I first checked with students about how they simulated the situation. I like to do this so students can hear about different ways to represent the given scenario and to identify any potential issues. Students presented using 0 or 1, 0-4 and 5-9, and flipping a coin to represent the situation. These all work and are easy to implement.
One girl presented her simulation by saying she picked individual random digits and whatever digit came up that was the size of the household. I asked the class what they thought of this representation.There were puzzled looks and some seemed unsure of what to think. I asked what the probability of a 9 child household would be using this technique and students responded 1 out of 9. I then asked them what the theoretical probability of having a 9 child household was. The response was it was a tiny number. I again asked if the model being discussed would work and the class replied that it wouldn't because the probabilities for the events didn't match.
This was a good opportunity to emphasize that the probability of occurrence should be accurately reflected by the random digit assignment. It is also important to think about how to apply this assignment to simulate the given scenario.
By this time, students were starting to understand the mechanics of finding the average number of children per household. Calculating the theoretical expected number was confusing to some students. I had asked these students to consider what they had done when calculating the average number of children per household. This helped students to think about what they had done and why.
During the class discussion, one student discussed how she had multiplied out number of children, an assumed household count of 500, and probability for each family size, added these values together, and then divided by 500 to get the average children per household. Another student presented essentially the same calculations but didn't bother to use the 500 value.
The class wanted to know which way was right. I told them they needed to consider what they were trying to find, the information they had to use, and what they had been doing. Some students took to this and had healthy discussions, performing various calculations to cross check results and techniques.
The final discussion centered on the algebraic equivalency of the two approaches. As one student put it, if you multiply each value by 500 and then divide the result by 500, you are cancelling out the 500 values.
The investigation concluded with providing the definition of expected value:
We then proceeded to work on the Shooting Free-Throw investigation. This investigation followed a similar tract to the One Girl Family Planning investigation. Students needed to simulate the situation, examine their results, calculate theoretical values, and compare results.
The first issue students face is how to simulate the scenario. Most students recognize that a 60% free-throw success rate could be simulated by using the digits 0-5 for a make and 6-9 for a miss. Some students, however, wanted to assign a 50-50 probability to making and missing. The thinking is that there are two outcomes and the inclination is that they should have equal probability. I simply ask these students what is the probability that a free-throw is made and they quickly realize they can't use a 50-50 split on their random numbers.
Students struggle with how to account for the one-and-one situation in the problem. The tendency is to say there are 3 outcomes: 0 points made, 1 point made, and 2 points made. The issue is that these events are not all equally likely. Referring students back to the One Girl Family Planning problem and asking them how the simulated a household having 3 children helped them to realize they needed to generate successive random numbers.
From here the next hurdle is calculating the expected point value. I asked students to refer back to the expected value definition. This helped students move beyond what to do and to begin recognizing how an expected value can be calculated.
Students also had questions about how to calculate the theoretical probability associated with the situation. Students are not good at representing the situation using a diagram or table. I encouraged students to try to make a visual representation of the problem. Many started to use tree diagrams, which helped them piece things together.
I asked students to complete the investigation as homework and that we'll discuss the results next class.
Visit the class summary for a student's perspective of the class and to view the lesson slides.
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