Intent In this activity, students conduct their first simulation, collecting, displaying, and summarizing data to test a conjecture about probability. The activity draws upon their intuitive sense of probability and engages them in thinking more precisely about the topic. This is one of several activities in which intuitive notions about probability can be at odds with experimental or theoretical analyses.

Mathematics When rolling two dice, what is the average number of rolls needed to roll a double? In this activity, students approach this question through collecting and analyzing data. By rolling two dice repeatedly, they collect many examples of how many rolls it can take to get a double. It could take a single roll, or it could take 30 rolls or more. (The distribution of their results will have an average value of about 6.)

Progression Up to now, students have been exploring the unit problem informally. Now they begin a series of explorations designed to help them develop the tools they will need to answer the unit problem. The in-class portion of this activity has several parts: reviewing the homework, gathering class data, and graphing the class data.

Approximate Time 5 minutes for introduction 20 minutes for activity (at home or in class) 25 minutes to collect and analyze class data

Classroom Organization Individuals, then small groups to review data, then whole class to gather and analyze data

Doing the Activity Students can do this activity for homework. When assigning the activity, you might have them try a few trials to be sure they know what they are counting and to consider the range of possible outcomes. Students are first asked to predict the average number of rolls it will take to get a double. Some students might have ideas about this based on experience with dice, while others might simply guess. Then they are to collect the outcomes of ten experiments. Stress that it is important to record the results exactly as they occur. Finally, students are asked to review their data, compute the average number of rolls for the ten experiments, and compare this average to their initial predictions. Through this process, students connect the average outcome of repeated trials with the expected number of rolls.

Discussing and Debriefing the Activity In small groups, have students compare their predictions with the results of their data collection. It is likely that many students will have at least one value as large as 15 rolls, and some may have values as large as 30 rolls or more. As a class, the ultimate goal is to create a display of the data collected by all of the students. Begin by giving students a chance to report to the class their initial predictions, how their experiments turned out (that is, the fewest and greatest number of rolls it took to get a double, and the average of their ten trials), and what they now think about their predictions. Next, ask the class how they might display the collected data. Students may have several ideas for graphing the outcomes. As a class, decide to make a frequency bar graph where each bar would represent a different number of rolls needed to get a double. The height of each bar would show the number of times that result occurred.

Once the class data have been graphed, ask students to make observations. Emphasize what a frequency bar graph can reveal about data, and refresh students’ memories about the concepts of mode and median. Ask students to compute the class average for the number of rolls needed to get a double. This value will be close to 6. Some students may be surprised to see far more results below 6 than above 6. Others may note that there are far more bars above 6 than below 6. As needed, help them understand that a single large result “balances” lots of small results, and that in general the average of all the frequencies does not equal the average of the actual results. Finally, ask students to review how their individual averages varied and how those averages compare to the class average. If you have access to the technology you might demonstrate or have students explore simulations. Class data can be simulated using dynamic data software such as Fathom [link to T010201]. Or you might use a graphing calculator [link to T010202].

Key Questions Before assigning the activity: How would you do this experiment? Why is it important to get genuine data when doing activities like this? How do you compute the average of a set of numbers? During the class discussion: How did you make your prediction? What were the highest and lowest number of rolls you got? How might you graph one student’s outcomes? Can you visualize the average in terms of the graph? How might we graph the outcomes of the entire class? Is 1 a likely outcome?

Supplemental Activities Average Problems (extension) has three tasks that delve into the meaning of the arithmetic average, or mean, of a set of data. Above and Below the Middle (extension) offers further experience with mean and introduces the concept of median.