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.

q (in degrees)

0

15

30

45

60

75

90

105

120

135

150

360

q (in radians)

 

 

 

 

 

 

 

 

 

 

 

 

 

coordinates (x, y)

 

 

 

 

 

 

 

 

 

 

 

 

 

sin q =

 

 

 

 

 

 

 

 

 

 

 

 

 

cos q =

 

 

 

 

 

 

 

 

 

 

 

 

 

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.

Strand(s):  Trigonometric Functions, Tools for Operating and Communicating with 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;

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.

x (in radians)

 

 

 

 

 

 

x (in degrees)

0

 

 

45

60

 

90

 

120

 

 

360

sin x

 

 

 

 

 

 

 

 

 

 

 

 

 

cos x

 

 

 

 

 

 

 

 

 

 

 

 

 

sin x ¸ cos x

 

 

 

 

 

 

 

 

 

 

 

 

 

tan x

 

 

 

 

 

 

 

 

 

 

 

 

 

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

·         Sunrise and sunset data for thousands of locations around the world is readily available at the U.S. Naval Observatory website, http://aa.usno.navy.mil/AA/. If students have access to a computer lab, the teacher may wish to direct students to the website to collect their own data. Alternatively, students that have access to a computer at home may collect data for the class.

·         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

·         Sunrise and sunset times for any location can be calculated if its longitude and latitude are known. In order to make meaningful conclusions from the data, the sunrise and sunset times are obtained for five different cities, located at approximately 20°N, 30°N, 40°N, 50°N, and 60°N.

·         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., 4:00 p.m. would be 16:00). Note: In the chart, Sunrise [: ] refers to the time in the standard format of hrs:min (e.g., 10:45), while Sunrise [ . ] refers to the time in decimal format (e.g., 10.75). The first couple of rows of the proposed table would appear as follows:

City:

Latitude:

Longitude:

Date

Day of Year

Sunrise [: ]

Sunrise [ . ]

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 Arctic Circle or south of the Antarctic Circle experience extended periods of 24-hour daytime and night-time, hence the hours of daylight function cannot be modelled using a sinusoidal curve.

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. Austin, TX: Texas Instruments, 1997. ISBN 1-886309-14-0

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.

Strand(s):  Trigonometric Functions, Tools for Operating and Communicating with 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;

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.

Ontario Catholic School Graduate Expectations

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 New Orleans, Louisiana, is given in the following table:

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(xs)], 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 – Sunrise/Sunset Tables (http://www.hia.nrc.ca/services/sunmoon/sunmoon.html)

The Herzberg Institute of Astrophysics, a division of the National Research Council of Canada, lists sun and moon data for hundreds of locations across Canada.

World Climate (http://www.worldclimate.com)

This site provides comprehensive climatological data for thousands of locations around the world.

 

 

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