Lesson 14

This number talk encourages students to rely on what they know about structure, patterns, decimal multiplication, and properties of operations to solve a problem mentally. Only two problems are given here so there is time to share many strategies and make connections between them.

Display one problem at a time. Give students 1 minute of quiet think time per problem, and ask students to give a signal when they have an answer and a strategy. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their explanations for all to see. If not mentioned by students, ask if or how the given factors impacted their strategy choice. To involve more students in the conversation, consider asking:

• 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 activity, students solve percentage problems in the context of shopping. Students consider how discounts described as percentages translate into reduced prices and the other way around. (In other words, they find \(A\) and \(C\) where \(A\%\) of \(B\) is \(C\)). In each problem, students need to first determine what value is associated with 100% and reason accordingly.

Students may choose to use double number lines or tables or simply reason without using a particular representation. For instance, to find 10% of $15 they may first find 50% by dividing $15 by 2, and then divide the resulting $7.50 by 5 to obtain 10% of $15. Those who have internalized the structure of percentages may be able to multiply or divide even more efficiently (e.g., \(15 \boldcdot \frac{1}{10} = 1.5\), or \(15 \div 10 = 1.5\)). These strategies will be investigated more in the next lesson. For now, encourage students to also explain their reasoning with a double number line or table.

As students work, identify students who use different strategies so that they can share later.

Display some coupons for all to see. Point out that some coupons specify amounts to be taken off in dollars (e.g., $5 off) and some specify percentages (e.g.,10% off). Tell students that they will solve a couple of shopping problems that involve discounts.

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

Since the first question asks students to find the dollar amount, some students may think that $6 is the answer to the second question and not realize that it is asking them to find the percentage. Also, some students may try to find the sale price on the first question and the percentage of the sale price on the second question, instead of the discount and the percentage of the discount. Encourage them to revisit the questions or clarify what the questions ask.

Select students who used different representations: first a tape diagram, then a double number line or a table (or both, time permitting). As students explain, illustrate and display those representations for all to see. If no students mention using a double number line or a table, demonstrate at least one of these methods. When discussing double number lines (or tables), ask students if the same double number line (or table) could be used to solve both problems and discuss why not. Emphasize the fact that two separate double number lines (or tables) are necessary because the value for 100% is different in each case.

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.