Lesson Plan | Traditional Methodology | Regular Polygons: Angles and Diagonals

Objectives

Duration: 10 to 15 minutes

The purpose of this stage is to provide a clear and detailed overview of the fundamental concepts related to regular polygons. This includes identifying their characteristics, calculating the diagonals that pass through the center, as well as determining the internal and external angles. This initial understanding is essential for students to follow along and actively participate in subsequent explanations, ensuring solid and structured learning.

Main Objectives

1. Recognize and identify regular polygons.

2. Calculate the number of diagonals in a regular polygon.

3. Determine the internal and external angles of a regular polygon.

Introduction

Duration: 10 to 15 minutes

The purpose of this stage is to provide a clear and detailed overview of the fundamental concepts related to regular polygons. This includes identifying their characteristics, calculating the diagonals that pass through the center, as well as determining the internal and external angles. This initial understanding is essential for students to follow along and actively participate in subsequent explanations, ensuring solid and structured learning.

Context

To start the lesson on regular polygons, it's important to contextualize students about what they are and where we can find these polygons in our daily lives. Regular polygons are geometric figures with all sides and angles equal. They are foundational in geometry and appear in various contexts, from architecture to art and nature. For example, the pentagon in the logo of the famous United States defense institution or the hexagons that make up honeycombs are examples of regular polygons.

Curiosities

Did you know that honeycombs are made of perfect hexagons? This occurs because hexagons are the most efficient shape for storing honey, using the least amount of wax and space. This structural efficiency is an example of how nature uses regular polygons in a practical and intelligent way!

Development

Duration: 45 to 50 minutes

The purpose of this stage is to deepen the students' understanding of regular polygons, providing them with tools to calculate the number of diagonals, as well as the internal and external angles of these figures. Through detailed explanations and practical examples, students will be able to apply the formulas and solve related problems, thus consolidating their understanding of the topic.

Covered Topics

1. Definition of Regular Polygons 2. Explain that a regular polygon is a geometric figure with all sides and angles equal. Provide examples such as equilateral triangle, square, regular pentagon, among others. 3. Characteristics of Regular Polygons 4. Detail the properties of regular polygons, such as symmetry, equality of internal and external angles, and the possibility of being circumscribed by a circle. 5. Calculation of the Number of Diagonals 6. Teach the formula for calculating the number of diagonals in a regular polygon: D = [n(n-3)]/2, where n is the number of sides of the polygon. Present practical examples for different polygons. 7. Calculation of Internal Angles 8. Explain the formula for calculating the sum of the internal angles of a regular polygon: S = (n-2) * 180°. Then teach the formula for calculating each internal angle: A = [(n-2) * 180°] / n. 9. Calculation of External Angles 10. Show that the sum of the external angles of any regular polygon is always 360°. Teach the formula for calculating each external angle: E = 360° / n.

Classroom Questions

1. How many diagonals does a regular hexagon have? Use the formula D = [n(n-3)]/2. 2. Calculate the measure of each internal angle of a regular octagon. 3. Determine the measure of each external angle of a regular decagon.

Questions Discussion

Duration: 20 to 25 minutes

The purpose of this stage is to review and consolidate students' understanding of the content covered, ensuring they comprehend and know how to apply the formulas to calculate diagonals and angles of regular polygons. Detailed discussion of questions and active student engagement promote deeper and more meaningful learning, allowing students to clarify doubts and reinforce their knowledge.

Discussion

•  How many diagonals does a regular hexagon have?

To calculate the number of diagonals in a regular hexagon, we use the formula D = [n(n-3)]/2, where n is the number of sides of the polygon. In the case of the hexagon, n = 6.

Applying the formula:

D = [6(6-3)]/2 D = [6 * 3]/2 D = 18/2 D = 9

Therefore, a regular hexagon has 9 diagonals.

•  Calculate the measure of each internal angle of a regular octagon.

To calculate the measure of each internal angle of a regular octagon, first we find the sum of the internal angles using the formula S = (n-2) * 180°, where n is the number of sides.

In the case of the octagon, n = 8.

S = (8-2) * 180° S = 6 * 180° S = 1080°

Now, we divide the sum of the angles by the number of sides to find the measure of each internal angle:

A = 1080° / 8 A = 135°

Therefore, each internal angle of a regular octagon measures 135°.

•  Determine the measure of each external angle of a regular decagon.

To calculate the measure of each external angle of a regular decagon, we use the formula E = 360° / n, where n is the number of sides.

In the case of the decagon, n = 10.

E = 360° / 10 E = 36°

Therefore, each external angle of a regular decagon measures 36°.

Student Engagement

1.  Question 1: If a regular polygon has 12 sides, how many diagonals does it have? 2.  Question 2: What is the measure of each internal angle of a dodecagon (12 sides)? 3.  Question 3: What is the measure of each external angle of an icosagon (20 sides)? 4. ❓ Reflection: Why is the sum of the external angles of any polygon always 360°? 5. ❓ Reflection: How does nature use regular polygons efficiently? Think of other examples besides honeycombs.

Conclusion

Duration: 10 to 15 minutes

The purpose of this stage is to review and consolidate the knowledge acquired during the lesson, ensuring that students have a clear and applied understanding of the discussed concepts. This reinforces learning and prepares students to use this knowledge in future contexts.

Summary

• Definition and characteristics of regular polygons.
• Calculation of the number of diagonals in a regular polygon using the formula D = [n(n-3)]/2.
• Calculation of the sum of the internal angles of a regular polygon with the formula S = (n-2) * 180°.
• Determination of each internal angle of a regular polygon using the formula A = [(n-2) * 180°] / n.
• Calculation of the external angles of a regular polygon through the formula E = 360° / n.

The lesson connected theory with practice by demonstrating how the mathematical concepts of regular polygons are applicable in real situations, such as in architecture and nature. Concrete examples, such as honeycombs and geometric shapes in logos, helped illustrate the applications of the calculations of diagonals and angles, making the theory more tangible and relevant for students.

The study of regular polygons is important for daily life as these shapes frequently appear around us. From the structural efficiency of honeycombs to the symmetrical beauty in architectural and artistic designs, understanding these mathematical concepts allows for a deeper appreciation and practical application in various fields, fostering critical thinking and problem-solving skills.