Lesson 8

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.)

The purpose of this activity is to introduce students to a context involving markups and markdowns or discounts, and to connect this to the work on percent increase and percent decrease they did earlier. The first question helps set the stage for students to see the connection to markups and percent increase. Look for students who solve the second question by finding 90% of the retail price, and highlight this approach in the discussion.

Tell students that a mark-up is a percentage that businesses often add to the price of an item they sell, and a mark-down is a percentage they take off of a given price. If helpful, review the meaning of wholesale (the price the dealership pays for the car) and retail price (the price the dealership charges to sell the car). Sometimes people call mark-downs discounts.

Provide access to calculators. Students in groups of 2. Give students 5 minutes of quiet work time, followed by partner then whole-class discussion.

It is important throughout that students attend to the meanings of particular words and remain clear on the meaning of the different values they find. For example, "wholesale price," "retail price," and "sale price" all refer to specific dollar amounts. Help students organize their work by labeling the different quantities they find or creating a graphic organizer.

For the first question, help students connect markups to percent increase.

Select students to share solutions to the second question. Highlight finding 90% of the retail price, and reinforce that a 10% discount is a 10% decrease.

Ask them to describe how they would find (but not actually find) . . . 

• "The retail price after a 12% markup?" (Multiply the retail price by 0.12, then add that answer to the retail price. Alternatively, multiply the retail price by 1.12.)
• "The price after a 24% discount?" (Multiply the retail price by 0.24, then subtract that answer from the retail price. Alternatively, multiply the retail price by 0.76.)

The purpose of this activity is to introduce students to the concept of a commission and to solve percentage problems in that context. Students continue to practice finding percentages of total prices in a new context of commission.

Monitor for students who use equations like \(c = r \boldcdot p\) where \(c\) is the commission, \(r\) represents the percentage of the total that goes to the employee, and \(p\) is the total price of the membership.

Tell students that a commission is the money a salesperson gets when they sell an item. It is usually used as an incentive for employees to try to sell more or higher priced items than they usually would. The commission is usually a percentage of the price of the item they sell.

Provide access to calculators. Students in groups of 2. Give students 2 minutes of quiet work time. Partner then whole-class discussion.

Students may find the percentage of an incorrect quantity. Ask them to state, in words, what they are finding a percentage of.

Students may not understand the first question. Tell them that a membership is sold for a certain price and the money is split with \$42 going to the gym and \$8 going to the employee.

Select students to share how they answered the questions.

During the discussion, draw attention to strategies for figuring out which operations to do with which numbers. In particular, strategies involving equations like \(c = r \boldcdot p\) where \(c\) is the commission, \(r\) represents the percentage of the total that goes to the employee, and \(p\) is the total price of the membership.

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?"

This activity gives students an opportunity to practice various vocabulary terms that come along with percentages. Students are asked to sort scenarios to different descriptors using the images, sentences or questions found on the scenario cards. The questions found on the scenario cards are intended to help students figure out which descriptor the scenario card belongs under.

As students work on the task, identify students that are using the vocabulary: tip, tax, gratuity, commission, markup/down, and discount. These students should be asked to share during the discussion.

Arrange students in groups of 2. Distribute the sorting cards, and explain that students will sort 8 scenarios into one of 6 categories. Demonstrate how students can take turns placing a scenario under a category and productive ways to disagree. Here are some questions they might find useful:

• Which category would you sort this under?
• What do you think this word means?
• What words can we use as clues about where to sort this card?

Students should use the question at the bottom of the card to help them if they get stuck sorting the scenarios.

Ask identified students to share which situations they sorted under each word. Ask them:

• "What made you decide to put these situations under this descriptor?"
• "Were there any situations that you were really unsure of? What made you decide on where to sort them?"

Consider asking some groups to order the situations from least to greatest in terms of the dollar amount of the increase of decrease and asking other groups to order them in terms of the percentage. Then, have them compare their results with a group that did the other ordering.

Answer students’ remaining questions about any of these contexts. Tell students there is a copy of this chart at the end of the lesson that they can use as a reference tool during future lessons. Allow them a space to take notes on their own to remember it or details from one of the activity examples.