Lesson 11

Percentage Contexts

11.1: Leaving a Tip (5 minutes)

Warm-up

The purpose of this warm-up 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 whole-class 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

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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 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.

Reading, Writing: MLR6 Three Reads. Use this routine to support reading comprehension of this word problem, without solving it for students. In the first read, students read the problem with the goal of comprehending the situation (e.g., A car dealership bought a car. The dealership wants to make a profit. They need to decide what price the car should be.). If needed, discuss the meaning of unfamiliar terms at this time (e.g., profit, wholesale, retail price, commission, etc.). Use the second read to identify the important quantities by asking students what can be counted or measured (e.g., wholesale price, profit or mark-up, and retail price). In the third read, ask students to brainstorm possible mathematical solution strategies to complete the task. This will help students connect the language in the word problem and the reasoning needed to solve the problem while keeping the intended level of cognitive demand in the task.
Design Principle(s): Support sense-making​

Student Facing

A car dealership pays a wholesale price of $12,000 to purchase a vehicle.

  1. The car dealership wants to make a 32% profit.

    1. By how much will they mark up the price of the vehicle?
    2. After the markup, what is the retail price of the vehicle?
    Cropped image of new cars.
  2. 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 whole-class discussion.

Student Facing

  1. 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?

  2. 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.

Reading, Writing: MLR3 Clarify, Critique, Correct. Present an incorrect response to the second question that reflects a possible misunderstanding from the class. For example, “For the family membership of $135, the employee would keep $8.” Prompt students to identify the error (e.g., ask, “Do you agree with the author’s reasoning? Why or why not?”), and then write a correct version. This will help students to understand that the employee's commission is always a rate of 16% and not a flat amount of $8.
Design Principle(s): Maximize meta-awareness

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?
Representation: Internalize Comprehension. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity by beginning with fewer cards. For example, give students a subset of the cards to start with and introduce the remaining cards once students have completed their initial set of matches.
Supports accessibility for: Conceptual processing; Organization
Speaking: MLR8 Discussion Supports. Show central concepts multi-modally by using different types of sensory inputs: acting out scenarios or inviting students to do so, showing videos or images, using gestures, and talking about the context of what is happening. This will help students to produce and make sense of the language needed to communicate their own ideas.
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: Cool-down - The Cost of a Bike (5 minutes)

Cool-Down

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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.