# Contextualization

## Introduction to Repeating Decimals

Repeating decimals, as the term suggests, are decimal numbers that have a pattern of one or more digits that repeat indefinitely after the decimal point. They are also known as recurring decimals or periodic decimals. Repeating decimals are a fascinating concept in mathematics that tie together the worlds of fractions and decimals.

In the base-10 number system that we use every day, some fractions can't be expressed exactly as a decimal, and we get a repeating decimal instead. For instance, the fraction 1/3, when converted to a decimal, becomes 0.3333... where the 3s repeat infinitely. Many fractions, even those that seem simple, can result in repeating decimals, such as 1/7 (0.142857142857...) or 2/11 (0.181818...).

Understanding repeating decimals is not just a theoretical exercise; it's a practical skill with many real-world applications. For instance, in the realm of finance, understanding repeating decimals can help us understand the concept of compound interest, which is essentially a repeating decimal calculation. In physics, many physical constants, like the speed of light, are known to an extremely high precision and are expressed as repeating decimals.

## Relevance and Real-World Applications

The concept of repeating decimals is not just limited to the classroom. It has numerous real-world applications, from calculating interest in financial transactions to understanding the behavior of physical constants in scientific research.

In the financial industry, banks and financial institutions often use repeating decimals when calculating compound interest. This type of interest is calculated on both the initial principal and the accumulated interest from previous periods. The use of repeating decimals ensures an accurate representation of the compound interest rate.

In the scientific field, repeating decimals are commonly used in physics, particularly in calculations involving physical constants. These are fixed, universal numbers that exist in nature, such as the speed of light or the gravitational constant. These constants are known to a high degree of precision and are often expressed as repeating decimals.

# Practical Activity

## Objective of the Project

The main goal of this project is to understand the concept of repeating decimals and their conversion into fractions. The students will explore, decode, and present different repeating decimals, focusing on how to convert them into their fractional form.

## Detailed Description of the Project

Students will form groups of 3 to 5. Each group will be tasked with researching and selecting five unique repeating decimal numbers. They will then explore the process of converting these repeating decimals into fractions and record their findings. Finally, they will create a presentation summarizing their work, showcasing their understanding of the topic, and explaining the process of converting repeating decimals to fractions.

## Necessary Materials

• Internet access for research
• Writing materials (notebook, pen, pencil)
• Calculator (optional, but helpful for calculations)
• Presentation tools (PowerPoint, Google Slides, etc.) for the final presentation

## Detailed Step-by-Step for Carrying Out the Activity

1. Formation of Groups (Estimated time: 10 minutes): Students form groups of 3 to 5. This project is designed to be completed in a group to encourage collaboration and teamwork.

2. Researching Repeating Decimals (Estimated time: 30 minutes): Each group will research and select five unique repeating decimal numbers. These can be from real-world examples or theoretical situations.

3. Conversion into Fractions (Estimated time: 2 hours): The groups will explore the process of converting their selected repeating decimals into fractions. They will record the steps they took and any patterns they noticed during the conversion process.

4. Creating a Summary Document (Estimated time: 1 hour): Using the data collected, the groups will create a summary document that includes their selected repeating decimals, the process they used to convert them into fractions, and any patterns they discovered.

5. Creating a Presentation (Estimated time: 1 hour): Based on the summary document, each group will create a presentation to explain their findings. The presentation should include the selected repeating decimals, the process of conversion, and any patterns or observations that the group made.

6. Presentation and Discussion (Estimated time: 1 hour): Each group will present their findings to the class. After each presentation, there will be a short discussion where students can ask questions or share their thoughts.

7. Writing the Report (Estimated time: 3-4 hours): Finally, the students will write a report following the guidelines provided, detailing the theory behind the topic, the process of the activity, the results obtained, and the conclusions derived.

## Project Deliverables

Each group will be responsible for delivering the following items:

1. Summary Document: This document will include the selected repeating decimals, the process used to convert them into fractions, and any patterns or observations made during the activity.

2. Presentation: A presentation summarizing their work and findings. The presentation should be engaging, informative, and clear.

3. Written Report: The report will be divided into four sections:

• Introduction: Here, students will contextualize the theme, its relevance, and real-world applications. They will also state the objective of the project.

• Development: This section will detail the theory behind repeating decimals and their conversion into fractions. It will also explain the activity in detail, the methodology used, and the results obtained.

• Conclusion: The students will revisit the main points of the project, explicitly stating what they learned, and drawing conclusions based on their observations and results.

• Bibliography: The students will list the sources they relied on for the project, such as books, web pages, videos, etc.

The activity is estimated to take around 10-12 hours per participating student, to be carried out over a week. The project will not only test the students' understanding of repeating decimals but also their research, problem-solving, and presentation skills.

IARA TIP

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