This Math Talk focuses on finding the percentage that one number is of another number. It encourages students to think about multiplicative comparisons and to rely on relationships between the dividend and divisor to mentally solve problems.
Next, these expressions are used as examples to introduce the concept of percent error. Students reason that an error that is 50% of the correct value is more problematic than an error that is 0.5% of the correct value.
As students compare the previous expression to the next expression and determine how to scale the percentage, students need to look for and make use of structure (MP7).
Launch
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies, and record and display their responses for all to see.
Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with access to sticky notes or mini whiteboards. Supports accessibility for: Memory, Organization
Activity
None
Find each percentage mentally.
3 is what percent of 6?
3 is what percent of 12?
3 is what percent of 120?
3 is what percent of 600?
Student Response
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Building on Student Thinking
If students try to figure out exact answers, encourage them to think about numbers that are close to the numbers in the problem in order to estimate the percentage for each question.
Activity Synthesis
To involve more students in the conversation, consider asking:
“Who can restate ’s reasoning in a different way?”
“Did anyone use the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”
After discussing students’ strategies, use these calculations as examples to introduce percent error, where an error is expressed as a percentage of the correct amount.
“If you bought a package that was supposed to contain 6 items, but it was missing 3 of them, how much would you care?” (3 is 50% of 6, so this is a significant error.)
“If you bought a package that was supposed to contain 600 items, but it was missing 3 of them, how much would you care?” (3 is only 0.5% of 600, so this is a relatively minor error.)
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I because . . . .” or “I noticed , so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. Advances: Speaking, Representing
11.2
Activity
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
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%.
11.3
Activity
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 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).
This activity uses the Critique, Correct, Clarify math language routine to advance representing and conversing as students critique and revise mathematical arguments.
Launch
Before students begin working, read the first question aloud. Ask several students to explain in their own words what information is given in the problem, and what it is asking them to find. The plant is supposed to get cup of water, but it is getting 25% more than that. We can think of this as cup increased by 25%.
Arrange students in groups of 2. Give students 3–5 minutes of quiet work time followed by time for partner discussion. Then hold a whole-class discussion.
Representation: Internalize Comprehension. Represent the same information through different modalities by using a tape diagram, double number line diagram, or a table to organize the information. Supports accessibility for: Conceptual Processing, Visual-Spatial Processing
Activity Synthesis
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 .
11.4
Activity
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 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).
This activity is optional in this course.
Launch
Arrange students in groups of 2. Give students 4–5 minutes of quiet work time followed by time for partner discussion. Then hold a whole-class discussion.
MLR6 Three Reads. Keep books or devices closed. Display only the problem stem, without revealing the questions. “We are going to read this situation 3 times.”
After the 1st read: “Tell your partner what this situation is about.”
After the 2nd read: “List the quantities. What can be counted or measured?”
For the 3rd read: Reveal and read the questions “List some board lengths that should be accepted. List some board lengths that should be rejected.” Ask, “What are some ways we might get started on this?”
Advances: Reading, Representing
Representation: Internalize Comprehension. Activate or supply background knowledge. Provide a paper which includes the meaning of percent error and how to convert a percentage to its decimal form for students to use as a reference. Supports accessibility for: Memory, Organization
Activity
None
A saw mill cuts boards that are 16 feet long. After they are cut, the boards are inspected and rejected if the length has a percent error of 1.5% or more.
List some board lengths that should be accepted.
List some board lengths that should be rejected.
The saw mill also cuts boards that are 10, 12, and 14 feet long. An inspector rejects a board that is 2.3 inches too long. What was the intended length of the board?
Student Response
Activity Synthesis
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.)
Lesson Synthesis
Share with students, “Today we saw how percentages can be used to describe 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.)
“When working with percent error, what quantity in the problem corresponds to 100%?” (The correct value corresponds to 100%.)
“Why is it helpful to express an error as a percentage of the correct value?” (It shows whether the error is relatively large or small in the context of the situation.)
“How is percent error similar to percent increase or percent decrease? How are they different?” (If the incorrect value is higher than the correct value, it is similar to percent increase. If the incorrect value is lower, it is similar to percent decrease. Instead of comparing before and after a change, percent error is comparing an error to the correct amount.)
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 .
Percent error can be used to describe any situation where there is a correct value and an incorrect value, and we want to describe the relative difference between them. For example, if a milk carton is supposed to contain 16 fluid ounces, and it only contains 15 fluid ounces:
The measurement error is 1 oz.
The percent error is 6.25% because .
We can also use percent error when talking about estimates. For example, a teacher estimates there are about 600 students at their school. If there are actually 625 students, then the percent error for this estimate is 4%, because and .
Standards Alignment
Building On
6.RP.A.3.c
Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
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.
If students fail to see the need for converting units of measure, ask, “How many feet are in 120 yards? How many inches?”
Activity
None
Instructions to care for a plant say to water it with cup of water every day. The plant has been getting 25% too much water. How much water has the plant been getting?
The pressure on a bicycle tire is 63 psi. This is 5% higher than what the manual says is the correct pressure. What is the correct pressure?
The crowd at a sporting event is estimated to be 3,000 people. The exact attendance is 2,486 people. What is the percent error?
Student Response
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Building on Student Thinking
Students might struggle with figuring out how to calculate how much water the plant has been getting. Ask students, “How much more water has the plant been getting? How do you calculate that total?”
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Building on Student Thinking
For the last question, some students may not realize that the board lengths were given in feet, while the amount of the error was given in inches. Consider asking “How can we convert these lengths so that they have the same units?”
