Objectives (5 - 10 minutes)

1. Understand the concept of a cylinder: Students should be able to identify and define a cylinder, recognizing its characteristics and properties. This includes the notion that a cylinder is a solid of revolution that has two parallel bases and a curved surface.

2. Calculate the volume of a cylinder: Students should learn how to calculate the volume of a cylinder using the formula V = πr²h, where V represents the volume, r the radius of the base, and h the height of the cylinder. To do this, they should be able to identify and measure the radius and height of the cylinder.

3. Calculate the surface area of a cylinder: In addition to volume, students should be able to calculate the surface area of a cylinder. This is done using the formula A = 2πrh + 2πr², where A represents the surface area, r the radius of the base, and h the height of the cylinder.

   Secondary Objectives:

   • Apply knowledge in real-world situations: Students should be able to apply the knowledge acquired in solving practical problems, such as calculating the volume of a gas cylinder or calculating the surface area of a soda can cylinder.

   • Develop critical thinking skills: When solving problems of volume and area of cylinders, students should develop critical thinking skills, including the ability to analyze, synthesize, and evaluate information.

Introduction (10 - 15 minutes)

1. Review of previous concepts: The teacher should start the lesson by reviewing the concepts of volume and area that were studied in previous classes. This includes defining volume as the space occupied by a three-dimensional object and area as the measure of the surface of an object. The teacher can use simple and familiar examples, such as shoe boxes and sheets of paper, to illustrate these concepts.

2. Problem situations: The teacher should then present two problem situations to instigate students' curiosity and prepare them for the new content. The situations can be:

   • Situation 1: 'Imagine you have a gas cylinder at home and need to know how much gas it contains. How could you use mathematics to calculate the volume of this cylinder?'

   • Situation 2: 'Suppose you work in a soda can factory and need to calculate the amount of aluminum needed to manufacture the cans. How could you use mathematics to calculate the surface area of these cans, which are cylinders?'

3. Contextualization: The teacher should explain how the concept of a cylinder and its properties are applied in everyday situations and in various areas, such as engineering, architecture, physics, and even in the production of food and beverages.

4. Introduction to the topic: To introduce the topic in an engaging way, the teacher can share interesting facts and applications about cylinders. Some examples could be:

   • Curiosity 1: 'Did you know that the cylinder is one of the oldest solids studied by humanity? It was first mentioned in mathematical texts from Babylon, over 4000 years ago!'

   • Curiosity 2: 'Did you know that the cylinder is one of the most efficient shapes for storing gas? This is because the cylindrical shape allows the gas to be stored in a compact and easy-to-transport space.'

   • Application 1: 'At NASA, rocket fuel tanks are designed in the shape of cylinders to optimize space and fuel efficiency.'

   • Application 2: 'In the food industry, many food and beverage packages, such as soda cans and ice cream tubs, are designed in the shape of cylinders to facilitate stacking and transportation.'

The teacher should ensure that students are engaged and curious by the end of the Introduction, ready to explore the topic in more depth.

Development (20 - 25 minutes)

1. Activity: Building a Cardboard Cylinder (10 - 12 minutes):

   • Description: The teacher should divide the class into groups of 3 to 4 students. Each group will receive a piece of cardboard, a ruler, scissors, and a marker. The challenge is to build a cylinder with the cardboard, measuring the radius and height, and then calculate the volume and surface area of the cylinder.
   • Step by step:
     1. The teacher should provide basic instructions for building the cylinder, such as rolling the cardboard around the ruler to form a tube and cutting the height of the cylinder.
     2. The students, in their groups, should build the cylinder following the instructions.
     3. Once built, the students should measure the radius and height of the cylinder and record the values.
     4. Using the measured values, the students should calculate the volume and surface area of the cylinder.
     5. Finally, the students should compare their calculations with the formula for the volume and area of the cylinder to verify if they are correct.

2. Activity: Solving Real-World Problems (10 - 12 minutes):

   • Description: In this activity, students will be challenged to apply the knowledge acquired in solving real-world problems involving cylinders.
   • Step by step:
     1. The teacher should provide students with a series of problems that involve calculating the volume and area of cylinders in practical situations. For example, 'Calculate the volume of gas that a barbecue gas cylinder, with a radius of 5 cm and a height of 30 cm, can store' or 'Calculate the amount of aluminum needed to manufacture a soda can, knowing that the radius of the base is 2 cm and the height is 10 cm'.
     2. The students, in their groups, should discuss and solve the problems, applying the correct formulas and making the calculations.
     3. The teacher should circulate around the room, guiding the groups when necessary and clarifying doubts.
     4. At the end of the activity, each group should present their solutions and the calculations made.

3. Discussion and Reflection (5 minutes):

   • Description: After the activities, the teacher should promote a classroom discussion for students to share their experiences, difficulties encountered, and lessons learned.
   • Step by step:
     1. The teacher should start the discussion by asking students about the difficulties they had in building the cylinder and solving the problems.
     2. Next, the teacher should ask students to share how they overcame these difficulties and what strategies they used to solve the problems.
     3. The teacher should then reinforce the concepts learned, highlighting the importance of accurate calculation and understanding the properties of the cylinder for problem-solving.
     4. Finally, the teacher should end the discussion by emphasizing the importance of the volume and area of cylinders in our daily lives and in various areas of knowledge.

Return (10 - 15 minutes)

1. Group Discussion (5 - 7 minutes):

   • Description: The teacher should facilitate a group discussion with all students to share the solutions found by each team and for each group to learn from the ideas of others.
   • Step by step:
     1. The teacher should ask each group to briefly present the solutions they found for the proposed problems.
     2. During the presentations, the teacher should encourage other groups to ask questions and share their own solutions or strategies.
     3. The teacher should highlight the most interesting solutions, the most common errors, and the most effective strategies, encouraging students to learn from each other.

2. Learning Verification (3 - 5 minutes):

   • Description: The teacher should conduct a learning verification to assess whether the lesson objectives were achieved and to identify possible comprehension gaps that need to be addressed.
   • Step by step:
     1. The teacher should ask quick questions to the groups about the main concepts of the lesson, such as the definition of a cylinder, the formulas to calculate the volume and surface area of a cylinder, and how these concepts were applied in solving the problems.
     2. The teacher should observe the students' responses and identify possible comprehension gaps or misunderstandings that need to be clarified.
     3. If necessary, the teacher should briefly review the concepts that were not understood by the majority of students or that generated many doubts.

3. Individual Reflection (2 - 3 minutes):

   • Description: The teacher should propose that students reflect individually on what they learned in the lesson and what questions have not been answered.
   • Step by step:
     1. The teacher should ask students to think for a minute about the following questions: 'What was the most important concept you learned today?' and 'What questions have not been answered yet?'.
     2. Next, the teacher should ask some students to share their answers with the class.
     3. The teacher should note the questions that have not been answered to address them in future lessons or review activities.

4. Feedback and Closure (1 minute):

   • Description: Finally, the teacher should thank the students for their active participation and effort during the lesson. The teacher should also encourage students to continue practicing and studying the content at home.
   • Step by step:
     1. The teacher should praise the students for their teamwork, creativity, and dedication during the activities.
     2. The teacher should remind students of the importance of practicing the concepts learned at home by solving more problems and reviewing the formulas.
     3. Finally, the teacher should provide overall feedback on the lesson, highlighting strengths and areas that need improvement.

Conclusion (5 - 7 minutes)

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

   • Description: The teacher should summarize the main points covered in the lesson, reviewing the concepts of cylinder, volume, and area, and the formulas to calculate the volume and area of a cylinder.
   • Step by step:
     1. The teacher should recall the definition and characteristics of a cylinder, emphasizing that it is a solid of revolution with two parallel bases and a curved surface.
     2. Next, the teacher should recap the formulas for calculating the volume (V = πr²h) and the surface area (A = 2πrh + 2πr²) of a cylinder, explaining each term of the formula.
     3. The teacher should also reinforce the importance of measuring the radius and height of the cylinder correctly to obtain accurate calculations.

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

   • Description: The teacher should explain how the lesson connected the theory of cylinders with the practice of construction and calculation of volume and area, and the applications of these concepts in real-world situations.
   • Step by step:
     1. The teacher should highlight that the activity of building the cylinder allowed students to visualize the characteristics and properties of a cylinder in a practical way.
     2. The teacher should also emphasize that solving real-world problems allowed students to apply the theoretical concepts of volume and area of cylinders in practical and relevant situations.

3. Supplementary Materials (1 minute):

   • Description: The teacher should suggest supplementary study materials for students who wish to deepen their understanding of the lesson topic.
   • Step by step:
     1. The teacher can recommend math books or educational websites that have detailed explanations and additional exercises on calculating the volume and area of cylinders.
     2. Additionally, the teacher can suggest explanatory videos or online animations that visually and interactively show the characteristics and properties of a cylinder.

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

   • Description: Finally, the teacher should emphasize the importance of the lesson topic for students' daily lives and for various areas of knowledge.
   • Step by step:
     1. The teacher should emphasize that understanding the concept of a cylinder and the calculation of volume and area is fundamental not only for mathematics but also for various areas of science, engineering, and technology.
     2. The teacher can cite practical examples, such as the application of calculating the volume of cylinders in the gas industry and the application of calculating the surface area of cylinders in the packaging industry.
     3. Finally, the teacher should encourage students to continue studying and practicing the subject, reminding them that mathematics, like cylinders, is present in many aspects of our daily lives.