Background

Introduction

Quadratic equations are a fundamental part of mathematics and have a wide variety of practical applications. They can be used to describe various real-world situations like the trajectory of an object thrown in the air, the optimization of an area or volume, or predicting the profit and loss in a business.

A quadratic equation is one that can be expressed as ax²+bx+c=0, where a, b, and c are real numbers and "a" is different from zero. It is named this way because the highest exponent of the unknown (which is x) is 2. The solution to a quadratic equation is given by what we know as the "Bhaskara Formula".

The detailed study of quadratic equations allows us to master a set of techniques that will be important in solving more complex problems later on. And understanding how the coefficients change the shape and position of the parabola in the Cartesian plane can help us interpret various phenomena.

Contextualization

Mathematics is the language we use to describe the universe and its laws. Therefore, by studying and understanding quadratic equations, you'll actually be learning how to "speak" to the universe in a more sophisticated way. You will be able to predict the behavior of natural phenomena, optimize processes, create more efficient strategies in games, and describe the physics of a scene.

Therefore, you will see that this mathematical tool, although somewhat abstract, has very real and tangible applications. And, finally, the interdisciplinary aspect of this project in the physics discipline will expand the usefulness of this important topic.

As a resource for further study, I suggest the book "Fundamentals of Elementary Mathematics: Vol. 1" by Gelson Iezzi and the website Só Matemática.

Practical Activity

Activity Title: "The Flight of the Parabola"

Project Objective

The challenge is to create a catapult game that applies concepts of quadratic equations and physics to solving practical and challenging problems. This work requires that participants work as a team, solve complex problems, be creative, master the subject, and have manual skills.

Detailed Project Description

The students should split into groups of 3 to 5 people. Each group should build a catapult using recyclable materials. The launch of the projectiles (small rubber balls or rolled up paper) must follow a parabolic trajectory.

After building the catapults, the students must conduct launches with different angles and forces, observe the trajectories, and record the results. The objective is to try to predict the landing site of the projectile using equations of the 2nd degree and concepts of physics.

This project will last 3 weeks. The students will have to manage their time efficiently to complete all of the activities by the established deadline.

Required Materials

• Recyclable materials to build the catapult (barbecue sticks, rubber bands, bottle caps, etc.)
• Projectiles (small rubber balls or rolled up paper)
• Measuring tape
• Notebook

Detailed Step-by-Step to Conduct the Activity

1. Form groups of 3 to 5 students.
2. Each group must plan and build a catapult using the supplied materials.
3. After building the device, each group must carry out a series of launches varying the strength and angle.
4. The results of the launches must be collected and plotted on a graph.
5. After the data is collected, the groups must try to determine the equation that describes the trajectory of the launches.
6. Finally, the groups must try to predict the trajectory and landing site of the projectile using their equations.

Project Delivery

Upon completion of the practical project, the groups must provide a detailed report containing the following topics:

1. Introduction: Detailed description of quadratic equations and their relationship with the proposed activity. Contextualization of the topic in the real world and presentation of the project objectives.
2. Development: Detailed explanation of how the catapult was built, the launches conducted, and the results collected. Presentation and discussion of the quadratic equations used to predict the trajectory of the projectile. Indication of the methodology used.
3. Conclusion: Discussion of the results, reflections on the difficulties encountered during the project and lessons learned. Critical review of the group's performance.
4. Bibliography: References of the materials and resources used throughout the project.

The students must connect the practice with theory, explaining how quadratic equations were used to predict the trajectory of the projectile, the influence of different factors (like the angle and force of the launch) on the equation and final trajectory.

The document must be written collaboratively with all the members of the group participating actively.