Lesson Plan | Active Learning | Function: Graphs

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 crucial to establish a clear foundation of what is expected for students to achieve by the end of the lesson. By defining objectives, the teacher directs students' attention towards the most important aspects of the study, ensuring that focus and effort are aimed at essential skills. This stage also serves to align expectations and prepare students for the practical activities that will follow, where they will apply the knowledge acquired in an active and collaborative way.

Main Objectives:

1. Empower students to interpret graphs of generic functions, identifying characteristics such as concavity, maximum and minimum points, and intercepts.

2. Enable students to construct graphs of basic functions, such as the graph of the function y = x, and recognize that it is an increasing line.

Side Objectives:

1. Develop students' logical and analytical reasoning skills through the manipulation and interpretation of graphs.

Introduction

Duration: (15 - 20 minutes)

The Introduction serves to engage students with the content they studied previously, utilizing problem situations that stimulate reflection and the application of function concepts in real contexts. Furthermore, the contextualization seeks to demonstrate the relevance of studying functions in practical and everyday situations, increasing student interest and establishing a bridge between theory and practice.

Problem-Based Situations

1. Imagine you are an urban planner and need to plan the expansion of a city. How could knowledge about functions and their graphs help you analyze and optimize urban growth in terms of sustainability and efficiency?

2. Consider a scenario where a scientist is studying the impact of climate change. They collect data on average temperatures and levels of carbon dioxide over decades. How could they use the construction and analysis of function graphs to predict future trends and plan mitigation strategies?

Contextualization

Understanding functions and their graphs is essential not only in mathematics but also in numerous practical applications in everyday life and across various fields of knowledge. For instance, in economics, function graphs are used to model and predict market behaviors. In engineering, they are vital for designing efficient structures. Additionally, the study of functions can be fascinating when applied to natural phenomena such as population growth or physical movements.

Development

Duration: (75 - 80 minutes)

The Development stage is designed to allow students to apply and deepen the knowledge acquired about functions and their graphs. Through practical activities, they are challenged to solve problems in contexts that simulate real situations, promoting active learning, collaboration, and creativity. This approach not only solidifies theoretical understanding but also develops critical thinking and problem-solving skills in an engaging and enjoyable environment.

Activity Suggestions

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

Activity 1 - The Graph Race

> Duration: (60 - 70 minutes)

- Objective: Apply knowledge of functions and graphs in a dynamic and collaborative manner, reinforcing interpretation and construction of graphs.

- Description: In this playful activity, students will be divided into groups of up to 5 people, and each group will represent a team in a math race. The objective is to 'navigate' through a giant board, representing a Cartesian plane, moving spaces according to the resolution of mathematical problems involving the interpretation and creation of function graphs. Each space on the board will have a challenge to be solved, and the team can only advance if they answer correctly and explain the reasoning behind their answer.

- Instructions:

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

• Explain that each group will be in a 'race' on the board and that they need to advance by correctly answering the mathematical challenges.

• Each space on the board will have a problem related to functions and graphs, such as identifying the concavity of a parabola or describing the behavior of an exponential function.

• When answered correctly, the team advances. Otherwise, they must remain in their space until they answer correctly or until another team overtakes them.

• The first group to reach the finish line correctly solving all challenges wins the race.

Activity 2 - Graph Detectives

> Duration: (60 - 70 minutes)

- Objective: Develop critical analysis and interpretation skills for function graphs, promoting collaboration and argumentation.

- Description: The students, organized into groups, will take on the role of math detectives in a mystery involving different functions. They will receive cards with partial information about function graphs and must use their knowledge to complete the information and solve the mystery. The mystery will include identifying functions from their graphs, analyzing specific behaviors, and solving contextualized problems.

- Instructions:

• Form groups of up to five students.

• Distribute cards with incomplete information about function graphs.

• Students must discuss in their groups and complete the information on the cards.

• After completing the information, each group must present their conclusions about the graph and justify their choices based on the studied concepts.

• Conclude with a class discussion about the different solutions presented by the groups.

Activity 3 - Builders of Mathematical Cities

> Duration: (60 - 70 minutes)

- Objective: Apply function concepts in practice, promoting critical thinking and creativity in solving fictional urban problems.

- Description: Using a model of a city, students, divided into groups, must plan and construct the city so that the streets and buildings follow mathematical patterns based on linear and nonlinear functions. Each group will receive a 'plot' with specific characteristics that must be met based on function concepts, such as the slope of the streets (representing linear functions) and the shape of the buildings (representing nonlinear functions).

- Instructions:

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

• Give each group a blank city map and some 'plots of land' with specific mathematical requirements.

• The groups must use these requirements to draw the streets and position the buildings, which must reflect graphs of linear and nonlinear functions.

• At the end, each group presents their city, explaining how function concepts were applied in urban planning.

• Hold a vote among students to elect the most creative city that best applied mathematical concepts.

Feedback

Duration: (10 - 15 minutes)

The purpose of this feedback stage is to consolidate students' learning through reflection and sharing of experiences. By discussing in groups, students have the opportunity to verbalize what they learned, hear different perspectives, and reflect on the learning process. This not only helps identify comprehension gaps that can be addressed, but also strengthens the students' communication and collaboration skills, essential abilities in any educational or professional context.

Group Discussion

To start the group discussion, the teacher should guide students to form groups of five and sit in a circle. The teacher will then introduce the topic, encouraging each student to share their experiences and learnings from the activities carried out. The teacher can start with an open question, such as 'What did you find most challenging in today's activities?' to stimulate participation. It's important for the teacher to circulate among the groups, listening carefully to each contribution and ensuring that everyone has the opportunity to speak.

Key Questions

1. What were the main difficulties you encountered when interpreting or constructing the graphs of the functions during the activities?

2. How would you use the knowledge acquired about functions and graphs in real situations or in other subjects?

3. Was there any group work strategy that proved particularly effective? Why?

Conclusion

Duration: (5 - 10 minutes)

The conclusion stage aims to consolidate students' learning, ensuring they can link theoretical concepts with the practical activities performed. Additionally, it serves to reinforce the importance and applicability of function concepts and their graphs in real contexts, preparing students to use these skills effectively in future situations.

Summary

In this final stage, the teacher should summarize the main points discussed during the class, reinforcing the interpretation of graphs of generic functions and the construction of graphs of basic functions, such as the function y = x. It's important to recap the characteristics observed in the graphs, such as concavity, maximum and minimum points, and intercepts, in order to solidify the knowledge acquired.

Theory Connection

The teacher should highlight how practical activities like 'The Graph Race', 'Graph Detectives', and 'Builders of Mathematical Cities' helped connect the theory studied at home with practice in the classroom. These activities allowed students to actively apply mathematical concepts, reinforcing understanding and the importance of function graphs in various contexts.

Closing

Finally, the teacher should emphasize the relevance of studying functions and their graphs in everyday and practical situations. Discussing how understanding these concepts can help students solve problems in various areas, from urban planning to predicting economic trends, reinforces the importance of mathematics as an essential tool in daily life and future careers.