Objectives (5 - 10 minutes)

1. Understand the concept of the volume of a cone and how it is calculated, using the formula V = 1/3πr²h.
2. Apply the formula for the volume of a cone to solve practical problems, where the measurements of the base radius and the height of the cone are provided.
3. Develop critical thinking skills and problem-solving abilities through the use of the cone volume formula in different contexts.

Secondary Objectives:

• Discuss the relationship between the volume of the cone and the height and radius of the base.
• Stimulate logical reasoning and abstraction skills when working with three-dimensional figures.
• Reinforce the application of mathematics in solving real-world problems, such as determining the volume of objects in the shape of a cone.

Introduction (10 - 15 minutes)

1. Recall of prior knowledge: The teacher will start the lesson by recalling the basic concepts of spatial geometry, especially about geometric solids, their characteristics, and volume formulas. Additionally, they will review the formula for the volume of a cylinder, as it is a concept that will be used to introduce the volume of a cone.

2. Problem situations: The teacher will present two situations that involve calculating the volume of a cone, but initially, the students will not have enough knowledge to solve them.

   • Situation 1: 'Imagine you work in an ice cream factory and need to calculate the amount of ice cream that fits in an ice cream cone. How would you do that?'
   • Situation 2: 'Suppose you are building a pyramid, and each of the smaller pyramids that compose it has the shape of a cone. How would you determine the volume of each cone to calculate the total volume of the pyramid?'

3. Contextualization: The teacher will explain that the volume of a cone has several practical applications beyond those mentioned in the problem situations. For example, in the engineering world, calculating the volume of a cone is essential for designing structures such as silos and storage tanks. Additionally, in areas like gastronomy and architecture, knowledge of the volume of a cone is useful for calculating quantities and proportions.

4. Introduction to the topic: The teacher will introduce the topic in a way that sparks the students' interest, making connections with the real world and mathematical curiosities.

   • Curiosity 1: 'Did you know that the formula for calculating the volume of a cone was discovered by the Greek mathematician Antipater of Thessalonica, who lived around 200 B.C.?'
   • Curiosity 2: 'Did you know that the famous monument Stonehenge, in England, has some of its stones carved in the shape of cones? Calculating the volume of these stones was essential to understand how they were transported and stacked to build the monument.'
   • Practical application: 'What if I told you that with the calculation of the volume of a cone, we can find out the amount of popcorn that fits in a microwave bag? Interesting, isn't it?'

The Introduction should stimulate the students' curiosity and show the relevance of the topic to their lives and the world around them.

Development (20 - 25 minutes)

1. Concept of the volume of a cone (5 - 7 minutes):

   • The teacher will start by explaining what a cone is and its characteristics. A cone is a geometric solid that has a circular base and a point called the vertex, which is not in the same plane as the base.
   • Next, the teacher will introduce the formula for the volume of a cone: V = 1/3πr²h, where V is the volume, r is the base radius, and h is the height of the cone.
   • To facilitate visualization, the teacher will draw a cone on the board, showing the base radius and height.
   • They will explain that the formula for the volume of the cone is similar to the formula for the volume of a cylinder (V = πr²h), but with the factor 1/3.

2. Application of the formula for the volume of a cone (10 - 12 minutes):

   • The teacher will show the step-by-step resolution of a problem involving the calculation of the volume of a cone. For example: 'A cone has a base radius measuring 5 cm and a height of 12 cm. What is its volume?'
   • The teacher will explain that to solve the problem, it is necessary to substitute the values into the formula for the volume of the cone: V = 1/3π(5²)(12).
   • They will perform the calculations on the board, showing how to simplify the expression and obtain the result (approximately 314.16 cm³).
   • The teacher will emphasize the importance of remembering to use the correct units of measurement and, if necessary, convert the units to ensure that all measurements are in the same unit.

3. Solving practical problems (5 - 6 minutes):

   • The teacher will present the problem situations proposed in the Introduction of the lesson.
   • Step by step, the teacher will guide the students in solving these problems, applying the formula for the volume of the cone.
   • The teacher will encourage the students to discuss their strategies and solutions, promoting participation and critical thinking.

During the Development of the lesson, the teacher should ask questions to check the students' understanding and correct any possible misconceptions. Additionally, it is important to encourage students to ask questions and share their difficulties, fostering a collaborative learning environment.

Return (10 - 15 minutes)

1. Group discussion (5 - 7 minutes):

   • The teacher will propose a group discussion about the solutions found by the students for the practical problems presented in the Development stage.
   • Students will be encouraged to share their problem-solving strategies and explain how they used the formula for the volume of the cone to arrive at the answer.
   • The teacher will ask questions to stimulate the students' reflection, such as: 'Why did you choose this strategy to solve the problem?' 'How could you have solved the problem in another way?' 'What were the difficulties you encountered when solving the problem?'
   • The teacher will guide the discussion, ensuring that all students have the opportunity to participate and that the answers are clear and precise.

2. Connection with theory (3 - 5 minutes):

   • The teacher will lead a conversation to help students make connections between the theory (the formula for the volume of a cone) and the practice (problem-solving).
   • The teacher may ask: 'How was the formula for the volume of a cone useful in solving the problems we saw today?' 'What are the main differences between calculating the volume of a cone and a cylinder?' 'Why does the formula for the volume of a cone have the factor 1/3 and not 1/2, for example?'
   • The teacher will explain that the formula for the volume of a cone is based on the idea that the volume is proportional to the base area and height, but with a ratio of 1/3. This means that for a cone with the same height and base as a cylinder, its volume is one-third of the cylinder's volume.

3. Individual reflection (2 - 3 minutes):

   • The teacher will suggest that students reflect individually on what they learned in the lesson.
   • The teacher will give a minute for students to think silently and then ask the following questions: 'What was the most important concept you learned today?' 'What questions have not been answered yet?'
   • The teacher will encourage students to write down their answers and share their questions and reflections in the next lesson.

The Return is an essential stage to consolidate learning, allow students to express their doubts and reflections, and for the teacher to assess the effectiveness of the lesson. The teacher should pay attention to the students' responses and the difficulties they express, to adapt teaching and ensure that all students are keeping up with the content.

Conclusion (5 - 7 minutes)

1. Summary of key points (2 - 3 minutes):

   • The teacher will summarize the key points covered in the lesson, reinforcing the concept of the volume of a cone and the formula to calculate it (V = 1/3πr²h).
   • They will review the application of the formula for the volume of a cone in solving practical problems, highlighting the importance of using the correct units of measurement.
   • The teacher will remind students that the volume of a cone is one-third of the volume of a cylinder with the same height and base, and that the formula for the volume of a cone is a generalization of this idea.

2. Connection between theory, practice, and applications (1 - 2 minutes):

   • The teacher will explain how the lesson connected theory (the concept of the volume of a cone and the formula to calculate it) with practice (problem-solving) and applications (determining the volume of cone-shaped objects in the real world).
   • They will highlight that understanding the volume of a cone and the ability to calculate this volume are fundamental in many fields, from engineering to culinary arts.

3. Extra materials (1 minute):

   • The teacher will suggest additional materials for students who wish to deepen their understanding of the volume of a cone. These materials may include explanatory videos, interactive math websites, textbooks, and online exercises.
   • For example, the teacher may recommend using dynamic geometry software, such as Geogebra, to visually explore the properties of the volume of a cone.

4. Relevance of the topic to everyday life (1 minute):

   • Finally, the teacher will emphasize the importance of the topic for students' everyday lives.
   • They will explain that the ability to calculate the volume of a cone can be useful in various practical situations, from determining the amount of ice cream in a cone or popcorn in a microwave bag, to designing structures like silos and storage tanks in engineering.