Lesson plan of Algorithms and Problems: Medium

## Objectives (5 - 10 minutes)

1. Understanding the concept of algorithms: Students should be able to understand what algorithms are and how they work. They should be able to identify algorithms in everyday situations and recognise their importance in problem solving.
2. Identifying and defining mathematical problems: Students should be able to identify and define mathematical problems that can be solved using algorithms. They should understand that a mathematical problem is a question or a statement that needs to be proven and that they can use algorithms to find the solution or the proof.
3. Applying algorithms to solve mathematical problems: Students should be able to apply algorithms to solve mathematical problems. They should understand that an algorithm is a sequence of steps that, when followed correctly, leads to the solution of the problem.

### Secondary objectives:

• Developing logical thinking: By working with algorithms and mathematical problems, students will naturally develop their logical thinking. This will be a side benefit for the Development of mathematical skills.
• Fostering collaboration and teamwork: By working in groups to solve problems, students will have the opportunity to develop collaboration and teamwork skills. This is key for active learning and for the Development of social skills.

Introduction (10 - 15 minutes)

1. Review of previous concepts: The teacher will start the lesson by reviewing previous concepts that are fundamental for the understanding of the topic of the lesson. This will include a quick review of basic algebra, such as operations with integers and variables. Additionally, the teacher could briefly review the concept of sequences and patterns, as they are key elements in the creation of algorithms. (3 - 5 minutes)

2. Problem situation 1: "The mystery box": The teacher will present the students with a closed box and will tell them that inside there is an object. The students will have to guess what this object is by asking questions that can only be answered with "yes" or "no". The teacher will explain that this is an example of how an algorithm can be used to solve a problem. (3 - 5 minutes)

3. Contextualization of the importance of the subject: The teacher will explain that algorithms are used in many aspects of everyday life, not only in mathematics, but also in technology, in engineering and in many other areas. He could mention examples of how algorithms are used in social media apps, internet search engines, GPS, etc. (2 - 3 minutes)

4. Problem situations 2: "The labyrinth" and "The thieves": The teacher will present two problem situations. In the first one, students will have to create an algorithm to get out of a labyrinth. In the second one, they will have to create an algorithm to catch the thieves in a city, knowing only the initial location of the thieves and of the patrols. These situations will serve to trigger students' curiosity and interest in the subject. (2 - 3 minutes)

5. Introduction of the topic of the lesson: After presenting the problem situations, the teacher will formally introduce the topic of the lesson: "Algorithms and Problem-solving: Medium". He will explain that in this lesson the students will deepen their understanding of algorithms and will learn how to apply them to solve mathematical problems of medium difficulty. (1 - 2 minutes)

Development (20 - 25 minutes)

Activity 1: "The sequence challenge" (10 - 12 minutes)

1. Forming the groups: The teacher will divide the class into groups no larger than 5 students. Each group should receive a sheet of paper, pencils and an eraser.

2. Problem situation: The teacher will present the following situation: "Imagine that you are scientists who have discovered a communication device with an alien civilization. They have sent a sequence of numbers that, if correctly deciphered, will reveal a great secret. You need to create an algorithm to decipher the sequence."

3. Developing the algorithm: The teacher will explain that the algorithm must be able to decipher the sequence of numbers. He can give some hints, such as looking if there is a pattern in the sequence, if the numbers follow a certain order, etc. The students will have to work in their groups to develop the algorithm.

4. Implementing the algorithm: After developing the algorithm, each group will implement it, that is, they will apply it to the sequence of numbers. The teacher will provide a sequence for each group.

5. Presenting the solutions: Finally, each group will present their algorithm and the deciphered sequence (if they managed to decipher it). The teacher will discuss the efficacy of each algorithm and the accuracy of the deciphering.

Activity 2: "The virtual labyrinth" (10 - 12 minutes)

1. Preparing the scenario: The teacher will prepare the "Virtual Labyrinth". He can do so by drawing a labyrinth on a large piece of paper, or by using an interactive labyrinth app in a tablet or a laptop connected to a projector. Each group of students will receive a set of colored markers (for example, colored buttons) and a dice.

2. Problem situation: The teacher will explain that the students are in a virtual labyrinth escape game. The goal of the game is to find the way out of the labyrinth in the smallest number of moves possible.

3. Developing the algorithm: The teacher will explain that the students need to create an algorithm to guide their character through the labyrinth. Each color of marker represents a possible move (for example, red to go right, blue to go left, etc.). The dice will be rolled to decide which move is made at each step.

4. Implementing the algorithm: The students will implement their algorithm, rolling the dice and moving their character through the labyrinth. The teacher will time how long it takes each group to find the exit.

5. Presenting the results: After the activity, each group will present their algorithm and discuss its efficacy. The teacher will lead a discussion on how different algorithmic approaches can lead to different outcomes.

Important:

• During the activities, the teacher should move around the classroom, observing the work of the groups, giving guidance when necessary and clarifying any doubts.
• The teacher should emphasize that the problem-solving process is as important as the final solution. The students should be encouraged to think critically about their problem-solving processes and to make adjustments when necessary.

Closing (10 - 15 minutes)

1. Group Discussion (5 - 7 minutes): The teacher will gather all the students and promote a group discussion. Each group will have up to 3 minutes to share their solutions or conclusions of the activities "The sequence challenge" and "The virtual labyrinth".

   1.1. The teacher will ask a representative of each group to briefly present the algorithm they created, the steps they followed and the difficulties they found.

   1.2. After each presentation, students will be encouraged to ask questions and make constructive comments.

2. Connection with Theory (3 - 4 minutes): The teacher will then make a connection between the activities carried out and the theory presented in the Introduction of the lesson. He will explain how creating and implementing algorithms to solve mathematical problems fits in the concept of algorithm and how these skills are useful in everyday life and in several professional areas.

3. Individual Reflection (2 - 3 minutes): The teacher will ask the students to reflect individually for one minute on the following questions:

   3.1. What was the most important concept learnt today?

   3.2. What questions have not been answered yet?

4. Sharing and Doubt-solving (2 - 3 minutes): The students will be invited to share their reflections with the class. After this, the teacher will open a space for the students to express their remaining doubts. He will be able to answer some questions right away and write down the ones that will need more time to be answered in a future lesson.

5. Closing the Lesson (1 minute): To conclude the lesson, the teacher will reinforce the main points covered and the importance of the topic for everyday life. He will also encourage the students to keep exploring the subject on their own, suggesting further reading or online practice activities.

This Closing moment is essential to make sure that the students have understood the concepts presented, have had the opportunity to express their doubts and reflections, and feel motivated to keep learning about the topic.

Conclusion (5 - 10 minutes)

1. Summary of the Contents (2 - 3 minutes): The teacher will do a brief summary of the main contents covered during the lesson. He will remind the students of the concept of algorithms, their importance in problem-solving and the steps to create and implement an algorithm. The teacher will also highlight the skills developed during the hands-on activities, such as logical thinking and team working skills.

2. Connection between Theory and Practice (1 - 2 minutes): The teacher will then explain how the lesson connected theory to practice. He will reinforce that the theoretical understanding of algorithms allowed the students to create and implement real algorithms to solve mathematical problems. The teacher will also highlight that the practical skills developed during the activities reinforce the theoretical understanding of the concept of algorithm.

3. Complementary Materials (1 - 2 minutes): The teacher will suggest some complementary study materials for those students who want to deepen their knowledge on algorithms. This could include mathematics books, online courses, educational games and algorithm practice websites. The teacher could also recommend that the students research about the application of algorithms in different areas, such as computer science, engineering and physics.

4. Relevance of the Subject for Everyday Life (1 - 2 minutes): Finally, the teacher will emphasize the importance of algorithms for everyday life. He will highlight that algorithms are used in many everyday situations, such as when navigating in a GPS, when using an internet search engine or when interacting in social media. The teacher could also mention that the ability to create and implement algorithms to solve problems is a valuable skill not only in mathematics, but also in many professions and fields of study.

This Conclusion will allow the students to consolidate what they learnt, understand the connection between theory and practice, and feel motivated to keep studying the topic.