Authors: Kelsey Shannon, Caroline Beshara, Jamie Jeske
November 12/14, 2013
LC DRAFT revision
Pre-AP Pre-Calculus 10 th , 11 th , &12 th Grade
Students will have knowledge of Trigonometric functions sine, cosine, tangent, cotangent, secant, and cosecant.
They will be able to define and graph these trigonometric functions as well as be able to transform them. They will be able to determine amplitude, frequencies, period, maximum, and minimum from trigonometric equations. The students will also be able to connect the concept of the unit circle to trigonometric functions.
Students will be able to make connections from real-world data to their previous knowledge of the previously mentioned topics.
Students will use critical thinking skills to solve real-world problems using their knowledge of trigonometry.
Students will model real-life data to trigonometric functions.
Trigonometric Functions
Amplitude
Frequency
Period
Rotation
Linear/Nonlinear
Transformation
Maximum/Minimum
Use functions such as logarithmic, exponential, trigonometric, polynomial, etc. to model real-life data
Materials:
Pencil & paper (Students will already have)
Student Activity Handout (Already created, will print copies the day before)
Student pre-/post-test (Already created, will print copies the day before)
Ferris wheel created from an erector set
Pretest (10 minutes)
(Fun Ferris wheel GIF will be displayed on the projector to peek student interest) http://25.media.tumblr.com/tumblr_lki7os9eS21qjcwy3o1_r1_500.gif

Introduce Problem (1-2 minutes)
There will be a Ferris wheel made from an erector set in the front of the classroom. Present question on the projector
“How can you determine how high a certain car is from the ground at any given time?”
Brainstorming (5-10 minutes)
Encourage them to make their own Know-Need to Know chart
Teacher Questions
What do you know? What do you need to know?
Why do we need to know that…?
Possible Student Responses
We only know that it’s a ferris wheel. We need to know the measurements.
We need measurements to make an equation and calculations
Discuss as a class the Know-Need to Know chart (5-10 minutes)
Instructors will answer any student questions. Students will come to the conclusion that they will need to graph the points. They will realize that they need to make measurements to go any farther. One volunteer will come up in front of the class and measure what the students deemed necessary. (Teacher will supply time it takes for the
Ferris wheel to make one rotation.)
While students create the Know-Need to Know chart, teacher assesses student understanding and addresses misconceptions and confusion as needed. Instructor will use Ferris wheel model to demonstrate the dimensions that the numbers are representative of. For example, the instructor will show why the vertical shift is that specific number, because that number is the distance from the ground to the center of the Ferris wheel (circle).
Possible Student Questions
Why do we need to do this?
What does it matter how high we are?
How do you figure this out?
What happens if we graph these points?
Teacher Responses
You need to know this because it helps you learn how to solve problems in the real world.
It doesn’t matter how high you are, it just matters that you have the knowledge to figure it out
Well think about what we have recently learned. What have we learned recently that may be applicable to this problem
Great question! Let’s do that and see.
Continue Ferris wheel problem (10-20 minutes)
Students will draw their own model of the Ferris wheel at the front of the class with the measurements labeled. Then students will graph the points and visually see that a sine function is demonstrated.

Discussion (5-10 minutes)
Instructor will ask each group to describe what they have observed on the graph.Instructors will answer students’ questions. Instructors will help students to explain why the problem displays a sinusoidal graph. Teacher will facilitate a class discussion where the students define the parts of the equation they are working to derive.
Possible Student Questions Teacher Responses
Can the Ferris wheel keep going? Do Ferris wheels in real life keep going or always stop? What do you think?
What happens when the Ferris wheel isn’t spinning at a constant rate?
Why does that motion make a Sine graph?
What do you think? Can we still calculate the distance from the ground of a certain car on the
Ferris wheel?
Because the car moves in a circular motion and the max height is constant. (Show a GIF on overhead that displays this.) http://upload.wikimedia.org/wikipedia/commons/7/7d/Sin_drawing_process.gif
Continue Ferris wheel problem (10 minutes)
Students will derive the equation for the problem given. They will define the following for the equation they just derived; Amplitude, frequency, period, max, min, rotations, transformation. They should have already found these values but now will applying these terms.
Review (5-10) minutes
Instructors will recap the concepts that the students should have gained during this activity. The instructor will have the class work together to solve another problem with different dimensions. By observing the students’ approach to this similar problem allows the students to get more practice and also allows the instructor to assess the students’ understanding after the day’s activity.
Homework (Assessment):
Students will be asked to write a summary of what they learned during class this day. They should write about the different components of their equation and what they were representative of in the real world model.

Warm up (10 minutes)
The teacher will ask the students to recap the lesson from the class before. Some students will share their homework summary. The teacher will also take this time to review important concepts and correct any student misconceptions.
The instructor will ask the students if they can think of anything other than a circle can represent a sinusoidal graph? Some acceptable answers are pendulums, springs, etc. Pull out a slinky and ask the students if they think this could represent a sinusoidal function. Have students demonstrate their ideas. Tell students that now you will apply what they learned the day before to new situations like the ones just discussed.
Group Problems (45 minutes)
Students work on individual group problems . Teacher will ask probing/leading questions based on the needs of an individual group.
Teacher will address the class if she finds that most or all of the class is struggling with a particular concept.
Possible Student Questions Teacher Responses
How is this the same material as last time?
What formula do we use? /Is this the right formula?
These are all sinusoidal functions. They can be expressed as either a sine or cosine function.
Think back to yesterday, what formula did we use to figure that out? Can you do something similar?
How do we calculate this? / Am I doing this right? Why don’t you tell me what you think? Why did you calculate this the way you did?
Instructor will respond to questions however she sees fit.
Mid-point assessment: teacher will stop students after 20 minutes of work. She will have students explain what they think they know based on their problems (concepts, not numbers i.e. we know the amplitude and period not we know height is 2.4). Then the instructor will ask the students what they think they need to figure out. Then how will these things help you? The students’ answers will let the instructor assess what the students know.
Discussion/ Presentations
Students, as a group, will explain how they approached and solved their individual group problems in front of the class. The teacher will assess the students understanding as the group explains their process in solving their individual problem.
Post Test (10 minutes)
Homework:
Packet of all the questions from class this day and a few more problems. For teacher to evaluate student understanding.

[Author name]
[Date]
Lamaj rode his bike over a piece of gum. Lamaj continued riding his bike at a constant rate. At time t = 1.25 seconds, the gum was at a maximum height above the ground and 1 second later the gum was at a minimum. If the wheel diameter is
68 cm, find a trigonometric equation that will find the height of the gum in cm at any time t. a. Find the height of the gum when Lamaj gets to the corner at t = 15.6 seconds if he maintains a constant speed. b. Find the first and second time the gum reaches a height of 12 cm while Lamaj is riding at a constant rate.

[Author name]
[Date]
Lamaj rode his bike over a piece of gum. Lamaj continued riding his bike at a constant rate. At time t = 1.25 seconds, the gum was at a maximum height above the ground and 1 second later the gum was at a minimum. If the wheel diameter is
68 cm, find a trigonometric equation that will find the height of the gum in cm at any time t. a. Find the height of the gum when Lamaj gets to the corner at t = 15.6 seconds if he maintains a constant speed. b. Find the first and second time the gum reaches a height of 12 cm while Lamaj is riding at a constant rate.

www.123rf.com

Solve the following problem. Then create a poster that outlines the steps to the solution. Be sure to include the reasoning behind each step.
The temperature during the day can be approximated by a sinusoidal function. At 4 a.m. the temperature was at a low of 65°F. At 4 p.m. the temperature hit a high of 103°F. a.
Write an equation which will find the temperature t hours after midnight. h = 19sin( (t + 5)) + 84 h = 19cos( + 84 b.
Find the temperature at 11 a.m.
19sin ( ) + 84 = approx. 65.65cm c.
Find the first time in the day when the temperature reaches 98°F.
98 = 19sin ( t + 5) + 84 t = + t = approx. 0.69

Solve the following problem. Then create a poster that outlines the steps to the solution. Be sure to include the reasoning behind each step.
The amount of air in a person’s lungs varies sinusoidally with time under normal breathing.
When full, Karen’s lungs hold 2.8 liters of air. When ‘empty’, her lungs hold 0.6 liters of air. Her brother starts timing her breathing. At t = 2 seconds she has exhaled completely and at t = 5 seconds she has completely inhaled. a.
Find a function that will find the amount of air in Karen’s lungs at any time.
H = 1.1
+ 1.7 b.
Find the amount of air in Karen’s lungs if she starts holding her breath 3.5seconds into the timing.
H = 1.1
) + 1.7 = 2.7 liters c.
Find the first time Karen has 2.3 liters of air in her lungs.
2.3 = ) + 1.7 t = = 1.23s

Solve the following problem. Then create a poster that outlines the steps to the solution. Be sure to include the reasoning behind each step.
The height of a piston in a cylinder can be modeled by a sine or cosine function. A piston is at its lowest point in a cylinder, 8 cm from the bottom, at t = 3.2 seconds. The piston is at its highest position, 39 cm from the bottom, at t = 3.6 seconds. a.
Find an equation for the height of the piston, in cm, at any given time t.
H = 15.5sin(7.85t – 1.57) +23.5 b.
Find the height of the piston 15 seconds after the engine has started.
15.5sin(116.18) +23.5 = 24.41cm c.
Find the first time the piston reaches 13 cm from the bottom
13 = 15.5sin(7.85
t – 1.57) + 23.5 t = 0.11s

Solve the following problem. Then create a poster that outlines the steps to the solution. Be sure to include the reasoning behind each step.
Sean got a new yo-yo and noticed that the height of the yo-yo follows a sine or cosine curve. At time = 3 seconds the yo-yo is at its lowest height of 40 cm above the ground. The string is 62 cm long and one cycle takes 2 seconds. a.
Find an equation that will determine the height of the yo-yo at any time t.
H = 31sin( ) + 71 b.
Find the height of the yo-yo after 20 seconds.
31sin( (20) – ) + 71 = 40cm c.
Find the first time the height of the yo-yo reaches 52 cm above the ground.
52 = 31sin( ( t ) – ) + 71 t = = 0.08s

Solve the following problem. Then create a poster that outlines the steps to the solution. Be sure to include the reasoning behind each step.
Cooper Toy Company has designed a new toy that uses a spring that follows a sinusoidal curve after you wind it up and start it. At t = 5 seconds, the end of the spring is at its highest point, 18 cm above the ground. Four seconds later, the spring is at its lowest point, which is 6 cm above the ground. a.
Find an equation that will determine the height of the spring at any time t.
H = 6 t – ) + 12 b.
Find the height of the spring after 26 seconds.
6cos( ( 26) - ) + 12 = 6cm c.
Find the first time the height of the spring reaches 16 cm above the ground
16 = 6cos( t - ) + 12 t = = 2.57s