Lesson Plan | Active Learning | Second Degree Inequality

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 aims to establish the goals that students must achieve by the end of the lesson, clearly outlining what is expected of them. This definition is crucial to guide both the teacher's instruction and the students' engagement in the proposed activities. By specifying the objectives, a clear focus for classroom discussions and practices is created, maximizing time utilization and learning effectiveness.

Main Objectives:

1. Empower students to solve quadratic inequalities, focusing on the value of the coefficient 'a', distinguishing between cases where 'a' is positive and negative.

2. Develop critical analysis and logical reasoning skills in interpreting and solving problems involving quadratic inequalities.

Side Objectives:

1. Encourage active participation from students in group discussions and problem-solving.

Introduction

Duration: (20 - 25 minutes)

The Introduction serves to engage students in the lesson by making them connect the theoretical content studied previously with real or hypothetical situations. By presenting problems based on real situations and contextualizing the importance of the topic, students can visualize the relevance of quadratic inequalities in practical applications, thus increasing interest and motivation to learn. This stage also helps activate students' prior knowledge and prepares them for the practical activities that will follow.

Problem-Based Situations

1. Imagine you are in a race and need to calculate in what time interval you should accelerate to overtake an opponent. This situation can be modeled by a quadratic inequality. How would you solve this inequality to determine the time limits in which you should accelerate?

2. Consider that a company manufactures and sells t-shirts. To make a profit, the quantity of t-shirts sold must be greater than a certain value, which varies according to fixed costs. This condition can be expressed by a quadratic inequality. How would you determine the sales interval that would ensure the company's profit?

Contextualization

Quadratic inequalities are not just an abstract mathematical concept, but have practical applications in many aspects of real life, such as in economics, engineering, and even sports. For example, in engineering, when analyzing the stability of a structure, it is essential to understand and apply quadratic inequalities to determine under what conditions the structure can bear the applied loads. Additionally, understanding these inequalities can aid in making more informed decisions in situations involving resource optimization and strategic planning.

Development

Duration: (65 - 75 minutes)

The Development stage is designed to allow students to apply their theoretical knowledge of quadratic inequalities in practical and meaningful contexts. By solving problems in groups, students not only solidify their understanding of the content but also develop collaboration and communication skills. This hands-on approach aims to maximize student engagement and ensure effective application of theoretical content, preparing them for real situations where the concept of quadratic inequalities may be necessary.

Activity Suggestions

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

Activity 1 - The Great Race of Inequalities

> Duration: (60 - 70 minutes)

- Objective: Develop the ability to solve quadratic inequalities in contexts that simulate real situations and promote collaboration and critical reasoning.

- Description: In this activity, students will be challenged to solve a series of quadratic inequalities that model situations in an obstacle course. Each obstacle represents a condition that must be met for the runner to advance. Students must determine the intervals in which the runner can accelerate or decelerate, according to the proposed inequalities.

- Instructions:

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

• Hand each group an activity sheet containing a series of quadratic inequalities, each representing an obstacle.

• Students must solve the inequalities to determine the time intervals in which the runner can overcome each obstacle.

• Each group must present their solutions and justify the reasoning used to arrive at them.

• Conduct a class discussion about the different approaches and solutions presented by the groups.

Activity 2 - The Market of Inequalities

> Duration: (60 - 70 minutes)

- Objective: Apply knowledge of quadratic inequalities in an economic context, developing analytical and presentation skills.

- Description: Students will work in groups to solve quadratic inequalities that represent profit conditions in a fictional market. They will need to determine the sales intervals that guarantee profit for different types of products, considering fixed and variable costs.

- Instructions:

• Form groups of up to 5 students.

• Distribute a market scenario to each group, containing fixed and variable costs and the quadratic inequality that models profit.

• Students must solve the inequality to determine the sales intervals that generate profit.

• Each group prepares a presentation of their results, including graphs and explanations of the solutions obtained.

• Conduct a question-and-answer session to discuss the different strategies used by the groups.

Activity 3 - Inequalities in Nature

> Duration: (60 - 70 minutes)

- Objective: Utilize quadratic inequalities to solve environmental problems, promoting ecological awareness and the development of applied mathematical skills.

- Description: In this activity, students will use quadratic inequalities to model and solve ecological challenges. They will need to determine the time intervals or environmental conditions in which certain natural phenomena occur or are sustainable.

- Instructions:

• Organize students into groups of up to 5 participants.

• Provide each group with a set of inequalities representing different environmental conditions or natural phenomena.

• Students solve the inequalities to identify the time intervals or necessary conditions.

• Each group prepares a visual presentation of their results, highlighting the importance of the solutions found.

• Conduct a question session where each group explains how the solutions influence the understanding of the natural phenomenon.

Feedback

Duration: (15 - 20 minutes)

The purpose of this stage of the lesson plan is to consolidate students' learning by allowing them to reflect on the practical activities carried out and share insights with peers. Through group discussion, students have the opportunity to articulate what they have learned, hear different perspectives, and receive feedback from peers and the teacher. This not only reinforces mathematical content but also promotes communication and critical thinking skills.

Group Discussion

Start the group discussion with a brief recap of the lesson objectives and the practical importance of quadratic inequalities. Ask each group to share the solutions they found and the thought processes they used to arrive at them. Encourage students to discuss the different approaches and solutions presented, highlighting the most interesting or challenging aspects of each activity. Use this moment for students to learn from each other and to reinforce learning through the exchange of ideas and perspectives.

Key Questions

1. What were the biggest challenges in solving the quadratic inequalities in the proposed activities?

2. How did the variation in the coefficient 'a' influence the solutions of the inequalities? Is there any observed trend?

3. How would you apply the knowledge acquired about quadratic inequalities in other areas or real-life situations?

Conclusion

Duration: (10 - 15 minutes)

The purpose of the Conclusion stage is to ensure that students have fully understood the concepts discussed and can relate them to real and theoretical situations. This moment of synthesis and reflection helps to consolidate learning and ensure that students can apply the acquired knowledge in various contexts. Additionally, it reinforces the importance of mathematics in daily life, motivating students to continue exploring and using mathematical concepts in their academic and professional lives.

Summary

In this final stage of the lesson, the teacher should summarize and recap the main points discussed about quadratic inequalities, emphasizing the importance of the value of the coefficient 'a', both positive and negative. It should review the resolution techniques and the practical applications discussed, reinforcing students' learning.

Theory Connection

It's essential for the teacher to connect the theoretical content studied at home with the practical activities carried out in class. Explaining how the hypothetical and real situations addressed during the lesson relate to the theory reinforces the understanding and applicability of mathematical concepts.

Closing

To conclude, the teacher should emphasize the relevance of quadratic inequalities in everyday life, demonstrating how these concepts are applied in various areas, from financial planning to engineering. This final moment serves to consolidate learning and motivate students, showing the importance of mathematics in solving practical problems.