Intent

In this activity, students use their developing intuition about measurement variation to make a decision.

Mathematics

Given the weights of a collection of real coins, is another coin of a particular weight real or counterfeit? In other words, is the weight of the suspect coin what you would expect, given the variability in the weights of real coins? These are essential statistical questions. At this point in the unit, students will use frequency bar graphs and calculations of mean, median, and mode to address these questions, creating the need to develop a new tool later in the unit: standard deviation. Students will revisit Question 2 of this activity in Penny Weight Revisited, with the tool of standard deviation in mind.

Progression

Students will work on this activity individually and then share their results in a class discussion.

Approximate Time

5 minutes for introduction

15 minutes for activity (at home or in class)

20 minutes for discussion

Classroom Organization

Individuals, followed by whole-class discussion

Materials

Graph paper

Transparency of graph [link to BLM pdf of Penny Weight, p. 7]

Doing the Activity

Read through the activity as a class. Suggest that students may want to use statistics like the mean, median, and mode, as well as frequency bar graphs, to analyze the situation and to support their opinions.

Discussing and Debriefing the Activity

To begin the discussion, ask students if they think that the weights of pennies are normally distributed. (This specific set of data is not, but it is likely that the variation among pennies, though slight, creates an approximately normal distribution.)

Then ask, Do you think Uncle Jack’s coin was counterfeit? What is your reasoning? Of those who do not think the coin was counterfeit, ask what weight it would have to be to convince them that it was.

Explain that although statistics will not tell whether the coin actually was counterfeit, statistics can reveal something about the probabilities involved. That is, it will tell them how rare such a weight is for a legitimate coin.

Bring out that there is no mathematical way to take Uncle Jack’s personal reliability into account. Even without this sort of factor, there is often a subjective component to interpreting statistics.

Key Questions

Do you think the weights of pennies are normally distributed?

Do you think Uncle Jack’s coin was counterfeit? What is your reasoning?