Project: Treasure Hunt - Systems of Equations

Contextualization

Theoretical Introduction

A system of equations is a set of two or more equations in which the same variables appear. These systems are an indispensable mathematical tool for describing and solving problems in various areas, such as physics, economics, engineering, and many others. Manually, we can solve a system of equations using various methods such as the substitution method, the elimination method, or the graphical method.

The graphical representation of a system of equations in the Cartesian plane is a visual way to understand the relationship between the variables. Each equation in the system represents a line in the plane and the solution to the system is the point where all these lines intersect. Our focus will be on systems of linear equations with two variables, where each equation can be represented by a straight line.

Solving a system of equations is finding the values of the variables that satisfy all the equations of the system at the same time. This means that these values, when substituted into the equations, make all the equations true. If a system of equations has a solution, we say that it is consistent. Otherwise, we say that it is inconsistent.

Contextualization

Systems of equations are everywhere in our lives. When you go to the supermarket with a limited amount of money and want to buy different types of products within your budget, you are basically solving a system of equations. When you are playing a video game and need to calculate the trajectory of a shot to hit a moving target, you are using a system of equations. When companies adjust their prices to maximize profit, they are solving complex systems of equations.

Systems of equations are not limited to finance or physics. They are also used in fields such as biology, where they can help predict the growth of a population, or in chemistry, where they are used to balance chemical equations. Virtually any problem involving the interaction of different variables can be modeled and solved using systems of equations.

Practical Activity

Activity Title: Treasure Hunt - Systems of Equations

Project Goal:The objective of this project is to apply in a practical and playful way the knowledge acquired during the study of the system of equations. Students will have to create and solve systems of equations, interpret their results, and also learn how to work as a team.

Detailed Project Description:This is a group project, where each group of 3 to 5 students will create a "treasure map" for the other group. The treasure is a location in the school or on a map generated by the students themselves where they want to take the other group. The clues to find the treasure must be in the form of a system of equations. For example, a clue might be: "The treasure is located at a point where the sum of the coordinates is 10 and the difference between the coordinates is 2". Thus, students will have to solve the system of equations to find the coordinates of the treasure.

Required Materials:1. Paper and pen to draw up the system of equations and the map.

2. Calculator.
3. Internet (for research if needed).
4. A place to hide the treasure (optional).

Step-by-Step Activity Guide:1. Form groups of 3 to 5 students. Each group will be responsible for creating a system of equations that will lead the other group to the "treasure".

2. The "treasure" can be a location in the school or a point on a map.
3. Groups should create a system of equations and represent it graphically on a "treasure map". They should ensure that the system has a unique solution, i.e., the two lines intersect at a single point.
4. The groups exchange their "treasure maps" and each group tries to solve the system of equations created by the other group.
5. Students must record the steps of solving the systems and the conclusions obtained.
6. Each group should present the result of the system of equations (the coordinates where the "treasure" is located).
7. Check if the solution proposed by each group is correct, according to the treasure map they received.
8. Gather all groups to discuss the different strategies employed, the difficulties encountered, and possible solutions.

Project Deliverables:After completing the practical activity, students will have to produce a written report containing the following topics:

1. Introduction: Here, students should make a brief introduction on systems of equations and their relevance in the real world, as well as contextualizing the proposed activity and its objective.

2. Development: The students should explain the theory of systems of equations, explain the activity in detail, indicate the methodology used to solve the system of equations and present the results obtained. They should also include the tools they used to solve the systems, the challenges encountered during the solution, and how they overcame them.

3. Conclusion: The students should conclude the work by summarizing their main points, explaining the lessons learned, the conclusions drawn about the project, and how this activity improved their understanding of systems of equations.

4. Bibliography: Students should indicate the sources they used to work on the project as books, web pages, videos, etc.