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 are printed 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 for Geometry Volume 1: Geometry Susan Cantey © 2011 Shapes are everywhere, From circles to squares, Even the planets Follow elliptical orbits, Geometry! Computer graphics, Aerodynamics, Swimming pools, And every tool, Geometry! Building, making old or new, What’s the math that’s often used? Geometry! Make kitchen cabinets, Or design a new dress, Build the interstates, Are there walls you want to paint? Geometry! Building, making old or new, What’s the math that’s often used? Geometry! Geometry is used by every person, In the home or in their profession Building, making old or new, What’s the math that’s often used? Geometry! Geometry! The Dimensions Song Susan Cantey © 2011 We live in three dimensions, I hesitate to mention There are other dimensions, Journey with me and see. Zero dimensions is a singularity, Just a point without any size, (no width at all – it’s infinitely small) One dimension is a line that stretches on, Left and right or up and down. (a line – straight line) Two dimensions is plane to see, A flat surface as thin as it can be, East, West, North and South, We can draw on the surface of a plane. We live in three dimensions, I hesitate to mention There are other dimensions, Journey with me and see. In three dimensions we can move around, East or West, North or South, Up or Down, Some people think the fourth dimension’s time, We move through time. We live in three dimensions, I hesitate to mention There are other dimensions, Journey with me and see. One Dimensional Figures Susan Cantey © 2011 (Music – Dance of the Sugar Plum Fairy by Peter Ilyich Tchaikovsky) One dimension is a line, you know, Left or right, up or down, But only one way, Or a line segment, just a little piece, Little piece of a line with two endpoints, Or a ray which is half of a line With one point, just one point, Attached to one end, It’s a mystery when you cut a line in half, What happens to the point in the middle? Maybe you get half a line And a ray, and a ray. One dimension is a line, you know, Left or right, up or down, But only one way, Or a line segment, just a little piece, Little piece of a line with two endpoints, Or a ray which is half of a line With one point, just one point, Attached to one end. Two Dimensional Figures Susan Cantey © 2011 (Music – March from Egmont Overture by Ludwig Van Beethoven) Triangles have three sides, Quadrilaterals have four sides, A pentagon has five, Hexagons are next And they have six sides, yes. Seven sided heptagons, Eight sided octagons, Nonagon, decagon, These are all polygons. Polygon means many sides, I guess you’ve realized, Two dimensional figures Also come with curved sides, Circles as well as polygons, All their names must be memorize, Two dimensional figures, all are well defined.* Quadrilaterals have four sides, And they are categorized by parallel sides, Two sides parallel is a trapezoid, Two pairs parallel is a parallelogram, Four right angles make a rectangle, Four equal sides, a rhombus, Four equal sides and angles will make a square. A kite is also defined where Two sets of adjacent sides are equal. These are the common quadrilaterals That you’ll learn about in school. Every day in some way Quadrilaterals are used. (repeat from start to *) Three Dimensional Figures Susan Cantey © 2011 A rectangular solid has six sides (called faces), All of which are rectangles (hence the name), A cube is a rectangular solid in which Each face is a square (yes, that’s it), A prism is quite similar (go on), Except the ends can be any polygon (any polygon), Pyramids can have any polygonal base, But triangles for every other face (that’s their way). Three D shapes have length and width And they have height too, You will see them everywhere Not just in school, Everywhere you look, Coffee cups and books, Ice-cream cones and volleyballs, You have seen them all. A cylinder has two circular bases (circular bases), And a lateral side but no faces (no faces), Only one base and a point at the top, Then the solid is a cone (that’s what we thought), Then there is a sphere or a ball (it’s round), Cut them in half and two hemispheres are formed (profound), If you tilt solids sideways The volumes will not change (and why should they?) (chorus) Tetrahedron! Tetrahedron! Four triangular sides Polyhedron! Polyhedron! N-polygonal sides (chorus) Angles Susan Cantey © 2011 A circle has 〖360〗^°, Why this is depends on what you believe, The diameter cuts a circle in half, It’s a straight angle, 180 in fact, A straight angle looks like a line But there’s a vertex in the middle every time. Acute, obtuse, right and straight, Reflex angles, those are great, When you know the names of each, Angles are within your reach. A right angle is 〖90〗^°, Like the floor and the wall in the place where they meet, An acute angle is smaller than that, An obtuse angle is kind of fat, Acute angles are between zero and ninety, Obtuse angles between ninety and one eighty. (chorus) Two angles that add to 〖90〗^° are complementary, Two angles that add to 〖180〗^° are supplementary. (chorus) Perpendicular Susan Cantey © 2011 (Music – Nocturne Op. 55, No. 1 by Frederic Chopin) What does perpendicular mean? What would the definition be? When lines are perpendicular They meet at 〖90〗^°. Where the wall meets the floor, Need we say more? That’s perpendicular, That’s perpendicular, Or right angles if you prefer. What does perpendicular mean? What would the definition be? When lines are perpendicular They meet at 〖90〗^°. When driving round the town, It is usually found, That most intersections, Are perpendicular Phenomenons. What does perpendicular mean? What would the definition be? When lines are perpendicular They meet at 〖90〗^°. Geometric Mean Susan Cantey © 2011 The arithmetic mean, Something I’m sure you’ve seen, a + b divided by two, The geometric mean Is also easy to do, Multiply a times b, Then take the square root. If y over m equals m over z, Then m is the geometric mean Of y and z Right triangle with altitude h Dividing the hypotenuse, Segment x next to a leg, Call that leg a if you choose, Then a is the geometric mean Of x and hypotenuse c, I bet you can prove it quite easily. If y over m equals m over z, Then m is the geometric mean Of y and z If y over m equals m over z, Then m is the geometric mean Of y and z Every Triangle Has Three Sides Susan Cantey © 2011 Every triangle has three sides, Line segments joined at their ends, And every point where they connect, Is called a vertex, my friend, One side is the base, And in every case, The altitude goes straight up, The height you might say, At right angles to the base, Stops at the vertex on top. A median connects the vertex To the midpoint of the opposite side, All three medians meet at the center Called the centroid, inside, The intersection of the angle bisectors Is called the incenter, The intersection of the perpendicular bisectors Of the sides is the circumcenter, The intersection of the altitudes Is the orthocenter. (chorus) In an equilateral triangle The centroid and the incenter, The circumcenter and the orthocenter are all the same point. (chorus) Triangle Property Susan Cantey © 2011 (Music – Pizzicati from Sylvia by Leo Debiles) (long instrumental introduction) The Triangle Property will help decide If a triangle exists or if it’s just a lie, Each pair of sides must be long enough to fit, The relationship between the sides goes just a little bit like this: The longest side, we’ll call it “a”, Should always be less than The sum of b + c, the other sides, oh yeah, Also if you take the shortest side, Let’s call it “c”, It will be greater than a – b Every pair of sides simply has to stretch Across the other side with room to spare, and yet, A lot of people can be fooled, for example, 1 and 2 and 3 simply will not work, But 3 and 4 and 5 is quite a famous set. So if you’re ever faced with some possibilities, Add the smallest sides and see If bigger’s what you get. (repeat from start) Congruent Triangles Susan Cantey © 2011 Congruent triangles, congruent triangles, Have the same shape and size, Pick one up, you’ll realize, It will fit right on top Of the other one, now stop, And think about what attributes Will be needed to prove Congruent triangles, congruent triangles. If all three sides are equal, yes, There’s congruency by SSS, Given 2 sides and the angle between them, SAS is a second theorem, Angle side angle is also good, ASA is the third and last rule. (repeat from start) Some folks think that double A S Is another theorem and that’s nice, But once you have two angles The third one follows, So ASA will suffice. (repeat chorus) Similar Figures Susan Cantey © 2011 (Music – The Happy Farmer by Robert Schumann) Congruent figures have the same size and shape, But similar figures have a different way, Corresponding parts must be proportional, One figure will be big, the other one small, Expand or shrink it, Round or linear, And you have created A figure similar For instance take a 3 – 4 – 5 right triangle, Now double all the sides but preserve the angles, What you have created is the same shape only bigger, What you have created is a similar figure. Expand or shrink it, Round or linear, And you have created A figure similar Find out if they’re proportional, Corresponding sides should give the very same ratio. Expand or shrink it, Round or linear, And you have created A figure similar Corresponding parts must be proportional, One figure will be big, the other one small. Types of Triangles Susan Cantey © 2011 Triangles are classified By their angles or their sides, Equilateral, all sides equal, Isosceles, two sides equal, Scalene, that’s the name, When no two sides are the same…scalene! If one angle’s over 〖90〗^°, Then call the triangle obtuse, please, Right triangle means One angle of 〖90〗^°, An acute triangle is what you’re finding If all the angles are less than ninety. (repeat from start) Pythagorean Theorem Susan Cantey © 2011 When two sides of a triangle meet At an angle of 〖90〗^°, These two sides are called the legs, The longest side - hypotenuse (that’s correct). Label the legs “a” and “b”, Label the hypotenuse with a “c”, a^2+b^2=c^2 a^2+b^2=c^2 Right triangles when you see them, Always use the Pythagorean Theorem. a^2+b^2=c^2 a^2+b^2=c^2 When all three sides are whole numbers, That’s so sweet, ai carumba! Three such measures are called triples, 3 – 4 – 5 is the very simplest. In this case 5 is c, The legs are labeled 4 and 3. (chorus) There are many famous triples, Such as 10 and 8 and 6, 5 – 12 – 13 is also nice, As is 7 – 24 – 25, Many more like this exist, Try to find them, Make a list. (chorus) If you need to find a leg, Use subtraction instead, b^2=c^2-a^2 a^2=c^2-b^2 Right triangles when you see them, Always use the Pythagorean Theorem. a^2+b^2=c^2 a^2+b^2=c^2 Special Right Triangles Susan Cantey © 2011 There are two special right triangles, 30 – 60 – 90, 45 – 45, There are two special right triangles, 30 – 60 – 90, 45 – 45 30 – 60 – 90 is half an equilateral, The short leg is half the hypotenuse, Root three times the short leg equals the long leg, Which the Pythagorean Theorem helps us deduce. (chorus) A 45 – 45 – 90 is isosceles, Pythagorean Theorem to the rescue! x^2+x^2=h^2 Hypotenuse is the leg times root two. (chorus)* Of all the things you learn In geometry, These triangles will return, In every course you ever see… (chorus) (repeat from start to *) How Many Degrees? Susan Cantey © 2011 When you lie awake in the night, Turning triangles over in your mind, Colors swirling round you like rainbows, You won’t rest until you know… Tear the corners off any triangle, Put them all together and you will see, They form a perfect line every time, Their angles always add to 〖180〗^°. In the night time when you cannot sleep, Questions into your consciousness creep, Hexagons and pentagons around your bed, Twirling, spinning, moving in your head. (chorus) (short instrumental interlude) A quadrilateral can be split into two triangles For a total of 〖360〗^°, A pentagon can be split into three triangles For a total of 〖540〗^°, Hexagon – four triangles, Septagon – five triangles, Every n-gon it seems, n – 2 times 〖180〗^° . (chorus) Parallel Lines Susan Cantey © 2011 When two lines are parallel, Oh, they never ever meet, And the distance between them Stays the same most certainly, If you draw a line across them A transversal it will be, And the angles created Have several properties. Adjacent angles are supplementary, That means they add to 〖180〗^°, Vertical angles have the same measure, But that was already known before, Alternating interior angles are congruent, Alternating exterior angles are congruent too, It’s pretty obvious, looking at the diagram, Which angles are acute and which are obtuse. (repeat from start) When two lines are parallel, Oh, they never ever meet, And the distance between them Stays the same most certainly, If you draw a line across them A transversal it will be, And the angles created Have amazing properties. Elevation and Depression Susan Cantey © 2011 (Music – slumber Song by Robert Schumann) Imagine you are walking along the street, When you glance up and what do you see? A big black crow sitting next to a window, How far did you tilt your head to look? The angle made by the ground and your line of sight Is called the angle of elevation. Imagine you are in an airplane, Flying parallel to the ground, And you glance down and see a town, How far did you tilt you head to look? The angle made by the plane and your line of sight Is called the angle of depression. Ahhhh….. Look up…it’s elevation, Down…depression. Perimeter Susan Cantey © 2011 Measure around a figure, that’s perimeter, Measure around a figure, that’s perimeter, Add up all the sides, What that signifies Is the distance around the figure, That’s perimeter. Take a walk around the block, How many steps did you count off? That’s perimeter. That’s perimeter. Measure around a circular fence, that’s circumference, Measure around a circular fence, that’s circumference, A problem will arise, Since we cannot measure its sides, Simply multiply The radius times two pi. Take a walk around a lake, How many steps did you take? That’s circumference, That’s circumference, Perimeter, circumference, Both measure distance around, One for straight things, Another for curved things, There’s nothing more profound. Measure around a figure, that’s perimeter, Measure around a figure, that’s perimeter, Add up all the sides, What that signifies Is the distance around the figure, That’s perimeter. That’s perimeter. That’s perimeter. Area Susan Cantey © 2011 Area always needs two measurements To multiply together, Area always needs two measurements To multiply together. Take a square, Side times side, Take a rectangle, Width times height. (chorus) Parallelogram – base times height, Triangle – one half base times height. (chorus) Trapezoid – a half the height times sum of the bases, Circle is pi radius times another radius. It’s all about little squares, How many fit into the area there? It’s all about little squares, How many square units are there? (chorus)