Lesson Plan | Active Learning | Probability: Sample Space

Assumptions: This Active Lesson Plan assumes: a 100-minute class, prior student study with both the Book and the start of Project development, and that only one activity (among the three suggested) will be chosen to be conducted during the class, as each activity is designed to take up a significant portion of the available time.

Objectives

Duration: (5 - 10 minutes)

The Objectives stage is fundamental to focus students and the teacher on the central aspects of the topic of Probability: Sample Space. This section aims to clearly establish what students should learn and apply during the lesson. By defining specific and clear objectives, students can better orient their prior studies and classroom participation, thus maximizing the effectiveness of learning time.

Main Objectives:

1. Teach students to recognize the sample space of a specific event and determine the number of elements contained in that space.

2. Develop the ability to apply the concept of sample space in practical and theoretical problems.

Side Objectives:

1. Encourage critical and analytical thinking about how different events can influence the sample space.
2. Promote discussion and exchange of ideas among students to deepen their understanding of the topic.

Introduction

Duration: (15 - 20 minutes)

The purpose of the Introduction stage is twofold: to engage students and to revitalize their prior knowledge through problem situations that stimulate reflection on the sample space. Additionally, contextualizing the importance of the subject with practical and historical examples helps establish a link between theory and the real world, motivating students to perceive the relevance of what they are learning in their daily lives and future applications.

Problem-Based Situations

1. Imagine you are organizing a football tournament with 8 teams. Each team can play against all the others once. How many games will be played? Use the concept of sample space to determine the answer.

2. Consider a common six-sided die. If you roll the die twice, what is the total number of possible outcomes? Discuss how the sample space can be visualized and calculated in this case.

Contextualization

Understanding the sample space is crucial not only in mathematical theory but also in practical applications such as game statistics, weather forecasts, and even in financial decision-making. Knowing how to determine the number of possible outcomes of an event enables individuals to evaluate risks and make more informed decisions. For example, understanding the probability of events in gambling can influence a player's strategy. The history of probability theory, which began with gambling games and sharing betting problems in the 17th century, shows how mathematical concepts can arise from practical needs and human curiosities.

Development

Duration: (75 - 80 minutes)

The Development stage is designed for students to apply the concepts of sample space they studied previously in a practical and interactive way. The suggested activities aim to consolidate theoretical understanding through tangible examples and stimulate collaboration and discussion among students. This approach not only helps in consolidating mathematical knowledge but also promotes critical reasoning and teamwork skills.

Activity Suggestions

It is recommended to carry out only one of the suggested activities

Activity 1 -  The Great Dice Tournament

> Duration: (60 - 70 minutes)

- Objective: Apply the concept of sample space in a practical situation and understand the variation between theoretical and practical results.

- Description: In this activity, students will be divided into groups of up to 5 people. Each group will receive two common six-sided dice and must simulate rolling the dice 30 times. The goal is for them to record the results of each roll and, based on this data, create a sample space representing all possible combinations of the two dice.

- Instructions:

• Divide the class into groups of no more than 5 students.

• Distribute two dice to each group.

• Each group must roll the dice 30 times, recording each result.

• After the rolls, each group must list all possible combinations of results of the two dice (from (1,1) to (6,6)).

• Ask each group to calculate the frequency of each observed combination and compare it with the expected theoretical frequency.

• Each group must present their results table and discuss any discrepancies between the observed and expected results.

Activity 2 -  Festival of Combinations

> Duration: (60 - 70 minutes)

- Objective: Understand combinations and apply the concept of sample space in a visual and creative way.

- Description: Students will work in groups to solve a problem involving the combination of ice cream flavors. The ice cream parlor 'Delícias Geladas' offers 5 different flavors, and customers can choose combinations of 2 flavors for each cone. The students' challenge is to determine the sample space of the possible combinations and present it in the form of a creative graph.

- Instructions:

• Form groups of up to 5 students.

• Explain that each group must consider the 5 available ice cream flavors.

• Each group must list all possible combinations of 2 flavors.

• Ask them to create a graph or visual diagram to represent the combinations.

• Each group must calculate the total number of possible combinations and verify that all have been considered.

• Groups present their graphs and explain the creation process and the results obtained.

Activity 3 -  Mathematical Lottery

> Duration: (60 - 70 minutes)

- Objective: Apply concepts of combinations and sample space to understand probabilities in a gaming context.

- Description: In this activity, students will create a lottery model. They will be given 10 numbers (from 1 to 10), and students will need to choose 3 numbers to create a lottery ticket. The challenge is to calculate the total sample space of possible combinations and discuss the probabilities of different combinations.

- Instructions:

• Divide the room into groups of up to 5 students.

• Distribute cards with the numbers from 1 to 10 to each group.

• Each group must choose 3 numbers to form their 'lottery ticket.'

• Groups must calculate the total sample space of possible combinations of 3 numbers.

• Group discussion about the probabilities of winning with different combinations.

• Each group presents their findings and discusses strategies to increase the chances of 'winning' in the mathematical lottery.

Feedback

Duration: (15 - 20 minutes)

The purpose of this section is to allow students to consolidate their learning through reflection and sharing of experiences. Discussing in groups helps to identify and correct potential misunderstandings, as well as allow students to see how different approaches can be applied to the same problems. This stage also reinforces the importance of dialogue and collaboration in mathematical learning processes.

Group Discussion

Encourage a broad discussion in the classroom, prompting each group to share their findings and reflections on the experiments conducted. Start the discussion by raising questions about the students' expectations before the activity and how these changed after the practical execution. Encourage students to discuss discrepancies between theoretical and practical results and what this can teach about probability theory and its application in the real world.

Key Questions

1. What were the biggest surprises when comparing theoretical results with practical ones?

2. How can understanding the sample space help in everyday decision-making?

3. What difficulties did you encounter when calculating the sample space and how did you resolve them?

Conclusion

Duration: (5 - 10 minutes)

The goal of this stage is to consolidate the learning achieved during the lesson, ensuring that students can connect theory and practice and recognize the utility of sample space concepts in their lives. By summarizing and recapping, we ensure that the key points were understood and that students have a clear view of how to apply the knowledge acquired.

Summary

In this final phase of the lesson, we recap the essential concepts of sample space, exploring both theory and practical applications. We discussed the calculation of sample space in different contexts, such as dice games, combinations of ice cream flavors, and lottery models, and how these concepts apply to everyday life and decision-making.

Theory Connection

Today's lesson connected probability theory with practice through activities that simulated real situations and games, allowing students to visualize and calculate sample spaces interactively. This not only solidified students' mathematical understanding but also highlighted the relevance of mathematics in solving everyday problems and predicting events.

Closing

Finally, we emphasize the importance of studying probability and sample space in students' daily lives. Understanding these concepts helps in evaluating risks and benefits, improving decision-making skills in various situations, from simple choices to complex financial and strategic decisions.