frameteachers

Featured in June, 2004 ...

This 7-page work, first written in 1997 as a Mathwright WorkBook, emerged from a student's question in a Calculus II course taught by the author. It is one of our new free Visualization Studio Interactive Web Books in the Math Cafe, and all visitors are invited to check it out before joining the Library to get a glimpse of what is possible with Mathwright. It exemplifies the applications of object-oriented design to geometry by placing the student in a carpentry shop where she cuts the wood to build a boat that will carry two children across Monet Pond.

The aim is to take account of Archimedes' bouyancy principle, and to solve the resulting optimization problem, i.e. guarantee that the boat will not sink, while testing the resulting 3D model of the boat. This hands-on approach to optimization succeeds well by making the task a challenge in a game-like environment. Later, we "do the math" so that students can appreciate the role of implicit differentiation in solving the problem.

The goals of structured discovery learning worked so well that we have translated it and extended it to our Microworld format so that you may view it either in your browser or offline.

Introduction to Linear Programming and the Simplex Algorithm

This 12-page microworld is aimed at a range of different levels. Solution of inequalities and of systems of inequalities can be studied at Intermediate Algebra and College Algebra. Solving systems of equations and linear programming problems in two unknowns using geometrical methods can be studied at the College Algebra and Pre-Calculus levels.

Solving systems of equations and the Simplex Algorithm can be studied at the Linear Algebra level. Originally, this microworld was aimed at College Algebra and Pre-Calculus levels. That is why it started from a very elementary consideration of what an inequality is, and how to solve an inequality or a system of inequalities.

Once we began solving linear programming problems using a geometrical method, it was natural to develop the simplex algorithm. We have considered only standard linear programming problems, although page 10 can be used to manually solve nonstandard problems also.

However, automating the simplex algorithm on pages 11 and 12 is really geared towards standard linear programming problems. It is our intention to follow up this microworld with another microworld, which would consider the nonstandard case. We believe that it will be appropriate to call such a work “dantzig”.

This Microworld contains two types of online Help. Most pages contain the mathematical explanations under the "Help for this page" button. The simplex algorithm needs to be considered in special detail. So, you may click the button “Math for this page” on pages 10 and 12 for an in-depth explanation of the Simplex Algorithm.

Featured in May, 2004 ...

HiFi: Personal Household Finance Manager

HiFi is a fully functional object-oriented, LISP-based Expert System that can make managing and planning your household finances easy and fun. To see for yourself what it can do, please visit the title page. All visitors are welcome to read about this fascinating technology. Either read the introduction online, or download the PDF File.

If you are curious to try HiFi out for yourself, then Library Members may click Add to your Collection, and it will be ready to use offline with Mathwright32 Reader. The downloaded program contains all of the documentation, along with a sample (realistic) session to get you started.

While you may actually use HiFi to manage your finances, HiFi is mainly a demonstration of Artificial Intelligence technology -- a teaching program

.Applications of Integration

This 9-page microworld explores arc length of a curve, area under a curve, and surface area and volume of revolution. For simplicity we explore only those surfaces of revolution that can be obtained by revolving a curve about x-axis.

Arc length, area, surface area, and volume can be found by dividing the arc, region, or solid into tiny portions in Riemannian spirit. You will be living in Riemannian spirit as you conduct explorations on the following interactive pages.

The theory will be briefly explained on the help pages that can be viewed by pressing the button “math for this page”. Often an example or two may be used to explain the theory. When a page of the microworld contains a button named “instructions”, you can press it to view instructions for using the interactivity of the page in order to make explorations.

Featured in February, 2004 ...

Calculus in Action

A Story of Calculus...

This 500 page book, which consists of 38 lectures and 43 Interactive Explorations, is presented in 10 Mathwright Microworlds. It is designed to illuminate, and to give readers the chance to explore in some depth, the basic ideas of Calculus within the context of its first triumphal scientific success: Isaac Newton's deduction of Kepler's three laws of planetary motion from his single hypothesis of Universal Gravitation. This book differs from a Calculus textbook in several ways. While it assumes a basic understanding of Geometry and Algebra, it is designed around the theme of Gravitation, rather than any particular syllabus. It is an Interactive Story that invites you to explore a selected range of ideas from the Calculus that were inspired by this theme.

Unlike a text, the book does not attempt to give an encyclopedic account of all the standard techniques of calculation and problem-solving that readers might someday be called upon to know. We develop in some detail a great variety of techniques, but only as we require them in the telling of the story. To that end, each Microworld Section of each Chapter of the book first discusses a problem that we need to solve to deepen our understanding of the gravitation theme, and then recruits and explains the techniques that Calculus can supply to help us solve it. The problems are not easy ones, but we attempt in the lectures and interactive explorations to bring them to life, so that readers can experiment, and become familiar with them.

The book is written for readers who enjoy mathematics, and have the curiosity and the desire to see the small part of it that we develop here, as a whole: roots, branches, and leaves. As the title implies, this book is not meant to be read like a text, but is designed for you to learn by acting on the various stages of the story. If you would like to know what Calculus is really about, and how it came to be, this book may be for you.

But we begin with a word of caution. This book is a gradual ascent to a high place. We have tried to write it so that the early part will be immediately meaningful to the thoughtful and curious high-school student. The Pre-Calculus Introduction is fairly non-technical, dealing mainly with the circle of ideas that led to the invention of Calculus in the 17th century. Those ideas are what really matter, and will form the basis for all that follows in later chapters. And as we gather the analytic tools and techniques that Isaac Newton invented to solve Kepler's problem of planetary motion, we will adopt an increasingly "rigorous" tone. The level of discourse may then not be accessible to beginners, no matter how dedicated they are to the task. It is an unfortunate myth that a beginning student can master the calculus after only one or two years of study. When you have followed the story that this book tells, you will be well on your way to that mastery!

So we counsel patience. This book is a Story, not an encyclopedia. It is not a textbook. Take your time, and enjoy it. You will not find the answers neatly laid out in its pages, until you yourself ask the questions. And that requires both time and reflection. There is no "royal road" to the Calculus, and you may find that you return to this book many times in order to see a point clearly. But we believe that the book will offer you a new opportunity to formulate and to ask your questions. Each part of each section of each chapter invites you to experiment, and to bring the ideas to life in a way that is meaningful to you.

We introduce the key ideas of the Calculus only as we need them to solve the problem at hand, whether it is to calculate the trajectory of a baseball, to determine the escape velocity, or to place a satellite into geosynchronous orbit around the Earth. And like the Calculus itself, our problems all grow out of the questions: "How does an object fall ? How do the Moon and Planets move ?" That is the theme of the story, our main question, for which the Calculus is the language that provides our best answer.

In the hands of its creator, Isaac Newton, the Calculus was a musical instrument. He made it sing the song that told of Kepler's "harmony of the spheres," and he gave us an instrument of thought that he felt would reveal the deepest secrets of Nature. In this series of lectures, you will follow young Isaac, and will see for yourself what music Calculus can make in your own hands.

Featured in October, 2003 ...

Color Portraits of Complex Mappings

by James White, Director Mathwright Library

Augustin-Louis Cauchy

Born: 21 Aug 1789 in Paris, France
Died: 23 May 1857 in Sceaux (near Paris), France

Analytic functions can be the source of beautiful geometric patterns and forms, precisely by virtue of their rich geometric structure. Even polynomial and rational functions disclose graceful symmetries, tensions and attractions among their zeros and poles that have the allure of fine art. It is in the spirit of developing an eye for the beauty and grace of analytic functions, rather than "analyzing" them that we develop this Microworld. If the heuristic of this book (which is to represent analytic functions with characteristic "pictures" or "color portraits") leads to a deeper appreciation of the mathematics, that will be a good thing. But what we are after is a style of visualizing them that is impressionistic rather than precise, and that leads to a variety of experiments for the reader.

We thus explore the heuristic value of representing such functions as "pictures." As always, the pictures can convey at a glance many of the qualitative geometric properties of such mappings, especially the polynomial and rational functions, but also a few transcendental functions. Among those qualitative features, we shall be especially interested in the interactions among zeros and poles, the orders of zeros and poles, and the relation in simple cases, between a complex mapping and its derivative. For the latter, we will illustrate and prove Marden's Theorem, which discloses a surprising geometric fact about zeros of the derivatives of cubic polynomials and their relation with the original polynomial.

The Microworld features a new capability of Mathwright (available since version 2.13, Sept 14, 2003) that makes use of 3D graphics and the built-in complex number type to draw fast color portraits of complex maps, thereby giving the reader the chance to experiment and to explore the properties of complex functions at her own pace, and with her own questions.

Featured in August, 2003 ...

A Dynamical Systems Primer
by Samad Mortabit, Metropolitan State University

This 23-page work, written in 1995 as a Mathwright WorkBook, was the substrate for a college-level course in discrete dynamics taught by the author. It exemplifies the goals of structured discovery learning so well that we have translated it essentially without change to our Microworld format so that you may view it either in your browser or offline. It is one of our new free Visualization Studio Interactive Web Books in the Math Cafe, and readers are invited to check it out before joining the Library to get a glimpse of what is possible with Mathwright.

This book also contains a "movie" that the user can generate, either online or off, that illustrates the "flip" and "tangent" bifurcations of the logistic map. At other places in the book, the reader may create cobweb diagrams using a bifurcation diagram (that she may also create for any dynamic she chooses) to select parameter values from the screen. The diagram below is an example.

The Microworld features a new capability of Mathwright that makes use of Windows Compiled HTML Help to tell the mathematical story on each information page. These help pages use publisher quality formatted mathematical text and illustrations to tell their story. The activity pages offer the reader to experiment and to explore the properties of discrete systems at her own pace, and with her own questions.

Vectors in the Plane
by Mike Pepe, Seattle Central Community College

This 8 page Mathwright Microworld is a visual introduction to the basic algebra and geometry associated with plane vectors. You may use it to experiment with vector addition and subtraction, vector spans and coordinate systems. It also includes interactive demonstrations of the vector representation of a line and of vector projections

Featured in June, 2003 ...

A World of Curves
by Ravinder Kumar, Alcorn State University

Graphing is a very important aspect of the teaching and learning of mathematics at all levels, particularly at the undergraduate level. The world of curves is full of wonders. But, with the advent of technology, we find ironically, that the practice of exploring this beautiful world is getting pushed into oblivion. This is despite the fact that these days books, particularly calculus books, emphasize the graphical aspect of concepts. Books, however, often do not talk about envelopes, evolutes, involutes, pedals, negative pedals etc., any more constructions. Historically, these and other ways of determining curves not only shed light on the curves and their characteristic properties, but also produce some fantastic curves, otherwise difficult to create and visualize.

It is the objective of this 14 page microworld to provide some approaches to gaining insight into the world of curves. This effort is by no means exhaustive or comprehensive. Here, we explore curves defined by parametric equations only. We also provide mechanism to understand and explore the envelope, pedal, negative pedal, and contrapedal. For a list of historically famous curves and their properties we refer to the website: MacTutor of History of Mathematics.

This book may be used for

Calculus I
Calculus II

Number of Pages: 14
Animation: Yes
Grade Level: 11-15

Mathwright Visualization Studio free demonstration Interactive Web Book:

Discrete Mathematics and Computational Structures Course, Part 1
by James White, Mathwright Library, and
Visiting Professor of Mathematics, Naval Postgraduate School

Now in the MATH Cafe, this 22 page Interactive Web Course is the first half of a 12-week course in Discrete Mathematics. It is freely available in the browser to all Library visitors. This first half of the course is an introduction to the theory of sets. It contains a set theory/propositional calculus language that will enable you to experiment with sets in a fresh and entertaining way.

In a series of 11 readings, you will learn the elements of a language and a methodology for the clear formulation of ideas in Set Theory. We recommend that you pursue the readings in the order in which they appear below, beginning with "Sets as Conceptual Tools." These readings now appear in the convenient form of compiled HTML Help files associated with each page.

Sets as Conceptual Tools
Sets and Logic
Operations on Sets
Boolean Algebra of Sets
Relations
Algebra of Relations
Functions
Composition of Functions
Permutations of Sets
Graphs and Directed Sets
Order and Cardinality

The readings contain many problems and exercises that illustrate its themes and test your understanding. These readings will direct you to our seven "activities," where you may test your ideas, and make them more tangible and concrete. You may, of course, visit those activities at any time, and you are encouraged to do so.

The aim of Sets, Functions, and Relations is to help the reader visualize, in a variety of ways, the basic properties of the objects in its title by making them concrete. While the purview of set theory is all of modern mathematics, our presentation begins with the consideration of simple small and discrete sets. Many of its central ideas may be captured in this realm -- and many may not. By "small" we mean finite. And by "finite", we mean "not infinite." But even that idea, the idea of some infinite thing, will have its clearest formulation in set theory itself, as we'll see in the last reading on "Order and Cardinality" by proceeding from the solid intuitions already built up from consideration of the properties of "small" sets. So no harm is done by proceeding from the familiar and concrete to the less familiar and less concrete.

The seven activities are designed to support your progress in visualizing sets, at each step of the way. These activities are:

Set Constructor
Set Viewer
Operations Lab
Relations Viewer
Workshop
Office
Set Safari Game

For each of these activities pages, you will find detailed instructions how to use the page under the Instructions button on that page.

The Set Constructor is the first place you go to build new sets for use throughout this Microworld. Later, in the Workshop, you may build sets more conveniently, but this is the place to start.

The Set Viewer is a place to view simple sets. Each set will have a name. 16 sets are predefined for you. They have the names: mammal, aquatic, lays_eggs, reptile, bird, hunts, snake, has_horns, feline, canine, primate, has_tusks, ungulate, fruit_eater, quadruped, and has_fur. As you create new sets, first in the Constructor, and later by command, using set operations and relations, new sets will be named: an1, an2, ... The sets of animals that you work with on projects may be saved to disk and later restored. In the Set Viewer, you may see the animals of any set that you name as colorful pictures.

The Operations Lab is the place to begin experimenting with the basic operations on sets: Union, Intersection and Complementation. These operations are discussed in the Reading "Operations on Sets." There you may use the Show command to show pictures of sets obtained by combining old sets using these operations, the Build command to create new sets in this way, the Size Command to tell the size of any set, and the IsEqual and Subset commands to compare sets.

The Relation Viewer is the first and simplest place to define and view relations between sets, and functions between sets. Here, you choose two sets: a Source set and a Target set, and define any relation or function from Source to Target. The properties of relations and functions are discussed in several of the readings, notably, "Algebra of Relations," and "Composition of Functions" and while the algebraic manipulation of these is best done in the Workshop, here you may create and view them with pictures. Like Sets, each relation and function created in the Microworld has a name such as: re1, re2,.. and these may also be saved to disk and later restored.

The Workshop and Office have similar functions. The Workshop is more graphical, and the Office more text-oriented. In these environments, you may use the Show command to show Sets, Relations, the results of operations on sets, the results of algebraic operations (such as composition and inversion) on relations or functions, the images and preimages of relations between sets, the effects of permutations (as invertible functions) and so on, interactively. You may use the Build Command to create new sets and relations by any combination of the mentioned operations. You may test sets so constructed for equality or for subsets. We also explore in the Workshop the structure of elementary groups, such as dihedral groups, the properties of ordering and equivalence relations, and the notion of cardinality.

Finally, the Set Safari Game brings it all together with an amusing exploration of the relations between set operations and the propositional calculus of logic. The rules of the game are simple (but described in more detail there). The computer creates a "hidden" set of animals, telling you its size only. It creates this set by generating randomly a proposition from the primitive 16 proposition-sets listed above. Such a proposition might be: "the union of all reptiles and animals that both hunt and are not quadrupeds" This is a definite set, and the computer can generate 2^17 such propositions randomly (over 100,000). This corresponds to a somewhat smaller number of actual sets (There are 2^46-1 sets possible).

The player then proposes propositions, such as: "the intersection of mammals and animals that do not hunt" The computer replies by informing the player of the number of animals in its set that satisfy the proposition. Using this information, the player proceeds to the next guess. If the player's proposition produces the identical set, the computer congratulates the player, and then shows its proposition and the set it produced. The player may of course use any of the commands (Show, Build, Subset, IsEqual and Size) to help her along. Often, with care, the player can find the precise set within 10 guesses, but the propositions are seldom identical.

The reasoning employed is, of course, precisely the reasoning that we formalize and develop in the readings, and so this exercise is a useful accompaniment to the course of readings.

Mathwright Visualization Studio free demonstration Interactive Web Book:

Discrete Mathematics and Computational Structures Course, Part 2
by James White, Mathwright Library, and
Visiting Professor of Mathematics, Naval Postgraduate School

Also a new member of the MATH Cafe, this 20 page Interactive Web Book is the second half of a 12-week course in Discrete Mathematics. It is freely available in the browser to all Library visitors. This second half of the course extends the set theory language, and combines it with Prolog, to provide a programming environment in which you may construct sophisticated sets based on propositions, relations, functions, their images, preimages, and algebraic combinations of these. It goes on to develop various strategies of search (dubbed artificial intelligence techniques) that allow the reader to build relational databases (as relations on sets represented as colorful graphs) and solve interesting problems in logic on them.

The term "Discrete Mathematics" in this Microworld will refer loosely to the collection of techniques, ideas, and constructions that have evolved over the years to describe artificial systems, and in particular, those systems from which the modern theories of computing have evolved. So, for example, an excellent model for our subject is the Turing Machine, or, equivalently, the lambda-calculus of Alonzo Church. In fact, the lambda-calculus is the conceptual basis for the computer language, LISP, which is the language in which this Microworld is written.

We recommend that you follow the 9 readings of this Microworld in the order in which they appear below, beginning with "Iteration and Recursion." These readings now appear in the convenient form of compiled HTML Help files associated with each page.

Those readings are:

Iteration and Recursion
Sets defined by Propositions
The Art of Counting
Relations and Functions
Digraphs as Relations
The Art of Searching
Critical Path Analysis
Automatic Problem Solving
Graphs and Logic

The readings contain many problems and exercises that illustrate its themes. They often cover some of the same ground as those of the first Microworld, but they do it in more depth, and so will develop and test your understanding. These readings will also give you the opportunity to experiment in the 8 Activity Labs with the ideas they develop. They will direct you to those activities where appropriate, and of course, you may visit those any time you like.

While the first Microworld worked with finite sets from a universe of 46 animals, in this Microworld we work with two types of sets: Finite sets of numbers, and subsets of products of finite sets of numbers. We examine three ways to define sets. The first is by recursive definition, the second is by propositions, and the third is by algebra, that is, through the use of Boolean Operations, Products, and the construction of images and preimages of functions and relations.

An example of the construction of sets by recursion is the famous Fibonacci sequence. Another example generates the nth level in Pascal's triangle, and so defines the basis for a large number of combinatorial and counting arguments. Examples of the second type are familiar from the "set builder notation." Thus "The set of positive whole numbers less than 100 that are congruent to their squares modulo 7" is an example. We have seen instances of the third type in the Set Safari game. But now, we may define interesting functions and relations for example that can help us formulate and solve problems in probability, game theory, and even computational logic.

The 8 activities are designed to support your progress in working with discrete models. These activities are:

Recursion Lab
Number Set Constructor
Relations/Functions Lab
Graph Constructor
Path Finder Lab
Rule-Based System Lab
Workshop
Fly-by-Night Airline

For each of these activities pages, you will find detailed instructions how to use the page under the Instructions button on that page.

The Recursion Lab allows you to construct sequences (ordered sets) of numbers by means of recursion formulas, such as: a(1) = 1, and a(k) = a(k-1)*k which defines the "factorial function," so important in combinatorics. We will discover there a number of beautiful relations that have their simplest and most elegant formulations as recursions.

The Number Set Constructor is the place to build sets of numbers defined by propositions:

{ x in MySet | p(x) is true }

to build products of these sets, and to build subsets of products (e.g. relations) by propositions of the form:

{ v in MySet | ( p(v(1)) is true ) or ( q(v(2)) is true ) }

and so on.

You may use any of the functions in the list below to build elaborate and decorative sets: The sets you create in this lab are available for use on all of the other pages of the Microworld. Each set will have a name. As you create new sets, first in the Constructor, and later by command, using set operations and relations, new sets will be named: Set1, Set2, ...

The Relation/Function Lab is where you define relations, functions, and propositions for use elsewhere in the Microworld. These relations may be composed, inverted, applied to sets and used in other ways.

The Graph Constructor is where you will build and study graphs for deductive information retrieval using Prolog in the Rule-Based Systems Lab. Each graph is conceptually a set of nodes N, together with a set of relations Ri :N -> N. Graphs are constructed in three steps:

Name and define the nodes (This defines the set)
Create the relations (These relations are associated to the set as arrows)
Assign "attributes" for the arrows of the relations.

Once a graph is created, it may be saved to disk. It is referred to by whatever name the system assigned to the set, and when the environment is restored, the graph is retrieved by referring to that set. For example, you may create a simple graph where the arrows of the relation represent kinship relations (each relation shown automatically in a different color). Or you may create more complex graphs such as the one in our "Fly-by-Night Airline".

The relations of these graphs may of course be manipulated algebraically. But they may also be the object of various kinds of "search" and of deductive information retrieval." For example, we could ask for all the paths from Washington D.C. to San Francisco that are under 3500 miles in length.

The Path Finder Lab is where you experiment with path analysis on a directed graph that we supply. You may do similar experiments on your own directed graphs in the WorkShop.

The Rule-Based System Lab uses a Prolog-style Automatic Problem Solver to enable you to create sophisticated relational databases, and to formulate rules for which the system will systematically search out all solutions to your queries. This illustrates an important and powerful application of search. We explain this search strategy in detail (backward chaining with recursive backtracking) because it is the main engine of a large class of "Expert Systems." You will in fact create your own Expert System and experiment with it here.

The WorkShop is the place to experiment with the basic Boolean operations on sets: Union, Intersection and Complementation. It is also the place to work with the set builder notation to build rich and illustrative new sets. In this environment, you may use the Show and Describe commands to show Sets, Relations, the results of operations on sets, the results of algebraic operations (such as composition and inversion) on relations or functions, the images and preimages of relations between sets, the effects of permutations (as invertible functions) and so on, interactively. You may also use the Build and Construct Commands to create new sets and relations by any combination of the mentioned operations. You may test sets so constructed for equality or for subsets.

We also explore in the Workshop the application of programs that you write to search directed graphs that you build. We look at the structure of elementary groups, such as dihedral groups, the properties of ordering and equivalence relations, and the notion of cardinality.

Finally, a visit to the Fly-by-Night Airline will introduce you to the science of path analysis using Prolog. It is an amusing discussion that will familiarize you with all options that are available to you in the Rules Based Systems Lab, and is an excellent preparation for your work there. Roughly speaking, you attempt to plan routes between the hub cities of the airline, and the system will then find all shorter routes (if any). You may also use the system to help you with your planning by asking certain questions about routes.

Featured in May, 2003 ...

Mathwright Visualization Studio free demonstration Interactive Web Book:

Exploring Quadratic Functions

Exploring Quadratic Functions
Samad Mortabit, Metropolitan State University, St Paul, Minnesota

Galileo Galilei, 1564-1642

This 9-page Interactive Web Book is a symphony of good mathematical pedagogy, artificial intelligence, and graphic art. It is an introduction to quadratic functions and quadratic growth. And it is full of experiments that will help you understand the ideas, beginning with Galileo's law of freely falling bodies. You control the simulations, so that you may step into Galileo's shoes and discover for yourself the mystery of "Natural Motion" - of uniform acceleration.
This Interactive Web Book tells its story on each page in a new way. Right-click where indicated on each screen to pop-up a menu, and choose Help, Help for this Page to view the Windows HTML Help for that page. These Help pages guide the reader through each topic with lively illustrations and with a discussion that presents the mathematics in textbook quality displayed formulas and charts.
Other experiments include tossing a ball up with various velocities, and applying the brakes to stop a car moving at various speeds. Different representations for quadratic functions are explored, and the relations of those representations to their graphs is developed through interaction.
Solving quadratic equations, both by graphing, and algebraically, is the central theme of this unique work. For this, the Book generates quadratic equations randomly, or allows you to make them up. In either case, it draws the graph and so displays the solution graphically, then explains, step-by-step, how to solve the quadratic equation algebraically. It does this for your problems also!
This is followed by a masterful treatment of quadratic inequalities. Whether you use an inequality it generates, or you make one up, it graphs the inequality, and shows which interval(s) in the line provide(s) the solution, by drawing those interval(s). Next, it gives a step-by-step explanation (using its built-in Expert System) of how the inequality can be solved algebraically.
All in all, it is a self-contained, animated exploration of the basic facts about quadratic functions. You may read it on the web in your browser, or off-line. Read at your own pace and enjoy this thoughtful discussion as a supplement to your studies. The book is full of questions (literally an unlimited number of them!) but the most important questions will be your own. And the book invites you to ask them. It will supply detailed answers for many questions that you may have about solving quadratic equations or solving quadratic inequalities.
The Windows HTML Help files can easily be printed, and they offer the familiar amenities. In particular, Help, Help for this Microworld provides a Table of Contents, so that readers can navigate smoothly from one topic to the next. It also explains how to prepare your machine to use Access Databases in case it is not ready to do so.

Fractals and the Mandelbrot Set
by Jim Swift, School District 70, Alberni, British Columbia

This 10 year old work began life in 1993 as a WorkBook for IBM's Toolkit for Interactive Mathematics. It exemplifies the goals of structured discovery learning so well that we have translated it without change to our Microworld format so that you may view it either in your browser or offline. It is one of our new free Visualization Studio Microworlds in the Math Cafe, and readers are invited to check it out before joining the Library to get a glimpse of what is possible with Mathwright.

There is much that has been said about the Mandelbrot iteration and its intriguing and colorful Fractal display. Jim Swift found something new to say, that he formulates as a conjecture for readers to pursue. The book steps through the strategy of complex iteration, so that the reader can get a clear visual understanding of the structure of the set. Once he sets the stage for his conjecture, he asks the reader to navigate the "bays" of the fractal design:

to discover in it the successive terms of the famed Fibonacci sequence! Now that is an unexpected connection indeed! But it is perhaps not a surprise that Fibonacci found his way to the unexpected Fractal shores of a future magical sea.

The Microworld features a new capability of Mathwright (available since version 2.10, May 12, 2003) that makes use of Windows Help to give pop-up information about each page if the reader desires it. This is new for MathwrightWeb, but was part of the original WorkBook.

Periodic Functions
by Jim Swift, School District 70, Alberni, British Columbia

This Classic 18 page Microworld also began life in 1993 as a WorkBook for IBM's Toolkit for Interactive Mathematics. We have translated it without change to our Microworld format so that you may view it either in your browser or offline. It is one of our new free Visualization Studio Microworlds in the Math Cafe, and readers are invited to check it out before joining the Library to get a glimpse of what is possible with Mathwright.

The aims of the Microworld are to help the novice learner visualize the "wrapping function" and its relation to the Sine and Cosine trigonometric functions. This is a venerable precalculus topic, and it is unfortunately often shrouded in a cloud of verbiage that can obscure the essential simple points.

Using the new page-by-page Windows Help Utility, the author guides the reader through a series of visual and dynamic animations (that the reader contols through her interactions) that illustrate the basic ideas in response to her own questions.

This "explanation" is supplemented on several pages with entertaining, game-like interactions in which the reader attempts to discover the formula that defines the graphs of periodic functions that the Microworld generates randomly. The overall effect is to encourage the reader to "play" as she learns each new idea, and in the process, to master the conventions and the notations that we wish to teach.