Math Articles

Here is a list of articles from various resource. This is just a resource list. The authors deserve all credit. Be sure to explore their sites.

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.

A Gateway to Modern Mathematics

This is a collection of short articles designed to provide an introduction to the areas of modern mathematics.

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 each other 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.