Number Cube

Getting Started

The student is unable to determine the probability of each outcome.

The student does not know how to represent or calculate probabilities. The student:

• Divides the total number of outcomes by the number of favorable outcomes (e.g., ).
  
  
  
   
• Divides the total number of outcomes by the number of favorable outcomes and appends a percent sign to the result (e.g., ).
  
  
  
   
• Writes the frequency as the probability.
  
  
  
   

How is probability determined?

How can you represent probability in fraction form? Should the total number of outcomes be in the numerator or denominator? Why?

If each outcome is equally likely, how should the frequencies of each outcome compare? How should the probabilities compare?

Review the meaning of probability and how it is calculated. Explain that the probability of an event is the number of outcomes favorable to that event compared to the total number of outcomes. Use a variety of manipulatives (e.g., coins, number cubes, and spinners) to demonstrate how probabilities are calculated. Clearly describe each possible outcome, the total number of outcomes, outcomes favorable to a particular event, and the number of outcomes favorable to that event. Guide the student to calculate specific probabilities and to write the probabilities in multiple forms: fraction, decimal, and percent. Remind the student that the probability of an event is a number between zero and one (or 0% and 100%). Consider implementing CPALMS Lesson Plan A Roll of the Dice (ID 34343) or Marble Mania (ID 4732), to help students understand probability of simple events.

Making Progress

The student is unable to determine whether the outcomes appear to be equally likely.

The student correctly records the probability of each outcome but:

• Thinks the outcomes must be exactly the same to be considered equally likely.
  
  
     
   
• Does not provide an explanation.

What does likely mean? What does equally likely mean?

Are all numbers on the number cube equally likely? Are the frequencies of the outcomes close to what you expected?

Would you expect each number to occur exactly the same number of times? Why or why not?

Make explicit that equally likely does not mean that the outcomes will occur with precisely the same frequency in an experiment. Guide the student to compare the theoretical probabilities of getting each number on a “fair” number cube to the experimental probabilities and to determine if they are reasonably close. Explain to the student that some deviation from the theoretical probabilities will occur even when the number cube is fair, but in the long run, the frequency of each outcome should be nearly the same. Consider implementing CPALMS Lesson Plan M & M Candy: I Want Green (ID 7021), a lesson in which theoretical and experimental probabilities are compared.

Have the student consider an experiment in which a coin is tossed. First, have the student calculate the theoretical probability of getting a heads and a tails [e.g., P(heads) = 0.5 and P(tails) = 0.5]. Then suggest that a coin was tossed twice and landed on heads both times. Have the student calculate the experimental probabilities [e.g., P(heads) = 1 and P(tails) = 0]. Ask the student to consider if this is enough evidence to conclude that the coin is not fair and that the outcomes are not equally likely. Emphasize that if the coin were tossed many times, the number of times that heads occurs should be very close to the number of times that tails occurs (if the coin is fair). Simulate tossing a coin using a graphing calculator. Determine the experimental probability of getting a heads after 1, 2, 3, 4, 5, 10, 50, and 100 trials. Guide the student to observe how the probabilities get closer and closer to 0.5.

Got It

The student provides complete and correct responses to all components of the task.

The student accurately calculates the probability of each outcome in either fraction, decimal, or percent form.

The student says that outcomes appear to be equally likely because all the probabilities are between 15% and 18%, which is very close together.

   

What would you expect to happen if the number cube had been tossed 600 times? Do you think the probabilities would be the same?

Ask the student to consider how different the probabilities would need to be in order for the student to confidently conclude that the number cube is not “fair.”

Consider implementing MFAS task Marble Probability to further assess the student’s understanding.