HCPSS Worthwhile Math Task
Ceiling Fan vs. The Earth
Common Core Standard
Goal. The student will demonstrate the ability to define trigonometric ratios and apply
trigonometry to solve real-world problems.
Objectives: The student will be able to:
g. Use trigonometric functions to model and solve real-world problems, including right
triangle relations, arc length, speed, and uniform circular motion.
MP2:
MP3:
MP4:
MP5:
MP7:
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Look for and make use of structure.
Trigonometry Honors, Unit 2
The Task
A blade of a ceiling fan rotates 30 times in one minute and Earth rotates once every 24 hours.
The radius of the fan is 2ft. and the diameter of Earth is approximately 8,000 miles. Which is
traveling faster- a particle moving on the fan blade or a particle moving on Earth?
Facilitator Notes
1. Students need a basic understanding of linear and angular speed.
2. Have students form a hypothesis individually before starting their calculations. (Look for
evidence of MP2.)
3. Start a class discussion where students discuss their hypotheses (Look for evidence of
MP3).
4. For consistency have students convert angular velocity to radians per hour and linear
velocity to miles per hour. (Look for evidence of MP5, MP4 and MP7.)
5. Students may work individually, in pairs, or small groups of three. (Look for evidence of
MP3 and MP4.)
6. Before students start the extension questions, have students compare their hypotheses
with their calculations in their groups and then with the class. Were you correct? Why or
why not? (Look for evidence of MP3.)
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.
HCPSS Worthwhile Math Task
7. After students complete the extension questions have them share their findings within
their small groups to determine how practical their fan is. (Look for evidence of MP5,
MP4 and MP7.)
Follow-Up Questions
1. Was your hypothesis correct? Why or why not?
2. Is it possible to design a fan where the particle on it has the same linear velocity as the
particle of the earth?
Solutions
Sample Hypothesis: I think that the earth will have a larger angular and linear velocity because it
is a larger object.
Fan angular velocity:
Earth angular velocity:
  Rotations  2 
 = 30  2 
 = 60 


=
  Rotations  2 
1
 2
24
2

=
or
24
12

t
60 
=
1
  60  radians per minute
t

 = 12
1




12
radians per hour
To convert to radians per mile:
  60  60
  3600  radians per hour
Fan linear Velocity:

Earth linear Velocity:

Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.
HCPSS Worthwhile Math Task
v  r
v  r
v  2  3600 
v  7200  feet per hour
7200 
v=
miles per hour
5280
15
v =  or 4.28mph
11
v  4000 

12
1
v  333  miles per hour
3
v = 1047.20mph


Follow Up Questions:
My hypothesis was partially correct. The angular velocity of the fan was larger since the radius
does not affect it. The linear velocity of the earth is greater since the radius of the earth is so
much greater.
To design a fan that could have the same linear velocity we need to work backwards and
determine the possible rotations per minute. Lets start with a fan that has the same radius.
v = 1047.20mph
1
v  333  mph
3
v  1,760,000  feet per hour
1,760,000 

2
  880,000  radians per hour
880,000 
rotations per hour
2
440,000 rotations per hour
1
7333 rpm
3
For a fan with a radius of 2 would require an extremely fast rpm. Have students try various blade
lengths to lower the rpm’s.

It is possible to design a fan that has the same linear speed as the earth, but the amount of stress
on the motor would break the fan.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.