Objectives
- Understand the concept of arithmetic progression (AP) and its application in everyday life, focusing on term identification.
- Learn to identify and calculate the general term of an AP, developing mathematical reasoning skills.
- Practice solving problems involving the general term of an AP, stimulating critical thinking and problem-solving abilities.
Introduction
1. Review of Previous Content (3-5 minutes): The teacher will begin the lesson by briefly reviewing the concepts of numerical sequences and progressions, reminding students about the meaning of term, common difference, and the sum of the terms of a sequence. This review can be done through direct questions to the students, aiming to activate their prior knowledge and prepare them for the new content.
2. Problem Situation (5-7 minutes): The teacher will present two problem situations to introduce the topic of arithmetic progression. The first can be the situation of a salesperson who earns R$100.00 more each month in commissions. The question for the students would be: "How can we represent this situation mathematically?". The second situation can be the example of an athlete who runs an additional fixed distance each day. The question for this situation would be: "How can we calculate the distance the athlete ran on a given day?"
3. Contextualization (2-3 minutes): The teacher will then explain how arithmetic progression is a very useful mathematical tool in various areas of knowledge and everyday situations. For example, in physics, AP can be used to calculate the velocity of an object in uniform motion. In economics, AP can be used to understand the growth of an amount of money over time.
4. Topic Introduction (2-3 minutes): To arouse the students' interest, the teacher can share some curiosities about arithmetic progression. For example, the most famous arithmetic progression is the Fibonacci sequence, which is an AP where each term is the sum of the two previous terms. Another curiosity is that the formula for the general term of an AP was discovered by the German mathematician Carl Friedrich Gauss when he was only 9 years old.
5. Objective Presentation (1 minute): Finally, the teacher will present the objective of the lesson: to understand the concept of the general term of an arithmetic progression and learn how to calculate it.
Development
1. Activity "Salary Increase" (8-10 minutes): The teacher will propose a practical activity so that students can understand and apply the concept of arithmetic progression. The activity consists of simulating a salary increase scenario over several years.
- Step 1: The teacher will divide the class into groups of 4 to 5 students.
- Step 2: Each group will receive a table that represents an employee's salary in a company over 10 years, with a fixed increase each year.
- Step 3: The challenge for the students will be to identify whether the presented situation is an arithmetic progression, and if so, to calculate the common difference of the progression and the general term.
- Step 4: After the calculations, the groups should present their conclusions to the class.
2. Activity "Athlete's Run" (8-10 minutes): Next, the teacher will propose a new activity for the students to deepen their understanding of the general term of an arithmetic progression.
- Step 1: The teacher will present a scenario where an athlete increases the distance covered each day by a fixed amount.
- Step 2: The students, again in groups, must calculate the total distance covered by the athlete in a given number of days.
- Step 3: After the calculations, the groups should present their answers and the process used to arrive at them.
3. Discussion and Reflection (4-5 minutes): After carrying out the activities, the teacher will promote a discussion in the classroom, encouraging students to share their solutions and the strategies used. The teacher will guide the discussion, asking questions to stimulate critical thinking and deepen students' understanding of the subject.
- Discussion questions may include: "How did you apply the concept of arithmetic progression to solve the problems?", "What were the main difficulties encountered?", "How could you use what you learned today in everyday situations or in other subjects?"
- The teacher should encourage the participation of all students, valuing the different strategies used and promoting a collaborative learning environment.
4. Review and Conclusion (3-5 minutes): Finally, the teacher will review the main concepts learned during the activities and emphasize the importance of the general term of an arithmetic progression. The teacher can summarize the calculations made by the groups and present the formula for the general term of an AP, explaining how it can be used to solve problems more quickly and efficiently.
Feedback
1. Group Discussion (3-4 minutes): The teacher will promote a group discussion, where each team will have up to 2 minutes to share their conclusions or solutions to the activities "Salary Increase" and "Athlete's Run". The teacher should encourage students to explain the reasoning behind their answers and how they applied the concepts of arithmetic progression. During this discussion, the teacher can also correct any conceptual errors and provide feedback to the students.
2. Connection with Theory (2-3 minutes): Next, the teacher will make the connection between the practical activities and the theory taught. The teacher can ask the students how they applied the concept of the general term of an arithmetic progression to solve the proposed problems. The objective is to ensure that students understand the relevance and applicability of the theoretical concepts in solving real-world problems.
3. Individual Reflection (2-3 minutes): The teacher will propose that students reflect individually on what they learned during the lesson. The teacher can ask questions such as: "What was the most important concept learned today?" and "What questions remain unanswered?" Students will have a minute to think about these questions.
- The teacher may ask students to write down their reflections in a notebook or on a piece of paper, which can be useful for the teacher to assess students' understanding and plan future lessons.
4. Sharing Reflections (1 minute): After reflection time, the teacher will invite some students to share their answers with the class. Students should be encouraged to express their doubts or difficulties, as well as their moments of greatest understanding. The teacher should value all answers, even if they differ from expectations, as this can promote a rich and meaningful discussion.
5. Conclusion (1 minute): Finally, the teacher will conclude the lesson, reinforcing the main concepts learned and emphasizing the importance of arithmetic progression and the general term. The teacher can also provide students with some study tips and support materials to deepen their understanding of the subject. The teacher should encourage students to continue practicing and exploring mathematics outside the classroom.
- For example, the teacher can suggest that students solve more AP problems at home, watch explanatory videos on the subject, or use math apps or games to practice.
Conclusion
1. Content Summary (2-3 minutes): The teacher will recap the main points covered during the lesson. This includes the definition of arithmetic progression, term identification, progression common difference, and calculating the general term. The teacher should reinforce the importance of correctly understanding and applying these concepts to solve problems involving APs.
2. Connection between Theory, Practice, and Applications (1-2 minutes): The teacher will highlight how the lesson connected the theory of terms of an arithmetic progression with the practice of solving problems. The teacher can recall the activities carried out and how they helped to solidify students' understanding of the topic. In addition, the teacher should emphasize the practical applications of the concepts learned, showing how they can be useful in everyday situations and in other subjects.
3. Supplementary Materials (1 minute): The teacher will suggest some additional study materials for students who want to deepen their knowledge of arithmetic progressions. This may include math books, educational websites, explanatory videos, math games, and learning apps. The teacher can also provide extra exercises for students to practice at home, and make himself available to answer questions via email or during virtual office hours.
4. Importance of the Subject (1 minute): Finally, the teacher will emphasize the importance of the learned content for the students' lives. The teacher can explain that the ability to identify and calculate terms of an arithmetic progression is a valuable skill in many areas, such as science, engineering, economics, and finance. Additionally, the teacher can emphasize that mathematics itself is a powerful tool for developing logical reasoning, problem-solving, and abstraction skills, which are useful in all areas of life.
