In this Warm-up, students use two rulers that have different levels of precision to measure the same lines. Students notice how differences in recorded measurements can result from the level of precision of the measuring device. This prepares students for expressing measurement error as a percentage of the correct amount.
Students will need to use the rulers again later in this lesson, so make sure they keep track of them.
Student Lesson in Spanish
Launch
Arrange students in groups of 2. Distribute one set of 2 pre-cut rulers to each group.
Give students 3–5 minutes of partner work time, then follow with a whole-class discussion.
Activity
None
Student Task Statement
Your teacher will give you two rulers and three line segments labeled A, B, and C.
Use the centimeter ruler to measure each line segment to the nearest centimeter. Record these lengths in the first column of the table.
Use the millimeter ruler to measure each line segment to the nearest tenth of a centimeter. Record these lengths in the second column of the table.
line segment
length (cm) as measured
with the first ruler
length (cm) as measured
with the second ruler
A
B
C
Student Response
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Building on Student Thinking
Students might not line up the edge of the ruler with the end of the line. Remind students that we need to line up the 0 mark on the ruler (in this case, the edge of the ruler) with the beginning edge of the line being measured.
Activity Synthesis
Ask students to describe what they notice about the lengths they got when they used the two different measuring devices. Explain to students that one source of measurement error can be the precision level of their measuring device. Ask students, “Assuming the measurements to the nearest tenth of a centimeter are exact, how much error was in each measurement when you used the centimeter ruler?” (7 cm was 0.3 cm too long, 0.1 cm too long, and 0.3 cm too short, respectively.)
13.2
Activity
10 mins
Measuring a Soccer Field
Standards Alignment
Building On
Addressing
7.RP.A.3
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
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.
This activity uses the Compare and Connect math language routine to advance representing and conversing as students use mathematically precise language in discussion.
Launch
Keep students in the same groups. Tell students that a soccer field is 120 yards long, and ask them how they can use a 30-foot-long tape measure to measure that length. (Note the use of two different units of measure, yards and feet.) If not mentioned by students, suggest measuring off 30 feet, making a mark, measuring off another 30 feet, and so on. Ask the class if they would all get exactly the same answer by this method.
Give the students 3–4 minutes of quiet work time to calculate the amount of the error and the percent error, followed by time for partner discussion. Then hold a whole-class discussion.
Select work from students with different strategies, such as those described in the Activity Narrative, to share later. Look for a mixture of correct and incorrect solutions.
Action and Expression: Develop Expression and Communication. Invite students to demonstrate their ideas for the last question using a physical ruler as a substitute for Han’s tape measure. Supports accessibility for: Language, Visual-Spatial Processing
Activity
None
Student Task Statement
A soccer field is 120 yards long. Han uses a 30-foot-long tape measure to measure the length of the field and gets a measurement of 376 feet 6 inches.
What is the amount of the error?
Express the error as a percentage of the actual length of the field. Explain or show your reasoning.
What are some possible causes for this error?
Student Response
Activity Synthesis
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%.
13.3
Activity
20 mins
Measuring Your Classroom
Standards Alignment
Building On
Addressing
7.RP.A.3
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
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.
This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.
Launch
Keep students in the same groups. Make sure that students still have access to the two rulers from the Warm-up of this lesson.
Assign each group three objects to measure. Explain that after they have measured their objects, you will tell them the actual lengths of the objects so that they can complete the last two columns of the table. Give students 5–6 minutes of partner work time, then follow with a whole-class discussion.
Use Collect and Display to direct attention to words collected and displayed from an earlier lesson. Collect the language students use to describe the measurement error, and express it as a percentage of the actual length. Display words and phrases such as “measurement error,” “measured length,” “actual length,” “more precise,” “closer to the actual value,” and “smaller markings on the ruler.”
Representation: Access for Perception. Begin by enacting a physical demonstration of measuring an item with both the centimeter ruler and millimeter ruler to support understanding of the context. Supports accessibility for: Conceptual Processing, Language
Activity
None
Student Task Statement
Your teacher will tell you which three items to measure. Keep using the paper rulers from the earlier activity.
Between you and your partner, decide who will use which ruler.
Measure the three items assigned by your teacher, and record your measurements in the first column of the appropriate table.
Using the cm ruler:
item
measured
length (cm)
actual
length (cm)
difference
percentage
Using the mm ruler:
item
measured
length (cm)
actual
length (cm)
difference
percentage
After you finish measuring the items, share your data with your partner. Next, ask your teacher for the actual lengths.
Calculate the difference between your measurements and the actual lengths in both tables.
For each difference, what percentage of the actual length is this amount? Record your answers in the last column of the tables.
What do you notice about the percentages?
Student Response
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Building on Student Thinking
Activity Synthesis
There are two desired outcomes of this activity:
To develop a procedure that makes sense to students for computing percent error
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.
Lesson Synthesis
Share with students, “Today we saw how percentages can be used to describe measurement error.”
To review these concepts, consider asking:
“What is measurement error?” (It is the difference between a measurement of an object and its actual quantity.)
“What causes measurement error?” (It may exist due to human error in using a measurement tool or because the tool itself is not precise.)
“How can we minimize the amount of error?” (Use precision tools, and use them carefully.)
“How can measurement error be expressed as a percentage?” (Divide the measurement error by the actual quantity.)
Student Lesson Summary
When we use a ruler or measuring tape to measure a length, we can get a measurement that is different from the actual length. This could be because we positioned the ruler incorrectly, or it could be because the ruler is not very precise. There is always at least a small difference between the actual length and a measured length, even if it is a microscopic difference!
Here are two rulers with different markings.
Two rulers, of equal length, each measuring 19 centimeters long. The top ruler has centimeter markings only. The bottom ruler has centimeter markings with 10 milimeter markings in between each centimeter marking.
The second ruler is marked in millimeters, so it is easier to get a measurement to the nearest tenth of a centimeter with this ruler than with the first. For example, a line that is actually 6.2 cm long might be measured to be 6 cm long by the first ruler because we measure to the nearest centimeter.
The measurement error is the positive difference between the measurement and the actual value. Measurement error is often expressed as a percentage of the actual value. We always use a positive number to express measurement error and, when appropriate, use words to describe whether the measurement is greater than or less than the actual value.
For example, if we get 6 cm when we measure a line that is actually 6.2 cm long, then the measurement error is 0.2 cm, or about 3.2%, because .
Have feedback on the curriculum?
Help us improve by sharing suggestions or reporting issues.
Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.
Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
