Lesson plan of Polygons: Diagonal

Objectives (5 - 10 minutes)

1. Comprehend the concept of a diagonal of a regular polygon: Students should be able to define what a diagonal is, especially in the context of a regular polygon. They should understand that a diagonal is a line segment in a closed figure that connects two non-consecutive vertices. They should also be able to distinguish between a diagonal and a side of a polygon.

2. Calculate the number of diagonals in a regular polygon: Students should be able to apply the correct formula to calculate the number of diagonals in a regular polygon. They should understand that the formula is D = n(n-3)/2, where D is the number of diagonals and n is the number of sides of the polygon.

3. Apply the concept of a diagonal to solve practical problems: Students should be able to apply what they have learned to solve problems that involve calculating the number of diagonals in a regular polygon. This includes the ability to adapt the formula to solve problems that may not be so straightforward.

Subsidiary objectives:

- Develop critical and analytical thinking skills when solving mathematical problems.
- Encourage active student participation during practical activities.
- Promote collaboration among students when working in groups to solve problems.

Introduction (10 - 15 minutes)

1. Review of previous concepts: The teacher should start the class by reviewing the concepts of polygons, vertices, and sides. They can do this through a short class discussion, asking students to share what they remember about these concepts. The teacher should ensure that all students have a solid understanding of these concepts, since they are fundamental to understanding the topic of the lesson.

2. Introductory problem situations: The teacher should then present two problem situations that will serve as a starting point for the discussion on diagonals. For example:

   • "If a regular polygon has 8 sides, how many diagonals does it have?"
   • "And if the polygon has 10 sides, how many diagonals will it have?"

3. Contextualisation of the importance of the subject: The teacher should then explain the importance of the topic, highlighting that the concept of a diagonal is fundamental to understanding the properties and calculations of regular polygons. They can mention how the diagonal is used in diverse areas, such as architecture, design, games, and even in nature.

4. Curiosities and stories: To pique the students' interest, the teacher can share some curiosities and stories related to the topic. For example:

   • "Did you know that the Great Pyramid of Giza, one of the Seven Wonders of the Ancient World, is actually a regular polygon with 4 sides (a square) and 4 diagonals?"
   • "And have you heard of the famous problem of drawing all the diagonals of a polygon? This is an old mathematical problem that has challenged many mathematicians throughout history."

5. Introduction of the topic: Finally, the teacher should introduce the topic of the lesson - the diagonal of a regular polygon. They can do this by explaining that a diagonal is a line segment that connects two non-consecutive vertices of a polygon. The teacher should then present the formula for calculating the number of diagonals in a regular polygon - D = n(n-3)/2 - and explain how it can be applied to solve the introductory problems.

Development (20 - 25 minutes)

1. Practical activity with folding: The teacher should distribute square sheets of paper to each student. They should then instruct the students to fold the sheets of paper in order to form different regular polygons, starting with a triangle and going up to a nonagon (a 9-sided polygon). As the students form each polygon, the teacher should encourage them to count the number of sides and vertices, and to draw all the diagonals. The teacher should circulate around the room, assisting students and clarifying any doubts. (10 - 15 minutes)

   • Step by step activity:
     1. Distribute the square sheets of paper to each student.
     2. Instruct the students to fold the sheets of paper in order to form different regular polygons.
     3. Encourage the students to count the number of sides and vertices, and to draw all the diagonals.
     4. Circulate around the room, assisting students and clarifying doubts.

2. Classroom discussion: After the practical activity, the teacher should lead a classroom discussion. They can start by asking the students what they observed when folding the sheets of paper. Then, they should ask the students if they noticed any patterns in the number of sides, vertices, and diagonals in each polygon. The teacher should then explain that, in fact, there is a pattern, which is given by the formula D = n(n-3)/2. (5 - 7 minutes)

3. Solving the introductory problem situations: The teacher should then return to the introductory problem situations and ask the students to try to solve them with the new understanding that they have acquired. They should encourage the students to use the formula D = n(n-3)/2 to solve the problems. The teacher should circulate around the room, assisting students and clarifying doubts. (5 - 7 minutes)

   • Step by step problem situation solving:
     1. Recall the introductory problem situations.
     2. Ask the students to try to solve them with the new understanding acquired.
     3. Circulate around the room, assisting students and clarifying doubts.

4. Application activity: To conclude the Development stage, the teacher should propose an application activity. They can present the students with a more complex problem that involves calculating the number of diagonals in a regular polygon, and ask them to solve it in groups. The teacher should circulate around the room, assisting groups and clarifying doubts. At the end, each group should present their solution to the class. (5 - 7 minutes)

   • Step by step application activity:
     1. Present the students with a more complex problem that involves calculating the number of diagonals in a regular polygon.
     2. Ask the students to solve the problem in groups.
     3. Circulate around the room, assisting groups and clarifying doubts.
     4. Each group should present their solution to the class.

Feedback (10 - 15 minutes)

1. Group discussion (3-5 minutes): The teacher should promote a group discussion on the solutions found by each of the groups. Each group will have up to 3 minutes to present their solution to the class, explaining the reasoning used and the steps taken to arrive at the answer. During the presentations, the teacher should encourage the other students to ask questions and give constructive feedback.

   • Group discussion step by step:
     1. Each group has up to 3 minutes to present their solution to the class.
     2. During the presentations, the teacher should encourage the other students to ask questions and give constructive feedback.

2. Connection with the theory (2-3 minutes): After all the presentations, the teacher should return to the theoretical concepts presented at the beginning of the class and relate them to the practical activities carried out. They can, for example, highlight how the formula D = n(n-3)/2 was applied to calculate the number of diagonals in each polygon and solve the problems proposed.

3. Individual reflection (3-5 minutes): The teacher should then propose that the students individually reflect on what they have learned during the class. They can do this through a series of questions, such as:

   1. "What was the most important concept you learned today?"

   2. "What questions still haven't been answered?"

   3. "How can you apply what you learned today to real-life situations?"

   • Individual reflection step by step:
     1. Propose that the students individually reflect on what they have learned during the class.
     2. Ask a series of questions to guide the students' reflection.
     3. Give the students one minute to think about each question.
     4. If there is time, ask some students to share their answers with the class.

4. Feedback and clarification of doubts (2-3 minutes): Finally, the teacher should request feedback from the students on the class, asking what they liked the most and what they would like to see more of in the next classes. They should also clarify any doubts that may still exist on the topic. The teacher should end the class by reinforcing the importance of what was learned and encouraging the students to continue studying the subject on their own.

   • Feedback and clarification of doubts step by step:
     1. Request feedback from the students on the class.
     2. Clarify any doubts that may still exist on the topic.
     3. End the class by reinforcing the importance of what was learned and encouraging the students to continue studying the subject on their own.

Conclusion (5 - 10 minutes)

1. Summary of the contents (2 - 3 minutes): The teacher should start the Conclusion stage by summarising the main contents covered in the class. They should recall the definition of a diagonal in a regular polygon, the formula for calculating the number of diagonals in a regular polygon (D = n(n-3)/2), and how this concept was applied to solve the problems proposed. The teacher should emphasise that the diagonal is a fundamental element for understanding the properties and calculations of regular polygons.

2. Connection between theory, practice and applications (2 - 3 minutes): Next, the teacher should explain how the class managed to establish the connection between theory, practice and applications. They should highlight how the practical folding activity allowed the students to visualise and count the diagonals of different polygons, reinforcing the theoretical concept. Furthermore, the teacher should stress how solving practical problems helped the students to apply the formula for calculating the number of diagonals in a regular polygon. Finally, the teacher should mention some real-world applications of this concept, such as in architecture and design.

3. Extra materials (1 - 2 minutes): The teacher should then suggest some extra materials for the students who wish to deepen their knowledge on the subject. These materials may include mathematics books, educational websites, explanatory videos and interactive games. The teacher should briefly explain each of the materials and how they can help the students to consolidate what they have learned in class.

4. Relevance of the subject (1 - 2 minutes): Finally, the teacher should highlight the importance of the subject for everyday life. They can explain that, although calculating the number of diagonals in a regular polygon may seem like a purely theoretical exercise, the ability to think in terms of patterns and formulas is a valuable skill that can be applied in many other areas. Furthermore, the teacher can again mention the practical applications of this concept, reinforcing that mathematics is present in our daily lives in ways that we may not always realise.