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Mathematics Articles


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


How to Add, Subtract, Multiply, and Divide Natural Numbers
This is a review mainly for adults with some historical background andexplanations of how the algorithms work. A child might be able to learnfrom 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 commonmultiples of numbers
How to Add, Subtract, Multiply, and Divide Integers
plus and minus number arithmetic, including some good reasons that minustimes 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 studentsand 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 articleattempts to explain what rational and irrational numbers are really about,ratios.


 Translating Verbal Expressions to Variable Expressions
This article will hopefully help algebra students learn how to translatewords 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 onevariable linear equations.
Translating Sentences into Equations
This is about a technique that helps algebra students translate sentencesinto equations.
How to Solve Percent Increase and Decrease Problems
This is the advice that I give my students on how to solve percent increaseand decrease problems. Included is a good explanation on why you can'tdo a percent decrease and then a percent increase and get back what youstarted with.
How to Add, Subtract, Multiply, and Divide Polynomials
polynomial arithmetic with comparison to natural number arithmetic andworked out and explained examples
 Factoring Polynomials
explanantion of the techniques of factoring polynomials taught in beginningand intermediate algebra classes
Rational Expressions
an explanation of how you can deal with algebraic fractions the same wayas 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 thequadratic formula
A Geometrical Approach to Completing the Square
an alternate approach for those who like the visual including in additionanother 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 thedifficulties 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 oftwo 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 degreehigher than 2 using synthetic division, the rational zeros theorem, theupper and lower bound theorem, and Descartes Rule of Signs. I'm tryingthe best I can to explain how approach these problems in hopes that itwill gives some ideas to students. These examples take a while to workout, 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 tellyou that logs and exponentials undo eachother without referring to inversefunctions. By thinking through the logic of these carefully they can becomeso obvious that you could never forget them.
The Three Big Properties of Logarithms
an explanation of how these properties following from the correspondingexponential properties and some worked out examples of how to use them
Some Techniques for Solving Exponential and LogarithmicEquations

Unit Conversion

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


Pythagorean Theorem Puzzle
an easy to understand proof of the Pythagorean Theorem using a puzzle thatyou 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 wrotean internet pi tour to help mathematics students and others find the bestmaterial about it for whatever their interests are.

Analytical Geometry

 How to Find Equations of Lines
slopes, intercepts, etc., including equations of parallel and perpendicularlines
How to Test a Relation for Symmetries
help for algebra students on this aid to the graphing of relations andfunctions
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 purposeis to explain how to find equations for circles and also how to use thisknowledge to provide a shortcut for graphing such equations.


Calculator Techniques for Trigonometry
help with some of the problems that students commonly have
  Graphing the Sine and CosineFunctions
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 articleis just the beginning.
 Double Angles and HalfAngles
how to get these trigonometry formulas from the sum formulas, etc.
 Simplifying Compositions of TrigonometricFunctions and Inverse Trigonometric Functions
an explanation of the less formal method of simplifying expressions likesin(arccosx) by using triangles and the Pythagorean Theorem instead oftrigonometric 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

Books Store

 Swetz, Frank J. FromFive Fingers to Infinity: A Journey through the History of Mathematics
This is a very nice and very readable collection of essays on the historyof mathematics.
Smith, D. E. History of Mathematics VolumeI and VolumeII
This is one of the few books on the history of mathematics with an emphasison elementary mathematics.

Bunt, Lucas N. H.,  Jones Phillip S., and Bedient, Jack D. TheHistorical Roots of Elementary Mathematics
A short book with just the things that I would like students to know about.
Schmandt-Besserat, Denise HowWriting Came About
Read this if you want to find out more about the tokens that the ancientBabylonians used to keep track of their goods before they knew much aboutnumbers.
Nahin, Paul J. AnImaginary Tale
This is a nice little book about the history of the complex numbers. Itis also a good place to find out what they are good for.
TheRhind Mathematical Papyrus : An Ancient Egyptian Text
You don't have to read ancient Egyptian for this, because it has been translated. Unfortunately it is out of print, but if you click on the link, they might be able to find a used one for you. Among other things, there is a worked out multiplication example where they show how to multiply 12 times 12. There are also some distribution problems, but they are harder than division problems, because they deal with unequal distribution.