Contextualization

Theoretical Introduction

Proportionality is a basic concept in mathematics that describes the relationship between two quantities. When two quantities are proportional, there is a proportionality constant that can be used to predict one quantity based on the other. There are two types of proportionality that we will explore in this project: direct proportionality and inverse proportionality.

In direct proportionality, as one quantity increases, the other quantity also increases in the same proportion. For example, if you are filling a container with water at a constant rate, the volume of water in the container (one quantity) and the time you are filling it (the other quantity) are directly related.

In contrast, in inverse proportionality, as one quantity increases, the other decreases in the same proportion. For example, the speed of a car (one quantity) and the time it takes to travel a certain distance (the other quantity) are inversely related.

Contextualization

Proportionality has numerous real-world applications, from sharing a cake among friends, calculating interest rates, adjusting recipes, to modeling physical phenomena such as constant speed or the emptying of a reservoir.

In the cake example, if you want to divide it equally among a number of friends, you will have to use the rules of direct proportionality. That is, if you have 8 pieces of cake and 4 friends, each will get 2 pieces.

The concepts of proportionality are also fundamental in the fields of economics, engineering, physics, chemistry, and many other related disciplines. In economics, for example, the concept of proportionality is used to calculate inflation, interest rates, the cost of living, among other aspects.

Practical Activity: "Proportional Journey"

Activity Title: Proportional Journey

Project Objective:

This project aims for students to apply their knowledge of direct and inverse proportionality in real-life scenarios.

Detailed Project Description:

Students will be divided into groups of 3 to 5 people and will work together to plan a fictional trip. Each group will have to take into account the distance to be traveled, the means of transportation to be used, the time it will take, as well as the cost of fuel or tickets. To make things more interesting, each group will have a different constraint to deal with: some will have a limited budget, others will have limited time, and so on.

Students will have to apply concepts of direct and inverse proportionality to solve these problems. They will have to figure out questions like: if we increase the speed, how does it affect the travel time? If we decide to travel by bus instead of car, how does it affect the total cost? And so on.

The activity will be concluded with a written report detailing each group's travel plan, explaining how they arrived at their conclusions and what challenges they faced.

Required Materials:

• Internet access for research
• Paper and pen for sketches and calculations
• Calculator
• Presentation software (optional, for the final project presentation)

Step by Step:

1. Form groups of 3 to 5 students.
2. Each group will choose a destination for their fictional trip.
3. The groups should then research the distance to the destination, possible means of transportation, and their associated costs.
4. Each group will be given a different constraint (for example, limited budget, limited time) that they will have to take into account when planning their trip.
5. Using the concepts of direct and inverse proportionality, the groups must solve the questions associated with their trip.
6. At the end of the project, the groups should create a report detailing how they planned the trip, how they applied the concepts of proportionality, and what challenges they faced.

Project Deliverables:

• Trip Presentation: Each group will make a presentation simulating the planned trip. The group can choose to create a digital presentation or simply explain verbally.
• Written Report: Each group must submit, at the end of the presentation, a report presenting the planning of the fictional trip. The report should include:
  • Introduction: Description of the chosen destination, means of transportation, and the context of the trip (for example, the limited budget or limited time).
  • Development: Detailed application of direct and inverse proportionality concepts in solving the trip's problems. It should include the description of challenges, decisions made, and the reasons behind them.
  • Conclusion: Recap of the main points of the project, explicit conclusions about the trip, learnings, and how proportionality was useful in solving the problems.
  • Bibliography: References used for research, including web pages, videos, and books.
• Project Work Time: This project is of difficult difficulty and should take more than twelve hours per student to complete.