Intent

In this activity, students gather, graph, and summarize data concerning sums from rolling two dice. They continue to explore the usefulness of displaying data in frequency bar graphs. The collected data will confirm that some two-dice sums are more likely than others.

Mathematics

Students design a method to keep track of experimental results. Then, they devise a way to organize that information into graphical form.

Progression

As a follow-up to The Counters Game, students first discuss this activity as a class. They then collect data and complete the tasks individually, likely for homework. Once returning to class, they create a frequency bar graph of the class data and make observations. Finally, students calculate the empirically derived percentage for each two-dice sum for comparison with the theoretical analysis that will be done in The Theory of Two-Dice Sums.

Approximate Time

5 minutes for introduction

20 minutes for activity (at home or in class)

25 minutes for discussion

Classroom Organization

Individuals, followed by whole-class discussion

Materials

Dice

Grid chart paper or grid-paper transparency

Doing the Activity

Stimulate interest in this activity by asking what techniques have been developed that may help students learn more about how frequently each two-dice sum occurs in the roll of a pair of dice. Some students will likely suggest actually doing the experiment and collecting data. Use this suggestion to engage students in developing a plan to collect data individually and then pool it together, as a large collection of data will provide a good sense of the theoretical probabilities.

Discussions about how to record and graph the data can occur before or after students begin their work. Involving students in the design of the data collection will help to alleviate the problem of some students not bringing legitimate data to class, if this activity is assigned as homework.

Discussing and Debriefing the Activity

You might begin the discussion by asking how students kept a record of their results. Some may have written down the results in the order they occurred, while others may have listed the possible outcomes and used tally marks to keep track. If no one mentions the second method, you might suggest it yourself.

Have volunteers share their graphs and any general conclusions they reached. If some students portrayed their information using something other than a frequency bar graph, ask them to share their representations as well. Presumably students’ data contain only a few very high sums (10, 11, or 12) and a few very low sums (4, 3, or 2), with most of the sums in the middle (5 through 9).

The next stage is to gather all the data together to create a class frequency bar graph. You might ask, What will we need to do to complete a frequency bar graph of all the data collected by our class? To whatever degree is appropriate, engage students in designing and creating the class graph, using grid chart paper or a grid-paper transparency. Having each group compile its members’ data and then send a representative to report to the class can be an efficient method. Or use a technology such as Fathom surveys. [Link to a technology document]

Students will need to think about the scale for the vertical axis, which will indicate how many times each outcome occurred. (You can expect about 250 7s in a class of 30 students.) The graph will be easier to read if column totals are written on the bars, as done below. This graph is from a simulation giving 50 results for each of 30 students, for a total of 1500 results.

Figure 1

Generally, it is a good technique to begin conversations about data by asking, So, what do you see? Additionally, you might turn groups back for some discussion among themselves. After some group discussion, it is likely that students will see that the class data confirm many of their suspicions and may also reveal some things they didn’t expect.

Use this opportunity to ask, How does our class graph compare to your individual graphs? This will probably elicit the observation that the class graph is “more even,” “smoother,” or “more symmetrical” than the individual graphs.

Finally, ask, What do you think the sum of the column totals should be? (It should be 50 times the number of students in class.) The class can verify that number and then use the total to find the percentage of occurrence for each sum. If the totals do not sum to a multiple of 50, some students will want to track down the error, a pursuit that may not be worth the necessary time.

Save the class frequency bar graph for comparison with the theoretical analysis in The Theory of Two-Dice Sums.