In this activity, students determine the amount of error in a measurement and express the error as a percentage of the correct value. They brainstorm possible sources of error and discuss how real-world limitations on humans using measuring devices can introduce measurement errors. As students interpret these results, they consider sources for error and improvement in a mathematical model (MP4).

The actual length is given in yards, while the measured length is given in feet and inches. Students must choose which unit to use and convert the other measurement before they can calculate the measurement error. Regardless of which unit they use, the percentage that the error is of the correct amount should be the same. Monitor for students who choose to use different units for their calculations: yards, feet, or inches. 

Also monitor for mistakes students may make in their solution process, such as:

• Failing to convert the lengths to the same unit.
• Dividing by the measured length instead of the actual length, resulting in the percentage 4.38%.

The key focus is distinguishing between different solution processes that give the same answer versus solution processes that give an incorrect answer.

The goal of this discussion is to contrast different methods that get the correct answer with methods that get an incorrect answer for the percentage that the error is of the correct amount. Display 2–3 approaches from previously selected students for all to see. If time allows, invite students to briefly describe their approach. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “Why do these different approaches lead to the same outcome?
• “Why does this approach lead to a different outcome?”

The key takeaway is that it doesn't matter which unit students chose. The percentage of the error should be the same as long as their units match on both the measured length and the actual length (and as long as students divided by the actual length, not the measured length).

Explain to students that measurement error is the positive difference between the measurement and the actual value. It is often expressed as a percentage of the actual value. We can use words to describe whether the measurement is greater than or less than the actual value. In this case, we might say that the measured length is more than the actual length with an error of about 4.6%.

In this activity, students work with their partner to measure three different things around the classroom using the two different rulers from the Warm-up. After both students in a group have measured their three objects, the teacher provides them with the actual measurements of those items. The students then calculate the measurement errors as percentages of the actual measurements.

As students measure multiple items and generalize that using the less precise ruler results in greater percent error, they are making use of repeated reasoning (MP8).

Monitor for students whose tables clearly show that the measurements taken with the centimeter ruler have greater percent error than the measurements taken with the millimeter ruler.

There are two desired outcomes of this activity:

1. To develop a procedure that makes sense to students for computing percent error
2. To reinforce that less precise measuring devices result in greater percent error

Direct students’ attention to the reference created using Collect and Display. Ask students to share what they noticed about their percentages. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. Ask students to suggest ways to update the display: “Are there any new words or phrases that you would like to add? Is there any language you would like to revise or remove?”

To highlight the effects of using a less precise measuring device, select a pair of students whose tables clearly show that the measurements taken with the centimeter ruler had greater percent error than the measurements taken with the millimeter ruler. Display their work, and ask students to share what they notice about these percentages. Invite students to explain why the measurements taken with the centimeter ruler had greater measurement error.