Lesson 11
Percentage Contexts
11.1: Leaving a Tip (5 minutes)
Warmup
The purpose of this warmup is to help students connect their current work with percentage contexts to their prior work on percent increase and efficient ways of finding percent increase.
Launch
Consider telling students that these questions may have more than one correct answer. Students in groups of 2. 2 minutes of quiet think time followed by partner and then wholeclass discussion.
Student Facing
Which of these expressions represent a 15% tip on a $20 meal? Which represent the total bill?
\(15 \boldcdot 20\)
\(20 + 0.15 \boldcdot 20\)
\(1.15 \boldcdot 20\)
\(\frac{15}{100} \boldcdot 20\)
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
For each expression, ask a few students to explain whether they think it represents: the total bill, the tip, or neither. For each expression, select a student to explain their reasoning.
11.2: A Car Dealership (10 minutes)
Activity
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.
Launch
Tell students that a markup is a percentage that businesses often add to the price of an item they sell, and a markdown 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 markdowns discounts.
Provide access to calculators. Students in groups of 2. Give students 5 minutes of quiet work time, followed by partner then wholeclass discussion.
Design Principle(s): Support sensemaking
Student Facing
A car dealership pays a wholesale price of $12,000 to purchase a vehicle.

The car dealership wants to make a 32% profit.
 By how much will they mark up the price of the vehicle?
 After the markup, what is the retail price of the vehicle?
 During a special sales event, the dealership offers a 10% discount off of the retail price. After the discount, how much will a customer pay for this vehicle?
Student Response
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Student Facing
Are you ready for more?
This car dealership pays the salesperson a bonus for selling the car equal to 6.5% of the sale price. How much commission did the salesperson lose when they decided to offer a 10% discount on the price of the car?
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
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.
Activity Synthesis
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.)
11.3: Commission at a Gym (10 minutes)
Activity
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.
Launch
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 wholeclass discussion.
Student Facing

For each gym membership sold, the gym keeps $42 and the employee who sold it gets $8. What is the commission the employee earned as a percentage of the total cost of the gym membership?

If an employee sells a family pass for $135, what is the amount of the commission they get to keep?
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
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.
Activity Synthesis
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.
Design Principle(s): Maximize metaawareness
11.4: Card Sort: Percentage Situations (10 minutes)
Optional activity
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.
Launch
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?
Supports accessibility for: Conceptual processing; Organization
Design Principle(s): Optimize output (for explanation)
Student Facing
Your teacher will give you a set of cards. Take turns with your partner matching a situation with a descriptor. For each match, explain your reasoning to your partner. If you disagree, work to reach an agreement.
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
Students should use the question at the bottom of the card to help them if they get stuck sorting the scenarios.
Activity Synthesis
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.
paid to:  how it works:  

sales tax  the government  added to the price of the item 
gratuity (tip)  the server  added to the cost of the meal 
interest  the lender (or account holder)  added to the balance of the loan, credit card, or bank account 
markup  the seller  added to the price of an item so the seller can make a profit 
markdown (discount)  the customer  subtracted from the price of an item to encourage the customer to buy it 
commission  the salesperson  subtracted from the payment the store collects 
Lesson Synthesis
Lesson Synthesis
In this lesson, we studied lots of different situations where people use percentages.
 “What are some situations in life in which people encounter percentages?”
 “Give examples of situations where you would encounter tax, tip, markup, markdown, commission.” (Lots of possible answers.)
 “When an item is marked down 10%, why does it make sense to multiply the price by 0.9?” (Since there is 10% off of the price, the new cost is 90% of the original.)
 “When an item is marked up 25%, why does it make sense to multiply the price by \(1.25\)?” (Since the item now costs 100% plus an extra 25%, the new item costs 1.25 times the original.)
11.5: Cooldown  The Cost of a Bike (5 minutes)
CoolDown
For access, consult one of our IM Certified Partners.
Student Lesson Summary
Student Facing
There are many everyday situations where a percentage of an amount of money is added to or subtracted from that amount, in order to be paid to some other person or organization:
goes to  how it works  

sales tax  the government  added to the price of the item 
gratuity (tip)  the server  added to the cost of the meal 
interest  the lender (or account holder)  added to the balance of the loan, credit card, or bank account 
markup  the seller  added to the price of an item so the seller can make a profit 
markdown (discount)  the customer  subtracted from the price of an item to encourage the customer to buy it 
commission  the salesperson  subtracted from the payment that is collected 
For example,
 If a restaurant bill is \$34 and the customer pays \$40, they left \$6 dollars as a tip for the server. That is 18% of $34, so they left an 18% tip. From the customer's perspective, we can think of this as an 18% increase of the restaurant bill.
 If a realtor helps a family sell their home for \$200,000 and earns a 3% commission, then the realtor makes \$6,000, because \((0.03) \boldcdot 200,\!000 = 6,\!000\), and the family gets \$194,000, because \(200,\!000  6,\!000 = 194,\!000\). From the family's perspective, we can think of this as a 3% decrease on the sale price of the home.