INTRODUCTION

The Relevance of the Topic

• Basis for Future Concepts: Double, half, triple, and one-third are key mathematical concepts that help build a solid foundation for understanding fractions, integers, and proportions.
• Everyday Tools: These mathematical ideas are frequently used in daily life — for example, when doubling a recipe or dividing a pizza among friends.

Contextualization

• First Steps in Algebra: Studying double, half, triple, and one-third is one of the first steps to understanding equations and more complex formulas that will be learned in the following years.
• In the Curriculum: This topic is located at the beginning of the study of mathematics in Elementary School, providing students with the necessary understanding to deal with numbers and basic operations.
• Connection to Other Areas: It applies to other disciplines and activities, such as arts (symmetry and proportions), sciences (measurements and transformations), and even physical education (point counting and team division).

Through this theme, students begin to see mathematics as an integral part of the world around them and as a tool to solve practical problems.

THEORETICAL DEVELOPMENT

Components

• Double: Means multiplying a number by 2. When talking about double, we are doubling the initial quantity.
  • Contribution to Understanding: Understanding double prepares for multiplication and also to understand the concept of symmetry.
• Half: Dividing a number by 2. Discovering the half is an essential part of understanding division and fractions.
  • Characteristics: Half of a number is always less than the whole number, and if the number is even, the half will be a whole number.
• Triple: Multiplying a number by 3. It means having three times the quantity of something.
  • Practical Importance: Helps in understanding counting patterns and in introducing the idea of multiples in mathematics.
• One-Third: Dividing a number by 3. Relates to the concept of dividing equally into three parts.
  • Connection to Fractions: Introduces the concept of a number being part of a whole, an essential step for understanding complex fractions.

Key Terms

• Multiplication: Mathematical operation where a number is added to itself several times.
• Division: Mathematical operation that consists of dividing one number by another, resulting in how many times the second number fits into the first.
• Even Number: An integer that can be divided by 2 without leaving a remainder.
• Odd Number: An integer that, when divided by 2, leaves a remainder of 1.

Examples and Cases

• Double Example: If we have 3 apples and want the double, we multiply 3 by 2. The result is 6 apples.
  • Theory: 3 (original quantity) × 2 (double) = 6 (result).
• Half Example: We have 10 candies and want to give half of them. We divide 10 by 2. Each friend receives 5 candies.
  • Theory: 10 (original quantity) ÷ 2 (half) = 5 (result for each friend).
• Triple Example: If a box has 4 balls and we want the triple, we multiply 4 by 3. We will have 12 balls.
  • Theory: 4 (original quantity) × 3 (triple) = 12 (result).
• One-Third Example: A ribbon of 9 meters will be cut into 3 equal parts. We divide 9 by 3. Each part of the ribbon is 3 meters.
  • Theory: 9 (original length of the ribbon) ÷ 3 (one-third) = 3 meters (length of each part).

These examples show the practical application of the concepts and help reinforce the understanding of double, half, triple, and one-third.

DETAILED SUMMARY

Relevant Points

• Understanding Double: When doubling a quantity, we have twice as much. It's like having two lines of the same size!
  • Memorization: Imagining two hands where each has the same number of candies helps remember what double is.
• Discovering Half: Half is like dividing a chocolate bar equally between two friends. Both get equal parts, which are smaller than the whole bar.
  • Visualization: Cutting papers into two equal pieces to observe that each piece is half of the original.
• Exploring Triple: Having triple means having three times a quantity. It's like having three identical triplets, all with the same number of toys!
  • Association: It can be associated with three meals a day - breakfast, lunch, and dinner - each with the same number of fruits.
• Understanding One-Third: One-third is a way to divide something into three equal pieces, like when dividing a pizza among three friends.
  • Representation: Using a three-piece puzzle to show that each piece is one-third of the whole puzzle.

Conclusions

• Multiplication and Division Are Inverses: When working with double and triple, we use multiplication; with half and one-third, we use division.
• Equality is Key: When dividing to find the half or one-third, it is important that the parts are equal.
• Even and Odd Numbers: We understand that the half of an even number is always a whole number, but the half of an odd number is not.
• Applicable Concepts: All these concepts are not only for numbers but also for everyday objects, such as food, toys, and even time.

Exercises

1. Doubling Numbers: If you have 5 pencils, how many pencils will you have if you take the double?
   • Expected Answer: 10 pencils.
2. Finding the Half: There are 12 chocolates in the box. If we divide them equally between two friends, how many chocolates will each have?
   • Expected Answer: 6 chocolates for each friend.
3. Dividing into One-Thirds: A rope is 15 meters long. If we divide it into three equal parts, what is the length of each part?
   • Expected Answer: 5 meters for each part of the rope.

These exercises are formulated to reinforce the practical application of the concepts of double, half, triple, and one-third. They encourage visualization and intuitive understanding of basic mathematical operations.