Measurement Error
15 min
Teacher Prep
Setup
Narrative
In this activity, students measure approximate lengths and decide possibilities for actual lengths. There are two layers of attending to precision (MP6) involved in this task:
- Deciding how accurately the pencils can be measured, probably to the nearest mm or to the nearest 2 mm, but this depends on the eyesight and confidence of the student
- Finding the possible percent error in the measurement chosen
Launch
Arrange students in groups of 2. Provide access to calculators. Give students 4–5 minutes of quiet work time, followed by partner and whole-class discussion.
Advances: Speaking, Conversing
Student Task
- Estimate the length of each pencil.
- How accurate are your estimates?
- For each estimate, what is the largest possible percent error?
Sample Response
- Sample response: The shorter pencil appears to be between 5.3 and 5.5 cm, perhaps 5.4 cm to the nearest mm, while the longer pencil appears to be 17.7 cm to the nearest mm (it is between 17.6 and 17.8 cm).
- Sample response: The estimate is accurate to within 1 mm. For the short pencil, it is more than 5.3 cm and less than 5.5 cm, but it is not possible to tell which is closer. Similarly, the longer pencil is more than 17.6 cm and less than 17.8 cm.
- For the shorter pencil, taking 5.4 cm as the measured length, the actual length x is at least 5.3 cm and at most 5.5 cm. The percent error if x is as small as possible is 5.30.1≈2%, and if x is as big as possible, then the error is 5.50.1≈2%. The first of these gives the greatest percent error, although they are close. For the longer pencil, the percent error is smaller. The biggest it can be is 17.70.1≈0.6%. This makes sense because 0.1 cm is a bigger percentage of the length of the small pencil.
Activity Synthesis (Teacher Notes)
The goal of this discussion is for students to practice how they talk about precision.
Discussion questions include:
- “How did you decide how accurately you can measure the pencils?” (I looked for a value that I was certain was less than the length of the pencil and a value that I was certain was bigger. My estimate was halfway in between.)
- “Were you sure which mm measurement the length is closest to?” (Sample responses: Yes, I could tell that the short pencil is closest to 5.4 cm. No, the long pencil looks to be closest to 17.7 cm, but I’m not sure. I am sure it is between 17.6 cm and 17.8 cm.)
- “Were the percent errors the same for the small pencil and for the long pencil? Why or why not?” (No. I was able to measure each pencil to within 1 mm. This is a smaller percentage of the longer pencil length than it is of the smaller pencil length.)
Other possible topics of conversation include noting that the level of accuracy of a measurement depends on the measuring device. If the ruler were marked in sixteenths of an inch, one would only be able to measure to the nearest sixteenth of an inch. If it were only marked in centimeters, one would only be able to measure to the nearest centimeter.
Anticipated Misconceptions
If students try to find an exact value for the length of each pencil, consider asking:
- “How did you decide the length of the pencil?”
- “Assuming you measured the pencil accurately to the nearest millimeter, what is the longest the actual length of the pencil could be? What is the shortest it could be?”
If students do not remember how to calculate percent error, consider asking:
- “How did you estimate the length of the pencil?”
- “What is the biggest difference possible between the estimated and actual lengths? What percentage of the actual length would that be?”
Standards
- 7.RP.3·Use proportional relationships to solve multistep ratio and percent problems.
- 7.RP.A.3·Use proportional relationships to solve multistep ratio and percent problems. <span>Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.</span>
15 min
15 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Measurement Error
15 min
Teacher Prep
Setup
Narrative
In this activity, students measure approximate lengths and decide possibilities for actual lengths. There are two layers of attending to precision (MP6) involved in this task:
- Deciding how accurately the pencils can be measured, probably to the nearest mm or to the nearest 2 mm, but this depends on the eyesight and confidence of the student
- Finding the possible percent error in the measurement chosen
Launch
Arrange students in groups of 2. Provide access to calculators. Give students 4–5 minutes of quiet work time, followed by partner and whole-class discussion.
Advances: Speaking, Conversing
Student Task
- Estimate the length of each pencil.
- How accurate are your estimates?
- For each estimate, what is the largest possible percent error?
Sample Response
- Sample response: The shorter pencil appears to be between 5.3 and 5.5 cm, perhaps 5.4 cm to the nearest mm, while the longer pencil appears to be 17.7 cm to the nearest mm (it is between 17.6 and 17.8 cm).
- Sample response: The estimate is accurate to within 1 mm. For the short pencil, it is more than 5.3 cm and less than 5.5 cm, but it is not possible to tell which is closer. Similarly, the longer pencil is more than 17.6 cm and less than 17.8 cm.
- For the shorter pencil, taking 5.4 cm as the measured length, the actual length x is at least 5.3 cm and at most 5.5 cm. The percent error if x is as small as possible is 5.30.1≈2%, and if x is as big as possible, then the error is 5.50.1≈2%. The first of these gives the greatest percent error, although they are close. For the longer pencil, the percent error is smaller. The biggest it can be is 17.70.1≈0.6%. This makes sense because 0.1 cm is a bigger percentage of the length of the small pencil.
Activity Synthesis (Teacher Notes)
The goal of this discussion is for students to practice how they talk about precision.
Discussion questions include:
- “How did you decide how accurately you can measure the pencils?” (I looked for a value that I was certain was less than the length of the pencil and a value that I was certain was bigger. My estimate was halfway in between.)
- “Were you sure which mm measurement the length is closest to?” (Sample responses: Yes, I could tell that the short pencil is closest to 5.4 cm. No, the long pencil looks to be closest to 17.7 cm, but I’m not sure. I am sure it is between 17.6 cm and 17.8 cm.)
- “Were the percent errors the same for the small pencil and for the long pencil? Why or why not?” (No. I was able to measure each pencil to within 1 mm. This is a smaller percentage of the longer pencil length than it is of the smaller pencil length.)
Other possible topics of conversation include noting that the level of accuracy of a measurement depends on the measuring device. If the ruler were marked in sixteenths of an inch, one would only be able to measure to the nearest sixteenth of an inch. If it were only marked in centimeters, one would only be able to measure to the nearest centimeter.
Anticipated Misconceptions
If students try to find an exact value for the length of each pencil, consider asking:
- “How did you decide the length of the pencil?”
- “Assuming you measured the pencil accurately to the nearest millimeter, what is the longest the actual length of the pencil could be? What is the shortest it could be?”
If students do not remember how to calculate percent error, consider asking:
- “How did you estimate the length of the pencil?”
- “What is the biggest difference possible between the estimated and actual lengths? What percentage of the actual length would that be?”
Standards
- 7.RP.3·Use proportional relationships to solve multistep ratio and percent problems.
- 7.RP.A.3·Use proportional relationships to solve multistep ratio and percent problems. <span>Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.</span>