Course Profile
Functions and Relations, Grade 11, University Preparation, Catholic and
Public
Unit
4: Trigonometric Functions
Time: 19 hours
Activity 4.1 | Activity
4.2 | Activity 4.3 | Activity 4.4 | Activity
4.4a | Activity 4.4b | Activity 4.4c | Activity
4.5
Unit Description
Students
investigate the periodic nature and graphical properties of the primary
trigonometric functions. Using technology, students explore the effects of
simple transformations on their graphs and equations. Students apply these
concepts to model authentic problems.
Activity 4.1: Surf’s Up! Let’s
Catch the Sine Wave
Time:
75 minutes
Description
Students
investigate the shape of the graphs of y = sin x and y =
cos x by plotting the values of these functions as determined from the
unit circle.
Strand(s): Trigonometric Functions
Overall
Expectations
TFV.02 -
demonstrate an understanding of the meaning and application of radian measure;
TFV.03 -
determine, through investigation, the relationships between the graphs and the
equations of sinusoidal functions.
Specific
Expectations
TF2.07 -
demonstrate facility in the use of radian measure in solving and graphing
equations;
TF3.01 -
sketch the graphs of y = sin x and y = cos x, and
describe their periodic properties.
Ontario
Catholic School Graduate Expectations
CGE3c - a
reflective and critical thinker who thinks reflectively and creatively to
evaluate situations and solve problems.
Prior Knowledge &
Skills
·
Students
should be proficient in the use of both radian and degree measure.
·
The
meaning of q and its use in representing angle
measures should be introduced to the students prior to this activity.
Planning Notes
·
The
teacher requires an overhead projector, acetate sheet and markers, an overhead
projection tablet, a graphing calculator with motion sensor attachment (such as
a CBR), and a piece of string (40 to 100 cm in length) and a mass (large
washers will do) to construct a pendulum.
·
The
teacher should also provide a unit circle, preferably superimposed on graph
paper. The students require graph paper.
Teaching/Learning
Strategies
This
activity is comprised of two sections. The teacher should facilitate a
discussion of periodic and cyclical behaviour, and demonstrate sinusoidal
motion in the swinging of a pendulum. The students then derive the graphs of y
= sin x and y = cos x from the unit circle.
A. Teacher Facilitation - Demonstration
·
As
an introduction to the graphs of sinusoidal functions, the teacher should
first, by directed questioning, acquaint students with the cyclical character
of the unit circle, specifically the repetitive nature of the values of y
= sin x and y = cos x. The students should then be asked
to provide everyday examples that exhibit similar behaviour (i.e., the hands of
a clock, rocking chairs, pendulums, Ferris wheels, the tides, etc.).
·
As
a demonstration, the teacher should set up a simple pendulum in the class and
use the motion sensor to measure the position of the bob (mass) as a function
of time. For best results, the mass should be pulled back about 30 cm from the
rest position and the motion sensor should measure at least 4 full
oscillations. Display the graph on the overhead. The graph will be sinusoidal
and should give the students an indication of the type of graphs that they will
be exploring throughout this unit. The periodic characteristics of the graph
(specifically the amplitude, period, domain, and range) should be discussed at
this point.
B. Student Activity
Suggestions
for teacher facilitation are included throughout this activity in italics.
1. Convert the angles given in the table below
into radian measure and enter their values into the table.
2. Starting from the positive x-axis
(calling this q = 0°) and rotating
counter-clockwise in 15° increments, determine the coordinates of the points on the unit circle
for each angle, and enter these coordinates in the table. Teachers may need
to demonstrate how to determine co-ordinates.
3. Determine the values of sin q and cos q for each of the given angles to two decimal places. It is assumed
that the students have been acquainted with the relationship of sin q and cos q and the unit circle in Unit 3:
Trigonometry.
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q (in degrees) |
0 |
15 |
30 |
45 |
60 |
75 |
90 |
105 |
120 |
135 |
150 |
… |
360 |
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q (in radians) |
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coordinates (x, y) |
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sin q = |
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cos q = |
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4. On the same set of axes, plot the graphs of y
= sin q and y = cos q.
C. Follow-Up Skills
Using the
graphing calculator or a sample of students’ work transferred to an acetate
sheet, the teacher should elicit from the students pertinent characteristics of
the graphs of y = sin q and y = cos q. These characteristics should include the
period, amplitude, roots, symmetry, domain, and range. The similarity of the
graphs of y = sin q and y = cos q should also be noted, and in particular the
translation that would map one graph onto the other. The teacher should
introduce the term phase shift at this point.
Assessment &
Evaluation of Student Achievement
Learning
skills can be assessed visually as the students are completing their graphs.
Their ability to work independently can be assessed using as criteria
accomplishing a task independently and self-direction. Time management skills
can be used to assess organization. Initiative can be assessed using the
students’ self-motivation and responses to prompts by the teacher as criteria.
Activity 4.2: Transformations of
Trigonometric Functions
Time: 150 minutes
Description
Students
investigate the effects of simple transformations on the graphs of y =
sin x and y = cos x through the use of graphing
technology.
Overall
Expectations
TFV.02 -
demonstrate an understanding of the meaning and application of radian measure;
TFV.03 -
determine, through investigation, the relationships between the graphs and the
equations of sinusoidal functions;
OCV.02 -
demonstrate an understanding of inverses and transformations of functions and
facility in the use of function notation.
Specific
Expectations
TF2.07 -
demonstrate facility in the use of radian measure in solving and graphing
equations;
TF3.02 -
determine, through investigation, using graphing calculators or graphing
software, the effect of simple transformations (e.g., translations,
reflections, stretches) on the graphs and equations of y = sin x
and y = cos x;
TF3.03 -
determine the amplitude, period, phase shift, domain, and range of sinusoidal
functions whose equations are given in the form y = a sin(kx +
d) + c or y = a cos(kx + d) + c;
TF3.04 -
sketch the graphs of simple sinusoidal functions [e.g., y = a sin
x, y = cos kx, y = sin(x + d),
y = a cos kx + c];
TF3.05 -
write the equation of a sinusoidal function, given its graph and given its
properties;
OC2.06 -
represent transformations (e.g., translations, reflections, stretches) of the
functions defined by
f(x) = x, f(x)
= x2, f(x) = 
, f(x) =
sin x, and f(x) = cos x, using function notation;
OC2.07 -
describe, by interpreting function notation, the relationship between the graph
of a function and its image under one or more transformations;
OC2.08 -
state the domain and range of transformations of the functions defined by f(x)
= x, f(x) = x2,
f(x) = , f(x) = sin x,
and f(x) = cos x.
Ontario
Catholic School Graduate Expectations
CGE3c - a
reflective and critical thinker who thinks reflectively and creatively to
evaluate situations and solve problems;
CGE5a -
works effectively as an interdependent team member;
CGE5e -
respects the rights, responsibilities and contributions of self and others;
CGE5f -
exercises Christian leadership in the achievement of individual and group
goals.
Prior Knowledge &
Skills
·
Students
should be proficient in the use of radian measure.
·
The
graphs and characteristics of y = sin x and y = cos x
should be familiar to all students.
·
Students
should be proficient in the use of graphing calculators to plot functions.
Students should also be able to graph manually.
·
The
meaning of q and its use in representing angle
measures should be introduced to the students prior to this activity.
Planning Notes
·
The
teacher requires an overhead projector, transparencies, and markers. (An
overhead projection tablet could be used as well.) In addition, the teacher
must prepare a student worksheet for this activity.
·
The
students require graphing calculators (or dynamic graphing software such as Zap-A-Graph
or Geometer’s Sketchpad), and graph paper.
·
The
first part of the activity is a discovery exercise, after which the students
summarize their results. The second part of the activity includes a similar
investigation involving more complex transformations.
·
This
activity may also be done without the use of technology, if required.
Part 1: Transformations: More than Meets the Eye
Time: 75 minutes
Teaching/Learning
Strategies
A. Teacher Facilitation
·
Students
are placed into groups of two or three. Each group is assigned a set of four
particular examples of sinusoidal functions of the following types:
Group 1 y
= a sin x Group 2 y = a cos x
Group 3 y
= sin kx Group 4 y = cos kx
Group 5 y
= sin(x + s) Group 6 y = cos(x + s)
Group 7 y
= sin x + c Group 8 y = cos x + c
·
This
activity is described for Group 1 only. Groups 2 to 8 answer similar questions
using a variety of values for the given parameters. In particular, it is
recommended that both positive and negative values be used, and that both whole
numbers and fractions are included. For example, Group 3 might be assigned the
functions y = sin 3x, y = sin(-
x), y =
sin 
, and y = sin(-4x).
·
As
trigonometric functions are traditionally considered to be functions of radian
measure, it should be noted that the values given for the constant s
(Groups 5 and 6) should be multiples of p.
·
Students
are asked to describe the graphs of the assigned functions. The teacher should
expect these descriptions to include at least the maximum and minimum points,
the amplitude, the roots, the period, symmetry, the phase shift, and the domain
and range.
B. Student Activity
Suggestions
for teacher facilitation are included throughout this activity in italics.
1. a) Plot
a graph of y = sin x on the graphing calculator, using a window
large enough to display two complete
cycles, one on either side of the y- axis. Reproduce this sketch on the
graph paper.
b) Graph
the following curves on the calculator, stopping after each to reproduce the
plot on the graph paper, using the same set of axes as in question 1a: y =
2 sin x, y = –sin x, y = 
sin x, and y = 
sin x.
c) In what ways are the graphs similar? In what
ways are they different?
2. a) Clear
the screen, then plot a graph of y = cos x on the graphing
calculator, using a window large enough
to display two complete cycles, one on either side of the y- axis.
Reproduce this sketch on the graph
paper using a new set of axes.
b) Use
the calculator to graph each of the functions y = -3 cos x, y = 5 cos x, y = cos x, and
y = 
cos x. After
graphing each equation, reproduce the plot on the graph paper, using the same
set of axes as in question 2a.
c) In what ways are the graphs similar? In what
ways are they different?
d) Compare and contrast these graphs to those
sketched in question 1.
3. Without
plotting, compare and contrast the graphs of the following equations:
a) y
= sin x and y = 12 sin x b)
y = sin x and y = 
sin x c) y = cos x and y = -3 cos x
After
each group has completed their respective investigation, they present their
results to the rest of the class, drawing attention to the role of the
constants in each transformation, and thus making connections between the
function and its graph.
C. Follow-up Skills
Time:
30 minutes
The
teacher should assign additional questions to the students in class to
reinforce the new learning. The teacher could facilitate a class discussion to
confirm responses, and use the opportunity to provide remediation as necessary.
The given questions should include a variety of constants, as outlined above.
Part 2: Give Me A Sine
Time: 75 minutes
Teaching/Learning
Strategies
A. Teacher Facilitation
·
The
second part of the activity proceeds in much the same manner as the first. The
following sinusoidal functions will be assigned to each group.
Group 1 y
= a sin kx Group
2 y = a cos kx
Group 3 y
= a sin(x + s) Group
4 y = a cos(x +
s)
Group 5 y
= sin kx + c Group
6 y = cos kx + c
Group 7 y
= sin k(x + s) Group
8 y = cos k(x +
s)
B. Student Activity
Once
again, each group is asked to present their results to the rest of the class,
drawing attention to the effect of each numerical value in the given functions.
C. Follow-up Skills
Time:
45 minutes
To
prepare students for subsequent activities, the teacher should further explore
some of the concepts learned in this activity:
·
Functions
of the form y = sin(x + s) and y = cos(x + s)
have, until now, included only multiples of p for the value of s. The
teacher should have students explore the effect of other real values of s,
as they play a most important role in the modelling of authentic problems later
in this unit.
·
A
connection must be established between the functions y = sin k(x
+ s) (studied in this activity) and y = sin(kx + d)
(an alternate form), and between the functions y = cos k(x +
s) and y = cos(kx + d). Specifically, it is
imperative that students recognize the roles of k, s, and d in
determining the period and the phase shift of a given function. Both of these
forms are examined in subsequent activities, so students should be comfortable
working with them both.
·
In
this activity, the students are asked to compare and contrast the equations and
graphs of simple transformations of y = sin x and y = cos x. The teacher should provide additional
questions to reinforce these skills. In addition, students should be asked to
sketch a variety of given sinusoidal functions, and to determine the equation
of various sinusoidal functions given their graphs. These skills are of great
importance to achieve success in the rest of this unit.
·
Suitable
textbook questions should be assigned to consolidate these concepts.
Assessment & Evaluation
of Student Achievement
The students’ oral reports may be
assessed using an appropriate group presentation rubric, with emphasis paid to
the assessment of Communication skills, particularly the use of mathematical
symbols and conventions. Other criteria may include the computation and
construction of graphs (Knowledge and Communication), and reasoning skills
(Inquiry and Communication). After completing the recommended textbook
exercise, a short pencil-and-paper task could be used to assess the core knowledge
and concepts. Learning skills, specifically initiative, could be assessed as
the teacher facilitates the classroom discussion to conclude the first part of
the activity. Teamwork skills can be assessed visually as the groups complete
their investigation. The students’ work habits and organization can be assessed
during their presentations.
Accommodation
The
teacher should ensure that those students with difficulties in understanding
concepts be placed into groups in which they can receive support from other
students.
Activity 4.3: Don’t Go Off on a
Tangent
Time: 75 minutes
Description
Students
identify the defining characteristics of the function y = tan x,
and establish connections with the functions y = sin x and y =
cos x.
Strand(s): Trigonometric Functions
Overall
Expectations
TFV.03 -
determine, through investigation, the relationships between the graphs and the
equations of sinusoidal functions.
Specific
Expectations
TF3.01 -
sketch y = sin x and y = cos x, and describe their
periodic properties;
TF3.06 -
sketch the graph of y = tan x; identify the period, domain, and
range of the function; and explain the occurrence of asymptotes.
Ontario
Catholic School Graduate Expectations
CGE3c - a
reflective and creative thinker who thinks reflectively and creatively to
evaluate situations and solve problems;
CGE4b - a
self-directed, responsible, life long learner who demonstrates flexibility and
adaptability;
CGE4f - a
self-directed, responsible, life long learner who applies effective
communication, decision-making, problem-solving, time and resource management
skills.
Prior Knowledge &
Skills
·
Students
should possess a comprehensive knowledge of the division of rational numbers,
and specifically the concept of division by 0.
·
The
primary trigonometric ratios should be familiar to all students.
·
Students
should understand the concept of an asymptote, as defined in Unit 1: Algebraic
Manipulation of Functions and Unit 2: Function Notation, Inverses, and
Transformations.
·
Students
must be able to use the radian and degree modes of a scientific calculator.
Planning Notes
·
Students
must have graph paper and a scientific calculator for use in the graphing of
the primary trigonometric functions.
·
The
teacher has to prepare the tables to be filled out by the students in this
activity.
Teaching/Learning
Strategies
A. Teacher Facilitation
·
Students
use characteristics of the graphs of y = sin x and y = cos
x and the relationship of the primary trigonometric ratios to determine
the graph of y = tan x, after which they make observations about
its properties.
·
The
teacher may pair or group students for this activity, or they may have the
students work independently.
·
As
the class is completing this investigation, the teacher should be addressing
students’ concerns individually or, if necessary, in small groups. In addition,
students should be encouraged to seek help from other students.
B. Student Activity
Suggestions
for teacher facilitation are included throughout this activity in italics. Some
solutions are included to aid in the flow of the activity.
1. Find the missing angle measures in the given
table.
2. Using a scientific calculator, determine the
values of sin x and cos x for the given angles to the nearest
thousandth.
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x (in radians) |
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x (in degrees) |
0 |
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60 |
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90 |
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120 |
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360 |
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sin x |
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3. Use the values in the table to sketch the
graphs of y = sin x and y = cos x.
4. List as many characteristics as possible
about these graphs. Students should include in their list the domain, the
range, the roots, the period, and the amplitude.
5. Using the values found in question 2,
calculate the ratio 
to the nearest
thousandth, and enter the results in the table.
6. Use the calculator to compute values for tan x
to the nearest thousandth, using the angles given in the table. Compare
these values with those found in question 5.
7. Plot the values of tan x. There may be
some difficulties for certain values of x. Why? The graph of
y = tan x is said to have an asymptote at these points,
and is to be represented on the graph by a dotted vertical line. What happens
to the graph as the angle values approach these asymptotes? Some values of x
(90°, for example) produce undefined answers for y = tan x. The concept
of an asymptote is introduced in Unit 1: Algebraic Manipulation of Functions
and Unit 2: Function Notation, Inverses, and Transformations.
8. Sketch the graph of y = tan x,
including all asymptotes.
9. Describe the graph of y = tan x,
calling attention to as many characteristics as possible. Any description
should include the following information: domain is {x 
R, 0 # x # 360°, x ¹ 90°, 270°}, range is {y
R}, asymptotes at x
= 90° and x = 270°, graph repeats itself every 180° (the period), roots are x =
0°, x = 180°, x = 360°.
10. Using
the primary trigonometric ratios, prove that 
.
11. Using the table and the graph, predict values
for the following:
a) tan 420° b)
c) tan(-135°) d) 
.
Confirm these values with a scientific
calculator.
C. Follow-up Skills
Time: 30 minutes
The
teacher may wish to facilitate a classroom discussion incorporating some of the
results of this investigation. The fact that tan x can be defined as the
quotient of sin x and cos x is significant, as are the properties
that y = tan x has the same roots as y = sin x, and
undefined values for the roots of y = cos x. The behaviour of y
= tan x around its asymptotes should also be reviewed. Suitable
textbook exercises should be used to reinforce concepts introduced in this
investigation. The teacher can then remediate as necessary.
Assessment &
Evaluation of Student Achievement
This
investigation can be used to assess the students’ independent work skills or
teamwork skills, particularly their ability to stay on task. The teacher can
use an appropriate observational rubric to assess the students’ progress with
the mathematical content. The teacher may have the students submit questions 7,
8, and 9 as a journal topic. Questions 7 and 8 address Knowledge and Inquiry.
Assessment criteria could include the accuracy of calculations, the
justification of the asymptotes, and the accuracy of the graph. Question 9
emphasizes Knowledge and Communication, specifically the use of mathematical
vocabulary and symbols, and the inclusion of all desired defining
characteristics.
Activity 4.4: Applications of
Trigonometric Functions
Time: 450 minutes
Description
Both
natural phenomena and manufactured devices exhibit sinusoidal motion.
Rotations, oscillations, and waves all exhibit sinusoidal motion. In three
different activities, the motion of a spring, a bicycle wheel, and the sun are
used to connect the transformations of trigonometric functions to the practical
world. During these investigations, students become more familiar with the
manipulation and properties of the general sine function.
Strand(s): Trigonometric Functions, Tools for Operating and
Communicating with Functions
Overall
Expectations
TFV.03 -
determine, through investigation, the relationships between the graphs and the
equations of sinusoidal functions;
TFV.04 -
solve problems involving models of sinusoidal functions drawn from a variety of
applications;
OCV.02 -
demonstrate an understanding of inverses and transformations of functions and
facility in the use of function notation.
Specific
Expectations
TF3.02 -
determine, through investigation, using graphing calculators or graphing
software, the effect of simple transformations, (e.g., translations, reflections,
stretches) on the graphs and equations of y = sin x and y =
cos x;
TF3.03 -
determine the amplitude, period, phase shift, domain, and range of sinusoidal
functions whose equations are given in the form y = a sin(kx +
d) + c or y = a cos(kx + d) + c;
TF3.04 - sketch the graphs of simple sinusoidal
functions (e.g., y = a sin x, y = cos kx, y
= sin (x + d),
y = a cos kx + c);
TF3.05 -
write the equation of a sinusoidal function, given its graph and given its
properties;
TF4.01 -
determine, through investigation, the periodic properties of various models
(e.g., the table of values, the graph, the equation) of sinusoidal functions
drawn from a variety of applications;
TF4.02 -
explain the relationship between the properties of a sinusoidal function and
the parameters of its equation, within the context of an application, and over
a restricted domain;
OC2.06 –
represent transformations (e.g., translations, reflections, stretches) of the
functions defined by
f(x) = x, f(x)
= x2, f(x) = , f(x) =
sin x, and f(x) = cos x, using function notation;
OC2.08 –
state the domain and range of transformations of the functions defined by f(x)
= x, f(x) = x2, f(x) = , f(x) = sin x,
and f(x) = cos x.
Ontario
Catholic School Graduate Expectations
CGE 2c -
an effective communicator who presents information and ideas clearly and
honestly and with sensitivity to others;
CGE 3c -
a reflective and creative thinker who thinks reflectively and creatively to
evaluate situations and solve problems;
CGE 4b -
a self-directed, responsible, life long learner who demonstrates flexibility
and adaptability;
CGE 5a -
a collaborative contributor who works effectively as an interdependent team
member;
CGE 5g -
a collaborative contributor who achieves excellence, originality, and integrity
in one’s own work and supports these qualities in the work of others.
Prior Knowledge &
Skills
·
Students
should possess competent manual graphing skills, and be proficient in the use
of a graphing calculator or computer software specifically in the compiling and
graphing of data.
·
Students
must be able to collect data using a variety of methods (motion sensor/sensory
probe, research, measurement).
·
Students
should possess a thorough understanding of the general properties of sinusoidal
functions and their graphs.
·
Radian
measure, as it pertains to trigonometric equations, should be familiar to all
students.
Planning Notes
·
These
activities have been developed with the intention that they would be conducted
sequentially. It is possible, however, for them to be conducted concurrently,
with groups of students rotating through each of them.
·
The
activities differ enough that they may cater to the various learning styles and
abilities present in the class. Activity 4.4B is probably the most tactile, and
Activity 4.4C is probably the most abstract. The teacher may wish to assign
students to a specific activity.
Activity 4.4A: It’s a Spring
Thing
Time: 150 minutes
Description
Students
study the motion of a mass on a spring using a graphing calculator with a
motion sensor attachment.
Planning Notes
·
As
students are placed into groups for this activity, the teacher should ensure
that there are enough graphing calculators and motion sensor units for each
group. Each group should have an acetate sheet, a spring holder, a given mass
and a different spring.
·
This
activity has strong ties to several topics in physics, particularly Hooke’s
Law. The teacher may wish to consult and collaborate with the physics teacher
in order to emphasize the cross-curricular tie.
Teaching/Learning
Strategies
A. Teacher Facilitation
·
Students
should be placed in groups of two or three.
·
When
collecting data with the motion sensor, the calculator should NOT be in realtime
mode, and should be set long enough to record at least four cycles. The
calculator should also be set for heavy smoothing to make the graphs
easier to match.
·
The
mass should not be closer than 50 cm to the motion sensor at any point during
the collection of data.
·
The
fact that the groups have different springs gives each group different periods.
If the teacher does not have different springs, a similar result can be
achieved if each group has the same spring but different masses.
·
The
results of this investigation are recorded on acetate sheets. Once each group
has completed the investigation, these acetate sheets should be discussed in
class.
·
This
investigation uses the general sine function y = a sin(kx +
d) + c, which differs from other forms studied thus far in this
unit (refer to Activity 4.2 – Follow-Up Skills).
B. Student Activity
Suggestions
for teacher facilitation are included throughout this activity in italics. Some
solutions are included to aid in the flow of the activity.
1. Set up the apparatus such that the spring and
mass are secure and oscillate freely (refer to Figure 1, below). Measure the
distance between the motion sensor and the mass and call this the Equilibrium
Position.
2. Start the spring in motion by pulling it down
approximately 10 cm from the Equilibrium Position.
3. Once the spring is in motion, begin
collecting data using the appropriate program on the graphing calculator.
4. Once the data is collected and plotted, it
should take the form of a sinusoidal curve. If it does not, resample the data.
5. As the motion sensor measures the position of
the mass over time, the time data will automatically be stored in L1,
and the distance data in L2.
6. Using previous knowledge of transformations,
determine the values of a, c, d, and k so that the
graph of y = a sin(kx + d) + c matches the plot created by the motion sensor. For this form
of the curve, a is the amplitude, c is the vertical translation, d ¸ k is the phase shift, and 2p ¸ k is the period.
7. Using the data collected, determine the
average position of the mass, and the difference between the highest and lowest
positions of the mass. Compare these values to a, c, d,
and k. Here the average position should be approximately c, and the
difference should be approximately 2a.
8. Find
the time between successive maxima, and the time between successive minima.
Average these values. What is this average called? Multiply this average with
the value of k, and explain the significance of the result. The
values are averaged in order to produce a more accurate measurement. The
average value is called the period. Multiplying the period and k yields a
number approximately equal to 2p, which should be so, because
mathematically the period can be calculated using the formula p = 2p ¸ k.
9. By
using the same values for a, c, d, and k, graph the function y
= a cos(kx + d) + c. Discuss the similarities
and differences between the sine and cosine graphs. The function y = cos
x is the same as the function y = sin x, but shifted to the left by 
units (i.e., cos
x = sin(x + )).
So a cos(kx + d) = a sin(k(x +) + d), or a sin(kx + d + 
). Thus, the cosine function is the same as the sine
function, but shifted to the left by units.
10. Repeat the experiment by pulling the spring a
different distance from the Equilibrium Position.
11. How do the values of a, c, d,
and k compare to those found in the first trial? In subsequent trials
the value of the constant a should be different (bigger or smaller depending on
how the pull of the spring compares to the first). There should be no
correspondence for the value of d, since it will depend on the initial position
of the mass when the data began to be sampled. Both k and c should be the same.
12. Discuss what aspect of the experiment controls
the value of d.
13. Summarize the data neatly and concisely on an
acetate sheet.
Figure 1
Assessment & Evaluation
of Student Achievement
Observing
and conferencing can be used to assess the students’ Knowledge and
Understanding while the activity is in progress. Testing different values of a,
c, d, and k to obtain a theoretical equation to match the
data demonstrates Inquiry/Problem Solving skills. The students’ acetate sheets
can be used to assess Communication skills, and should form the basis of a
classroom discussion facilitated by the teacher to confirm and summarize
results. The teacher may wish to assign tasks to each member of a group and
assess students on the performance of their task. Sample tasks may include
“copy producer” (scribe, assessed on written Communication skills), “experiment
engineer” (performs experiment, assessed on Knowledge and Inquiry skills),
“public relations specialist” (representative to explain results, assessed on
oral Communication skills), etc.
Activity 4.4B: Ferris Fair
Time: 150 minutes
Description
Students
study the height of a rider in a gondola on a Ferris wheel as the wheel
rotates. To simulate the rotation of a Ferris wheel, a bicycle wheel is rolled
along the ground.
Planning Notes
·
Students
are placed into groups for this activity. A bicycle wheel (or any other
circular object, such as a hula-hoop, a paint can, etc.) should be provided to
each group to simulate a Ferris wheel.
·
Tape
and metre sticks are needed to measure the height of the rider.
Teaching/Learning
Strategies
A. Teacher Facilitation
·
Students
should be put into groups of two or three.
·
Be
sure each group has sufficient room to roll their wheel so that its motion is
not obstructed.
·
The
teacher should note that when students are asked to compare experimental values
to theoretical values, they should not only be looking for values that are
similar, but also values that are multiples of each other.
·
As
an alternative to using a physical model, the Resources section includes a link
to a website to simulate a Ferris wheel using Geometer’s Sketchpad.
·
The
Ferris wheel could also be modelled by placing a bicycle upside down and
marking one of its wheels with a piece of chalk to represent the rider.
B. Student Activity
Suggestions
for teacher facilitation are included throughout this activity in italics. Some
solutions are included to aid in the flow of the activity.
1. Place one piece of tape on the ground
(starting point) and one on the rim of the wheel (the rider). Roll the wheel
away from the starting point (see Figure 2, above) and stop the wheel in at
least 16 different positions, including points at which the rider is at the top
of the wheel and bottom of the wheel. At each position, measure the distance
from the starting point to the bottom of the wheel (x) and the height (h)
of the rider. Care should be taken to make sure that the wheel rolls in a
straight line, and that the wheel makes at least one complete rotation. If
the top and bottom positions are not included, the students may get an
inaccurate measurement for the amplitude.
2. Repeat part 1 three more times, using
different starting positions for the rider each time.
3. Create a table of values and plot the graph
of the distance (x) vs. height (h) for each set of data, using a
graphing calculator or graphing software.
4. List as many characteristics as possible of
each graph. Characteristics should include the fact that the graph is
sinusoidal, the number of maxima and minima, the difference between the heights
of the maxima and minima, the difference between successive maxima, the
difference successive minima, the period, amplitude, phase shift, and vertical
translation.
5. Determine the values of a, c, d,
and k in the equation y = a sin(kx + d) + c
that models the data and matches the graph.
6. Determine the radius of the wheel. How does
this value relate to the equation? The radius should correspond to the value
of a. For this investigation it will also correspond to the value of c.
7. Find the circumference of the wheel. What is
the significance of this value? Students should recognize this value as
being the period of the function. Thus, this value should be equal to 2p ¸ k.
8. Describe how this model of a Ferris wheel
differs from a real Ferris wheel. For this model, what does the value of c
correspond to? For a real Ferris wheel, would the value of c correspond
to the same thing? Explain. For a real Ferris wheel, the value of c
corresponds to the distance from the ground to the centre of the wheel. This
result may be better visualized using the inverted bicycle model (see Teacher
Facilitation). Setting the bicycle on the floor or the desk will produce
different values of c, but the value of a will remain constant.
9. What is the significance of the value of d?
Different values of d occur depending on the starting position of the
gondola.
10. Using the same data, model the movement of the
rider using the general cosine function
y = a cos(kx + d) + c. Answer questions 5 to
9 again, using this new equation for reference.
11. Compare
the equations and graphs of the sine and cosine functions, stating any similarities
and differences. The curves are exactly the same, except for the fact that
the cosine function is shifted to the left. The teacher may need to remind
students that the standard cosine function
y = cos x is shifted to the left by units from the
standard sine function y = sin x. With a little facilitation, the
students should be able to determine that their cosine function is shifted to
the left by k ´ units.
Figure
2
C. Follow-up Skills
Time:
75 minutes
The
teacher should supplement the preceding activities with textbook exercises.
Types of questions that should be assigned include:
·
Given
the graph of a sinusoidal function, determine its equation.
·
Given
a general sinusoidal equation of any form (i.e., y = a sin(kx +
d) + c, y = a sin k(x + s) + c,
y = c + a sin k(x - s), etc.), describe and sketch the graph of the
equation by hand.
The
teacher should also review the role of all of the constants in the various
forms of the sinusoidal equation.
Assessment &
Evaluation of Student Achievement
As in
Activity 4.3A, students continue to develop confidence in the manipulation of
sinusoidal equations. It is suggested that at this point that
Knowledge/Understanding be assessed by a paper-and-pencil task, such as a quiz.
Activity 4.4C: Let the Sine Shine
In
Time: 150 minutes
Description
Students
collect sunrise and sunset data and use it to model an hours of daylight
function for a particular location.
Planning Notes
·
·
Graph
paper should be made available to the students, as two sets of axes are
required.
·
This
activity has strong ties to several topics in astronomy. Suggested topics for
further student research are included throughout this activity.
Teaching/Learning
Strategies
A. Teacher Facilitation
·
·
The
students form groups and each is assigned a latitude.
·
Groups
may use an atlas, if available, to choose a city, or they may use the following
suggestions: 60°N - St. Petersburg (Leningrad), Russia; 50°N - Winnipeg,
Canada; 40°N - Philadelphia, U.S.A.; 30°N - Cairo, Egypt; 20°N - Santiago,
Cuba.
·
This
activity uses the general sine function y = c + a sin k(x
- s). Increased exposure to different forms will
help the students develop confidence in their algebraic manipulation skills.
B. Student Activity
Suggestions
for teacher facilitation are included throughout this activity in italics. Some
solutions are included to aid in the flow of the activity.
Part 1 – Table for Sun,
Please
After
obtaining the sunrise and sunset data for the given location, complete the
following table. To make future calculations easier, use 24-hour metric time,
(e.g.,
|
City: |
Latitude: |
Longitude: |
||||
|
Date |
Day of Year |
|
|
Sunset [: ] |
Sunset [ . ] |
Hours of Daylight [ . ] |
|
01 01 |
0 |
|
|
|
|
|
|
01 15 |
14 |
|
|
|
|
|
|
… |
… |
|
|
|
|
|
Note
that January 1 is designated as day 0, to aid in the construction of an
equation. Include in the table March 20, June 21, September 23, and December
21, and at least two additional days from each month.
Part 2 – Can You See the
Light?
The
intent of this section is to have the students determine sinusoidal equations
by inspection. Using graphing techniques, students identify defining
characteristics of a sinusoidal curve, and use them to determine its equation.
It is anticipated that not all students will have access to graphing
calculators. Several questions have thus been presented in two formats, one for
use with graphing calculators (denoted [C]), the other with paper-and-pencil
(denoted [P]).
1. If it
has not already been done, compute and insert the number of hours of daylight
(in decimal format) into the last column.
2. [C] Construct a scatter plot by entering the day of
the year in L1, and the hours of daylight in L2, using a
window large enough to display all of the data.
[P] Construct a scatter plot of the
day of the year vs. the hours of daylight on the first set of axes.
3. Visually estimate the amplitude, period,
phase shift, and vertical translation of this graph.
4. Using the general form h(t) = c
+ a sin k(t - s), determine an equation that models this data.
This is yet another form of the general sinusoidal equation. The constants
c, a, and s will represent the vertical translation, amplitude, and phase
shift, respectively. The constant k is 360° (or 2p) divided by the period.
5. [C] Store
this equation in Y1 and plot its graph using the dashed line. How
well does it match the scatter plot?
[P] Using a table of values, graph
this curve on the same set of axes. How well does it match the scatter plot?
Part 3 – Straining
Functions Through a Calendar
Students
use the table to algebraically determine the equation of the curve.
1. a) What
are the longest and shortest days of the year (i.e., what days receive the most
and the least amount of daylight)?
Estimate the length of the longest day and the shortest day. What is the range of daylight hours of the course
of the year? The longest and shortest days of the year are June 21 and December 21.
b) What two days receive an equal
amount of daytime and night-time? The days of equal daytime and night-time are March 20 and September 23.
c) Explain the significance of
these four days. These four days are used to indicate the changing of the seasons.
2. Estimate the daily average number of hours of
daylight over the entire year.
3. From previous knowledge, it can be easily
verified that y = sin x passes through the origin. In order to
match the hours of daylight function to the general function y = sin x,
where should the origin be placed? The function y = sin x passes
through the origin, which can be considered the “middle” or “average” value of
the graph. To match the hours of daylight function to the sine function a
similar point should be found. The origin should be placed at March 20, because
this day serves as a “middle” or “average” value for the hours of daylight
function (having an equal amount of daytime and night-time). This question will
determine the phase shift.
4. Estimate the period of the daylight curve.
Justify your answer.
5. Using the answers to questions 1 through 4,
determine the equation that can be used to model the number of hours of daylight.
Question 1 relates to the amplitude, question 2 will provide the vertical
translation, question 3 will be used to determine the phase shift, and question
4 will provide the period. Incidentally, the sinusoidal regression function on
most graphing calculators would be able to compute this equation.
6. [C]
Store this new equation in Y2 and plot its graph using the
thin solid line. Compare this graph with the graph plotted in part 2. By what
method (visual or computational) was the best fit achieved?
[P] Graph the new equation on the
same set of axes as the previous graph using a table of values. Compare the two
graphs. By what method (visual or computational) was the best fit achieved?
Part 4 – Dare to Share and
Compare
For
this part of the activity, the students collect data from two other groups in
order to compare results. They need their second set of axes for this section.
1. [C]
Clear Y1 from the list of equations. Keep the second
equation.
[P] Replot the second graph on a
new set of axes.
2. Obtain the hours of daylight data from two
other groups. Take special note of the location of their chosen city.
3. Find
the amplitude, period, phase shift, and vertical translation of both sets of
data, and determine their respective equations.
4. [C] Store the two equations in Y3 and
Y4. Graph the equations using the dashed line and the thick solid
line, respectively, to make it easy to identify each graph.
[P] Graph the equations on the
same set of axes. Be sure to label your plots.
5. Compare and contrast the three equations.
6. Compare and contrast the three graphs. For
what reasons would the graphs be similar or different?
C. Follow-up Skills
Time: 60
minutes
Supplemental
textbook exercises could be used to reinforce learning. In addition, the
following questions could be used to consolidate concepts and further
understanding of this particular model.
1. For an hours of daylight curve of the form h(t)
= c + a sin k(t - s), how were a, c,
k, and s determined?
2. What is the least amount of data needed to
determine a daylight curve? Two points: the maximum and minimum points will
provide the amplitude and phase shift. The vertical translation is always 12
hrs, and the period is always 1 year.
3. Hours of daylight functions can also be
modelled using cosine functions. In what ways would this function differ from
the sine equation? Determine the cosine equation that could be used to model
the data in the table found in Part 1 – Table for Sun, Please.
D. Supplemental Research
The following questions provide the teacher
with some topics for supplemental student research.
1. Find other cities with the same latitude as
those given in this activity. Would you expect the hours of daylight curves to
be different or similar? Confirm your prediction with some research.
2. The longest and shortest days of the year and
the days of equal daytime and night-time have special names. What are they
called and why? The longest and shortest days of the year are called the
vernal and autumnal equinoxes, respectively. The days of equal daytime and
night-time are called the winter and summer solstices.
3. Can hours of daylight data be modelled as a
sinusoidal function for every location on earth? Explain. Latitudes north of
the
4. From your original table, graph the sunrise
and sunset data on the same set of axes using the given days as your points of
reference. Are these curves sinusoidal? For any curves that do not look
sinusoidal, describe the way in which it fails to be sinusoidal. Why would
these curves not be sinusoidal? Due to the tilt of the earth’s axis, sunrise
and sunset curves are skewed, and cannot be modelled using sinusoidal functions.
5. Find other natural phenomena that can be
modelled using sinusoidal functions.
6. Longitude and latitude are measured in
degrees, minutes, and seconds. This is quite similar to our measurement of
time. Why is this so? Both the measurement of time and the measurement of
longitude and latitude derive from the ancient Babylonian number system, which
was base-60.
Assessment &
Evaluation of Student Achievement
All knowledge and skill categories
can be assessed in this activity. Parts 2 and 3 emphasize Knowledge and
Application skills, particularly in the modelling of functions using
information collected by the students themselves. Part 4 focuses primarily on
Knowledge and Communication skills, specifically the students’ ability to
compare and contrast graphs and equations. The follow-up section contains some
questions requiring students to use their Inquiry, Application, and
Communication skills. Criteria for assessment would include the ability to
hypothesize and justify reasoning, and the ability to apply their knowledge in
an unfamiliar setting. Learning skills, particularly initiative, organization,
and teamwork, can be assessed using appropriate rubrics. Part 4 of the activity
lends itself to group presentations, which can be assessed using a suitable
oral report rubric. The follow-up and extension questions could be assigned as
journal topics.
Accommodations
Because
of the long list of instructions in this activity, the teacher should ensure
that students with comprehension or communication difficulties are grouped with
students that can assist them.
Resources
Antinone,
L., S. Gough, and J. Gough. Modeling Motion: High School Math Activities
with the CBR.
Data
Services (http://aa.usno.navy.mil/AA/)
The
Astronomical Applications Department of the U.S. Naval Observatory produces
almanacs, software, and web services to provide precise astronomical data for
practical applications, serving the defence, scientific, commercial, and
civilian communities.
Ferris
Wheel Rides
(http://curry.edschool.virginia.edu/curry/centers/partnership/honalg2.htm)
This
is a Ferris wheel demonstration that includes some sample data as well as a
suggested rubric.
Modelling a Ferris Wheel Using Translations and
Animation (http://mathforum.com/dynamic/jrk/ferris_dir/)
This
is a tutorial to create a Ferris Wheel animation using Geometer’s Sketchpad.
Activity 4.5: Summative Assessment
Time: 150 minutes
Description
Students
demonstrate their ability to apply the skills and knowledge acquired in this
unit. A summative assessment is used to determine how the students have met the
expectations of this unit.
Overall
Expectations
TFV.02 -
demonstrate an understanding of the meaning and application of radian measure;
TFV.03 -
determine, through investigation, the relationships between the graphs and the
equations of sinusoidal functions;
TFV.04 -
solve problems involving models of sinusoidal functions drawn from a variety of
applications;
OCV.02 -
demonstrate an understanding of inverses and transformations of functions and
facility in the use of function notation.
Specific
Expectations
TF2.07 - demonstrate facility in the use of
radian measure in solving equations and in graphing;
TF3.01 - sketch y = sin x and y
= cos x, and describe their periodic properties;
TF3.02 -
determine, through investigation, using graphing calculators or graphing
software, the effect of simple transformations (e.g., translations,
reflections, stretches) on the graphs and equations of y = sin x and
y = cos x;
TF3.03 -
determine the amplitude, period, phase shift, domain, and range of sinusoidal
functions whose equations are given in the form y = a sin(kx +
d) + c or y = a cos(kx + d) + c;
TF3.04 -
sketch the graphs of simple sinusoidal functions [e.g., y = a sin
x, y = cos kx, y = sin(x + d),
y = a cos kx + c];
TF3.05 -
write the equation of a sinusoidal function, given its graph and given its
properties;
TF3.06 -
sketch the graph of y =
tan x; identify the period, domain, and range of the function; and
explain the occurrence of asymptotes;
TF4.01 -
determine, through investigation, the periodic properties of various models
(e.g., the table of values, the graph, the equation) of sinusoidal functions
drawn from a variety of applications;
TF4.02 -
explain the relationship between the properties of a sinusoidal function and
the parameters of its equation, within the context of an application, and over
a restricted domain;
TF4.03 -
predict the effects on the mathematical model of an application involving
sinusoidal functions when the conditions in the application are varied;
TF4.04 -
pose and solve problems related to models of sinusoidal functions drawn from a
variety of applications, and communicate the solutions with clarity and
justifications, using appropriate mathematical forms;
OC2.06 -
represent transformations (e.g., translations, reflections, stretches) of the
functions defined by
f(x) = x, f(x)
= x2, f(x) = , f(x) =
sin x, and f(x) = cos x, using function notation;
OC2.07 -
describe, by interpreting function notation, the relationship between the graph
of a function and its image under one or more transformations;
OC2.08 -
state the domain and range of transformations of the functions defined by f(x)
= x, f(x) = x2, f(x) = , f(x) = sin x,
and f(x) = cos x.
CGE2b -
an effective communicator who reads, understands and uses written materials
effectively;
CGE3c - a
reflective and creative thinker who thinks reflectively and creatively to
evaluate situations and solve problems;
CGE3e - a
reflective and creative thinker who adopts a holistic approach to life by
integrating learning from various subject areas and experience.
Prior Knowledge &
Skills
Students
should possess a comprehensive knowledge of the concepts introduced and
extended throughout this unit.
Teaching/Learning
Strategies
·
It
is intended that this evaluation provide the teacher with a variety of
assessment instruments, including a short activity, group work, and individual
work. This assessment could, however, take on several other forms. If the
teacher desires, this sample evaluation could be used as a unit test, to be
completed individually by students. Teachers may wish instead to have pairs or
groups of students complete the given tasks and be assessed collectively.
Selected activities and questions could even be delivered in the form of an
assignment.
·
It
is recommended that this summative assessment take place over two days to allow
for a thorough evaluation of student performance.
Part 1 – Me Tarzan, You
Bob
Description
Students
measure and determine the equation governing the sinusoidal motion of a
swinging pendulum.
Prior Knowledge &
Skills
·
Students
should have some experience in the modelling of sinusoidal equations.
·
Students
should be able to describe sinusoidal equations and their respective graphs
using characteristics such as the domain, range, period, phase shift, vertical
translation, and amplitude.
Planning Notes
·
Students
are paired for this activity, but they produce individual reports.
·
The
teacher should provide one piece of string of length 50 cm to each group, along
with one washer of at least 100 g weight and substantial surface area for use
as a bob. The washer should be tied to the end of the string to construct the
pendulum.
·
A
graphing calculator and a motion sensor (such as a CBR) are required for each
group, as are measuring tapes or rulers.
Teaching/Learning
Strategies
A.
Teacher Facilitation
·
This
activity ties the pendulum demonstration at the beginning of the unit with the
modelling of sinusoidal equations.
·
Provide
each pair of students with a pendulum. For best results, the pendulum should be
suspended from a fixed height. Alternatively, one student could hold the string
while the other records the data.
·
Students
record the motion of the bob for three different trials. The motion sensor
should be set to collect data for approximately 10 seconds, and the smoothing
should be set to light.
·
There
will be some “dead time” at the beginning of the students’ plot. The user can reset
the domain of the plot to eliminate this space.
B.
Student Activity
To aid
in the evaluation process, some solutions have been included in italics.
1. Extend the bob 10 cm from the rest position.
Start the motion sensor, and set the pendulum in motion. List the
characteristics of the resulting graph. Determine its equation. Explain the
reasoning by which this equation was arrived at.
2. Repeat this experiment by extending the bob
20 cm and 25 cm from the rest position.
3. Compare and contrast the graphs and their
respective equations. The period will be the same, regardless of the
starting position of the bob. The amplitude will increase as the bob is
extended further from the vertical. The vertical translation should be
constant, because of the fixed position of the ranger. If the domain has been
reset on the calculator, the phase shift will also be constant.
4. Predict with justification the equation that
could be used to model the motion of the pendulum if the bob was extended 15 cm
from the rest position. Repeat the experiment to confirm the prediction.
Part 2 – Group Assessment
A. Teacher Facilitation
·
Each
task in this group assessment is preceded by a suggestion of the skill
categories that would be most applicable to the given task ([K] indicates
Knowledge/Understanding, [I] indicates Thinking/Inquiry/Problem-Solving, [C]
indicates Communication, and [A] indicates Application).
B. Student Activity
To aid in the evaluation process, some
solutions have been included in italics.
1. The
average monthly temperature in
|
Month |
J |
F |
M |
A |
M |
J |
J |
A |
S |
O |
N |
D |
|
°C |
16.5 |
18.3 |
21.8 |
25.6 |
29.2 |
31.9 |
32.6 |
32.4 |
30.3 |
26.4 |
21.3 |
18.0 |
a) [K] What is the range of this
function?
b) [K] What is the average yearly
temperature?
c) [K,
C] Is this function sinusoidal? Fully explain your answer. This function is
periodic, with a period of 12 months. It is therefore sinusoidal.
d) [K]
Graph the data using a scatter plot. Does this confirm the answer from question
1C?
e) [K,
A] Given the general sinusoidal function T (t) = c + a sin[k(x
– s)], what do a, c, k, and s represent?
f) [K,
I] What characteristics of the function correspond to the constants a, c, k,
and s? The value of c corresponds to the average yearly temperature,
the value of a relates to the range of temperatures, k relates to the period (k
will have a value of 2p ¸ 12 in this case), and s corresponds
to the phase shift.
g) [I]
Determine the temperature function T(t). [Hint: Consider January
to be Month 0, February to be Month 1, etc.]
2. In 2001, Windsor, Ontario will receive its
maximum amount of sunlight, 15.28 hrs, on June 21, and its least amount of
sunlight, 9.08 hrs, on December 21.
a) [I,
A] Due to the earth’s revolution about the sun, the hours of daylight function
is periodic. Determine an equation that can model the hours of daylight
function for Windsor, Ontario.
b) [K] On what day(s) can Windsor expect 13.5 hours
of sunlight?
3. [K, I, A] Tides are cyclical phenomena caused
by the gravitational pull of the sun and the moon. On a particular retaining
wall, the ocean generally reaches the 3 m mark at high tide. At low tide, the
water reaches the 1 m mark. Assume that high tide occurs at 12:00 p.m. and at
12:00 a.m., and that low tide occurs at 6:00 p.m. and 6:00 a.m. What is the
height of the water at 10:30 a.m.?
4. The largest Ferris Wheel in the world is the
London Eye in England. The height (in metres) of a rider on the London Eye
after t minutes can be described by the function h(t) = 70 + 67
sin
(t – 30).
a) [A] What is the diameter of this Ferris
wheel?
b) [A,
C] Where is the rider at t = 0? Explain the significance of this value. This
position indicates the height of the boarding platform.
c) [A] How high off the ground is the rider at
the top of the wheel?
d) [A] At what time(s) will the rider be at the
bottom of the Ferris wheel?
e) [A] How long does it take for the Ferris
wheel to go through one rotation?
5. At Canada's Wonderland, a thrill seeker can
ride the Xtreme Skyflyer. This is essentially a large pendulum of which the
rider is the bob. The height of the rider is given for various times:
|
Time (s) |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
Height (m) |
55 |
53 |
46 |
36 |
25 |
14 |
7 |
5 |
8 |
15 |
a) [K] Create a graph of the position of the
pendulum with respect to the time.
b) [K,
I, A] Find the amplitude, period, vertical translation, and phase shift for
this function. [Note: that the table does not follow the bob through
one complete cycle, so some thought will be required to answer this question.]
c) [I]
Determine the equation of the function in the forms y = a sin k(x
+ s) + c and
y = a sin(kx + d) + c.
d) [K,
C] How could the amplitude be determined without creating the graph or finding
the function? The amplitude is half the range of the data.
e) [A]
What would the rest position of the pendulum be?
f) [A,
I] What is the maximum displacement for this pendulum? The maximum
displacement is the range.
g) [K]
The time for one complete cycle is the period. How long would it take to
complete 15 cycles?
6. [K, I, A] A mass suspended on a spring will
exhibit sinusoidal motion when it moves. If the mass on a spring is 85 cm off
the ground at its highest position and 41 cm off the ground at its lowest
position and takes 3.0 s to go from the top to the bottom and back again, determine
an equation to model the data.
Part 3 – Individual
Assessment
A.
Teacher Facilitation
Each task
in this group assessment is preceded by a suggestion of the skill categories
that would be most applicable to the given task ([K] indicates Knowledge/Understanding,
[I] indicates Thinking/Inquiry/Problem-Solving, [C] indicates Communication,
and [A] indicates Application).
B.
Student Activity
1. [K] Graph the function y = 3 sin x,
-2p # x # 2p.
2. [K, C] Compare and contrast the
characteristics of the graphs of:
a. y = sin x and y
= cos x b. y = sin x and y = tan x c.
y = cos x and y = tan x
3. [K, C] Does sin(-x) = -sin x? Explain by describing
and sketching their graphs.
4 [K] Given the graph y = cos x,
use transformations to sketch the following:
a. 
y = cos(x- ) b. y = cos(x + 1) c. y = -cos 4x
Assessment &
Evaluation of Student Achievement
In this summative assessment, several
opportunities exist for the evaluation of all of the knowledge and skill
categories. Criteria to be assessed in the activity might include:
·
ability
to follow the steps outlined in the investigation (Knowledge and
Communication);
·
ability
to compare characteristics of graphs (Communication and Knowledge);
·
ability
to determine the equation of a sinusoidal function (Inquiry and Knowledge);
·
ability
to predict results (Application);
·
proficiency
in calculator usage (Knowledge).
In the group and individual assessments, criteria might
include:
·
the
use of limited information to determine a sinusoidal equation (Inquiry and
Knowledge),
·
the
manipulation of information to answer indirect questions (Application and
Communication),
·
the
proper use of mathematical vocabulary in the justification of conclusions
(Communication, Knowledge, and Inquiry),
·
graphing
techniques (Knowledge and Communication).
All
learning skills (initiative, organization, work habits, teamwork, and the
ability to work independently) can be evaluated at some point during this summative
assessment.
Resources
NRC
CNRC –
The
Herzberg Institute of Astrophysics, a division of the National Research Council
of Canada, lists sun and moon data for hundreds of locations across
World
Climate (http://www.worldclimate.com)
This site
provides comprehensive climatological data for thousands of locations around
the world.
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