Lesson no. 9 (Situational Problems Involving Graphs of Circular Functions)

LESSON NO. 9
SITUATIONAL PROBLEMS
INVOLVING CIRCULAR
FUNCTIONS

Lesson No. 9| Situational Problems Involving Circular Functions
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Topic:
• Simple Harmonic Motions

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Introduction
• Repetitive or periodic behavior is common in nature. As
an example, the time-telling device known as sundial is
a result of the predictable rising and setting of the sun
everyday.

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Introduction
• It consists of a flat plate and a gnomon. As the sun
moves across the sky, the gnomon casts a shadow on
the plate, which is calibrated to tell the time of the day.

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ENGAGE

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Engagement Activity 1
Interactive Simple Harmonic Motion illustrations

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Questions:
1. What can you say about the interactive simple
harmonic motion?
2.What happens to the weight when it is suspended on
a spring, pulled down and released?
3. Neglecting resistance, what will happen to the
oscillatory motion of the weight?
4. Is the height of the oscillatory motion periodic with
respect to time?Why do you say so?

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Simple Harmonic Motion - FerrisWheel
Author: jeromeawhite
Topic:Trigonometry
Simple Harmonic Motion as displayed through the motion of a Ferris wheel
Reference: https://www.geogebra.org/m/XhKqBhvx#material/qyfesjMV

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Simple Harmonic Motion – Pendulum
Topic:Trigonometric Functions,Trigonometry
Simple Harmonic Motion as displayed through the swinging of a pendulum.
Reference: https://www.geogebra.org/m/XhKqBhvx#material/zFfuZWNt

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Simple Harmonic Motion – Piston
Topic: Sine,Trigonometry
Simple Harmonic Motion as displayed through the motion of a piston
Reference: https://www.geogebra.org/m/XhKqBhvx#material/GmnRkHsc

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Simple Harmonic Motion - Mass on a Spring
Topic: Function Graph,Trigonometry
Reference: https://www.geogebra.org/m/XhKqBhvx#material/Ne3TwAdW

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Questions:
1. What can you say about the interactive simple
harmonic
motion illustrations using GeogebraApplet?
2. What does each of the sliders represent in the
respective simple harmonic motion illustrations?

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Questions:
3. How will you describe the graphs considering the
horizontal position of the point at the top as a function
of time in the respective simple harmonic motion illustrations?
4. How does the graphs of circular functions concept, particularly
sine and cosine functions are applied in simple harmonic
otion illustrations?

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Engagement Activity N0. 2
Small-Group Interactive Discussion
Situational Problems Involving Graphs of
Circular FunctionsJ

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Simple Harmonic Motion
Questions:
• What are the examples of real-life occurrences that
behave in simple harmonic motion considering
unimpeded movements?
• What are the equations of simple harmonic motions?
• How will you describe the information on equations
of simple harmonic motions?

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Simple Harmonic Motion

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Exploration Activity 1
• The class will be divided into 8 groups (5-6
members).
• Each group will be given a problem-based task
card to be explored, answered and presented to
the class.
• Inquiry questions from the teacher and learners
will be considered during the explore activity.

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Explore 1
Rubric/Point System of theTask:
0 point – No Answer
1 point – Incorrect Answer/Explanation/Solutions
2 points – Correct Answer but No Explanation/Solutions
3 points – Correct Answer with Explanation/Solutions
4 points – Correct Answer/well-Explained/with
Systematic Solution

Explore 1
Assigned Role:
Leader – 1 student
Secretary/Recorder – 1 student
Time Keeper – 1
Peacekeeper/Speaker – 1 student
Material Manager – 1-2 students
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Explore 1
Task 1 (Group 1 & Group 2): Ferris Wheel
Problem:
A carnival Ferris wheel with a radius of 14 m makes one
complete revolution every 16 seconds. The bottom of the
wheel is 1.5 m above the ground. If a person is at the top of the
wheel when a stop watch is started, determine how high above
the ground that person will be after 1 minute and 7 seconds?
Sketch one period of this function.

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Explore 1
Task 2 (Group 3 & Group 4): OceanTides
Problem:
The alternating halfdaily cycles of the rise and fall of the ocean are called
tides. Tides in one section of the Bay of Fundy caused the water level to rise 6.5m
above mean sealevel and to drop 6.5m below. The tide completes one cycle every
12 hours.
Assuming the height of water with respect to mean sealevel to be modelled
by a sine function,
(a) draw the graph for a the motion of the tides for one complete day; (b) find an
equation for the graph in (a).

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Explore 1
Task 3 (Group 5 & Group 6): Roller Coaster
Problem:
John climbs on a roller coaster at Six Flags Amusement Park. An observer st
arts a stopwatch and observes that John is at a maximum height of 12 m
at t = 13.2 s.
At t = 14.6 s, John reaches a minimum height of 4 m.
a) Sketch a graph of the function.
b) Find an equation that expresses John's height in terms of time
c) How high is John above the ground at t = 20.8 s?

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Explore 1
Task 4 (Group 7 & Group 8): Spring Problem
Problem:
A weight attached to a long spring is being bounced up and down by an
electric motor. As it bounces, its distance from the floor varies periodically
with time. You start a stopwatch, when the stopwatch reads 0.3 seconds,
the weight reaches its first high point 60 cm above the ground. The next
low point, 40 cm above the ground, occurs at 1.9 seconds.
a) Sketch a graph of the function.
b) Write an equation expressing the distance above the ground in terms of
the numbers of seconds the stopwatch reads.
c) How high is the mass above the ground after 17.2 seconds?

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EXPLAIN

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Explain
• Group Leader/Representative will present
the solutions and answer to the class by
explaining the problem/concept explored
considering the given guide questions.

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members).
• Each group will be given a project-based task
card to be explored, accomplished and presented
to the class.
will be considered during the exploration activity.

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Explain
Guide Questions:
• What is the problem-based task all about?
• What are the given in the problem-based task?
• What are the things did you consider in solving/answering the problem-based
task?
• What methods did you use in solving/answering the given problem-based task?
• How did you solve/answer the problem-based task using that method?
• Are there still other ways to answer the problem/task? How did you do it?
• Are there any limitations to your solution/answer?
• What particular mathematical concept in trigonometry did you apply to solve the
problem-based task?

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Explore 2
0 point – No task explored, accomplished and presented
1 point – Correct Output but No Explanation/Solutions & Presentation
2 points - Correct Output with Explanation/Solutions & Presentation
3 points - Correct Output with well Explained/with Systematic Solution &
Presentation
(Note: Adapted Project-Based Output Rubric will also be utilized)

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Explore 2
Assigned Role:
Time Keeper – 1

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Explore 2
GeneralTask
Objective:To formulate, solve and present accurately situational
problems involving circular functions.
Specific Tasks
Objectives:
1.To create two situational problems involving circular
functions particularly simple harmonic motion concept application
2. Prepare interactive powerpoint presentation of the situational
problems created.
3. Discuss the situational problems created within the group.
4. Present the situational problems created to the class.

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members).
• Each group will be given a design-based task
card to be explored, created and presented to
the class.
will be considered during the exploration
activity.

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Explore 3
0 point – No Output Explored, Created and Presented
1 point – Correct/Accurate Output but No Explanation/Solutions
2 points - Correct/Accurate Output with Explanation/Solutions
3 points - Correct/Accurate Output with well- Explained/with
Systematic Solution
(Note: Adapted Design-Based Output Rubric will also be utilized)

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Explore 3
Assigned Role:
Time Keeper – 1

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Explore 3
GeneralTask
Objectives:
1.To formulate and graph circular functions model.
2.To design a work of art using the graph of circular functions
Specific Tasks
Objectives:
1.To create circular functions model.
2.To graph circular functions model.
3.To design/make creative output using the graphs of circular
functions.
4.To present the design-based output to the class.

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Explore 3
DBL Goal: Use at least 3 different trigonometric parent function graphs to create a
design-based art output.
Requirements:
(Modified guidelines from:
http://thefischbowl.weebly.com/uploads/6/2/8/2/62829617/trig_art_project_1.doc
1. Art must include a minimum of 3 different trig parent functions showing 2 or
more periods of each function. At least two of the functions must include shifts
(vertical or horizontal or both). At least one function must have a period other
than 2π.
2. Use one-fourth of large poster board.
3. Draw graphs on graphing paper or transparency sheets with the x and y axis units
the same for ALL graph.

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Explore 3
DBL Goal: Use at least 3 different trigonometric parent function
graphs to create a design-based art output.
Requirements:
4. One section of the poster board must display 3 separate overlays –
one for each trig function. You must label the axes and units on at
least one of the overlays or on the poster board. Each overlay must
have the equation of the trig function.
5. The chart below must be completed and attached to the front of your
poster or drawn and completed neatly on the front of your poster. It
may be computerized or hand written. The domain/range listed
should be only what was needed for your art work not the whole
function.

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Explore 3
DBL Goal: Use at least 3 different trigonometric parent function
graphs to create a design-based art output.
Requirements:
6.The bottom half of the poster should be an art work combining the
3 or more graphs. It should have a title, use at least 5 colors, and
be neat. The artwork should consume at least half of your poster
board.
7. Creativity, neatness, and originality will be graded as well as
content of the design-based art output.

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Explore 3
Function Amplitude Period
Vertical
Shift
Horizontal
Shift
Domain Range
y =
y =
y =

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ELABORATE

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Elaborate
Think-Pair-Share Activity

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Task:
Answer the following problems:
1. In Canada’s wonderland there is a roller coaster that is a continuous
series of identical hills that are 18m high from the ground. The platform to
get on the ride is on top of the first hill. It takes 3 seconds for the coaster
to reach the bottom of the hill 2m off the ground .
a) Sketch a graph which expresses the path of
the roller coaster.
b) What is the sinusoidal equation (sine and cosine) that
best reflects this roller coaster's motion?

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Task:
Answer the following problems:
2. Mr. Keeping, disguised as Mathman, a costumed crime fighter, is
swinging back and forth in front of the window to Ms. Gibbons’s math class.
At t = 3s, he is at one end of his swing and 4m from the window. At t = 7s,
he is at the other end of his swing and 20m from the window.
a) Sketch the curve. Use the distance from the window on the
vertical axis and the time in seconds along the horizontal axis.
b) What is the equation (in terms of sine and cosine), which represents
Mathman's motion?

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Generalization of the Lesson:
• What are the things to be considered in
solving situational problems involving
graphs of circular functions?
• How do you solve situational problems
involving graphs of circular functions?
• What are the steps in solving situational
problems on circular functions?

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Elaborate
Integration of PhilosophicalViews
In this part, the teacher and learners will relate the
term(s)/content/process learned in Lesson 6 about
Situational Problems Involving Graphs of Circular
Functions in real life situations/scenario/instances
considering the philosophical views that can be
integrated/associated to term(s)/content/process/ skills of
the lesson.

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Elaborate
Questions :
• What are the things/situations/instances that you can
relate with regard to the lesson about Situational
Problems Involving Graphs of Circular Functions?
• Considering your philosophical views, how will you
relate the terms/content/process of the lesson in real-life
situations/instances/scenario?

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• Simple harmonic motion occurs when a particle or object
moves back and forth within a stable equilibrium position
under the influence of a restoring force proportional to
its displacement.
• Simple Harmonic Motion is common in nature, from the
ups and downs of a roller coaster to the girl on a swing.
• The motion of a dog’s tail and the oscillations of the
strings that make up the quarks and gluons - all these
follow the simple harmonic motion.

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• The experiences we have in life follow a simple
harmonic motion.
• We are bound to encounter the ups and downs in life
like the ups and downs in a roller coaster.
• When we are on top, we feel happy, satisfied, and
contented while when we are in our downfall, we feel
sad, dissatisfied, and discontented.
• As we experience these things, we need to be
resilient and optimistic.

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• Our experience in life can be described as a simple
harmonic motion. It is up for us to scream and enjoy the
ride.
• In some situations, we cannot control the track, but we
do have a choice on how we will feel about the journey.
• We can scream and hate it as we go back and forth, or we
can throw our hands up in the air with a beautiful smile
on our face and yell out joy at the thrill of the ride.

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EVALUATE

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Evaluate
1. A pendulum on a grandfather clock is swinging back and forth as it
keeps time. A device is measuring the distance the pendulum is
above the floor as it swings back and forth. At the beginning of the
measurements the pendulum is at its highest point, 36cm high
exactly one second later it was at its lowest point of 12cm. One
second later it was back to its highest position.
a) Use the information above to sketch a diagram of this
sinusoidal movement.
b) Write the sinusoidal equation (sine and cosine) that
describes this situation.

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Evaluate
2. Sam is riding his bike home from school one day and
picks up a nail in his tire. The nail hits the ground every 2
seconds and reaches a maximum height of 48 cm (assume
the tire does not deflate).
a) Use the information above to sketch a diagram of
this sinusoidal movement.
b) Write the sinusoidal equation (sine and cosine) that
describes the situation in part a.

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Evaluate
3. At high tide the water level at a particular boat
dock is 9 feet deep. At low tide the water is 3 feet
deep. On a certain day the low tide occurs at 3 A.M.
and high tide occurs at 9 A.M. Find an equation for
the height of the tide at time t, where t=3 is 3 A.M.
What is the water level at 2 P.M.?

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Assignment:
Answer the following:
1.What are the fundamental trigonometric
identities?
2. How do you establish/prove/verify problems
on trigonometric identities?
Reference: DepED Pre-Calculus Learner’s Material, pages 171-172
-GNDMJR-

Lesson no. 9 (Situational Problems Involving Graphs of Circular Functions)

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