Objectives (5 - 7 minutes)

Main Objectives:

1. Understand the concept of independent events and their application in probability.
2. Develop skills to calculate the probability of independent events.
3. Apply the concept of independent events to solve real-world problems.

Secondary Objectives:

1. Stimulate students' logical and critical thinking when working with probability situations.
2. Encourage collaboration and communication among students during group activities.
3. Reinforce the importance of mathematics in the context of the real world, demonstrating how probability is used in various everyday situations.

Introduction (10 - 15 minutes)

1. Review of Previous Concepts: The teacher should start the lesson by reviewing the basic concepts of probability that were discussed in previous classes. This includes the definition of probability, the use of fractions and percentages to represent probability, and the difference between independent and dependent events. This review can be done through questions and answers with the students or through a quick interactive activity.

2. Problem Situations: Next, the teacher should present two problem situations involving independent events to spark students' interest and demonstrate the relevance of the topic. For example:

   • "If you flip a coin in the air twice, what is the probability of getting heads both times?"
   • "If you randomly choose two cards from a deck without replacement, what is the probability that both are kings?"

   These questions should be posed so that students begin to think about how they can use what they have learned about probability to solve these problems.

3. Subject Contextualization: The teacher should then contextualize the subject, explaining how the probability of independent events is applied in various areas of life. For example, in weather forecasting, when playing a game of chance, or when predicting results in opinion polls.

4. Topic Introduction: To introduce the topic and capture students' attention, the teacher can share some curiosities or stories related to the probability of independent events. For example:

   • "Did you know that if you roll a fair die six times, the probability of getting a specific number, like a six, on all rolls is only 1 in 46,656?"
   • "And that in chess, the probability of an amateur player making the best move on each turn is about 1 in 10,000?"

   These curiosities can help spark students' interest and show the relevance of the probability of independent events in real-world contexts.

Development (20 - 25 minutes)

1. Theory Presentation (10 - 12 minutes):

   • Definition of Independent Events (3 - 4 minutes): The teacher should start by explaining that two events are independent if the occurrence or non-occurrence of one event does not affect the probability of the other event occurring. For example, when flipping a coin twice, the result of the first flip does not affect the result of the second flip.

   • Product Rule (3 - 4 minutes): Next, the teacher should present the "Product Rule," which is used to calculate the probability of two independent events occurring in sequence. The product rule states that the probability of two independent events occurring in sequence is the product of their individual probabilities. For example, the probability of getting heads on a fair coin is 1/2, so the probability of getting heads on two consecutive flips is 1/2 x 1/2 = 1/4.

   • Practical Examples (2 - 3 minutes): The teacher should then present several practical examples of calculating the probability of independent events. This may include coin flips, dice rolls, card selections from a deck, etc. The examples should be solved step by step, with the teacher explaining each step of the process.

   • Theory Review (2 - 3 minutes): To conclude the theoretical part, the teacher should briefly review the concepts presented and clarify any doubts students may have.

2. Group Practical Activity (10 - 13 minutes):

   • Group Division (2 - 3 minutes): The teacher should divide the class into groups of 3 to 4 students. Each group will receive a worksheet that includes several problems of probability of independent events to solve.

   • Problem Solving (5 - 7 minutes): The teacher should guide the groups in solving the problems. Students should discuss among themselves and apply what they have learned to calculate the probability of independent events in each problem. The teacher should circulate around the room, monitoring the groups' progress and providing guidance and clarifications as needed.

   • Solution Presentation (3 - 4 minutes): After the designated time, each group should present their solutions to the problems. The teacher should guide the discussion, highlighting key points and correcting any errors or misunderstandings.

3. Connection with Theory (3 - 5 minutes): To conclude the Development stage, the teacher should revisit the practical examples solved, explaining how they relate to the problems in the group activity. This will help consolidate students' understanding of the concept of independent events and the product rule.

Return (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes): The teacher should start this stage by promoting a group discussion on the solutions or conclusions each team reached during the practical activity. Each team should briefly share their findings, explaining how they applied the theory and the reasoning behind their answers. The teacher can ask guiding questions to stimulate the discussion and ensure that all key concepts have been understood.

2. Connection with Theory (2 - 3 minutes): The teacher should then connect the groups' conclusions with the theory presented at the beginning of the lesson. He can highlight how the product rule was used to calculate the probability of independent events in each of the group activity problems. This will help reinforce the concept of independent events and the importance of knowing how to calculate their probability.

3. Individual Reflection (2 - 3 minutes): The teacher should ask students to reflect individually on what they learned in the lesson. He can propose the following questions to guide the reflection:

   1. "What was the most important concept you learned today?"
   2. "What questions have not been answered yet?"
   3. "How can you apply what you learned today in everyday situations or in other disciplines?"

   Students should write down their answers on a piece of paper or in their notebooks, which will be collected by the teacher at the end of the lesson. This will allow the teacher to assess students' understanding of the topic and identify any areas that may need reinforcement in future classes.

4. Closure (1 minute): To conclude the Return stage, the teacher should thank the students for their participation and effort. He should encourage them to review the lesson material, complete any assigned homework, and prepare any questions they may have for the next lesson. The teacher should also remind students of the topic of the next lesson and any materials or preparations needed.

Conclusion (5 - 7 minutes)

1. Content Summary (2 - 3 minutes): The teacher should begin the Conclusion of the lesson by summarizing the main points covered. He should review the definition of independent events, the product rule for calculating the probability of independent events, and how to apply these concepts to solve problems. Additionally, the teacher should highlight how the probability of independent events is used in everyday situations and in various areas of life.

2. Theory-Practice Connection (1 minute): Next, the teacher should emphasize how the lesson connected theory to practice. He should reinforce how the practical examples and the group activity helped students understand and apply the theoretical concepts of independent events and probability calculation.

3. Extra Materials (1 - 2 minutes): The teacher should then suggest additional reading and study materials for students who wish to deepen their understanding of independent events and probability. This may include math books, educational websites, explanatory videos, and online practice exercises.

4. Topic Relevance (1 - 2 minutes): Finally, the teacher should reiterate the importance of the topic for students' everyday lives. He should remind them of how the probability of independent events is used to make informed decisions in various situations, from gambling games to weather forecasts. Additionally, the teacher should emphasize how the logical and analytical thinking developed when working with probability is a valuable skill that can be applied in many aspects of life.

5. Closure (1 minute): To conclude the lesson, the teacher should thank the students for their participation and effort. He should encourage them to review the lesson material, complete any assigned homework, and prepare any questions they may have for the next lesson. The teacher should also remind students of the topic of the next lesson and any materials or preparations needed.