Maryland CCRG Algebra Task Project
Round and Round We Go!
Common Core Standard
F.TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude,
frequency, and midline.
Common Core Traditional Pathway: Algebra II, Unit 2
Common Core Integrated Pathway: Mathematics III, Unit 3
The Task
A group of eight friends attend a summer carnival and decide to ride the Ferris wheel. John, one
person from the group, observed that the ride traveled in a counterclockwise motion at a constant
speed and took 36 seconds to make a complete revolution. Since the wheel will be revolving at
a constant speed, your team will be developing a model for the height of the Ferris wheel based
on the number of seconds the ride is in motion.
Materials needed to construct a physical model: foam disk, floral stem wire (sturdier (20
gauge)), ruler, and borrow a table vice grip from science dept or technology education
department
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Maryland CCRG Algebra Task Project
1. Hook the vice grip to the table and secure the floral stem wire with the grip so that the wire
will be through the center of the foam disk. (You may need to bend the floral stem wire at a
90 degree angle to produce an appropriate height off the table. Next insert another wire into
the center of the foam disk and bend at a 90 degree angle to form a crank). Mark the edge of
the foam disk that will be used to measure heights off the ground. Set the initial position of
the marker on the disk to be at the bottom of the disk. Measure the height of the mark from
the table/surface.
2. A standard carnival Ferris wheel takes 36 seconds to make one full rotation. Turn the disk
counterclockwise one quart of a revolution. This would represent the position after 9
seconds of motion. Measure the height of the mark from the table. Continue move the disk
and measure heights based on time the Ferris wheel is in motion. Create an equation to
model the height in feet above the ground a rider is based on time in seconds.
3. The standard carnival Ferris wheel has a diameter of 120 feet. The bottom of the wheel
stands 10 feet off of the ground. Use the equation and data from the model Ferris wheel
created and write an equation to model the full sized Ferris wheel.
Facilitator Notes
1. Have students work in small groups to complete the lab investigation. If you are unable
to complete a hands-on lab, provide groups with data sets to analyze and generate
models.
2. Allow groups to compare data, checking to see how close the data sets are. Have
students discuss potential sources of error in measurements.
3. After students have determined a model and are able to justify why their model is
appropriate, have students complete the extension activity.
Extension Activities
After the ride John and his friend Sarah remembered what they learned in Algebra 2 that
previous school year regarding periodic functions. They wanted to write an equation that
represented their vertical distance in feet above the ground as a function of time in seconds
during the ride. Before leaving the carnival Sarah asked the operator of the ride a few questions.
Sarah and John both worked on their problem during the night. John called Sarah the next
morning with his results.


John came up with the function h  20 sin  x    25 , where t represents the time in
 18
2
seconds and h is the height above the ground in feet. Sarah was confused with John’s answer
and explained to him that she derived a different solution. Sarah’s equation is
 
h  20 sin  x   25 .
 18 
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Maryland CCRG Algebra Task Project
1. Did one of the students make a mistake in their work? Explain how they could have
arrived different solutions if they were on the same ride.
2. What questions did Sarah ask the operator? Why were these questions necessary in
order for her and John to write their equations?
3. Sketch what the Ferris wheel ride at the carnival looked like. Include any important
information you would need to know to get the equations the two students found.
4. Mark, another one of the friends who rode the Ferris wheel, entered a pie eating
contest that begins at 8:30. The friends were loading the Ferris wheel at 8:21 and the
ride was to begin its constant movement at 8:23. If the ride makes ten full revolutions
and it takes 2 minutes to unload riders, will Mark make it off the ride in time for the
contest? Explain your answer.
5. Sarah and John told their friend Elizabeth about their equation writing. Elizabeth
went to a different carnival the following weekend and did the same thing. Elizabeth
 
came up with the equation h  15 cos  x   30 . What was different about the
 18 
Ferris wheel Elizabeth rode in comparison to the one John and Sarah rode? Where
was Elizabeth sitting when the ride started? Which Ferris wheel was moving at a
faster speed?
Solutions
While there are many approaches students may use to solve this task, here are a few
sample answers:
Students will need to have knowledge of the Unit Circle, the basics of sine and cosine curves, as
well as the definitions to amplitude, period, and midline in order to answer the questions
following this task. Students can answer these questions in a variety of ways and have to make
connections between what is occurring on the ride in relation to these functions. Students will
use their knowledge regarding transformations of functions learned in Algebra 1 to answer
certain questions.
Part I: Data Gathering Lab Investigation
1. Students can work independently or in groups for this part of the assignment.
2. The answers will vary for this question depending on the size of the foam disc that is
used. Students should recognize that heights will repeat every 360 degrees or 2π
radians. Students will need to use their knowledge of diameter of a circle and relate it
to amplitude and midline of a periodic function. Students will also need to use the fact
that a wheel takes 36 second to make one full rotation and relate this to the period a
trigonometric function.


3. Students should get the equation y  60 sin  x    70
 18
2
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Maryland CCRG Algebra Task Project
The answers can be found using technology or by hand. If students used technology they can get
the answers using the steps below.
Students will use the data from the table in question #2 but change the heights based on the
actual size of a Ferris wheel.
Students can also use their knowledge of period, amplitude, and midline to find the equation of
the function by hand. In order to do this, students must have prior knowledge on trigonometric
functions.
.
Extension Activity:
1. Did one of the students make a mistake in their work? Explain how they could have
arrived different solutions if they were on the same ride. One of the students would
have made a mistake if John and Sarah were sitting in the same seat. Students should
recognize that since John and Sarah arrived at different solutions they were at different
starting points on the Ferris wheel. Further investigation will show that John was sitting
at the bottom seat of the Ferris wheel and Sarah was a quarter of a rotation on the wheel
when it started.
2. What questions did Sarah ask the operator? Why were these questions necessary in
order for her and John to write their equations? In order to find a trigonometric
function of a circle in motion, students need to know the period, amplitude and midline of
this periodic function. Therefore Sarah would have needed to ask the operator how long
it takes one revolution of the wheel, the diameter of the wheel, and the height of the
center of the wheel. The diameter will give us the rider’s height after a certain amount of
seconds.
3. Sketch what the Ferris wheel ride at the carnival looked like. Include any important
information you would need to know to get the equations the two students found.
Answers here will vary but students should include that the wheel has a diameter of 40
feet, the wheel’s center is 25 feet above ground, and the bottom of the wheel is 5 feet
above the ground. They should also draw an arrow showing that the Ferris wheel is
moving in a counter clockwise direction and completes one full revolution in 36 seconds.
Students can use the equations given from Sarah and John to find this. Two methods
below are ways students can go about this.
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Maryland CCRG Algebra Task Project
Method 1: Using knowledge of period and amplitude students can use the equations
to find this.
2
2
Period: Using the formula Period =
students can calculate
 36 . This means

b
18
the period of the function is 36 which means the Ferris wheel will complete one
revolution in 36 seconds.
The value of 25 being added on at the end of the equation will vertically shift our
graphing by 25 units. Students will recognize that this will move the midline of our
periodic function to 25. Making connections previously learned the center of the wheel
stand at 25 feet.
Amplitude: The amplitude is defined by the highest point subtract the lowest point
and divide that answer by 2. Given the equation, the amplitude is defined by the “a”
value which is 20. Multiplying that value by 2 will give us the diameter of the wheel
which is 40 feet. Or students could add half the diameter to the center of the circle.
Method 2: Students can use technology in order to find the period and amplitude.
Once students recognize that the period is 36 seconds, changing the scale to 9 gives a good
representation of a rider’s height every 90◦ or π radians.
The tables give students a good idea of a rider’s height at different times. This will help students
make their sketch.
4. Mark, another one of the friends who rode the Ferris wheel, entered a pie eating
contest that begins at 8:30. The friends were loading the Ferris wheel at 8:21 and the
ride was to begin its constant movement at 8:23. If the ride makes ten full
revolutions and it takes 2 minutes to unload riders, will Mark make it off the ride in
time for the contest? Explain your answer. This question can be yes or no depending
on how student look at it. If the ride starts at 8:23 and each revolution takes 36 seconds
the ride will last for 360 seconds or 6 minutes. The ride will not be over until 8:29. The
unloading of individual riders depends on where they were seated at the start of the ride.
If Mark was sitting at the bottom of the ride with John, he would be the first to be let off.
It he was anywhere to the counterclockwise of that seat most likely he will not make it.
This question can promote class discussion on what sections of the wheel he could be on
to have a better chance of making the contest.
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Maryland CCRG Algebra Task Project
5. Sarah and John told their friend Elizabeth about their equation writing. Elizabeth
went to a different carnival the following weekend and did the same thing. Elizabeth
 
came up with the equation h  15 cos  x   30 . � What was different about the
 18 
Ferris wheel Elizabeth rode in comparison to the one John and Sarah rode? Where
was Elizabeth sitting when the ride started? Which Ferris wheel was moving at a
faster speed? Students will first recognize that Elizabeth used a cosine function instead
of a sine function. Students will have previously learned that sine and cosine are very
similar and that a horizontal shift of either curve will make them the same. Below are two
method students can use to answer the questions for #5.
Method 1: This function has amplitude of 15 and has the same period as the first
function. Students can use the same methods as used for #3 to find this. Since the
graph of this function has a vertical shift of 30, the center of the wheel will be 30 feet
in the air. Since the amplitude is 15 the max height of the wheel is 45 feet and the
minimum height is 15 feet.
Elizabeth will have been sitting at the top of the Ferris wheel, 45 feet in the air.
Students can find this by substituting 0 in for t.
The Ferris wheel that John and Sarah rode on was going faster. The diameter of John
and Sarah’s Ferris wheel was 40 feet and the diameter of Elizabeth’s Ferris wheel
was 30 feet. Since both wheels made one revolution in 36 seconds, the one with the
larger diameter had to be going faster.
Method 2: Students can use technology to answer the questions based on the equation
given.
Using technology, students can use the table to see that the period is 36 seconds, the amplitude is
15 feet, and the midline is 30 feet.
MSDE has licensed this product under Creative Commons Non-Commercial. For more
information on this license, refer to: http://creativecommons.org/licenses/by-nc/3.0/.