Mrs. Callens and I were talking about board games the other day. She said she and her daughter and son were playing Monopoly® over the weekend. In Monopoly®, you get an extra turn when you throw a double. She said she was really lucky in those types of games because she rolled doubles about 1/3 of the time. Show Mrs. Callens how much you know about probability and either have evidence that proves she was lucky that day or that it was not luck - that anyone would expect to throw doubles 1/3 of the time.
We were studying probability in my sixth grade class. With partners in class we experimented with pulling a colored chip from a bag that had three blue chips, two red chips and one yellow chip. We talked about the experimental and theoretical probabilities and how to record each probability. We also predicted and experimented with tossing two coins. We discussed why HT came up twice as much as TT or HH. I wanted a problem that required students to apply their knowledge of probability. I also wanted to assess their understanding that like the coin toss, sometimes you need to distinguish between the two coins. In the "Double Dilemma" they need to find the total possibilities and remember that rolling a one and a two is different from rolling a two and a one. I also wanted to see how comfortable they were with both experimental and theoretical probabilities. I was hoping that they would include both experimental and theoretical probabilities in their solutions. Some students began listing all the possible throws to find the theoretical probability. Others began with rolling two dice and finding the experimental probability. The question of whether to consider one or two ways of throwing a three came up. When they asked me if there was a difference, I simply told them that was a decision that had to be made, but that I was not going to make it for them. Two, 45-minute periods This task can be used with a science unit on weather. For example, how does a meteorologist decide the probability of rain? I happen to have colored die - this fact helped some students remember that a three could be thrown with a one and two as well as a two and one. As I gave out the die - some students specifically asked for two different colors. Graph paper should be available for graphing, as well as a compass and protractor, in case some students want to make a circle graph.
- Colored die
- Graph paper
- Compass
- Protractor
The theoretical probability of throwing doubles is six out of 36 combinations so:
P (doubles) = 6/36 = 1/6 = .166... = 16 2/3%. Mrs. Callens was lucky that day since she rolled doubles 1/3 = .333... = 33.33...% or 33 1/3% - about double the expected rolls.
This student experimented with two die and kept track of the doubles that were thrown. However, there is no evidence that this student understood the problem. They do not appear to connect their results to Mrs. Callen's results. There is no probability language; the chart to keep track of their throws is minimal. There is no explanation of their strategy or why this experiment would help solve the problem.
This student's response is a typical Apprentice response. They did not understand that throwing a three could be a one and two or a two and one. They only found 21 possible combinations. This student even found an experimental probability very different than his/her theoretical probability and did not realize that something must have gone wrong. Their strategy - finding the theoretical and experimental probabilities - is good, but their lack of understanding of probability, and their lack of understanding of their results, showed poor reasoning skills. There is no mathematical representation, although their mathematical language is good.
This student has a good understanding of theoretical and experimental probabilities. Their strategy leads to a solution. They use effective reasoning. They compare their experimental probability to Mrs. Callens's results. Then they figure the theoretical probability correctly. Their last statement compares these results and concludes that Mrs. Callens rolled twice as many doubles as expected. They use accurate probability language throughout although there is no mathematical representation. The explanation is clear.
This student has taken the problem and found a generalization for harder cases. They are finding the probability for throwing doubles, triples, quadruples, etc. This student has a deep understanding of probability. By finding a generalization, they are finding an efficient and sophisticated strategy that leads directly to the solution. This student is using refined and complex reasoning. There is a clear and effective explanation detailing how the problem was solved. There also is precise and sophisticated mathematical representation and notation (use of exponents and circle graphs). This response is certainly a high Expert. I was thinking a student that pulled together theoretical probability, experimental probability, good language and made a circle graph (just a little harder than a bar graph) comparing the theoretical, experimental and Mrs. Callen's results would probably have the making of an Expert.
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