# Lesson 11

Error Intervals

## 11.1: A Lot of Iron Ore (10 minutes)

### Warm-up

The purpose of this warm-up is to begin to understand the idea that a maximum percent error defines an interval of values that a quantity can lie within. Students are asked to give possible readings on a scale that has a possible error of up to 1%. Different answers are possible.

### Launch

Give students 3 minutes quiet work time, followed by whole-class discussion.

### Student Facing

An industrial scale is guaranteed by the manufacturer to have a percent error of no more than 1%. What is a possible reading on the scale if you put 500 kilograms of iron ore on it?

### Activity Synthesis

Draw a blank number line and then put three tick marks in the middle and label them 490, 500, 510. Poll the class for possible measurements, and plot each on the number line. Some students may give answers outside the error interval. Record those with the others and flag them mentally for discussion. Make sure everyone agrees with all of the possible measurements. Students might limit their answers to the extreme values 495 and 505, which have an error of exactly 1%. Be sure to solicit other answers if those are the only two offered.

Summarize the results using the greatest and least possible measurements (495 and 505, respectively) and point out how all of the possible measurements fall between these two values.

## 11.2: Saw Mill (10 minutes)

### Optional activity

This activity uses a quality control situation to work with percent error. It is very common that products in a factory are checked to make sure that they meet certain specifications. In this case, boards are cut to a specific length. If they are too long or too short they are rejected. Students should make sense of the decision to be based on percent error and not on measurement error. If we know the correct length and an acceptable percent error, then we can find out which lengths are acceptable and which should be rejected.

### Launch

Arrange students in groups of 2. Allow students 3--5 minutes of quiet work time followed by partner and whole-class discussion.

Representation: Internalize Comprehension. Activate or supply background knowledge of the meaning of percent error and how to convert a percentage to its decimal form. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Design Principle(s): Support sense-making

### Student Facing

1. A saw mill cuts boards that are 16 ft long. After they are cut, the boards are inspected and rejected if the length has a percent error of 1.5% or more.

1. List some board lengths that should be accepted.
2. List some board lengths that should be rejected.
2. The saw mill also cuts boards that are 10, 12, and 14 feet long. An inspector rejects a board that was 2.3 inches too long. What was the intended length of the board?

### Activity Synthesis

The purpose of the discussion is for students to understand why a tolerance based on percent error may be acceptable.

• "A 16 foot board is also 192 inches long. What is the maximum number of inches allowed for an acceptable board of this length?" (2.88 inches)
• "Why should the mill accept boards that are longer or shorter than 16 feet exactly?" (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 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.)

## 11.3: Info Gap: Quality Control (20 minutes)

### Optional activity

This info gap activity uses a quality control situation working with percent error. It is very common that products in a factory are checked to make sure that they meet certain specifications. In this case the odometer of a car is tested and the amount of liquid in a bottle that is automatically filled is checked. It makes sense that the decision should be based on percent error and not on absolute error.

The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of the cards for reference and planning:

### Launch

Provide access to calculators. Tell students they will continue to work with percent errors in realistic scenarios. Explain the Info Gap structure and consider demonstrating the protocol if students are unfamiliar with it. There are step-by-step instructions in the student task statement.

Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After you review their work on the first problem, give them the cards for a second problem and instruct them to switch roles.

Engagement: Develop Effort and Persistence. Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their own words. Keep the display of directions visible throughout the activity.
Supports accessibility for: Memory; Organization
Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary to solve problems involving percent error. Display questions or question starters for students who need a starting point such as: “Can you tell me . . . (specific piece of information)”, and “Why do you need to know . . . (that piece of information)?"
Design Principle(s): Cultivate Conversation

### Student Facing

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the problem card:

3. Explain how you are using the information to solve the problem.

Continue to ask questions until you have enough information to solve the problem.

4. Share the problem card and solve the problem independently.

If your teacher gives you the data card:

2. Ask your partner “What specific information do you need?” and wait for them to ask for information.

If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information.

3. Before sharing the information, ask “Why do you need that information?” Listen to your partner’s reasoning and ask clarifying questions.

4. Read the problem card and solve the problem independently.

5. Share the data card and discuss your reasoning.

### Activity Synthesis

The purpose of the discussion is to recognize what information is needed when dealing with tolerances based on percent error.

After students have completed their work, share the correct answers and ask students to discuss the different ways they solved this problem. Some guiding questions:

• "What information did you and your partner have to figure out?"
• "What different calculations did you have to make for the two situations?"
• "Why might a car manufacturer not want to sell a car that shows a speed lower than what the car is actually going?" (It could be dangerous for the driver if they think they are going slower than they actually are. They are also more prone to getting speeding tickets if they think they are going under the speed limit based on the speedometer, but are actually going faster.)
• "Why might a juice manufacturer not want to have too much more juice in the bottle than it says on the label?" (If too many bottles are overfilled, the company may be losing money by giving away extra juice without charging the customer more.)
Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary to solve problems involving equivalent ratios.
Design Principle(s): Cultivate Conversation

## Lesson Synthesis

### Lesson Synthesis

This lesson was about error intervals.

• “Why might companies accept a percent error in measurement for their products?” (It may be very difficult to get the measurements exact in an efficient way, so some small error is allowed.)
• “Why might a percent error be used rather than an absolute measurement error?” (The same absolute measurement error may have a much greater impact for smaller measurements than for larger measurements. A percent error allows the company to use a single rule for multiple measurements rather than writing a new rule for each measurement.)
• “How do you find the range of values that are acceptable when you know the target measurement and a percent error that is allowed?” (Multiply the target measurement by the percent error, then add and subtract that value from the target measurement to get the two endpoints for the interval that is allowed.)

## 11.4: Cool-down - An Angler's Dilemma (5 minutes)

### Cool-Down

Percent error is often used to express a range of possible values. For example, if a box of cereal is guaranteed to have 750 grams of cereal, with a margin of error of less than 5%, what are possible values for the actual number of grams of cereal in the box? The error could be as large as $$(0.05) \boldcdot 750 = 37.5$$ and could be either above or below than the correct amount.