Susan Cantey has rounded out her collection of Math Songs with the release of 2 CD's full of Geometry Songs. The listener will find a smorgasbord of song style from classical to Spanish guitars to in your face rock and roll. Mathematics teachers will have music pertaining to almost any topic. These songs will add joy and life to almost any lesson. According to Lynell Burmark in "They Snooze You Lose": "Music connects to 39 places in the brain." She also states that music (as well as images) is a sure fire way to "break down any resistance to your message so that your content can flow straight into their long term memory." Students who purchase these songs will have a complete venue for reviewing their formulas and theorems. The lyrics areprinted below. Sing along and learn your math the fun way. Enjoy! (Be sure to check out Algebra Vol. 1 , Algebra Vol. 2, Pre-Calc!, and The Tree of Calculus) Lyrics to Geometry Vol. 2: Hero’s Formula Susan Cantey © 2011 There was a hero, who lived long ago, A hero who lived long ago, He wrote a formula to compute the area, Of a triangle, a triangle… Here comes hero formula, Based on the semi perimeter, s is half of the quantity a + b + c Area is the square root of s times s – a Times s - b times s – c, Oh yes indeed! First century in Alexandria, In Egypt long, long ago, A mathematician and engineer, Made many many things in his career, What a career! Here comes hero formula, Based on the semi perimeter, s equals half of the quantity a + b + c Area is the square root of s times s – a Times s - b times s – c, Oh yes indeed! Surface Area Susan Cantey © 2011 Have you ever seen a stamp? Well just think of that, Paste stamps onto something flat, Make no gaps or overlaps, Then count those stamps, Area is just like that. A cube has six flat surfaces, They are called its faces, Each one has the same area fixed, So measure one and multiply by six, Every flat surface of every solid, Has its own surface area. (chorus) Rectangular solids, each surface has a match, Left and right, top and bottom, front and back, Add them up to get the total surface area, Then move on to more complicated figures, A sphere’s surface is 4 pi r squared, But it takes calculus to get your there. (chorus) Cylinders have two circular bases And rectangular lateral faces (chorus) Volume Susan Cantey © 2011 In its very basicness, Volume makes perfect sense, Little cubes in a container, How many cubes make a single layer? The next question is How many layers fit? Fill up all of the room, Count the cubes, Then sing this tune. Every volume is 3-D, Length times width times height ‘twill be, In every formula you use, Multiplying’s what you’ll do. A rectangular solid is a box, Even kids in bobby socks, Know V is lwh, At a very early age, Cylinders are nice and round, Here’s how the volume’s found, One layer is pi r squared, The height is how many layers are there. (chorus) Volume, volume, how many cubes, Fill the container with exactitude, Count them, count the, when you do, The volume is what you’ll come to. Area of the base times height, Is another formula you might Like to use when in a pinch, Go right ahead, if you wish, If the sides don’t go straight up, If the shape has a pointed top, Area gets half a cut, But a cone or a pyramid’s a third as much. (chorus) Measurement Susan Cantey © 2011 Measurement, measurement, Linear, squared or cubed, Measurement, measurement, A ruler’s only one tool we use, Measurement comes in two systems, Standard and metric, The first one came from the kings of old, The second one’s more copacetic. Twelve inches make a foot Three feet make a yard, 5-2-8-oh feet in a mile, Why’d they make it so hard? Sixteen ounces in a pound, Two thousand pounds in a ton, You may have a headache Before you’re done. (chorus) The metric system is based on ten, Ten millimeters make a centimeter, One hundred centimeters make a meter, And a thousand meters in a kilometer, A thousand grams make a kilogram, A thousand milligrams a gram, All the world uses the metric system, But not all Americans can. (chorus) Sixteen ounces is a pint, Two pints in a quart, A bushel equals thirty two quarts, Four quarts in a gallon of course, I’d rather use liters and milliliters. (chorus) Tessellations Susan Cantey © 2011 Tessellate, tessellate, what shapes tessellate? Tessellate, tessellate, what shapes tessellate? Put them all together; cover all the area, If there’s no gapping and no overlapping Then the shapes tessellate, tessellate, tessellate. The shapes tessellate, tessa…tessa… tessellate If only one regular polygon is used, The tessellation is called regular too, Squares and hexagons are a snap, Equilateral triangles don’t leave gaps, Their interior angles happen to be Factors of 〖360〗^°. (chorus) If at every vertex the same pattern’s formed, Then the tessellation is “uniform”, If it uses more than one regular figure, It’s called semi-regular, You may have heard of the greatest tessellator, The Dutch artist M. C. Escher. (chorus) Constructions Susan Cantey © 2011 Constructions are the heart of Geometry, An ancient art from antiquity, What you can do is truly wondrous, With just a pencil, a ruler and a compass. Arc, arc, make your mark, Draw it lightly not too dark, Use the arcs’ intersections, Make appropriate connections, Always think of why it works, Concentrate, stay alert, Understanding, you’ll remember, Constructions, forever. __ Begin with line segment XY, Place the compass point on X and draw a nice big arc, Now put the compass point on Y keeping the same radius, And draw another big arc, And connect the intersection points. What you have made is a perpendicular bisector. Next we’re gonna bisect an angle, Place your compass point on the vertex And draw an arc across the angle, Now put your compass point on each of the intersections, Make two arcs in the middle of the angle, Now connect the vertex to where the arcs intersect And that line, the angle will bisect. (chorus) Coordinates Susan Cantey © 2011 (Music – Gavotte by George Frederic Handel) The coordinate plane, invented by Descartes, Was intended to replace geometry, Descartes was trying to express Every shape algebraically. The x-axis is a number line, y-axis, a number line perpendicular, Start at the origin, move left or right, Then move up or down for y. An ordered pair represents each vertex Of any polygon you see, In fact, any figure geometry delivers Can be expressed in x, y and z. Distance Formula Susan Cantey © 2011 The distance formula’s an application Of the Pythagorean Theorem, Which as you recall Isn’t hard at all, So let’s proceed, Find distance “d”. Plot two points on the coordinate plane, x one, y one, is the first one’s name, x two, y two’s the second name we use, The distance between them The hypotenuse. (chorus) One leg’s the difference: x_2-x_1, The second leg measures: y_2-y_1, Square and sum those quantities, And take the square root, Distance “d” ‘s the length Of the hypotenuse. (chorus) d = the square root Of the quantity x_2-x_1 in parentheses Squared… Plus y_2-y_1, also in parentheses Squared… The distance formula’s an application Of the Pythagorean Theorem, Which as you recall Isn’t hard at all, Yes, it’s easy to find Distance “d”. Midpoints Susan Cantey © 2011 Given two points on the coordinate plane, How do you locate the point half way between? The point we call the midpoint? A right triangle might help us see. Middle, average, mean, Half way in between, x_1+x_2 divided by two, Comma y_1+y_2 divided by two, Ah…. What number is half way Between x_1 and x_2 ? The same thing can be asked Concerning the two y-values. (chorus) There is a connection Between geometry and algebra, The man who made this possible Was Rene Descartes. (chorus) Slope Susan Cantey © 2011 A horizontal line doesn’t rise or fall, It won’t go up or down at all, We call that zero slope. A line that goes up from left to right Has a positive slope if we generalize. Any line that goes downhill Has a negative slope and always will. Rise over run, Rise over run, That’s the layman’s definition, Rise over run, Rise over run, Units up or down divided by Units going left or right. On the Cartesian Plane, A formula will help us name The slope of any line, y_2-y_1 over x_2-x_1 , Rise over run, That’s how it’s done, Unless we find a vertical line, Then the slope is undefined, No slope can be assigned. (chorus) Linear Equations Susan Cantey © 2011 (Music – Symphony in B flat (second movement) by Franz Schubert) Equations of lines come in many forms, The favorite one is this: y = mx + b, which is called Slope intercept, you may recall. In calculus you’ll find Point/slope is used all the time, y-y_1 equals “m” times The quantity -x_1 , And last but not least Standard form: Ax + By = C Another form which isn’t used much That I like a lot Is the intercept form, Take the standard form And divide both sides by C, Reduce the coefficients, And place them underneath, When you are done You’ll see this phenomenon Underneath the x The x-intercept, Underneath the y The y-intercept, Which makes graphing the line So sublime. Parts of the Circle Susan Cantey © 2011 (Music – Symphony Pathetique by Peter Ilyich Tchaikovsky) Each point on a circle is the same distance From the center point, Any line segment that goes across the circle Is called a chord and if that chord goes through The center of the circle, Then that chord is called a diameter. The distance from the center to any point On the circle’s called the radius, And the distance around the circle’s Called its circumference, If you only take a part, You have an arc. Don’t confuse a circle with the area it encloses, That’s called a disc, And each slice: a sector. Circles are Magical Susan Cantey © 2011 Circles are magical, Drawing them gives me chills, Perfect in every way, Round and round they go. Circles are magical, They make my heart grow still, Contemplating theorems Few people know. Like the measure of inscribed angles Are half of their central angles, Like a triangle inscribed Using the diameter will be right. The tangent line in particular, To the radius, Is perpendicular. (chorus) The product of pieces of intersecting chords Are always equal of course, Secant lines from the same exterior point are Not so inferior, “a” times the sum “a + b” Is the same as “c” times “c + d” (chorus) Ah…. Transformations Susan Cantey © 2011 Transformations, Transformations, Translations, reflections, Rotations, dilations Isometries are transformations That create congruent figures, Translations, reflections And also rotations (chorus) Three kinds of reflections Are quite popular, x-axis, y-axis And y = x (chorus) To get a translation Also known as a slide, Add a constant To x and/or y (chorus) Dilations make figures That are similar, Make the shape smaller, Or make it bigger (chorus) Vector Susan Cantey © 2008 An arrow has a fixed position, a single photograph, A vector is allowed to move, change its stop and start, It must not change its length or the angle that it makes, It must maintain its essence, the same path delineate. A vector is a directed line segment, Use an arrow to show the way it went, Its length is called its norm, Standard position’s when it’s drawn, Emanating from the origin. Force or distance traveled represented by the length, Direction is the angle, along which the object’s sent, When you add them the resultant goes straight from start to stop, The short cut as the crow flies that gives the same result. (repeat chorus) Standard position, “a” comma “b”, Length is the square root, “a” squared plus the square of “b”, Direction: tangent inverse of “b” over “a”, But you might need to add pi, To get the quadrant right! (repeat chorus) SOHCAHTOA Susan Cantey © 2011 Sohcahtoa was an Indian chief, He was very brave, but he had bad feet, After the hunt they’d hurt him, so he’d go Down to the river to soak his toes. Chief SOHCAHTOA, Chief SOHCAHTOA, Chief SOHCAHTOA! Sohcahtoa was very shy, It worried the tribe that he had no wife, He needed an heir, so they chose a bride for him, Made her a teepee from buffalo skins. (chorus) Years passed, (Chief Sohcahtoa), No child, (Chief Sohcahtoa) Die cast, (Chief Sohcahtoa), Second wife (Ugh! Ugh! Ugh!) Second wife’s teepee was made from deer skin, Time went on, still no baby making, Then Sohcahtoa fell in love! Made her a teepee from hippopotamus. (chorus) Nine months later, a miracle! Wife number three birthed a boy and a girl, Wife number one had a little boy, Wife number two, a girl, oh what joy! (chorus) That’s when (Chief Sohcahtoa) Wrote the Theorem (Chief Sohcahtoa) Known as Pythagorean: (Chief Sohcahtoa) The squaw of the hippopotamus is equal to The sum of the squaws of the other two hides. Sine is the opposite over hypotenuse, Cosine is adjacent over hypotenuse, Tangent is the opposite over adjacent. (Sohcahtoa, Sohcahtoa, Sohcahtoa) The Law of Sines Susan Cantey © 2011 (Music – Ballet Music from Rosamunde by Franz Schubert) Draw a triangle, arbitrarily, Label the angle on the left big “A”, The one on the right, call it big “B” Corresponding opposite sides little “a” and “b” Now drop an altitude, call it “h” Using the two right triangles formed h = a sin B which equals b sin A Now divide by “a” times “b” sin B over “b” equals sin A over “a” Since the orientation was arbitrary The ratio also equals sin C over “c” In any given triangle, The ratio of the sine to the opposite side Will remain the same. (instrumental interlude) Given any triangle, the ratio of the sine Of an angle divided by the length of the opposite side Is the same for all three possibilities: sin A over “a” will equal sin B over “b” which equals sin C over “c” That’s the way it will always be. Law of Cosines Susan Cantey © 2011 The Law of Cosines can be proved Using the Cartesian Plane as a tool Along with trigonometry, That’s far beyond plane geometry, So I guess you’ll have to wait, Waiting is your fate. Oh, it’s useful, (useful!) So here goes, (go!) The Law of Cosines’ Good to know, (know!) a^2=b^2+c^2-2bc cosA (Or another way) b^2=a^2+c^2-2ac cosB Don’t you see? The choice of letters is arbitrary. The Law of Cosines can be proved By splitting a triangle in two, Employing the base and the altitude, This is something you CAN do Pythagoras, substitution, That will get the solution. (chorus) Proofs Susan Cantey © 2011 (Music – Polovtsian Dances by Alexander Borodin) In Geometry, you must write proofs in two columns, And you are expected to be clairvoyant, But most proofs are written in paragraph form, Writing in two columns is just plain wrong. Before you start, An idea must be hatched, You should explore the facts, Before you act, You will need a plan, A plan of attack, First try to understand, Then write your draft. Direct proofs start out with what is given, Which leads to this and that and then to the conclusion, Indirect proofs start with the conclusion’s negation, Then slowly lead the reader to a contradiction. (chorus) Proofs can never leave any holes, And they are challenging for both young and old. Non Euclidean Geometry Susan Cantey © 2011 Geometry as taught in school’s Not the only kind, it’s true, Euclid thought the world was flat, But now we know better than that. Non Euclidean Geometry, Here’s how it goes! Instead of drawing on a plane, Use a sphere or a bowl. On a sphere we redefine, What we mean by a line, A great circle will suffice, Each pair of lines intersect twice. (chorus) Draw a triangle on a sphere, A strange thing happens here, The sum of the angles will exceed One hundred eighty degrees. (chorus) (short instrumental interlude) Draw inside a bowl, that’s hyperbolic, Given a line and a point not on it, Through that point, we have this time, Infinitely many parallel lines. (chorus)