What Are Percentages?
10 min
Teacher Prep
Setup
Narrative
This Warm-up introduces the term “percent” in a geometric context and elicits students’ initial ideas about its meaning. The work here enables students to approach the idea of “per 100” visually before they reason about it more abstractly later.
Students first quantify the area of a shaded region in several large gridded squares. Then they describe each area relative to 100 square units, the area of a single large square, using the phrase “percent of the large square.” The quantity being compared to 100 includes values less than, equal to, and greater than 100.
During class discussion, students offer their interpretations of the term “percent,” which they will revisit in subsequent work.
Launch
Arrange students in groups of 2. Give students 2 minutes of quiet time to work on the first question. Ask them to pause for a discussion before completing the last question.
Display Diagrams A–E for all to see. Ask students to share the area of each shaded region and record it near the diagram. Tell students, “We can say that in Diagram A, 1 percent of the large square is shaded, and in Diagram B, 15 percent of the large square is shaded.”
Ask students, “What percent of the large square is shaded in Diagram C?” (30 percent) Repeat the question for Diagrams D and E (100 percent and 108 percent of the large square is shaded, respectively). Then ask students to complete the last question.
Student Task
-
A large square represents an area of 100 square units. How many square units are shaded in each diagram?
A B C D E Pause here so your teacher can review your work.
-
We can say that “1 percent of the large square” is shaded in Diagram A and “15 percent of the large square” is shaded in Diagram B.
Shade 50 percent of the area of the large square.
Sample Response
- A: 1 square units, B: 15 square units, C: 30 square units, D: 100 square units, E: 108 square units
-
Sample responses:
Activity Synthesis (Teacher Notes)
Invite students to share their depictions of 50 percent of the area of the large square and how they knew how much to shade.
Then ask partners to discuss what they think the term “percent” means and invite them to share their initial ideas with the class.
Because the percent values in the sample statements match the number of shaded square units in the diagrams, some students may infer that “percent” is another name for square units. Others may think in terms of the ratio of the area of the shaded region to the area of a large square, relating the former to 100 square units.
Use Collect and Display to create a shared reference that captures students’ developing mathematical language. Collect the language that students use to describe what “percent” means in the context of area. Display words and phrases, such as “compared to 100,” “out of 100 square units,” “the ratio of shaded squares to 100 squares,” or “how many shaded squares in a larger square of 100 squares.”
Tell students that they will check and refine these interpretations in upcoming activities and lessons.
Standards
- 2.MD.8·Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.
- 2.MD.C.8·Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using \$ and <span class="math">\(¢\)</span> symbols appropriately. <span>Example: If you have 2 dimes and 3 pennies, how many cents do you have?</span>
- 5.NBT.3·Read, write, and compare decimals to thousandths.
- 5.NBT.A.3·Read, write, and compare decimals to thousandths.
- 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.
10 min
15 min
Knowledge Components
All skills for this lesson
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What Are Percentages?
10 min
Teacher Prep
Setup
Narrative
This Warm-up introduces the term “percent” in a geometric context and elicits students’ initial ideas about its meaning. The work here enables students to approach the idea of “per 100” visually before they reason about it more abstractly later.
Students first quantify the area of a shaded region in several large gridded squares. Then they describe each area relative to 100 square units, the area of a single large square, using the phrase “percent of the large square.” The quantity being compared to 100 includes values less than, equal to, and greater than 100.
During class discussion, students offer their interpretations of the term “percent,” which they will revisit in subsequent work.
Launch
Arrange students in groups of 2. Give students 2 minutes of quiet time to work on the first question. Ask them to pause for a discussion before completing the last question.
Display Diagrams A–E for all to see. Ask students to share the area of each shaded region and record it near the diagram. Tell students, “We can say that in Diagram A, 1 percent of the large square is shaded, and in Diagram B, 15 percent of the large square is shaded.”
Ask students, “What percent of the large square is shaded in Diagram C?” (30 percent) Repeat the question for Diagrams D and E (100 percent and 108 percent of the large square is shaded, respectively). Then ask students to complete the last question.
Student Task
-
A large square represents an area of 100 square units. How many square units are shaded in each diagram?
A B C D E Pause here so your teacher can review your work.
-
We can say that “1 percent of the large square” is shaded in Diagram A and “15 percent of the large square” is shaded in Diagram B.
Shade 50 percent of the area of the large square.
Sample Response
- A: 1 square units, B: 15 square units, C: 30 square units, D: 100 square units, E: 108 square units
-
Sample responses:
Activity Synthesis (Teacher Notes)
Invite students to share their depictions of 50 percent of the area of the large square and how they knew how much to shade.
Then ask partners to discuss what they think the term “percent” means and invite them to share their initial ideas with the class.
Because the percent values in the sample statements match the number of shaded square units in the diagrams, some students may infer that “percent” is another name for square units. Others may think in terms of the ratio of the area of the shaded region to the area of a large square, relating the former to 100 square units.
Use Collect and Display to create a shared reference that captures students’ developing mathematical language. Collect the language that students use to describe what “percent” means in the context of area. Display words and phrases, such as “compared to 100,” “out of 100 square units,” “the ratio of shaded squares to 100 squares,” or “how many shaded squares in a larger square of 100 squares.”
Tell students that they will check and refine these interpretations in upcoming activities and lessons.
Standards
- 2.MD.8·Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.
- 2.MD.C.8·Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using \$ and <span class="math">\(¢\)</span> symbols appropriately. <span>Example: If you have 2 dimes and 3 pennies, how many cents do you have?</span>
- 5.NBT.3·Read, write, and compare decimals to thousandths.
- 5.NBT.A.3·Read, write, and compare decimals to thousandths.
- 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.