Summary of Comparison of Object Areas

INTRODUCTION

The Relevance of the Theme

Comparing areas is essential to understand how different objects occupy space. It's like comparing pieces of land to know which one is larger. In mathematics, learning to compare areas helps us solve everyday problems, such as choosing the best size of gift paper or deciding which rug fits best in the classroom.

Contextualization

In the vast world of Mathematics, area is a key concept that begins to be explored in the early years of Elementary School. The ability to compare areas builds the foundation for more complex understandings that will be learned in the following years, such as calculating area in units, spatial geometry, and even in physics, when exploring concepts like pressure. Comparing areas helps us 'measure' space visually and make smarter choices when dealing with spaces and objects around us.

THEORETICAL DEVELOPMENT

Components

• Area: Space occupied by a shape or surface, measured in square units (e.g., cm², m²). Comparing areas is like looking at two rugs and deciding which one covers more ground.
  • Importance: Knowing the area helps us understand the size of flat surfaces.
  • Characteristics: Area is always positive and depends on the shape of the object.
  • Comparing areas: Involves looking at two or more objects and identifying which one has a larger or smaller area.

Key Terms

• Larger and Smaller: Used to talk about size. Larger is when something has more space, smaller is the opposite.

  • Larger: More covered area.
  • Smaller: Less covered area.

• Geometric Shapes: Objects like squares, rectangles, and circles that have area.

  • Square: Four equal sides and the same covered area in each square unit.
  • Rectangle: Two longer sides equal and two shorter sides equal; area is the product of the base by the height.
  • Circle: Has no sides! Its area is a bit more complicated, but it is the region inside its round border.

Examples and Cases

• Comparing two squares: One with sides of 2 cm and another with sides of 3 cm.

  • 2 cm Square: Area = side × side = 2 cm × 2 cm = 4 cm².
  • 3 cm Square: Area = side × side = 3 cm × 3 cm = 9 cm².
  • Which one is larger? The 3 cm square has a larger area (9 cm² > 4 cm²).

• Comparing a rectangle and a square: Rectangle with sides of 2 cm by 4 cm and a square of 3 cm on each side.

  • Rectangle: Area = base × height = 2 cm × 4 cm = 8 cm².
  • Square: Area = side × side = 3 cm × 3 cm = 9 cm².
  • Which one has more area? The square, with 9 cm².

• Comparing irregular objects: Drawings of two lakes on graph paper, count the filled squares to know the area.

  • Lake A: Has 15 filled squares.
  • Lake B: Has 20 filled squares.
  • Which lake is larger? Lake B, with 20 filled squares.

Each example demonstrates the process of visually comparing areas, helping to understand which object occupies more space on a flat surface.

DETAILED SUMMARY

Relevant Points

• Occupied Space: Understanding that area is the space a shape occupies.
• Square Units: Knowing that area is measured in little squares or square units, such as cm² or m².
• Visual Comparison: Learning to compare areas by looking and imagining how many little squares of a certain size fit inside each object.
• Shapes and Sizes: Identifying that different shapes can have the same area, and that equal shapes can have different areas depending on the size.
• Larger vs. Smaller Area: Knowing that the larger area covers more space and the smaller one, less space.

Conclusions

• Important Visualization: We conclude that comparing areas can be done visually, without necessarily measuring.
• Area and Shape: We discovered that the shape of the object (square, rectangle, circle) influences how we calculate the area.
• Practical Use: We understand that knowing how to compare areas is useful in practical everyday activities.

Exercises

1. Comparing Rugs: Draw two rugs on graph paper: rug A with 4 squares wide by 3 squares long, and rug B with 5 squares wide by 2 squares long. Which rug has a larger area?

2. Book Covers Area: Take two hardcover books, one small and one large. Without measuring, try to decide which cover has a larger area. Then, use a ruler to check your estimate.

3. Little Squares and Colors: Paint 10 little squares blue and 15 green on graph paper. Which color represents a larger area? Now, create a drawing that has exactly the same area as the green squares, but with a different shape.

Each exercise promotes the application of area comparison in practical and visual contexts, reinforcing the understanding of the concept.