Objectives (5 - 10 minutes)

1. Understand the definition of a rhombus:

   • Identify the characteristics of a rhombus: congruent sides, perpendicular diagonals, and bisecting of internal angles.
   • Differentiate a rhombus from other quadrilaterals.

2. Calculate the area of a rhombus:

   • Use the formula for the area of a rhombus: Area = (larger diagonal x smaller diagonal) / 2.
   • Solve practical problems involving the calculation of the area of a rhombus.

3. Solve problems involving the perimeter of a rhombus:

   • Identify the formulas for the perimeter of a rhombus: Perimeter = 4 x side or Perimeter = 2 x (side 1 + side 2).
   • Apply the perimeter formulas to solve problems.

Secondary Objectives:

• Develop logical and deductive reasoning skills when working with the properties and formulas of the rhombus.
• Apply the acquired knowledge about rhombuses in everyday situations, reinforcing the importance of mathematics in daily life.

Introduction (10 - 15 minutes)

1. Review of previous concepts:

   • The teacher will start the lesson with a quick review of the concepts of quadrilaterals and their properties. Students will be reminded that quadrilaterals are flat figures with four sides and that their properties include the sum of internal angles, the sum of external angles, sides, angles, and diagonals. This review is crucial for understanding the rhombus, which is a specific type of quadrilateral.

2. Problem situation:

   • The teacher will present two problem situations to arouse students' interest and introduce the topic in a practical way:
     • Situation 1: A jeweler has a precious stone in the shape of a rhombus. He needs to calculate the area of the stone to determine its value. How can he do that?
     • Situation 2: An architect is designing a new facade for a building and wants to use tiles in the shape of rhombuses. He needs to know how many tiles he will need. How can he determine that?

3. Contextualization:

   • The teacher will explain that the study of rhombuses is not only theoretical but also has practical applications in real life. For example, in situation 1, the jeweler needs to understand the concept of rhombus area to determine the value of the precious stone. In situation 2, the architect needs the knowledge of the rhombus perimeter to calculate the amount of tiles needed. In addition, students may encounter rhombuses in various contexts, such as in logos, flags, and clothing designs.

4. Presentation of the topic:

   • The teacher will formally introduce the topic, explaining that a rhombus is a quadrilateral with congruent sides and perpendicular diagonals that bisect. He will discuss the origin of the term 'rhombus' and how this shape fits into the general classification of quadrilaterals. The teacher will also emphasize that rhombuses have unique properties that differentiate them from other quadrilaterals, which will be explored in the theoretical part of the lesson.

Development (20 - 25 minutes)

1. Rhombus theory (10 - 12 minutes):

   • Definition: The teacher will present the formal definition of a rhombus, emphasizing that it is a quadrilateral with congruent sides. He will also explain that the internal angles of the rhombus are bisected by the diagonals, and that the diagonals are perpendicular to each other.
   • Properties: The teacher will discuss the main properties of the rhombus, such as the sum of internal angles (360°), the sum of external angles (180°), the bisecting of internal angles by the diagonals, and the perpendicularity of the diagonals.
   • Difference between rhombus and other quadrilaterals: The teacher will highlight the differences between the rhombus and other quadrilaterals, such as the parallelogram (which also has congruent sides, but its diagonals are not perpendicular) and the rectangle (which has internal angles of 90°, but its diagonals are not perpendicular).

2. Calculation of the rhombus area (5 - 7 minutes):

   • Area formula: The teacher will present the formula for the area of the rhombus: Area = (larger diagonal x smaller diagonal) / 2. He will explain how the diagonals are measured and how the formula is applied.
   • Practical examples: The teacher will solve practical examples of calculating the area of the rhombus, step by step, explaining each stage. Examples may include the jeweler's situation from the Introduction and other similar problems.

3. Calculation of the rhombus perimeter (5 - 7 minutes):

   • Perimeter formula: The teacher will present the formulas for the perimeter of the rhombus: Perimeter = 4 x side or Perimeter = 2 x (side 1 + side 2). He will explain how the formula is applied, considering that all sides of the rhombus are congruent.
   • Practical examples: The teacher will solve practical examples of calculating the perimeter of the rhombus, step by step, explaining each stage. Examples may include the architect's situation from the Introduction and other similar problems.

4. Consolidation exercises (3 - 4 minutes):

   • The teacher will provide some exercises for students to solve individually, applying what they have learned about the definition, properties, area, and perimeter of the rhombus. He will circulate around the room, offering help as needed.

5. Discussion and clarification of doubts (2 - 3 minutes):

   • The teacher will conclude the theoretical part of the lesson by inviting students to share their solutions and express any doubts they may have. He will clarify doubts and reinforce important concepts as needed.
   • The teacher may also take this opportunity to discuss the practical applications of the rhombus, reinforcing the relevance of the topic to the real world.

Return (10 - 15 minutes)

1. Review of concepts (5 - 7 minutes):

   • The teacher will ask students to share their answers to the consolidation exercises. He will encourage students to explain their solutions, allowing them to apply rhombus concepts in a practical way. The teacher will correct and provide feedback for each answer, ensuring that students fully understand the concepts.
   • The teacher may also ask students to explain the formulas for calculating the area and perimeter of the rhombus in their own words. This will help verify students' understanding of the formulas and allow the teacher to clarify any misunderstandings.

2. Theoretical-practical connection (3 - 5 minutes):

   • The teacher will emphasize how the theory of the rhombus, including the properties, definition, and formulas for calculating the area and perimeter, was applied in solving practical exercises. He will highlight how understanding these concepts allowed students to solve real-world problems, such as calculating the value of a precious stone or determining the number of tiles needed for an architectural project.
   • The teacher may also provide additional examples of how rhombus concepts are applied in other areas of mathematics and science, as well as in everyday situations. For example, he may talk about how the property of bisecting internal angles is used in geometry to prove theorems, or how the property of congruent sides is used in physics to calculate the resistance of a material.

3. Individual reflection (2 - 3 minutes):

   • The teacher will ask students to spend a minute reflecting on what they learned in the lesson. He will ask questions like:
     1. What was the most important concept you learned today?
     2. What questions do you still have about rhombuses?
   • After a minute of reflection, the teacher will ask some students to share their answers. He will provide additional clarifications, if necessary, and encourage students to continue thinking about rhombus concepts after the lesson.

4. Teacher feedback (1 - 2 minutes):

   • The teacher will end the lesson with brief feedback, highlighting students' strengths and areas that may need more practice. He will reinforce the importance of the rhombus in mathematics and other disciplines, and encourage students to continue exploring the topic on their own.

5. Final questions (1 - 2 minutes):

   • The teacher will open a space for final questions, where students can clarify any issues that have not been addressed. He will commit to answering these questions at the beginning of the next lesson or at an appropriate time.
   • The teacher may also ask students to bring examples of rhombuses they find in their homes or at school for the next lesson, providing an opportunity for practical and contextualized learning.

Conclusion (5 - 10 minutes)

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

   • The teacher will recap the main points covered during the lesson, reinforcing the definition of a rhombus, its properties, the formula for calculating the area and perimeter, and the difference between a rhombus and other quadrilaterals.
   • He will also recall the problem situations presented at the beginning of the lesson and how rhombus concepts were applied to solve them.
   • The teacher may use visual aids, such as a diagram of the rhombus and its properties, to assist in the recap.

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

   • The teacher will highlight how the lesson connected the theory, practice, and application of rhombus concepts.
   • He will reinforce how understanding the properties of the rhombus enabled the resolution of the problem situations presented, which had practical applications in real life.
   • The teacher may refer to specific examples of calculating the area and perimeter of the rhombus presented during the lesson to illustrate this connection.

3. Additional Materials (1 - 2 minutes):

   • The teacher will suggest some materials for additional study, allowing students to deepen their knowledge of rhombuses.
   • These materials may include math books, educational websites, explanatory videos, and interactive games that address the topic of rhombuses.
   • The teacher may provide a list of these materials, which students can access at home or in the school library.

4. Relevance of the Subject (1 - 2 minutes):

   • To conclude, the teacher will highlight the importance of studying rhombuses. He will explain how understanding these geometric figures is not only useful for mathematics but also has applications in other disciplines, such as physics and engineering.
   • The teacher may mention examples of how rhombuses are used in designing logos, flags, jewelry, and architecture, emphasizing the relevance of the topic to students' daily lives.
   • He may also emphasize how studying rhombuses helps develop valuable skills, such as logical reasoning, problem-solving, and spatial visualization.

5. Closure (1 - 2 minutes):

   • Finally, the teacher will end the lesson by thanking the students for their participation and reinforcing the importance of continuous study. He may remind students to bring their questions to the next lesson and encourage them to explore the topic on their own.
   • The teacher may also use this moment to make a quick announcement about the topic of the next lesson, creating anticipation and maintaining students' interest.