Unit 3 Measuring Circles — Unit Plan

TitleTakeawaysVisual / Anchor ChartAssessment
Lesson 1
How Well Can You Measure?
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Examining Relationships (1 problem)

For each situation, explain whether the measurements shown on the graph could represent a proportional relationship.

  1. The height of a plant was measured every ten days.

    Graph of 7 plotted points for plant height.
    Graph of 7 plotted points, origin O, with grid. Horizontal axis, time in days, scale 0 to 60, by 10’s. Vertical axis, plant height in centimeters, scale 0 to 25, by 5’s. Plotted points at approximately, 0 comma 0, 10 comma 4, 20 comma 8, 30 comma 11, 40 comma 17, 50 comma 19, 60 comma 23.

    Could the relationship between the number of days and the height of the plant be proportional? Explain your reasoning.

  2. The height of the snow was measured every hour.

    Graph of 7 plotted points on grid.
    Graph of 7 plotted points, origin O, with grid. Horizontal axis, time, hours, scale 0 to 6, by 1’s. Vertical axis, height of snow, inches, scale 0 to 4, by 1’s. Plotted points at,  0 comma 0, 1 comma 1.5, 2 comma 3, 3 comma 3, 4 comma 3, 5 comma 3, 6 comma 4.

    Could the relationship between the number of hours and the height of the snow be proportional? Explain your reasoning.

Show Solution
  1. Yes, there may be a proportional relationship. Sample reasoning: The point (0,0)(0,0) is on the graph, the points are close to being on a line, and there could be measurement error. However, it is also possible that the relationship is not proportional. It is not possible to decide for sure from the graph.
  2. No, there is not a proportional relationship. Sample reasoning: For several hours there was no snow falling while some time at the beginning and toward the end there was some snowfall. 
Lesson 2
Exploring Circles
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Comparing Circles (1 problem)

Here are two circles. Their centers are AA and FF.

Two circles
The first figure is a circle with center A and points E, C, B, and D lie on the circle. A line segment extends from A to point D and a second line segment extends from A to point C, where line segment AC is labeled 4 centimeters. A third line segment is extends from point E to point B, where line segment EB goes through point A. The second figure is a circle with center F and points H and G lie on the circle. A line segment is extends from point H to point G where line segment HG goes through point F and is labeled 8 centimeters.

  1. What is the same about the two circles? What is different?

  2. What is the length of segment ADAD? How do you know?

  3. On the first circle, what segment is a diameter? How long is it?

Show Solution
  1. Because they are both circles, they are both round figures, without corners or straight sides, enclosing a two-dimensional region, that are the same distance across (through the center) in every direction. Both circles are the same size. They have the same diameter, radius, and circumference. The only difference is which additional segments (radii) are drawn.
  2. Segment ADAD is 4 cm long because it is also a radius of the circle.
  3. The diameter, segment EBEB, is 8 cm long.
Lesson 3
Exploring Circumference
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Identifying Circumference and Diameter (1 problem)

Select all the pairs that could be reasonable approximations for the diameter and circumference of a circle. Explain your reasoning.

  1. 5 meters and 22 meters.
  2. 19 inches and 60 inches.
  3. 33 centimeters and 80 centimeters.
Show Solution
  1. does not work, because 22÷5>422 \div 5 > 4
  2. does work, because 60÷19≈3.15860 \div 19 \approx 3.158
  3. does not work, because 80÷33 <2.580 \div 33 < 2.5
Lesson 4
Applying Circumference
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Circumferences of Two Circles (1 problem)

Circle A has a diameter of 9 cm. Circle B has a radius of 5 cm.

  1. Which circle has the larger circumference?
  2. About how many centimeters larger is it?
Show Solution
  1. Circle B has the larger circumference. Circle A has a diameter of 9 cm, and Circle B has a diameter of 5⋅25 \boldcdot 2, or 10 cm. Since Circle B’s diameter is larger than Circle A’s diameter, and circumference is proportional to diameter, that means Circle B’s circumference is also larger.
  2. The difference is about 3.14 cm because the circumference of Circle A is 9π9\pi, or about 28.26 cm, and the circumference of Circle B is 10π10\pi, or about 31.4 cm. The difference is 31.4−28.2631.4 - 28.26, or about 3.14 cm.
Lesson 5
Circumference and Wheels
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Biking Distance (1 problem)

The wheels on Noah's bike have a circumference of about 5 feet.

  1. How far does the bike travel as the wheel makes 15 complete rotations?
  2. How many times do the wheels rotate if Noah rides 40 feet? 
Show Solution
  1. 75 feet, because 5⋅15=755 \boldcdot 15 = 75
  2. 8 rotations, because 40÷5=840 \div 5 = 8
Section A Check
Section A Checkpoint
Problem 1

Select all the equations that correctly state a relationship between the radius (rr), diameter (dd), and circumference (CC) of a circle.

Show Solution
D, E, F
Problem 2
A wagon wheel has a radius of 21 inches. What is the circumference of the wheel? Explain or show your reasoning.
Show Solution
About 132 inches. 21⋅2=4221 \boldcdot 2 = 42, and 42⋅3.14≈13242 \boldcdot 3.14 \approx 132.
Lesson 6
Estimating Areas
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The Area of Alberta (1 problem)

Estimate the area of Alberta in square miles. Show your reasoning.

<p>A map of Alberta. Lengths of sides starting at top and clockwise direction, 410 mi, 760 mi, 180 mi, unknown, 470 mi.</p>

Show Solution

About 250,000 square miles. Sample reasoning: Alberta can be surrounded with a 410-mile-by-760-mile rectangle with a 290-mile-by-230-mile triangle removed in the lower left corner. The answer has been rounded because the part missing in the lower left is not exactly a triangle.

Lesson 7
Exploring the Area of a Circle
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Areas of Two Circles (1 problem)
  • Circle A has a diameter of approximately 20 inches and an area of 300 in2.
  • Circle B has a diameter of approximately 60 inches.

Which of these could be the area of Circle B? Explain your reasoning.

  1. About 100 in2
  2. About 300 in2
  3. About 900 in2
  4. About 2,700 in2
Show Solution

D. About 2,700 in2. Sample reasoning: The diameter of Circle B is 3 times bigger than the diameter of Circle A, so the area of Circle B is larger than the area of Circle A. The pattern shows that the area grew quickly, so 900 is probably not large enough. The radius of Circle B is 30 inches, so the area is about 3⋅302 in23 \boldcdot 30 ^ 2 \text{ in}^2 (and is definitely more than 30230^2 because a square of side 30 inches fits inside the circle with a lot of space left).

Lesson 8
Relating Area to Circumference
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A Circumference of 44 (1 problem)

A circle’s circumference is approximately 44 cm. Complete each statement using one of these values:

7, 11, 14, 22, 88, 138, 154, 196, 380, 616

  1. The circle’s diameter is approximately ‾\underline{\hspace{.5in}} cm.
  2. The circle’s radius is approximately ‾\underline{\hspace{.5in}} cm.
  3. The circle’s area is approximately ‾\underline{\hspace{.5in}} cm2.
Show Solution
  1. 14
  2. 7
  3. 154
Lesson 9
Applying Area of Circles
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Area of an Arch (1 problem)

Here is a picture that shows one side of a child's wooden block with a semicircle cut out at the bottom.

<p>The face of an arch-shaped block.</p>
The face of an arch-shaped block. The horizontal side of the block is labeled 9 centimeters and the vertical side of the block is labeled 4.5 centimeters. A semi circle with diameter labeled 5 centimeters is removed from the block.

Find the area of the side. Explain or show your reasoning.

Show Solution

The area of the side of the block is about 30.68 cm2. The area of the rectangle is 9⋅4.59 \boldcdot 4.5, or 40.5 cm2. The area of a circle with a diameter of 5 cm is 6.25π6.25\pi cm2. The front face of the wooden block is a rectangle missing half of circle with diameter 5 cm, so its area in cm2 is 40.5−3.125π40.5 - 3.125 \pi or about 30.68.

Section B Check
Section B Checkpoint
Problem 1

Lin measured the diameter and circumference of a circle. Then she used her measurements to calculate the area.

Han measured the diameter and circumference of a different circle.

diameter (in) circumference (in) area (in2)
Lin’s circle 6 19 28.5
Han’s circle 3 9.5 ?

Han thinks the area of his circle is 14.25 in2. Do you agree? Explain or show your reasoning.

Show Solution

Sample responses:

  • No, the area of a circle is not proportional to the diameter. Since the diameter of Han’s circle is one-half the diameter of Lin’s circle, the area of Han’s circle will be (12)2(\frac12)^2 the area of Lin’s circle.
  • No, the area of Han’s circle is about 7 in2. Possible strategies:
    • The radius is 1.5 inches because 3÷2=1.53 \div 2 = 1.5.
      Area=12circumference⋅radius\text{Area} = \frac12 \text{circumference} \boldcdot \text{radius}
      Area=12(9.5)⋅(1.5)\text{Area} = \frac12 (9.5) \boldcdot (1.5)
      Area=7.125\text{Area} = 7.125
    • The radius is 1.5 inches because 3÷2=1.53 \div 2 = 1.5.
      A=πr2A = \pi r^2
      A=3.14∗(1.5)2A = 3.14 * (1.5)^2
      A=7.065A = 7.065
Problem 2

What is the area of the shaded region?

A.$3\pi$ square units
B.$9\pi$ square units✓
C.$12\pi$ square units
D.$36\pi$ square units
Show Solution
B
Lesson 10
Distinguishing Circumference and Area
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Measuring a Circular Lawn (1 problem)

A circular lawn has a row of bricks around the edge. The diameter of the lawn is about 40 feet.

An image of a circular lawn with a row of bricks that completely go around the edge without gaps or overlap.

  1. Which is the best estimate for the amount of grass in the lawn?

    1. 125 feet
    2. 125 square feet
    3. 1,250 feet
    4. 1,250 square feet

  2. Which is the best estimate for the total length of the bricks?

    1. 125 feet
    2. 125 square feet
    3. 1,250 feet
    4. 1,250 square feet
Show Solution
  1. D. 1,250 square feet
  2. A. 125 feet
Lesson 11
Stained-Glass Windows
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No cool-down
Unit 3 Assessment
End-of-Unit Assessment