Contextualization

In this part of the course, we will explore one of the fundamental concepts of mathematics: relations and equations of quantities. But what are quantities? In simple terms, quantities are everything that can be measured. Integers, fractions, decimal numbers, and even variables represent different types of quantities. And when we talk about relations and equations of quantities, we are referring to how these quantities interact and relate to each other.

Now, let's contextualize this in our daily lives. Imagine you are cooking and the recipe calls for 2 cups of flour and 1 cup of sugar. If you wanted to double the recipe, you would need to double the amount of each ingredient. This is a relation of quantities! If you double the amount of flour, you will have to double the amount of sugar. This is a direct or directly proportional relation.

On the other hand, think about a car trip. The faster you drive, the less time it takes to reach your destination. This would be an inverse or inversely proportional relation. If you double the speed, the travel time would be halved.

Introduction

Relations and equations of quantities are used in various areas such as Physics, Chemistry, Engineering, Economics, and many others. Understanding how these relations work and how we can represent them through equations is essential for solving various problems.

In mathematics, when we talk about directly proportional quantities, we are referring to quantities that increase or decrease in the same proportion. For example, if we double the amount of flour in a recipe, we will have to double the amount of sugar to maintain the proportion. This can be represented by the equation y = kx, where k is the constant of proportionality.

Conversely, inversely proportional quantities are those that, as one increases, the other decreases in the same proportion. As in the example of the car trip, the faster you drive, the less time it takes to arrive. This can be represented by the equation y = k/x.

Practical Activity

Activity Title: "Building Proportional Spreadsheets"

Project Objective

The objective of this project is for students to apply the concepts of directly and inversely proportional quantities in building graphs and equations using a spreadsheet tool, such as Microsoft Excel or Google Sheets.

Detailed Project Description

The student groups will be tasked with creating two spreadsheets: one to demonstrate a relation of directly proportional quantities and another for an inversely proportional relation. Each spreadsheet must contain a data table, a representative equation, and a graph.

Each group should choose two quantities for each spreadsheet. For example, for the directly proportional quantities spreadsheet, they could choose the relationship between the number of hours spent studying and the number of exercises solved. For the inversely proportional quantities spreadsheet, an example would be the relationship between a car's speed and the time to reach the destination.

After compiling the data and creating the graphs, students should interpret the results, identify the type of relation (directly or inversely proportional), and write a report on their findings and conclusions.

Required Materials

1. Access to a computer with an Internet connection.
2. Google account to use Google Sheets or Microsoft Excel software.
3. Paper and pencil for notes and drafts.

Detailed Step-by-Step for Activity Completion

1. Group Formation: Students should divide into groups of 3 to 5 people, and each group should choose a leader.

2. Choosing Quantities: Each group should choose two quantities they believe are directly proportional and two quantities they believe are inversely proportional.

3. Creating Spreadsheets and Graphs: Using Google Sheets or Microsoft Excel, students should create two spreadsheets. One spreadsheet should contain the data table, equation, and graph for directly proportional quantities. Another spreadsheet should contain the same elements but for inversely proportional quantities.

4. Analysis of Results and Report Writing: After creating the spreadsheets and graphs, students should interpret the results, identify the type of relation (directly or inversely proportional), and write a report on their findings and conclusions.

Project Deliverables

Students must deliver two spreadsheets (one for directly proportional quantities and one for inversely proportional quantities) containing data tables, equations, and graphs. Each spreadsheet must be accompanied by a written report on the activity.

The report should be divided into four sections: Introduction, Development, Conclusions, and Bibliography.

The Introduction should present the theme, relevance, real-world application, and the project's objective.

In the Development section, the student should explain the theory behind directly and inversely proportional quantities, the activity in detail, the methodology used, and finally present and discuss the results obtained.

The Conclusions should contain the learnings and conclusions drawn from the project.

The Bibliography should list the sources consulted during the project, whether books, websites, videos, among others.

Students will be evaluated both on the technical content of their spreadsheets and reports and on the socio-emotional skills developed during the project, such as time management, communication, problem-solving, creative thinking, proactivity, etc.