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Oscillation

Teacher’s Guide

55

Mathematics: Modeling Our World Teacher’s Guid

*The following Supplemental Activities are found in theTeacher’s Guide on the listed pages.

S5.2 The Circular Sine 60

56

Video Support

LESSON ONE 57

Life’s Ups and Downs

LESSON TWO 59

A Sine of the Times

LESSON THREE 62

Connections

LESSON FOUR 65

Fade Out

LESSON FIVE 67

Now We’re Cookin’

TEACHER’S GUIDE 55–70

HANDOUTS H5.1–H5.5 203–212

*SUPPLEMENTAL

ACTIVITIES S5.1–S5.13 289–322

ASSESSMENT A5.1–A5.10 487–508

TRANSPARENCY T5.1 556

5

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56 VIDEO SUPPORT UNIT FIVE: OSCILLATION

Teacher’s Guide Mathematics: Modeling Our World

B efore viewing the video for this unit, give eachstudent a copy of Handout H5.1, Video Viewing

Guide, that has questions for students to answer. Thequestions below may be used for discussion after stu-dents have seen the video.

1. During what months of the year do you thinkyour household’s electric bills (if they are notcombined with rent) are highest?

Sample answers:

In the summer because it’s hot and we turn on the air

conditioner.

In the winter because we have electric heat. Also, in

the winter the days are shorter so we turn on lights

for longer periods.

2. When was the last time we had a power outage

in our area? What caused it?

Sample answers:

It was so hot that everyone turned up their air condi-

tioners until the power went out.

An ice (or wind) storm took down some power lines.

3. What times of day do you think are peak hoursfor electrical usage? Do you think peak and slowdemand times occur at approximately the sametime every day? Are they different on weekdaysthan on weekends?

Peak times are probably in the morning, when peo-

ple are getting up to go to work and in the eveningaround supper time. In the morning, irons are heat-

ing, breakfast is cooking, and in the winter the heat

gets turned up, etc. In the evening, supper is cook-

ing, if it’s summer the house may need some extra

cooling, refrigerators are being opened, the television

is turned on, etc. Slow times are probably in the

early morning, say from 12 midnight to 4.00 AM. Most

people are sleeping at this time.

Video SupportVIDEO

SUPPORT

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57UNIT FIVE: OSCILLATION LESSON ONE


PREPARATION READING

To Oscillate

See Annotated Teacher’s Edition.

ACTIVITY 1

Would You Repeat That?

T he U. S. Navy has two Internet sites that provideinteresting resources for students:

Times of sunrise/sunset, moonrise/moonset, twilight,and other astronomical data are available from theUnited States Navy Observatory’s (USNO’s)Astronomical Applications Department athttp://aa.usno.navy.mil//AA/data/

The site: http://tycho.usno.navy.mil/srss.html providesvirtual reality moon-phase pictures. You only have to

specify the century, year, month, day, hour, and timezone.

According to the U. S. Naval Observatory:

The Moon’s phases are not technicallydefined in terms of fraction illuminated.However, the phase of the Moon can beidentified from the fraction illuminated. Tothe accuracy given in the table, the fractionilluminated at New Moon is 0.00, at Firstand Last Quarter it is 0.50, and at FullMoon it is 1.00. First and Last Quarter can be distinguished by noting whether the

fraction illuminated is increasing (Moonwaxing; in evening sky) and Last Quarteroccurs when the fraction illuminated isdecreasing (Moon waning; in morning sky).

INDIVIDUAL WORK 1

Play Ball!


ACTIVITY 2

Circle Game


INDIVIDUAL WORK 2

Nights at the Amusement Park


SUPPLEMENTAL ACTIVITY S5.1

The Changing Moon

Internet access or Handout H5.2

In this activity, students use the Internet to access dataon the fraction of the moon that is illuminated from January–March of the present year in your time zone.

The following two U.S. Naval Observatory Internetsites can be used to gather information for this activity:

For data on the portion of the moon illuminated—Astronomical Applications Department:http://aa.usno.navy.mil//AA/data/

For virtual reality pictures of the moon—Time ServiceDepartment: http://tycho.usno.navy.mil/srss.html

If you choose not to use the Internet for this activity,have the students use the data from Handout H5.2 tocomplete Item 1. For Item 2, have students sketchwhat they think the moon would look like on thesedates based on their data.

LESSON ONE

Life’s Ups and Downs

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Note that Assessment Problems are listed in theAnnotated Teacher’s Edition with the specific activi-ties and individual works in which the assessed con-cepts first appear. Although use of an assessment atthat point is appropriate, be cautious about assessingtoo soon. You may prefer to use the assessment afterconcepts have been revisited.

ASSESSMENT A5.1

Record Deal

T his assessment checks that students understandthe relationship between the rate of circular

motion, measured in revolutions per minute, andvelocity. In addition, it asks students to representgraphically the distance between a fixed point and anobject moving in a circle over time. It should beassigned at the end of Lesson 1.

Note: This problem assumes that students are familiarwith records and record players. Most students are

more familiar with CD players than with record play-ers. You might have to do some “historical education!”

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59UNIT FIVE: OSCILLATION LESSON TWO


PREPARATION READING

Round and Round It Goes,Where It Stops Nobody Knows


ACTIVITY 3

We’re Just Rolling Along


INDIVIDUAL WORK 3

Going in Circles


ACTIVITY 4

In the Light of the Moon


INDIVIDUAL WORK 4

What’s Your Sine?


ACTIVITY 5

Ferris Wheel Fun

For classes without access to spreadsheet software,graphing calculators may be used for this investi-

gation.

Items 1–4 may be completed directly, using table andgraph features of the calculator. (You may wish to usea function graph instead of a scatter plot for Item 4.)

A full investigation of symmetric difference valuesover the one- and two-revolution domains of Items 8and 9 requires a bit more effort on calculators. Oneapproach is to use lists, if your calculator will do listoperations. Perhaps a simpler approach is to definethe symmetric difference function using an expressionsuch as

Y2 = (Y1(X + 0.1) – Y1(X – 0.1))/(0.2)

and displaying its table or graph. The increment (0.1in this example) can be changed for greater precisionas needed. If this option is used, be sure to checkstudent understanding of the geometry behind thisformula!

INDIVIDUAL WORK 5

Bicycles, Daylight, and Tides


LESSON TWO

A Sine of the Times

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The Circular Sine

CD-ROM: Sin.GSP

T his activity has no student pages. Its purpose isto allow students to develop a visual and kinetic

feel for the unit-circle definition of the sine function.

The CD-Rom file Sin.GSP displays a dragable pointon the unit circle, simultaneously with the corre-sponding radian location of that point, the verticalsegment from that point to the horizontal axis (defin-ing the sine), the length of that segment (the sine),and the corresponding point on the graph of the sinefunction.

Provide a few minutes for students to explore thesedisplays in an unstructured manner. After they seemto have developed some comfort with the connectionsamong the various displays, have students revisitItems 3 and 6(d) of Individual Work 3, using the pro-gram to check previous answers. Units in both itemsand in the simulation are in radians. However, sincesine measurements are associated with rotations from(1, 0), students will have to adjust the simulationinformation in order to interpret it in the tire-rollingcontext.

The ferris wheel items can provide additional usefulfamiliarization in a similar fashion, but additionalwork will be needed since those have not yet beenconverted to radian measure.


Playing with Parameters

CD-ROM: SinGraph.GSP

T his assignment helps students connectmotion on a unit circle with graphs of sinu-

soidal functions. Students can set the values for A, B, C, and D to see how these values affect thegraph of the model y = Asin(B(x – C)) + D.

Use this activity as a replacement Item 1,Activity 4 (Option 2). Or use it as a whole-classactivity to reinforce ideas from Item 1, Option 1.


Directed Investigationof Sinusoidal Functions

T his activity is a more directed version of Item 1of Activity 4. Students investigate how the con-

stants in sinusoidal models of the form

y = Asin(B(x – C)) + D affect the graph.


Another Lunar Mission

S tudents investigate a “best” fitting model for themoon data according to the least squares criteria.

Remind students that according to a least squarescriterion, the “best fitting” sinusoidal function has thesmallest sum of squared residual errors. In Course 1,Unit 4, Prediction, students used spreadsheet and/orgraphing calculators to estimate the equation of a

least squares line. The principle here is the same.

You can use Excel to aid in fitting a sinusoidal modelto these data. Figure 1 shows a worksheet that illus-trates the set-up. The data have been entered intocolumns A and B. C2 contains the formula= 0.5 x sin($B$1 x (A2 – $C$2)) + 0.5. (In Excel x = *.)Column D contains a formula that computes theresiduals, the differences between the entries in col-umn B and C. Column E contains the squared residu-als. Column F contains the sum of the squared errorsfor this model. Students may change the values ofconstants B and C in order to find a model that has a

small value in F2.

Figure 1. Excel worksheet.

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Once students have fit a model to the moon data,point out that the model will not hold forever. Useyour model to predict moon data 10 years into thefuture. Then have students access that informationfrom the Internet. The fact that the fit is not perfectindicates that the illumination data are not exactlysinusoidal. The U. S. Naval Observatory model

adjusts for non-circularity of orbits and other factors.For Item 2(b), check that students rewrite

y = 0.49sin(0.21x + 2.84) + 0.53 in the form y = 0.49sin(0.21(x + 13.52 )) + 0.53 so that they canread off the phase shift from their formula.

In Item 2(c), remind students that residual plots areuseful tools for assessing the fit of a model. Do thepoints in the residual plot appear to be randomlyscattered? Are approximately half of the residualspositive and half negative?


High Noon

T his activity gives students an opportunity topractice fitting sinusoidal models to graphs. In

addition, it asks students to use their model to answerquestions related to the context of the model.

This activity may be used after Activity 5 for review,reinforcement, or assessment.

ASSESSMENT A5.2

Rodents

In this assessment, students are given a graph andmust determine a sinusoidal model that describes

the graph. For this graph, the phase shift in a modelof the form y = Asin(B(x – C)) + D is 0 (this makes theproblem somewhat easier for students).

This assessment can be assigned any time after stu-dents have completed Activity 4, “In the Light of theMoon.”

ASSESSMENT A5.3

Design a Formula

For this assessment, students write a sinusoidalmodel that describes a wave on the screen of an

oscilloscope. Note that the units of measure are given, but “zero points” are not. Thus many answers are

possible, differing by translations.Assign this problem after students have completedIndividual Work 4. A good follow-up to this assess-ment would be to team teach a lesson with a scienceteacher and actually perform this experiment.

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62 LESSON THREE UNIT FIVE: OSCILLATION


PREPARATION READING

Sine Off


ACTIVITY 6

What are Your Rates?

I f the calculators your students are using have anumeric derivative command, you may wish to

hand out Supplemental Activity S5.7, “Using nDerivon the TI-83 to Graph Rate Functions,” to ease the cal-culations for symmetric difference quotients. (Provideappropriate adjustments if you are not using TI-83’s.)However, do not use the supplement if students donot understand the symmetric difference quotient asan approximation of slope. Otherwise the nDerivcommand will seem like magic without any meaning.

INDIVIDUAL WORK 6

What’s My Equation?


ACTIVITY 7

Point of Reference

Precursors to the Sine and Cosine

In early (Asian) Indian mathematics, trigonometryformed an integral part of astronomy and refer-ences to trigonometric concepts are found as early as400 A.D. The arc of a circle was pictured as a bow(Figure 2) and referred to as capa.

Its full-chord (line segment AMB) was visualized asthe bow-string (samasta-jya). Indian mathematicians,in their study of trigonometry, generally used the

half-chord (line segment AM or MB) called jya-ardha,which lead to the abbreviated name jya, the precursorof our modern sine function. The precursor of ourcosine function was kojya θ = OM. The modernequivalents to these functions are:

jya θ = AM = r sin(θ ), and

kojya θ = OM = r cos(θ ).

Of course, the values of jya θ and kojya θ depend onthe radius chosen for the circle. Different mathemati-cians chose different-sized circles. For example,Varahamihira, Aryabhata, and Brahmagupt used cir-

cles of radii 120, 3438, and 3270 units, respectivelyand calculated tables of values of jya for variousangles θ . If only they had divided their entries bytheir chosen radii, their estimates could have beencompared directly to modern approximations (using acalculator) of sin(θ ).

LESSON THREE

Connections

A

B

O C M

r

θ

Figure 2. The bow.

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Geometer’s Sketchpad file SinCos.GSP

You will need the Course 3 CD-ROM file, SinCos.GSP,for the final items in this activity. This file generatesan animation of a point as it moves around a unit cir-cle and of the corresponding legs of the right trianglethat define the x- and y-coordinates of the point (seeFigure 3). The lengths of the legs are simultaneously

plotted against the radian measure of the portion ofthe circle through which the point has moved.

The file can be used for a class demonstration or forsmall-group investigations. When the program isopen, the screen will resemble Figure 3. Students canclick on the Graph Fast or Graph Slow buttons, orthey can drag the point around the unit circle manual-ly. Manual operation provides the most direct controlover the animation.

Figure 3. One frame from the animation SinCos.GSP.

INDIVIDUAL WORK 7

The Angle’s Central



Using nDeriv on theTI-83 to Graph Rate Functions

T his activity can be interwoven with Activity 6.

Use Item 1 in this activity to ease the calculationsof the symmetric difference quotients in Item 3(a),Activity 6. Use Item 2 to complement Item 3(d),Activity 6. Use Item 3 to extend the explorations inItem 8, Activity 6. Finally, use Item 4 to remind stu-dents that they have studied rate functions before. Inparticular, students studied the rate of change func-tions corresponding to linear and quadratic functionsin Course 2, Unit 7, Motion.


Practice Makes Perfect

T his activity asks students to explain how variousmodifications to the sine and cosine functions

will affect the graphs. In addition, given graphs, stu-dents determine at least two equations (one based on

the sine function and the other based on the cosine)that describe the graphs.


Project 1: Swing Your Partner

Motion detector apparatus (for example, CBL, motiondetector, calculator)

Plastic soda bottle or water jug

Stopwatch

String

CD-ROM: Pendulum.83p (or .82p)

Students use motion detectors to collect data on aswinging pendulum, then fit sinusoidal models totheir data.

This is a great project to assign any time during thislesson. In fact, you could assign this project at the endof Lesson 2. However, because it provides a goodlead-in for damped oscillations, introduced in Lesson4, it may be best used during Lesson 3. For the shorttime interval that the pendulum swings in this activi-ty, its graph appears periodic. The question that links

this activity to Lesson 4 is: If you let the pendulumswing for a long period of time, how would yourgraph change over time?

SUPPLEMENTARY ACTIVITY S5.10

An Early Sine

In this activity, students use calculations from twoIndian Astronomers (500 A.D.) to estimate the value

of sin(θ ) for six different angles θ .

63UNIT FIVE: OSCILLATION LESSON THREE


–2 21–1

–1

–2

1

2

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ASSESSMENT PROBLEM A5.4

Oranges

In this assessment, students must model the price oforanges over a year, using both sine and cosine

functions. Based on their model, students must deter-mine the months when prices are rising or falling

most rapidly.This assessment can be assigned any time after stu-dents have completed Activity 6.


Satellite Orbits

S tudents are presented with a projection of a satel-lite as it circles the earth. The projections shift

each time the satellite orbits the earth, and studentsare asked to model three of the projected patternswith a sinusoidal function.

This assessment provides a real-world application forsinusoidal models with inputs in degrees (°). Note,however, that the period for each orbit is slightly(about 30°) less than 360°. In fact, if that were not thecase, each orbit would be directly coincident with theprevious orbit and there would be no shift. The shiftis caused by the earth’s rotation under the satellite; inreality, the satellite orbit does not shift, but the earthshifts relative to the orbit! The amount of shift isdetermined by how long one full orbit takes. Sincethis satellite orbit moves about 30° each pass, one tripmust take approximately 2 hours:

(30°) x (24 hours/360°).

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65UNIT FIVE: OSCILLATION LESSON FOUR


PREPARATION READING

The Nature of Sound

See Annotated Teachers Edition.

ACTIVITY 8

Do You Hear What I Hear?


INDIVIDUAL WORK 8

Music Fills the Air


ACTIVITY 9

Diminishing Returns


INDIVIDUAL WORK 9

Name that Graph


ACTIVITY 10

It’s Slinky® Time


ACTIVITY 11

Cowabungee, Dude!


SUPPLEMENTAL ACTIVITIES S5.11

How Do You Fit aSquare Peg in a Round Hole?

In this activity, students try to approximate a squarewaveform by summing sinusoidal functions. This

is an excellent means of helping students visualize thegraph of the sum of two sinusoidal functions from thegraphs of the individual functions.

You may want to have students write a program thatgenerates the models formed from sums of asequence of sine functions.


Playing Around

For this assessment, students examine two func-tions that result from composing the sine and

cosine with functions that involve the cosine and sine,respectively.

This assessment could be used any time after the sineand cosine functions are introduced. However, itmakes sense to use it in this lesson, because it falls inthe category of a compound function. The curves

produced by each of these functions have interestingcharacteristics, one of which is a period of 2π.

LESSON FOUR

Fade Out

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A Vibrating Platform

In this assessment, students look at a graph pro-duced by a vibrating oil-drilling platform. Students

are asked to focus on the amplitude of the oscillationand to create a graph representing the amplitude ver-

sus time.You may assign this assessment any time afterActivity 8. However, it is probably best to wait at leastuntil after Activity 9. This gives students a chance toplay with a set of real data that exhibits damped oscil-lation before assessing related skills.


Predator-prey Cycle

S

tudents are given a joint predator-prey popula-tion graph and must extract population-versus-

time information for both the predator and the prey.In both cases, the population-versus-time graphs pro-duce damped oscillating patterns.

You can assign this assessment any time after dampedoscillation has been introduced (Activity 8). However,it probably is the most effective as an end-of-lessonassessment.

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67UNIT FIVE: OSCILLATION LESSON FIVE


PREPARATION READING

Growing Concerns


ACTIVITY 12

Do(ugh) Re Mi

Y ou may wish to permit students to use built-insinusoidal regression to obtain or check their

models in this activity. Sample answers for that caseare given below.

1. a) Using the regression features on a TI-83gives y = 9.67 sin(0.547x – 3.000) + 102.9.

Below is a graph of this model and thescatter plot of these data. The modelappears to do a reasonably good job in

summarizing the flow of these data.

1. b) Sample answer based on the regressionmodel in the sample answer to (a). Belowis the residual plot.

The data appear to be randomly

scattered. The only problem might be thatmore of the residuals are positive in thesecond year than in the first year.

2. e) Below is a residual plot based on theregression model from sample answer toItem 1.

2. f) Regression model: y = 0.278x – 3.417.

3. a) Regression model:

y = [9.67 sin(0.547x – 3.000) + 102.9] +

[0.278x – 3.417] ≈

9.67 sin(0.547x – 3.000) + 0.278x + 99.5.

LESSON FIVE

Now We’re Cookin’

Window [–1.3,26.3]x[88.79,118.81]

Window [–1.3,26.3]x[88.79,118.81]

Window [–3.7,52.7]x[–7.74,20.36]

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The revised model appears to have arising oscillating pattern similar to the

data. Below is a graph of the revisedmodel and a scatter plot of the data.

3. b) Regression model:

The residual plot below appears to befairly randomly scattered. The revisedmodel appears adequate to describe the

pattern in these data.

4. b) Regression model:

Use the sinusoidal regression feature onyour calculator to fit a model describingthe oscillating pattern of the residual plotcorresponding to the linear model in (a).

The model for the oscillating pattern ofthe residuals is:

y = 9.73 sin(0.532x – 2.877) + 0.10.

4. c) Final model based on regression model:

y = [9.73 sin(0.532x – 2.877) + 0.10] +

[0.32x + 98.13] ≈

9.73 sin(0.532x – 2.88) + 0.32x + 98.23.

Below is a residual plot based on thismodel.

The residual plot looks reasonably goodexcept for one extreme outlier of –7.4%

(at month 23).

5. a) Regression models:

Model from 3(a): y = 9.67 sin(0.547x – 3.000) + 0.278x +

99.5.

Model from 4(c): y = 9.73 sin(0.532x – 2.88) + 0.32x + 98.23.

Both models have the same form: y = Asin(BCx – C) + Dx + E. The values ofthe constants A, B, C, D, and E differ only

slightly between the two models.However, the slope of the linear trend inmodel 4(c) is slightly larger than formodel 3(a).

INDIVIDUAL WORK 10

The Remains of the Data


ACTIVITY 13

Calendar Effects


68 LESSON FIVE UNIT FIVE: OSCILLATION


Window [–3.7,52.7]x[87.36,128.64]

Window [–3.7,52.7]x[–9.89,10.41]

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Housing Starts

T his activity returns to the housing-starts dataintroduced in Item 7, Individual Work 1, Lesson

1. It gives students an opportunity to apply whatthey have learned in Activities 12 and 13 to another

context.You may wish to allow students to use their calcula-tors’ regression feature when they fit a sinusoidalmodel to these data. However, you may also want toinsist that they correct the model so that it has the cor-rect period of 12. If students get an error messagewhen trying to perform sinusoidal regression, explainthat they may need to tell their calculator (as part ofits regression command) what the period of this func-tion should be before the calculator is able to fit asinusoidal model.

You can extend this activity by asking students to

search for calendar effects in the housing data usingan analysis similar to that suggested in Activity 12.For example, because beginning a house is such alarge project, you might expect more houses to bestarted on Mondays than on any other days; thatturns out to be fairly visible in the data.

If you would like your students to work on morerecent housing-starts data, you can access this infor-mation on the internet. The U.S. Census mainatinsrecords of such information. Either search under thephrase “housing starts and building permits,” or godirectly to the site

http://www.census.gov/pub/const/c20_hist.htm for themenu “Housing Starts and Building Permits HistoricPress Releases.”


Project 3: Eclipsing Binary Stars

CD-ROM: EcBiStar.GSP

In this activity, students run a simulation that gener-ates light curves associated with eclipsing binary

stars. During the simulation, students select the sizesand brightnesses of the stars in the eclipsing pair.Then they create a classification system for the vari-ous types of light curves generated.

Here is some background information.

Eclipsing binary stars are two or more stars that are inclose orbit about each other. The stars are often soclose that they appear to be one point of light in thenight sky. If the plane of their orbit is oriented to con-

tain the earth, and if their sizes and luminosities areappropriate, the brightness of the star system variesover time. Eclipsing binary stars that exhibit this phe-nomenon belong to the class of variable stars.Astronomers can learn much about variable stars bystudying their luminosity or light curves. A lightcurve is a graph of the brightness of the star system

over time. Eclipsing binary stars have periodic lightcurves.

The CD-Rom file EcBiStar.GSP contains a simulationof two eclipsing binary stars. You can vary the sizes ofthe two stars, their relative brightness, and the orien-tation of their orbital plane. After setting these para-meters, you can run the simulation and see a graph ofthe light curve that these stars would produce ifviewed at a great distance.

Because the simulation is simple, a couple of impor-tant characteristics of the model must be made clear.As the simulation runs, the first passing of the two

stars represents the yellow star passing behind the redstar as it moves left to right. The second passing ofthe two stars represents the yellow star passing in

front of the red star as it moves right to left. Rememberthis sequence of events when studying the lightcurves.

Also, although the light curve is correct, the apparentmotion of the yellow star with respect to the red star isnot completely correct. In reality, if the yellow starappears to pass behind the red star off-center (forexample below the center of the red star) in one direc-tion, it would pass the red star off-center in the other

direction when it passes in front of the red star duringthe second half of the orbit (see Figure 4). The dis-tance the star is off-center, that is above or below theother star, is constant, regardless of the direction ofmotion. Therefore, the progress of the overlapping ofthe two stars is identical whether the yellow star ispassing behind or in front of the red star. Many calcu-lations are simplified by taking advantage of thissymmetry and having the yellow star simply move back and forth along a straight line.

The simplicity gained in the linear transit model doesintroduce a small error into the light curves. Thiserror is negligible, however, if your intent is to studythe large-scale characteristics of luminosity curves. Adetailed study is beyond our purposes here andwould require a more accurate model. Perhaps aninterested student could construct such a model as aspecial project.

Figure 4 illustrates the yellow star’s orbit about thered star. The left column of pictures show the positionof the two stars at various times during a complete

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orbit. The right column shows the model’s representa-tion of this motion. Instead of the yellow star’s orbit-ing the red star in a circular path, it moves along aline. As it moves right, it passes behind the red starand on its return left, it passes in front of the red star.

Figure 4. Comparison of actual orbit

and linear transit model used by simulation.


Electrocardiogram

In this assessment, students are presented a graphof an electrocardiogram. Then they must extract

information from a smoothed version of the electro-cardiogram and the residual plot.

This assessment can be assigned any time after you begin this lesson.


Milk Production

S tudents are shown data on quantity of milk ver-sus time. These data exhibit both oscillation and

trend. However, the situation is made more complex by the fact that the graph is split into two equal time

intervals.Use this any time after Activity 12.

70 LESSON ONE UNIT FIVE: OSCILLATION


In front

Behind

YellowRed

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