Lesson 15

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.

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

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.

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.

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

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

Consider asking these discussion questions:

• "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.)

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:

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.

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.)