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Estimating the value of Pi

Lesson Overview

Embark on an interactive journey in Scratch to explore and understand the concept of Pi (π), the mystical ratio that has fascinated mathematicians for centuries. This project is designed to provide students with an intuitive grasp of π by exploring it through two distinct mathematical approaches—the perimeter method and the area method. By engaging in these activities, students will not only enhance their computational thinking and problem-solving skills but also deepen their understanding of Pi and its significance in mathematics.

Learning Objectives

  • Learn about Pi as the ratio of a circle’s circumference to its diameter.

  • Explore the concepts of perimeter and area, focusing on circles and polygons.

  • Understand the idea of limits and how a polygon can approximate a circle as its sides increase.

  • Apply basic trigonometry for calculating the lengths of polygon sides.

  • Estimate Pi using the formula for the area of a circle (A=πr²).

  • Utilize loops and conditional statements in programming

Task Description and Resources

Task 1: Perimeter Approximation (Archimedean Algorithm)

Problem Statement: Approximate a circle by drawing its inscribed regular polygon and increase the number of sides. What is the relationship between the perimeters of the polygons and the circumference? With the answer to the previous question, can you estimate the value of pi?

Objective: Understand the relationship between the perimeters of polygons inscribed in a circle, the circle’s circumference, and the estimation of the value of pi using its geometric meaning.

Suggested Steps:

  1. Begin with a polygon inscribed in a unit circle (e.g., a hexagon) and progressively increase the number of sides (to 12, 24, etc.).

  2. Calculate the side lengths using basic trigonometry (for a polygon with n sides inscribed in a unit circle, the length of each side can be calculated as 2sin(π/n)).

  3. Use the perimeter of these polygons (n× side length) to approximate Pi, comparing it to the circumference of the unit circle (which should be 2π for a unit circle).

  4. Reflect on how increasing the number of sides makes the polygon's perimeter closer to the circle's circumference.

Reference Code:

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Task 2: Estimation Using the Ratio of Areas

Problem Statement: Grasp the relationship between the area of a circle and Pi, demonstrating the application of the area formula A=πr² through experimental activities.

Key Concepts: Simulation using scatter plot , Ratio and proportion, Application of the area formulae of polygon and circle, Statistical estimation

Suggested Steps:

  1. Use Scratch’s stamp tool to generate a multitude of points within a preset area, simulating a scatter plot. This can be done by randomly positioning a sprite within a square that encloses a unit circle.

  2. Calculate the proportion of points that land inside the unit circle versus the total number of points. Points are considered inside the circle if their distance from the center is less than or equal to 1 (the radius of the unit circle).

  3. Approximate Pi using the ratio of the circle's calculated area (number of points inside the circle) to the square of its radius (r 2 , which is 1 for a unit circle). The formula to approximate Pi becomes π≈4×(points inside circle/total points).

  4. Discuss how this method provides an estimation of Pi and the significance of using a large number of points for accuracy.

Reference Code:

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Summary

Through these tasks, students will not only discover the value of Pi in a fun and engaging manner but also gain insights into fundamental mathematical and programming concepts. By blending the exploration of Pi with Scratch programming, students will advance their mathematical reasoning, computational skills, and understanding of pi, setting a solid foundation for future learning in STEM fields.

Acknowledgement 

The author would like to thank Hui Yan YE for designing this lesson and appreciate all the anonymous teachers and students who participated in this research.

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