Expansion and Reduction of Figures | Active Summary

Objectives

1. 🎯 Understand and apply the concept of enlargement and reduction of figures proportionally, calculating areas and perimeters.

2. 🔍 Develop spatial visualization and logical reasoning skills when working with geometric transformations in figures.

3. 🌍 Recognize the practical importance of these concepts in various fields such as engineering, architecture, and design.

Contextualization

Did you know that the same principle of enlargement and reduction of figures that we use in Mathematics is applied in engineering to create scaled models of large constructions? For example, to design a new skyscraper, engineers often build a smaller model that exactly represents what the final building will look like. This not only helps visualize the project but also tests its stability and efficiency. Therefore, understanding these concepts is not just useful in the classroom but also in real projects that shape our world!

Important Topics

Enlargement of Figures

Enlarging a figure means proportionally increasing its dimensions. This concept is crucial not only in mathematics but also in many practical applications such as architecture and design. When enlarging a figure, all sides and angles are increased in the same proportion, maintaining similarity with the original figure.

• All sides of a figure are multiplied by the same factor to perform the enlargement.

• The area of the enlarged figure is the square of the enlargement factor in relation to the original area.

• The perimeter of the figure is multiplied by the enlargement factor because all sides are increased proportionally.

Reduction of Figures

Reducing a figure means decreasing its dimensions proportionally. This concept is essential for various applications, such as map scales and technical drawings. In reduction, all sides and angles are decreased in the same proportion, preserving similarity with the original figure but with a smaller size.

• All sides of a figure are multiplied by a factor less than 1 to perform the reduction.

• The area of the reduced figure is the square of the reduction factor in relation to the original area.

• The perimeter of the figure is multiplied by the reduction factor because all sides are decreased proportionally.

Calculation of Area and Perimeter

Calculating area and perimeter is fundamental when working with enlargements and reductions of figures. The area of an enlarged or reduced figure is calculated by applying the square of the change factor to the original area. The perimeter is calculated by multiplying the change factor by the original perimeter.

• Area = (Enlargement or reduction factor)² x Original area.

• Perimeter = Enlargement or reduction factor x Original perimeter.

• These calculations are essential for projects that involve scale changes, such as in civil engineering and cartography.

Key Terms

• Proportional Enlargement: Increase of all sides of a figure while maintaining the same proportion of increase.

• Proportional Reduction: Decrease of all sides of a figure while maintaining the same proportion of reduction.

• Enlargement or Reduction Factor: Number by which the sides of a figure are multiplied to perform the enlargement or reduction.

To Reflect

• How can the ability to work with enlargement and reduction of figures be applied in your daily life outside the classroom?

• Why is it important for all sides of a figure to be altered proportionally during an enlargement or reduction?

• In what ways can knowledge about calculating areas and perimeters influence decisions in design or architecture projects?

Important Conclusions

• Today, we explored the fascinating world of enlargements and reductions of figures, learning how all dimensions of a figure must be proportionally altered to maintain its shape and characteristics. This is not only crucial in mathematics but also in various practical applications, from engineering projects to interior design.

• We understood that when enlarging a figure, the area is multiplied by the square of the enlargement factor, while the perimeter is simply multiplied by the factor. In reduction, the calculations follow a similar pattern but with reduction factors.

• These skills not only expand our mathematical knowledge but also stimulate critical and analytical thinking that is essential in many professions and everyday situations.

To Exercise Knowledge

1. Select an object in your home that can be reproduced on paper. Draw it in a smaller and larger size, calculating the dimensions to maintain proportionality. 2. Create a treasure map where you shrink a map of a local park and hide a 'treasure' (a small item) at the reduced location. Challenge a friend or family member to find the treasure using your instructions. 3. Use drawing software to practice enlarging and reducing more complex figures, such as a floor plan of a house.

Challenge

Designer Challenge: Imagine you are an interior designer and need to redesign a room on a smaller scale for a project. Use your enlargement and reduction skills to create the new layout of the room, including furniture and accessories, while maintaining functionality and aesthetics. Present your project with the appropriate proportions and explain your design choices.

Study Tips

• Use graph paper to practice drawings and calculations of enlargement and reduction. This will help visualize scale changes better.

• Explore design and 3D modeling apps that allow you to adjust the proportions of figures. Many of these apps are free and can be fun to use.

• Regularly review the concepts of area and perimeter to strengthen your understanding and application in enlargement and reduction problems.