This Math Talk focuses on finding the factor that scales 50 to another number. It encourages students to think about increase and decrease in terms of fractions, decimals, or percentages and to rely on properties of operations to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students write equations to represent percent increase or decrease situations.
As students use results from the previous equation to help solve the next equation, they are making use of repeated reasoning (MP8).
Student Lesson in Spanish
Launch
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies, and record and display their responses for all to see.
Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with access to sticky notes or mini whiteboards. Supports accessibility for: Memory, Organization
Activity
None
Solve each equation mentally.
Student Response
Loading...
Building on Student Thinking
Activity Synthesis
To involve more students in the conversation, consider asking:
“Who can restate ’s reasoning in a different way?”
“Did anyone use the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”
If all students give their answers as fractions, encourage them to express the fraction as a decimal. This will help prepare students for working with equations where decimals are used to represent percent increase or decrease.
If any student expresses their reasoning in terms of percentages, for example, by saying that 60 is a 20% increase from 50, highlight this strategy for all to see.
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I because . . . .” or “I noticed , so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. Advances: Speaking, Representing
8.2
Activity
Standards Alignment
Building On
Addressing
7.EE.A.2
Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, means that “increase by ” is the same as “multiply by .”
In this activity, students match equations to verbal descriptions of a percent increase or decrease situation. To help identify the correct equation, students first draw a diagram to represent the situation and make sense of what percentage the new amount is of the original amount. As students relate verbal descriptions of the situations with equations, they reason quantitatively and abstractly (MP2).
Launch
Arrange students in groups of 2. Give students 4–5 minutes of quiet work time followed by time for partner discussion. Then hold a whole class discussion.
Activity
None
For each situation:
Draw a diagram to represent the situation.
Match an equation to the situation.
Solve the equation to find the initial value, .
The water level in a reservoir is now 52 meters. If this was a 32% increase, what was the initial depth?
Diagram:
Equation:
Solution:
The snow is now 52 inches deep. If this was a 68% decrease, what was the initial depth?
Diagram:
Equation:
Solution:
Write a story for one of the equations that doesn’t have a match.
Student Response
Loading...
Building on Student Thinking
Activity Synthesis
Ask one or more students to share which equation they matched with each situation, and resolve any discrepancies. Once the matches are agreed upon, ask students how they would solve the equation to find the amount without actually solving the equations.
Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. Provide sentence frames for students to use when they share the equation they matched to each situation: “Situation ____ matches with equation ____ because ____.” Call students attention to how the percent increase is represented in the equation and the situation. Design Principle(s): Support sense-making; Optimize output (for explanation)
Representation: Develop Language and Symbols. Make connections between representations visible. Draw students’ attention to correspondences between the diagrams and equations. Supports accessibility for: Language, Conceptual Processing
8.3
Activity
Standards Alignment
Building On
Addressing
7.EE.A.2
Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, means that “increase by ” is the same as “multiply by .”
In this activity, students write equations that represent situations involving percent increase or decrease. They apply the distributive property to find the coefficient that represents the percentage that the new value is of the original value. They solve the equations to find the new value or the original value. As students use the equations to determine whether to multiply or divide to find the unknown value, they are making use of structure (MP7).
Launch
Introduce the context of this activity by asking students, “Have you ever taken a plain piece of fabric and decorated it somehow? What are some ways you can think of to decorate fabric?” Then explain that zardozi (zar-DOO-zee) and molas (MOH-lah) are two different ways that some people decorate fabric. Zardozi is a Persian art form that involves sewing metal threads and beads onto silk or other fabrics. Molas are made by the Guna people. They sew multiple layers of fabric together and cut out pieces to make a colorful pattern.
Give students 4–5 minutes of quiet work time followed by time for partner discussion. Then hold a whole-class discussion.
MLR2 Collect and Display. Collect the language students use to describe the differences between each problem. Display words and phrases such as “original amount,” “new amount,” “percent increase,” and “percent decrease.” During the Activity Synthesis, invite students to suggest ways to update the display: “What are some other words or phrases we should include?” Invite students to borrow language from the display as needed. Advances: Conversing, Reading
Representation: Internalize Comprehension. Activate or supply background knowledge. Show a video for students who are unfamiliar with embroidery or sewing. Supports accessibility for: Conceptual Processing, Language
Activity
None
Write an equation to represent each situation. Then, solve the equation.
A piece of fabric weighed 15 ounces. After it was decorated with zardozi, the weight had increased by 82%. What is the weight of the finished piece?
Another finished zardozi piece weighs 22 ounces. This is a 76% increase from the original weight of the fabric. What was the original weight?
Before making a mola, the layers of fabric weighed 4.7 ounces. When the mola was finished, the weight had decreased by 17%. What is the weight of the finished mola?
Another finished mola weighs 4.9 ounces. This is a 21% decrease from the original weight of the fabric. What was the original weight?
Student Response
Loading...
Building on Student Thinking
Activity Synthesis
The purpose of this discussion is to contrast the structure of the equations: those with an unknown value of the original weight and those with an unknown value of the new weight. Invite students to share the equations they wrote for each situation. To help students contrast the equations, consider asking:
“What is the same about the equations? What is different?” (They all involve multiplication. Two have an unknown factor, while the other two have an unknown product.)
“How is the percent increase or percent decrease represented in each equation?” (For the percent increase situations, the coefficient represents 100% plus the percent increase. For the percent decrease situations, the coefficient represents 100% minus the percent decrease.)
“Which two equations represent situations where the finished weight was unknown?” (the two with the unknown product)
“Which two equations represent situations where the original weight was unknown?” (the two with the unknown factor)
“How did you determine the unknown value for each equation?” (I multiplied to find the unknown product and then divided to find the unknown factor.)
8.4
Activity
Optional
Standards Alignment
Building On
Addressing
7.EE.A.2
Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, means that “increase by ” is the same as “multiply by .”
In this activity, students practice writing equations to represent situations involving percent increase or decrease. The situations are the same ones that students previously represented with double number line diagrams in a previous lesson.
In this partner activity, students take turns sharing their initial ideas and first drafts. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3). As students revise their writing, they have an opportunity to attend to precision in the language they use to describe their thinking (MP6).
This activity uses the Stronger and Clearer Each Time math language routine to advance writing, speaking, and listening as students refine mathematical language and ideas.
Launch
Show students the double number line from the activity in the previous lesson.
Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to the first question, about the equation . In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.
If time allows, display these prompts for feedback:
“ makes sense, but what do you mean when you say . . . ?”
“Can you describe that another way?”
“How do you know . . . ? What else do you know is true?”
Close the partner conversations and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer.
After Stronger and Clearer Each Time, give students 3–4 minutes of quiet work time to answer the rest of the questions.
Action and Expression: Provide Access for Physical Action. Activate or supply background knowledge. Provide students with access to double number lines to support information processing. Supports accessibility for: Visual-Spatial Processing, Organization
Activity
None
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?
Explain why this situation can be represented by the equation . Make sure that you explain what represents.
Write an equation to represent each of the following situations.
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?
After a 25% discount, the price of a T-shirt was $12. What was the price before the discount?
Compared to last year, the population of Boom Town has increased by 25%.The population is now 6,600. What was the population last year?
Student Response
Loading...
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
Select students to share the different equations they came up with. Points of discussion could include:
How the distributive property is useful for finding the percentage that corresponds with the new value instead of the percentage of the change.
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%.
How solving problems about percent change may require either multiplying or dividing numbers. It can be confusing, but it helps to first express the relationship as an equation and then think about how the unknown number can be found.
Lesson Synthesis
Share with students, “Today we wrote equations to represent situations involving percent increase or percent decrease.”
To review the different types of problems that students solved, for each of these situations, consider asking, “How could you write an equation to represent the situation? What does the variable represent?”:
“The original amount was 50. There is a 35% increase.” (, where represents the new amount)
“The new amount is 50. There was a 35% increase.” (, where represents the original amount)
“The original amount was 50. There is a 35% decrease.” (, where represents the new amount)
“The new amount is 50. There was a 35% decrease.” (, where represents the original amount)
Student Lesson Summary
We can use equations to express percent increase and percent decrease.
For example, if is 15% more than , we can represent this by using any of these equations:
So if someone makes an investment of dollars, and its value increases by 15% to reach \$1,250, then we can write the equation to find the value of the initial investment.
Here is another example: if is 7% less than , we can represent this by using any of these equations:
So if the amount of water in a tank decreased 7% from its starting value of to its ending value of 348 gallons, then we can write .
Often, an equation is the most efficient way to solve a problem involving percent increase or percent decrease.
Standards Alignment
Building On
5.NF.B
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
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.
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.
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.
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.
