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Hero’s Formula
Surface Area
Volume
Measurement
Tessellations
Constructions
Coordinates
The Distance Formula
Midpoints
Slope
Linear Equations
Parts of the Circle
Circles Are Magical
Transformations
Vectors
SOHCAHTOA
The Law of Sines
The Law of Cosines
Proofs
Non Euclidean Geometry
Cantey
Cantey
Cantey
Cantey
Cantey
Cantey
Cantey/Handel
Cantey
Cantey
Cantey
Cantey/Schubert
Cantey/Tchaikovsky
Cantey
Cantey
Cantey
Cantey
Cantey/Schubert
Cantey
Cantey/Borodin
Cantey
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: 𝑥2 − 𝑥1 ,
The second leg measures: 𝑦2 − 𝑦1,
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
𝑥2 − 𝑥1 in parentheses
Squared…
Plus 𝑦2 − 𝑦1, 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,
𝑥1 + 𝑥2 divided by two,
Comma
𝑦1 + 𝑦2 divided by two,
Ah….
What number is half way
Between 𝑥1 and 𝑥2 ?
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,
𝑦2 − 𝑦1 over 𝑥2 − 𝑥1 ,
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 − 𝑦1 equals “m” times
The quantity −𝑥1 ,
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!)
𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 𝑐𝑜𝑠𝐴
(Or another way)
𝑏 2 = 𝑎2 + 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)