Objectives (5 - 7 minutes)

1. Understand the concept of difference of squares: The teacher should introduce the concept of difference of squares, explaining that it is the difference between two perfect squares. Students should understand that factoring a difference of squares results in a product of two conjugate expressions.

2. Apply the factoring of difference of squares: Students should learn to apply the concept of difference of squares to factor algebraic expressions. This will involve identifying perfect squares and correctly factoring the expression.

3. Solve problems involving difference of squares: In addition to applying the factoring of difference of squares, students should be able to solve mathematical problems that involve the use of this concept. This may include simplifying algebraic expressions or solving equations.

Secondary Objectives:

• Develop critical thinking and problem-solving skills: By working with the factoring of difference of squares, students will have the opportunity to develop their critical thinking and problem-solving skills. They will need to identify the best method to factor the given expression and apply this method effectively.

• Promote classroom collaboration: The flipped classroom strategy allows students to work in small groups, which promotes collaboration and discussion. This can help students better understand the concept of difference of squares and factoring, as they can share ideas and different approaches.

Introduction (10 - 15 minutes)

1. Recall previous content: The teacher should start the lesson by recalling the concepts of perfect squares and factoring. This can be done through a quick review, either through direct questions to the students or a brief quiz. This step is crucial as understanding these concepts is fundamental to understanding the difference of squares.

2. Problem situation 1: The secret of factoring

   • The teacher should then present the students with a complex algebraic expression, such as x^2 - 9.
   • Students should be challenged to factor this expression. The teacher can ask: "How can we factor this expression? Is there any way to simplify this process?"
   • The goal here is to spark students' curiosity and prepare them for the new concept that will be presented.

3. Contextualization: Applications of difference of squares factoring

   • The teacher should then contextualize the importance of the concept of difference of squares, explaining that it is widely used in various fields, such as cryptography, for example.
   • It can be mentioned how the factoring of difference of squares is used in the creation of certain cryptographic keys, which are used to protect confidential information.

4. Problem situation 2: The code to be deciphered

   • The teacher should present the students with a problem situation related to the previous example. For example: "Imagine you received a secret code to decipher. The algebraic expression that represents the code is x^2 - 9. How can we factor this expression to decipher the code?"
   • Students should be challenged to apply the concept of difference of squares to solve the problem.
   • This activity serves to engage students and show the applicability of the concept that will be addressed in the lesson.

Development (20 - 25 minutes)

1. Activity 1: "Perfect Square Hunt" (10 - 12 minutes)

   • The teacher should divide the class into groups of up to 5 students each. Each group will receive a large sheet of paper and colored markers.
   • The teacher should then announce a number, for example, 25. Students, in their respective groups, should work together to find all the algebraic expressions that result in 25 when factored (for example, 5^2, -5^2, 1^2 * 25, -1^2 * 25, 2.5^2, -2.5^2, etc.).
   • When a group finds an expression, they should write it on their sheet of paper and color the corresponding "square" area on the sheet.
   • The game continues until all groups have exhausted their possibilities or time runs out.
   • This playful and interactive activity will allow students to visualize and internalize the concept of perfect squares, which is fundamental for the factoring of difference of squares.

2. Activity 2: "Deciphering the Enigma" (10 - 12 minutes)

   • The teacher should provide each group with an envelope containing a series of algebraic expressions that represent codes. These expressions are examples of difference of squares.
   • Students should work in their groups to factor the expressions and thus decipher the codes. They should also write the original expression that corresponds to each code on the envelope.
   • The teacher should circulate around the room, providing guidance and clarifying doubts as needed.
   • The goal of this activity is to allow students to practice the factoring of difference of squares in a fun and challenging context. At the end of the activity, the group that deciphers the most codes correctly will be the winner.

3. Group Discussion and Reflection (5 - 8 minutes)

   • After the activities are concluded, the teacher should gather all the students for a group discussion. Each group should share their solutions and strategies, and the teacher should clarify any misunderstandings and reinforce key concepts.
   • The teacher should then lead a reflection, asking students to think about what they learned during the lesson. They should be encouraged to make connections between the theory (the concept of difference of squares) and practical applications (factoring expressions and solving codes).
   • This final discussion and reflection are essential to consolidate learning and ensure that students have understood the lesson's objectives.

Return (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes)

   • The teacher should gather all the students and promote a group discussion about the solutions and strategies found by each team during the practical activities.
   • Students should be encouraged to share their experiences, challenges, and achievements. This will allow them to learn from each other and see different ways to approach the factoring of difference of squares.
   • The teacher should intervene, clarifying doubts, reinforcing concepts, and highlighting the most effective strategies.

2. Connection between theory and practice (2 - 3 minutes)

   • The teacher should then guide students to reflect on how the practical activities relate to the theory presented at the beginning of the lesson.
   • Students should be asked: "How did today's activities help you better understand the concept of difference of squares and factoring?" and "What were the main learnings of the day?"
   • The teacher should encourage students to make direct connections, explaining how the practice of factoring expressions during the activities is a direct application of the concept of difference of squares.

3. Individual Reflection (3 - 4 minutes)

   • Finally, the teacher should propose that students make an individual reflection. They should think for a minute about the following questions:
     1. "What was the most important concept learned today?"
     2. "What questions remain unanswered?"
   • After the reflection time, students should be encouraged to share their answers with the class.
   • The teacher should listen carefully, note the main doubts and difficulties, and plan how to address them in future lessons.

This Return is a crucial step in the learning process, as it allows students to consolidate what they have learned, reflect on the learning process, and identify areas that have not yet been fully understood. Additionally, it provides the teacher with valuable feedback on the lesson's effectiveness and the students' learning needs.

Conclusion (5 - 7 minutes)

1. Summary of Content (2 - 3 minutes):

   • The teacher should start the Conclusion by recapping the main points covered in the lesson. This includes the concept of difference of squares, the identification of perfect squares, and the factoring of difference of squares.
   • They can reinforce this recap through a visual summary on the board or slides, highlighting the necessary steps for factoring and the importance of identifying perfect squares.
   • The goal of this step is to ensure that students have a clear and complete understanding of the lesson's content.

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

   • The teacher should then explain how the lesson connected theory to practice. They can highlight how the group activities allowed students to apply the concept of difference of squares in factoring expressions.
   • Additionally, the teacher should emphasize how solving the proposed problems during the practical activities reflects the real-world applications of difference of squares factoring, such as cryptography.
   • This connection between theory and practice is essential to ensure that students understand the usefulness and relevance of what they are learning.

3. Extra Materials (1 - 2 minutes):

   • The teacher should then suggest extra materials for students who wish to deepen their knowledge of difference of squares factoring. These materials can include textbooks, online educational videos, math websites, and additional exercises.
   • For example, they can recommend an explanatory video on difference of squares factoring, a website with various practice exercises, and a math book with a section dedicated to this topic.
   • These extra materials will allow students to study the content at their own pace and explore the topic more thoroughly.

4. Importance of the Subject (1 minute):

   • Finally, the teacher should emphasize the importance of difference of squares factoring for everyday life and other areas of knowledge.
   • They can mention, for example, how this concept is used in fields such as engineering, physics, and cryptography. This will help motivate students by showing that math is not just an abstract subject but has practical and relevant applications.
   • Additionally, the teacher should encourage students to ask any questions that may have arisen during the lesson and reinforce that learning is a continuous process and that it is normal to have doubts.