Representing Percentages with Double Number Line Diagrams
10 min
Narrative
In this Warm-up, students make sense of percentages when 100% corresponds to 10. Students can likely reason mentally about 50% of 10, 75% of 10, and relate 15 to 10, but are prompted to use a double number line diagram to represent the situation. Doing so encourages them to see percentages as rates and in terms of equivalent ratios. It also promotes the use of a double number line diagram as a tool for reasoning about percentages, which will be useful in subsequent activities and lessons.
The percentages are limited to multiples of 25 and 50, and the number lines are partitioned but only partially labeled. As students represent quantities and identify percentages on the diagram, they practice reasoning quantitatively and abstractly (MP2).
Launch
Remind students that in the previous lesson, we found percentages of 100 and of 1 using double number line diagrams. Explain that in this lesson we will find percentages of other numbers.
Arrange students in groups of 2. Give students 2–3 minutes of quiet work time, followed by 1–2 minutes to discuss their responses with a partner.
Student Task
Priya was saving money to buy a $10 hat.
- After one week, she had saved 50% of the cost of the hat.
- After two weeks, she had saved 75% of the cost of the hat.
- After three weeks, Priya had $15.
- How much money is 100% of the cost of the hat?
- Label the double number line diagram to represent the amounts and percentages in this situation.
Sample Response
- $10
- Sample response:
Activity Synthesis (Teacher Notes)
This discussion is to highlight two ideas: that 100% can correspond to an amount other than 100 or 1, and that percentage as a rate per 100 still applies just the same.
Display the blank diagram for all to see. Invite students to share how they found the percentages and how they went about labeling the diagram to represent the situation. Annotate the diagram based on students’ explanations. Make sure students see that in this case 100% corresponds to $10, which is the cost of the hat and the target amount that Priya wanted to have.
Then, discuss how the definition of “rate per 100” is still in play when comparing an amount to 10 units. If 100% of the cost of the hat is $10, then:
- 50% of the cost is $5 because the ratio of $5 to $10 is equivalent to 50 to 100.
- 75% of the cost is $7.50 because the ratio of $7.50 to $10 is equivalent to 75 to 100.
- 150% of the cost is $15 because the ratio of $15 to $10 is equivalent to 150 to 100.
Consider referring to the completed double number line diagram to illustrate these equivalent ratios.
Standards
- 6.RP.3.c·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.
- 6.RP.A.3.c·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.
15 min
10 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Representing Percentages with Double Number Line Diagrams
10 min
Narrative
In this Warm-up, students make sense of percentages when 100% corresponds to 10. Students can likely reason mentally about 50% of 10, 75% of 10, and relate 15 to 10, but are prompted to use a double number line diagram to represent the situation. Doing so encourages them to see percentages as rates and in terms of equivalent ratios. It also promotes the use of a double number line diagram as a tool for reasoning about percentages, which will be useful in subsequent activities and lessons.
The percentages are limited to multiples of 25 and 50, and the number lines are partitioned but only partially labeled. As students represent quantities and identify percentages on the diagram, they practice reasoning quantitatively and abstractly (MP2).
Launch
Remind students that in the previous lesson, we found percentages of 100 and of 1 using double number line diagrams. Explain that in this lesson we will find percentages of other numbers.
Arrange students in groups of 2. Give students 2–3 minutes of quiet work time, followed by 1–2 minutes to discuss their responses with a partner.
Student Task
Priya was saving money to buy a $10 hat.
- After one week, she had saved 50% of the cost of the hat.
- After two weeks, she had saved 75% of the cost of the hat.
- After three weeks, Priya had $15.
- How much money is 100% of the cost of the hat?
- Label the double number line diagram to represent the amounts and percentages in this situation.
Sample Response
- $10
- Sample response:
Activity Synthesis (Teacher Notes)
This discussion is to highlight two ideas: that 100% can correspond to an amount other than 100 or 1, and that percentage as a rate per 100 still applies just the same.
Display the blank diagram for all to see. Invite students to share how they found the percentages and how they went about labeling the diagram to represent the situation. Annotate the diagram based on students’ explanations. Make sure students see that in this case 100% corresponds to $10, which is the cost of the hat and the target amount that Priya wanted to have.
Then, discuss how the definition of “rate per 100” is still in play when comparing an amount to 10 units. If 100% of the cost of the hat is $10, then:
- 50% of the cost is $5 because the ratio of $5 to $10 is equivalent to 50 to 100.
- 75% of the cost is $7.50 because the ratio of $7.50 to $10 is equivalent to 75 to 100.
- 150% of the cost is $15 because the ratio of $15 to $10 is equivalent to 150 to 100.
Consider referring to the completed double number line diagram to illustrate these equivalent ratios.
Standards
- 6.RP.3.c·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.
- 6.RP.A.3.c·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.