What Are the Chances?

Lesson Content

• Lesson Plan Template:

  CTE Lesson Plan

• CTE Course Info and Benchmark Notes

  
  

  Web Applications Development and Programming

  Course Title: Procedural Programming (9007220)

  Career Cluster: Information Technology

  FOCUS Benchmark

  28.08 - Design and implement a simple simulation algorithm to analyze, represent and understand natural phenomena.

  SUPPORTING Benchmark

  27.07 - Compile, run, test and debug programs.

• Unit Outline

  
  

  This lesson is written as a piece in an escalating series of simple simulation programs for real-world events. The unit begins by having students program a realistic coin-flip application. In the second lesson, students extend the complexity of their simulation to the rolling of a single 6-sided die. Students take what they learn from the second simulation and apply it to this lesson as they attempt to realistically simulate the rolling of a pair of dice. Finally, students will share their programs and offer feedback to eachother based on realistic interactions and outcomes, including a comparison between theoretical and observed probabilities.

• Career Connection

  
  

  Many web and mobile applications model or simulate real-world interactions. This lesson allows students to experience the programming side of app development.

  Students will apply their mathematical knowledge to use and develop problem-solving, critical thinking, and logical thinking skills, all of which are pertinent for many careers in technology.

• Learning Objectives

  
  

  Students will be able to:

  • Design a simple simulation algorithm to represent and analyze a natural phenomenon.
  • Create a computational artifact, using reflection, analysis, and iteration to improve user interaction and accurately model real-life outcomes.
  • Analyze and interpret the probability of a pair of independent events in the context of rolling a pair of dice.
  • Describe events as subsets of a sample space by categorizing outcomes of sums rolled by a pair of dice.

• Prior Knowledge

  
  

  Students should be familiar with:

  • Adding and manipulating sprites in Scratch.
  • Using various types of blocks in Scratch, including events, controls, variables, and operators.
  • Applying the general multiplication rule to determine the probability of combinations of independent events.

• Guiding Questions

  
  
  • How can we use block programming to simulate the rolling (and repeated rolling) of a pair of dice?
  • How can knowledge of theoretical probabilities help us analyze the realism of a dice-roll simulation?

• Procedure

  
  
  1. To introduce the lesson, activate students’ prior knowledge.
     • Ask them to compare the simple experiments of flipping a 2-sided coin and rolling a single 6-sided die.
       • Students should be able to discuss:
         1. differences in probabilities (1 in 2 chance for each side of the coin and 1 in 6 chance for each face of the die).
         2. how observed outcomes for these simple experiments should approach theoretical probabilities closer and closer with repetition.
     • Ask students how programming a simulation of these simple experiments would be similar/different.
       • Students should be able to discuss ways to:
         1. use and manipulate sprites to show the various outcomes (ie – heads or tails, 1-6 on the die faces).
         2. use a random operator to select the outcome for each trial.
         3. use variables to track the number of times each outcome is observed and the experimental and theoretical probabilities of each outcome.
         4. use broadcast/receive events to trigger updates to the simulation for each roll.
  2. Split students into pairs and introduce the new task.
     • Ask students if rolling a pair of 6-sided dice is the same as rolling a single 6-sided die. Allow students to discuss ideas with their partners.
       • Consider providing pairs of students with dice as a manipulative to assist their thinking.
     • Tell students their task for this lesson is to create a program that accurately simulates rolling a pair of dice. Distribute the What Are The Chances Rubric (see attachment) and explain to students that their program must meet the following criteria:
       1. Simulate rolling a pair of 6-sided dice.
       2. Allow for repeated rolls.
       3. Calculate the total for each roll of the dice.
       4. Track and update the number of observations for each outcome.
       5. Track and update the experimental probability for each outcome with each roll.
       6. Provide a rich and engaging user experience.
     • Ask students to think about how they might be able to adapt elements of a single 6-sided die simulation for use in a simulation of rolling a pair of 6-sided dice.
       • What programming can carry over?
       • What sort of changes will be necessary?
  3. Allow partner-groups to begin working in Scratch.
     • Circulate to provide assistance where needed. Encourage partners to talk through their questions and try multiple solutions before asking for teacher help.
     • Encourage partner-groups to use the rubric as a guideline to ensure their program meets all the project specifications.
  4. To bring the lesson to a close,
     • Have each partner-group respond to the What Are the Chances Reflection Questions (see attachment):
       1. How does rolling a pair of 6-sided dice compare mathematically to rolling a single 6-sided die?
          • How did you account for these similarities and differences when programming your simulation?
       2. How is the observed or experimental probability different than the theoretical probability for each outcome?
          • Do these change over time with repeated trials? If so, how? If not, why?
          • How does your simulation demonstrate this with repeated rolls?
     • Discuss the reflection questions as a class. Preview the following lesson by informing students their next task will be to assess each other’s projects, so it is important they understand the probabilities associated with each outcome for rolling a pair of 6-sided dice and are familiar with the specifications of the app design project.

• Teacher Notes

  
  
  • Students will need Scratch accounts for this lesson. Teachers may choose to use the Scratch Teacher account feature. Visit https://scratch.mit.edu/educators/#teacher-accounts for more information.
  • This lesson is designed for intermediate or advanced experience in Scratch. Depending on student experience with Scratch, teachers can modify the requirements and/or provide students with more or less scaffolding.
  • To download the Scratch example project, use Firefox or Safari to download the attached sb3 file. Then, import the file into Scratch by creating a new project and loading the file from its saved location. The Scratch project can also be viewed using the following link: https://scratch.mit.edu/projects/800807922/
  • During reflection and discussion, students should be able to explain:
    • How each die in the pair is independent of the other, making each one a duplicate of the single die roll. This may influence their programming of each die.
    • There are 6 possible outcomes for a single die roll (1-6), but there are 36 possible combinations when two dice are rolled (different combinations of 1-6 for each die). The combinations resulting in different totals will influence the theoretical probabilities of each total outcome.
    • When the pair of dice are totaled together there are 11 possible outcomes (2-12), compared to just 6 for a single die.
    • Each outcome for a single die roll has the same probability (1 in 6), but the combined totals for the pair of dice produce unequal probabilities due to multiple combinations that produce the same total. For example, only one of the 36 possible combinations result in a total of 2 (1 and 1), but six different dice combinations result in a total of 7 (1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, 6 and 1). This should be reflected in the programming for the theoretical probability.
    • The number of observations for each outcome and the total number of rolls should be tracked by the program. A ratio of these values should be used to generate an observed or experimental probability for each outcome that can be compared with the theoretical probability. Assuming the random number generator is truly random, we would expect the experimental probability to approach the theoretical probability for each outcome as the number of trials increases.
  • There are a wide variety of ways for students to accomplish this task using Scratch. Allow students the freedom to explore alternative solutions to each part of the task. Consider using probing questions such as these to encourage student exploration:
    • How can you set up the program so the user controls when and how many times the dice are rolled?
    • How can you show the user the value of each die when rolled in a way that simulates this action in real life?
    • How can you set up the program to calculate the total for the dice after each roll?
    • How is the theoretical probability calculated? How can you program the simulation to generate this value?
    • What values do you need in order to calculate the observed or experimental probability? How can you program the simulation to track and/or generate these values?
    • How can you enhance the user experience to more accurately simulate the action of rolling a pair of dice?
    • How can you program the simulation to reset so the user can start over?

• Optional

  
  

  Formative Assessment Ideas:

  • Collect the reflection questions from each partner-group prior to the class discussion.

  Summative Assessment Ideas:

  • Have students respond to an exit ticket such as:
    • Pick three of the following block types and describe how you used them in your simulation:
      • Motion
      • Looks
      • Events
      • Control
      • Operators
      • Variables
    • Does your simulation accurately model the rolling of a pair of 6-sided dice? Provide evidence and explain your reasoning.

  Accommodation Suggestions:

  • Provide coins and dice for students to manipulate as they think through the introductory questions and plan their simulation program.

  Extension Ideas:

  • Have students use their program in place of actual dice to play a board game.
  • Have students develop their program into a mobile application.