Intent

Students examine the results from three previous experiments to help them develop intuition about what is “ordinary” and what is “rare,” which will help them to understand the concept of standard deviation later in the unit.

Mathematics

Standard deviation is one of many measures of the variability in a set of data. With normally distributed data, the standard deviation can be used to determine whether a particular result is an ordinary or a rare occurrence. For example, when flipping 10 coins, getting 4, 5, or 6 heads is an ordinary result, as each is within one standard deviation of 5, the mean value. However, getting 9 or 10 heads, or 0 or 1 heads (each more than two standard deviations from the mean), would be rare. In this activity, the notions of “ordinary” and “rare” are introduced in an intuitive way that will lay the foundation for understanding the meaning of standard deviation later in the unit. [Link to math maps]

Progression

Students will work on the four parts of this activity in their groups. The first three parts refer to results of earlier experiments, and the fourth asks students to step back and review their work across experiments.

Approximate Time

25 minutes

Classroom Organization

Groups, followed by whole-class discussion

Materials

Frequency bar graphs from Time Is Relative, What’s Your Stride? and Pulse Analysis

Doing the Activity

Tell the class that in addition to providing numeric answers to the questions in this activity,they should keep track of the reasoning behind their answers.

The two categories of ordinary and rare need not cover all possibilities; students may decide that some results fall between the two.

Discussing and Debriefing the Activity

Begin the discussion by asking, How did you decide what to call ordinary and what to call rare? How did you decide on the percentages? The distinction between ordinary and rare is an arbitrary one, so have groups share how they decided to define these terms.

Now ask, Were your percentages similar for all three situations? Discuss whether the percentages defining ordinary and rare for one set of data hold up for another set of data—that is, whether there is some consistency in what students hold these terms to mean. If so, you will be able to build on this result when students look at standard deviation.

Finally, you may also want to discuss the question of whether it was the timing instruments or the person doing the timing that was responsible for the variation in the results for the experiment in Time Is Relative.

Ask students how their work of the last few days relates to the unit problem. They should mention that they have been gaining insight into the normal distribution—in particular, what level of variation can be considered ordinary and what level can be considered rare in normally distributed data.

Assure students that they will be applying everything they are learning to the pendulum question.

Key Questions

How did you decide what to call ordinary and what to call rare?

How did you decide on the percentages?

Were your percentages similar for all three situations?

How are these ideas connected to the unit problem?