Lesson 12

In this warm-up, students are reminded of the tape diagram method for understanding parts of a whole. The tape diagrams are used in the context of tips, taxes, and discounts.

Students in groups of 2. Allow students 2 minutes quiet work time followed by partner then whole-class discussion.

The purpose of the discussion is for students to recognize that a tape diagram can be useful for working with percentages as part of a whole.

Consider asking these discussion questions:

• "The tax on the car is what fraction of the car price before tax was added on?" (One fourth)
• "With the tip added on, how is the length of the entire bar related to the length of the bar that just represents the food cost?" (It is 1.2 times as long.)
• "How does the value \(\frac{2}{3}\) relate to the last tape diagram?" (It represents the fraction of the original cost of the shirt that is the price of the shirt after the discount.)

In this activity, students continue to practice finding percentages from dollar amounts including commission, tip, and markdown. Some of the questions require multiple steps to solve for the percentage needed.

As students work, monitor for students who take different approaches to solving the multiple steps involved in these problems. For example, some students may subtract the discounted price from the original price to find the amount that the item has been marked down before finding the percentage of markdown. Other students may find the percentage of the original price used for the discounted price and then subtract that from 100%.

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

The purpose of the discussion is for students to share methods for finding percentages in problems that require multiple steps.

Select the previously identified students to share their different methods for solving the last 2 questions. Sequence so strategies using diagrams come first, and those using equations come second.

The purpose of this info gap activity is for students to identify the essential information needed to determine the total savings after various discounts are applied to different items.

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 percentages and their similarities to proportional relationships. 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.

Students might fail to notice that Elena and Andre buy multiple cans of tennis balls and packages of socks, respectively. Ask students to figure out how much 2 packages of socks (or 3 cans of tennis balls) will cost.

If students automatically give the 15% discount on all of Elena's purchases, ask students which of Elena's items fall under the discount.

Students might apply the discount after the adding in the sales tax.  Remind students that the discount gets applied to the subtotal before the tax is calculated.

Some students may include the sales tax when calculating the percentage of Andre's savings. Remind them that the problem specifies "before tax."

The purpose of the discussion is for students to recognize what information may be needed to solve problems involving percentages for prices of items.

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?"
• "How did you determine the cost of Elena's tennis racket?" (Multiply the original cost by 0.85 or multiply by 0.15 and subtract from the original cost.)
• "How did you determine the total cost after tax for Elena's purchases?" (Multiply the total by 1.085 or multiply by 0.085 and add to the original cost.)
• "What different calculations did you have to make for Andre and Elena’s situations?"
• "Was there information given that you did not need to use?"