Objectives (5 - 7 minutes)

1. Review basic probability concepts: Students should be able to recall and understand the fundamental concepts of probability, including the definition of probability, sample space, event, and probability of an event.

2. Understand probability properties: Students should be able to understand and apply probability properties, including the probability of the complementary event, the probability of the union of two events, the probability of the intersection of two events, and conditional probability.

3. Solve problems involving probability properties: Students should be able to apply probability properties to solve practical problems. This includes the ability to identify and interpret problems involving probability properties, as well as the ability to apply appropriate strategies to solve them.

Secondary objectives:

• Develop critical thinking and problem-solving skills: In addition to learning mathematical concepts, students should be encouraged to develop critical thinking and problem-solving skills. This involves the ability to analyze situations, formulate problem-solving strategies, and evaluate the effectiveness of these strategies.

• Promote collaboration and discussion in the classroom: Students should be encouraged to work in groups and discuss problem solutions. This helps promote collaboration and effective communication, important skills both in mathematics and everyday life.

Introduction (10 - 15 minutes)

1. Review of previous concepts: The teacher should start the lesson by reviewing the basic probability concepts that were previously taught. This may include the definition of probability, sample space, event, and probability of an event. This review serves to prepare students for the new material that will be presented and to ensure that everyone is on the same page in terms of understanding.

2. Problem situation 1 - 'The card game': The teacher should present students with a practical problem involving probability. For example, 'If a deck of cards is shuffled and a card is drawn without looking, what is the probability that the card is an ace?'. The teacher should ask students how they would solve this problem and which probability concepts they would apply. This activity serves to engage students and demonstrate the relevance of probability concepts.

3. Problem situation 2 - 'The coin toss': The teacher should present another problem situation involving probability. For example, 'If two coins are tossed, what is the probability that both will land with the same face up?'. Again, students should be invited to discuss and propose solutions to this problem. This activity helps illustrate the application of probability concepts in real-world situations.

4. Contextualization: The teacher should then explain how probability is used in various areas of life, including gambling, weather forecasting, medicine, social sciences, etc. This helps students understand the relevance and importance of probability.

5. Introduction of the topic: Finally, the teacher should introduce the topic of the day - probability properties. The teacher can explain that these properties are mathematical rules that help us calculate the probability of complex events. The teacher should emphasize that understanding and applying these properties is crucial for effectively solving probability problems.

This Introduction should serve to engage students, show the relevance and application of probability, and prepare them for the new material that will be presented.

Development (20 - 25 minutes)

1. Activity 1 - 'The dice game' (10 - 12 minutes): The teacher should divide the class into groups of 3-4 students. Each group will receive a set of dice of different colors. The teacher should then instruct the groups to roll their dice 30 times and record the results. The results should be recorded in a table, indicating the number of times each color was rolled. After collecting the data, students should calculate the probability of each color being rolled. Then, the teacher should guide the students to apply the probability properties to calculate the probability of two events occurring together. For example, the probability of rolling a 5 and a 6 in two consecutive rolls. This practical activity helps students understand and apply probability properties in a fun and engaging way.

   • Step 1: Each group rolls the die 30 times and records the results.
   • Step 2: Each group calculates the probability of each color being rolled.
   • Step 3: Students apply the probability properties to calculate the probability of two events occurring together.

2. Activity 2 - 'The card experiment' (10 - 12 minutes): The teacher should provide each group with a deck of cards. Each group should draw a card from the deck without looking and record the suit and number of the card. After recording the results, each group should calculate the probability of drawing an ace, a king, a queen, a jack, or a specific number. Then, students should calculate the probability of drawing a red card and an ace. This activity helps students apply probability properties in a practical context.

   • Step 1: Each group draws a card from the deck and records the results.
   • Step 2: Each group calculates the probability of drawing an ace, a king, a queen, a jack, or a specific number.
   • Step 3: Students calculate the probability of drawing a red card and an ace.

3. Group discussion (5 - 7 minutes): After the conclusion of the activities, the teacher should gather all students and promote a group discussion. Each group should share their results and the challenges they faced when applying the probability properties. The teacher should clarify any doubts and highlight the key points of the discussion. This discussion helps reinforce students' understanding of probability properties and the application of these properties to solve problems.

Return (8 - 10 minutes)

1. Classroom discussion (3 - 4 minutes): The teacher should invite each group to share their solutions or conclusions from the activities. Each group will have a maximum of 2 minutes to present. During the presentations, the teacher should intervene, if necessary, to correct any misconceptions and reinforce the correct concepts. The goal of this stage is to ensure that all students have understood the probability properties and are able to apply them correctly.

2. Connection with theory (2 - 3 minutes): After the presentations, the teacher should provide a quick review of the probability properties and how they were applied in the activities. The teacher should highlight how the theory connects with practice, reinforcing the importance of understanding probability properties to solve real problems. This will help students consolidate the knowledge acquired and realize the relevance of what they have learned.

3. Individual reflection (2 - 3 minutes): To conclude the lesson, the teacher should propose that students reflect individually on what they have learned. The teacher should ask the following questions and give students a minute to mentally respond:

   1. What was the most important concept you learned today?
   2. What questions do you still have about probability properties?

   After the reflection, the teacher can ask some students to share their answers with the class. This may reveal any gaps in students' understanding and provide valuable feedback to the teacher. Additionally, this reflection helps students internalize what they have learned and identify any areas that may need further review or additional study.

4. Feedback and clarification of doubts (1 minute): Finally, the teacher should encourage students to seek clarification for any remaining doubts they may have. The teacher can inform students about office hours outside of class, where they can seek additional help if needed. The teacher should also praise students' efforts and encourage them to continue practicing and studying to strengthen their understanding of probability properties.

Conclusion (5 - 7 minutes)

1. Lesson summary (2 - 3 minutes): The teacher should provide a brief summary of the key points covered in the lesson. They should recap the definition of probability, sample space, event, and probability of an event, emphasizing that these are fundamental concepts that should be remembered. Then, the teacher should recall the probability properties that were learned and applied during the practical activities. The teacher should emphasize that these properties are mathematical rules that help us calculate the probability of complex events.

2. Connection between Theory, Practice, and Applications (1 - 2 minutes): The teacher should explain how the lesson connected theory, practice, and applications. They should highlight how the practical activities helped students understand and apply probability properties in a concrete and applied manner. The teacher should also reinforce the importance of probability in various areas of life, including gambling, weather forecasting, medicine, social sciences, etc.

3. Extra Materials (1 minute): The teacher should suggest extra materials for students who wish to deepen their knowledge of probability and its properties. These materials may include textbooks, educational websites, explanatory videos, and practice exercises. The teacher may also recommend participation in math olympiads or math clubs, where students will have the opportunity to solve challenging probability problems and interact with other students interested in mathematics.

4. Applicability in Everyday Life (1 - 2 minutes): Finally, the teacher should reinforce the applicability of what was learned in everyday life. They should explain that the ability to calculate the probability of events is useful in various life situations, from making informed decisions to understanding and analyzing statistical information. The teacher can give concrete examples of how probability is used in everyday situations, such as weather forecasting, event planning, risk analysis, and more. This will help students realize the relevance and importance of what they have learned and motivate them to continue studying and applying their knowledge of probability.