Contextualization

Theoretical Introduction

A prism is a solid geometric figure, formed by a polygon at the base and straight lines (edges) that connect all the vertices of the base to a single point outside the base plane, called the apex. In this context, the study of metric relations in prisms involves understanding and calculating dimensions such as height, volume, surface area, sides, and diagonals.

Metric relations are the formulas that establish the relationships between the measurements of the parts of a figure, whether it is flat or spatial. In the case of a prism, the main metric relations involve calculating the volume (product of the base area by the height), the total area (sum of the areas of all faces), and the lateral area (sum of the areas of the lateral faces).

Understanding these concepts becomes fundamental for mathematics, as many of its branches, such as spatial geometry, trigonometry, and calculus, use these foundations. Understanding the metric relations in prisms enables the handling of more advanced mathematical concepts, as well as providing a more complete and profound view of mathematics as a whole.

Contextualization

Mathematics has a strong presence in our daily lives, and the study of geometric solids is not limited to school classes alone. They are present in many places, such as in architecture, design, civil engineering, mechanics, and even in art.

Considering the prism, we can identify its presence, for example, in juice boxes, soda cans, in the shape of a building, or even in the light prism of famous rock album covers. Knowing the metric relations of a prism is understanding how much of what surrounds us works.

The importance of unraveling the metric relations in a prism is, therefore, the ability to understand and interact with the physical world around us in an informed and efficient manner. From this perspective, mathematics ceases to be just numbers on a page and becomes an instrument of interpretation and transformation of the spaces that surround us.

Activity

Activity Title: Prism in Action - Building and Deconstructing Metric Relations

Project Objective

The objective of this project is to provide students with a practical experience with the concept of a prism, its characteristics, and metric relations. Students will learn to calculate the volume, surface area, and height of the prism. Additionally, they will work in teams, divide tasks, manage time, solve problems, and develop presentation skills.

Detailed Project Description

I. Planning and Research

Students, in groups of 3 to 5, should research the topic using the provided references and other materials. Each group should choose a specific type of prism to focus on (square base prism, triangular base prism, etc) and create a detailed plan for building a model of this prism using recyclable materials.

II. Prism Construction

Following their plan, students should build a real model of the chosen prism. The model should be large enough to allow for precise measurement of its dimensions.

III. Calculations of Prism Metric Relations

After building the prism, students must make a series of measurements and calculations to determine the prism's metric relations: surface area, volume, and height.

IV. Presentation and Final Report

Each group should present their prism and the results of their calculations to the class. Each group should also submit a detailed final report on the project.

Required Materials

1. Recyclable materials (cardboard boxes, toilet paper tubes, PET bottles, etc)
2. Tape measure
3. Pen / Marker
4. Glue
5. Scissors
6. Paper
7. Computer with internet access for research

Detailed Step-by-Step Guide for Activity Execution

Step 1: Divide students into groups of 3 to 5.

Step 2: Each group chooses a type of prism to research and build.

Step 3: Groups research the chosen prism and create a plan for building the model.

Step 4: Following the plan, groups build the prism model.

Step 5: Groups measure and calculate the prism's metric relations (surface area, volume, and height).

Step 6: Each group prepares a presentation of their prism and the calculations made for the class.

Step 7: Groups write a detailed report on the project, including:

1. Introduction: Definition of the chosen prism, importance, and real-world applications.

2. Development: Detailed description of the research process, planning, model construction, calculations performed, and results obtained.

3. Conclusion: Reflections on the project, what was learned, and how it can be applied in the future.

4. Bibliography: References used for research and project development.

Project Deliverables

1. Prism built according to the group's specifications;

2. Presentation to the class on the constructed prism and the calculations of metric relations;

3. Final written report with project details, including introduction, development, conclusion, and bibliography.

Your report will be a demonstration of the knowledge acquired about the prism and its metric relations and will also be an opportunity to show how you and your group organized, researched, planned, and executed the project, managing time and available resources, as well as dealing with the challenges that arose.