Contextualization

Theoretical Introduction

In mathematics, volume is a quantity that expresses the extension of a three-dimensional object or the capacity of a closed container. To conceptualize this, it is useful to think of a cube. A cube is a rectangular block, with all sides of the same measure. The capacity of this cube is considered as its volume. Volume is measured in cubic units, which are represented by a number raised to the third power, such as cubic centimeters (cm³), cubic meters (m³), and others.

To calculate the volume of a cube, we use a simple formula: Edges³, where the edges represent the length, width, and height of a rectangular block. In the case of our cube, since all sides have the same measure, the formula becomes Side³. The idea of unit cubes combines these two concepts. A unit cube is a cube whose sides measure 1 (unit). Therefore, its volume will always be equal to 1 (unit)³.

The concept of unit cubes is used to teach the volume of three-dimensional shapes in an intuitive way. Instead of worrying about the exact measurements of an object, you can simply count how many unit cubes would fit in the object to determine its volume.

Contextualization

Volume is a measure of great importance in our daily lives. Whether in engineering, architecture, chemistry, cooking, etc.; understanding volume improves our relationship with the world, whether to know how much concrete will be needed in a construction or how much liquid a bottle can hold.

Unit cubes are equally important. These small blocks can be used to help understand the size of an object in terms of other shapes. This allows complex things to be broken down into smaller and more understandable parts. For example, an engineer can use unit cubes to estimate the volume of material needed for a construction, or a teacher can use them to teach students about volume and area.

Practical Activity: "Building Cities with Unit Cubes"

Project Objective

The objective of this project is to provide practice of volume concepts using unit cubes in a fun and collaborative context. Students will have the opportunity to apply theoretical knowledge of volume in the construction of a mini city, while improving their teamwork, communication, time management, and problem-solving skills.

Project Description

Teams (of 3 to 5 members) will design and build a miniature city using unit cubes, playing "The Cube City Game" for this purpose. The city must include at least four different structures (e.g., houses, buildings, hospitals, schools, etc.) and each structure must be made with a different number of unit cubes, varying in shape and height.

Teams will have to plan the city, calculate the volume of each structure and the total volume of the city, as well as keep an accurate record of their decisions and calculations.

After the city is built, students will have to prepare an oral presentation for the class, explaining how the volumes were calculated and how the project decisions were made.

The key theoretical concepts addressed in the project are:

• Volume of cubes and rectangular blocks
• Volume measurement units
• Conversion between measurement units
• Addition and multiplication of volumes
• Understanding and manipulation of three-dimensional shapes

The disciplines involved are Mathematics and Arts, with the latter being used in an interdisciplinary way during the city construction.

Required Materials

• Unit cubes made of wood, plastic, or styrofoam (each cube should have edges of 1 cm, 1 dm, or 1 m, depending on the available space)
• Millimeter paper sheet
• Pens, pencils, and erasers
• Ruler
• Compass
• Calculator

Step by Step

Planning (4 hours)

1. Each team should start by sketching their city plan on the millimeter paper sheet, ensuring it has at least four different structures.
2. Next, they should determine how many unit cubes will be needed for each structure and make an estimated calculation of the total volume of the city.

Construction and Calculation (6 hours)

3. Teams will start building the city, adjusting the plan as necessary.
4. At the same time, they should keep a detailed record of how many unit cubes are being used for each structure and verify if the estimated total volume is correct.
5. They will also have to calculate the volume of each structure and the total volume of the city.

Documentation and Presentation (2 hours)

6. Teams should prepare a detailed report explaining the city plan, how they calculated the volume of each structure and the city as a whole, and how they made their decisions.
7. The report should include diagrams or photos of the structures and the complete city.
8. Finally, teams will give an oral presentation to the class, explaining their project and answering questions.

Project Deliverables

• City built with unit cubes.
• Written report following the structure of Introduction, Development, Conclusions, and Bibliography.
  • The Introduction should contextualize the theme, its relevance and real-world application, and the project objective.
  • The Development should explain the theory behind the key theoretical concepts, detail the activity, indicate the methodology used, and present and discuss the results obtained.
  • The Conclusion should summarize the main points of the project, explain the learnings obtained, and draw conclusions about the project.
  • The Bibliography should indicate the sources used for the project, including books, videos, web pages, etc.
• Oral presentation of the built city, where students will explain how they applied volume concepts, how they made their decisions, and will answer class questions.

Students should focus on how their volume calculations connect to the creation of the city, including how decisions were made based on these calculations. The report and presentation should show a clear understanding of the theoretical concepts addressed in the project and how they were applied in practice.