Lesson 10
Distinguishing Circumference and Area
Let’s contrast circumference and area.
Problem 1
For each problem, decide whether the circumference of the circle or the area of the circle is most useful for finding a solution. Explain your reasoning.
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A car’s wheels spin at 1000 revolutions per minute. The diameter of the wheels is 23 inches. You want to know how fast the car is travelling.
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A circular kitchen table has a diameter of 60 inches. You want to know how much fabric is needed to cover the table top.
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A circular puzzle is 20 inches in diameter. All of the pieces are about the same size. You want to know about how many pieces there are in the puzzle.
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You want to know about how long it takes to walk around a circular pond.
Problem 2
The city of Paris, France is completely contained within an almost circular road that goes around the edge. Use the map with its scale to:
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Estimate the circumference of Paris.
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Estimate the area of Paris.
![Map of the city of Paris, France.](https://staging-cms-im.s3.amazonaws.com/Q9b9Eb5sv4QCnqp76Mg4ppSy?response-content-disposition=inline%3B%20filename%3D%227-7.3.B.PP.Image.10.png%22%3B%20filename%2A%3DUTF-8%27%277-7.3.B.PP.Image.10.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T143126Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=08003eb52c3d779f08a7046a6e2cc818cc047f5439495cea640c4dd639e2c15e)
Problem 3
Here is a diagram of a softball field:
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About how long is the fence around the field?
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About how big is the outfield?
![A quarter-circle that represents the diagram of a softball field.](https://staging-cms-im.s3.amazonaws.com/nozCebVY7MkGx9DAbEEDFLEP?response-content-disposition=inline%3B%20filename%3D%227-7.3.B.PP.Image.13.png%22%3B%20filename%2A%3DUTF-8%27%277-7.3.B.PP.Image.13.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T143126Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=9dcd5ff8da18024e92c0bbf94e6c8013fea084afcaadb6e995b2e16e7cb797e6)
Problem 4
While in math class, Priya and Kiran come up with two ways of thinking about the proportional relationship shown in the table.
\(x\) | \(y\) |
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2 | ? |
5 | 1750 |
Both students agree that they can solve the equation \(5k = 1750\) to find the constant of proportionality.
- Priya says, “I can solve this equation by dividing 1750 by 5.”
- Kiran says, “I can solve this equation by multiplying 1750 by \(\frac15\).”
- What value of \(k\) would each student get using their own method?
- How are Priya and Kiran's approaches related?
- Explain how each student might approach solving the equation \(\frac23 k=50\).