# Lesson 14

Percent Error

## 14.1: Number Talk: Estimating a Percentage of a Number (5 minutes)

### Warm-up

The purpose of this number talk is for students to reason about a percentage of a number based on percentages they already know or could easily find. The percentages and numbers were purposefully chosen so that it would be cumbersome to calculate the exact answer and encourage making an estimate. During the whole-class discussion, highlight the percentages students found helpful and ask them to explain how they used these percentages. For example, if a student is estimating 9% of 38 and says, "I know 10% of 38 is 3.8..." ask the student to explain how they found 10% of 38.

### Launch

Display each problem one at a time. Give students 30 seconds of quiet think time followed by a whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Estimate.

25% of 15.8

9% of 38

1.2% of 127

0.53% of 6

0.06% of 202

### Anticipated Misconceptions

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

Ask students to share their responses for each question. Record and display student responses for all to see. After each response, ask students:

• "What benchmark percentages do you find it helpful to think about, when estimating?"
• "Is your estimate more or less than the actual answer? How do you know?"
Speaking: 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.
Design Principle(s): Optimize output (for explanation)

## 14.2: Plants, Bicycles, and Crowds (10 minutes)

### Activity

The purpose of this activity is to give students practice using the language of percent error. The problems here are similar in structure to percent increase or decrease problems, but the language is different. Students may need some help interpreting the language used for percent error and drawing parallels to the language used for percentage increase and decrease. Students should use similar strategies they used to calculate percentage increase or decrease. This activity includes one of each type of problem:

• finding the erroneous amount given the correct amount and the percent error
• finding the correct amount given the erroneous amount and the percent error
• finding the percent error given the erroneous amount and the correct amount

Students may need help understanding that a different approach is needed for each question. As students work, monitor for those who use strategies similar to those used for percentage increase and decrease, and ask them to share during the whole-class discussion.

### Launch

Before students begin working, read the first question aloud. It is helpful to tie the language used in the task to the phrase "percentage increase." 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 $$\frac34$$ cup of water, but it is getting 25% more than that. We can think of this as $$\frac34$$ cup increased by 25%.

Arrange students in groups of 2. Give students 3–5 minutes of quiet work time. Partner then by whole-class discussion.

Representation: Internalize Comprehension. Activate or supply background knowledge. Represent the same information through different modalities. If students are unsure where to begin, suggest that they draw a double number line to help organize the information provided.
Supports accessibility for: Conceptual processing; Visual-spatial processing

### Student Facing

1. Instructions to care for a plant say to water it with $$\frac34$$ cup of water every day. The plant has been getting 25% too much water. How much water has the plant been getting?
2. 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?
3. 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 Facing

#### Are you ready for more?

A micrometer is an instrument that can measure lengths to the nearest micron (a micron is a millionth of a meter). Would this instrument be useful for measuring any of the following things? If so, what would the largest percent error be?

1. The thickness of an eyelash, which is typically about 0.1 millimeters.

2. The diameter of a red blood cell, which is typically about 8 microns.

3. The diameter of a hydrogen atom, which is about 100 picometers (a picometer is a trillionth of a meter).

### Anticipated Misconceptions

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?"

### Activity Synthesis

Select students who used these strategies for each problem to share:

• Calculate a quarter of $$\frac34$$ and add it to $$\frac34$$.
• Use 1.05 to divide into 63.
• Calculate percent error using $$\frac{514}{2,486}$$.

After each student has shared, ask the class if they agree or disagree or if they had a different way to calculate the solution. If students use strategies similar to ones they did calculating percentage increase or decrease, ask students if they see a connection. If no student brings it up, ask students how the solution strategies here are similar to the ones used with percentage increase and decrease.

Speaking: MLR8 Discussion Supports. As students describe their strategies for calculating the erroneous amount, correct amount, and percent error, revoice student ideas to demonstrate mathematical language use. Press for details in students’ explanations by requesting that students challenge an idea, elaborate on an idea, or give an example. This will help students to produce and make sense of of the language needed to communicate their own ideas.
Design Principle(s): Support sense-making; Optimize output (for explanation)

## 14.3: Measuring in the Heat (10 minutes)

### Activity

In this activity students use what they have learned about percent error in a multi-step problem.

Monitor for students who multiply 0.0000064 by 50 to answer the second part of the problem rather than using the calculation from the first part of the problem. These students should be asked to share during the whole-class discussion.

### Launch

Display the image of the metal measuring tape for all to see. Ask if any students have used a tool like this before, and for what purpose. Tell them that many measuring tapes like this are made out of metal, and that some metals expand or contract slightly at warmer or colder temperatures.

In this problem, a metal measuring tape gets 0.00064% longer for every degree over $$50^\circ$$ Fahrenheit. Ask students what would happen if a measuring tape was used to measure 10 feet, and then got 0.00064% longer. How much longer is that? Can they show the difference between two fingers? The difference would be very, very small, only 0.0000064 feet (about the width of a hair) . . . barely perceptible! For most uses, this difference wouldn't matter, but if someone needed a very, very precise measurement, they would want to know about it.

Arrange students in groups of 2. Give students 5 minutes of quiet work time. After 5 minutes, give students 3 minutes to discuss with a partner the ways they approached this problem.

Action and Expression: Internalize Executive Functions. To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Check to make sure students have converted the percentage $$0.00064%$$ to its decimal form $$0.0000064$$ prior to calculating how much longer the measuring tape is than its correct length of 30 feet.
Supports accessibility for: Memory; Organization
Representing, Speaking: MLR1 Stronger and Clearer Each Time. Use this routine to give students a structured opportunity to refine their explanation of their strategy for calculating the added length of the measuring tape and the percent error. Ask each student to meet with 2-3 other partners in a row for feedback. Provide listeners with prompts for feedback that will help teams strengthen their ideas and clarify their language. For example, “What did you do first?”, “How did the example help you?”, “How did you use 0.00064%?”, etc. Students can borrow ideas and language from each partner to strengthen their final response.
Design Principle(s): Optimize output (for explanation)

### Student Facing

A metal measuring tape expands when the temperature goes above $$50^\circ\text{F}$$. For every degree Fahrenheit above 50, its length increases by 0.00064%.

1. The temperature is 100 degrees Fahrenheit. How much longer is a 30-foot measuring tape than its correct length?

2. What is the percent error?

### Anticipated Misconceptions

If students struggle with how to calculate the length increase, ask students

• "How many degrees over $$50^\circ$$ is $$100^\circ$$?"
• "How do you calculate the length increase knowing that there is a $$50^\circ$$ increase?"
• "What is the actual length of the tape measure?"

### Activity Synthesis

Select previously identified students to share how they obtained their answer to the second part of the activity.  Ask students who multiplied 0.0000064 by 50 why that method works and if there is another method (using the answer from the first part of the activity). Encourage students to make the connection between the two methods.

Ask students how big the error is in inches, and to show approximately how big they think it is with their thumb and forefinger. 0.0096 feet is almost 3 millimeters. When measuring 30 feet, this may not seem like very much, but the importance of the error may depend on what is being measured. The length of a car may not be as important as a precise scientific instrument to measure the speed of light, for example.

## Lesson Synthesis

### Lesson Synthesis

Students should feel confident calculating percent error given different contexts and information. Ask students:

• “What strategies did we use to solve percent error problems?” (diagrams, tables, equations)
• “How are these strategies similar to the ones we used while solving percent increase/decrease problems?” (the same)

## Student Lesson Summary

### Student Facing

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, and
• the percent error is 6.25% because $$1 \div 16 = 0.0625$$.

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 was 4%, because $$625 - 600 = 25$$ and $$25 \div 625 = 0.04$$.