feedback (8 - 10 minutes)

1. Group Discussion (3 - 5 minutes): The teacher should organize a group discussion with all students. Each group will have up to 3 minutes to share their solutions or conclusions with the class. The teacher should ensure that each group addresses the issues of polynomial properties, pattern identification, and problem-solving strategies. This is an opportunity for students to learn from each other, see different ways to approach a problem, and consolidate their own understanding.

   • Preparation: The teacher should ensure that all groups have a chance to share their solutions or conclusions. The teacher should also prepare some additional questions to stimulate the discussion, if necessary.
   • Rules: The teacher should establish rules for the discussion, such as respecting the opinions of others, listening attentively, and asking relevant questions.

2. Theoretical Connection (2 - 3 minutes): After the group discussion, the teacher should recap the main theoretical points of the class and relate them to the solutions or conclusions presented by the groups. The teacher should highlight how the theory was applied in practice and how the properties of polynomials were used to solve the problems.

   • Preparation: The teacher should prepare a brief recap of the most important theoretical points of the class before starting the group discussion.
   • Rules: The teacher should establish rules for the recap, such as staying focused on the most important theoretical points and the connections with practice.

3. Individual Reflection (3 - 5 minutes): Finally, the teacher should propose that the students reflect individually on what they learned in the class. The teacher can do this by presenting some questions to guide the reflection, such as "What was the most important concept you learned today?" and "What questions have not yet been answered?". Students should write down their reflections in a notebook or on a piece of paper. The teacher can collect these notes to evaluate the students' understanding and to identify any areas that need to be revisited in future classes.

   • Preparation: The teacher should prepare the reflection questions in advance and remind the students to write down their answers.
   • Rules: The teacher should establish rules for the reflection, such as the need to be honest and respectful in their notes.

By the end of the class, students should have a solid understanding of the properties of polynomials and how to apply them to solve problems. They should also have had the opportunity to practice their problem-solving skills, to work in groups, and to reflect on their own learning.

Conclusion (5 - 7 minutes)

1. Content Summary (2 - 3 minutes): The teacher should begin the Conclusion stage by summarizing the main points covered in the class. This can include the definition of polynomials, the properties of polynomials (such as the number of terms, degree, and dominant term), and the techniques for simplifying, adding, and subtracting polynomials. The teacher should ensure that students have a clear understanding of these concepts before moving on.

   • Relevance: The teacher should reinforce the importance of these concepts, highlighting how they are fundamental for solving problems involving polynomials and how they are widely applied in diverse areas of knowledge.

2. Connection between Theory, Practice, and Applications (1 - 2 minutes): Next, the teacher should highlight how the class connected the theory, practice, and applications of polynomials. The teacher can mention how practical activities, such as the domino game and the group problem-solving activity, allowed students to apply theory to practice. In addition, the teacher should reinforce the applications of polynomials, mentioning again examples of how they are used in different areas of knowledge.

   • Relevance: The teacher should emphasize that the ability to connect theory, practice, and applications is a valuable skill that will help students to better understand mathematical concepts and apply them effectively in different situations.

3. Extra Materials (1 - 2 minutes): The teacher should then suggest some extra materials for students who want to deepen their understanding of the topic. These materials can include reference books, math websites, educational videos, and additional exercises. The teacher should encourage students to explore these materials at their own pace as a way to review and consolidate what was learned in the class.

   • Relevance: The teacher should explain that the use of extra materials is an effective way to complement classroom learning, allowing students to study the topic in more detail and at their own pace.

4. Importance of the Topic (1 minute): Finally, the teacher should highlight the importance of the topic presented for daily life. It can be mentioned, for example, that the ability to manipulate and solve polynomials is useful in several professions, such as engineering, computer science, physics, and economics. Furthermore, the teacher can emphasize that the study of polynomials helps to develop problem-solving skills, logical thinking, and mathematical reasoning, which are transferable and valuable skills in many aspects of life.

   • Relevance: The teacher should explain that by understanding the practical and real-world importance of the topic, students will be more motivated to study it and to make an effort to learn.