Algebra Skill Building
Links for Review & Practice

 

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These are the Web sites that Algebra Connections directs readers to for further study, practice, and information.  This online portal organizes the comprehensive set of links that provide alternative presentations, explanations, tutorials, and interactive lessons about the basic skills that prepare students for higher-level math classes in the Mathematics.

Main Factors of Factoring.

The Primes Pages.  You can learn EVERYTHING you could ever imagine about prime numbers here!

Prime Factoring.  To prime factor a number, begin dividing by the smallest possible prime and continue until the quotient is a prime number.  Links on this site provide a "mini-lesson, " worksheets, and an interactive factoring program that returns the prime factorizations and solutions to any number that is entered.

Decimals & Percents

Percents. These pages teach percent skills.  Each page has an explanation, interactive practice and challenge games about percents and ratios.

The Three Percentage Cases.  To explain the cases that arise in problems involving percents, it is necessary to define the terms that will be used. Rate (r) is the number of hundredths parts taken. This is the number followed by the percent sign. The base (b) is the whole on which the rate operates. Percentage (p) is the part of the base determined by the rate.  These three cases are the key to all percentage story problems.

Working With Fractions  

Fractions, Decimals, Percentages: Explanations.  Great presentation of these concepts and how they relate -- highly recommended!

Simplifying Fractions.  Simplifying fractions is really wimple, when you follow the rules.  Simplifying fractions is often required when your answer is not in the form required by the assignment. As a matter of fact, most math instructor will demand that you always simplify results.

Fraction Links. Maintained by Math League, this set of links contains EVERYTHING you would ever want to know about fractions.

Reducing Fractions & the Least Common Denominator.  This Sparks Notes study guide covers reducing fractions and the LCD.  There is also a "pull-down" menu for more helpful topics and links to other math and test taking resources.  

Fractions Fast Facts.  As the name implies, another good site for an easy review of fractions.  This site includes a wide variety of  fraction resources, as well as a direct link to Mrs. Glosser's Math Goodies.

Finding the Lowest Common Denominator.  Here is animated PowerPoint presentation that reviews important concepts about working with fractions. I recommend that students look at this presentation as a quick, easy, visual review of fractions -- its really well done.

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Properties of Real Numbers

Properties of Real Numbers:  First Glance.   In pre-algebra, you learned about the properties of integers. Real numbers have the same types of properties, and you need to be familiar with them in order to solve algebra problems.  Here is a wonderful interactive review.

Properties of Real Numbers:  In Depth.  In this lesson we look at some properties that apply to all real numbers. If you learn these properties, they will help you solve problems in algebra. Let's look at each property in detail, and apply it to an algebraic expression.

Real Numbers:  Properties.  Here is a short, concise, and clear explaination of the properties of real numbers.  

Properties of Real Numbers.  This site provides lessons, short interactive practice activities, and even teacher resources to review properties of real numbers.  

Negative Numbers

Ask Dr. Math.  This award-winning math site at Drexel has answers to almost any math related question.  If not, you can always ask!  Be sure to check out what the Doctor has to say about adding positive and negative numbers and negative times a negative.

Adding and Subtracting Positive and Negative Numbers.  Remember! Adding a positive and a negative number is like subtracting. If the greater number is positive, the answer is positive. If the greater number is negative, the answer is negative.  Try this interactive exercise.

Positive and Negative Numbers.  This set of links from Math League covers almost anything you need to know of the topic.

Combining Like Terms & Simplifying Expressions

Combining Like Terms.  This lesson explains the process of combining like terms to simplify an expression or an equation using addition and subtraction of the coefficients of terms.

Combining Like Terms and Solving.  Simplify each equation by combining like terms, then solve. Enter exponents using the ^ symbol. For example, "five x squared" should be entered as "5x^2".

Graphical Universal Expression Simplifier and Solver.  Check this one out -- you supply the expression and the site returns the answer, a short animated solution, and detailed step-by-step instructions.  Pretty amazing stuff!

Expressions Calculator.  When you enter an expression into the calculator, the calculator will simplify the expression by expanding multiplication and combining like terms. Additional capabilities including factoring will be added with future updates. Use the following rules to enter expressions into the calculator.

Equation Calculator.  When you enter an equation into the calculator, the calculator will begin by expanding (simplifying) the problem. Then it will attempt to solve the equation by using one or more of the following: addition, subtraction, division, taking the square root of each side, factoring, and completing the square.

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Problem Solving With Story Problems

Translating Word Problems.  The hardest thing about doing word problems is taking the English problem and translating it into math. Usually, once you get the math equation, you're fine.  This site suggest tips and tricks to learn how to do word problems.

Solving Math Word Problems.  This study guide breaks the process of working with story problems into two steps: (1).  Translate the wording into a numeric equation, (2).  Solve the equation!

Solving Linear Equations in 1 Variable 

Addition Property of Equality.  This is where we start getting into the heart of what algebra is about,  solving equations.  In this tutorial we will be looking specifically at linear equations and their solutions using the addition and subtraction properties of equality.

Multiplication Property of Equality.  This Web picks up where the one above left off, we look at linear equations and their solutions using the multiplication and division properties of equality.  

Solving Linear Equations.  In this this tutorial, we will be solving linear equations by using a combination of simplifying (combining like terms)  and various properties of equality.  Knowing how to solve linear equations will open the door to being able to work a lot of other types of problems that you will encounter in your various algebra classes.

Solving Linear Equations.  This online practice session challenges students to solve some linear equations in 1 variable and lets you see hints and step-by-step solutions to the problems.

MORE! Solving Linear Equations.  Here is another site that presents examples of linear equations, asks you to solve them, and presents step-by-step solutions.

Solving Linear Inequalities

Solving Linear Inequalities.  Wiki Books says, "Think free, learn free!"  Here is an excellent presentation on solving linear inequalities.

Inequalities.  Solving'' an inequality means finding all of its solutions. A "solution'' of an inequality is a number which when substituted for the variable makes the inequality a true statement.  

Inequalities:  In Depth.  Solving an inequality is very similar to solving an equation. You follow the same steps, except for one very important difference. When you multiply or divide each side of the inequality by a negative number, you have to reverse the inequality symbol!

Solving Linear Inequalities.  This is a straight-forward, "no-frills" presentation on solving linear inequalities.  

ThinkQuest. Solving Linear Equations.  Inequalities are solved in ALMOST the same way as equations.  Let's see why.

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Exponents

Exponents:  First Glance.  In algebra, you'll often be working with exponents. Here are some rules.

Exponents:  In Depth.  Exponents are used in many algebra problems, so it's important that you understand the rules for working with exponents. Let's go over each rule in detail, and see some examples.

Exponents.  Here is a set of lessons about what exponents are, how they work, and rules to use them in expressions and equations.

Scientific Notation

Scientific Notation.  While not hard, working with scientific notation is a fundamental skill in most college science classes.  This "refresher" is from ChemTutor.

Scientific Notation Lesson.  Scientific notation is simply a method for expressing, and working with, very large or very small numbers.  It is a short hand method for writing numbers, and an easy method for calculations.  Numbers in scientific notation are made up of three parts: the coefficient, the base and the exponent. 

Understanding Scientific Notation.  Do you know this number, 300,000,000 m/sec.t's the Speed of light !  Do you recognize this number, 0.000 000 000 753 kg. ?  This is the mass of a dust particle!  Scientists have developed a shorter method to express very large numbers. This method is called "scientific notation" . Scientific notation is based on powers of the base number 10.

More On Scientific Notation.  Astronomy deals with big numbers. Really big numbers. Itís impossible to talk about the distance to the Sun or the speed of light without thinking about the tremendously huge. But big numbers intrude on all aspects of our lives and as responsible citizens, we ought to have a way of dealing with them.  WOW, this site makes it all sound so important.  They're right, it is.

Why Use Scientific Notation?  Or "Why your wrist (or keyboard) will thank you for not writing all those zeros."

What Is Scientific Notation And How Is It Used?  The title of this page says it all.

Interactive Practice:  Scientific Notation.  This page is an exercise in scientific notation. When you press "New Problem", either the scientific notation or non-scientific notation representation of a number will be shown. Put the corresponding value(s) in the empty cell(s) and press "Check Answer". The results will appear in the second table.

Polynomials

Polynomial Basics.  Here is a good primer to review and get us started.  

Polynomials.  We have covered variables and exponents and have looked expressions.  Polynomials are sums of these expressions. Each piece of the polynomial, each part that is being added, is called a term;. Polynomial terms have variables to whole-number exponents; there are no square roots of exponents, no fractional powers, and no variables in the denominator.

Ask Dr. Math:  Polynomial Basics.  Let's see what the Doctor has to say.

Online Self-Check Quiz: Polynomial Basics.  Here is a short, interactive quiz to check understanding.

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Adding & Subtracting Polynomials

Like Terms.  Adding or subtracting polynomials is really all about collecting like terms.  Let's start with a quick review.

Polynomials:  Adding & Subtracting.  This tutorial looks at looking at the different components of polynomials.  Then moves on to evaluating polynomial functions as well as adding and subtracting them.

Adding & Subtracting Polynomials. Adding and subtracting polynomials is all about collecting like terms and looking at the order of operations.  

Adding and Subtracting Using Algebra Tiles.  Here is an alternative, visual presentation that let's you "see" how to add and subtract polynomials.

Online Self-Check Quiz: Adding and Subtracting Polynomials.  Here is a short, interactive quiz to check for understanding.  

More Links for Polynomial ExpressionsLinks to review all kinds of important concepts about polynomials, including adding and subtracting them.

Solve Your Math Problem:  Add or Subtract Polynomials.  This page will show you how to add and/or subtract polynomials.  You supply the probem and this site generates your answer.  Note:  You need to use and "MS Excel" style exponential notation.  To square "x," enter x^2.

Multiplying & Dividing Polynomials

How to Add, Subtract, Multiply, and Divide Polynomials .  A one variable polynomial, the kind you will most often see, is an algebraic expression made up from adding, subtracting, and multiplying of the variable and numbers. 

Multiplying Polynomials. This online tutorial looks at at multiplying polynomials together with and emphasis on the distributive property of multiplication. 

Polynomials:  Multiplying.  This tutorial looks at the process of multiplying polynomials, from simplest to more complex.

Dividing PolynomialsIn this tutorial we are dividing polynomials, but it follows the same steps and thought process as when you apply it numbers and is a good review of long division. 

Polynomials:  Dividing.  There are only 2 methods to dividing polynomials.  The first is actually just simplification, reduction of a fraction.  The second method uses long division.

Polynomial Long Division (also called synthetic division).  Here is another review of division.  Everything here is also a review for working with numbers.

Division of Polynomials and Synthetic Division (Word Document).  There is a summary of the steps used to divide polynomials and long (synthetic) division of polynomials.

Polynomials Worksheets Huge collection of links for free worksheets to review virtually EVERYTHING about operations with polynomials including multiplication and division.  

Solve Your Math Problem:  Multiplying Polynomials.  This page will show you how to multiply polynomials together.   Here are some example you could try: (x+5)(x-3), (x^2+5x+1)(3x^2-10x+15), or (x^2+5)(x^2-19x+9).

Solve Your Math Problem:  Dividing Polynomials.   This page will show you how to divide polynomials together.  You supply the problem and the Web site returns the solution.

Graphical Universal Expression Simplifier and Solver.  This link is also presented above under Simplifying Expressions, but this site does so much more, including:  reduction of constants, removal of unneeded parentheses, reduction of similar factors and terms, linear equations, quadratics, some polynomial reductions, associative property, and some substitutions.  You supply the expression and the site returns the answer, a short animated solution, and detailed step-by-step instructions.  

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Factoring Polynomials

Greatest Common Factor.  The first thing to do when factoring any expression is to look for a greatest common factor.  Let's look at how this applies to polynomials.

Factoring Out GCF.  The basics of factoring polynomials is presented here.  

GCF & Factoring by Grouping.  Factoring is to write an expression as a product of factors.  We can also do this with polynomial expressions. In this tutorial we look at two ways to factor polynomial expressions, factoring out the greatest common factor and factoring by grouping.

GCF & Factoring by Grouping Lesson.  Here is another presentation of these two important factoring strategies.

Factoring & Polynomials.  This is a short, concise presentation factoring polynomials.

Factoring Polynomials.  Factoring a polynomial is the opposite process of multiplying polynomials. Recall that when we factor a number, we are looking for prime factors that multiply together to give the number.  When we factor a polynomial, we are looking for simpler polynomials that can be multiplied together to give us the polynomial that we started with.

Solve Your Math Problem:  GCF & Polynomials.  This page will show you how to factor a GCF out of a of a polynomial.  You supply the problem and the Web site returns the solution.

Factoring Trinomials

Simple Trinomials as Product of Binomials.  (MS Word document, many examples, printable.  Factoring a trinomial results in a product of 2 binomials.  The simplest example is when the "lead" coefficient is 1 (x2+bx+c).

Factoring Trinomials where a=1 (x2+bx+c). Whether we use the FOIL method, or line up the factors vertically to multiply, the answer  is a trinomial.   To factor a trinomial of this form, we need to reverse the multiplication process.

Factoring Trinomials where a is not 1 (x2+bx+c).  The tutorial above showed us patterns when factorting trinomials with a lead coefficient of 1.  Now let's look at when the lead coefficient is other than 1.

Undoing FOIL:  Factoring Trinomials.  Because factoring is the reverse of multiplication, factoring skills depend on multiplication skills.  Let's look at how this works.

Factoring Polynomials.  Animated PowerPoint presentation about factoring polynomials & trinomials. 

Factoring Trinomials in the form  ax2+bx+c.  Another animated PowerPoint presentation about factoring polynomials. 

Factoring & Dividing Polynomials:  Undoing FOIL.  This is a clever presentation of factoring polynomials that looks at the process as the reverse of FOIL.

Factoring Trinomials.  This is another presentation that shows how factoring trinomials is really just the revers of FOIL.

Solve Your Math Problem:  Factoring Polynomials.   This page will show you how to factor trinomials.  You supply the problem and the Web site returns the solution.

Algebrator:  Factoring.  Enter an expression and click the FACTOR button.  This Web will return the solution.

Quadratic Equations

Introduction to Quadratic Equations.  This tutorial is designed to give a brief introduction to the concepts of Quadratic Equations.

Ask Dr. Math: Quadratic EquationsAs usual, Drexel's Dr. Math has a great presentation to understanding algebraic concepts.

Quadratic Equations.  Here's a simple review that addresses basic questions students often have about quadratic equations.

The World of Quadratic Equations.  An equation of the form ax2+bx+c=0 is called a quadratic equation, where a,b,c are known values (i.e. constants), a is non-zero and x is the unknown value (i.e. a variable). For ex: 5x2+7x+3=0, 4x2+2=0, 3x2+8x+4=8, are all quadratic equations

College Algebra Tutorial on Quadratic EquationsThis tutorial looks at solving a specific type of equation called the quadratic equation.  The methods of solving these types of equations that we will take a look at are solving by factoring, by using the square root method, by completing the square, and by using the quadratic equation.

101 Uses of a Quadratic Equation.  It isn't often that a mathematical equation makes the national press, far less popular radio, or most astonishingly of all, is the subject of a debate in the UK parliament. However, in 2003 the good old quadratic equation, which we all learned about in school, was all of those things.

101 Uses of a Quadratic Equation: Part II  In 101 Uses of a Quadratic Equation: Part I, we took a look at quadratic equations and saw how they arose naturally in various simple problems. In this second part we continue our journey. We shall soon see how the humble quadratic makes its appearance in many different and important applications.

Solving Quadratic Equations by Factoring.  There are different ways to solve quadratic equation.  This online lesson starts with factoring, but then covers other methods too if you continue to "click" to the next level.

Quadratic Equations: Solutions by Factoring.  Sometimes it is easier to find solutions or roots of a quadratic equation by factoring. Indeed, the basic principle to be used is: if a and b are real or complex numbers such that ab=0, then a=0 or b=0 .

Solving Quadratic Equations.  This program solves Quadratic Equations. Enter the coefficients in appropriate boxes and click Solve. It will show the results in boxes Root1 and Root2.

Quadratic Equation Calculator.  This site solves quadratic equations in standard form (ax2+bx+c=0).  Enter the values of the coefficients of a quadratic equation and an answer is generated.

Quadratic Equation Solver.   Quadratic equations have the form ax2 +bx+c=0.  They will generally have two solutions; that is, two different values of x that make the equation true.  It can happen that both solutions are the same number, and it is possible that the solutions will be complex or imaginary numbers.  To use this utility, you type in values for a, b, and c in the boxes below, and press the Solve button.

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Rational Expressions

Preliminaries:  Rational  Expressions  A rational expression is nothing more than a fraction in which the numerator and/or the denominator are polynomials. 

Algebraic Techniques:  Rational Expressions.  Rational expressions are represented as the quotient of two algebraic expressions. Thus, they can be manipulated like fractions. This tutorial explains how properties of fractions can be used to deal with rational expressions:

Rational Expressions.  A rational expression is an algebraic expression that can be written as the ratio of two polynomial expressions. A rational function is a function whose value is given by a rational expression.

Rational Expressions Lesson.  This 2 part lesson Explains what rational expressions are, and describes how to find their domain.

Rational Algebraic Expressions.  This interactive lesson reviews rational rexpressions.

SparkNotes: Rational Expressions.  This site contains a set of links to review almost anything you would want to know about rational expressions.

Simplifying Rational Expressions (pdf).  Here is a direct, brief explanation of simplifying rational expression.

Rational Expressions Self-Check.  This interactive quiz will allow you to see if you are comfortable with the concepts the rest of this unit is based on.

Multiplying & Dividing Rational Expressions

Multiplying Rational Expressions.  This tutorial presents the process of multiplying rational expressions as a series of simple steps.  

Multiplying Rational Expressions Lesson.  Most students find multiplying and dividing fractions easier than adding or subtracting fractions with different denominators.  The same is true with rational expressions -- let's see why.

Dividing Rational Expressions.  Dividing fractions is done by leaving the first fraction as it is and then multiplying by the reciprocal of the fraction following the division sign.  Multiply by the reciprocal is all you have to do.

Dividing Rational Expressions Lesson.  Just like dividing fractions, remember to "flip-n-multiply."  

Multiplying & Dividing Rational Expressions.  This tutorial reviews factor, simplify rational expressions and multiply polynomials to be able to complete the multiplication or division problems.

Multiplying & Dividing Rational Functions.  For our purposes, a rational expression is a rational function.  When we call it a function, we are simply referring to the set of numbers that result in a fraction that does not have a denominator of "0."  This set of numbers is called the "domain."

Math 101:  Multiplying and Dividing Rational Expressions.  Here is a simple, short, direct review of simplifying, multiplying, and dividing rational expressions.

Rational Expressions:  Multiplying, Dividing, Adding, and Subtracting.  This lesson is starting to get ahead of ourselves, but if you are ready for the next section, then this is a good place to start.

Adding & Subtracting Rational Expressions

Fraction Review & Lowest Common Denominator.  Adding and subtracting rational expressions is really like working with fractions.  Perhaps it will be  appropriate to start with a review the basics.

Adding and Subtracting Rational Expressions With Like Denominators.  Just like working with fractions, this is the simplest case when adding or subtracting rational expressions.  Perhaps this is a good place to start.

Adding or Subtracting Algebraic Fractions. Here is another tutorial that looks at numeric fractions with fractions with variables to get us ready for rational expressions -- these concepts are really all the same. 

Rational Expressions:  Adding or Subtracting.  This tutorial starts with the simplest case, common denominators, and then moves to more complex cases.

Addition and Subtraction of Rational Expressions.  Whenever fractions are added or subtracted, we must find a common denominator. The same is true for rational expressions.

Adding and Subtracting Rational Expressions.  If you have ever felt dazed and confused when working with fractions, you are not alone.  This tutorial is devoted to rational expressions (fractions).  In this tutorial we will be looking at adding and subtracting them.

Graphing Linear Equations and Inequalities

The Coordinate Plane and Graphing Linear Equations. In this unit we'll be learning about equations in two variables. A coordinate plane is an important tool for working with these equations. 

Tutorial: Graphing Linear Equations. When you graph linear equations, you will end up with a straight line. Let's see what you can do with these linear equations.

Graph of a Line. We can graph the linear equation defined by y = x + 1 by finding several ordered pairs. For example, if x = 2 then y = 2 + 1 = 3, giving the ordered pair (2, 3). Also, (0, 1), (4, 5), (-2, -1), (-5, -4), (-3, -2), among many others, are ordered pairs that satisfy the equation.

Graphs of Linear Equations: Lines and Slope. Lessons and some practice sets.  

Graphing Inequalities in 2 Variables. The solution set for an inequality in two variables contains ordered pairs whose graphs fill an area on the coordinate plane called a half-plane. An equation defines the boundary or edge of the half-plane.

Worksheets: Linear Equations and Inequalities. Need more practice? These worksheets will help. 

ThinkQuest: Graphing Linear Equations and Inequities. Let's review basics of graphic equations and inequalities.

Automatic Graph Solutions: Equations. Enter the equation you want to plot, in terms of the variables x and y, set the limits and click the Plot button. 

Automatic Graph Solutions: Inequalities. Enter the polynomial inequality you want to plot, in terms of the variables x and y, set the limits and click the Plot button.

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