BASIC ALGEBRA LESSONS
Whether needing help with basic algebra homework or reviewing for tests, Mr. X can help you, your child or your students better understand Basic Algebra. Our lessons are designed to reinforce the instructor's message. We also have a library of sample algebra problems with examples of solved problems for each basic algebra lesson. Check out our free samples below, as well as the basic algebra curriculum.
Basic Algebra Sample Lesson 1
Basic Algebra Sample Lesson 2
Basic Algebra Curriculum
Algebraic Expressions
Title DescriptionPineapples Classic Lesson  A Classic Lesson in learning how to write Algebraic Expressions.  
Cake and Pie Classic Lesson  A Classic Lesson in learning how to write Algebraic Expressions. We may use either two variables or just one. 

Algebraic Notation  You need a good understanding of a variable, as x.  
Dr. X no Longer Operates  At some point we need to leave the symbol for the operation of multiplication behind, as "X" has become a variable, and not a symbol to multiply.  
A Bridge from Arithmetic to Algebra Part II  Some simple problems lend themselves to solutions with both Arithmetic and Algebra. Visualizing these type of problems provide a bridge into the world of Basic Algebra.  
Learning how to Setup an Algebraic Equation  Our suggestion is to pick letters that represent something easily understood. In this example M represents the number of pineapples Mike begins with. L represents the number of pineapples that Laura has.  
Equal Parts mean Equal Fractions  These word problems can be solved with either Arithmetic or Algebra. Practice with fractions in word problems to learn how to write algebraically.  
A Bridge from Arithmetic to Algebra Part I  Some simple problems lend themselves to solutions with both Arithmetic and Algebra. Visualizing these type of problems provide a bridge into the world of Basic Algebra.  
Adding Like Terms  Terms combine when they have the same variables to the same powers.  
Coefficients Add Efficiently  To do algebra easily and effectively, understand that coefficients add efficiently.  
Like Terms  Reinforcement on combining like terms. Terms combine when they have the same variables to the same powers. 

Negative Multiplication  A Lesson on Handling Negative Multiplication with Algebraic Terms.  
Sum Groups are Worth Joining  Grouping terms according to the Associative Law is exceptionally easy to see.  
Ignorance of the Law is No Excuse  The Commutative Law and the Distributive Law are discussed.  
Using Algebraic Notation  A lesson in using Algebraic Notation and Evaluating Algebraic Expressions. Understanding how to work with variables and equations is a key skill to succeed in an Algebra course. 

X is Handy, In (or Out)  We evaluate algebraic expressions whether x is inside or outside parenthesis. We use the distributive property when x is inside parenthesis. 
Basic Skills
Title DescriptionA Bridge from Arithmetic to Algebra Part I  Some simple problems lend themselves to solutions with both Arithmetic and Algebra. Visualizing these type of problems provide a bridge into the world of Basic Algebra.  
A Bridge from Arithmetic to Algebra Part II  Some simple problems lend themselves to solutions with both Arithmetic and Algebra. Visualizing these type of problems provide a bridge into the world of Basic Algebra.  
Learning how to Setup an Algebraic Equation  Our suggestion is to pick letters that represent something easily understood. In this example M represents the number of pineapples Mike begins with. L represents the number of pineapples that Laura has.  
Algebraic Notation  You need a good understanding of a variable, as x.  
Equal Parts mean Equal Fractions  These word problems can be solved with either Arithmetic or Algebra. Practice with fractions in word problems to learn how to write algebraically.  
Dr. X no Longer Operates  At some point we need to leave the symbol for the operation of multiplication behind, as "X" has become a variable, and not a symbol to multiply.  
Pineapples Classic Lesson  A Classic Lesson in learning how to write Algebraic Expressions.  
Cake and Pie Classic Lesson  A Classic Lesson in learning how to write Algebraic Expressions. We may use either two variables or just one. 

Arithmetic Operators  We introduce the operators of arithmetic, the signs (or symbols) for addition, subtraction, multiplication, and division. These are generally read "plus," "minus," "times," and "divided by," respectively. We also officially introduce the equal sign. 

Using Algebraic Notation  A lesson in using Algebraic Notation and Evaluating Algebraic Expressions. Understanding how to work with variables and equations is a key skill to succeed in an Algebra course. 

X is Handy, In (or Out)  We evaluate algebraic expressions whether x is inside or outside parenthesis. We use the distributive property when x is inside parenthesis. 

Like Terms  Reinforcement on combining like terms. Terms combine when they have the same variables to the same powers. 

Negative Multiplication  A Lesson on Handling Negative Multiplication with Algebraic Terms.  
Ignorance of the Law is No Excuse  The Commutative Law and the Distributive Law are discussed.  
Adding Like Terms  Terms combine when they have the same variables to the same powers.  
Sum Groups are Worth Joining  Grouping terms according to the Associative Law is exceptionally easy to see.  
Coefficients Add Efficiently  To do algebra easily and effectively, understand that coefficients add efficiently.  
Percent of Change 2  Practice with the concepts of Percents and Percentages as we build a bride to Basic Algebra.  
Percent of Change 1  A lesson that emphasizes building a bridge from Arithmetic to Algebra with an understanding of Percentage Calculations. 
Equations
Title DescriptionX is the Unknown  We begin with a variable representing a "fillintheblank" to make a statement true.  
Cancel my Policy, Please  We may subtract the same value from each side of an equality and maintain the equality. We may divide the same value (a factor) from each side of an equality and maintain the equality. 

X Fills in the Blank  Don't let the concept of a variable throw you. You just "fill in the blank" with what you already know. Each "x" is unique to a math statement. We do not use different values of x "in the same problem." 

One Good Idea Top to Bottom  Dividing a fraction with like coefficients in the numerator and the denominator simplifies to unity. This is used frequently when solving equations. 

X Can Be Anything  The value that x stands for can be any value. Typically, we seek the value for x the makes a statement true. 

OneStep Equations with Fractions, Lesson 1  We explain the logic and the procedures associated with moving fractions and a variable in an Algebraic Expression. It's easy. 

Anything Goes with X  Variables can be negative, positive, irrational, rational... all kinds of things. Anything goes... Try to make the statement true. A statement that is always true is called an identity. 
Equations
Title DescriptionX is the Unknown  We begin with a variable representing a "fillintheblank" to make a statement true.  
X Fills in the Blank  Don't let the concept of a variable throw you. You just "fill in the blank" with what you already know. Each "x" is unique to a math statement. We do not use different values of x "in the same problem." 

Cancel my Policy, Please  We may subtract the same value from each side of an equality and maintain the equality. We may divide the same value (a factor) from each side of an equality and maintain the equality. 

One Good Idea Top to Bottom  Dividing a fraction with like coefficients in the numerator and the denominator simplifies to unity. This is used frequently when solving equations. 

X Can Be Anything  The value that x stands for can be any value. Typically, we seek the value for x the makes a statement true. 

Anything Goes with X  Variables can be negative, positive, irrational, rational... all kinds of things. Anything goes... Try to make the statement true. A statement that is always true is called an identity. 

OneStep Equations with Fractions, Lesson 1  We explain the logic and the procedures associated with moving fractions and a variable in an Algebraic Expression. It's easy. 

Square Roots and Absolute Values  A review of square roots and absolute values. 

Solving Proportions  We use all five types of proportion problems from MathAids.com. Solve for the value of the unknown (or variable) that makes the statement true. 

A Lesson in Concentrations  Parts per Million and Parts per Billion are explained in a way to let it soak in, as it were. It is a very fluid idea. 
Exponents
Title DescriptionAn Overview of Rules for Exponents  We take a look at the MathAids handout for rules of exponents.  
An Introduction to Negative Exponents  An introduction to negative exponents. We reference reciprocals without detail about them. 

A Lesson in Evaluating Exponential Functions  We examine the nature of functions before we jump into the evaluation of Exponential Functions.  
A Lesson with Functions Decreasing Exponentially  Two basic problems for functions decreasing exponentially.  
Multiplying with Exponents  An introductory lesson for multiplication of terms with exponents.  
Scientific Notation with Positive Exponents 2  This lesson in Scientific Notation uses only Positive Powers of Ten.  
Scientific Notation with Negative Exponents 2  This lesson in Scientific Notation uses only Negative Powers of Ten.  
Scientific Notation with Positive and Negative Powers  A lesson in Scientific Notation discusses the difference between Positive Exponents and Negative Exponents.  
Scientific Notation, Positive and Negative Exponents  A lesson in Scientific Notation discusses the difference between Positive Exponents and Negative Exponents.  
Scientific Notation with Positive and Negative Exponents  A lesson where we multiply numbers in Scientific Notation with a little multiplication and a smidgen of addition.  
Scientific Notation with both Positive and Negative Exponents  We raise values expressed in Scientific Notation to both Positive Exponents and Negative Exponents.  
Scientific Notation and Operations  Mr. X does the HARD Long Division and the student gets the EASY Long Division.  
Scientific Notation Operations  We show all answers and work toward them. As we do we emphasize that reliance on a calculator may not be entirely helpful. 
General Topics
Title DescriptionAdding Like Terms  From a First Book in Algebra, 1895, by Wallace C. Boyden. Terms combine when they have the same variables to the same powers. 

Algebraic Notation  From A First Book in Algebra, Wallace C. Boyden, 1895. You need a good understanding of a variable, as x. 

Algebraic Notation, many terms  From A First Book in Algebra, Wallace C. Boyden, 1895. Take your time and understand the nature of a variable, as x. 

An Interesting Relationship, with Algebra  From our multiplication table, an interesting relationship between arithmetic and algebra. We observe that x²  1 = (x1)(x+1), or, x² = (x1)(x+1) + 1. 

Arithmetic within Functions  We may do basic arithmetic operations within and between functions.  
Basic Algebra Lesson 01  X is the Unknown  We begin with a variable representing a "fillintheblank" to make a statement true.  
Basic Algebra Lesson 02  X Fills in the Blank  Don't let the concept of a variable throw you. You just "fill in the blank" with what you already know. Each "x" is unique to a math statement. We do not use different values of x "in the same problem." 

Basic Algebra Lesson 03  X Can Be Anything  The value that x stands for can be any value. Typically, we seek the value for x the makes a statement true. 

Basic Algebra Lesson 04  Anything Goes  Variables can be negative, positive, irrational, rational... all kinds of things. Anything goes... Try to make the statement true. A statement that is always true is called an identity. 

Basic Algebra Lesson 05  One Good Idea Top to Bottom  Dividing a fraction with like coefficients in the numerator and the denominator simplifies to unity.  
Basic Algebra Lesson 06  Coefficients Add Efficiently  To do algebra easily and effectively, understand that coefficients add efficiently.  
Basic Algebra Lesson 07  Dr. X Can No Longer Operate  At some point we need to leave the symbol for the operation of multiplication behind, as "X" has become a variable, and not a symbol to multiply.  
Basic Algebra Lesson 08  X is Handy, In (or Out)  X is a handy thing, whether it is inside or outside of parentheses.  
Basic Algebra Lesson 09  Sum Groups are Worth Joining  Grouping terms according to the Associative Law is exceptionally easy to see.  
Basic Algebra Lesson 10  Ignorance of the Law is No Excuse  The Commutative Law and the Distributive Law are discussed.  
Basic Algebra Lesson 11  One Identity, One Additional Identity  Zero is the Additive Identity. One is the Multiplicative Identity. 

Basic Algebra Lesson 12  Inverses, Additive and Reciprocal  Discussed are necessary elements from arithmetic. Students should master arithmetic before embarking on algebra. 

Basic Algebra Lesson 13  Division and Subtraction Defined  Differences and quotients are explained with our most basic operations. We may define both subtraction and division as addition and multiplication, respectively. 

Basic Algebra Lesson 14  Rearrange the Furniture  The terms of a sum may be arranged in any order. The terms of a product may be arranged in any order. 

Basic Algebra Lesson 15  Cancel My Policy, Please  We may subtract the same value from each side of an equality and maintain the equality. We may divide the same value (a factor) from each side of an equality and maintain the equality. 

Basic Algebra Lesson 16  Negative Multiplication  When you have a single minus sign in a multiplication operation you're almost surely ending up on the negative side of the ledger.  
Basic Algebra Lesson 307  Appropriate for both Basic Algebra and Advanced Algebra, we solve two inequalities in one variable joined with the conjunction OR.  
Basic Algebra Lesson 308  This lesson is appropriate for both Basic Algebra and Advanced Algebra. We solve two inequalities in one variable joined with the conjunction AND. 

Basic Algebra Problem 302  This lesson is appropriate for the "end of a basic algebra course" as well as "the beginning of an advanced algebra course."  
Fast Lesson in Slope  This is a fast lesson in slope.  
Graphing Linear Inequalities 315  We make a quick sketch of two linear inequalities, then graph the region of intersection.  
Like Terms 18  From A First Book in Algebra, Wallace C. Boyden, 1895. Watch your minus signs. 

Multiplying Polynomials  When we multiply polynomials we sum individual products. We multiply each term in one polynomial times each term in the other polynomial. 

Solve Linear Eqns with Fractions 05  These problems are also found in Solve Linear Eqns. Problem Set 05. 

Solve Linear Equations 01  Practice with basic algebra at its finest. These skills are important to build upon for more challenging problems that come later. 

Solve Linear Equations 02  Good, basic practice with simple equations in x.  
The Ferry Problem  This little quadratic problem lends itself to Basic Algebra, Advanced Algebra, and the Calculus. In this video, we look at a simple spreadsheet to answer the question, "What fare should you charge to maximize your daily revenue for your ferry?" 
Inequalities
Title DescriptionProperties of Inequalities  An overview of the Inequality Properties Handout and some notions about onevariable inequalities. 
Inequalities
Title DescriptionProperties of Inequalities  An overview of the Inequality Properties Handout and some notions about onevariable inequalities.  
Compound Inequalities with the Conjunction "OR"  We solve two inequalities in one variable joined with the conjunction OR.  
Compound Inequalities with the Conjunction "AND"  We solve two inequalities in one variable joined with the conjunction AND.  
Inequality in One Variable with Absolute Value  We solve a single variable inequality that includes an absolute value.  
Square Roots and Absolute Values  A review of square roots and absolute values. 
Linear Equations & Inequalities
Title DescriptionFinding Slope from a Graphed Line Lesson 2  A lesson in Slope. We can read slope from a graph, or calculate it from given coordinates. 

Finding Slope from a Graphed Line Lesson 1  A lesson in Slope. We can read slope from a graph, or calculate it from given coordinates. 

Finding Slope from a Pair of Points Problem Set 1  Simple calculations for Slope using a pair of Ordered Pairs.  
A Fast Lesson in Slope  A visual demonstration and lesson on slope. We use Cartesian coordinates for this demonstration. 

Graphing Lines Given YIntercept and an Ordered Pair  Given two points, where one point is the yintercept, write the equation of the line as y = mx + b; and graph the line (segment) on the 10 x 10 grid.  
Graphing Two Linear Inequalities  What if we have two inequalities? In this lesson we address this scenario by graphing the intersection of two linear inequalities.  
Square Roots and Absolute Values  A review of square roots and absolute values. 
Linear Functions
Title DescriptionFinding Slope from a Graphed Line Lesson 2  A lesson in Slope. We can read slope from a graph, or calculate it from given coordinates. 

Finding Slope from a Graphed Line Lesson 1  A lesson in Slope. We can read slope from a graph, or calculate it from given coordinates. 

Finding Slope from a Pair of Points Problem Set 1  Simple calculations for Slope using a pair of Ordered Pairs.  
A Fast Lesson in Slope  A visual demonstration and lesson on slope. We use Cartesian coordinates for this demonstration. 

Graphing Two Linear Inequalities  What if we have two inequalities? In this lesson we address this scenario by graphing the intersection of two linear inequalities. 
Monomial and Polynomials
Title DescriptionNegative Multiplication  A Lesson on Handling Negative Multiplication with Algebraic Terms.  
Multiplying Polynomials  When we multiply polynomials we sum individual products. We multiply each term in one polynomial times each term in the other polynomial, then add the products. 

An Interesting Relationship, with Algebra  From our multiplication table, an interesting relationship between arithmetic and algebra. We observe that x²  1 = (x1)(x+1), or, x² = (x1)(x+1) + 1. 

Dividing Polynomials (Simplifying)  We show how being able to factor terms allows to divide polynomials. The result is akin to simplifying the rational expression. 

An Interesting Relationship  It turns out that x² = (x1)(x+1) + 1. We look at this algebraic relationship with the standard multiplication table. If you prefer, equivalently, x²  1 = (x1)(x+1). 

Dividing Polynomials with Long Division Lesson  Quotients and remainders result from these division problems where both dividend and divisor are polynomials in one variable. 
Monomials and Polynomials
Title DescriptionAn Interesting Relationship  It turns out that x² = (x1)(x+1) + 1. We look at this algebraic relationship with the standard multiplication table. If you prefer, equivalently, x²  1 = (x1)(x+1). 

Multiplying Polynomials  When we multiply polynomials we sum individual products. We multiply each term in one polynomial times each term in the other polynomial, then add the products. 

Negative Multiplication  A Lesson on Handling Negative Multiplication with Algebraic Terms. 
Quadratic Function
Title DescriptionGraphing a Parabola  Graph a Parabola As y = f(x), we have a simple quadratic that graphs to a parabola. We employ y = a(xh)Â² + k. 

Graphing a Sideways Parabola  We look at a "sideways" function with the relation x = f(y) instead of our usual y = f(x).  
Graphing Quadratic Functions  We solve for the zeroes (roots) of a polynomial by factoring and by graphing, with assistance from the good folks at desmos.com.  
Graphing a Simple Parabola  We walk through a simple sketch of a graph of y = xÂ²  5. The parabola is a conic section. 

Solving Roots of Quadratics by Completing the Square  We solve for the zeroes (roots) of a polynomial with C.T.S. (Completing the Square), by use of the Quadratic Formula, then by Factoring. No words, just fast writing and nice music. 
Radical Expressions
Title DescriptionSimplifying Square Roots  A few words about the simplifcation of radicals and square roots. Sometimes simpler is not so simple. 

A Lesson in Simplifying Radicals  Preaching the need to learn the Facts of Multiplication. Learn the facts of the Times Table. Please. 

Additional Examples with Simplifying Radicals  Simplifying radicals (square roots) is really a test of your mastery of basic facts of multiplication. Identifying the factors of the radicand will make it easy to work these problems. 

Detailed Examples with Simplifying Radicals  Simplifying radicals (square roots) is really a test of your mastery of basic facts of multiplication. Identifying the factors of the radicand will make it easy to work these problems. 

More Detailed Examples with Simplifying Radicals  Simplifying radicals (square roots) is really a test of your mastery of basic facts of multiplication. Identifying the factors of the radicand will make it easy to work these problems. 

Square Roots and Absolute Values  A review of square roots and absolute values. 

A Lesson in Multiplying Radical Expressions  Hard Problems: Now we are making our way from Algebra toward Advanced Algebra. We multiply binomials with radical expressions. 

Dividing Radical Expressions (Level: Medium)  Level: Medium. Simplify fractions by cancelling factors top and bottom. Make sure to rationalize the denominator. 

Dividing Radical Expressions (Level: Easy)  Level: Easy. We simplify fractions that contain radicals in the numerator and the denominator. We cancel out factors simultaneously from topandbottom. 

Dividing Radical Expressions (Level: Hard)  Level: Medium. Simplify fractions by cancelling factors top and bottom. Make sure to rationalize the denominator. 

The Distance Formula and the Pythagorean Theorem  We find the distance between two points in rectangular (or Cartesian) coordinates. We also look at the familiar Pythagorean Theorem for the relationship between sides of a right triangle. 
Rational Expressions
Title DescriptionDividing Polynomials (Simplifying)  We show how being able to factor terms allows to divide polynomials. The result is akin to simplifying the rational expression. 

How to Divide Rational Expressions  Dividing Rational Expressions is made easier by changing the operator to multiplication. Multiply the First Fraction by the Reciprocal of the Second Fraction. 
Systems of Equations
Title DescriptionSystems of Equations  Substitution Method  Isolate one variable by itself. Then rewrite the other equation with the Substitution.  
Systems of Equations  Graphing Method  When we graph two lines, the point of intersection of those lines is the simultaneous solution to both linear equations.  
Systems of Equations  Cramer's Rule  Cramer's Rule is a handy way to solve systems of linear equations. We use determinants of square matrices for the topandbottom of a fraction. 

Systems of Equations  Elimination Method  We may eliminate a variable by adding two linear equations with opposite coefficients on the same variable. So you "look ahead a bit" to figure out how to get those opposite coefficients. 

Solving System of Equations Example 2  We solve for the simultaneous solution to two linear equations. We find where the lines intersect. 

Solving System of Equations Example 1  We solve two linear equations and examine the graph of their intersection. 
Systems of Equations
Title DescriptionSystems of Equations  Substitution Method  Isolate one variable by itself. Then rewrite the other equation with the Substitution.  
Systems of Equations  Graphing Method  When we graph two lines, the point of intersection of those lines is the simultaneous solution to both linear equations.  
Systems of Equations  Cramer's Rule  Cramer's Rule is a handy way to solve systems of linear equations. We use determinants of square matrices for the topandbottom of a fraction. 

Systems of Equations  Elimination Method  We may eliminate a variable by adding two linear equations with opposite coefficients on the same variable. So you "look ahead a bit" to figure out how to get those opposite coefficients. 

Solving System of Equations Example 2  We solve for the simultaneous solution to two linear equations. We find where the lines intersect. 

Solving System of Equations Example 1  We solve two linear equations and examine the graph of their intersection. 
Trigonometry
Title DescriptionReciprocal Trigonometric Ratios  Quickly learn the pairs of reciprocal trig functions: sine and cosecant are reciprocal trig functions; cosine and secant are reciprocal trig functions; tangent and cotangent are reciprocal trig functions.  
Cofunctions  The names of the six basic trig functions make it easy to follow: sine and cosine functions are cofunctions; tangent and cotangent functions are cofunctions; secant and cosecant functions are cofunctions.  
The Basic TRIG Functions  While we have six basic trigonometric functions, in this lesson we concentrate on "the first three," namely, sine, cosine, and tangent.  
Tangent is Just Like Slope  We introduce x, y, and r around a circle centered at the origin. the radius of the circle is r, and points around the circle are the familiar (x, y). 

Inverse TRIG Functions  Inverse TRIG functions return an angle.  
The Distance Formula and the Pythagorean Theorem  We find the distance between two points in rectangular (or Cartesian) coordinates. We also look at the familiar Pythagorean Theorem for the relationship between sides of a right triangle. 

The Classic LadderAgainsttheWall  Every trig course includes at least one problem of a ladder of fixed length resting against a vertical wall.  
Pythagorean Relations with Sines and Cosines  We solve for the sine and cosine of an acute angle within a right triangle.  
Pythagorean Relations with Irrational Lengths  We solve right triangles with various lengths of the sides. The Pythagorean relation holds, where the sum of the squares of the legs (the perpendicular sides) is equal to the square of the hypotenuse (the square of the longest side). 

The Classic Ladder Before and After Moving the Base  Just as we always have a static ladder problem in every trig course, we always have a problem where the base of the ladder is moved in a second scenario. How far does the top of the ladder rise when we move the ladder's base a fixed distance toward the wall? 

The Classic Ladder Before and After Moving the Base  Alternate Approach  In this solution to the Ladder Before and After problem, we use a Pythagorean relation instead of a trig function to derive the answer.  
Sines and Cosines of Acute Angles in Right Triangles  We practice with irrational numbers as we find the sine and cosine of acute angles within right triangles.  
Area of a Triangle  There are many ways to calculate the area of a triangle. Trigonometry helps when the altitude of the triangle is unknown. We look at Heron's (Hero's) Formula, as well as a 3x3 determinant. 
Word Problems
Title DescriptionA Lesson in Consentrations  Parts per Million and Parts per Billion are explained in a way to let it soak in, as it were. It is a very fluid idea. 