Contextualization

Mathematics is a discipline with wide scope and applications. It is found in almost every context and situation of our lives. Among the various mathematical concepts that we study, the concepts of divisors and multiples of a number play a central role. They are basic and essential tools not only for mathematics, but for several other disciplines and everyday situations.

The idea of divisors and multiples is quite simple, yet powerful. We say that a number is a multiple of another if it is possible to divide the first by the second without any remainder. Similarly, a number is considered a divisor of another if the second can be divided by the first without any remainder as well. It is as if these numbers maintain a relationship of "kinship", where the divisors are the "parents" and the multiples are the "children".

For example, the divisors of 12 are 1, 2, 3, 4, 6, and 12. This means that if we divide 12 by any of these numbers, we will get a result with no remainder. The multiples of 3 are numbers such as 3, 6, 9, 12, 15, 18, 21, and so on. Note that 12 is both a multiple of 3 and has 3 as one of its divisors!

Understanding these concepts is crucial not only for our mathematical education, but for understanding the world around us. It is the basis for our understanding of fractions, divisions, factorizations, and even more advanced concepts such as the Fundamental Theorem of Arithmetic. Furthermore, they appear in several everyday situations. When we divide a pizza into equal pieces, we are using the idea of divisors. When we plan a trip and divide the costs among the participants, we are working with the notion of multiples.

Thus, it is very important that we master these concepts so that we can use mathematics effectively and accurately. And that is what this project sets out to do. Our goal is to deepen our understanding of these concepts and develop our calculation and logical reasoning skills.

To this end, we suggest the following reliable sources to further explore the topic:

1. Brasil Escola: What are Multiples and Divisors
2. Mundo Educação: Divisors of a Number
3. Khan Academy: Divisors, Multiples, and Factorization

Practical Activity: “Discovering Family Relations between Numbers – Divisors and Multiples”

Project Objective

The objective of this project is to allow students to apply and improve their knowledge of divisors and multiples through a fun and engaging activity. It is expected that, at the end of the project, students will be able to identify, differentiate, and apply the concepts of divisor and multiple of a number.

Detailed Description of the Project

Students will be divided into groups of 3 to 5 and each group will receive a random main number between 1 and 100. Each group must investigate and list all the divisors and multiples of this number up to 500.

After that, each group must create a "genealogical map" or a "family tree of numbers". This map should show the main number in the center, its divisors on its left (as "parents"), and its multiples on its right (as "children"). The map can be drawn on paper or digitally.

Next, each group must create a "family book", where a story will be written about the main number, its "ancestors" (divisors), and its "progeny" (multiples). This story must be creative and engaging, telling how the main number interacts with its divisors and multiples.

Finally, each group will give a presentation of their work to the class, displaying the map, telling the story, and explaining how they reached their conclusions.

Required Materials

• Paper and cardboard.
• Pencils, pens, and colored markers.
• Ruler and compass (to draw the genealogical map).
• Access to the internet for research.
• Cameras or webcams (for the virtual presentation, if necessary).

Detailed Step-by-Step Instructions for the Activity

1. Form groups of 3 to 5 students.
2. Distribute a random main number between 1 and 100 to each group.
3. Each group must research and list all the divisors and multiples (up to 500) of the main number.
4. Each group must create a "genealogical map" or "family tree of numbers" with the main number in the center, its divisors on the left, and its multiples on the right.
5. Each group must create a "family book" with a creative story about the main number, its divisors, and its multiples.
6. Each group must prepare a presentation for the class, explaining the map and the story.
7. Give the presentations.

Project Deliverables

Each group must deliver:

1. The genealogical map of the main number, with all divisors and multiples clearly identified.

2. A digital or physical copy of the "family book".

3. A report documenting the project, structured as follows:

   a) Introduction: Contextualize the project, explain the importance of the topic, the objective of the project, and the main number assigned to the group.

   b) Development: Explain the theory of multiples and divisors and how they were identified for the main number. Detail the process of creating the genealogical map and the family book. Discuss the difficulties encountered and how they were overcome.

   c) Conclusions: Conclude the work, revisiting the main points of the project, reaffirming the learning acquired, and highlighting the practical application of the concepts of multiples and divisors.

   d) Bibliography: Indicate the sources of information used for the work, including website links, book titles, etc.

The expected duration of the project is approximately 3 hours per student, and the deadline is one week from the start date.

At the end of the project, students should have understood and be able to apply the concepts of divisors and multiples, in addition to having developed socioemotional skills such as time management, communication, problem solving, creative thinking, and proactivity, among others.