MA.8.DP.2.3

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.8.NSO.1.2
• MA.8.NSO.1.7

Terms from the K-12 Glossary

• Theoretical Probability

Vertical Alignment

Previous Benchmarks

• MA.7.DP.2

Next Benchmarks

• MA.912.DP.4

Purpose and Instructional Strategies

• Students should understand that results from an experiment do not always match the theoretical results, but if they do a large number of trials, they should be close.
• When determining experimental probabilities, students should understand that this may be done by performing a simple experiment more than once and also by performing a repeated experiment more than once.
  • For example, to determine the experimental probability of tossing four heads in a row, the repeated experiment of tossing a coin four times will need to be performed many times (see also Benchmark Example 2).
• Instruction includes $P$($e$$v$$e$$n$$t$) notation.
• Instruction includes opportunities for students to run various numbers of trials to discover that the increased repetition of the experiment will bring the experimental probability closer to the theoretical. Use virtual simulations to quickly show higher and higher volumes of repetition that would be difficult to create with physical manipulatives (MTR.5.1).
• When comparing theoretical to experimental probability, it is important to not just compare the number of times the event occurs, but the probabilities themselves.

Common Misconceptions or Errors

• Students may incorrectly assume the theoretical and experimental probabilities of the same experiment will always be the same. To address this misconception, provide multiple opportunities for students to experience simulations of different situations, with physical or virtual manipulatives, in order to find and compare the experimental and theoretical probabilities.
• Students may incorrectly expect to see every possible outcome occur during a simulation. While all may occur in a simulation, it is not certain to happen. Students may inadvertently let their own experience with an experiment affect their response.
  • For example, during an experiment if a student never draws an ace from a standard deck of cards, this does not indicate it could never happen.
• Students may incorrectly believe the theoretical probability of an event is the proportion of times that event will actually occur.

Strategies to Support Tiered Instruction

• Teacher reviews the root words theoretical (theory) and experimental (experiment) and discusses the difference between a theoretical probability and experimental probability. Teacher provides graphic organizer to keep as reference for root words.
  • For example, experimental probabilities are from simulations whereas theoretical probabilities are from calculations.
• Teacher provides opportunities to discuss the difference between simulating simple experiments and simulating repeated experiments.
  • For example, students could discuss tossing a coin 150 times as a simulation of the simple (single) experiment “tossing a coin” and tossing a coin 150 times as a simulation of the repeated experiment “tossing a coin three times.” In the first case, the simulation has 150 trials whereas the second simulation has 50 trials.
• Teacher sets up a simulation with several trials to work through and discuss what ‘we think’ should happen (Theoretical Probability) and what actually happens when the experiment is completed (Experimental Probability). Then, the teacher models how to find experimental probability, showing how the more trials that are done, the closer the results should get closer to the theoretical probability. Teacher provides instruction focused on color-coding when setting up a proportional relationship to ensure corresponding parts are placed in corresponding positions within the proportion.
  • For example, if Jason choses a card from a standard deck of cards 104 times and replaces the card each time, what is a reasonable prediction on how many times he will choose a heart?
    
    
    
• Instruction includes providing multiple opportunities to experience simulations of different situations, with physical or virtual manipulatives, in order to find and compare the experimental and theoretical probabilities.

Instructional Tasks

• Part A. If the repeated experiment is to spin the spinner twice, predict how many times the event of landing on the same number twice will occur in 100 simulations of the repeated experiment.
• Part B. Using technology, perform a simulation of the experiment described in Part A.
• Part C. Compare the number of times you actually landed on the same number twice during the simulation to your prediction from Part A.

Instructional Items

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