Contextualization

Introduction

Approximation is a fundamental concept in mathematics. It allows us to work with numbers that are not exact, but are close enough for practical purposes. One class of numbers that are often approximated are irrational numbers. These are numbers that cannot be expressed as a simple fraction, and they have an infinite number of non-recurring digits after the decimal point.

In this project, we will focus on two specific irrational numbers: π (pi) and √2 (square root of 2). Pi is the ratio of a circle's circumference to its diameter, and it is approximately equal to 3.14159. The square root of 2 is a number that when multiplied by itself gives the result of 2, and it is approximately equal to 1.41421. These numbers, like all irrational numbers, have an infinite number of decimal places, so we can only ever approximate them.

Importance of Approximation

Approximation is a fundamental tool used not only in mathematics but also in a variety of other disciplines like physics, engineering, computer science, and even in everyday life. For example, engineers use approximations when designing a building, computer scientists use approximations in algorithms, and we use approximations when estimating the cost of groceries.

The ability to understand and work with approximations of irrational numbers is a key mathematical skill. It allows us to make sense of the world around us and solve complex problems. So, let's dive into this project and explore the fascinating world of irrational number approximation!

Resources for Further Study

1. Khan Academy: Approximating Irrational Numbers
2. Math is Fun: Irrational Numbers
3. Math Planet: Approximating irrational numbers
4. Wolfram MathWorld: Approximation

These resources provide a comprehensive overview of the topic, and students are encouraged to explore them to gain a deeper understanding of irrational number approximation.

Practical Activity

Activity Title

Approximating Irrational Numbers in the Real World

Objective of the Project

The main goal of this project is to apply the concepts of irrational number approximation in real-life situations. Students will work in groups to estimate the value of pi (π) and the square root of 2 (√2) using different methods and then apply these approximations in various problem-solving scenarios.

Detailed Description of the Project

In this project, each group will be divided into two subgroups. The first subgroup will focus on approximating the value of π, while the second subgroup will work on approximating the value of √2. Each subgroup will use two different methods for approximation: the first method is the geometric method, and the second method is using the infinite series.

The geometric method involves inscribing a regular polygon inside a circle and then calculating the perimeter of the polygon. The more sides the polygon has, the closer its perimeter is to the circumference of the circle, which is equal to 2π. The infinite series method uses a mathematical series that converges to the value of π or √2. Students will calculate the sum of the series up to a certain point, which will give them an approximation of the irrational number.

After approximating the values, each subgroup will work together to apply their approximations in problem-solving scenarios. These scenarios will be based on real-world situations where approximation of irrational numbers is used. For example, calculating the area of a circle or the diagonal of a square.

Necessary Materials

1. Rulers
2. Compasses
3. Protractors
4. Calculators
5. Paper and pencils

Detailed Step-by-Step for Carrying out the Activity

1. Research and Discussion (2 hours): Each subgroup should start by researching and discussing the two methods of approximation for their respective irrational number. They should also discuss the real-world applications of these irrational numbers.

2. Approximation (3 hours): Using the methods they researched, each subgroup should approximate the value of their irrational number using both the geometric method and the infinite series method. They should record their results and discuss any discrepancies.

3. Problem Solving (2 hours): The two subgroups should work together to solve a set of problem-solving scenarios. Each scenario will involve using their approximations of π and √2 to solve a problem. For example, calculating the volume of a sphere or the hypotenuse of a right-angled triangle.

4. Report Writing (2 hours): Each group will then write a report detailing their process, findings, and solutions to the problem-solving scenarios.

Project Deliverables

1. Approximations: Each group should deliver their approximations of π and √2 using both the geometric and infinite series methods. These approximations should be clearly documented in the report.

2. Problem-Solving Solutions: Each group should deliver their solutions to the problem-solving scenarios. These solutions should be based on their approximations of π and √2.

3. Written Report: Each group will write a report documenting their process and findings. The report should contain the following sections:

   • Introduction: Contextualize the theme, its relevance, and real-world application. State the objectives of the project.

   • Development: Detail the theory behind irrational number approximation, explain the activity in detail, indicate the methodology used, and present and discuss the obtained results.

   • Conclusions: Revisit the main points of the work, explicitly state the learnings obtained, and draw conclusions about the project.

   • Bibliography: Indicate the sources used to work on the project, such as books, web pages, videos, etc.

The project duration is set at one week, and each student should expect to dedicate about 10 hours to the project. At the end of the week, each group will present their findings to the class, explaining their approximations and how they applied them in the problem-solving scenarios.