These are little self-contained articles, many of them taken from my class notes. The original reason for isolating them was to submit them to Themestream, a site where they payed for articles based on many many people viewed them.  They were highly rated there, so when Themestream went out of business, I put together this site so that people could still be able to view them. This is also why their html format is fairly simple. You couldn't do much more than text with Themestream, and also I wrote them before I knew how to do much with web pages. I may eventually improve on their format, but I'm not sure when that will be, because it would be a lot of work. Similar to the class notes, they are a way to write up more than you can normally fit in a lecture, and to hopefully put together enough explanation so that there's something for a great variety of different ways of thinking. With this in mind, you should in no way feel you need to read all of the articles in order to understand the topic. The idea is to read what helps, and ignore what doesn't. Perhaps what doesn't help you will help someone else.

Also I have just recently transfered them from another server, done some alterations in the process. I have checked the links and pictures somewhat, but there still may be things I have missed. Please let me know if you find things that don't work in them.

History

Egyptian Addition
how to use Egyptian hieroglyphics to make addition of natural numbers more concrete
Egyptian Subtraction
how to use Egyptian hieroglyphics to make subtraction of natural numbers more concrete
Egyptian Multiplication
how to use Egyptian hieroglyphics to make multiplication of natural numbers more concrete
Egyptian Division
how to use Egyptian hieroglyphics to make division of natural numbers more concrete
How to Add and Subtract with a Counting Board
Learn how the medieval merchants would have done their additions and subtractions and at the same time get a better understanding for the meaning of these operations.
How to Multiply with a Counting Board
Here's a challenge for those who have mastered adding and subtract with a counting board.
Gelosia Multiplication
an alternate method of multiplying natural numbers that was commonly done in earlier times and has some advantages

Basics

How to Add, Subtract, Multiply, and Divide Natural Numbers
This is a review mainly for adults with some historical background and explanations of how the algorithms work. A child might be able to learn from it too, though, with some help from a parent or a teacher.
How to Prime Factor a Number
help for math students including several worked out examples
GCFs and LCMs
help for math students with finding greatest common factors and least common multiples of numbers
How to Add, Subtract, Multiply, and Divide Integers
plus and minus number arithmetic, including some good reasons that minus times minus is plus
Integer Exponents
an explanation for beginning math students of what integer exponents mean, positive, negative, and zero
Rational Numbers
a review of all the things we do with fractions
 The Order of Grouping Agreement
a better way to think about order of operations for mathematics students and instructors
Complex Fractions
how to deal with fractions that have fractions in them
 Square Roots
the meaning, notation, and properties
Irrational Numbers
Why does anyone care whether the decimals terminate or repeat? This article attempts to explain what rational and irrational numbers are really about, ratios.

Algebra

 Translating Verbal Expressions to Variable Expressions
This article will hopefully help algebra students learn how to translate words into algebraic expressions.
 Getting Rid of Parentheses
help for algebra students on simplifying expressions where there are parentheses
Simplifying Exponential Expressions
explanation and worked out examples for mathematics students
Linear Equations
This is the advice that I give to my algebra students about solving one variable linear equations.
Translating Sentences into Equations
This is about a technique that helps algebra students translate sentences into equations.
How to Solve Percent Increase and Decrease Problems
This is the advice that I give my students on how to solve percent increase and decrease problems. Included is a good explanation on why you can't do a percent decrease and then a percent increase and get back what you started with.
How to Add, Subtract, Multiply, and Divide Polynomials
polynomial arithmetic with comparison to natural number arithmetic and worked out and explained examples
 Factoring Polynomials
explanantion of the techniques of factoring polynomials taught in beginning and intermediate algebra classes
Rational Expressions
an explanation of how you can deal with algebraic fractions the same way as you do number fractions with worked out and explained examples
 How to Rationalize a Denominator
help for algebra students with several worked out examples
 Quadratic Equations
solving quadratic equations by factoring, completing the square, and the quadratic formula
A Geometrical Approach to Completing the Square
an alternate approach for those who like the visual including in addition another explanation of why minus times minus is plus
Complex Number Arithmetic
a quick lesson on dealing with the imaginary in mathematics
Using Function Notation
This article will provide help for both students and instructors with the difficulties that students have with dealing with expressions like f(x+1). It can also be helpful for teaching and learning about composition of functions.
 Polynomial and Rational Inequalities
This is what I think is the simplest way to approach solving these inequalities.
 How to find Compositions of Functions
some help with dealing with the notation when finding the composition of two functions and some worked out examples
 How to find the Inverse of a Function
explanation and some worked out examples
Synthetic Division
This is an explanation of how to use synthetic division why it works.
Polynomial Equation Examples
Here are some worked out examples of solving polynomial equations of degree higher than 2 using synthetic division, the rational zeros theorem, the upper and lower bound theorem, and Descartes Rule of Signs. I'm trying the best I can to explain how approach these problems in hopes that it will gives some ideas to students. These examples take a while to work out, so there is never enough time in class to give enough of them.
The Inverse Properties of Logarithms
I think it is helpful for students to understand the properties that tell you that logs and exponentials undo eachother without referring to inverse functions. By thinking through the logic of these carefully they can become so obvious that you could never forget them.
The Three Big Properties of Logarithms
an explanation of how these properties following from the corresponding exponential properties and some worked out examples of how to use them
Some Techniques for Solving Exponential and Logarithmic Equations

Unit Conversion

Chasing Cheetahs
some interesting examples of dimensional analysis to help math students and others, including figuring out how fast Maurice Greene can run in mph

Geometry

Pythagorean Theorem Puzzle
an easy to understand proof of the Pythagorean Theorem using a puzzle that you can print out and cut up
Pi on the Web
The number pi has played a big role throughout history in mathematics. I have also found that it plays a major role on the internet, so I wrote an internet pi tour to help mathematics students and others find the best material about it for whatever their interests are.

Analytical Geometry

 How to Find Equations of Lines
slopes, intercepts, etc., including equations of parallel and perpendicular lines
How to Test a Relation for Symmetries
help for algebra students on this aid to the graphing of relations and functions
Graphing Quadratics
How to graph quadratic functions by stretching, shrinking, pushing  around, and completing the square.
Equations of Circles
This article is taken from the notes for my Math 107 web page. It's purpose is to explain how to find equations for circles and also how to use this knowledge to provide a shortcut for graphing such equations.

Trigonometry

Calculator Techniques for Trigonometry
help with some of the problems that students commonly have
  Graphing the Sine and Cosine Functions
These are some notes on graphing sine and cosine from my Math 108 web pages.
 Starting with the Cosine Difference Formula
This one identity opens up a whole world in trigonometry. This article is just the beginning.
 Double Angles and Half Angles
how to get these trigonometry formulas from the sum formulas, etc.
 Simplifying Compositions of Trigonometric Functions and Inverse Trigonometric Functions
an explanation of the less formal method of simplifying expressions like sin(arccosx) by using triangles and the Pythagorean Theorem instead of trigonometric identities
 Solving Equations Involving Cosine Plus Sine
One method of solving these equations is to square both sides of the equation and use some identities. This is a thorough explanation of all of the details involved in this.

Graphing Calculator

Solving Equations

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