Alg2 Lesson 4T.4: A Model for Circular Motion

Lesson 4T.4: A Model for

Circular Motion

Obj: I can construct a representation of a
sinusoidal function.

EQ: How can I approximate the value of a trig
expression?

Lesson Goals:
● Creative Thinking
● Talk through controversies and conflict
● Recognize and reduce ambiguity
● Encourage thinking based on formulas and prior info
● Help explain ideas to each other
● Own your ideas and work
● Record ideas in your journal
● Answer Questions on Slides
● Follow your team roles

Roles:
Facilitator
Scribe
Resourcer
Includer

Facilitator

• Make sure that all peers are staying on task.

• Give advice or suggestions to resolve the problem.

• Be sure everyone is able to explain.

Scribe

• Make sure peers organize their results on their own papers.

• Remind peers to use color, arrows, and other math tools to
communicate your mathematics, reasons, and connections.

• Be ready to join the teacher for a huddle.

Resourcer

• Make sure peers are getting the materials needed.

• Make sure that all materials are put away neatly.

• Make sure that peers are logged in to the needed site.

• Help troubleshoot any technology difficulties that may arise.

Includer

• Make sure that all peers are talking about their work.

• Helps keep peers’ voice volume low.

• Encourages everyone to ask questions.

• Communicates conflicts or questions to the teacher.

● Check off tasks & skills on calendar.

● Select skills to work on.

● Work on Deltamath.

Remember to work on the following too…

Part 1: Creating a Model of

a Ferris Wheel Car

Multiple Choice

What does it mean for an angle to measure 2π radians?

• An angle of 2π radians represents a full rotation (360 degrees) around a circle.
• An angle of 2π radians is half a rotation (180 degrees) around a circle.
• An angle of 2π radians measures 45 degrees.

• An angle of 2π radians indicates a quarter rotation (90 degrees) around a circle.

Drag and Drop

The degree measure of an angle is ​

Placeholder

proportional to the radian measure of an angle. The constant of proportionality is π/​

Placeholder

Drag these tiles and drop them in the correct blank above

directly

inversely

jointly

180

90

360

45

Match

Match the radians and degrees.

• 2π/5

• 3π/4

• 5π/2

• 4π/3

• π/2

72°

135°

450°

240°

90°

• 72°

• 135°

• 450°

• 240°

• 90°

Fun-Time Amusement Park is building Ferris wheels in an unusual configuration: The lower
half of each Ferris wheel will be underground, while the upper half will be above ground.
Each Ferris wheel has 12 passenger cars equally spaced around its outer edge. The
amusement park has decided to build two such Ferris wheels at the park, with the smaller
one for children and the larger one for teens and adults. The loading platform for the Ferris
wheel will be at ground level. This engineer’s model for the Ferris wheel is not complete. You
will draw a model of one of these Ferris wheels. Your drawing should include the height of
each car when the Ferris wheel is in a stationary position as shown in the figure.

​

Considerations for the Ferris Wheel Problem

What are the quantities you are investigating? The quantities we are investigating the angle measures of the Ferris wheel car and the corresponding heights of the car relative to the ground.

The angle measure of the Ferris wheel car should be measured in radians counterclockwise from the loading platform, and the height of the car relative to the ground should be measured in centimeters.

Since the circular model has a radius of r cm, the maximum height of a Ferris wheel car is r cm and the minimum height of the car is –r cm.

It makes sense to draw a curve through the points because the Ferris wheel car achieves every angle measure from 0 to 2π radians (or from 0° to 360°) as it turns. The heights of the Ferris wheel car would follow a predictable pattern along the curve.

​

​h(2) is the height of the Ferris wheel car when the angle of the car is 2 radians. The value of h(2) is positive because an angle measure of 2 radians sweeps out an arc length of 2 radii. A 2-radii arc length has a height that is above ground. Examining the graph suggests that a value for h(2) is approximately 2.7 cm for a circular model with a radius of 3 cm.

Dropdown

Our table of values shows the heights of the car when it is at various
angles within one full rotation of the Ferris wheel. Suppose that we

Select answer

the
heights of the car in the first rotation for coterminal angles. This is because

Select answer

.

Part 2: Defining the Sine

Function

Suppose that we measured the height of the Ferris wheel car
in a nonstandard unit of measure, like the radius of the circle,
instead of in centimeters. Complete each of the following:

(a) Create your own custom ruler whose length is 2 radii, and
then partition each radius length into at least 8 equal parts.

(b) Use your new ruler to measure the height of the Ferris
wheel car at the 12 angle measurements from your original
table of values.

(c) Plot the new data and connect the data points with a
curve.

(d) Compare your new graph to your old graph. How are they
the same? How are they different?

https://www.desmos.com/calculator/v9g6mwgyv0

Part 3: Defining the Cosine

Function

Wind power is a source of renewable energy. The principle of how a wind turbine works is straightforward: Wind turns a mechanism that looks like an airplane propeller. The blades of the mechanism are connected to a rotor, which is simply a machine that rotates. The rotor spins an electric generator, which creates electricity. A modem wind turbine usually has three long blades equally spaced around the rotor, as shown in the following figure:

https://www.desmos.com/calculator/jnyqphdf9j

The sine function graph intersects the
y-axis at (0, 0) while the cosine function

graph intersects the y-axis at (0, 1).

Handout 4T.4: Practice Using Sine and Cosine

Random Question of the Day Time

https://wheelofnames.com/4ke-epz We’ll spin
the wheel as a class and spend a minute or
so discussing our answers.