# Lesson 9

Measurement Error (Part 2)

## 9.1: Measurement Error for Area (20 minutes)

### Optional activity

This activity examines how measurement errors behave when quantities are multiplied. In other words, if I have a measurement $$m$$ with a maximum error of 5% and a measurement $$n$$ with a maximum error of 5%, what percent error can $$m \boldcdot n$$ have?

Monitor for students who use different methods to solve the problem, such as trying out sample numbers or using expressions with variables.

### Launch

If desired, suggest that students try out several different sample numbers for the length and width of the rectangle, calculate the maximum percent error, and look for a pattern. Give students 4–5 minutes of quiet work time, followed by time to discuss their work with their partner and make revisions, followed by whole-class discussion.

Conversing, Representing: MLR3 Clarify, Critique, Correct. Present an incorrect response that reflects a possible misunderstanding from the class such as: “Since both length and width have maximum errors of 5%, I multiplied 5% by 5% to get 25%, because area is length times width.” Prompt pairs to clarify and then critique the incorrect response, and then write a correct version. This provides students with an opportunity to evaluate, and improve on, the written mathematical arguments of others.
Design Principle(s): Maximize meta-awareness; Support sense-making

### Student Facing

Imagine that you measure the length and width of a rectangle and you know the measurements are accurate within 5% of the actual measurements. If you use your measurements to find the area, what is the maximum percent error for the area of the rectangle?

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

Students may think that the maximum error possible for the area is 5% because both the length and width are within 5% of the actual values. Encourage these students to make calculations of the biggest and smallest possible length and the biggest and smallest possible width. Then have them make calculations for the biggest and smallest possible area.

### Activity Synthesis

Have students trade papers with a partner and check their work.

Invite students to share their solutions, especially those who looked for a pattern or used variables. Consider discussing questions like these:

• “Did you calculate the maximum percent error for any specific sample measurements? What did you find?” (The maximum percent error for the largest and smallest possible values were not the same: 9.75% and 10.25%.)
• “How do you know that this pattern is true for any possible length and width of the rectangle?” (I used variables to express the unknown measurements.)
• “How could you use variables to help solve this problem?” (I can use variables to represent the length and width of the rectangle and write expressions in terms of these variables to represent the largest and smallest possible areas.)

## 9.2: Measurement Error for Volume (25 minutes)

### Optional activity

This challenging activity examines how measurement errors behave when 3 quantities are multiplied (versus 2 quantities in the previous activity). In other words, if I have measurements $$a$$, $$b$$, and $$c$$ each with a maximum error of 5%, what percent error can $$a \boldcdot b \boldcdot c$$ have? The arithmetic and algebraic demands of this task are high because students take a product of three quantities that each have a maximum percent error of 5%.

### Launch

Arrange students in groups of 2. Provide access to calculators. Make sure students realize that the first question gives the measured values, not the actual values, for each dimension.

Give students 10 minutes to discuss with their partners, followed by whole-class discussion.

Conversing: MLR8 Discussion Supports. Before students determine the maximum measurement error for the volume of a prism, invite pairs to make a prediction and justification. Use a sentence frame such as: “In a prism, if each dimension has a 5% maximum measurement error, we predict the volume’s maximum percentage error is ___ because….”. This will help students use the mathematical language of justifications to begin reasoning about the measurement error for volume.
Design Principle(s): Cultivate conversation; Support sense-making

### Student Facing

1. The length, width, and height of a rectangular prism were measured to be 10 cm, 12 cm, and 25 cm. Assuming that these measurements are accurate to the nearest cm, what is the largest percent error possible for:

1. each of the dimensions?
2. the volume of the prism?

2. If the length, width, and height of a right rectangular prism have a maximum percent error of 1%, what is the largest percent error possible for the volume of the prism?

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

Students may think that the maximum error possible for the volume is 1% because the length, width, and height are within 1% of the actual values. Encourage these students to make calculations of the biggest and smallest possible length, width, and height. Then ask them to make calculations for the biggest and smallest possible volumes.

### Activity Synthesis

Some discussion points include:

• The first problem gives measurements and errors (but no percent error), while the second problem gives no measurements but does give the percent error. This makes the calculations notably different for the two problems.
• In the first problem, we are given measurements and the possible size of error. We need to find the greatest percent error and, as we have seen in other cases, this happens for the smallest possible value of the measurement. If we were to find the percent error of each measurement, we would find that the error for the volume is a larger percent error than for any of the individual measurements.
• In the second problem, we are given the maximum possible percent error but no measurements, and we need to find the largest possible error for the volume, that is, for the product of the three unknown measurements. The greatest percent error possible for the volume occurs when the measured value is as large as possible.
• A unifying feature in these two problems is that we notice that the largest percent error occurs when the actual measurements are smaller than the measured values, as much smaller as possible.
Representation: Internalize Comprehension. Use color and annotations to illustrate connections between representations. As students share their diagrams and reasoning, use color and annotations to scribe their thinking on a display of each problem so that it is visible for all students.
Supports accessibility for: Visual-spatial processing; Conceptual processing