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Expecting the Unexpected

Module by: Interactive Mathematics Program

Intent

This activity gives students experience with experimental variation as they continue to develop the skills of collecting and organizing data. Students also make predictions and compare their predictions with their observations.

Mathematics

Gathering data about coin flips will help students get a sense of experimental variation. Students combine data generated by the class and make a frequency bar graph of the results, practicing the skill of keeping track of experimentally generated data. They also examine different ways to group data to make a frequency bar graph.

Progression

This activity engages students’ intuitive ideas about probability. Later in this unit, students will consider random events and equally likely outcomes more closely.

Approximate Time

5 minutes for introduction

30 minutes for activity (at home or in class)

15 minutes for discussion

Classroom Organization

Individuals, then groups, followed by whole-class discussion

Materials

Coins

Doing the Activity

To introduce the activity, ask, How many heads would you expect to get if you flipped a coin 50 times? Encourage discussion. It is likely that some students will immediately respond with 25 and that others will suggest a range. You might also have students share their initial predictions for Questions 2a and 2b.

This activity will likely be done as homework. Remind students that the discussion at the next class meeting will depend on all students bringing in the results of their experiments.

Discussing and Debriefing the Activity

Have students share their data with the class. Students might be surprised that the most likely outcome—25 heads out of 50 flips—is not particularly likely, happening only about 11% of the time. If you didn’t bring up the “most likely versus likely” distinction earlier in the unit, this is a good time to do so. (In a class of 30 students, with the 50-flip experiment done a total of 60 times, the highest result would be approximately 33 heads, and about 7 of the 60 experiments would result in exactly 25 heads.)

Ask, How might we graph the data?

Students may have several ideas about graphing the data. This is a good time to pursue the idea of grouping data using a frequency bar graph, which the class encountered in Waiting for a Double.

To encourage students to begin thinking about how to organize the data so that each student’s data are included, ask, What planning will we need to do as a class so that each of us can have a frequency bar graph that shows all our data? Depending on students’ level of autonomy, and what you wish to challenge them with, consider how much input into the organization of this data collection and representation you wish to have.

Ask, How might you group the data?

Have each group choose its own way of grouping the results and making the corresponding frequency bar graph. Display some of the bar graphs that groups create. As an example, students may produce a graph something like this.

Figure 1
Figure 1 (graphics1.jpg)

You might ask if the graph can be redrawn with fewer bars to make the “big picture” more obvious. How might you modify the graph to see the “big picture” better? You can use the term bin width to describe the size of each grouping section. For instance, a bar representing results from 23 through 26 has a bin width of 4.

If someone suggests making a graph in which the bin widths are not all the same, you might ask whether this organization could be misleading. For example, if results 27 and over were combined in one bar and all other results represented as single cases, the graph would seem to indicate that the most likely outcome is a result of at least 27. For this reason, we typically make all groups the same “width.” For example, the graphs below combine frequencies for two outcomes at a time (bin width = 2) and three outcomes at a time (bin width = 3).

Figure 2
Figure 2 (graphics2.jpg)
Figure 3
Figure 3 (graphics3.jpg)

Students may debate the merits of various grouping methods. For instance, some may argue that if they are combining three bars at a time, they should work symmetrically around the value 25, so that the middle group is 24–26.

To close the activity, summarize how some of students’ initial ideas about what is most likely to occur when flipping 50 coins seem to appear within the frequency bar graph. You may want to point out that, generally speaking, all of these graphs are higher in the middle and lower at the ends, whereas the frequency bar graph of the data from Waiting for a Double started out high and gradually got lower.

If you have the necessary technology you can show or have students explore a Fathom simulation of this activity [link to T010203].

Key Questions

How many heads would you expect if you flipped a coin 50 times?

What planning will we need to do as a class so that each of us can have a frequency bar graph that shows all our data?

How might we graph the data?

How might we group the data?

How might we modify the graph to see the “big picture” better?

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