This concept-based, learner-centered, mathematics book, is intended for 8th grade and/or algebra students. This course is the result of twenty years (and counting) of sharing my passion for mathematics with middle level students, with the hopes of them becoming independent thinkers, lifelong learners, and people who are able to experience inner peace.
Three of the main goals of this course, are to align the Common Core State Standards Initiative with the Maine Learning Technology Initiative, to more effectively leverage the use of laptops in the mathematics class, and to instill in kids, a deep understanding of, and appreciation for, algebra and mathematics in general.
The picture above foreshadows how these three goals will be met. On the left is an example of an algebraic polynomial equation. On the right is an example of a geometric representation. These two examples in turn, are part of a book whose title includes both of the words, theory and experiment. The main premise of this book, is that algebra can be thought of as the logical and creative endeavor of translating, transforming, and discovering polynomial patterns. It is through this common theme of polynomials, that kids will be able to gain a deeper understanding of algebra and the Common Core State Standards. This approach is made possible due to the dual nature of algebraic and geometric ideas, which allows each to be represented by the other through technology. Using interactive applets and programs to experiment and to theorize, will also allow kids to realize that mathematics is alive and well, and that new math is being created every day.
Experiencing 8th grade math/algebra through the study of polynomials, will not only allow learners to make the connections necessary to truly understand the mathematics presented, but it will also give them a taste of what it’s like to do mathematics, and the chance to appreciate the beauty and fulfillment the world of mathematics has to offer.
Chapters 0 through 9 are designed for making deep connections, and need to be covered thoroughly, and in order, starting with chapter 0. Chapters 10 through 12 should be done in order as well, but teachers are free to decide which material learners should cover. This course is not about answers and being right or wrong. This course is about asking questions, exploring ideas, making connections, and defending discoveries. It’s important for learners to realize, that getting stuck is not a bad thing. Knowing what to do when one doesn’t know what to do, and being able to sustain a determined effort over a long period of time, are two important skills to develop. Learners should not be tempted to read ahead when they find themselves challenged. To help learners make deeper connections, much of the reading is designed to be transitional or summative, but not prefatory.
Regardless of the task at hand, learners should be encouraged to collaborate with each other, by participating in constructive mathematical discourse. It’s important to establish a classroom culture where mathematical discourse is welcoming, respectful, and thought-provoking. This begins by having a classroom layout which provides a natural environment for mathematical discourse to freely take place. Teachers should also keep in mind, this is a zero-lecture course. Learners should be free to move around, interact with each other, and work at their own pace. Teachers should be free to facilitate and to conduct ongoing formative assessments.
One possibility, would be to have the desks facing out, in a circular configuration. This way learners could group themselves in twos or threes, and be able to move directly to other learners. This would allow teachers to circulate as they conduct formative assessments (on small groups while inside the circle, or on the class as a whole while outside the circle).
Each chapter comes with GeoGebra applets and Chipmunk Basic programs, which will allow learners to interactively explore mathematical ideas and/or randomly generate problems for learners to solve. Grapher will also be used.
All three of these applications are preloaded on MLTI laptops. GeoGebra and Chipmunk Basic can be downloaded for free, and used on machines running OSX.
Teachers familiar with GeoGebra and/or writing in BASIC, are free to tailor the included applets and/or programs to better meet the needs of their students. No experience on the part of teachers or students is necessary however, to begin using the applets and programs which come with this course.
The first applet and the first two programs in chapter 0, come with instructional movies, which will show how to both access and use them.
For most applets and programs, learners will be asked to use copies of pdfs, for recording work, defending answers, showing examples, and recording discoveries on paper. These pdfs can be found in the “notes templates” folder, so teachers can provide copies for their students to use.
Many of the programs in each chapter randomly generate math problems and also provide immediate feedback. As a result, learners can have as much practice as they need and work at their own pace, both inside and outside of school. These programs can also be used by teachers to create additional assessments.
For additional practice and exploration, chapters 1 through 12, also come with six types of problem sets.
- Algebra: 25 cummulative algebra problems are provided to go along with this course.
- Basics: 10 basics problems and 10 recreational math problems are provided for basics practice.
- Cogitation: 5 cogitation problems are provided for word problem and math contest practice.
- Discovery: a math topic is presented where the learner has to discover how to create a mathematical object.
- Everything: 25 everything problems are provided for further vocabulary, skills, and math contest practice.
- Fun: a game or puzzle is presented as a fun challenge.
The recreational math problems, discovery, and fun topics, come with additional Chipmunk Basic programs. The pdfs for the algebra, basics, cogitation, and everything problems, provide spaces for showing work and defending answers.
Chapter 0: Software and Transformations
Polynomials have a geometric side and an algebraic side. By exploring polynomials geometrically, we are able to understand them better algebraically. To do this however, we need two things, a way of seeing this geometric side, and a way of describing it. The included software will allow us to quickly and interactively see representations, which would be difficult if not impossible to render by hand. Transformations of the plane will give us the vocabulary needed to describe and better understand these representations.
In Chapter 0, learners will have the opportunity to do the following.
- Become familiar with the two main interactive applications used throughout the book, GeoGebra and Chipmunk Basic.
- Discover and explore the four transformations of the plane, which will allow kids to better understand algebraic ideas, and take those ideas to a higher degree of abstraction.
- Explore ambigrams and Ford spheres.
- Use the included 5 applets, 5 programs, 2 pdf’s, and templates for recording and defending discoveries.
Chapter 1: Introduction To Polynomials
Algebra can be thought of as the logical and creative endeavor of translating, transforming, and discovering polynomial patterns. In other words, the common theme of polynomials can be used to make the connections needed to truly understand algebra. Chapter 1 will introduce the learner to polynomials, which will make the difference between taking algebra and learning algebra.
In Chapter 1, learners will have the opportunity to do the following.
- Learn about polynomials and their related vocabulary.
- Learn about functions and function notation.
- Explore the graphs and characteristics of polynomials using Chimpmunk, GeoGebra, and Grapher.
- Explore examples of polynomials to get a better feel for their diversity (math, sports, money, physics, fractals).
- Learn about complex numbers in order to better understand the fractal geometry examples throughout the book.
- Discover how to add, subtract, and multiply polynomials.
- Take the function challenge.
- Learn how to simplify expressions with monomials.
- Explore prime numbers, play Battle Boats, and explore John Conway’s game of Life.
- Use the included 10 applets, 33 programs, 10 pdf’s, and templates for recording and defending discoveries.
Chapter 2: Factoring Quadratic Polynomials
Traditionally, in an algebra book, one will find chapters on quadratics, toward the back. As it turns out, it’s makes much more mathematical sense to begin with quadratics. Being able to add, multiply, and factor quadratics, requires and reinforces, in context, the main pre-algebra skills necessary to be successful in algebra. Also, many of the ideas and formulas in algebra, can be derived and understood with a basic knowledge of quadratic polynomials. In other words, the order in which material is covered, and the connections which are made, make a big difference when it comes to appreciating and understanding algebra.
In Chapter 2, learners will have the opportunity to do the following.
- Explore monic quadratic polynomials.
- Explore area and graphical models of quadratic polynomials, and discover their factored forms.
- Discover how to factor quadratic polynomials.
- Use what was learned about factoring, to simplify rational polynomial expressions.
- Discover how to factor out the greatest common monomial of a polynomial.
- Discover patterns in calendars, make conjectures, and write algebraic proofs.
- Take the bivariate input function challenge, and discover how it can give the steepness formula of a polynomial.
- Discover it’s possible to create parabolas with lines.
- Discover how light rays are reflected, and try to figure out how the scientific toy, Mirage, works.
- Explore deficient, perfect, and abundant numbers, solve shikaku puzzles, and create para’boo’las (like the one shown).
- Use the included 13 applets, 15 programs, 6 pdf’s, and templates for recording and defending discoveries.
Chapter 3: Polynomial Equations
By using transformations to explore the geometric side of polynomials, and to truly understand what solutions are, we can give meaning to the algebraic steps behind solving linear equations. In other words, by understanding the algebraic steps behind solving linear equations, rather than just memorizing the steps to get answers, we can make the connections necessary to understand and solve more complicated equations.
In Chapter 3, learners will have the opportunity to do the following.
- Explore univariate polynomials.
- Discover what a solution of a polynomial is, and how many solutions a polynomial can have.
- Discover how to use transformations of the plane, to find solutions of linear polynomial equations.
- Understand the algebraic steps of solving a linear equation, by discovering the geometric meanings behind them.
- Discover how scalings, reflections, and translations, can also explain how to multiply or add negative integers.
- Use this geometric/algebraic connection to solve multiple step linear equations, inequalities, and absolute values.
- Learn about iterating polynomial functions algebraically and geometrically, and explore their possible orbits (iterate a square root polynomial expression and a mortgage polynomial, and use cobweb diagrams).
- Discover the politeness of a number, try the order of operations challenge, and explore Julia set fractals.
- Use the included 12 applets, 50 programs, 16 pdf’s, and templates for recording and defending discoveries.
Chapter 4: Roots Of Polynomial Equations
By exploring quadratics as polynomial functions, we can convert any quadratic equation into a monic one, which is a form we can always solve. In other words, we can derive the quadratic formula for finding the two roots of any quadratic equation. Better still, by understanding this formula, rather than just memorizing it for finding roots, we can derive a formula for the slope of the best fit line for a scatterplot, prove perpendicular lines have negative reciprocal slopes, and solve certain max/min problems which are traditionally solved using first year college calculus. In other words, when truly understood, algebra is much more than just a gateway course to higher mathematics.
In Chapter 4, learners will have the opportunity to do the following.
- Discover how to geometrically solve factorable quadratic equations, and discover the zero product property.
- Be able to determine the rational roots of quadratic equations, or given the roots, be able to determine the polynomials.
- Discover how to geometrically turn any quadratic polynomial equation, into a monic quadratic, set equal to zero.
- Discover how to solve a quadratic polynomial equation, when the coefficient of the x-term is zero.
- Discover how to geometrically and algebraically solve any quadratic polynomial equation with real roots, and solve a general quadratic polynomial equation, in order to derive the quadratic formula.
- Learn about the quadratic relationship between time and distance of an object in free fall, near the earth’s surface.
- Learn how to build a gravity ruler, and explore the quadratic relationship of trajectories near the earth’s surface.
- Discover how to use synthetic division to determine if an input value is a root.
- Explore vampire numbers, play the Tax Agent game, and go on a baby Mandelbrot hunt.
- Use the included 15 applets, 16 programs, 6 pdf’s, and templates for recording and defending discoveries.
Chapter 5: Linear Polynomials In One Variable
Given a linear polynomial in one variable, it’s not too hard to explore it further, by creating an xy table and/or graphing the line. To convert a table or a line, into a linear polynomial however, requires understanding the slope of a linear polynomial. Truly understanding the connection between slope and linear polynomials, means we can describe the linear relationships between mathematical objects exactly, and model the linear relationships between real world phenomena. When the linear relationships represent tangent lines, understanding linear polynomials in one variable will allows us to further explore and understand more complicated functions as well.
In Chapter 5, learners will have the opportunity to do the following.
- Discover what an xy table is, and discover how to graph a linear polynomial, with and without one.
- Be able to graph linear polynomials in slope-intercept form, and determine linear polynomials from their graphs.
- Take the slope-intercept challenge.
- Discover the formula for slope, and learn about the different ways of describing it.
- Learn about trig ratios, and discover the connection between similar/congruent triangles and slope.
- Determine if a wheelchair ramp meets ADA specifications.
- Discover how to convert xy tables into linear polynomials, convert Fº to and from Cº, and take the xy table challenge.
- Discover best fit points, squared error, and how to find (and derive) the best fit line for a scatter plot.
- Explore happy numbers, play a game of Triplets, and explore line art.
- Use the included 14 applets, 12 programs, 5 pdf’s, and templates for recording and defending discoveries.
Chapter 6: Linear Polynomials In Two Variables
Using the idea of squared error from the last chapter, allows us to derive a formula for the distance between two points on a plane. Once we realize this formula is just an application of the Pythagorean Theorem, we enter the world of polynomials in two variables, since the Pythagorean Theorem is also the quadratic polynomial in two variables for circles. Exploring general quadratics in two variables, leads us to discovering conic sections and linear polynomials in two variables. This in turn, gives us a way to represent vertical lines, gives us the intercept form of a line which will help us understand conic sections, and gives us the standard form of a line which is more useful for solving systems of equations.
In Chapter 6, learners will have the opportunity to do the following.
- Discover how to determine the area of lattice polygons.
- Discover how to find the distance of a line segment using squared error.
- Fill in the words for a ‘proof without words’ of the Pythagorean Theorem.
- Discover a connection between the Pythagorean Theorem and polynomials in two variables, and explore conic sections.
- Explore linear polynomials in two variables, and how to graph lines in standard and intercept form.
- Show how to geometrically and algebraically convert slope-intercept, to and from, standard form.
- Create digital string art parabolas and a paper folded parabola.
- Explore pandigital fractions, play speedmath, and create spirolaterals.
- Use the included 9 applets, 8 programs, 1 pdf, and templates for recording and defending discoveries.
Chapter 7: Parallel and Perpendicular Linear Polynomials
If you are indoors right now, look around at all of the human-made objects. Most likely, their designs and/or outlines, involve parallel and/or perpendicular line segments. Many mathematical objects can be described and/or created using parallel and/or perpendicular line segments as well. Of course, creating human-made and mathematical objects, which form angles with measures less than ninety degrees, is also practical and important. Now that we know we can write polynomials implicitly, we can better model and understand these objects, by deriving a more powerful form of a line, based on slope and a given point,
In Chapter 7, learners will have the opportunity to do the following.
- Discover how to find the middle of a line segment.
- Discover the connection between the slopes of perpendicular lines, and show why it’s true.
- Determine the coordinates of a point on the paper folded parabola from the last chapter.
- Geometrically explain the connection between the slopes of parallel lines, and take the parallel/perpendicular challenge.
- Show how using the definition of slope, can be used to determine parallel or perpendicular lines.
- Discover how the point-slope form of a linear polynomial, can determine the line through any two given points.
- Discover the sum of the measures of the 3 angles of any given triangle, and show why with parallels and transversals.
- Discover how to determine the circumcenter of 3 points, and find the best location for a radio tower serving 3 towns.
- Explore narcissistic numbers, solve the lattice puzzle, and create Poincare disks.
- Use the included 12 applets, 14 programs, 6 pdf’s, and templates for recording and defending discoveries.
Chapter 8: Solving Systems Of Linear Polynomials Geometrically
Being able to solve systems of linear polynomial equations, opens the doors to a lot of interesting mathematics. The good news is, solving systems of linear polynomial equations by graphing, is rather straightforward. The bad news is, geometric representations have limitations when it comes to accuracy. Understanding polynomials and the difference between variables and unknowns however, will allow us to derive a way to solve systems of linear polynomial equations, with perfect accuracy.
In Chapter 8, learners will have the opportunity to do the following.
- Discover what a solution of a system of linear equations is.
- Discover the three possible outcomes, when looking for a solution of a system of linear equations.
- Solve systems of two linear polynomial equations with integer and estimated solutions, by graphing.
- Learn how knowing the difference between a variable and an unknown, means we can solve any system of two linear polynomial equations, written in slope-intercept form.
- Solve systems of linear equations by substitution, with integer and rational solutions.
- Learn the connection between finding roots and solving systems of equations.
- Create an Archimedean triangle, and determine the area of a section of the paper folded parabola from chapter 6.
- Determine a formula for the angle between any two lines.
- Explore the persistence of numbers, play the ladders game, and create Lissajous curves.
- Use the included 1 applet, 20 programs, 6 pdf’s, and templates for recording and defending discoveries.
Chapter 9: Solving Systems Of Linear Polynomials Algebraically
As we discovered before, linear polynomials can be written in one variable or in two variables, and both ways have their pros and cons. As it turns out, there is a more powerful method of solving systems of linear polynomial equations, when they are written in standard form, and it avoids having to use fractions as well. One of the reasons this method is more powerful, is we can also use it to derive a degree n polynomial, from any n of its solutions, by converting those solutions into a system of n, degree 1 equations with n unknowns.
In Chapter 9, learners will have the opportunity to do the following.
- Geometrically discover what happens when you multiply both sides of an equation written in standard form, and algebraically show why it happens.
- Discover under what conditions, will adding two linear equations of a system, written in standard form, allow us to solve for one of the unknowns.
- Be able to solve systems of two linear equations, written in standard form, by addition, which have integer or rational solutions.
- Explore and solve systems of 3 degree 1 equations with 3 unknowns, including systems in the style of Mensa problems.
- Learn how treating 1 equation with 1 unknown, as a system, we can convert any terminating or repeating decimal to its fractional form, and also learn how to find rational estimates for square roots.
- Learn how to convert rational and irrational numbers to continued fractions, and use the partial quotients to find the best possible rational estimates for square roots.
- Learn how to use the three known points, to determine the quadratic polynomial of the paper folded parabola.
- Use solutions of an unknown polynomial, to create a system of degree 1 equations, in order to derive the polynomial.
- Discover how to create and solve a system of two linear equations, to solve certain equilibrium problems in physics.
- Discover how to create and solve a system of two linear equations, to solve certain fulcrum problems in physics.
- Determine a formula to solve a system of two general linear polynomial equations, written in standard form.
- Learn how to use Crame’rs Rule, to solve a system of two linear polynomial equations with two unknowns.
- Learn how to solve a system of two linear inequalities with two unknowns, and solve a linear programming problem.
- Explore Harshad numbers, play Break The Code, and go on a Mandelbrot bulb hunt.
- Use the included 6 applets, 24 programs, 4 pdf’s, and templates for recording and defending discoveries.
Chapter 10: Linear Polynomial Transformations Of The Plane
Throughout the book, we’ve been using transformations of the plane, to better understand linear polynomials. Returning the favor, linear polynomials will now allow us to better understand transformations of the plane. By using linear polynomials to describe lines of symmetry, we can show all rotations and translations, are really just two reflections. Also, by using matrices, we can see why they are referred to as linear transformations of the plane.
- Chapter 10 is still under construction.
Chapter 11: Applying Polynomials
Applying polynomials could easily be a whole other book in itself. This chapter will just focus on geometic formulas (specifically the volumes of spheres, cones, and cylinders) and word problems. By realizing geometric formulas are usually just polynomials, we can better understand what they represent, and better understand the geometric objects as well. This in turn, will help us solve word problems, by understanding how to model them mathematically. The examples in this chapter will serve as practice for solving word problems, by first translating them into geometric or algebraic polynomial models.
- Chapter 11 is still under construction.
Outside of a degree 1,825 polynomial example in Chapter 1, and a few degree 4 and degree 3 examples, our focus has been on degree 2 and degree 1 polynomials. There’s nothing however, preventing the existence of infinite degree polynomials. In fact, one type is quite common, and otherwise known as geometric series. One of the many applications for geometric series, is to describe mathematical fractals, which require an infinite number of iterations to be created. Outside of the best fit line example in Chapter 5, our focus has also been on mathematical objects with exact polynomial patterns. As it turns out, any finite pattern of integers, whether polynomial or not, can be modeled perfectly by an infinite number of polynomials of finite degree.
- Chapter 12 is still under construction.
Screen recordings are currently being made to go along with this book, which may be used by learners and/or teachers.
Text movies: Text movies for groups of pages, narrate the text in the book to help with comprehension, and making the directions clearer. Text movies for a particular page, help make the intent of the applets and programs clearer.
Feedback movies: Feedback movies show and explain possible ways of defending the work and discoveries, as a result of using the applets and programs. Although the feedback movies are always available, they should not be viewed by collaborating learners until their teachers give them the go-ahead. It’s important to realize the feedback movies are post discovery tools, for helping learners critique their own work, and modeling good learning habits. Adults at school or at home who wish to support learners however, should feel free to view the movies anytime.
Text movies ps: The text movies for the “discovery” and “fun” problem sets, narrate the text to help with comprehension, and making directions clearer.
Feedback movies ps: The feedback movies for the “algebra”, “basics”, “cogitation”, and “everything” problem sets, show and explain possible ways of defending answers. Again, these movies are always available, but they should not be viewed by collaborating learners until their teachers have given them the go-ahead. It’s important for learners to realize the focus is on staying with problems for a long time, making connections, and defending work, not the answers. It’s also important for collaborating learners to use these movies as learning tools, not answer sheets.
Common Core State Standards
The following two sections show in which chapter(s) the 8th grade Common Core State Standards are addressed.
Descriptions of the CCSS can be found at http://www.maine.gov/education/lres/math/standards.html#ccss .
Below is the rest of the chart, showing in which chapters the Grade 8 Mathematics CCSS are addressed.
Note, parts of this book are still under construction. When finished, standard 8.G.9 will be addressed in chapter 11, and standards 8.EE.3, 8.EE.4, and 8.SP.4 will be addressed using material found in the folder “the appendix”.
Below are the chapters in which the following High School Common Core State Standards are addressed.
Common Core State Standards For Mathematical Practices:
Due to the nature of this book, there will be plenty of opportunities to connect the standards for mathematical practices with the content.
1 Make sense of problems and persevere in solving them.
For example, in chapter 2, learners will be challenged to create a ghost made solely out of parabolas, where they will have to “analyze givens, constraints, relationships, and goals”, “make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt”, “try special cases and simpler forms of the original problem in order to gain insight into its solution”, and “monitor and evaluate their progress and change course if necessary.
2 Reason abstractly and quantitatively.
For example, in chapter 1, learners will be challenged to discover what it means to add and multiply polynomials by each other, where they will have to “bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize – to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to the referents – and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved”, and create a “coherent representation of the problem at hand”, “attending to the meaning of quantities”, and use “different properties of operations and objects.”
3 Construct viable arguments and critique the reasoning of others.
For example, in chapter 3, collaborating learners will be challenged to discover how many solutions a polynomial equation can have, where they will have to “understand and use stated assumptions, definitions, and previously established results in constructing arguments”, “make conjectures and build a logical progression of statements to explore the truth of their conjectures”, “recognize and use counterexamples”, “justify their conclusions”, and “respond to the arguments of others.”
4 Model with mathematics.
For example, in chapter 7, learners will be challenged to solve a civil engineering problem, where they will have to “apply the mathematics they know to solve problems”, “identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas”, “interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.”
5 Use appropriate tools strategically.
For example, in chapter 8, learners will be challenged to discover the three possible outcomes for the solution of a system of two linear equations, where they will have to “analyze graphs of functions and solutions”, “visualize the results of varying assumptions, explore consequences, and compare predictions with data”, and “use technological tools to explore and deepen their understanding of concepts.”
6 Attend to precision.
For example, in chapter 6, collaborating learners will be challenged to write a proof for the Pythagorean Theorem, where they will have to “communicate precisely to others”, “use clear definitions in discussion with others and in their own reasoning”, “state the meaning of the symbols they choose”, and “examine claims and make explicit use of definitions.”
7 Look for and make use of structure.
For example, in chapter 4, learners will be challenged to derive a method for determining the roots of any quadratic equation, where they will have to “look closely to discern a pattern or structure”, “step back for an overview and shift perspective”, and “see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects.”
8 Look for and express regularity in repeated reasoning.
For example, in chapter 9, learners will be challenged to convert square roots to continued fractions, where they will have to “notice if calculations are repeated, and look for general methods and for shortcuts”, “maintain oversight of the process, while attending to the details”, and “continually evaluate the reasonableness of their intermediate results.”
The folder, “the appendix”, will contain additional projects and/or programs, which teachers and/or students may use if and when they see fit. Note, this folder is a work in progress, and as additional projects and/or programs are completed, they will be added to the folder. To see the current list of additional material, in alphabetical order, open the “READ ME” pdf inside the folder “the appendix”.
I would like to thank my wife Dani, whose unconditional love and support over the years, has made any and every success I’ve had as a teacher possible. I would like to thank Jeff Mao, State of Maine, Department of Education, Learning Technology Policy Director, for noticing my efforts in the classroom, and giving me the opportunity to write this book.
I would like to thank Steve Garton, State of Maine, Department of Education, Coordinator of Educational Technology,
Dr. Ruben Puentedura, Founder and President of Hippasus, Michele Mailhot, State of Maine, Department of Education, Mathematics Specialist, and Pam Buffington, Education Development Center, for their roles as reviewers. I would like to thank Dr. Len Brin, Mathematics Professor, Southern Connecticut State University and Pete Brin, Software developer for their support. I would like to thank Amy, Jenn, Brittany, Zach, Cecelia, Victorija, Bill, Laurette, Ken, Dave, Steve, Bob, and my parents for their support as well. I would like to thank Shawn, Amy and Kate for their tireless support in the classroom, and the administration, faculty, parents, and students of Freeport Middle School for their confidence.