This activity examines how measurement errors behave when they are added together. In other words, if there is a measurement with a maximum error of 1% and a measurement with a maximum error of 1%, what percent error can have? In addition to examining accuracy of measurements carefully (MP6), students work through examples and look for patterns (MP8) in order to hypothesize, and eventually show, how percent error behaves when measurements with error are added to one another.

Monitor for students who look for patterns, recognize the usefulness of the distributive property, or formulate the problem abstractly with variables.

The goal of this discussion is for students to generalize from their specific examples of measurements to understand the general pattern and express it algebraically.

Poll the class on the measurements they tried and the maximum percent error they calculated. Invite students to share any patterns they noticed, especially students who recognized the usefulness of the distributive property for making sense of the general pattern.

Guide students to use variables to talk about the patterns more generally.

• If a board is supposed to have length with a maximum percent error of 5%, then the shortest it could be is and the longest it could be is .
• If another board is supposed to have length , it could be between and .
• When the boards are laid end-to-end, the shortest the total length could be is , which is equivalent to .
• The longest the total length could be is , or .
• Because of the distributive property, it is seen that the maximum percent error is still 5% after the board lengths are added together.

One interesting point to make, if students have also done the previous activity about measuring pencils, is that the sum of the board lengths could be measured with a lower percent error than each individual board (assuming the tape measure is long enough), just like an error of 1 mm was a smaller percentage of the length of the longer pencil.

This activity examines how measurement errors behave when quantities are multiplied. In other words, if there is a measurement with a maximum error of 5% and a measurement with a maximum error of 5%, what percent error can have? Students reason abstractly and quantitatively when they decide how to represent this situation and interpret the meaning of their solution (MP2).

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

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). 

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response by correcting errors, clarifying meaning, and adding details. 

• Display this first draft:

“Both the length and width have maximum errors of 5%. I know that area is length times width, so I multiplied 5% by 5% to get 25%.”

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 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.

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 , , and each with a maximum error of 5%, what percent error can 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%. Students repeat their process for each measurement and use this repetition to build their understanding of measurement errors (MP8).

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, measurements and the possible size of error are given. The task is to find the greatest percent error and, as seen in other cases, this happens for the smallest possible value of the measurement. If one were asked to find the percent error of each measurement, they would find that the error for the volume is a larger percent error than for any of the individual measurements.
• In the second problem, the maximum possible percent error is given, but no measurements, and the task is 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 the largest percent errors occur when the actual measurements are smaller than the measured values.