Contextualization

Introduction

Sequences are a crucial part of mathematics and are used in a variety of topics and disciplines. At its core, a sequence is an ordered list of numbers that follows a specific pattern. This pattern determines which numbers are in the sequence and in what order they appear.

There are two main types of sequences that we will explore in this project: non-recursive sequences and recursive sequences. Non-recursive sequences, also known as arithmetic sequences, follow a constant pattern - each subsequent number in the sequence is determined by adding (or subtracting) the same amount to the previous number. On the other hand, in recursive sequences, each subsequent number in the sequence is determined in a more complex way, usually depending on one or more of the previous numbers in the sequence.

However, a common point in both cases is that each element of the sequence is determined by an algorithm, a set of instructions, or rules to follow. In this project, our goal is to identify the rule or algorithm that defines a given sequence and use this rule to determine the next numbers in the sequence.

Contextualization

Sequences are incredibly useful and have numerous real-world applications. In finance, sequences are used to calculate compound interest and loan amortizations. In computer science, sequences are used to sort data and solve complex problems. In biology, sequences are used to analyze DNA and study evolution.

Furthermore, learning about sequences will help you develop important skills such as logical thinking, problem-solving skills, and the ability to follow precise instructions - all highly valued skills in today's society.

Therefore, by understanding sequences and being able to work with them, we are not only learning mathematics, but also developing skills that are applicable to many other fields and life situations.

Practical Activity

Activity Title: "The Mysterious Sequence"

Project Objective

Create and solve numerical sequences, building algorithms represented by flowcharts.

Detailed Project Description

Divided into groups of 3 to 5 students, you will have one month to complete this project, which should require between five to ten hours of work per participating student. The project will be divided into two main parts:

Part 1: Creating a Mysterious Sequence

Each group must create two numerical sequences - one non-recursive and one recursive. You must also develop an algorithm, represented by a flowchart, that describes the logic of the created sequence. The sequence must have at least 10 numbers.

Part 2: Solving the Mysterious Sequence

The groups will then exchange their created sequences. Each group must solve the received sequences using the algorithms represented by the flowcharts. It is important to understand and follow the algorithm's logic to discover the next numbers in the sequence.

Required Materials

• Paper and pen for drafting
• Computer with internet access for research
• Software for creating flowcharts (can be online and free like Draw.io)

Detailed Step-by-Step for Activity Completion

1. Divide into groups of 3 to 5 students.
2. Research different types of mathematical sequences, whether recursive or non-recursive.
3. Each group should create two sequences: one non-recursive and one recursive.
4. For each sequence, create a flowchart (algorithm) that describes the sequence's logic. This flowchart should be able to generate the sequence when followed.
5. Exchange the created sequences with another group.
6. Based on the received flowchart, try to identify the sequence rule and determine the next numbers.
7. Compare the results obtained with the original sequence.
8. Develop a detailed project report following the format specified in the introduction of this activity.

Project Deliverables

At the end of the project, each group must deliver:

1. The two created sequences (non-recursive and recursive) with their respective flowcharts.

2. The sequences obtained when solving the challenges proposed by other groups, with an explanation of the logic followed.

3. A written report describing the process of creating and solving the sequences and algorithms. The report should include:

   • Introduction: Contextualization about sequences, their importance and application in the real world, and the objective of this project.

   • Development: Explanation of the theory behind sequences and algorithms, detailing the activity performed, methodology used, and discussion of the results obtained.

   • Conclusion: Recapitulation of the main points of the project and reflection on the learnings and conclusions drawn from the project.

   • Bibliography: References used for the project.

Remember that the quality of the sequences and flowcharts, as well as the clarity and depth of the report, will be key points in the project evaluation.