Distinguishing Circumference and Area
5 min
Teacher Prep
Setup
Narrative
In this Warm-up, students apply what they have learned about finding the area of a circle to estimate the area of a circular plate in terms of a smaller circle. Students see a plate with a single marble and, from this information, they are asked to make a reasoned estimate of the number of marbles required to cover the plate
Launch
Student Task
About how many marbles can fit on the plate in a single layer? Be prepared to explain your reasoning.
Sample Response
Sample responses:
- By comparing the plate to the lines of marbles off to the side, the diameter appears to be about 11 marbles. The radius would be 5.5 marbles. 52=25 and 62=36, so 5.52≈30. The area of the plate is about 3.14⋅30 times the area of a marble, so about 94 marbles should fit.
- By replicating the marble on the plate, the radius appears to be about 5 marbles. The area of the plate is about 3⋅52 times the area of a marble, so about 75 marbles should fit.
Activity Synthesis (Teacher Notes)
Poll the class on their estimates for the number of marbles that would fit.
Invite students to share their estimation strategies. 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?
Standards
- 7.G.4·Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
- 7.G.B.4·Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
15 min
Knowledge Components
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Distinguishing Circumference and Area
5 min
Teacher Prep
Setup
Narrative
In this Warm-up, students apply what they have learned about finding the area of a circle to estimate the area of a circular plate in terms of a smaller circle. Students see a plate with a single marble and, from this information, they are asked to make a reasoned estimate of the number of marbles required to cover the plate
Launch
Student Task
About how many marbles can fit on the plate in a single layer? Be prepared to explain your reasoning.
Sample Response
Sample responses:
- By comparing the plate to the lines of marbles off to the side, the diameter appears to be about 11 marbles. The radius would be 5.5 marbles. 52=25 and 62=36, so 5.52≈30. The area of the plate is about 3.14⋅30 times the area of a marble, so about 94 marbles should fit.
- By replicating the marble on the plate, the radius appears to be about 5 marbles. The area of the plate is about 3⋅52 times the area of a marble, so about 75 marbles should fit.
Activity Synthesis (Teacher Notes)
Poll the class on their estimates for the number of marbles that would fit.
Invite students to share their estimation strategies. 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?
Standards
- 7.G.4·Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
- 7.G.B.4·Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.