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Mathematics Articles

History

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

Basics

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.

Algebra

 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

Geometry

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.

Trigonometry

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.

 

 

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