Contextualization

Exponentiation is a widely used mathematical concept and plays a crucial role in various areas of science, engineering, and technology. Basically, Exponentiation is an operation that involves two numbers, the first being the base number and the second being the exponent. When dealing with rational exponents, that is, exponents that are fractions, we enter the realm of radicals.

Our decimal system is based on powers of ten, and the domain of exponentiation and rational exponents allows us to understand and manipulate these numbers more effectively. Furthermore, exponentiation is a prerequisite for understanding more complex mathematical concepts such as exponential and logarithmic functions.

In a more practical context, exponentiation and specifically exponentiation with rational exponents are used in a variety of everyday and professional contexts. For example, in engineering and physics, exponentiation is often used to calculate measures of energy, electricity, and force. In economics and finance, it is used to calculate compound interest. In computer science, it is used to calculate speed and processing capacity.

Understanding the concept of exponentiation with rational exponents allows expanding problem-solving skills, helping to develop mathematical thinking skills and the application of mathematical concepts in a variety of real-world situations.

As supporting material for the study of this concept, you can use:

1. Book: BONJORNO, José Roberto. Matemática: volume único. São Paulo: FTD, 2004.
2. Website: Só Matemática. Exponentiation. Available at: www.somatematica.com.br/fundam/potenciacao.php
3. Video: YouTube. Matemática Rio with Prof. Rafael Procopio. Explaining Power with fractional exponent. Available at: www.youtube.com/watch?v=H_6q8qH_6q8

Practical Activity

Title: Demystifying Rational Exponents through the Construction of Geometric Solids

Project Objective:

This activity aims to allow students to:

1. Understand the concept of exponentiation with rational exponents and the relationship between exponentiation and radicalization.
2. Relate the mathematical concepts learned to real-life situations through the construction of models of geometric solids.
3. Develop teamwork skills, problem-solving, and critical thinking.

Detailed Project Description:

In this project, students will be challenged to build models of geometric solids (cubes, spheres, prisms) using cardboard or recyclable materials. Each model should have a specific volume or surface area determined using the concept of exponentiation with rational exponents.

Students will need to calculate the dimensions of the solids (height, width, length, radius) using exponentiation with rational exponents, build the models, and then verify if their calculations are correct by measuring the volume and surface area of the constructed models.

This is a group project, and each group should consist of 3 to 5 students. The project duration is one month.

Required Materials:

• Cardboard or other recyclable materials.
• Tape measure or ruler.
• Pen for marking.
• Scissors.
• Glue.

Detailed Step-by-Step for Activity Execution:

1. Form groups of 3 to 5 students.
2. Each group should choose which geometric solid will be built (cube, sphere, prism).
3. Determine the surface area or specific volume for each geometric solid using exponentiation with rational exponents.
4. Students should calculate the necessary dimensions (height, width, length, radius) for their model using the concept of exponentiation with rational exponents.
5. Students will build their models using recyclable materials.
6. Students will verify if their calculations are correct by measuring the volume and surface area of the constructed model.
7. At the end, each group should present their model and explain how the calculations were performed.

Project Delivery:

At the end of the activity, in addition to the construction of the models, each group must submit a report containing:

1. Introduction: In this section, students should contextualize the concepts of exponentiation and radicalization, their relevance and application in the real world, and the objective of the work carried out.
2. Development: Here, they should explain the theory behind the concepts studied, describe in detail the activities carried out, the methodology used to develop the model, and discuss the results obtained.
3. Conclusion: In this part, the group should summarize the main points of the work, explain the lessons learned, and the conclusions drawn through the project.
4. Bibliography: Resources (books, web pages, videos, etc.) used to support the construction of the project should be indicated.

The project evaluation will take into account both the realization of the three-dimensional model and the quality of the report submitted.