Intent In this activity, students conduct experiments in a situation in which the results are normally distributed.

Mathematics Students began to collect and analyze experimental data in The Game of Pig. Now they are reminded that there is a difference between experimental and theoretical results, but that these differences are reduced as the number of trials increases. Students will encounter the coin-flip distribution in later years of IMP. At this time, it is enough that the discussion supports their understanding that the distribution is approximately normal. [Link to math maps]

Progression Students will work individually to collect data and then pool their results with their group mates and, ultimately, with the entire class.

Approximate Time 5 minutes for introduction 25 minutes for activity (at home or in class) 20 minutes for discussion

Classroom Organization Individuals, then groups, followed by whole-class discussion

Materials 10 coins per student

Doing the Activity Demonstrate that a single experiment consists of flipping a set of 10 coins and counting the number of heads. Emphasize again, as students learned in The Game of Pig, the importance of gathering genuine data on this and similar assignments.

Discussing and Debriefing the Activity Ask students for verbal descriptions of their frequency bar graphs. They will probably offer a simple description, such as that most results are “in the middle” with fewer results at the extremes. (Note that because the total number of trials is relatively low, coin-flip results will not always match the theoretical probabilities.) Ask each group to total the number of times each possible result occurred among its members (how many 0s, how many 1s, and so on). Have group representatives read the totals out loud while you record a master list on the board. Then create a frequency bar graph of the class data. It will be interesting to see whether any 0s (no heads) or 10s (all heads) occurred, as well as what students’ expectations about the results were. (In a class of 30 students, the probability is about 36% that at least one person will get a 0, about 36% that at least one person will get a 10, and about 59% that at least one of these two extreme cases will occur.) For Question 4, students are likely to think that the graph will look normal. In fact, the theoretical distribution of the coin flips is quite close to normal. The graph below, created using Fathom, shows that with 10,000 trials the results approximate a normal distribution. [Link to flip_flip.ftm]

Question 5 is important as it exposes students to the idea of what conclusions might be drawn from unusual results. Ask students to discuss how much deviation you would have to see in order to decide that the coin was really unbalanced. [Link to math maps] If anyone got a result in the coin-flipping experiment of 0 or 10 (or even 1 or 9), you might ask if that person suspects that the coins he or she used might have been unbalanced. Through these experiments and discussion, students should be realizing that they can generally expect some deviation from the average when they conduct an experiment. They should also be gradually developing intuition about what level of deviation should be regarded as significant.

Key Questions What does your frequency bar graph look like? How does the class graph compare to what you predicted in Question 4?