Lesson 10

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.

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

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.

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?"

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.

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.

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?"

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.

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.

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.

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?"

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.