# Performing Calculations using Measured Values that Include Uncertainty

## Summary

In this activity, students practice performing calculations using measured values that include uncertainty. Students measure the mass (using an electronic balance) and volume (by water displacement) of pennies and use the values to calculate the density of the pennies. All measured values include uncertainty, and students practice using the rules for making calculations using numbers that include uncertainty.

As the students increase the number of pennies they use, the relative uncertainty of their calculated density decreases. Students will see factors that affect the uncertainty of each measurement and also how the uncertainty of each measurement contributes to the uncertainty of their calculated results.

Provided here is a data set for copper pennies (pre-1982) and zinc/copper pennies (post 1982). Students can use the data to identify what these pennies are made of, but only when the uncertainty of their calculated density is lower than the difference between the density of copper and the density of zinc.

## Learning Goals

## Context for Use

## Description and Teaching Materials

If the activity is done in the lab, the worksheet provided here can be used as an assessment for the lab.

## Teaching Notes and Tips

The composition of US pennies changed in 1981-82. Pre-1982 pennies are mostly copper and pennies made after 1982 are mostly zinc. Sort pennies into pre- and post- 1982 piles. Pennies made in 1982 could be either alloy, depending which mint made them. The US mint in Denver continued to make pennies of mostly copper through 1982. The instructor should sort the pennies into pre-1982, post-1982, and, for an extra mystery treat, 1982 D (Denver). This way, students will get different results and reach different conclusions, which makes for interesting class discussions.