01 Whole Numbers
Mathematics is more than numbers
It is within most things we do
So much more than adding or algebra
And knowing one and one is two.
It’s about the study of patterns
And the relationships that they contain
A fabulous body of knowledge
That is humanity’s great gain.
Mathematics is more than numbers
And over sixty centuries old
A story of ideas and thinking
Imagination and passions bold.
To do it, you must know its language
Symbols, definitions, and rules
It’s also worth learning times tables
They are useful, everyday tools.
Mathematics is more than numbers
It is the universe’s code
Seen in flowers, atoms, and galaxies
A logical, mystical ode.
Believe in yourself, you can do it
Have the right mindset, oh be involved
Some of the problems are a challenge
But your worries can be resolved.
Believe in yourself, you can do it
Have the right mindset, oh be involved
Some of the problems are a challenge
But your worries can be resolved.
Mathematics is more than numbers
And it can set your mind aflame
So answer questions, find the patterns
And play the mathematics game!
© 2016 M Murphy & H Prochazka
02 Fractions
The Egyptians invented fractions
For other than whole number actions
Fraction confusion is everywhere
But these numbers need not cause despair.
To add or subtract them
You must have the bottoms the same
And with a bit of multiplying
That becomes an easy aim
Fractions are simply parts of the whole
Mastering them is an achievable goal.
So take a chance and tackle fractions
Try these additions and subtractions
Some simple procedures are the key
To work with fractions confidently.
To multiply these numbers
Times the tops and the bottoms too
To divide, flipping the divisor
Is the correct thing to do
And where mixed numbers are concerned
Into improper fractions, they must be turned.
But if you still can’t deal with fractions
Calculators can do these actions
So just relax and press a few keys
Use their built in fraction expertise.
Fractions are simply parts of the whole
Mastering them is an achievable goal
Fractions are simply parts of the whole
Mastering them is an achievable goal.
©2016 M Murphy & H Prochazka
03 Decimals
Infinity is a powerful concept
That has puzzled many a mind
Think of it as the opposite of zero
But know that it is undefined.
In the beginning zero was a marker
That merely held an empty space
Then the Hindus made it a proper number
And gave zero its own place.
This made possible a new number system
Created by thinkers in the East
And later adopted by Baghdad scholars
Zero’s value had increased.
Every whole number could now be written
With the use of only ten digits
In positions that determined their value
This clearly had benefits.
The decimal point was a smart addition
And the system was now complete
Meaning parts of whole numbers could be written
So innovative and so neat.
Fibonacci championed these numerals
And in Europe a fight broke out
Saracen symbols versus Latin letters
A paper and abacus bout.
The new number system won the long battle
A maths invention to astound
This discovery now used around the world
So simple, so simple, so simple, yet so profound.
©2016 M Murphy & H Prochazka
04 Geometry
Geometry is physical and practical.
Seen in nature and man-made formations
Peacock feathers and flags of all nations
In diamond crystals and rare orchid blooms
City skyscrapers and new crescent moons
The geometric is really wonderful
Multi-faceted and many-dimensional.
Geometry is conceptual and logical.
Euclid’s “Elements” brought him much fame
This was mathematics as a mind game
Training intellects throughout the ages
We can read it now on internet pages
The geometric is truly provable
Multi-faceted and many-dimensional.
Geometry is spiritual and magical.
Mystical shapes made with circles and lines
Underpin many symbolic designs
Meaningful archetypes that resonate
And which sacred geometers contemplate
The geometric is inspirational
Multi-faceted and many-dimensional.
Geometry is so playful and beautiful.
Have fun with Platonic polyhedra
Or make stellated icosahedra
Delight in drawing tessellating art
And let symmetrical shapes shift your heart
The geometric is so sensational
Multi-faceted and many-dimensional.
©2016 M Murphy & H Prochazka
05 Negative numbers
Mathematics caters for society’s needs
Negative numbers were one such invention
An idea formed by creative minds
That solved a problem but caused contention.
When multiplying and dividing
There is no need to fuss
Two opposites make a minus
Two negatives make a plus
When adding and subtracting
Make paired signs a single sign
Then it’s easy to work it out
By using a number line.
Education responds to society’s needs
Such as the space race that caused much concern
Then real life skills and problem solving
Became the new maths for students to learn.
There are four problem solving steps
The first one is to ask
The questions that will determine
What is required by the task
Then decide upon a method
And a plan of what to do
Use it to find the solution
Check your answer and review.
Mathematics teaches deductive thinking
And solving problems is more than routine
You can invent your own approaches
Or use methods you have already seen.
©2016 M Murphy & H Prochazka
06 Algebra
First done in Babylonia
Later by Persians in Arabia
Algebra is about communication
Very elegant and very precise
With its own rules and conventions
And letters that make it concise
It’s a smart mathematical device.
With the pronumerals we operate
And maths ideas we can translate
Now that we have gotten that straight
To algebra we can relate.
Now “2b plus b” equals “3b”
And “2c plus c” is the same as “3c”
But “2b plus c” is not “3bc”
For “2b plus c” stays as “2b plus c”
Hopefully, you might soon agree
That algebra can be so easy!
Ladies once thought it fashionable
Entertaining and quite respectable
For algebra is about clarifying
And describing various relations
It is thinking and abstracting
Not just mere manipulations
And it has wide-ranging applications.
With the pronumerals we operate
And new equations we can create
For thoughts we want to illuminate
Our algebra is really great.
Now “b plus c” equals “c plus b”
But “b minus c” is not “c minus b”
And “b times 3” equals “3 times b”
But “b over 3” is not “3 over b”
Hopefully, you will now agree
That algebra can be so easy!
©2016 M Murphy & H Prochazka
07 Powers & Roots
Pythagoras is known for a theorem
Matters of music and the soul
He founded a fraternity
In which women could enrol
They learned philosophy and science
In his progressive mystery school
Students were called “mathematikoi”
And mathematics became their tool.
Number properties were reckoned
By these mathematical Greeks
They wondered about their qualities
For they were number theorist geeks
The feminine numbers were even
The masculine, indivisible by two
While the cubic and triangular
Were geometrically true.
The universe they sought to explain
With whole numbers to show the way
But a radical discovery
A square root that caused dismay
For this number was irrational
Unable to be fractionalised
And to their utter embarrassment
A new number had materialised.
The Hellenic ideas developed
Into modern exponent modes
Such as scientific notation
And the base of binary codes
That are used in digital photos
Or in producing a digital tune
A techno-mathematical leap
From the days of that Greek commune
From the days of that Greek commune
Pythagoras
Pythagoras
Pythagoras….
©2016 M Murphy, H Prochazka & A Jacobson
08 Measurement
During the French Revolution
Many mathematicians combined
To work out a measurement solution
And the metre was defined
Many countries later agreed
Seventeen of them signed a deed.
It’s such a clever metric system
And it’s based on powers of ten
For measuring all things on earth
Mathematics has done it again.
With kilo-, mega-, giga-, tera-
The unit sizes increase
With deci-, centi-, milli-, micro-
The unit sizes decrease
For any measurements big or small
Twenty prefixes handle them all.
It’s such an elegant metric system
And it’s based on powers of ten
For measuring the universe
Mathematics has done it again.
A kilo of water occupies a litre
A tonne takes up a metre cubed
One gram occupies a millilitre
Or a centimetre cubed
And a hundred metre sided square
Has an area of one hectare.
It’s such a marvellous metric system
And it’s based on powers of ten
For measuring things everyday
Mathematics has done it again.
It’s such a marvellous metric system
And it’s based on powers of ten
For measuring things everyday
Mathematics has done it again
Mathematics has done it again.
©2016 M Murphy & H Prochazka
09 Percentages & Ratios
A number has three equivalent forms
Percentage, fraction, and decimal
Divide or multiply by a hundred
For changing forms is arithmetical.
Percentages are central to finance
And with them you can compare
Sports scores and numerous other things
By transforming figures to make them fair.
The rules are quite easy to understand
And here are three facts to have on hand
A tenth or point one equals 10%
A fifth or point two equals 20%
And a half or point five equals 50%
Yeah a half or point five equals 50%.
There are two types of percentage problems
A fact worth taking into account
Changing from one form to another
Or finding a percentage of an amount.
The rules are quite easy to understand
And here are more facts to have on hand
To double, increase by a 100%
To triple, increase by 200%
To make it zero, decrease by a 100%
To make it zero, decrease by a 100%.
Ratios can be related to fractions
And with ratios you can compare
Quantities which are of the same kind
By transforming figures to make them fair.
And ratios in certain proportions
Can be found in nature’s curving lines
Golden shapes and the sounds of music
In company logos and sacred designs.
The rules are quite easy to understand
And here are three facts to have on hand
A tenth or point one equals 10%
A fifth or point two equals 20%
And a half or point five equals 50%
Oh, a half or point five equals 50%.
(Un dixième ou point un est égal dix pour cent
Un cinquième ou point deux est égal vingt pour cent
Et un demi ou point cinq est égal cinquante pour cent
Et un demi ou point cinq est égal cinquante pour cent.)
©2016 M Murphy & H Prochazka
10 Areas & Volumes
Get a sense of the formulaic
Formulas can show you the way
Written in letters algebraic
They are useful for us today.
For a rectangle or a triangle
Or a parallelogram case
We can formulate the area
Using height and length of base
But take care and be aware
That rounded shapes rely
On a radius and the number pi.
And pi is
3 point 1-4-1-5-9
2-6-5-3-5-8-9
On and on, endlessly
On and on, to infinity.
Formulas use symbols to summarise
The method that you might pursue
There are formulas you should memorise
And knowing them will help you.
For a cuboid or a cylinder
Indeed for any prism case
We can formulate the volume
Using height and area of base
But take care and be aware
That rounded shapes rely
On a radius and the number pi.
And pi is
3 point 1-4-1-5-9
2-6-5-3-5-8-9
On and on, endlessly
On and on, to
3 point 1-4-1-5-9
2-6-5-3-5-8-9
7-9-3-2-3-8-4-6-2-6-4-3-3-8-3
On and on, endlessly
On and on, to infinity.
Today computers fuel the races
To discover even more pi places
While all we need are three or four
Computers keep finding millions more
Millions more
Millions more!
©2016 M Murphy & H Prochazka
11 Equations
Mathematicians love to think
And to explore the world of the mind
Searching for beauty, pure or applied
Hoping for something new to find
Pure logic is their artistry
And equations are their poetry.
These thinkers write many equations
It’s essential for what they do
They set them up
They look at them
Sometimes they even solve them too.
Others also play in this mind world
Exploring with their patterning sense
Solving puzzles using imagination
Reason and life experience
Mathematics can make you feel glad
There is much enjoyment to be had.
School students solve many equations
Because their teachers want them to
Sometimes they sit
And just wonder
Exactly what they need to do.
First master the basic equations
The inverse undoes what’s been done
Always do the same thing to both sides
With balance, solutions are won
Learn the skills to find the unknown
Automate them and make them your own.
But equations aren’t always the answer
It’s good to think each problem through
Analyse the task
Seek different ways
There could be another avenue!
©2016 M Murphy & H Prochazka
12 Pythagoras & Trigonometry
A theorem is a mathematical truth
Irrefutable, permanent, and grand
A logical argument built step by step
To create a proof that will always stand
Pythagoras’ Theorem is the best-known one
Simple, intriguing, and with a touch of fun.
When using this famous triangle result
The basic strategy that maths provides
Is to identify the hypotenuse
Then go ahead and square all three sides.
Trigonometry deals with triangles too
And has numerous applications
Describing waves and making maps
And astronomical calculations
The hypotenuse, opposite, and the adjacent
Define the sine, cosine, and the tangent.
When doing a trigonometric task
The basic strategy that maths provides
Is to identify the hypotenuse
Then make a fraction with two of the sides.
Now to decide how to solve a right triangle
This is how to tackle the problem
If working with just the three sides
Then go with the Pythagorean theorem
But if working with two sides and one angle
Then trigonometry will the task untangle.
When using this famous triangle result
The basic strategy that maths provides
Is to identify the hypotenuse
Then go ahead and square all three sides.
©2016 M Murphy & H Prochazka
13 Graphs & Relationships
With just two numbers, it is well known
Positions of points can be shown
Placed inside brackets
To make little packets
Descartes’ coordinate zone.
We can describe a position
With just an “x” and a “y”
On only these two letters
We can rely.
Two coordinates can condense
Patterns in a number sequence
Turning the numeric
To nice algebraic
With rules of precise eloquence.
We can describe a pattern
With just an “x” and a “y”
On only these two letters
We can rely.
For lines that are drawn on a plane
Two pronumerals do it again
Straight or parabolic
Even hyperbolic
Using equations that explain.
We can describe a line
With just an “x” and a “y”
On only these two letters
We can rely.
They are used for animations
Video games and simulations
Fractal mathematics
In vector graphics
And modelling applications.
©2016 M Murphy & H Prochazka
14 Chance & Data
Mathematics has practical concepts
Such as median, mode, and mean
Averages that can lead us
Through an overwhelming data scene.
For to deal with chance and to manage risks
We need to understand statistics
And the laws of probability
Then we can see through a statistics lie
Decoding the numbers and the details
To find the truth that they imply.
And mathematics is the foundation
Of most things we do and see
For so much of modern life
It is the secret, it holds the key.
Mathematics is always there for us
This branch of knowledge, so fabulous
Helping predict and determine trends
Explaining and showing us what is real
And guiding us to make good decisions
Based on what the figures reveal.
But it is much more than number crunching
Calculating is not the goal
It is about brain training
While satisfying heart and soul.
You might even have a subtle feeling
Of something that could be appealing
Maybe mathematics is divine
It is mysterious and so immense
With harmony, order, and symmetry
Resonating with some inner sense.
Mathematics is much more than numbers
Come walk this wonderful road
Of beauty, ideas, and logic
That is the universe’s code.
©2016 M Murphy & H Prochazka