This Math Talk focuses on dividing by 100. It encourages students to think about place value and to rely on properties of operations to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students use decimals to represent percentages that are not whole numbers.
As students use results from the previous expression to help evaluate the next expression, 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
Find the value of each expression mentally.
Student Response
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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?”
The key takeaway is to remind students that numbers can have more than two decimal places. Up to this point, students have mostly worked mostly with percentages that are whole numbers, such as 14% and 7%. Representing these percentages as decimals always gives values with two decimals places, like 0.14 and 0.07. This could lead students to overgeneralize and think that 3.5% would be represented as 0.35. Focusing on the relative size of 3.5% compared to 7% can help students see why the decimal representation is actually 0.035.
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
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 activity, students reason about percentages that are not whole numbers. Students use what they know about place value and properties of operations to calculate fractional percentages of numbers, making use of structure (MP7). Similar to a Math Talk, the problems are sequenced so that students can use the result of the previous problem to help answer the next problem.
Monitor for students who:
Use 1% to find 0.1%.
Make substitutions with known quantities to help compute unknown quantities.
Use the result of the previous problem to help answer the next problem.
Launch
Give students 1–2 minutes of quiet work time for the first problem. Ask students to pause their work and share what they noticed about these expressions and their answers. A key takeaway is that students can use the result of the previous problem to help them reason about the next expression. For example, 3% is of 30%, and of 18 is 1.8. Avoid using the terminology “moving the decimal.”
Give students 4–5 minutes of quiet work time to finish the rest of the problems.
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
Find each percentage of 60. What do you notice about your answers?
30% of 60
3% of 60
0.3% of 60
0.03% of 60
20% of 5,000 is 1,000, and 21% of 5,000 is 1,050. Find each percentage of 5,000, and be prepared to explain your reasoning. If you get stuck, consider using the double number line diagram.
1% of 5,000
0.1% of 5,000
20.1% of 5,000
20.4% of 5,000
A double number line with 12 tick marks. The first tick mark is followed by a break and then 11 evenly spaced tick marks. For the top number line, the number 0 is on the first tick mark, 1000 on the second, and 1050 on the twelfth. For the bottom number line, the percentage 0% is on the first tick mark, 20% on the second, and 21% on the twelfth.
15% of 80 is 12, and 16% of 80 is 12.8. Find each percentage of 80, and explain your reasoning.
15.1% of 80
15.7% of 80
Student Response
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Building on Student Thinking
When students calculate the various percentages of 60, they may make mistakes in the place value of the answers. Refer students to the previous activity's discussion. You may also want to ask students to calculate 10% of 60 and use that answer to calculate 30%.
If students get stuck calculating various percents of 5,000, recommend that they use the double number line provided. Ask them:
“What percentages are visible in the bottom number line?“
“How much is that 1% in reference to the top number line?“
“How can we use that 1% to figure out the other percentages?“
Use these same questions if students get stuck calculating 15.1% and 15.7%.
Activity Synthesis
Invite students to share how they reasoned about the problems. Highlight strategies by students who recognized that they can use 1% of a number to calculate 0.1% of a number and make multiples of that to get, for example, 0.7% of a number.
MLR1 Stronger and Clearer Each Time. Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their first draft response to the last question, about 15.7% of 80. Invite listeners to ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing. Give students 3–5 minutes to revise their first draft based on the feedback they receive. Advances: Writing, Speaking, Listening
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 find the new value after a percent increase that involves a fractional percentage. The first problem asks about an 8% increase, and the second problem asks about a 0.8% increase, so students can use the relationship between these two percentages to help solve the second problem.
Monitor for students who:
Calculate the percentage first and then add that to the original amount.
Multiply the original amounts by 1.08 and 1.008, respectively.
Plan to have students present in this order to support moving them from less efficient to more efficient methods.
As students explain why either city could be considered to have grown by more, they are attending to precision (MP6).
Launch
Arrange students in groups of 2. Give students 4 minutes of quiet work time followed by time for partner discussion. Then hold a whole-class discussion.
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. Encourage students to begin partner discussions by reading their written responses for the last question aloud. If time allows, invite students to revise or add to their responses based on the conversation that follows. Advances: Conversing, Speaking
Engagement: Develop Effort and Persistence. Provide tools to facilitate information processing or computation, enabling students to focus on key mathematical ideas. For example, allow students to use calculators to support their reasoning. Supports accessibility for: Memory, Conceptual Processing
Activity
None
The population of City A was approximately 243,000 people. It increased by 8% in one year. What was the new population?
The population of City B was approximately 7,150,000. It increased by 0.8% in one year. What was the new population?
Lin says that City A grew by more. Andre says that City B grew by more. Do you agree with either of them? Explain your reasoning.
Student Response
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Building on Student Thinking
Students who want to multiply by may have trouble determining where to put the decimal. Have them think about the problem in steps. How can you find 8%? (Multiply by 0.08.) How can you find 0.8%? (Multiply by 0.008.)
Activity Synthesis
The purpose of this discussion is to compare and contrast the 8% increase with the 0.8% increase. First, invite previously selected students to share how they calculated the new populations for each city. 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 are these strategies the same? How are they different?”
“How does the 8% increase (or 0.8% increase) show up in each method?”
“Why do the different approaches lead to the same outcome?”
The key takeaway is that the same strategies work for finding 0.8% as for 8%. We can multiply first and then add, or we can add first and then multiply. The multiplier is just different by a factor of 10 (0.008 instead of 0.08).
Next, invite students to share their reasoning about which city grew by more. Either city could be the correct answer depending on whether students are viewing the growth additively or multiplicatively. Make sure that some students support each perspective. If needed, assign some students to support Lin’s statement and others to support Andre’s statement. The goal is for students to articulate more precisely what is meant by the phrase “grew by more” in each interpretation. To involve more students in the conversation, consider asking:
“Do you agree or disagree? Why?”
“Who can restate ’s reasoning in a different way?”
“Does anyone want to add on to ’s reasoning?”
Lesson Synthesis
Share with students, “Today we worked with percentages that were not whole numbers.”
To review the reasoning developed in this lesson, consider asking students:
“How are these percentages related to each other: 40%, 4%, 0.4%, 0.04%?” (Each percentage is of the previous one.)
“How can we use 40% of a number to help calculate the other percentages?” (4% of that number will be of what 40% of that number was. 0.4% of that number will be , and so on.)
“If we know 1% of a number, how can we use that to help us calculate 0.5% of a number?” (0.5% of a number is half of 1% of that number.)
Student Lesson Summary
A percentage, such as 30%, is a rate per 100. To find 30% of a quantity, we multiply it by , or 0.3.
The same method works for percentages that are not whole numbers, like 7.8% or 2.5%. In the square, 2.5% of the area is shaded.
To find 2.5% of a quantity, we multiply it by , or 0.025. For example, to calculate 2.5% interest on a bank balance of \$80, we multiply , so the interest is \$2.
We can sometimes find percentages like 2.5% mentally by using convenient whole-number percentages. For example, 25% of 80 is one fourth of 80, which is 20. Since 2.5 is one tenth of 25, we know that 2.5% of 80 is one tenth of 20, which is 2.
Standards Alignment
Building On
6.RP.A.3
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
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.
