INTRODUCTION

The Relevance of the Topic

Hello, shape explorers! 🌟 Let's embark on an adventure through the world of geometric shapes. With the help of shape properties, we'll discover how objects around us are constructed and related.

Shape properties are like mathematical superpowers that help us:

• 🎨 Create art and design with symmetry and patterns.
• 🏠 Understand architecture and construction from castles to dog houses.
• 🗺️ Navigate through maps and solve puzzles.
• 💡 Develop logical reasoning and problem-solving skills.

They are fundamental because they give us tools to describe and compare different shapes, which is super important in mathematics and in various activities of our daily lives.

Contextualization

Imagine yourself as a math detective 🕵️‍♂️, seeking clues to understand the world of shapes. Shape properties are the clues we find in this search.

• We are in the universe of Geometry, a field of mathematics that studies sizes, shapes, and spaces.
• The shapes we will study are called polygons, and they are like puzzles that make up everything we see around us.
• We bring with us the knowledge of straight and curved lines, angles, and simple figures we learned in previous years.
• Today's adventure fits into our mathematical journey, preparing us for future explorations in more complex areas, such as area and perimeter.

Remember, young mathematicians 🔍: each shape has its story, its unique characteristics, and its place in the great map of Geometry. Ready to unravel these secrets? Let's go!

THEORETICAL DEVELOPMENT

Components of Shapes

• Sides of a polygon: These are the straight lines that form the edge of the figure.
  • A polygon can have several sides, and each side is an important part of the puzzle.
  • The number of sides helps to name the polygon: triangle (3), square (4), pentagon (5), and so on.
• Vertices (or corners): The point where two sides meet.
  • The number of vertices is always equal to the number of sides in a polygon.
  • Vertices are angular points, like the tips of a star.
• Internal angles: The space between two sides that meet at a vertex.
  • In a regular polygon, all internal angles are equal.
  • The angles give us clues about the shape of the polygon (sharp like an arrow or wide like a door).
• External angles: The space between one side of the polygon and the extended line of the adjacent side.
  • Important for understanding how shapes fit together with others around them.

Key Terms

• Polygons: Closed flat figures, formed by straight lines.
  • Come from "poly" (many) and "gon" (angles) in ancient Greek, meaning many angles.
• Regular Polygons: All sides and angles are equal.
  • Like a perfect square or an equilateral triangle.
• Irregular Polygons: Sides and angles of different sizes.
  • More common in real life, like a country's map.
• Diagonal: Straight line that connects two non-adjacent vertices.
  • Not a side, but crosses the interior of the polygon.
  • Helps to see how polygons can be divided into triangles.

Examples and Cases

• Triangle: The simplest form of polygon.
  • Has 3 sides, 3 vertices, and 3 internal angles.
  • The internal angles of any triangle add up to 180°.
• Square vs. Rectangle: Both have 4 sides, but...
  • Square: all sides and angles are equal.
  • Rectangle: opposite sides are equal, and all angles are right (90°).
• Regular Pentagon vs. Irregular:
  • Regular: 5 equal sides, equal angles, symmetrical.
  • Irregular: 5 sides and angles that may not be equal, varied shape.
• Calculation of Diagonals:
  • Formula to find the number of diagonals: Diagonals = n(n - 3)/2, where n is the number of sides.
  • A square has 4 sides, so diagonals would be 4(4 - 3)/2 = 2 diagonals.

DETAILED SUMMARY

Relevant Points

• Identification of Polygons: Ability to recognize polygons by the number of sides and angles.
  • Importance of observing and counting sides and corners to correctly name each polygon.
• Difference between Regular and Irregular: Understanding that regular polygons have equal sides and angles and irregular ones do not.
  • Regulars are symmetrical and balanced; irregulars are more common and unique.
• Internal and External Angles: Learning how to identify and differentiate angles inside and outside polygons.
  • Knowledge that the sum of internal angles changes with the number of sides.
• Diagonals of a Polygon: Understanding how diagonals connect vertices and the formula to calculate how many there are.
  • Ability to apply the formula to discover new shapes within polygons.

Conclusions

• Polygons are Universal: Recognition that polygons can be found everywhere, from art to architecture.
  • Understanding the importance of polygons in describing and constructing the world around us.
• Each Polygon is Unique: Awareness that each polygon has specific properties that make them special.
  • Appreciation of the differences between polygons, which enriches our geometric understanding.
• Mathematics is a Tool: Perception that mathematical formulas and concepts are tools for solving real-world problems.
  • Encouragement in using mathematical logic to explore and create new shapes.

Exercises

1. Identification of Polygons: Draw 5 different shapes and name them based on the number of sides. Example: a pentagon has 5 sides.
2. Finding Angles: Draw an irregular quadrilateral and hexagon. Estimate the measure of the internal angles and define which are larger or smaller.
3. Mysterious Diagonals: Use the formula to calculate the number of diagonals in a 6-sided polygon (a hexagon). Then, draw the polygon and all possible diagonals.