9. Adjustable Spinner:
Experimenting with probability using a digital spinner
Look in the Classroom
Mr. Haddad recognizes that it is necessary for his students to have numerous experiences with probabilistic events in order to build an intuitive understanding of probability. He and his students have been conducting experiments using physical manipulatives, including the cardboard spinner. He knows that the Adjustable Spinner will help students to extend their understanding of the difference between theoretical and experimental probability and set the stage for introducing the "law of large numbers." By the end of the week-long investigation, Mr. Haddad wants his students to be able to use theoretical probability to make predictions about spinner outcomes and be able to explain how the number of trials affects the experimental probability.
As he begins the class, Mr. Haddad uses the classroom projector and the Adjustable Spinner to review the recent probability concepts they have studied. He adjusts the size of sections and challenges students to predict outcomes. (Figure 2) For his demonstration, Mr. Haddad steps through trials of ten spins at a time.
Following the introduction, Mr. Haddad distributes a set of investigations. During each investigation, the students consider theoretical probabilities and predict the number of trials necessary for the experimental probabilities to be within one percent of the theoretical probability. Pairs of students design spinners to explore a variety of scenarios and begin conducting experiments, observing the changes in the patterns in the experimental probabilities. They also keep careful logs of their work and record comparisons between experimental and theoretical probabilities. Mr. Haddad helps groups work through these investigations.
After their experimentation, Mr. Haddad conducts a class discussion where students reflect on what they learned.
Mr. H: How did the number of trials affect the experimental probability?
S1: The more you do, the more accurate it gets.
Mr. H: What do you mean by "more accurate"?
S1: I mean it is closer to what it should be.
Mr. Haddad seizes this opportunity and begins to help the students reframe their statements.
Mr. H: Good point! What does [S1] mean by "what it should be"?
S2: He means it gets closer to the theoretical probability.
Mr. H: Can someone use one of your spinner experiments as an example and explain how the experimental probability gets closer to the theoretical probability?
S3: Well, when I made 50% of the circle green, it should have landed on green half of the time. But when I only did five spins, I got green only once, and that is only 20%, which was not very close to half. But when I did 100 spins, I was much closer, like 40%. Then when I did 1000 spins, the number was really close, like 51.33%.
The students' comments increasingly reveal their deepening understanding of the law of large numbers. One student explains:
S4: When you spin the spinner just a few times, like five or ten spins, it's easy to get numbers that are really different from the theoretical probability. There is always a chance for any color to come up no matter how small its section is. But if we do it a whole lot, like over 1000 times, things sort of even out and the data from the experiment gets close to what it should be according to the math.
When most of the class agrees with this statement, Mr. Haddad is confident that most students have achieved the goal for the unit. He will examine their experiment logs after class to confirm his sense of their learning.
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