The purpose of this Warm-up is to help students connect their current work with percentage contexts to their prior work on efficient ways of finding percent increase.
Launch
Consider telling students that these questions may have more than one correct answer. Arrange students in groups of 2. Give 2 minutes of quiet think time followed by partner discussion. Then hold a whole-class discussion.
Activity
None
Which of these expressions represent a 15% tip on a \$20 meal? Which represent the total bill?
Student Response
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Building on Student Thinking
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. Invite other students to share whether they agree or disagree and why, or how they might explain it differently.
9.2
Activity
Standards Alignment
Building On
Addressing
7.RP.A.3
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
In this partner activity, students take turns matching situations to terms that describe applications of percent increase and decrease: sales tax, interest, gratuity/tip, markdown/discount, markup, commission, and depreciation. The situations are written on slips of paper and the terms are listed on a sorting mat.
It is not necessary for students to memorize the meanings of all these financial terms. The goal is for students to see that there are many different ways that percent increase and decrease are applied to money in the real world. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).
Launch
Math Community
Display the Math Community Chart for all to see. Give students a brief quiet think time to read the norms, or invite a student to read them out loud. Tell students that during this activity they are going to practice looking for their classmates putting the norms into action. At the end of the activity, students can share what norms they saw and how the norm supported the mathematical community during the activity.
Display the sorting mat for all to see. Consider reading the terms aloud and inviting students to chorally repeat them. However, it is not necessary to explain the meaning of the terms at this time. Tell students that each situation matches one of the terms on the mat. Some of the terms match more than one situation.
Explain how to set up and do the activity. If time allows, demonstrate the steps with a student as a partner. Consider demonstrating productive ways to agree or disagree, for example, by explaining mathematical thinking or asking clarifying questions.
Arrange students in groups of 2. Give each group a sorting mat and a set of slips cut from the blackline master. Give students 5–6 minutes of partner work time to sort the slips. When each group finishes sorting, check their work, and let them know which slips, if any, need to be revised. When their sorting is correct, instruct students to answer the rest of the questions.
MLR8 Discussion Supports. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frame for all to see: “I noticed , so I matched . . . .” Encourage students to challenge each other when they disagree. Advances: Reading, Conversing
Representation: Access for Perception. Ask students to read the situations aloud to their partner. Students who both listen to and read the information will benefit from extra processing time. Supports accessibility for: Language, Attention
Activity
Activity Synthesis
Invite 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?”
Invite students to share their sentences describing what the percent increase or decrease told us about the situation. If they do not specify what amount corresponds to 100%, ask them to revise their sentence to add in that information. For example, if a student says “Andre’s interest was 3%,” they could rephrase this as “Andre’s interest was 3% of his starting balance.”
Answer students’ remaining questions about any of these contexts. Tell students there is a chart at the end of the lesson that they can use as a reference tool during future lessons. The key takeaway is that these are all different ways that percent increase or percent decrease are applied to money in the real world.
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
depreciation
the buyer
subtracted from the price of an item as the item gets older
commission
the salesperson
subtracted from the payment the store collects
Math Community
Conclude the discussion by inviting 2–3 students to share a norm they identified in action. Provide this sentence frame to help students organize their thoughts in a clear, precise way:
“I noticed our norm ‘’ in action today, and it really helped me/my group because .”
9.3
Activity
Standards Alignment
Building On
Addressing
7.RP.A.3
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
In this activity, students apply their understanding of percent increase and decrease to a context involving markup and markdown. They see that a 32% increase followed by a 10% decrease of that new amount does not result in a 22% increase of the original amount. As students relate the percentages to the quantities they represent, they reason abstractly and quantitatively (MP2).
Monitor for students who use these different strategies to calculate the discounted price of the car:
Create a diagram to represent the situation.
Multiply the retail price by 0.1, and then subtract this amount.
Multiply the retail price by 0.9.
Multiply the wholesale price by 1.22.
Plan to have students present in this order to support moving them from the more concrete to the more abstract and efficient. Note that the last strategy on the list incorrectly oversimplifies the situation.
Launch
Arrange students in groups of 2. Give students 5 minutes of quiet work time followed by time for partner discussion. Then hold a whole-class discussion.
If needed, explain to students that profit is the amount of markup from the wholesale price of an item to the retail price of an item. The profit can be expressed as a percentage of the wholesale price.
Select students who used each strategy described in the Activity Narrative to share later. Aim to elicit both key mathematical ideas and a variety of student voices, especially from students who haven’t shared recently.
MLR8 Discussion Supports. Use multimodal examples to show the meaning of monetary terms like “wholesale price,” “retail price,” and “sale price.” Use verbal descriptions along with gestures, drawings, or concrete objects to show the relationships between these terms and other terms that students are already familiar with, such as “markup” and “discount.” Advances: Listening, Representing
Representation: Develop Language and Symbols. Support understanding of the problem, by inviting students to act it out. For example, have students enact the roles of the wholesaler, dealership, and customer by exchanging a toy car as well as different amounts of play money. Supports accessibility for: Conceptual Processing, Language
Activity
None
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?
After the discount, what percent profit will the dealership make?
Activity Synthesis
The purpose of this discussion is to compare different strategies for finding the sale price after both a markup and a markdown. Invite previously selected students to share how they found the price the customer pays for the car. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.
Connect the different responses to the learning goals by asking questions, such as:
“How does the 10% discount show up in each method?”
“Why do the different approaches lead to the same outcome?”
“Are there any benefits or drawbacks to one strategy compared to another?”
The key takeaway is that finding 90% of the retail price is equivalent to—and more efficient than—calculating the 10% discount and then subtracting that amount from the retail price.
If a student tried to find the sale price by calculating 122% of the wholesale price, ask “Why does this approach lead to a different outcome?” If no student tried to find the sale price by calculating 122% of the wholesale price, consider asking “Jada thought that the final markup percentage would be 22%, because . Why is this incorrect?”
Lesson Synthesis
Share with students, “Today we studied lots of different situations where people use percentages.”
To review the new contexts, consider asking students:
“What are some situations in life in which people encounter percentages?”
“Give examples of situations where you would encounter tax, tip, markup, markdown, commission.” (buying things; eating at a restaurant; a company wants to make money; a store is having a sale; a salesperson sells something)
“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.)
Student Lesson Summary
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
depreciation
the buyer
subtracted from the price of an item as the item gets older
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, this can be thought of 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 , and the family gets \$194,000, because . From the family's perspective, this can be thought of as a 3% decrease on the sale price of the home.
Standards Alignment
Building On
6.EE.A.2.b
Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression as a product of two factors; view as both a single entity and a sum of two terms.
Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
Your teacher will give you a set of cards and a sorting mat. Take turns with your partner to match a situation to a category.
For each match that you find, explain to your partner how you know it’s a match.
For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.
Pause here so your teacher can review your work.
Choose one situation that involves a percent increase.
Find the percentage that the increase is of the original amount.
What does this percentage tell us about the situation?
Choose one situation that involves a percent decrease.
Find the percentage that the decrease is of the original amount.
What does this percentage tell us about the situation?
Student Response
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Building on Student Thinking
Student Response
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Building on Student Thinking
Throughout this activity, it is important 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.
