Projeto: Wheel of Fortune

Context

Circular Permutation is an interesting and relevant concept in Mathematics, specifically in the area of Combinatorics, which deals with the different ways in which we can organize or group objects. In circular permutation, we consider arrangements of items in a circle, where it is the relative position of each item that matters, not its absolute position. The concept is essential for solving problems where the order of arrangement in a circular structure is involved.

There are some fundamental rules to keep in mind when it comes to circular permutation. The first one is that, in a circular organization, there is no first or last position. Additionally, two arrangements are considered to be the same if they can be obtained by rotating one into the other. These facts have a great impact on counting the number of possible arrangements.

In the real world, circular permutation finds its uses in a number of situations. It can be applied in arranging events, like weddings or corporate dinners, where the seating of participants around a table might be crucial. It is also used in designing certain types of games, arranging products in a store, or composing teams where the relative positions of the members can have a significant impact.

To aid you in your study of this topic, we suggest you use the following resources:

1. Book: Discrete Mathematics by Richard Johnsonbaugh, Chapter 4, Section 4.2 – Permutations
2. Website: Khan Academy, Advanced Math course, Permutations and Combinations topic
3. Video: Youtube channel "The Mathemusician", Combinatorics playlist, video "Circular Permutations"

All these sources are rich in practical examples and exercises for you to practice and delve deeper into the concept of Circular Permutation.

Practical Activity: "Wheel of Fortune"

Project Objective

The purpose of this project is to apply the theory of circular permutations in solving real-world problems. Students will use their knowledge to create a "Wheel of Fortune", a game inspired by the famous TV show, where each position in the wheel represents a prize or a loss. They will have to calculate the number of possible combinations of prizes in the wheel, considering that the prizes are distinct and that spinning the wheel does not change the combination (circular permutation!).

Project Description

Students, in groups of 3 to 5, will create the game using stationery materials. The game wheel should have 6 distinct prize positions, which will be chosen by the students themselves, but they cannot be repeated.

The group will also calculate the number of possible permutations of the prizes in the wheel, considering the circular nature of the arrangement, and write a detailed explanation of how they arrived at that result. This will include a discussion of the principles of circular permutation and a demonstration of how these principles were applied in calculating the permutations in the game wheel.

Students will have one week to complete the project, with an estimated time of 2 to 4 hours of work per student. At the end, each group should present their "Wheel of Fortune" to the teacher and the class, as well as hand in a detailed report describing the whole project, including the theory of circular permutation and the calculation of the wheel's permutations.

Required Materials

• Cardboard or poster board to build the wheel.
• Colored markers to highlight the positions in the wheel.
• Fictional prizes to fill in the positions (creativity is the limit).

Step-by-Step Guide to Carry Out the Activity

1. Form groups of 3 to 5 students.
2. Review together the theory of circular permutation using the suggested resources.
3. Choose 6 distinct prizes to fill in the positions in the game wheel.
4. Build the game wheel using the cardboard or poster board and the markers. Each position in the wheel should be clearly marked and identified with a prize.
5. Calculate the number of possible permutations of the prizes in the wheel, explaining in detail the reasoning and methodology used.
6. Write a detailed project report, including the theory of circular permutation, the calculation of the permutations, and the creation of the wheel. The report should follow the structure Introduction, Development, Conclusions, and References.
7. Present the "Wheel of Fortune" to the class, explaining how it was built and how the permutations were calculated.
8. Hand in the "Wheel of Fortune" and the project report to the teacher.

Project Deliverables

The project completion will involve two main deliverables:

1. Wheel of Fortune: A physical game wheel that the students have created, with distinct prizes in each position. This wheel will be presented and explained to the class.

2. Project Report: A written document that details the theory of circular permutation, how the game wheel was created, how the permutations were calculated, and the students' experience during the project. Students should strive to clearly connect the theory to practice in describing how they applied circular permutation when calculating the combinations in the game wheel.

These deliverables are interconnected, as the project report details the creation of the Wheel of Fortune and how the theory of circular permutation was applied in practice. The report should be seen as an opportunity for the student to demonstrate their understanding of the concepts learned and how they were applied in a real-life scenario.