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.
What is the same about the two circles? What is different?
What is the length of segment AD? How do you know?
On the first circle, what segment is a diameter? How long is it?
Show Solution
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.
Segment AD is 4 cm long because it is also a radius of the circle.
The diameter, segment EB, is 8 cm long.
Lesson 4
Applying Circumference
Circumferences of Two Circles
Circle A has a diameter of 9 cm. Circle B has a radius of 5 cm.
Which circle has the larger circumference?
About how many centimeters larger is it?
Show Solution
Circle B has the larger circumference. Circle A has a diameter of 9 cm, and Circle B has a diameter of 5⋅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.
The difference is about 3.14 cm because the circumference of Circle A is 9π, or about 28.26 cm, and the circumference of Circle B is 10π, or about 31.4 cm. The difference is 31.4−28.26, or about 3.14 cm.
Lesson 5
Circumference and Wheels
Biking Distance
The wheels on Noah's bike have a circumference of about 5 feet.
How far does the bike travel as the wheel makes 15 complete rotations?
How many times do the wheels rotate if Noah rides 40 feet?
Show Solution
75 feet, because 5⋅15=75
8 rotations, because 40÷5=8
Section A Check
Section A Checkpoint
Problem 1
Select all the equations that correctly state a relationship between the radius (r), diameter (d), and circumference (C) of a circle.
A.C=πr
B.d=πr
C.C=2d
D.C=πd
E.r=21d
F.C=2πr
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=42, and 42⋅3.14≈132.
Lesson 6
Estimating Areas
The Area of Alberta
Estimate the area of Alberta in square miles. Show your reasoning.
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 9
Applying Area of Circles
Area of an Arch
Here is a picture that shows one side of a child's wooden block with a semicircle cut out at the bottom.
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⋅4.5, or 40.5 cm2. The area of a circle with a diameter of 5 cm is 6.25π 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 40.5−3.125π 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 (21)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=1.5. Area=21circumference⋅radius Area=21(9.5)⋅(1.5) Area=7.125
The radius is 1.5 inches because 3÷2=1.5. A=πr2 A=3.14∗(1.5)2 A=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
$9\pi$ square units
Lesson 10
Distinguishing Circumference and Area
Measuring a Circular Lawn
A circular lawn has a row of bricks around the edge. The diameter of the lawn is about 40 feet.
Which is the best estimate for the amount of grass in the lawn?
125 feet
125 square feet
1,250 feet
1,250 square feet
Which is the best estimate for the total length of the bricks?
125 feet
125 square feet
1,250 feet
1,250 square feet
Show Solution
D. 1,250 square feet
A. 125 feet
Lesson 11
Stained-Glass Windows
No cool-down
Unit 3 Assessment
End-of-Unit Assessment
Problem 1
A circle has a radius of 50 cm. Which of these is closest to its area?
A.
157 cm2
B.
314 cm2
C.
7,854 cm2
D.
15,708 cm2
Show Solution
7,854 cm2
Problem 2
The shape is composed of three squares and two semicircles. Select all the expressions that correctly calculate the perimeter of the shape.
A.
40+20π
B.
80+20π
C.
120+20π
D.
300+100π
E.
10+10+10π+10+10+10π
Show Solution
A, E
Problem 3
Select all of the true statements.
A.
π is the area of a circle of radius 1.
B.
π is the area of a circle of diameter 1.
C.
π is the circumference of a circle of radius 1.
D.
π is the circumference of a circle of diameter 1.
E.
π is the constant of proportionality relating the diameter of a circle to its circumference.
F.
π is the constant of proportionality relating the radius of a circle to its area.
Show Solution
A, D, E
Problem 4
A class measured the radius and circumference of various circular objects. The results are plotted on the graph.
Does there appear to be a proportional relationship between the radius and circumference of a circle? Explain or show your reasoning.
Why might the measured radii and circumferences not be exactly proportional?
Graph of 4 plotted points, origin O, with grid. Horizontal axis, radius in cm, scale 0 to 10, by 1’s. Vertical axis, circumference in cm, scale 0 to 45, by 5’s. Plotted points at 3 comma 18, 4 comma 25, 6 comma 38, 7 comma 44.
Show Solution
Yes. Explanations vary. Sample explanation: If you divide each circumference by its radius, you get the numbers 6, 6.25, approximately 6.33, and approximately 6.29. These numbers are close enough that they are evidence of a proportional relationship between circumference and radius.
The measurements were taken using rulers that have only so much accuracy. Students needed to round their answers to the nearest ruler marking, or perhaps rounded even less accurately than that. Also, students probably didn’t hold the rulers perfectly still or perfectly straight.
Minimal Tier 1 response:
Work is complete and correct.
Acceptable errors: in Part A, writing or implying that the points are collinear.
Sample:
(With accompanying line drawn in) Yes, because the points are on a line that goes through (0,0).
The points are not exact because of error in measurement.
Tier 2 response:
Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Division errors make it look as if the ratios are not similar enough to indicate a constant of proportionality, or explanation in Part B does not appeal to measurement error in some way.
Tier 3 response:
Significant Errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: errors types from Tier 2 response on both problem parts, or explanation in Part A does not appeal to facts students know about proportional relationships.
Problem 5
For each quantity, decide whether circumference or area would be needed to calculate it. Explain or show your reasoning.
The distance around a circular track.
The total number of equally-sized tiles on a circular floor.
The amount of oil it takes to cover the bottom of a frying pan.
The distance your car will go with one rotation of the wheels.
Show Solution
Circumference. The distance around the track is the circumference of the circular track.
Area. The number of tiles it takes to cover the floor times the area of each tile is the area of the floor.
Area. The pan is circular and the entire circular surface is being covered in oil. To know how much oil is used, we need to know the area of the circle (as well as the thickness of the layer of oil).
Circumference. The distance the car goes in one rotation is the distance around (circumference of) the tires.
Minimal Tier 1 response:
Work is complete and correct.
Acceptable errors: An illuminating drawing can take the place of a verbal explanation.
Sample:
Circumference, because around the track means around the circle.
Area, because covering a surface is about area.
Area, because you cover the inside of the pan with oil, not just the rim.
Circumference, because when a tire rolls it’s only the outside that counts.
Tier 2 response:
Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Correct answers with no explanation or misguided explanation, one incorrect answer.
Tier 3 response:
Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Two or more incorrect answers, two or more answers with very poor explanation.
Problem 6
This figure is made from a part of a square and a part of a circle.
A figure composed of a part of a square and a part of a circle. Top, horizontal segment = 10 units. Left side, vertical segment = 10 units. Bottom, from left to right, horizontal segment = 5 units, then curved portion connecting the bottom to the right side. Right side, from bottom to top, curved portion, vertical segment = 5 units. The center of figure has a right angle with dotted line segments connecting to the bottom and right sides of the figure in the bottom right quadrant of the figure.
What is the perimeter of this figure, to the nearest unit?
What is the area of this figure, to the nearest square unit?
Show Solution
38 units (The quarter-circle’s perimeter is 41⋅2⋅π⋅5 units. The rest of the perimeter is 30 units. The total perimeter is approximately 37.9 units.)
95 square units (The quarter-circle’s area is 41⋅π⋅52 square units. The rest of the area is 75 square units. The total area is approximately 94.6 square units.)
Problem 7
A groundskeeper needs grass seed to cover a circular field that is 290 feet in diameter.
A store sells 50-pound bags of grass seed. One pound of grass seed covers about 400 square feet of field.
What is the smallest number of bags the groundskeeper must buy to cover the circular field? Explain or show your reasoning.
Show Solution
4 bags. The field’s size is π⋅1452 square feet, just over 66,000 square feet. Each pound of seed covers 400 square feet; each 50-pound bag covers 20,000 square feet. The number of bags needed is given by:
20,000π⋅1452≈3.30
It is not possible to purchase 3.3 bags, and 3 bags is not enough. It takes 4 bags of grass seed to cover the field.
Minimal Tier 1 response:
Work is complete and correct, with complete explanation or justification.
Sample: The area of the field is π⋅1452 square feet, and each bag covers 20,000 square feet. 3 bags cover 60,000 square feet and that's not enough. 4 bags cover 80,000 square feet and that is enough.
Tier 2 response:
Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Correct calculation (about 3.3 bags) but incorrect use of context gives answers of 3 or 3.3 bags; incorrect calculation of the number of square feet per bag but otherwise correct work, including correct use of contextual rounding; calculation errors, but not errors in formula application, when determining the size of the field or the number of bags; incorrect calculations in determining the number of bags when using a strategy that does not involve dividing.
Tier 3 response:
Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: Incorrectly determining the area of the field using circumference, or using 290 as the radius; incorrect type of calculation performed on the bags, such as dividing in reverse order; two or more error types from Tier 2 response.
Acceptable errors: Any response giving the correct area of the field earns at least a Tier 3 response, regardless of other work. Any response giving an incorrect area of the field, but correct work on the proportional relationship based on that incorrect area, earns at least a Tier 3 response.
Tier 4 response:
Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Incorrectly determining the area of the field, along with incorrect work on the proportional or contextual relationship; answer without explanation, regardless of accuracy.