algebra through modeling

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Algebra Through Modeling

This module is designed to familiarize teachers with a new and very innovative

advanced algebra course. This course presents algebra from the perspective ofmodeling and data analysis using the basic algebraic functions and a graphing

calculator with data analysis capabilities such as those of any of the TI-83/84family of graphing calculators including the TI-83 83! 83! "ilver #dition 84!

and 84! "ilver #dition. This new approach will be useful in high school andcollege for students who have completed two years of algebra but are not ready

for precalculus. It reviews algebra without repeating earlier course wor$ and itclearly demonstrates the importance of algebra to the solution of real world

problems. Credit:% grad. sem. hr.

Common Core Standards for Mathemtical Practicethat are emphasizedinclude&

' (. )eason abstractly and *uantitatively.

' 4. +odel with mathematics.' ,. se appropriate tools strategically.

' . oo$ for and ma$e use of structure.

Algebra through Modeling with the TI-83 family of ra!hing Calculators

was written by Tony Peressiniand "ohn #u$erof the niversity of Illinoisduring 0ovember %112. It was revised by Peressiniand #u$erduring 0ovember%11 and by Peressini and Tom Andersonin 0ovember (3 and une (,.

Chec$ out some sam!le !ro%ects for this module in the MT# Classroom

Pro%ects to the right&

'etailed 'escri!tion

The !ur!ose of this module is to familiari(e mathematics teachers at the

high school and lower di)ision college le)el with the content and teaching

strategies for a !romising new a!!roach to teaching ad)anced algebra&

This a!!roach stresses modeling and sol)ing real world !roblems andde)elo!s s$ills and conce!ts of algebra as needed for this modeling !rocess&

In this course* the students wor$ in a collaborati)e learning en)ironment

and ma$e e+tensi)e use of a gra!hing calculator with data analysis andse,uence ca!abilities such as those found on the TI-8* TI-83* TI-83. and

TI-83. Sil)er /ditionAfter com!leting this module* a teacher will $now the details of this

a!!roach and will be !re!ared to use or ada!t the course or some to!ics of

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the course to his or her own instructional setting.

This course, which has been used at the University of Illinois quite successfullyfor six years, was based on preliminary editions of the text, 'Functioning in theReal orld', by !heldon "ordon and #en Fusaro which was published by

$ddison%esley &ublishing ompany in (ovember, )**+ and in a secondedition in --./ 0!ee the Required 1aterials button at the bottom of the 1odule 2

home page for a detailed description of the text and ordering information/3 Thistext presents algebra, trigonometry and other topics from a non%traditional

perspective that emphasi4es the use of the basic algebraic and trigonometricfunctions to model interesting and current real%world problems/ 5ata analysis is

used to develop and test these models and to draw inferences from them/1anipulative s6ills are taught 7as needed7 for the model development and

analysis and not as 7stand alone7 topics as is typical in traditional algebra textsand courses/

The Illinois version of the course is taught using the following teaching strategies

and tools81)The !mall%"roup Instruction 9or ollaborative :earning; 1ethod

2)1astery Testing of $lgebraic !6ills3)"raphing calculators with data analysis and sequence capabilities 9forexample, any member of the TI%


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In addition to the general requirementsfor participating in Math Teacher Link,this module also has the following requirements:

Any of the TI-83 family of calculators (including TI-82, TI-83,TI-83+

and TI-83+ Silver Edition, TI-8 !lus"Each participant needs to have access to either of these Texas Instruments

raphing calculators! Functioning In The Real World#y Sheldon !$ %ordon

Each participant must o"tain a cop# of the second edition of the text

Functioning In The Real World"# $heldon %! ordon, et al!, &'ddison()esle# Longman* I$+- .(/.0(12123(4*! 5or 6rdering Information,

please call the pu"lisher 'ddison )esle# Longman at 0(2..(3//(.783!9sed copies of the "ook are also availa"le at 'maon!comat significantl#

reduced prices! This "ook is an excellent resource to have availa"le in an#high school mathematics li"rar#!

Ste& #y Ste& Instructions

)e estimate that the work for this module will take participating teachers anaverage of a"out ;7 hours to complete! Technicall#, #ou have a 1(month

enrollment period to complete the module after #ou enroll! export>html>40


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Step #2

After you have obtained a copy of the text Functioning in the Real World(Second Edition) by Gordon, et al, you should read the sample sections in the

text for each of the Chapters , 2, and !, and then print and read the"nstructors $otes %or Chapters , 2, and !& 'hen print and complete thesample student orsheet (see belo) for that chapter&

The sample sections of the text that you should read are Sections 1.1 through 1.5of Chapter 1, Sections 2.2 through 2.9 of Chapter 2, and Sections 4.5 through 4.7of Chapter 4.

"t is important to do Step one chapter at a time. That is, after you hae readthe sample sections of the text and the !eading "otes for that chapter for one of

these chapters, print and complete the Sample Student #or$sheets %attached&elo'( for each of these three chapters as )ssignment 5.1.

*f you are enrolled for graduate or continuing education credit, mail the threecompleted 'or$sheets for Chapters 1, 2, and 4 to+

Tom )nderson15721 a$eside -rie

Sterling, * 1/01

e sure to ma$e a copy of student 'or$sheets &efore you mail them

Attachment Si*e

Ch 1 'or$sheet 1.pdf 04.50 3

Ch 2 'or$sheet 2.pdf 50.1 3

Ch 4 'or$sheet 4.pdf 221. 3

"nstructors $otes on Chapters , 2,

and !

"ntroduction+

#e hae as$ed you to read carefully the follo'ing sample sections of the text+

1( Sections 1.1 through 1.5 of Chapter 1 2( Sections 2.2 through 2.0 of Chapter 2

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( Sections 4.5 through 4.7 of Chapter 4.

Chapter eadin- $otes

The principal topic of the sample sections for Chapter 1 is functions of one

aria&le as defined &y formulas, graphs and ta&les. *t includes the usual :rule:definition of function and the recitation of the standard function terminology+domain, range, dependent and independent aria&les, etc.

These are standard topics in irtually any adanced alge&ra course. The onlyindication that this &oo$ might hae a different emphasis than other adanced

alge&ra &oo$s is in the pro&lems for Section 1.5 'hich repeatedly as$ the studentto dra' a reasona&le graph of a function that is descri&ed &y some :real 'orld:

situation. -rill pro&lems coering the terminology and definitions are included,&ut emphasi;ed less than in traditional alge&ra &oo$s.

Chapter 2 eadin- $otes

The principal topics of the sample sections, linear, exponential, logarithmic andpo'er functions and their properties, are certainly familiar topics in irtually any

adanced alge&ra course. xponential functions are those 'ith a constantpercentage rate of increase %ordecrease(.

6o'er functions hae no similar description in terms of rates of increase ordecrease.


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3. The values of a power function with a large positive power will alwayseventually become and stay larger than the values of a power function with

a smaller positive power, regardless of the relative sizes of their (positive)initial values. The practical implications of these comparisons are exploredin a variety of contexts including population growth and comparative

economic growth.

The section on logarithmic functions makes similar comparisons about therelative growth of power and logarithmic functions.

Chapter 4 Reading Notes

hapter ! deals with polynomial functions, a standard topic in advancedalgebra, and with transformations of functions (vertical and horizontal

shifts, vertical and horizontal dilations, etc.), an important topic that isfre"uently neglected in advanced algebra courses.

#ou are not re"uired to read $ections !.% through !.! for any of the work

specified for this module, but you may find it interesting to scan the contentbecause you will find that it focuses much more on the graphical

characteristics of polynomial functions than typical advanced algebra

books.

The focus of the sample section, &inding 'olynomial 'atterns, is on

recognizing polynomial functions by successive differences, an importanttopic for applications that is normally omitted in standard advanced algebracourses.

The two sample sections on transformations of functions, uilding ew

&unctions &rom *ld, are a bit misplaced in this chapter because the contentis not restricted to polynomial functions. *ur experience with teaching this

material is that it is "uite accessible to +ust average students, and that ithelps all students to begin to recognize and use relationships between an

algebraic formula for a function and its graph.

Step #3

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Read the sample text sections for Chapter 3. Then complete the Fitting Data WithThe TI-82 tutorial.

The sample sections for Chapter 3 are Sections 3.2 through 3.6

Reading these sections should familiarize you with the basic ideas and

terminology of linear, eponential and power regression analysis of two !ariabledata. The calculator tutorial will show you how to carry out regression

calculations and displays on the T"#$2 calculator in the contet of a concreteproblem in cancer data analysis. The modifications that are necessary if you are

using any other T"#$3 family calculator are relati!ely minor.

Step #4

Reie! the "ample "tudent Wor#sheet for Chapter 3 $attached %elo!&. Then printand do the sample !or#sheet.

The wor% that you ha!e done in Step 3 will pro!ide the mathematical bac%groundand calculator s%ills to do the sample wor%sheet and de!elop similar ones foryour classes.

If 'ou are enrolled for graduate or continuing education credit( mail thecompleted !or#sheet for Chapter 3 as )ssignment *.2 to+

Tom &nderson

'()2' *a%eside +ri!eSterling, "* 6'$'

Attachment Size

wor%sheet3.pdf '2(.)) -

Step #5Read the sample text sections and theInstructor,s otes for Difference /uations

0Chapters * and 12. Reie! the t!o "ample "tudent Wor#sheets for thesechapters attached %elo!. rint and complete them as )ssignment *.3.

The sample sections are Sections (.', (.2 and (./ of Chapter ( and Sections '2.',

'2.2, '2.3 and '2./ of Chapter '2.

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If you are enrolled for graduate or continuing education credit, mail the twocompleted worksheets for these chapters to the mailing address printed in Step

#2.

If you are enrolled as an MTL guest or for continuing education credit, you haveust completed the module. !ongratulations" Those of you enrolled for

continuing education credit will e sent feedack on the three assignments thatyou were re$uired to sumit.

Those of you enrolled for graduate credit still need to complete an approved %inal

&roect. 'o to Step ( to find out how to do that.

Attachment Size

worksheet).pdf **+ -

worksheet*2.pdf 2*./0 -

Instructor's Notes for Difference

Equations (Chapters 5 & 12)

Introuction

1ur course at the niversity of Illinois was developed using a preliminaryversion of %unctioning in the 3eal 4orld 5 6 &recalculus 78perience y Sheldonet al and was taught for several years using !hapters * through ( and later using

the first edition of the ook. The second edition of the ook was pulished in2++9. 4e decided to modify this course module in 2++) so that it can e used

with the second edition ecause the first edition is no longer in print and usedcopies of the first edition are increasingly difficult to otain from te8took

resellers.

%ortunately, the innovative approach to algera and modeling with functions thatfirst attracted us to this course and that was ased on the preliminary version ofthe ook was retained and enhanced in the first and second editions. In fact, many

of the changes that we recommended to the preliminary and first editions of theook were adopted in the second edition. 6s a result, the latest edition is very

well suited for use in this course module.

78cept for relatively minor organi:ational changes and some new or revisedprolems, the content and approach of !hapters *, 2, / and 9 of %unctioning in

the 3eal 4orld 5 6 &recalculus 78perience have remained relatively constant

through the various editions of the ook. ;owever, the material in !hapters ) and( on modeling with difference e$uations in the preliminary and first edition haveundergone sustantial change in the latest edition. The content of !hapter ( in the

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preliminary and first editions has been moved to Chapter 12 in the secondedition, which is a chapter that is only available for download from the book web

site at http://www.aw.com/ggts

In our course at Illinois, we found it necessary to cover the material in a

somewhat different order than that found in the book, partly because of the timeconstraints of our course and partly because we believe that the order of the

content in the tet led to certain confusions in the presentation that can easily beavoided.

!he order of the content on difference e"uations in the tet is this: In Chapter # of

all versions of the book, difference e"uations of first and second order, linear andnonlinear, are introduced and used to model a number of interesting applications.

$owever, the systematic development of solution methods for these differencee"uations was delayed until Chapters % and & in the first edition and until Chapter12 in the second edition. 'olutions to difference e"uations arising from models

and applications introduced in Chapter # are usually either stated without formal(ustification or are obtained by special methods that do not generali)e well.

*lthough systematic solution methods are presented later +in Chapters % and & inthe first edition and in Chapter 12 in the second edition, we find this delay in

presenting systematic solution methods for difference e"uations to beunsatisfactory because students often do not grasp the important distinction

between a difference e"uation and its solution and, as a result, they are not able tounderstand the point of looking for a difference e"uation to model a given

application in the first place.

In our three semester-hour algebra course, we do not have time to cover much ofChapter % or any of Chapter &. Conse"uently, we limit our discussion ofdifference e"uations to a detailed presentation of first order linear difference

e"uations, their applications and their solution methods. e are only able todevote a day or two at the end of the semester to one or two applications that lead

to nonlinear or second order e"uations.* similar limitation would seem to be in order for most year-long high school or

one semester college courses based on this tet. !hese notes are intended to helpinstructors in such courses to cover this content successfully.

Instructors 0otes on Chapter #: odeling with ifference 3"uations

!his chapter begins with a discussion of applications of difference e"uations toproblems involving the elimination of medications from the bloodstream underrepeated drug dosages in 'ection #.1. ost of the basic terminology and solution

methods concerning difference e"uations are either discussed informally in termsof this application or delayed until the first two sections of Chapter 12. !hese

medication elimination problems are all modeled by se"uences specified by firstorder linear difference e"uations with given initial terms, although the definitions

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of first order linear difference equations are not introduced formally until Section12.1.

These problems are all modeled on the basis of the assumptions stated on Page366:

i) The idneys remo!e a fi"ed proportion 1 # a of the medication

from the bloodstream in e!ery time period.ii) The repeated dosage of the medication e!ery time period is $.

These assumptions about the amount %n of the medication in the bloodstream atthe beginning of the nth#time period is modeled by the sequence &%n'

determined by the follo(ing difference equation and initial !alue:

The solutions of such problems that is* the nth#term formulas for these sequencesare obtained by a +guess and chec, method in (hich a nth#term formula is

+guessed, on the basis of calculations from the initial !alue and differenceequation. The !alidity of the guessed nth#term formula for the sequence is then+checed, by demonstrating that the nth#term formula satisfies the difference

equation and has the correct initial !alue.-n the case of the elimination of medications from the bloodstream under

repeated drug dosages problems* the nth#term formula is gi!en by:

The number is called the maintenance le!el of the medication in the

bloodstream. is the limit of the sequence &%n' as n increases (ithout bound.

/hat are sequences0

Sequences are described !ery informally in Section .1 as +a set of numbers in aparticular order, and then more precisely as a function (hose domain is in the set

of non#negati!e integers and (hose range is in the set of real numbers. Thismaes a sequence +an infinite ordered set of numbers called terms of the

sequence. This does not mean that the range of a sequence must be an infinite

set. or e"ample* the sequence that assigns the number 2 to each non#negati!einteger has infinitely many terms that are all equal to 2* yet its range is the set &2'(ith only the single number 2 in it.

Throughout hapter * sequences are usually specified in one of the follo(ing

t(o different (ays:1) $y prescribing a formula* often called the nth#term formulas* (hich gi!es the!alue of the nth#term of the sequence directly in terms of n for any integer n in

the domain of the sequence.

2) $y a difference equation and initial !alues* that is* by gi!ing the first fe(terms of the sequence 4initial !alues) and a formula that allo(s you to compute a

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given term of the sequence from the values of one or more of its predecessors (adifference equation).

Some helpful definitions not found in Chapter 5:

If the difference equation for a sequence {an} expresses the next term an! of thesequence in terms of onl" the immediatel" preceding term an# then the difference

equation is of first order. If the difference equation expresses the next term an! of

the sequence in terms of onl" the t$o immediatel" preceding terms anand an%!#

the difference equation is of second order. &or

example#

are first order difference equations $hile

are second order equations. ' first order difference equation is linear if it can e$ritten in the form:

$here is a real numer and c(n) is a function of n. ' second order differenceequation is linear if it can e $ritten in the form:

$here and c are real numers and d(n) is a function of n. &or

example#are oth linear difference equations.

' first order linear difference equation

is homogeneous if c(n) * for all values of n. +iven a non%homogeneous firstorder linear difference (,)# the homogeneous first order linear difference equation

is called the associated homogeneous linear difference equation for (,).

Similarl"# a second order linear difference equation

is homogeneous if d(n) * for all values of n# and given a non%homogenouslinear second order difference equation (,) the homogeneous second order lineardifference equation

is called the associated homogeneous linear difference equation for (,).'s $e $ill sho$ later in these notes and as is sho$n in Section !-.- of Chapter

!-# the associated homogeneous linear difference equation for a given non%homogeneous linear difference equation pla"s a central role in constructing all

solutions of non%homogeneous linear difference equations

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In our course at the University of Illinois and in this course module, wedeal almost exclusively with first order linear difference equations and a few

selected equations that are not linear or that are second order linear differenceequations.

Why do we need two different ways to specify a sequence?The book does not say much about the followin very important point! The

difference equation and initial values approach to specifyin a sequence is mostuseful for usin sequences to model real"world problems while the nth"term

formula for specifyin a sequence is usually the result of #solvin$ a differenceequation with specified initial values.

%or very simple sequences, this solution process may be quite transparent. %orexample, the sequence &', (, ), *. '+, (, +), .........- of successive non"neative

inteer powers of ( can be specified by nth"term formula!

,or, with equal simplicity, by its initial value and difference equation!

owever, when sequences are used to model real world problems such as that

described in the followin example, it is usually much easier to specify theappropriate difference equation and initial values than to obtain the nth"term

formula./xample 01b2 of 3ection 0.( in 4hapter 0! Initially, +55 lbs. of a contaminant isalready present in a lake when a new industry beins dumpin'55 lbs. of the

contaminant per year into the lake. If the lake washes away '56 of thecontaminant present in the lake each year, find the number of pounds of the

contaminant in the lake at the end of n years.In this example, it is straihtforward to see that the number of pounds 4nof

contaminant in the lake at the end of the nth"year is modeledby the followin difference equation and initial value!

owever, the nth"term formula

is much less obvious.We can prove that this nth"term formula is correct for all n as follows!

i2 %irst show that the nth"term formula satisfies the difference equation and theinitial value.

1If , then!

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ii) Use i) and mathematical induction to prove that values given by the nth-term

formula are the only sequence that satisfies the difference equation and initialvalue.(We have already shown in i) that the proposed nth-term formula

gives the correct value for n = 0. uppose that the proposed nth-term formulagives the correct value of !"for some positive integer ". #hen

#herefore$ the proposed formula gives the correct value for "%&. 't follows by

mathematical induction the values given by the nth-term formula are the onlyones that satisfy the difference equation and have the correct initial value.)

#he applications of difference equations considered in ections . either dealwith population growth or e*ponential decay modeled by first order lineardifference equations or with the +ibonacci sequence which is a second order

linear difference equation. ,*ample (b) discussed above is typical of theproblems leading to first order linear differential equations.

lthough the +ibonacci sequence is historically based on a rather artificial andunrealistic model for the growth of a rabbit population$ the +ibonacci sequence

itself has many very interesting mathematical properties and seems to come up inmany une*pected settings. +or e*ample$

i) the probability nof two successive heads in n flips of a fair coin can be shownto be given by

/

ii) the limit of the ratios of successive terms of the +ibonacci sequence e*ists

and is equal to the golden ratio which psychologists claim is the ratioof length to width of a rectangle that is most pleasing to the eye.iii) an nth-term formula for the +ibonacci sequence is given by

in which the nth term of the +ibonacci is e*pressed in terms of the olden 1atio

and the so-called con2ugate of the olden 1atio.'t is an e*ample of a sequence determined by a homogeneous second order linear

difference equation.#he logistic difference equation

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provides one model for the study of population growth in an environment thatcan support only a limited population size.

Note that if b =0, this difference equation reduces to whichdescribes exponential growth with a 100a percent growth rate in the nthtime

period.

!ecause we are assuming b " 0 and b is much smaller than a in the logisticequation, it follows that b#a is a small positive number. $herefore, since the

logistic equation can be written in the following form

it follows that as the population change %n&1 %n in the nthtime interval

approaches 0. $hat is why the constant b#a is called the maximum sustainable

population for the given environment.

Teaching Strategies'ur strategy for teaching the content of (hapters ) and 1* that we cover in our

course is summarized by the following three teaching ob+ectives'b+ective 1 -xplore sequences

$he first thing to do is to have the students explore sequences by computing thefirst few terms, checing for convergence or divergence, checing for increasing

or decreasing sequences, etc. %roblems 1) */ at the end of ection ).1 can andshould all be done by hand to begin with but you may need to supplement these

with some favorites of your own to have enough for homewor, worsheets, andexamples.

Next, you will need to show them how to enter and compute sequences on theircalculators. $here are only minor differences between the capabilities of the $23 family of graphing calculators as far as computing and graphing sequences are

concerned. 4ou can use the equences with the $23 tutorial that is availableunder the $utorials button at the top of the 5ath $eacher 6in7s hort (ourses

8or 5athematics $eachers home page. %roblems *9 )/ at the end of ection).1 provide plenty of practice as well as interesting mathematical information on

using the calculator to explore sequences.'b+ective * 8ocus on first order linear difference equations.

$he applications of first order linear difference equations discussed in ections).1, ).*, and ).: really need to be covered together with the solution methods for

such equations that are found under 'b+ective 3 below or in ections 1*.1 and1*.* of (hapter 1*, which are available for download at www.aw.com#ggts. ;esip the detailed discussion of the 8ibonacci sequence model that occurs at the

end of ection ).* and the discussion of the logistic growth model in ection ).3until we have completed the discussion of all of the applications that lead to first

order linear difference equations. $hen, near the end of our course we devote 3 ) class days to the 8ibonacci sequence in ection ).* and the logistic growth

models in ection ).3.

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$he solution method for any homogeneous first order linear difference equationis developed at the beginning of ection ).* and can be summarized as follows

8act 1$he general solution of the homogeneous first order linear differenceequation

is

where < is a parameter that varies with the choice of the solution.

8act * $he general solution of the nonhomogeneous first order linear difference

equation

is given by the sum of the sequences an>& pn> where an> is the general

solution of the associated homogeneous equation

and pn> is any particular solution of the given nonhomogeneous equation ?@@A.

8act * follows from the linearity of the difference equation because

8or if an> is the general solution of the associated homogeneous equation ?@@A

and if pn> is any particular solution of the given nonhomogeneous equation

?@@A, then an>& pn> are solutions of ?@A by direct substitution. 'n the otherhand, if an@> is any solution of ?@A, then an@> pn> is a solution of ?@@A by

direct substitution and so an@> pn> must be among the solutions in an>.

'b+ective 3 6earn to solve certain types of first order linear difference equations

with constant coefficients.$he basic method of solution for such equations is simple to summarize for

people lie you who have considerable experience with mathematics andmathematics teaching. Bowever, it will not be an easy topic for your students.

$hat is probably the reason why the boo delays the discussion of the solutionmethods until (hapter 1*.

$he following summary of the method provides a brief alternative to readingections 1*.1 and 1*.*

ummary8acts 1 and * yield the following threestep procedure for solving a nonhomogeneous first order linear difference equation Civen a linear, non

homogeneous first order difference equation and initial value

,

proceed as follows

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Step 1: Find some (perhaps very special) solution of the given non-homogeneousdifference equation by "hook or crook"

(For e!ample it is pretty easy to see that the constant sequence #pn$ %hose terms

are all 1 is one very special solution of: )

Step &: 'he general solution of the associated homogeneous linear difference

equation is dd the particular

solution #pn$ of the non-homogeneous equation obtained in Step 1 to the general

solution of the associated homogeneous linear difference equation to obtain the

general solution of the non-homogeneous equation

(For e!ample the general solution of:

is given by:

in vie% of the particular solution observed in Step 1)

Step : *se the initial value to evaluate the constant in the general solutionobtained in Step &(For e!ample if %e are looking for the solution of the difference equation

that has initial value of %e %ould substitute this value and n +

, into the general solution:

to obtain + & 'hus the desired solution %ould be:

)

'he most difficult step in this procedure is Step 1 %hich requires that %e find aparticular solution #pn$ of the given non-homogeneous linear difference

equation e teach the students ho% to find a particular solution for the cases in%hich the non-homogeneous term c(n) is of one of the follo%ing special types:1) c(n) is a constant sequence

(Substitute a constant sequence c(n) + #.$ into the given non-homogeneous difference equation and solve for .)

&) c(n) is a geometric sequence #r skn$ for given r s and k

(Substitute a geometric sequence c(n) + #/ s

.n

$ into the given non-homogeneous difference equation and solve for / and .)

) a sequence #q(n)$ %here q(n) is a polynomial in n of order k

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(Substitute a sequence into thegiven non-homogeneous difference equation and solve for the polynomial

coefficients .)

Step #6

Complete an approved Final Project consisting of developing and teaching a

classroom unit for one of your classes based on one or two sections of the book.

You are given a great deal of latitude in the choice of topic for this unit becausewe want the choice to reflect your teaching situation and your interests. We

require only that the unit is based on the text and that you plan it and teach it as acollaborative learning unit. he unit should be sufficient to support at least one

class period but typically should cover two or more class periods.

he sections of the text that you select for your !inal "ro#ect should typicallyinclude at least one section not included among the sample sections reviewed in

the previous steps.

$fter you have selected the sections that you would li%e to cover and you havesome idea of how you want to develop your classroom unit& e-mail a brief

summary of your proposed unit to us for review and approval'algebramtl.math.uiuc.edu

Your classroom unit should include a lesson plan and the necessary student

wor%sheets for the unit. f practical& you are also required to teach the unit or partof the unit using the collaborative learning instruction method& and to prepare a

brief written evaluation of the unit.

Submit your final project though the Moodle hand in system and drop yourinsturctor an email letting them know you submitted the final project.

You are done* he +ath eacher ,in% instructional staff will review and provide

feedbac% on your assignments and lassroom "ro#ect.

"age / of /$lgebra hrough +odeling

algebra through modeling

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