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 solve three different problems involving percent error. The problems here are similar in structure to percent increase or decrease problems, but the language is different. Students interpret the language used for percent error, drawing parallels to the language used for percentage increase and decrease. This activity includes one of each type of problem:

• Finding the erroneous amount given the correct amount and the percent error—this is similar to finding the new value after a percent increase or decrease.
• Finding the correct amount given the erroneous amount and the percent error—this is similar to finding the original value before a percent increase or decrease.
• Finding the percent error given the erroneous amount and the correct amount—this is similar to finding the percentage that the increase or decrease is of the original amount.

Students connect the structure of percent increase and decrease to the situations involving percent error (MP7). In this activity, students critique a statement or response that is intentionally unclear, incorrect, or incomplete and improve it by clarifying meaning, correcting errors, and adding details (MP3). 

The purpose of this discussion is to highlight the connections between these problems involving percent error and the types of problems students solved previously involving percent increase and decrease. Invite students to share how they solved each problem. After each student shares, ask the class if they agree or disagree or if they had a different way to calculate the solution. Highlight efficient strategies, such as writing an equation. Ask students how the solution strategies here are similar to the ones used with percentage increase and decrease.

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response to the last question by correcting errors, clarifying meaning, and adding details. 

• Display this first draft:

  “To find the error, I subtracted to get 514. To find the percent, I divided 514 by 3,000 to get 17%.”

  Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.

• Give students 2–4 minutes to work with a partner to revise the first draft. 
• Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

The key takeaway is that percent error is the amount of the error expressed as a percentage of the correct amount. In this case, the correct amount is the exact attendance, not the estimated crowd size, so it should be .

In this activity, students apply percent error to a quality control situation. It is very common that products in a factory are checked to make sure that they meet certain specifications. This activity is about a saw mill cutting boards to specific lengths. Boards that are too long or too short are rejected. Students decide whether boards should be accepted or rejected based on the intended length and an acceptable margin of error. As students determine values that are within, above, or below an acceptable range, they are making sense of how percent error characterizes the situation. They also persevere in solving when working backwards to determine the intended length given the percent error (MP1).

The purpose of this discussion is to solidify students’ understanding of how percent error can be used to establish an error tolerance. Invite students to share their responses for the first question about lengths that should be accepted or rejected. Record their responses on a number line, displayed for all to see. Ask students to describe any patterns they notice in the values that are accepted or rejected.

To help students understand these values in context, consider asking:

• “Why should the mill accept boards that are a little longer or shorter than the intended length?” (It may be difficult to cut the boards to that exact length, especially as quickly as a saw mill probably needs to cut them, so some range of acceptable lengths is probably allowed.)
• “Why shouldn’t the mill accept boards that are a lot longer or shorter than the intended length?” (People who are using the boards expect to get a consistent product. It would make them harder to transport, sell, or use if there was a lot of variability in the lengths.)
• “Why does it make sense for the range of acceptable lengths to be listed as a percent error rather than on a fixed length?” (While 2.88 inches may be ok for a board that should be about 192 inches long, if the board was supposed to be 5 inches long and it was allowed to be 2.88 inches longer, it would be more than 50% longer than intended.)