# 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 of modeling and data analysis using the basic algebraic functions and a graphing calculator with data analysis capabilities such as those of any of the TI-83/84 family of graphing calculators including the TI-83, 83+, 83+ Silver Edition, 84+, and 84+ Silver Edition. This new approach will be useful in high school and college for students who have completed two years of algebra but are not ready for precalculus. It reviews algebra without repeating earlier course work, and it clearly demonstrates the importance of algebra to the solution of real world problems. Credit: 1 grad. sem. hr.

Common Core Standards for Mathemtical Practice that are emphasized include:

• 2. Reason abstractly and quantitatively.
• 4. Model with mathematics.
• 5. Use appropriate tools strategically.
• 7. Look for and make use of structure.

Algebra through Modeling with the TI-83 family of Graphing Calculators was written by Tony Peressini and John Luker of the University of Illinois during November 1996. It was revised by Peressini and Luker during November 1997 and by Peressini and Tom Anderson in November 2003 and June 2005.

Check out some sample projects for this module in the MTL Classroom Projects to the right.

# Detailed Description

The purpose of this module is to familiarize mathematics teachers at the high school and lower division college level with the content and teaching strategies for a promising new approach to teaching advanced algebra.
This approach stresses modeling and solving real world problems and develops skills and concepts of algebra as needed for this modeling process. In this course, the students work in a collaborative learning environment and make extensive use of a graphing calculator with data analysis and sequence capabilities such as those found on the TI-82, TI-83, TI-83+ and TI-83+ Silver Edition
After completing this module, a teacher will know the details of this approach and will be prepared to use or adapt the course or some topics of the course to his or her own instructional setting.

This course, which has been used at the University of Illinois quite successfully for six years, was based on preliminary editions of the text, 'Functioning in the Real World', by Sheldon Gordon and Ben Fusaro which was published by Addison-Wesley Publishing Company in November, 1996 and in a second edition in 2004. [See the Required Materials button at the bottom of the Module 5 home page for a detailed description of the text and ordering information.] This text presents algebra, trigonometry and other topics from a non-traditional perspective that emphasizes the use of the basic algebraic and trigonometric functions to model interesting and current real-world problems. Data analysis is used to develop and test these models and to draw inferences from them. Manipulative skills are taught "as needed" for the model development and analysis and not as "stand alone" topics as is typical in traditional algebra texts and courses.

The Illinois version of the course is taught using the following teaching strategies and tools:
1) The Small-Group Instruction (or Collaborative Learning) Method
2) Mastery Testing of Algebraic Skills
3) Graphing calculators with data analysis and sequence capabilities (for example, any member of the TI-83 family of graphing calculators).

The use of the graphing calculator is an essential requirement of the text. We believe that the use of the Collaborative Learning Method has been absolutely crucial to the success of our course at Illinois. Although we believe that the use of Mastery Testing has also been helpful, we have not discussed that aspect of this course in this module because it is rather independent of the other two strategies. Thus, this module will focus on features 1) and 3) of the Illinois course. This module will prepare you to implement and adapt this course or aspects of this course for your school if you choose to do so.

Although it is certainly possible to use other graphing calculators for this course, the TI-83 family of graphing calculator have features that are especially well suited to the required tasks. Consequently, we have used it throughout this module. If you or your students are not familiar with the operation of the TI-83 family of calculators, there are interactive tutorials available at the tutorial button under the Math Teacher Link logo at the top of this and other MTL Module pages.

# Required Material

In addition to the general requirements for 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-84 Plus)
Each participant needs to have access to either of these Texas Instruments Graphing calculators.
• Functioning In The Real World by Sheldon P. Gordon
Each participant must obtain a copy of the second edition of the text Functioning In The Real World by Sheldon P. Gordon, et al., (Addison-Wesley Longman) ISBN# 0-201-38389-6). For Ordering Information, please call the publisher Addison Wesley Longman at 1-800-922-0579. Used copies of the book are also available at Amazon.com at significantly reduced prices. This book is an excellent resource to have available in any high school mathematics library.

# Step by Step Instructions

We estimate that the work for this module will take participating teachers an average of about 45 hours to complete. Technically, you have a 3-month enrollment period to complete the module after you enroll. However, we recommend that you complete the module in eight weeks or less. That should easily be possible even when you are teaching if you set aside three or four hours each week to work on the module.

# Step #1

If necessary, complete the Basic TI-82 or TI-83, 83+ Calculator Tutorial appropriate for your calculator. These tutorials are available free from MTL. You can find a link to these tutorials under the Math Teacher Link banner on this and other MTL pages.

For the first part of this module (described in Step 2 below), you will need to use one of these TI calculators or some other graphing calculator to:
1. Enter and evaluate arithmetic expressions and the standard functions.
2. Enter and graph one or more functions on specified windows.
3. Create tables of values of one or more functions over specified ranges of the variable. If you already know how to do all of these things, you can skip this step and go on to Step 2.
4. If you are at all unsure of any of these skills, we recommend that you go through the appropriate tutorial until you are comfortable with them.

# 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 1, 2, and 4, and then print and read the Instructor's Notes For Chapters 1, 2, and 4. Then print and complete the sample student worksheet (see below) for that chapter.

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

It is important to do Step 1 one chapter at a time. That is, after you have read the sample sections of the text and the Reading Notes for that chapter for one of these chapters, print and complete the Sample Student Worksheets (attached below) for each of these three chapters as Assignment 5.1.

If you are enrolled for graduate or continuing education credit, mail the three completed worksheets for Chapters 1, 2, and 4 to:

 Tom Anderson15721 Lakeside DriveSterling, IL 61081

Be sure to make a copy of student worksheets before you mail them

AttachmentSize
Ch 1 - worksheet 1.pdf84.58 KB
Ch 2 - worksheet 2.pdf658.16 KB
Ch 4 - worksheet 4.pdf221.66 KB

# Instructor's Notes on Chapters 1, 2, and 4

### Introduction:

We have asked you to read carefully the following sample sections of the text:

• 1) Sections 1.1 through 1.5 of Chapter 1;
• 2) Sections 2.2 through 2.8 of Chapter 2;
• 3) Sections 4.5 through 4.7 of Chapter 4.

### Chapter 1 Reading Notes

The principal topic of the sample sections for Chapter 1 is functions of one variable as defined by formulas, graphs and tables. It includes the usual "rule" definition of function and the recitation of the standard function terminology: domain, range, dependent and independent variables, etc.
These are standard topics in virtually any advanced algebra course. The only indication that this book might have a different emphasis than other advanced algebra books is in the problems for Section 1.5 which repeatedly ask the student to draw a reasonable graph of a function that is described by some "real world" situation. Drill problems covering the terminology and definitions are included, but emphasized less than in traditional algebra books.

### Chapter 2 Reading Notes

The principal topics of the sample sections, linear, exponential, logarithmic and power functions and their properties, are certainly familiar topics in virtually any advanced algebra course. However, as you read these sample sections, you will find that the treatment of these topics is much different than that found in traditional advanced algebra courses. In standard courses, linear, exponential and power functions are treated as separate, self-contained packages, each with its own application and with its applications usually placed at the end of each package. In this book, these three types of functions are introduced as means to describe different types of growth (increase) and decay (decrease) of a quantity or function.
Linear functions are those with a constant rate of increase (or decrease).
Exponential functions are those with a constant percentage rate of increase (or decrease).
Power functions have no similar description in terms of rates of increase or decrease. However, the main points that made are:

1. The values of any exponential function with a positive growth factor will always eventually become and stay larger than the values of any power function (including any linear function) with a positive power, regardless of the relative sizes of their (positive) initial values.
2. The values of an exponential function with a large positive growth factor will always eventually become and stay larger than the values of an exponential function with a smaller positive growth factor regardless of the relative sizes of their (positive) initial values.
3. The values of a power function with a large positive power will always eventually 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 explored in a variety of contexts including population growth and comparative economic growth.

4. The section on logarithmic functions makes similar comparisons about the relative growth of power and logarithmic functions.

### Chapter 4 Reading Notes

Chapter 4 deals with polynomial functions, a standard topic in advanced algebra, and with transformations of functions (vertical and horizontal shifts, vertical and horizontal dilations, etc.), an important topic that is frequently neglected in advanced algebra courses.

You are not required to read Sections 4.1 through 4.4 for any of the work specified for this module, but you may find it interesting to scan the content because 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, Finding Polynomial Patterns, is on recognizing polynomial functions by successive differences, an important topic for applications that is normally omitted in standard advanced algebra courses.

The two sample sections on transformations of functions, Building New Functions From Old, are a bit misplaced in this chapter because the content is not restricted to polynomial functions. Our experience with teaching this material is that it is quite accessible to just average students, and that it helps all students to begin to recognize and use relationships between an algebraic formula for a function and its graph.

# Step #3

Read the sample text sections for Chapter 3. Then complete the Fitting Data With The 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, exponential and power regression analysis of two variable data. The calculator tutorial will show you how to carry out regression calculations and displays on the TI-82 calculator in the context of a concrete problem in cancer data analysis. The modifications that are necessary if you are using any other TI-83 family calculator are relatively minor.

# Step #4

Review the Sample Student Worksheet for Chapter 3 (attached below). Then print and do the sample worksheet.

The work that you have done in Step 3 will provide the mathematical background and calculator skills to do the sample worksheet and develop similar ones for your classes.

If you are enrolled for graduate or continuing education credit, mail the completed worksheet for Chapter 3 as Assignment 5.2 to:

Tom Anderson
15721 Lakeside Drive
Sterling, IL 61081

AttachmentSize
worksheet3.pdf125.77 KB

# Step #5

Read the sample text sections and the Instructor's Notes for Difference Equations [Chapters 5 and 12] . Review the two Sample Student Worksheets for these chapters attached below. Print and complete them as Assignment 5.3.

The sample sections are Sections 5.1, 5.2 and 5.4 of Chapter 5 and Sections 12.1, 12.2, 12.3 and 12.4 of Chapter 12.

If you are enrolled for graduate or continuing education credit, mail the two completed 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 have just completed the module. Congratulations! Those of you enrolled for continuing education credit will be sent feedback on the three assignments that you were required to submit.

Those of you enrolled for graduate credit still need to complete an approved Final Project. Go to Step 6 to find out how to do that.

AttachmentSize
worksheet5.pdf110 KB
worksheet12.pdf218.39 KB

# Instructor's Notes for Difference Equations (Chapters 5 & 12)

Introduction

Our course at the University of Illinois was developed using a preliminary version of Functioning in the Real World - A Precalculus Experience by Sheldon et al and was taught for several years using Chapters 1 through 6 and later using the first edition of the book. The second edition of the book was published in 2004. We decided to modify this course module in 2005 so that it can be used with the second edition because the first edition is no longer in print and used copies of the first edition are increasingly difficult to obtain from textbook resellers.

Fortunately, the innovative approach to algebra and modeling with functions that first attracted us to this course and that was based on the preliminary version of the book 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 the book were adopted in the second edition. As a result, the latest edition is very well suited for use in this course module.

Except for relatively minor organizational changes and some new or revised problems, the content and approach of Chapters 1, 2, 3 and 4 of Functioning in the Real World - A Precalculus Experience have remained relatively constant through the various editions of the book. However, the material in Chapters 5 and 6 on modeling with difference equations in the preliminary and first edition have undergone substantial change in the latest edition. The content of Chapter 6 in the preliminary and first editions has been moved to Chapter 12 in the second edition, 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 time constraints of our course and partly because we believe that the order of the content in the text led to certain confusions in the presentation that can easily be avoided.

The order of the content on difference equations in the text is this: In Chapter 5 of all versions of the book, difference equations of first and second order, linear and nonlinear, are introduced and used to model a number of interesting applications. However, the systematic development of solution methods for these difference equations was delayed until Chapters 6 and 8 in the first edition and until Chapter 12 in the second edition. Solutions to difference equations arising from models and applications introduced in Chapter 5 are usually either stated without formal justification or are obtained by special methods that do not generalize well. Although systematic solution methods are presented later (in Chapters 6 and 8 in the first edition and in Chapter 12 in the second edition), we find this delay in presenting systematic solution methods for difference equations to be unsatisfactory because students often do not grasp the important distinction between a difference equation and its solution and, as a result, they are not able to understand the point of looking for a difference equation to model a given application in the first place.

In our three semester-hour algebra course, we do not have time to cover much of Chapter 6 or any of Chapter 8. Consequently, we limit our discussion of difference equations to a detailed presentation of first order linear difference equations, their applications and their solution methods. We are only able to devote a day or two at the end of the semester to one or two applications that lead to nonlinear or second order equations.

A similar limitation would seem to be in order for most year-long high school or one semester college courses based on this text. These notes are intended to help instructors in such courses to cover this content successfully.

Instructor's Notes on Chapter 5: Modeling with Difference Equations

This chapter begins with a discussion of applications of difference equations to problems involving the elimination of medications from the bloodstream under repeated drug dosages in Section 5.1. Most of the basic terminology and solution methods concerning difference equations are either discussed informally in terms of this application or delayed until the first two sections of Chapter 12. These medication elimination problems are all modeled by sequences specified by first order linear difference equations with given initial terms, although the definitions of first order linear difference equations are not introduced formally until Section 12.1.

These problems are all modeled on the basis of the assumptions stated on Page 366:

i)                    The kidneys remove a fixed proportion 1 - a of the medication from the bloodstream in every time period.

ii)                   The repeated dosage of the medication every time period is B.

These assumptions about the amount Dn of the medication in the bloodstream at the beginning of the nth-time period is modeled by the sequence {Dn} determined by the following difference equation and initial value:

The solutions of such problems; that is, the nth-term formulas for these sequences are obtained by a “guess and check” method in which a nth-term formula is “guessed” on the basis of calculations from the initial value and difference equation.  The validity of the guessed nth-term formula for the sequence is then “checked” by demonstrating that the nth-term formula satisfies the difference equation and has the correct initial value.

In the case of the elimination of medications from the bloodstream under repeated drug dosages problems, the nth-term formula is given by:

The number L is called the maintenance level of the medication in the bloodstream.  L is the limit of the sequence {Dn} as n increases without bound.

What are sequences?

Sequences are described very informally in Section 5.1 as “a set of numbers in a particular order” and then more precisely as a function whose domain is in the set of non-negative integers and whose range is in the set of real numbers. This makes 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. For example, the sequence that assigns the number 2 to each non-negative integer has infinitely many terms that are all equal to 2, yet its range is the set {2} with only the single number 2 in it.

Throughout Chapter 5, sequences are usually specified in one of the following two different ways:

1) By prescribing a formula, often called the nth-term formulas, which gives the value of the  nth-term of the sequence directly in terms of n for any integer n in the domain of the  sequence.

2) By a difference equation and initial values, that is, by giving the first few terms of the sequence (initial values) and a formula that allows you to compute a given term of the sequence from the values of one or more of its predecessors (a difference equation).

If the difference equation for a sequence {an} expresses the next term an+1 of the sequence in terms of only the immediately preceding term an, then the difference equation is of first order. If the difference equation expresses the next term an+1 of the sequence in terms of only the two immediately preceding terms an and an-1, the difference equation is of second order.  For example,

are first order difference equations while

are second order equations. A first order difference equation is linear if it can be written in the form:

where b is a real number and c(n) is a function of n. A second order difference equation is linear if it can be written in the form:

where b and c are real numbers and d(n) is a function of n.  For example,

are both linear difference equations.

A first order linear difference equation

is homogeneous if  c(n) = 0 for all values of n.  Given a non-homogeneous first order linear difference (*), the homogeneous first order linear difference equation

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

Similarly, a second order linear difference equation

is homogeneous if  d(n) = 0 for all values of n, and given a non-homogenous linear second order difference equation (*) the homogeneous second order linear difference equation

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

As we will show later in these notes and as is shown in Section 12.2 of Chapter 12, the associated homogeneous linear difference equation for  a given non-homogeneous linear difference equation plays a central role in constructing all solutions of non-homogeneous linear difference equations

In our course at the University of Illinois and in this course module, we deal almost exclusively with first order linear difference equations and a few selected equations that are not linear or that are second order linear difference equations.

Why do we need two different ways to specify a sequence?

The book does not say much about the following very important point:  The difference equation and initial values approach to specifying a sequence is most useful for using sequences to model real-world problems while the nth-term formula for specifying a sequence is usually the result of “solving” a difference equation with specified initial values.

For very simple sequences, this solution process may be quite transparent.  For example, the sequence {1, 2, 4, 8. 16, 32, 64, .........} of successive non-negative integer powers of 2 can be specified by nth-term formula:

,

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

However, when sequences are used to model real world problems such as that described in the following example, it is usually much easier to specify the appropriate difference equation and initial values than to obtain the nth-term formula.

Example 5(b) of Section 5.2 in Chapter 5: Initially, 600 lbs. of a contaminant is already present in a lake when a new industry begins dumping100 lbs. of the contaminant per year into the lake. If the lake washes away 10% of the contaminant 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 straightforward to see that the number of pounds Cn of contaminant in the lake at the end of the nth-year is modeled

by the following difference equation and initial value:

However, the nth-term formula

is much less obvious.

We can prove that this nth-term formula is correct for all n as follows:

i) First show that the nth-term formula satisfies the difference equation and the initial value.

(If  , then:

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 initial value.

(We have already shown in i) that the proposed nth-term formula

gives the correct value for n = 0.  Suppose that the proposed nth-term formula gives the correct value of Ck for some positive integer k. Then

Therefore, the proposed formula gives the correct value for k+1. It follows by mathematical induction the values given by the nth-term formula are the only ones that satisfy the difference equation and have the correct initial value.)

The applications of difference equations considered in Sections 5.2 either deal with population growth or exponential decay modeled by first order linear difference equations or with the Fibonacci sequence which is a second order linear difference equation.  Example 5(b) discussed above is typical of the problems leading to first order linear differential equations.

Although the Fibonacci sequence is historically based on a rather artificial and unrealistic model for the growth of a rabbit population, the Fibonacci sequence itself has many very interesting mathematical properties and seems to come up in many unexpected settings.  For example,

i) the probability Pn of two successive heads in n flips of a fair coin can be shown to be given by

;

ii) the limit of the ratios of successive terms of the Fibonacci sequence exists and is equal to the golden ratio   which psychologists claim is the ratio of length to width of a rectangle that is most pleasing to the eye.

iii) an nth-term formula for the Fibonacci sequence is given by

in which the nth term of the Fibonacci is expressed in terms of the Golden Ratio G and the so-called conjugate of the Golden Ratio.

It is an example of a sequence determined by a homogeneous second order linear difference equation.

The logistic difference equation provides one model for the study of population growth in an environment that can support only a limited population size.

Note that if b =0, this difference equation reduces to  which describes exponential growth with a 100a percent growth rate in the nth-time period.

Because we are assuming b > 0 and b is much smaller than a in the logistic equation, it follows that b/a is a small positive number.  Therefore, since the logistic equation can be written in the following form

it follows that as  the population change Pn+1 -  Pn in the nth-time interval approaches 0. That is why the constant b/a is called the maximum sustainable population for the given environment.

### Teaching Strategies

Our strategy for teaching the content of Chapters 5 and 12 that we cover in our course is summarized by the following three teaching objectives:

Objective 1: Explore sequences

The first thing to do is to have the students explore sequences by computing the first few terms,  checking for convergence or divergence, checking for increasing or decreasing sequences, etc. Problems 15 - 26 at the end of Section 5.1 can and should 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 homework, worksheets, and examples.

Next, you will need to show them how to enter and compute sequences on their calculators. There are only minor differences between the capabilities of the TI-83 family of graphing calculators as far as computing and graphing sequences are concerned.  You can use the Sequences with the TI-83 tutorial that is available under the Tutorials button at the top of the Math Teacher Link’s Short Courses For Mathematics Teachers home page.  Problems 27 - 56 at the end of Section 5.1 provide plenty of practice as well as interesting mathematical information on using the calculator to explore sequences.

Objective 2: Focus on first order linear difference equations.

The applications of first order linear difference equations discussed in   Sections 5.1, 5.2, and 5.4 really need to be covered together with the solution methods for such equations that are found under Objective 3 below or in Sections 12.1 and 12.2 of  Chapter 12, which are available for download at  www.aw.com/ggts . We skip the detailed discussion of the Fibonacci sequence model that occurs at the end of Section 5.2 and the discussion of the logistic growth model in Section 5.3 until we have completed the discussion of all of the applications that lead to first order linear difference equations.  Then, near the end of our course we devote 3 - 5 class days to the Fibonacci sequence in Section 5.2 and the logistic growth models in Section 5.3.

The solution method for any homogeneous first order linear difference equation is developed at the beginning of Section 5.2 and can be summarized as follows:

Fact 1:The general solution of the homogeneous first order linear difference equation

is:

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

Fact 2: The general solution of the non-homogeneous 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 non-homogeneous equation (**).

Fact 2 follows from the linearity of the difference equation because

For if {an } is the general solution of the associated homogeneous equation (**) and if { pn} is any particular solution of the given non-homogeneous equation (**), then {an }+{ pn} are solutions of (*) by direct substitution.  On the other hand, if  {an*} is any solution of (*), then {an *}-{ pn} is a solution of (**) by direct substitution and so  {an *}-{ pn} must be among the solutions in {an }.

Objective 3: Learn to solve certain types of first order linear difference equations with constant coefficients.

The basic method of solution for such equations is simple to summarize for people like you who have considerable experience with mathematics and mathematics teaching. However, it will not be an easy topic for your students. That is probably the reason why the book delays the discussion of the solution methods until Chapter 12.

The following summary of the method provides a brief alternative to reading Sections 12.1 and 12.2:

Summary:

Facts 1 and 2 yield the following three-step procedure for solving a non-homogeneous first order linear difference equation: Given a linear, non-homogeneous first order difference equation and initial value:

,

proceed as follows:

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

(For example, it is pretty easy to see that the constant sequence {pn} whose terms are all 1 is one very special solution of: )

Step 2: The general solution of the associated homogeneous linear difference equation  is . Add 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 example, the general solution of:

is given by:

in view of the particular solution observed in Step 1.)

Step 3: Use the initial value to evaluate the constant A in the general solution obtained in Step 2.

(For example, if we are looking for the solution of the difference equation  that has initial value of 3, we would substitute this value and n = 0 into the general solution:

to obtain A = 2. Thus, the desired solution would be:  .)

The most difficult step in this procedure is Step 1 which requires that we find a particular solution {pn} of the given non-homogeneous linear difference equation. We teach the students how to find a particular solution for the cases in which the non-homogeneous term c(n) is of one of the following special  types:

1) c(n) is a constant sequence.

(Substitute a constant sequence c(n) = {K} into the given non-homogeneous difference equation and solve for K.)

2) c(n) is a geometric sequence {r skn} for given r,  s and k.

(Substitute a geometric sequence c(n) = {R sKn}  into the given non-homogeneous difference equation and solve for R and K.)

3) a sequence {q(n)} where q(n) is a polynomial in n of order k.

(Substitute a sequence  into the given 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 because we 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 a collaborative learning unit. The unit should be sufficient to support at least one class period but typically should cover two or more class periods.

The sections of the text that you select for your Final Project should typically include at least one section not included among the sample sections reviewed in the previous steps.

After you have selected the sections that you would like to cover and you have some 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: algebra@mtl.math.uiuc.edu

Your classroom unit should include a lesson plan and the necessary student worksheets for the unit. If practical, you are also required to teach the unit or part of 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 your insturctor an email letting them know you submitted the final project.

You are done! The Math Teacher Link instructional staff will review and provide feedback on your assignments and Classroom Project.