Lesson 8

The mathematical purpose of this lesson is for students to use data to support or oppose a mathematical claim. Students are presented with a model and asked to determine whether the model applies to the situation when the actual data differs from the model.

Tell students they will use data to support or oppose a mathematical claim.

The purpose of this discussion is for students to support or oppose Lin’s mathematical claim using evidence. Some questions for discussion:

• “Is there evidence to support Lin’s claim that they each have a 60% chance of making each shot? Explain your reasoning.” (There appears to be some evidence from Lin’s observations, presumably of their previous performance, that they have about the same ability as she does. However, the data shows that Kiran made 75% of the free throws and Diego made about 35% of the free throws which is not consistent with Lin’s claim.)
• “What would you estimate that Kiran and Diego’s chances of make a shot is?” (I would estimate that Kiran has about a 75% chance and Diego has a 35% chance.)
• “How could you test your estimate?” (I could ask Kiran and Diego to shoot another 100 free throws and test my prediction.)
• “If Diego took another 100 free throws and made 61 of them, would that provide evidence for the claim that Diego has a 60% chance of making each shot?” (I think it would provide evidence for the claim because 61% is really close to 60%. Predictions are estimates so there should be some room for error when talking about evidence to support predictions.)

The mathematical purpose of this activity is to make, critique, and justify claims using the data-generating process. Students flip a coin to determine the number of heads that show up for several groups of 20 flips. Students should recognize that some variability is expected from the expected 10 heads that are predicted for 20 flips, but that some results are very unlikely even if they are possible. Students should begin to recognize when a statistical model is consistent with results and when the data is so unusual that a new model might be more appropriate.

In later lessons, students will need to recognize when a normal curve is an appropriate model for data so that the area under a normal curve can be used to get information about the data.

Some students may be uncomfortable with the less precise answers for the work in this unit and this activity in particular. Remind students that the actual world is messy and the work of statistics is to help make sense of that messiness. Math is not proving Priya has a fair coin or not, but can give us evidence to suspect that it is not and a decision can be made based on that evidence and anything else that is known about the situation.

The purpose of this discussion is to talk about making and justifying claims based on data. Here are some questions for discussion:

• “You performed 20 coin flips 5 times. Based on your data alone, what range of values would you predict for the number of times the coin would land on heads if it was flipped 20 times?” (I got 8, 9, 10, 12, 13 heads so I would predict that a coin would land on heads between 7 and 13 times. I included seven in my estimate because 13 is 3 above ten and 7 is 3 less than 10. The coin should land on heads an average of 10 times.)
• “How does your prediction compare to the range of values you described from the class's data? Why do you think it was the same or different? Explain your reasoning.” (My range was smaller than the range from the whole class data. Since there was much more data for the whole class, I expected there to be a broader range of values since there was likely to be more variability.)
• “If 1,000 people flipped a coin 20 times, how might that change your prediction from the rage for the class's data?” (If we had 1,000 data values, I would feel really confident about the range of values that I would expect from 20 coin flips. I know that it is possible to get between 0 and 20 heads, but I wonder how unlikely it really is to get 0 or 20. I think the 1,000 data values would help me to know a typical range of values.)

The mathematical purpose of this activity is to make, critique, and justify claims using the data-generating process. The situation involves questioning whether the apparent disproportionate representation of a sample is due to bias or random selection. Students perform a simulation to determine how often the given scenario is likely to happen through random selection. Finally, students determine whether it is likely the group was selected using a biased method or a random method.

Tell students that they are going to perform a simulation to determine if the results of an experiment are reasonable. Monitor students as they select names and collect data from the class to be included in a dot plot. Identify students who use measures of center or variability to support mathematical claims.

The goal of this discussion is for students to use data to support or oppose a claim. Ask previously identified students “How does using measures of center or variability help you to support or oppose the students’ complaint?” (I calculated the mean, approximately 3.33, and the standard deviation, approximately 1.50. Five is not within one standard deviation from the mean but it was very close so I think it was possible for the principal to select five students from the graduating class using the random method. Five is within two standard deviations of the mean.)

Here are some questions for discussion:

• “Based on the class data, what is the probability that 5 or more students in the graduating class were chosen?” (Sample response: It happened 15 out of 75 times, or 20%.)
• “What percentage of trials had fewer than 5 students selected from the graduating class?” (80%)
• “If the principal had selected 7 students in the graduating class to go on the trip, would there be enough evidence to prove that the principal was not being fair?” (The evidence would suggest that the principal was not being fair, but it would not prove it. It is still possible even if it is unlikely, so the statistical reasoning cannot prove this claim.)
• “How many students from the graduating class would you typically expect to be drawn from 25 applications if 8 students were selected at random? Explain your reasoning.” (I think 2, 3, or 4 would be drawn typically. Since 10 out of 25 students are in the graduating class, and 8 names are drawn, and \(\frac{10}{25}\) of 8 is 3.2 which is approximately 3. When I look at the data in the dot plot, I see that 2 and 4 are typical as well.)
• “Do you think a normal curve would be a good model for this data or the data from the previous problem? Explain your reasoning.” (Yes because the data is approximately symmetric and bell-shaped.)
• “How would fitting a normal curve help in understanding the results?” (I could find out how likely it is that between 2 and 4 students would be selected.)
• "Have you ever experienced a situation that seemed unfair, but might have been possibly due to random chance? What are some mathematical things you could do to help decide whether it was due to random chance or deliberately unfair?" (Each week, the adults at our house decide who will take the trash out between my brother and I. I've had to do it 4 weeks in a row, which seems unfair. I could flip a coin 52 times to represent a year of taking out the trash and see how often either heads or tails comes up 4 times in a row. Repeating this many times might give me an idea of how often it might happen by chance or whether the adults are picking on me on purpose.)