Contextualization

The Least Common Multiple, better known by the acronym LCM, is a very important mathematical concept for operations involving integers and fractions. The LCM of two or more numbers is the smallest integer that is a multiple of all of them. Thus, a number is a multiple of another when the remainder of the division between them is zero.

The LCM is one of those essential concepts that applies in various everyday scenarios, for example, when we want to synchronize tasks that repeat after different intervals. Imagine you have an alarm that goes off every 6 hours and another one that goes off every 8 hours. When will these two alarms go off at the same time again? Answering this question involves calculating the LCM between 6 and 8.

Furthermore, the LCM is fundamental for the addition and subtraction of fractions. When fractions have different denominators, it is necessary to find a common number for them, which is the LCM of the denominators. This way, we can perform the mathematical operation between the fractions.

The concept of LCM is widely used in various areas, such as engineering, exact sciences, and even in music, where the least common multiple between musical notes can be used to create harmonious rhythms.

To delve deeper into the concept of LCM and understand how it works, students can refer to the materials below:

• Article on Só Matemática about LCM
• Video lesson by Professor Ives Urquiza on YouTube about LCM
• Article on Mundo Educação about LCM

Practical Activity

Activity Title: "The Rhythm of Numbers"

Project Objective

The objective of this project is to apply the concept of LCM in the creation of a musical composition. For this, students should explore the multiples and the idea of rhythm and harmony, allowing them to understand the practical application of LCM.

Detailed Project Description

In groups of 3 to 5 students, each group should choose 3 different musical notes and relate each note to an integer representing the number of beats per minute (bpm) of that note in the composition's rhythm. The same note can be played at different tempos during the composition, with a minimum of 2 and a maximum of 10 bpm.

The first step will be to calculate the LCM of the different chosen bpm. Then, use this LCM to define the duration of the musical composition and to organize the notes to create a harmonious rhythm.

The project should be completed within 4 hours per participant, and the total duration, including the report preparation, should not exceed one week.

Necessary Materials

1. Paper and pen for notes and calculations.
2. Internet access for research and use of applications or websites that allow the creation of musical compositions, such as Flat or Soundtrap.
3. Mathematics books or online resources for consultation on LCM.

Detailed Step-by-Step

1. Gather in a group and discuss the activity, ensuring that everyone understands the concept of LCM and its application to the activity.
2. Choose three different musical notes and assign an integer representing the bpm (beats per minute) of that note, ranging from 2 to 10.
3. Calculate the LCM of the chosen bpm.
4. Use the found LCM as the number of beats for the musical composition.
5. Plan the musical composition so that all notes are played at the same time, providing a harmonious rhythm.
6. Create the musical composition using an application or website.
7. Record a video presenting the musical composition and explaining how the LCM was used.

Project Deliverables

The group must deliver the video of the musical presentation and a written report on the activity. The report should be structured as follows:

1. Introduction: Contextualize the theme, its relevance and real-world application, as well as the project's objective.
2. Development: Explain the LCM theory, detail the activity performed, the methodology applied, and discuss the results obtained.
3. Conclusion: Recap the main points, highlight the learnings and conclusions about the project.
4. Bibliography: Indicate the research and consultation sources for the project.

This report should contain the theory used (LCM), the description of the musical rhythm creation process, an explanation of how the LCM was used in the composition, an analysis of the results, and reflections on the project and teamwork collaboration.