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Classroom Case Studies, Grades 6-8 Part C: Inferences and Predictions (35 minutes)

The NCTM (2000) data analysis and probability standards state that students should “develop and evaluate inferences and predictions that are based on data.” In grades 6-8 classrooms, students are expected to develop and evaluate inferences and predictions in order to do the following:

• Use observations about differences between two or more samples to make conjectures about the populations from which the samples were taken
• Make conjectures about possible relationships between two characteristics of a sample on the basis of scatter plots of the data and approximate lines of fit
• Use conjectures to formulate new questions and plan new studies to answer them

When viewing the video segment, keep the following questions in mind:Inference and prediction are more advanced aspects of working with data, as they require an understanding of sampling. Students in grades 6-8 are developing an understanding of the idea of sampling. They often still expect their own intuition to be more reliable than the information they are obtaining from the data. Students begin to develop an understanding of these statistical ideas through conversations as they consider what the data are telling us, what might account for these results, and whether this would be true in other similar situations. Students’ early experiences are often with census data — e.g., the population of their class. When they begin to wonder what might be true for other classes and other schools, they begin to develop the skills of inference and prediction. In the later middle grades and in high school, students begin to learn ways of quantifying how certain one can be about statistical results.

• How does Mr. Sowden encourage students to make inferences and predictions?
• What are some of the students’ preliminary conclusions?
• Which of the students’ inferences are based on the data, and which are based on their own personal judgement?

Video Segment
In this video segment, Paul Sowden asks the students to look for patterns in the four line plots and to try to determine where the coins with no marking were minted. Students discuss the variance in the data and speculate on why the coins have no mint marks and about where those coins might have been minted.

Problem C1
Answer the questions you reflected on as you watched the video:

a. How does Mr. Sowden encourage students to make inferences and predictions?
b. What are some of the students’ preliminary conclusions?
c. Which of the students’ inferences are based on the data, and which are based on their own personal judgement?

Problem C2
In the video segment from Part A, students considered why this set of coins contained more coins from Philadelphia. One student hypothesized that this was because Philadelphia is the closest U.S. Mint. This student was beginning to think about the sample of coins the students were using. How might you facilitate a discussion with your students about bias in data and the extent to which a data set can be representative? What questions would you pose? What issues would you raise?

Problem C3
If you were teaching this lesson on investigating nickels and their mint marks, what questions might you ask to focus students’ attention on each of the following central elements of statistical analysis: See Note 5 below.

Defining the population
Defining an appropriate sample
Collecting data from that sample
Describing the sample
Making reasonable inferences relating the sample and the population

Problem C4
In the video, the students used circle graphs and line plots to examine the variation in their data. The teacher plans to continue analyzing the variation in the data, using different types of representations. What other types of representations might he use to examine the data?

Problem C5
How could you extend the discussion in this video segment to bring out more speculations about the nickels? How might you formalize these notions into stated conjectures that could be investigated further? What are some conjectures that might arise? How could you formulate them into new questions? How could these questions then be investigated?

Principles and Standards for School Mathematics (Reston, VA: National Council of Teachers of Mathematics, 2000). Standards on Data Analysis and Probability: Grades 6-8, 248-255.

Notes

Note 5
You might want to review the statistical ideas of samples and populations.
Session 1, Part D: Bias in Sampling

Solutions

Problem C1
a. The teacher asks the students to consider reasons why some coins do not have mint marks. Then he pushes them to think about which mints can likely be eliminated based on the information they see across the four line plots.
b. Some students think the coins with no mint marks were just mistakes. Others think that those coins are likely to be from Philadelphia, because so many nickels contain the Philadelphia mint mark. When thinking about which coins could be eliminated, the students eliminate Denver first and then San Francisco.
c. At first, students are reasoning more from their own personal judgment, such as when one student says that the missing mint marks are a mistake because the machines were working too fast. Other students’ comments were more grounded in the data.

Problem C2
Two questions you might ask are, “Where do you think I got these coins?” and “How might that affect our results?”

Problem C3
•
How well do you think our sample of nickels represents the population of nickels in this country?
•
If we lived near San Francisco and collected nickels, how might our results be different? How about if we lived near Denver?

Problem C4
The teacher might have the students make box plots for each category of coin mint mark.

Problem C5
One example of a conjecture that students might make is “Philadelphia produces more nickels than the other mints.” This could be formulated as a new question to be investigated: “Does Philadelphia produce more nickels than the other mints?” The students could investigate this question in several ways. They might want to just enlarge their own sample of nickels, with each student collecting nickels over the next week for further analysis. This could also evolve into an Internet project in which your students contact students in other parts of the country, especially those who live closer to the other mints. Each group of students could collect and analyze a sample of nickels. They could then compare across samples and finally combine them into one larger sample.