
1 The Mistake Everybody Makes

2 Adding Algebraic Fractions

3 Multiplying Algebraic Fractions

4 Dividing Algebraic Fractions

5 Long Division

6 Fractions Within Fractions

7 The Difference (Or Sum) of Cubes

8 Properties of Equality

9 Solving Systems of Equations

10 Inequalities

11 Absolute Value

12 Exponents

13 A to the Zero

14 A to the Negative N

15 Roots

16 Equations with Square Roots

17 Fractional Exponents

18 Graphing a Linear Equation

19 Graphing Quadratics

20 Sequences

21 Series (Sigma Notation)

22 The History of Algebra
Susan Cantey has taught mathematics for over thirty years at the high school and college levels. She has been writing songs, mainly Christian themed songs for church, since the 1970's. A few years ago she realized that in addition to visual learning, many students could benefit from auditory learning. That thought birthed her two former CD's: The Tree of Calculus and PreCalc! This CD of Algebra songs and it's companion, Algebra Vol. 1, contain songs about almost every conceivable algebra topic. Vol. 2 covers some of the more advanced topics that you might see in a second year algebra course as well as the rest of the basics in exponents, systems of equations and graphing. Teachers can use these songs to liven up their classes. Parents can purchase them for their kids to augment studying for exams. There are many styles of music from rap to classical  something for every taste. Susan has worked hard to make the melodies catchy and the lyrics memorable. Hundreds of students are enjoying (and learning with) Susan's Calculus and PreCalculus songs. Algebra is even more important and ubiquitous. Join in the fun! Buy both volumes today. Lyrics: The Mistake Everybody Makes by Susan Cantey When distributing a negative, it becomes imperative To apply the sign to everything inside, Parentheses may mesmerize, but you've got to realize The negative applies to everything inside. Often it's forgotten and it messes up the problem, The mistake everybody makes. Don't you do it, don't you do it, Don't you do it, don't you do it, Unh Uh! Unh Uh! Don't you do it, don't you do it, Don't you do it, don't you do it, Unh Uh! Unh Uh! Negative five times the quantity 2x + 3y Equals negative 10x minus 15y, Negative 2 times the quantity x  y, Equals negative 2x plus 2y, Change all the signs, Parentheses is why, You'll remember if you try, Don't make your teacher cry. (chorus) When you see a negative sign, In front of an expression, Tell yourself you'll distribute it, 'Cause it's just like a negative one. There's a list of lotsa kids, yeah, you know what they did, More than once in their lives, they kind of lost their minds, Ben and Jen, Nate and Kate, Bobby, Freddy, they don't rate, Dave and Jane, Little Wayne; in the sun and in the rain, Why does every one forget to change the middle sign? You've done it, I've done it, Happens all the time, (chorus) (repeat first verse) (chorus) Adding Algebraic Fractions by Susan Cantey When adding algebraic fractions The basic rules stay the same, You'll need a common denominator, Some or all of them must change, Factor what you've got to start with, Write the ones the fractions share, Include any extra factors The LCD will be there. LCD, LCD, that's the key, the LCD, It's what you need, the special fact, Before you add or subtract. The second step in adding fractions Is multiply each term by one, But use the form that makes the denominators All the same when you're done, For example 1 over x Plus 1 over 2x, You multiply the first term by 2 over 2, And the second by x over x. (chorus) Least Common Denominator, That's what we you use, Then add the numerators, And you are through... Here's a harder problem with trinomials, Made up just for you, 1 over x  4 plus x over x  2, All you need is the extra factor x + 2, Now what you do, Is multiply the second fraction by x + 2 over x + 2 (chorus) Multiplying Algebraic Fractions by Susan Cantey When you multiply, Factor everything, Before you dare to cross out anything, Quotients of factors will Sometimes equal one, But never cancel part of a sum. X over 15 times 9 over x, x over 3 times 5 times 3 times 3 over x times x, x over x is one, so is 3 over 3, 3 over 5x is all that's left. (chorus) x  9 over x 1 Times x +3x  4 Over x + x 12, Four quantities to factor. (chorus) x + 3 times x  3 over x + 1 times x  1, x  1 times x + 4 over x + 4 times x  3, uh huh! (chorus) Here comes the happiest part of all, Crossing out factors is so much fun, Here's what will be left, x + 3 over x + 1. (chorus) Factor first; factor first, Don't forget to factor first! (chorus) Dividing Algebraic Fractions by Susan Cantey Dividing by 2 is the same as multiplying by one half, Dividing by 4 is the same as multiplying by one fourth, 'cause The rules never change, From arithmetic to algebra still, To divide, we just rearrange, Multiply by the reciprocal. If you can multiply, oh, then you can divide, It's really the same thing, And in algebra, it's just as easy as, This song is to sing, Turn the second fraction on it's head, Multiply by that instead, It doesn't take a genius, Rules aren't made from meanness, Everything in math makes sense. 12 over x divided by 9 over x, Equals 12 over x times x over 9, Now simplify that as you should, To 4x over 3, The same way that you would Do it ordinarily. (chorus) If trinomials are involved, The same algorithm will evolve, Flip and then factor or factor and then flip, It doesn't matter one little bit. (chorus) Long Division by Susan Cantey Dividing monomials is easy, Just use the Distributive property, 8x + 6x + 2x Divided by 2x will be, 4x + 3x + 1. Dividing by monomials is never worrisome. Everybody loves long division, Every boy and every fraulein, Don't be paralyzed by indecision, Just watch your plus and minus signs, Long division, long division, Just looks long upon the page, Long division, long division, No need to be afraid. X + 2x + 5x Divided by x + 1, oh yeah, let's just give it a try, x goes into x, x times, Now multiply x times x and 1, yeah that's right, Subtract x from x and x from 2x, Leaving x + 5x, now don't go away. X goes into x, 1x times, Now multiply, get x + x, oh that's just fine, Subtracting leaves you with 4x, x goes into 4x four times, that's what's next, 4 times x + 1 is 4x + 4, Subtract and negative 4's the remainder, Can't divide no more. (chorus) Fractions Within Fractions by Susan Cantey If you've got fractions within fractions, You'd better take some action, Don't you leave your answer in a mess, Find the least common denominator, Multiply the top and bottom; simplification at it's best. For instance, if you have all over What you need to do is Times it by 12 over 12 because That's what the common denominator was, 4 + 2 over 3 + 6 in a word, 6 over 9 or two thirds. (chorus) For instance if you have all over You've got to multiply by xy over xy, To cancel the x and the y, y + x over 2y + 3x, That's what you will get. (chorus) Fractions within Fractions don't look nice, Eliminate them, take my advice, Do it in one step, I'll show you how, Just multiply by one, Listen now... For instance if you have all over 2 + y, Here's what you must try, Multiply by 5x over 5x, Use the LCD, don't guess, 5 + x over 10x you see Plus 5xy it will be. (chorus) The Difference (or Sum) of Cubes Music by Anton Rubinstein / by Susan Cantey How do you factor the difference of cubes? a cubed minus b cubed, Here's what you do, a minus b times the following quantity, a squared plus ab plus b squared or secondly, If the second factor you should forget, Use long division to determine it instead, Or use some SOAP, You should use SOAP, Same, opposite, always positive, Same, opposite, always positive. How do you factor the sum of cubes? a cubed plus b cubed, Here's what you do, a plus b times the following quantity, a squared minus ab plus b squared or secondly, (chorus) x cubed plus y cubed will always be x plus y times the following quantity, x squared minus xy plus y squared, I hope you see, Never a "two" will there be. (chorus) x cubed minus y cubed will always be x minus y times the following quantity, x squared plus xy plus y squared, I hope you see, Never a "two" will there be. Properties of Equality By Susan Cantey Reflexive (x4) a = a, b = b, Reflexive (x4) Property of Equality a = a, b = b Symmetric (x2) If b = a then a = b (x2) Symmetric Symmetric Property of Equality There are 3 properties of equality. (x2) Transitivity, If a = b and b = c, Then a = c, If a = b and b = c Then a = c, Transitive property of equality, Property of equality. Solving Systems of Equations By Susan Cantey Solving systems of equations In two variables, There are 2 techniques to learn, Both are valuable. Linear Combinations, Sometimes called elimination, Is when you multiply One or both equations. You make the coefficient Of x or y in both equations, The same so that when subtracted, One variable's eliminated. (chorus) The other way is substitution, That's when you solve one equation, For x or else for y, And here's the reason, You can take that quantity, And subtract for x or y, In the other equation, And solve it easily then. (chorus) Instrumental Interlude (chorus) Inequalities Music by Leo Delibes / lyrics by Susan Cantey An inequality Is just like an equation, Except for one little thing, If you multiply, Or divide both sides By a negative number, You must turn over The inequality sign, Turn it over To face the other way, Except for this little detail, you see, Treat inequalities the same as equalities. An inequality is the same As any equality, The same game, You can add anything, Or subtract the same thing, From both sides, Multiply or divide. An inequality is just like an equation Except for just one little thing, If you multiply or divide both sides By a negative number, You must flip The inequality sign, Face it the other way, Turn it around, That's the only Thing that's different From an equation, otherwise, It's just the same as solving an equality, Add or subtract the same quantity, Multiply or divide both sides... That's all there is! Absolute Value by Susan Cantey Absolute value has three definitions, They are all useful, The first one is the default definition, The distance from zero to x on the real number line, Or the square root of x squared, x if x's positive, opposite of x if x is negative. Absolute Value, Makes everything positive, Like super strong parentheses, It's often what you need, Absolute Value. The absolute value of a quantity, You've got to think this way, What's inside could be positive already Or it could be a negative just as easily, So use a plus or minus, Two equations find us, Two answers for x in the problem as you'll often see. (chorus) Instrumental bridge (chorus) Exponents by Susan Cantey Exponents, exponents, gotta know the rules, But just memorizing simply won't do, 'Cause you'll forget when the summer's thru, You've got to understand exponents too. X squared means x times x, x cubed means x times x times x, The exponent tells you how many times You multiply x by itself. Now think what it means to multiply two powers, x squared times x cubed is just what it seems, x times x and then x times x times x, How many x's are there now? Count them, take your time, Two plus three is five. (chorus) Now put x squared in parentheses, And raise that to the power of three, x squared times x squared times a third x squared, That makes, Count them, yes that's it, x to the sixth. Do you see the pattern, do you? x to the ninth times x to the third, Nine plus three, get x to the twelfth, x to the third raised to the ninth Is x to the 27th, multiplied. Don't just guess; think what's meant, When in doubt, write it all out. (chorus) A to the Zero By Susan Cantey Some are mystified, By a to the zero, don't know why, Others are confused, Cause they don't know what they should do, But you Can be a hero, If you know a to the zero Is always one unless a's zero itself, It's always one, never anything else When you do division With some powers of a, Like a cubed over a squared, Do it this way, over, Cancel 2 a's, that's what to do, In powers of a it's 3 minus 2, a to the one, it's true. (chorus) Now take it one step further, When all is said and done, a squared over a squared Has to equal one, But if you follow the pattern, The power is 2 minus 2, That gives up a to the zero, Now you know it's true. (chorus) But be careful, There's an exception, 0 to the 0 is undefined, This is because, It's illegal, To use 0 to divide! (chorus) A to the Negative N by Susan Cantey 3 is 27, 3 is 9, 3 is 3, 3 is 1 Notice that each number is one third smaller So 3 What should it be? Yes, it's one over three! 3 's 1/9 A negative exponent means you must divide. A to the negative n is one over a to the n, One over a to the negative n is back to a to the n again, la la, la la la, a to the negative n. 2 is 8, 2 is 4, 2 is 2, 2 is 1 Notice that each number is one half smaller So 2 What do we do? Yes, it's one over two! 2 's 1/4 A negative exponent means you must divide. (chorus) When you see a negative exponent, Just think "one over..." Make the power positive, But put it in the denominator. (chorus) Roots by Susan Cantey Roots, square roots, Cube roots and fourth roots, nth roots, The key is to factor What's under the radical, And group them by n. For example: the cube root of 8x, There is one group of three two's, There are 2 groups of three x's, So the answer is, it's true. (chorus) For example: the fourth root of 81y, There is one group of four threes, One group of four y's with one left over, So the answer is 3y times The fourth root of y. (chorus) Factor completely, Group the factors by n, Every group inside Makes one outside, Amen. (chorus) Equations with Square Roots Music by Wolfgang Amadeus Mozart / lyrics by Susan Cantey When there is a square root in an equation, Isolate the square root on one side of the equal sign, And if there is more than one square root, You may have to repeat the method one more time. Root x minus 5 equals one, Add 5 to both sides, Root x equals 6, Yes it is, All you need to do is square both sides, And you will get x equals 36. Root x plus 1 equals 10, Subtract one from both sides, Root x equals 9, One more time, All you need to do is square both sides, And you will get x equals 81. When there is a square root in an equation, Isolate the square root on one side of the equal sign, And if there is more than one square root, You may have to repeat the method one more time. Fractional Exponents By Susan Cantey 8 plus 8 equals 16, we say, That's why ½ times 16 is 8, 4 times 4 is also 16 as you know, So 16 to the ½ power will equal to 4. Fractional exponents, oh yeah, Fractional exponents, oh yeah, They are to multiplication as Multiplication is to addition, Fractional exponents, oh yeah. 9 plus 9 plus 9 gives us 27, So 1/3 of 27 is 9 and not 10, 3 times 3 is 27, you see, So 27 to the 1/3 power is equal to 3, (chorus) The exponent 1/2 is the same as square root The exponent 1/3 is the same as a cube root 81 to the ¼ power is equal to 3, 'Cause to make the product 81 it's 4,3's you need, 1 million to the 1/6 power is gonna be ten, 'Cause 610's multiplied together makes one million. (chorus) (chorus) Graphing a Linear Equation Traditional Gaelic Melody / lyrics by Susan Cantey Graphing a linear equation, y = mx + b, that's what you need, Put a point on the yaxis, That's the yintercept corresponding to b. Now for a second point on the line, Use the m in front of the x, In fractional form this is the slope, It will guide you to the point you need next. There's exceptions, this I know, y = b has zero slope, That's a horizontal line, y equals b all the time. Vertical's another case, Look for x = a, Now y can be whatever it wants, The line goes straight up and never stops. If the slope is a positive fraction, Move up from the yintercept using the numerator, Then move to the right parallel to the xaxis, Using the number in the slope's denominator. But if the slope is a negative fraction, Then move downwards instead of up, Still going right with the denominator, Connect the two points, with a ruler if you must. Graphing Quadratics Music by George Minor / lyrics by Susan Cantey Graphing Quadratics, graphing quadratics, They're always parabolas, always parabolas, Graphing quadratics is easy as can be. Quadratics, quadratics, Parabolas, parabolas, Graphing quadratics is easy as can be. If there are two zeros, place them on the xaxis, The vertex is always half way in between, x equals negative b all over 2a, That's where the vertex will always be. (chorus) If a is negative, Then parabola is upside down, If a is negative, The graph looks like a frown. If there's only one, one real zero, Then that's where the vertex certainly will be, No real zeros? Then just use the vertex, And plot a few more points, only two or three. (chorus) Sequences By Susan Cantey A sequence is an ordered list of numbers Separated by commas, In Math there's generally a rule That tells you what to do, A sequence is an ordered list of numbers 1, 11, 21, 31, is an arithmetic sequence, 'Cause to get to each next term We always add on ten, 6, 10, 14 etc. Is also arithmetic, Since we're adding 4 each time, 4 is called the common difference. (chorus) 2, 4, 8, 18, 32, is a geometric sequence, 'Cause to get to each next term you Must multiply by 2, 12, 4, 4/3, 4/9, is geometric also, This time multiply by 1/3, Called the common ratio. (chorus) Geometric, Arithmetic are the main types, But you may see other kinds in your life. (chorus) Series (Sigma Notation) by Susan Cantey Sigma Notation isn't all that complex, Just needs getting used to like anything else, The number at the bottom tells you where to start, Then substitute until you use the number at the top. I'm not going to bore you with all those Formulas you need to know, So write them down and do your best To learn them before the test. (chorus) Infinite Series can be very clever, Adding terms together forever, In algebra there's only one you can do, Geometric when one's greater than r's absolute value. (chorus) Sigma means add, just add. Geometric series using one half to the n Equals 1/2 + 1/4 + 1/8 and so on, It's surprising this infinite sum Is equal to one. (chorus) The History of Algebra by Susan Cantey It's no mystery, it's just history, In Babylon and China and in Alexandria, Letters of the alphabet in Mahavira's manuscripts, The father of algebra is Musa alKwarizmi. AlKaraji made us free from laborious geometry, Jia Xian polynomials, equations quite phenomenal, Cardano, Bombelli, Leibniz and Seki, Abel and Galois, all advancing algebra. (repeat first stanza) Six hundred BC in India, Five unknowns, equations linear, Three Hundred BC, Euclid writes in Greek, Two hundred AD in Alexandria, Diaphantus thinks he's the father of algebra, Eight hundred AD, there's an awakening, Algebra explodes, Starts accelerating. Algebra was finally translated into Arabic, The Mid East was elated. Musa alKwarizmi writes for the Arab Nations, Balancing, completing every calculation, Quintics to the east in India, Being solved by Mahavira, Finally, Persian Al Karaji Used arithmetic instead of geometry, Twelve hundred AD algebra finds Europe, Fibonacci is the white man's hope, Europeans all go crazy, Algebra from Al Karaji! It's no mystery, it's just history, It's no mystery, it's just history...