Dart Problem

Lesson Overview

This engaging lesson introduces students to the interplay between probability, geometry, and programming through a hands-on activity in Scratch. Participants will simulate throwing darts at a custom-designed dartboard, with each square representing a unique scoring zone. The main objective is to investigate the probabilities of darts landing in these zones and to understand the impact of these probabilities on game strategy and outcomes.

Learning Objectives

Gain an in-depth understanding of probability through the computation and analysis of experimental probabilities, sharpening mathematical reasoning.
Utilize the design of a dartboard to apply concepts of geometry, including area and position, to improve spatial awareness and grasp the impact of geometric principles on real-world outcomes.
Strengthen problem-solving, logical reasoning, and algorithmic design skills through programming. Students will dissect complex problems, employ structured logic, and craft algorithms, laying a robust groundwork for advanced studies in mathematics and programming.

Task Description and Resources

Task 1: Darts Throwing - Square Challenge

(1) Designing the Dartboard

Purpose: Engage in a creative process to design a dartboard using Scratch's drawing tools or sprites. The design phase allows for freedom in determining the size and placement of each square, enabling students to incorporate their strategic preferences into the board's layout.

(2) Simulating Dart Throws

Purpose: Create a Scratch program that simulates the action of throwing darts at the board. This can be achieved through mouse clicks or employing random functions to generate dart landing points. The program will identify the square in which each dart lands and maintain a tally of the results.

(3) Calculating Probabilities

Purpose: Analyze the probability of darts landing in each square by dividing the number of hits per square by the total number of throws. This calculation provides valuable insights into the likelihood of hitting specific zones, offering a practical exploration of experimental probability.

Questions to explore: Throw as many darts as possible. How do we know the probability of the dart falling in the area A, B, C, D and E? Use Scratch to help you solve!

Reference Code:

Link to Scratch: https://scratch.mit.edu/projects/999620933

Task 2: Darts Throwing - Circular Challenge

Purpose: Design a dart game to automatically throw a ball (onto the dartboard). Requirements for the game are as follows:

Size of the ball has to be small, with a pre-set of 20%.
Keep the costume of the circles remains unchanged.

Guiding Questions:

Use table to record the results (5 trials, no restriction on the number of throws)
Calculate the experimental probabilities according to the results.
Why do we need to set the size of the dart to be small?
We are now going to introduce a scoring system to the game.
Compare the two systems above, which is fairer? Why?
Design a scoring system and explain whether it is fairer than the above two.

Reference Code:

Link to Scratch: https://scratch.mit.edu/projects/555846758

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 Ka Ying LIU and Zhi Hao CUI for designing this lesson and appreciate all the anonymous teachers and students who participated in this research.