This Warm-up prompts students to compare four diagrams that represent percent increase or percent decrease. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the diagrams in comparison to one another.
Launch
Arrange students in groups of 2–4. Display the diagrams for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three diagrams that go together and can explain why. Next, tell students to share their response with their group and then together find as many sets of three as they can.
Student Lesson in Spanish
Activity
None
Which three go together? Why do they go together?
A
B
C
D
A double number line with 6 evenly spaced tick marks. On the top number line, starting with the first tick mark, zero, 5, 10, 15, 20 and 25 are labeled. On the bottom number line, starting with the first tick mark, zero percent, 25 percent, 50 percent, 75 percent, 100 percent and 125 percent are labeled.
Student Response
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Building on Student Thinking
Activity Synthesis
Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations, and ensure the reasons given are correct.
During the discussion, prompt students to explain the meaning of any terminology they use, such as “original amount,” “new amount,” “increase,” and “decrease,” and to clarify their reasoning as needed. Consider asking:
“How do you know . . . ?”
“What do you mean by . . . ?”
“Can you say that in another way?”
If not mentioned by students, ask them to discuss the amount that corresponds to 100% for each diagram.
7.2
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 activity, students complete double number line diagrams that represent situations involving percent increase and percent decrease. For each situation, two values are given. Students interpret the situation to determine where to label the given values on the double number line diagram. Then students can use the diagram to find the unknown, third value, whether it is the new amount, the original amount, or the percentage of the increase or decrease. The goal is to reinforce that the original amount pertains to 100%.
In this activity, students critique a statement or response that is intentionally unclear, incorrect, or incomplete and improve it by clarifying meaning, correcting errors, and adding details (MP3).
This activity uses the Critique, Correct, Clarify math language routine to advance representing and conversing as students critique and revise mathematical arguments.
Launch
Arrange students in groups of 2. Give 5–8 minutes of partner work time.
Action and Expression: Internalize Executive Functions. To support development of organizational skills in problem-solving, chunk this task into more manageable parts. For example, present one question at a time, and monitor students to ensure they are making progress throughout the activity. Supports accessibility for: Organization, Attention
Activity
None
For each problem, complete the double number line diagram to show the percentages that correspond to the original amount and to the new amount.
Last year, scientists counted 12 foxes in a conservation area. This year, they counted 50% more than that. How many foxes did they count this year?
A double number line for “number of foxes” with 4 evenly spaced tick marks. On the top number line, zero is on the first tick mark and the remaining tick marks are not labeled. On the bottom number line starting with the first tick mark zero percent, 50 percent, 100 percent, and 150 percent are labeled.
After replacing some grass with rocks, a business decreased its water usage by 20%. If their old water usage was 15,000 gallons per week, how much do they use now?
A double number line for “water usage in gallons” with 7 evenly spaced tick marks. On the top number line the number zero is on the first tick mark and the remaining tick marks are not labeled. The bottom number line, starting with the first tick mark, zero percent, 20 percent, 40 percent, 60 percent, 80 percent, 100 percent, and 120 percent are labeled.
A school had 1,200 students last year and only 1,080 students this year. What was the percent decrease in the number of students?
Double number line, top line number of people, evenly spaced tick marks labeled 0, 120, 240, 360, 480, 600, 720, 840, 960, 1080, 1200. Bottom line, corresponding tick marks, first labeled 0% and the rest blank.
One week, gas was \$1.25 per gallon. The next week, gas was \$1.50 per gallon. By what percentage did the price increase?
A double number line for “price of gas in dollars” with 9 evenly spaced tick marks. On the top number line starting with the first tick mark zero, zero point 2 5, zero point 5, zero point 7 5, one, one point 2 5, one point 5, one point 7 5 and 2 are labeled. On the bottom number line, zero percent is on the first tick mark and the remaining tick marks are not labeled.
After a 25% discount, the price of a T-shirt was \$12. What was the price before the discount?
A double number line for “price of shirts in dollars” with 6 evenly spaced tick marks. On the top number line, the number zero is on the first tick mark and the remaining tick marks are not labeled. On the bottom number line, starting with the first tick mark, zero percent, 25 percent, 50 percent, 75 percent, 100 percent and 125 percent are labeled.
Compared to last year, the population of Boom Town has increased 25%. The population is now 6,600. What was the population last year?
A double number line for "number of people" with 6 evenly spaced tick marks. For the top number line, the number 0 is on the first tick mark and the remaining tick marks are blank. For the bottom number line, starting with the first tick mark, the percentages 0%, 25%, 50%, 75%, 100%, and 125% are labeled.
Student Response
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Building on Student Thinking
Students may continue to struggle to recognize the original amount and new amount with the proper percentages on the double number line. Remind them that the original amount always corresponds to 100%.
Activity Synthesis
Invite students to share the values they identified as the original amount and the new amount for a few problems. Discuss how 100% always corresponds to the original value and when there is an increase in the value, the new value corresponds to a percentage greater than the original 100%.
Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response to the last question, about the population of Boom Town, by correcting errors, clarifying meaning, and adding details.
Display this first draft:
“Since the population has increased by 25%, that means last year’s population was 75% of this year’s population. , so last year’s population was 4,950 people.”
Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
Give students 2–4 minutes to work with a partner to revise the first draft.
Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.
7.3
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 interpret situations involving percent increase or percent decrease to determine whether they need to find the new amount or the original amount. No diagrams are given, so students may decide to create a tape diagram or double number line diagram to represent each situation. As students choose a solution strategy such as drawing a diagram, they are making sense of problems and persevering in solving them (MP1).
The first problem involves finding the new amount after a 10% increase. The second problem involves finding the original amount before a 10% decrease. By the end of this activity, students should come to see why these two situations call for different solution methods.
This activity uses the Three Reads math language routine to advance reading and representing as students make sense of what is happening in the text.
Launch
Use Three Reads to support reading comprehension and sense-making about this problem. Display only the problem stem and description of each beach, without revealing the questions.
For the first read, read the problem aloud, then ask, “What is this situation about?” (the number of sea turtles on a beach). Listen for and clarify any questions about the context.
After the second read, ask students to list any quantities that can be counted or measured (this year’s number of turtles; last year’s number of turtles; the percentage of the increase or decrease).
After the third read, reveal the two questions: “How many turtles came ashore to lay eggs in the sanctuary this year?” and “How many nesting turtles were at this sanctuary last year?” and ask, “What are some ways we might get started on this?” Invite students to name some possible starting points, referencing quantities from the second read. (creating a tape diagram or double number line diagram; identifying the amount in each situation that corresponds to 100%)
Give students 4–5 minutes of quiet work time followed by time for partner discussion. Then hold a whole-class discussion.
Representation: Internalize Comprehension. Represent the same information through different modalities by using a double number line diagram to help organize the information provided. Supports accessibility for: Conceptual Processing, Visual-Spatial Processing
Activity
None
Green sea turtles live most of their lives in the ocean, but they come ashore to lay their eggs. Some beaches where turtles often come ashore have been made into protected sanctuaries so that the eggs will not be disturbed.
One sanctuary had 180 green sea turtles come ashore to lay eggs last year. This year, the number of turtles increased by 10%. How many turtles came ashore to lay eggs in the sanctuary this year?
At another sanctuary, the number of nesting turtles decreased by 10% between last year and this year. This year there were 234 nesting turtles. How many nesting turtles were at this sanctuary last year?
Student Response
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Building on Student Thinking
For the percent decrease problem, students may calculate 10% of 234, getting a change of 23.4 turtles and an original number of 257.4 turtles. Remind them that the percent decrease describes the change as a percentage of the original value, not as a percentage of the new value. If needed, prompt students to use a double number line to represent the situation, placing 234 to the left by 10% of the quantity they want to find, which should be associated with 100%.
Activity Synthesis
The purpose of this discussion is to highlight why the solution processes are not the same for the two problems:
Finding the new amount after a 10% increase
Finding the original amount before a 10% decrease
The key point to remember is that 100% corresponds to the original amount.
Invite students to share how they reasoned about each problem. To help students make comparisons between the two problems, consider asking:
“How are the two problems the same? How are they different?” (They both involve a change of 10%. The first situation is an increase, while the second situation is a decrease.)
“What information is given in the first problem that is not given in the second? And in the second that is not in the first?” (The first problem gave the original amount. The second problem gave the new amount.)
“In which problem were you given the quantity that represented 100%?” (the first problem)
“For the second situation, the given number 234 represents what percentage of the original amount? How do you know?” (90%. It is the amount that was left after a 10% decrease.)
“How can you use that percentage to figure out the original number of green sea turtles?” (by finding the value that is 100% for which 234 is 90%.)
Lesson Synthesis
Share with students, “Today we used double number line diagrams to make sense of situations involving percent increase or percent decrease.”
To review the different types of problems that students solved, consider asking:
“What part of the situation is always 100%?” (the starting amount)
“If there was a 30% increase, where would you put the new amount on the double number line?” (at 130%)
“Where would you put the original amount?” (at 100%)
“If there was a 30% decrease, where would you put the new amount on the double number line?” (at 70%)
“Where would you put the original amount?” (still at 100%)
Student Lesson Summary
We can use a double number line diagram to show information about percent increase and percent decrease:
Double number line, 8 evenly spaced tick marks. Top line, cereal, grams. Beginning at first tick mark, labels: 0, 100, 200, 300, 400, 500, 600, 700. Bottom line, beginning at first tick mark, labels: 0 percent, 20 percent, 40 percent, 60 percent, 80 percent, 100 percent, 120 percent, 140 percent.
The initial amount of cereal is 500 grams, which is lined up with 100% in the diagram. We can find a 20% increase by adding 20% of 500:
In the diagram, we can see that 600 corresponds to 120%.
If the initial amount of 500 grams is decreased by 40%, we can find how much cereal there is by subtracting 40% of the 500 grams:
So, a 40% decrease is the same as 60% of the initial amount. In the diagram, we can see that 300 is lined up with 60%.
To solve percentage problems, we need to be clear about what corresponds to 100%. For example, suppose there are 20 students in a class, and we know this is an increase of 25% from last year. In this case, the number of students in the class last year corresponds to 100%. So the initial amount (100%) is unknown and the final amount (125%) is 20 students.
Double number line, 8 evenly spaced tick marks. Top line, number of students. Beginning at first tick mark, labels: 0, 4, 8, 12, 16, 20, 24, 28. Bottom line, beginning at first tick mark, labels: 0 percent, 25 percent, 50 percent, 75 percent, 100 percent, 125 percent, 150 percent, 175 percent. 16 and 100 percent are circled.
Looking at the double number line, if 20 students is a 25% increase from the previous year, then there were 16 students in the class last year.
Standards Alignment
Building On
Addressing
Building Toward
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.
