Objectives (5-7 minutes)

1. Understand the concept of a cylinder, its parts, and properties.

   • Recognize a cylinder as a three-dimensional geometric figure made of two parallel circular bases and a curved lateral surface.
   • Understand that a cylinder has a height, which is the distance between the two bases.
   • Identify the radius and the diameter of each base of the cylinder.

2. Calculate the volume of a cylinder.

   • Use the formula V = π * r² * h, where V represents the volume of the cylinder, π is an approximate constant of 3.14, r is the radius of the cylinder's base, and h is the cylinder's height.
   • Solve practical problems involving the calculation of the volume of a cylinder.

3. Calculate the surface area of a cylinder.

   • Use the formula A = 2π * r * (r + h), where A represents the surface area of the cylinder, π is an approximate constant of 3.14, r is the radius of the cylinder's base, and h is the cylinder's height.
   • Solve practical problems involving the calculation of the surface area of a cylinder.

Secondary objectives:

• Develop problem-solving, critical thinking, and mathematical reasoning skills.
• Apply the concepts learned to real-world situations, such as calculating the volume of a gas cylinder or the area of a cylinder of a soda can.

Introduction (10-15 minutes)

1. Review of previous concepts: The teacher begins by reviewing the concepts of area and volume of flat figures, such as circle, square, and rectangle. This review is critical to help students to understand how these concepts apply to a cylinder. The teacher can also pose questions that prompt the students to think about how these concepts can be extrapolated to a three-dimensional object, such as a cylinder.

2. Opening problem situations: The teacher presents two problem situations to instigate student interest in the lesson's topic. The first problem situation could be, "If we have a cylinder with a height of 10 cm and a circular base with a radius of 5 cm, how could we calculate the volume of that cylinder?" The second situation could be, "Imagine you have a gas cylinder at home, how could you calculate the amount of gas inside it, knowing only the dimensions of the cylinder and the amount of gas it contains when it is full?"

3. Contextualization: Next, the teacher contextualizes the importance of studying the volume and area of a cylinder. The teacher explains that these calculations are used in several different areas of daily life, such as in engineering, architecture, physics, and chemistry. For example, calculating the volume of a cylinder is fundamental to determine the amount of liquid or gas it can hold, while the calculation of the surface area is important to determine how much paint is required to paint a cylinder or how much heat it can exchange with the environment.

4. Introduction of the topic: The teacher introduces the topic of volume and area of a cylinder, giving a brief explanation that cylinders are common figures in daily life, present in objects such as soda cans, tubes, gas cylinders, among others. Also, the teacher can show images of cylinders and ask the students if they can identify the different parts of the cylinder (base, height, lateral surface), and how they think they could calculate the volume and the area of these figures.

5. Fun facts: To raise further interest from the students, the teacher shares some fun facts about cylinders. For instance, the teacher could mention that the formula used to calculate the volume of a cylinder (V = π * r² * h) is similar to the formula to calculate the area of a circle (A = π * r²), but with the addition of height. Also, the teacher can mention that the concept of a cylinder goes back to ancient times, being widely used by ancient Egyptians and Mesopotamian people in their constructions.

Development (20-25 minutes)

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

   • The teacher gives the students cardboard sheets, scissors, and glue.
   • Explains that they will build a 3D model of a cylinder, following the next steps:
     1. Draw two circumferences on the cardboard. The teacher can provide the radius values (3 cm for example) or let the students choose them.
     2. Cut out both circumferences.
     3. Cut a cardboard strip of the desired cylinder's height (example 6 cm).
     4. Glue the circumferences to the ends of the cardboard strip.
   • During this activity, the teacher should walk around, helping the students, and making sure they understand how the different parts of the cylinder are connected.
   • At the end of the activity, the students have made a cardboard cylinder that they can use as a reference to understand better the volume and area concepts of a cylinder.

2. Activity 2 - Calculating Volume and Area: (10-12 minutes)

   • Using the cylinder they built, the students measure the radius and the height of the cylinder using a ruler. The teacher explains that these measures are needed to do volume and area calculations.
   • Then, the students calculate the volume and the area of the cylinder, following the formulas explained in the lesson's Introduction.
   • After finishing the calculations, the teacher asks the students to compare the obtained volumes and areas with the real dimensions of the cylinder. This will reinforce the student's understanding of the volume and area's concepts' applications.
   • To encourage discussion, the teacher could select some students to share their results and explain the process used to arrive at their answers. The objective here is that the student feels comfortable to use these concepts in real situations.

3. Activity 3 - Real-world Problems with Cylinders: (5-10 minutes)

   • Now, the teacher presents the students with real-life problems that involve the volume and area of a cylinder. For instance, "If we have a cylindrical water tank with a height of 2 meters and a radius of 1 meter, how much water can it store?" or "We have a gas cylinder that is 30 cm high and has a radius of 5 cm, and it's filled until the middle, what is the amount of gas inside it?"
   • The students, divided into teams, solve these problems, applying the volume, and area concepts they've learned. The teacher helps the teams when they need it.
   • When the teams are finished, the teacher selects some of them to share their solutions with the class, this will allow the students not only to see different approaches for problem-solving but also helps the teacher check the students' comprehension of the lesson topic.

Closure (8-10 minutes)

1. Group Discussion: (3-4 minutes)

   • The teacher gathers the class in a circle to have a discussion. Each team will share its solutions and conclusions for the real-life problem they selected.
   • During this discussion, the teacher could ask each team how they used the volume and area concepts to solve the problems. That will allow the teacher to evaluate if the class understood the lesson and to identify concepts that need further explanation.
   • The teacher can also encourage students to ask questions between each other during the discussion, this fosters interaction, and the exchange of ideas between the students.

2. Theory Connection: (2-3 minutes)

   • After the discussion, the teacher makes the connection between the practical activities done in class and the theory explained in the lesson's introduction. The teacher explains how the theoretical concepts of volume and area of a cylinder were used in practice, using the students' cardboard cylinder and the real-world problems.
   • The teacher could also mention common mistakes the students made during the activities and explain to the students how to fix these errors.

3. Personal Reflection: (2-3 minutes)

   • To finish the class, the teacher asks the students to reflect individually about what they have learned. They will have time to think about answers to questions such as:
     1. What was the most important concept you have learned today?
     2. What questions do you still have about the topic?
   • The teacher gives one minute for the students to think about these questions, then the teacher chooses a couple of volunteers to share their answers with the class. This will allow the teacher to assess the effectiveness of the lesson and to identify areas that may need review or reinforcement in further classes.
   • In this way, the personal reflection helps students to reinforce the concepts and to notice if there is any knowledge gap on the topic, so they can fill this gap with further research or asking questions to the teacher.

4. Feedback: (1 minute)

   • The teacher ends the lesson by giving feedback to the students about the class performance, complimenting the class on its participation, good questions and on the creative solutions they found for the problems, also, the teacher could highlight the aspects in which the class performed well, as well as aspects that need more dedication or review.
   • This feedback helps to motivate students and encourages them to keep learning and improving.

Conclusion (5-7 minutes)

1. Summary of the Lesson's Contents: (2-3 minutes)

   • The teacher reviews the class' main points, reminding students what was learned. These include the definition of a cylinder, its parts (base, height, lateral surface), the formulas to calculate volume (V = π * r² * h) and surface area (A = 2π * r * (r + h)), and how these formulas can be used in real-life problems.
   • The teacher could use as a prop the students' cardboard cylinder model built in class to illustrate the concepts taught. For example, the teacher could demonstrate how the model's dimensions correspond to the variables in volume and area's formulas.

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

   • Now, the teacher highlights how the class connected theory (the mathematical concepts of volume and area of a cylinder) to practice (the construction of a cylinder model and the solving of practical problems), and real-world applications (the usage of these concepts in daily situations).
   • The teacher could mention how the volume formula of a cylinder (V = π * r² * h) is an extension of the formula of a circle's area (A = π * r²), and that volume and area calculations are used in real-life situations to solve problems such as how to calculate the amount of liquid contained in a cylindrical tank, or the amount of paint needed to cover a cylinder-shaped object.

3. Supplemental Study Resources: (1-2 minutes)

   • Now, the teacher shares extra materials that the student could access to deepen the knowledge about this topic. This could include math textbooks, educational websites, online videos explaining the concepts, among others.
   • The teacher could indicate, for example, the usage of 3D modeling software to explore further the properties of a cylinder or research examples of real-world problems involving cylinders.
   • Also, the teacher could leave some additional exercises for students to practice at home, and encourage them to bring their questions or doubts about these exercises for the next class where the teacher will solve and discuss the answers with the whole class.

4. Relevance of the Subject in Daily Life: (1 minute)

   • Lastly, the teacher emphasizes the relevance the subject has for the students' lives. For example, the teacher can mention how calculating volume and area of cylinders is utilized in many fields such as civil construction (when calculating the capacity of water tanks or a gas cylinder), in the industry (to determine the amount of raw material necessary for the manufacturing of a cylinder, for example), or even in household activities (such as cooking, where the capacity of pots is calculated using the cylinder's volume and surface area that compose it).
   • The teacher encourages students to observe the cylinders that surround them and think about how the concepts learned in class could be applied to these objects. This will reinforce the math and the real world are connected and show them how useful and practical the subjects they learn in class are.