Lesson 23

The purpose of this number talk is to help students multiply and divide decimal numbers by 100 in preparation for their work with percentages later in the lesson. 

Display one problem at a time. Give students 30 seconds of quiet think time for each problem, and ask them to give a signal when they have an answer and a strategy. Keep all previous problems displayed throughout the task. 

Select a couple of students to share their answer and strategies for each problem. Record and display their explanations for all to see. After evaluating all four expressions, ask students:

• How is multiplying by \(\frac{1}{100}\) related to division?
• What is important to remember about dividing a one digit number by 100?

To involve more students in the conversation, consider asking as the students share their ideas:

• Who can restate ___’s reasoning in a different way?
• Did anyone solve the problem the same way but would explain it differently?
• Did anyone solve the problem in a different way?
• Does anyone want to add on to _____’s strategy?
• Do you agree or disagree? Why?

In this info gap activity, students find both \(A\) and \(C\) (where \(A\%\) of \(B\) is \(C\)) in the context of buying a music device. The value of \(B\) is different in each of the two questions about the music device, so students who choose to draw diagrams or tables need to draw two. When answering the second question—expressing \$24 as a percentage of \$25—students may notice that drawing a complete double number line diagram with all 25 tick marks is rather time consuming. Encourage students to look for and make use of any noticeable structure (MP7) or pattern (MP8) to help them solve more efficiently.

Some students may, for example, see that if \$25 corresponds to 100%, then each dollar—and thus each tick mark—is \(100 \div 25\) or 4. They can then bypass drawing the rest of the tick marks and simply multiply \(4 \boldcdot 24\) to obtain 96. Some may notice that \$24 is \$1 away from \$25 and simply subtract the corresponding percentage from 100% (\(100 - 4 = 96\)). A table provides another efficient structure for reasoning about this. Monitor for students who use different strategies.

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:

Note: If time is short, the second set of cards can be considered optional. It would be better for students to thoroughly understand one of these problems than to rush through both of them with less understanding.

Arrange students in groups of 2. In each group, distribute the first problem card to one student and a data card to the other student. After debriefing on the first problem, distribute the cards for the second problem, in which students switch roles.

Select students with different strategies to share their approaches to the first question, starting with less efficient methods and ending with more efficient methods. Then, ask the class to predict how the same strategies might be used to solve the second question, and how the second problem could be solved more quickly.

In general, \(P\%\) of something is \(\frac{P}{100}\) times that thing. The purpose of this activity is to make this explicit. In this activity, students are asked to find 80% of several different values, generalize their process, and express their generalization in different ways. As they make repeated calculations, students look for and express regularity in their work (MP8) and see more explicitly that \(P\%\) of a number is \(\frac{P}{100}\) times that number. Once they arrive at one or more generalizations, students practice articulating why each generalization always works (MP3).

Give students quiet think time to complete the activity and then time to share their explanation with a partner.

Make sure students have the correct values in the table for the second question. Then, discuss the expressions, starting with \(\displaystyle \frac{80}{100} \boldcdot x.\)

Guide students to see that if we know the value of 100%, dividing that value by 100 (or equivalently, multiplying it by \(\frac{1}{100}\)) tells us the corresponding value for 1%. We can then multiply that value by the desired percentage. Consider using a table to organize this argument, as shown below:

After everyone understands where this expression comes from, ask students to discuss the remaining expressions with a partner. This is a good opportunity for just-in-time review on equivalent fractions, if needed.