Context

Before we dive into an exciting math project, let's understand the fundamental concept that this project involves - equations with two variables. These equations are an important way to represent mathematical relationships between two different quantities. The most common example of an equation with two variables is the linear equation y = mx + c, where m and c are constants and x and y are our variables. This equation represents a line on the Cartesian plane.

Equations with two variables have an interesting characteristic - they have infinite solutions! For example, if we take the simple equation y = 2x + 1, and substitute x with any number, we get a corresponding value for y. Therefore, x and y form an ordered pair (x, y), which is a solution to the equation. The set of all these solutions forms a line on the Cartesian plane. Through this project, we will explore this idea more deeply.

Now you might be wondering, 'why do we need to learn about equations with two variables?' and 'where do we use these equations in real life?'. The answer is: in many places! Equations with two variables are used in economics to understand the relationship between supply and demand. In physics, they are used to understand the relationship between the distance traveled by an object and the time it took. In computer graphics, they are used to draw shapes and lines. In fact, every time you see a relationship between two different things, it is likely that it can be represented by an equation with two variables!

This project challenges you to explore the concept of equations with two variables through a group project. Your goal will be to create a real-world problem that can be represented by an equation with two variables and solve it. Additionally, you will learn to work as a team, manage your time efficiently, and communicate your findings clearly. So, let's embark on this mathematical journey!

To get started, we recommend the following resources that you can use to understand the topic:

1. Khan Academy: Systems of Equations
2. SOS Professor: Equations with Two Variables
3. Book: Fundamentals of Elementary Mathematics - Gelson Iezzi and Carlos Murakami

Practical Activity

Activity Title: 'Keeping an Eye on the Bakery'

Project Objective

The objective of this project is to apply the concepts of equations with two variables in the scenario of a bakery. Students will be challenged to solve questions involving the production cost and selling price of cakes, considering different ingredients and sizes.

Detailed Project Description

Groups of 3 to 5 students will take on the role of bakery owners who produce two types of cakes: chocolate cakes and vanilla cakes. Each of these cakes requires a different amount of ingredients (flour, eggs, chocolate/vanilla) and each ingredient has an associated cost. Additionally, the cakes are sold at different prices.

Using the costs of the ingredients, the quantity of ingredients per cake, and the selling price of the cakes, students must establish two equations with two variables. One equation will be for the production cost of the chocolate cakes and the other for the vanilla cakes. They must then solve the equations to determine the quantity of each cake they should produce to maximize their profit.

Required Materials

1. Writing materials (pencils, erasers, paper)
2. Calculator
3. Graph paper for drawing the equation graph, if necessary.

Detailed Step-by-Step Guide for the Activity

1. Each group starts the project by researching or deciding on possible costs for the ingredients needed for each type of cake and the selling prices for each cake. They must also define quantities of ingredients for each type of cake.

2. Once each group has all costs and quantities defined, they must establish their two equations. The first equation should be the total cost of the chocolate cakes and the second equation should be the total cost of the vanilla cakes.

3. Each group solves their equations to find the optimal values for the number of each type of cake to be produced to maximize profit.

4. With the solutions found, the groups must then plot the equations on the Cartesian plane.

5. Each group then prepares a detailed project report, as explained below, and presents their solutions and graphs to the class.

Project Deliverables

1. Written report containing the following topics:

   • Introduction: Students must contextualize the theme, explaining the relevance and application of equations with two variables, highlighting the project's objective and how it applies to a bakery scenario.
   • Development: Students must explain the theory behind the concept of equations with two variables, discuss how they defined their costs and prices, describe the methodology used to solve the equations, and present their solutions and graphs.
   • Conclusion: Students must recap the main points, highlighting what they learned and how the project allowed the practical application of the concept of equations with two variables.
   • Bibliography: Students must list the sources they used during the project.

2. Project presentation to the class: Each group must present their equations, solutions, and graphs to the class, explaining their reasoning and the strategies they used to solve their equations.

Note: The report and the presentation should be considered as complementary pieces and should clearly show how students moved from theory to practice. They should also show how the final solution was obtained and what strategies were used to reach it.