standard_description: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
problem_activity: Activity
In the previous lesson, students learned the definition of amplitude and midline by examining sine graphs produced to model the vertical position of a point on a rotating windmill. This activity keeps the windmill context and asks students to interpret both cosine and sine equations. They will need to focus on three different aspects of the equations:
The amplitude of the sine graph (length of the windmill blade)
The midline of the sine graph (height of the center of the windmill blades)
The direction the windmill spins (clockwise or counterclockwise).
Launch
Arrange students in groups of 2. Display an image of the blades of a windmill for all to see, such as this one from a previous lesson:
[IMG]
Tell students that these blades extend 4 meters from the center and that the center of the windmill is 20 meters off the ground. The windmill rotates counterclockwise.
Ask students to write an equation to represent the height in meters of the point
\(W\)
, relative to the ground, as a function of the angle
\(\theta\)
of rotation (
\(h = 4\sin(\theta) + 20\)
). Select 2–3 students to share how they wrote their equation. Tell them that in this activity they are going to work the opposite direction. That is, they are going to start with an equation giving the vertical position of a point
\(W\)
at the tip of a windmill blade and use it to describe the windmill.
Conversing: MLR8 Discussion Supports.
Use this routine to help students describe the three different windmills based on their equations. In their groups of 2, invite Partner A to begin with this sentence frame: “The windmill _____ because . . .”  and “The positive of negative sign tells me _____ because . . . .” Invite the listener, Partner B, to press for additional details and ask clarifying questions, such as “What is the amplitude of the windmill?” and “What is the midline of the windmill?” Students should switch roles for each equation. This will help students interpret sine equations in the context of a windmill by referring to specific features of the equations.
Design Principle(s): Support sense-making; Cultivate conversation
Representation: Access for Perception.
Provide students with a physical copy of the image displayed and information provided during the Launch. Check for understanding by inviting students to rephrase the information in their own words. Invite students to label the image based on the information provided. Keep the display of the image and information visible throughout the activity.
Supports accessibility for: Language; Memory
Student Facing
Here are three equations for three different windmills. Each equation describes the height
\(h\)
, in feet above the ground, of a point at the tip of a blade of a different windmill. The point is at the far right when the angle
\(\theta\)
takes the value 0. Describe each windmill and how it is spinning.
\(h = 2.5 \sin(\theta) + 10\)
\(h = \frac{4}{5} \sin(\theta) + 3\)
\(h = \text-1.5 \sin(\theta) + 5\)
Student Response
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Anticipated Misconceptions
If students struggle with the negative amplitude in the final question, encourage them to substitute in a few angle values
\(\theta\)
and plot them.
Activity Synthesis
Display the 3 equations for all to see. Begin the discussion by asking, “What is the midline and amplitude of each function and what do these values mean in this context?” (The midline is the height of windmill blades’ center and the amplitude is the length of the blades.) Record student responses next to the equations.
Ask students how they interpreted the sign in front of the sine on the last equation (it means that the windmill is spinning in the clockwise direction as the angle
\(\theta\)
increases). To help students visualize the impact of the negative sign in front of the expression, note that while
\(\sin(\theta)\)
increases from 0 to 1 as
\(\theta\)
increases from 0 to
\(\frac{\pi}{2}\)
,
\(\text-\sin(\theta)\)
decreases from 0 to -1 as
\(\theta\)
increases from 0 to
\(\frac{\pi}{2}\)
. In terms of the windmill, this would mean that instead of going toward the highest spot on its circle of rotation, the point
\(W\)
begins by going to the lowest spot on its circle of rotation. This means that the windmill is rotating in a clockwise direction instead of a counterclockwise direction.


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