Introduction

Relevance of the Topic

Exploring the enlargement and reduction of figures is essential for understanding vital aspects of Mathematics and beyond. Through these operations, we learn to see and manipulate the world around us in a mathematically informed way. This allows us to comprehend complex phenomena, such as time dilation in physics or exponential growth in economics.

Contextualization

The enlargement and reduction of figures are relevant subjects that fit into the 6th grade of Elementary School, a stage where students are beginning to experience more abstract and complex Mathematics. This approach provides the foundation for future concepts, such as proportions and ratios, and is closely linked to geometry. By mastering this skill, students will be well-prepared to deepen their studies in the field of Mathematics and, at the same time, strengthen their logical thinking and problem-solving skills.

Theoretical Development

Components

• Enlargement of Figures: Enlarging a figure means increasing its dimensions without altering its shape. This is done by multiplying all its measurements by the same factor. For example, to enlarge a figure by a factor of 2, each side will be doubled.
• Reduction of Figures: Unlike enlargement, the reduction of figures involves decreasing the dimensions of a figure, again without altering its shape. To reduce a figure by a factor of 2, each side will be divided by 2.
• Enlargement and Reduction Factors: The enlargement or reduction factor is the proportion of increase or decrease in each dimension of the figure. This factor is usually expressed as a fraction or decimal, and is a positive real number.

Key Terms

• Figure: A figure is any two-dimensional shape, such as a rectangle, circle, triangle, etc.
• Enlargement: Proportional increase in the dimensions of a figure, while maintaining the shape.
• Reduction: Proportional decrease in the dimensions of a figure, while maintaining the shape.
• Enlargement/Reduction Factor: Number used to multiply (enlargement) or divide (reduction) the dimensions of the figure.

Examples and Cases

• Enlargement of Brazil: If we want to create an enlarged representation of the map of Brazil, we could use an enlargement factor of 2. In this case, the longitude and latitude of each border of Brazil will be multiplied by 2, maintaining the general shape of the country, but significantly increasing its area.
• Reduction of a Drawing: Suppose we have a drawing of a house that measures 10 centimeters in height. If we want to make a reduced version of this drawing that is half the original size, the height of the new version would be 5 centimeters. In this case, the reduction factor would be 0.5.

By understanding the theory and exemplifying the concept with everyday situations, students' ability to enlarge and reduce figures will be enhanced and their practical applications understood.

Detailed Summary

Key Points

• Enlargement and Reduction: A figure can be enlarged or reduced while maintaining its original shape. This means that, although its dimensions have been altered, the angles and proportions between the parts of the figure remain the same.

• Factors and Operations: The operations of enlargement and reduction are carried out using an enlargement or reduction factor. This factor is a positive real number and all measurements of the original figure are multiplied by this factor (enlargement) or divided by it (reduction).

• Practical Applications: The ability to enlarge and reduce figures has very practical applications, from creating maps and charts to reproducing drawings and projects in different scales.

Conclusions

• Proportion Manipulation: Enlarging and reducing figures are operations that allow us to manipulate the proportions between the parts of a figure. This is a fundamental skill that has applications in many different fields, from mathematics and physics to art and design.

• Geometric Connection: The ability to enlarge and reduce figures is directly linked to the study of geometry. It teaches us to consider the shape and proportions, rather than the absolute size, a critical skill for solving geometric problems.

• Flexible Thinking: The practice of enlarging and reducing figures also helps develop flexible thinking. This is because it requires students to see the same things in different ways and consider multiple perspectives.

Suggested Exercises

1. Enlargement and Reduction of Maps: Given a map of a city, ask students to create an enlarged or reduced version of that map using an enlargement or reduction factor of their choice. They should explain how they arrived at their enlargement or reduction factor and what will be the impact of this factor on the size of distances on the map.

2. Reduction of Drawings: Ask students to make a reduced copy of a given drawing, using a reduction factor of their choice. They should explain how they arrived at their reduction factor and what will be the final size of the drawing in relation to the original size.

3. Proportions in Practice: Given a rectangle, ask students to create an enlarged and reduced version of this rectangle. They should explain what their enlargement or reduction factor was and what the measurements of the new rectangle are in relation to the original one.