Teach Number Mandala with Cyclic Addition Mathematics
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Transcript of Teach Number Mandala with Cyclic Addition Mathematics
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By Jeff Parker
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Teach Number Mandala
with Cyclic Addition Mathematics
© Copyright 2015 Jeff Parker
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Contents
Introduction … 5Create Number Mandala… 9
Sphere #7 Teach Number Mandala… 11
Problems with Current-Day Number
Solutions using Cyclic Addition Mathematics
Cyclic Addition as a Complete Number System
Cyclic Addition ToolKit
Cyclic Addition Laws on 5 Steps and Cylinder
A Teacher’s Study of the Sphere’s #1 to #6
The Role of Circle with Number Mandala
Advantages and Improvements to plain ‘ol Number
Teaching, Learning and Curriculum for Cyclic Addition Mathematics
A pretend story of Cyclic Addition and Current-Day Number Teach Sphere #1 to Sphere #6
Sphere #1 The Book of Wheels… 61
Wheels Reference Page with Common Multiple 1 to 7
Rational Number
Pure Circular Fractions
Exponentials
Match Whole Number to Pure Circular FractionsWheels
Sphere #2 The Wheels within the Wheel … 96Object Count
Mini-Wheels onto NumberGridsJourney from Current-Day Base 10 Number to Cyclic Addition Number
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Contents cont…
Sphere #3 The Creation of a Whole Number… 112
Cyclic Addition 5 StepsStep 1: Counting
Step 2: Place Value
Step 3: Move Tens
Step 4: Remainder
Step 5: 7×Multiple
How Circle completes Whole Number
Sphere #4 The Count Sequence… 165
The Role of Count Sequence with Cylinder
Count Spacing with Wheel and Cycle
Remainder Pattern with Count Cycle and Cylinder Fibonacci Number together with Cyclic Addition
Cylinder with Common Multiple 7
Sphere #5 The Cylinder… 183
Shape, Structure and Design…
Cylinder with Common Multiple 1, 49 and 343
Patterns with Remainder and Cylinder
Cylinder Hierarchy and Tier Unity
Sphere #6 Patterns with Number, Circle and Common Multiple… 206
A Beginning for Number 1 to 7Completion with Units 9
Hexagon PatternsMisinterpreting Number 533 and proving Number true
Reference Page with Common Multiple, Wheel and Tier
Practical Tiers 1, 2, 3 and 4 with 1 Cycle on the Cylinder for
Common Multiple 2, 7, 9, 10, 11, 13, 15, 16, 17, 19, 21 and 23
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5
Introduction of ‘Teach Number Mandala with Cyclic Addition’
What is a Mandala ? Mandala is a Sanskrit word meaning “Circle” and is a spiritual and ritual
symbol, in Eastern Religion, representing the Universe. Following mystical and spiritual
traditions, Mandalas are used to focus attention and as a spiritual guidance tool, and to create
a sacred space and as an aid to meditation.
Today, Mandalas have become more common describing Circular diagrams, charts or
geometric Patterns that represent the cosmos metaphysically or symbolically.
There are many Mandala pictures or images on the internet. Showing the beauty and harmony
with varied types of Circular Patterns. Children’s colouring in books could be filled with
Mandalas showing infinite possibilities for the Circle.
Circle can be of many forms. Both natural and personmade. Like flowers, a water droplet into
a pond, a tree trunk, hexagonal crystals, a candle flame, a soap bubble and the equator around
the Earth. In the material realm Circle is often found. Like a car or bicycle tyre, a bowl, a
plate, a CD or DVD, an electrical globe or chandelier or lampshade with cone, a button, a fan,
most power plugs, a cap or hat, a cup with cylinder and sphere, a coaster, a compass, a bottle
lid, a bangle, a clock or analogue watch face, a pen or texta with cylinder again, a volume
control and relevant to this book is the Cyclic Addition mathematical ‘Cylinder’.
This book delves into a universe of Circular Number, thus the title “The Number Mandala”.
Showing natural Laws and Patterns that last forever no matter where you are in the cosmos.
Practically speaking, coming down to earth, there are significant discoveries of Number
shared in the following pages. Proving the home of our 1400 year old Number is with Circle.All types of Number – Whole or Natural, Rational, Exponential, Fibonacci all show their best
and original form with Circle. Number within Circle allows exploration into all mysteries and
patterns possible. There by firing up a child’s or young adult’s creative awe and wonder, to
follow the paths of ancients, and to show a genuine curiosity for such a subject as Number.
Cyclic Addition is the Mathematical pathway and framework used to communicate the
Number Mandala. Cyclic Addition has 5 books on Google Play. Each of these show a unique
facet of Number. As this book is a ‘complete’ work on Number it draws upon all of these 5
books. Let’s brief each of these books.
The first book “Wheels” discovered around 2002 shows Circles of 6 Numbers. Each Wheelhas a hierarchy of Tiers. Each Wheel belongs to a Common Multiple. These Wheels connect
to Exponentials and Rational Number with Mathematics. The Wheel is the foundation of
Cyclic Addition Number. All Mathematics stems from the Wheel’s Number and Circle.
The second book “A New Invention: Cyclic Addition Mathematics that repairs and perfects
An old Invention: Number” presents a reference book of Cyclic Addition. The Mathematics
of Cyclic Addition is called the 5 Steps –
Step 1: Counting
Step 2: Place Value
Step 3: Move Tens to Units
Step 4: Remainder Step 5: 7×Multiple.
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All of which act with the Wheel. To place a framework of Mathematical nature around these
5 Steps the ToolKit was invented as well. There are 7 facets of the ToolKit – Wheel,
Sequence, Operation +× ̶ ÷, Pattern, Circle, Common Multiple and Cylinder. All of these act
in part or whole with the Wheel to perform each Mathematical Step.
This book “A New Invention: Cyclic Addition” was made in 2012–2013. Showing the nutsand bolts of the Mathematics to master Cyclic Addition. There are early mapping techniques
to navigate with Wheels of the hierarchy. This mapping began the tireless search for order
and form of Number. Constantly striving to find beauty, perfection and natural Law from
Number and with Number. Also the influence and Maths from Rational Number. These are
termed Pure Circular Fractions. Pure as only 1 number creates the whole fraction and
Circular because all of the generated sequences are a perfect Circle of Number.
A New Invention gives a place to 4 types of Number – Whole, Rational, Exponential and
Fibonacci. These 4 types are woven together with Mathematics. As the high order of this new
invention: Cyclic Addition commands an overall authority with Number by its unity and
perfection of all 4 types. The exploration into Whole Number serves Mathematics of theother types of Number.
Right in the middle of Cyclic Addition 5 Steps there was the need to stop and search for other
Mathematics surrounding the Count Sequence. These were eventually called ‘A Collection of
Emphasis’. An example is how the Count Sequence serves the next higher Tier 7×Multiple.
This was the beginning of the modern Tool termed ‘Cylinder’. Without this preparation in A
New Invention: Cyclic Addition perfect Number from the Cylinder may well have been
overlooked. There are subtle vibrations of perfect Mathematics that crop up and one is
required to fit them into the big picture. Such is the ‘completeness’ of Cyclic Addition.
The Third Book “A Prophetic Design for Number from Cyclic Addition Mathematics” is a
contrast between the ‘current-day’ Number and Cyclic Addition Number. There are 12 topics
in the book that show dangers in the current-day use of Number. The book looks at the basic
fundamental Base 10 Place Value System in use. The whole system relies on fragile beliefs of
Number that have persisted over centuries. These beliefs are questioned with Mathematics
rather than lengthy discourses of argument and meaningless debate.
There is a brief look at the current-day teaching of the Mathematics Number Strand for
Primary years. This at first may seem irrelevant, however to forge a new Number System
from scratch requires a detailed examination of current-day techniques used to prop up the
existing System.
Cyclic Addition answers in this book most of the big questions that determine how Number is
translated into the school or workplace. For example ‘Is Number merely 10 symbols smashed
together side by side to form Number ?’ big questions, right at the very heart of how we
interpret Number. This book A Prophetic Design as the title says is the way forward for
Number to remain with mathematical qualities. The future of mathematical Number requires
all of the Mathematics of Cyclic Addition Number.
The Fourth book “The Complete Mathematics of the Cyclic Addition Cylinder” looks solely
at the completely numerical ‘Cylinder’. This book was also the first year of the ToolKit:
‘Cylinder’. From its numerical perfection and mathematical Mandala the Cylinder was madethis year, 2015, into a small 67 page book in its own right. Showing how to map any Count
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Sequence spiralling around the Cylinder. Perfecting the Common Multiple and Tier by
concentrating energy of one Common Multiple and one Tier on one Cylinder at a time. The
Patterns show how the Cylinder submits all Counts to the next Tier 7×Multiple.
The Cylinder is an amazing feat of mathematical engineering. Bringing together all the
Mathematics of Cyclic Addition 5 Steps into a glorious spiralled array of Number. Thismeshing of Counts all the way down the Cylinder makes for a strong Reference Tool that can
be used anytime. Proving all possible Patterns and Mathematics for a Common Multiple.
The Pattern Making with Number on a Cylinder is without equal. Literally this is the pinnacle
of Mathematics with Whole Number. Nothing else can stand side by side and offer the unity
and perfection and teaching of Whole Number amongst all Number.
Mathematical Laws guide and steer a direction amongst a sea of Number, aid navigation with
the ToolKit and the Cyclic Addition 5 Steps, invite exploration of Patterns freely without any
barrier. To benefit those who climb higher to see further with practical Number. This
pathway of Laws prevents getting lost or drifting aimlessly around the Cylinder.
The Fifth Book “Cylinder PDF’s” is found on Google Play Books and on the CD-Rom in a
user friendly form. All possible and perfect Cylinders for the first 5 Tiers of the Cyclic
Addition Hierarchy are presented. This is to allow an infinite exploration into the Cyclic
Addition Cylinder. There is no limit to the Cyclic Addition Hierarchy. One can even create
your own Cylinder for higher Tiers.
Thus we return to this book “The Number Mandala with Cyclic Addition”. The major
emphasis is ‘The Circle’ and its complete encompassing nature with Whole Number. The
Number Mandala begins with the Cyclic Addition Wheel. From where Cyclic Addition
began. Proving all Number evolves from Circle.
This book is the first study of Number and Circle. Consequently the mathematician strives for
order, form, creation, perfection, and harmony with previous works of Cyclic Addition. Like
the recent Cylinder book broke new ground with Mathematics of Cyclic Addition, so too the
Number Mandala binds the Circle with Number forever.
In amongst the perfection of Number and Circle forming this text ‘The Number Mandala’ is
the great and momentous task of improving the original invention of ‘Number’. The first 5
text iterated above show this and this text brings to light all Cyclic Addition knowledge
within a view of the limitless Circle.
Here are some of these improvements to plain ‘ol historical Number.
Strengthen Place Value positions of a Number.
Strengthen sequence of Numerals forming a Number.
Unify Operations +× ̶ ÷ with Number.
Bring together an ordered collection of Number on a Cylinder for Mathematics.
Unify knowledge of a Common Multiple and Hierarchy with this Cylinder of Number.
Discover infinite Pattern making and Order to Whole Number, Rational Number, Exponential
Number and Fibonacci Number.
Bring Number together with the geometry of Circle and Spiral.
Improve the original invention whilst maintaining existing order of Whole Number.Lay a Mathematical foundation to broaden the strand of Number in school/college.
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The Number Mandala is a Mathematical picture of Cyclic Addition. Each Sphere #1 to #7 is
with a collection of illustrations. These are labelled as “Sphere #1A” where the #1 is the
Sphere chapter and the A to Z is the ordered page by page collection of diagrams.
This book is a little experimental. Where the three previous books of “A New Invention”, “A
Prophetic Design” and “The Complete Mathematics of the Cyclic Addition Cylinder” weretargeted to show an aspect of Cyclic Addition. This book “The Number Mandala” asks the
mathematician to recreate their world of Whole Number, Rational Number, Exponential
Number and Fibonacci Number. This journey searches for a path of discovery and
exploration into Number Mandala.
There are many pages of illustrations thrown in to create a ‘Whole’ aspect of Cyclic
Addition, possibly unseen by the approach of the above three text. The author this time, in the
Number Mandala, puts English second to the Mathematics. Allowing the Teacher, Student or
interested budding mathematician, to discover for themselves how perfect and awesome these
Natural Laws to Number are.
Often the illustrations prove the Circle and Cyclic Addition Mathematics as being simple yet
all encompassing. Finally Laws on how Number works amongst all Number has a benchmark
of “the Number Mandala” and the CD-Rom with other texts and practicals.
All along the journey one might question: Why have this Cyclic Addition discovery now
rather than several hundred years ago ? Is not it easier to incorporate Mathematics of
centuries ago into a curriculum for school to teach ? This puzzle of purity and perfection is
left to the reader of “the Number Mandala”.
This is the first work of Mathematics proving Number and Circle are one. Thus the author
asks all to create along with this text. All new all fascinating Mathematics aimed at
expressing all inter-relationships of any Number with all Number. Thus each participant in
this Cyclic Addition Mathematics can do anything with these Natural Numerical Laws !
A Quote from Carl Friedrich Gauss:
“Mathematics is the queen of the sciences and number theory is the queen of mathematics.”
A quote from the author:“To bring Number and Circle together with Mathematics is like two facets shining light uponeach other”
A little History of our Hindu-Arabic Number
Leonardo of Pisa, commonly known as Fibonacci, wrote ‘Liber Abaci’ in 1202. This book
was to be a milestone and a reference point for our modern day Hindu-Arabic Numerals.
From this point onward, and a battle over the next 300 years+, the ten numerals (0, 1, 2, 3, 4,
5, 6, 7, 8, 9) and the way Fibonacci wrote about them were to be received by the Commercial
and Schooling World of the Day. There ease of use, efficiency and mathematical perfection,
over particularly the Roman Numerals and worded numerals, was to win the day, eventually,
across the known world right through to the Renaissance Era. Many Schools trained students
in the Italy of the Day using Leonardo’s mathematics. There were over a thousand books
written like his book over the next 300 years+. Thus this is marked as a historical moment for
our modern day Number. An ounce of history and man’s reaction to change, illumes lightupon problems of the day, to enable us to receive a kilogram of future.
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Create Number Mandala
Let’s consider the Number Mandala on multiple spheres of Cyclic Addition Mathematics. No
matter the direction of viewpoint onto a sphere a Circle is always seen.
The first sphere is the book of Wheels which covers all Common Multiples. Showing theCircular nature of the whole Reference Page. Circular Pure Circular Fraction 69 Sequence,
Circular Exponentials and corresponding Circular Wheels. Also Rational Number and how to
link Whole Number with Pure Circular Fractions. The two are inextricably joined together
with equality numeral by numeral. And the small relevance of other Remainder Sequences.
The second sphere is the wheels within the Wheel. By chopping up the Wheel ‘1 3 2 6 4 5’
into 60 mini-wheels we receive the simple basics of Cyclic Addition. There is even an early
learning Object Count preserving the Circle, Sequence, Object and Number with Count.
Progressing to a standard NumberGrid for each Common Multiple focuses learning. A simple
believable pathway to join Base 10 Number to Cyclic Addition Number. Thus forming the
Cyclic Addition Wheel from ‘current-day’ Number.
The third sphere is the creation of a Whole Number from the 6 Number Wheel and the Cyclic
Addition 5 Steps. All 5 Steps: Count, Place Value, Move Tens, Remainder and 7×Multiple
utilise the Circle of the Wheel. This is the heart of Cyclic Addition presenting each Number
to a higher Order of the same Common Multiple. Each Step, in sequence, applies a
Mathematical Order to a Count. Yielding a place, position and Order for all Whole Number.
Each of the 5 Steps improve ones knowledge of Place Value positions on a Number. Each of
the 5 Steps is with Circle. Each of the 5 Steps is with Sequence of Number and Numeral.
Each Count has a Remainder to link it to the next higher Tier Order. Learning these 5 Steps,with practise, one acquires perfect Whole Number. Many illustrations detail these 5 Steps.
The fourth sphere is the Count Sequence. The previous Sphere generates a continuous Count
Sequence. This Count Sequence increments by the Circling Wheel. The next Sphere joins the
Mathematics of all possible Count Sequence with a Wheel together. Thus this Sphere
emphasises the relevance of the single Count Sequence. Each Sequence is roughly 1 to 7
Cycles long. Each Cycle is a complete Circle around the Wheel. To navigate with Whole
Number is to steer along the Count Sequence to discover Whole Number and a Common
Multiple. The Fibonacci Sequence of Number connects to Pure Circular Fractions. This
ancient Sequence shows its relevance to Whole Number Cyclic Addition Mathematics.
The fifth sphere is the Cylinder . Each Cylinder is Circular by its mathematical shape. All
Counts belonging to a Wheel, for Tier 1 Cylinders, are presented on one Circular Cylinder.
All Count Sequences Spiral down and around the Circular Cylinder. Forming Complete
knowledge about the Common Multiple. There are Circular Patterns of Remainder that bind
the Spiral Count Sequences all the way down the Cylinder. These Circular or Ring Patterns
equal the 6 Number Wheel. Thus the Circle of the Cylinder submits all Counts in return to the
Counting Wheel.
Amongst all Counts on a Cylinder are end of Cycle 7×Multiple. These are always found
united together on the same Ring or Circle of the Cylinder. This shows the nexus of actual
Counts on the Cylinder with the next higher Tier Wheel. The Cylinder(s) orchestrated by thehigher Tier Wheel show these 7×Multiple from the lower Tier Wheel with higher Order.
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The Cylinder has a Hierarchy that follows the Wheel Common Multiple. The lower Tier
connects to the next higher Tier with all Counts via the 7×Multiple. All 7×Multiple are
shown upon the higher Tier Cylinder. Tier 2 and above have 4 Cylinders to present a Tier 2
and above Wheel. All Cylinders of the hierarchy share the same Common Multiple. Each
successive Tier put the lower Tier to a higher Order. Across all 4 Cylinders for a certain
Common Multiple there are Circular Patterns of equality that link all 4 Cylinders together.
The sixth Sphere is Patterns with Whole Number . Beginning with perfecting Place Values by
Completion of a Number with a Common Multiple units 9. This improves ones Place Value
position movement within a Whole Number. A great and simple resource. This Sphere
follows the previous Cylinder Pattern and looks at entirely a Common Multiple and Count
Pattern. Each Common Multiple illustration, generously sampled from the 69 Common
Multiples present how to navigate and search for Patterns within a Whole Number. There is a
mystical, strange and yet to be explored way of reading a Common Multiple ̶ this Sphere
starts that exploration.
This Sphere shows the Wheel and corresponding Cylinder present Circular Patternscompletely. Every mathematical action forming a Pattern with the Wheel and the Cylinder is
proved to be Circular. Thus mastering Cyclic Addition Mathematics one must see the Circle
and Number as One Whole.
The seventh Sphere is Teaching and Learning Cyclic Addition. This is split into 3 stages.
The first stage is Current-Day Base 10 Place Value Number. What it is and how it works
today. Looking at Whole Number and a snapshot of its problems and weaknesses. The
Design and how its taught. The role the Zero plays in Base 10 Number. Place Value Names.
Older style arithmetic. Lack of Order, Law and Form. Using Number as 10 symbols rather
than scaled Whole Number.
The second stage is Cyclic Addition as a Complete Number System. Presented as a collection
of solutions to the inadequacies of Current Day Base 10 Place Value Number. Incorporating
the previous chapters Sphere #1 to Sphere #6 to solve these problems. Sub-Topics include
Properties of a Number System. Major Cyclic Addition Laws in brief. Operations +× ̶ ÷ and
order. Scale of Number from a Cylinder. The role of Circle within Number Mandala.
Patterns, multiples and factors with Whole Number. Creativity and purpose with Cyclic
Addition. Uniting Rational, Whole and Exponential Numbers. Advantages and improvements
to plain ‘ol Number.
The third stage is a Method and Sequence to Teach and Learn Cyclic Addition Mathematics.
How Cyclic Addition can sit alongside of the existing ‘Current-Day’ Number. How the two
can coexist. Teaching the components of Cyclic Addition with interactive tools. Teaching the
5 Steps in Sphere #3. Teaching Common Multiple and higher Tiers. Teaching the wow and
fun with the mathematical Cylinder. A Top helicopter View to simplify Cyclic Addition.
The seventh Sphere is taught at the beginning of this book, rather than the end. The main
reason is so Teachers are able to question and search for there own answers to ‘deciding how
to Teach Number ?’ The Number Mandala is new to many, thus one requires proof, scientific
proof, that this work of Number Mandala with its knowledge, wisdom and practical
application is eventually included in a School Curriculum.
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Sphere #7 Teach Number Mandala
The seventh Sphere is Teaching and Learning Cyclic Addition. This is split into 3 stages.
The first stage is Current-Day Base 10 Place Value Number. What it is and how it works
today. Looking at Whole Number and a snapshot of its pr oblems and weaknesses. TheDesign and how its taught. The role the Zero plays in Base 10 Number. Place Value Names.
Older style arithmetic. Lack of Order, Law and Form. Using Number as 10 symbols rather
than scaled Whole Number.
Current-Day Base 10 Place Value Number
Number is formed from 10 universal Numerals. These were established hundreds of years
ago. These numerals (1, 2, 3, 4, 5, 6, 7, 8, 9, 0) are put into an order and sequence to make a
Number. This Sequence joins numeral running together forming a Number, usually read and
written from left to right. For example a three digit Number using just ‘1’, ‘2’ and ‘3’ once
presents ‘123’, ‘132’, ‘213’, ‘231’, ‘312’, and ‘321’. Six different 3 digit Numbers in any
order, all still make a Number.
Each of these 6 numbers using a 1, 2 and 3 numerals form a unique Number. This uniqueness
is from each Number’s Place Value positions within a Number. For example Place Value
Notation says 123=100+20+3= (1×102 )+(2×101 )+(3×100 )=123 each numeral has a base 10
assignment. These powers or exponentials of 10n × numeral in that space signify the value of
the numeral. All the numerals are in essence, applying simple Addition to each numeral×10n ,
join to form the Number 123. In fact Current-Day Number can be taught in this fashion for
any Number.
These successive numerals, from right to left, increasing in powers of 10 have simple Names.
100 =1 units, 101 =10 Tens, 102 =100 Hundreds, 103 =1000 Thousands, 104 =10000 TenThousands and so on right up to gigantic Number of 10100 =1 with a hundred zeros called a
Googol. Normally these Names are grouped into 103n or three zeros.
From Sphere #2F the Top Table shows four rows. If one picks at most 1 number from each
row one can form the base 10 structure of a Number. For example 1+20+300+4000=4,321.
Any 4 digit or 4 numeral Number from 1 to 9,999 can be formed. Taught up to year 4+.
This Table shows all possible Base 10 Number needed to form any 4 digit Number. Step back
and just notice the dominance of the Zero in the Table. There is a burdensome and heavy
reliance on the Zero to communicate a 4 digit Number.
Number, at school, is taught with groupings for small 1, 2 and 3 digit Number. These
groupings put units from a choice of 1 to 9 or 0 in a column, then longs or Tens from a choice
of 10 to 90 in a column and then flats or grids 10×10 Hundreds from a choice of 100 to 900
or omitting the column all together. This pictorial relationship of cubes, longs and flats or
other diagrammatic forms of 1’s, 10’s and 100’s establish an early learning form of any three
digit or 3 numeral Number.
Is it fair to say that there is a concentration of teaching effort on what is a (3 digit) Number
rather than Mathematics to connect all the (3 digit) Number. So learning becomes a statement
of singular Picture, Word and Numerals to form a Number. Addition is taught later rather
than with two Number forming a third Number.
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Number is taught to grasp these numerals from 1 to 9 and then trading (carrying) 10 of a
numeral for 1 of the next numeral left. This barter system of trading (carrying) is essentially
the real glue that sticks all the numerals together. Before our base 10 example above.
Number is taught with a count to 100. Utilizing a 10×10 grid for just about any mathematical Number is a standard we live by. Counting by 1’s from left to right along the Grid reaching
the next Ten and starting from the next row at 11 progressing by 1’s to 20 and the next row
from 21 to 30 and so on. In fact there is a dominant Pattern of +1 across the Grid and +10
down the Grid. Many other Patterns rely on this Grid for example teaching Step Counting by
1’s, 2’s, 5’s and 10’s.
If one teaches a Number, like our example numerals 1, 2 and 3 above how does the child or
teacher differentiate between the 6 possible Numbers. By Picture and cubes, longs and flats
other than that one requires Mathematics. And the 100 Grid is too small.
If one teaches 4 digit Number like our table in Sphere #2F one reverts back to 1 to 9 with theapplicable number of Zeros. Thus Counting from 1 to 10 is performed over and over for each
numeral from right (units) to left (tens, hundreds and thousands). This develops a scale or
magnitude of Number from 1 to 10 ×10n where n is from 0 to 3, from units to thousands.
Little else is gained by repeatedly performing Number creation from the Base 10 Table.
Thus Mathematics is required to transcend this Base 10 reliance and begin association of
Number with Number using our familiar Operations +× ̶ ÷. Number as taught with Cyclic
Addition heavily relies on the Operations +× ̶ ÷ in Order and following simple Laws.
Number, later, is brought together with Addition and Multiplication. In fact Number
Sentences are formed to enable the Operations +× ̶ ÷ to be fully understood rather than their
effect upon Number. For example Repeated Addition 3+3+3+3=12 or 4 lots of 3 or the
‘equation’ 4×3=12. The relationship between Addition and Subtraction is brought together.
For example 4+3=7, 3+4=7, 7 ̶ 4=3, 7 ̶ 3=4 from one number sentence others are formed.
Simple Number Multiplication is introduced with qualities of the 0 and 1 upon the Operations
+× ̶ ÷. 5×1=1×5=5, 4+1=1+4=5, 5×0=0×5=0, 4+0=0+4=4 and so on to inculcate the
intricacies of Number, numeral and Operations. This, later, becomes commutative and
associative Laws which may take preference over Number, Pattern and Operations +× ̶ ÷ .
Cyclic Addition quickly establishes simple diagrammatic Number and continuous Pattern and
Operation to allows speedy mastery for small 1 to 3 digit Number.
The Whole Number journey right through Years 1 to 7 uses Base 10 Place Value Number as
a fall-back to misunderstanding or misinterpretation of a Number. The Addition Algorithms
of the trading (carrying) 10 units for 1 ten continues and remains the basis for Adult Number.
If Number is too large to manually calculate on paper, the Calculator takes over the labour
and mathematics of Adding two numbers together. After all that investment in Primary Years
with Base 10 Place value Number one is encouraged to replace most of there numerical skills
with an automated approach to the Calculator.
Very rarely, surprisingly, Counting Number (or Whole Number) is shown against adescription of Counting something. Varying in magnitude, some hundreds, some thousands
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and even some millions, showing children a sense of real world proportion and scale with
Counting Number.
Number is hammered against itself with terminology and symbol rather than Number with
Pattern and creativity. For example ascending and descending choices. Also greater than and
less than comparisons. These tug of war exercises with Number prevail right up to 5, 6 and 7numeral Number or up to millions.
Number Sequences are shown with 3 numbers with a Pattern followed by 3 blanks to fill in.
Only a 6 number sentence in all. These sequences concentrate on shifting a numeral in a
place value up or down by 1, 2 or 5. So a complex 6 digit number might increase by 5 tens
and the pattern follows. Again asking a child to unravel their learning from the 100 Grid
taught years ago to follow Number up or down, which way is of no concern. Cyclic Addition
can be considered the perfect way to investigate and learn about Patterns with just Number.
Rounding off of other Numerals to zeros. For example Round to the nearest million, ten
thousand, hundred and so on. This abbreviation of mathematical Number allows for faster communication of Number. For example 3 579 246 = 3.6 million (3 600 000). Note all the
effort of operational mathematics with Number is partially destroyed for Zero base 10
representation of a Whole Number. We conclude that the mathematics to reach that 7 digit
Number is questionably unable to be supported by ‘Current-Day Base 10 Place Value
Number’. Having to reach for a mathematical tool that destroys the effort of creating the
Number in the first place. Cyclic Addition makes no use or rounding as any length Number,
with practise, is made perfect.
Exponentials of 10 are mixed with Number Sequences to move a decimal point from left to
right making a bigger and bigger Number. Base 10 exponentiation is the primary tool to
interrelate numeral forming number.
A History and introduction to other Number Systems. Like Roman Numerals, Binary Base 2
System, Counting in Chinese, Codes and even Ancient Egyptian. These number systems
show just how easy it is to perform speedy Operations + × ÷ even by hand. And how the
side by side numerals forming a Number is very compact and efficient even for large
Number. Allowing us to use Number for a myriad of things. History showed for hundreds of
years a reluctance and resistance to fully celebrating our 10 Hindu-Arabic Numerals.
The foundation of the Hindu-Arabic Number system is based on Place Value. The way it’s
taught is basically repetition of Base 10 numeral aside numeral mathematics. There is very
little evidence, up until now, to show that Whole Number can be simply given to just Number and Mathematics. Although large Number like phone number, credit card number and
product codes are used solely with just numerals. Mathematics with these numbers is thrown
out the window and replaced with 10 logical symbols.
The Current Day Mathematics thus shows little else to rely on gluing the Numerals of a
Number. Thus our Adult Number, our Teaching Number and our communicating Number is
stuck in this norm. Cyclic Addition: A New Invention asks “to repair and perfect, preserve
and protect our Old Invention: Number”. A title of a previous Cyclic Addition text.
Thus there are no illustrations on Current Day Base 10 Place Value Number. Any modern
Australian Mathematics Text Book or Workbook for Primary and Secondary Years showhow Number is taught and learnt.
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Searching for parallel teaching between Number Mandala and Current Day Base 10 Place
Value Number is left to the third stage of this Chapter. Be especially vigilant in your search
for how Current Day Number expresses Pattern with Number, Laws of Number, and a
Wholeness to Number. Search far and wide for an overall repetition of teaching that ‘seems
to’ constantly destroy previous work to create new work. This skill in teaching a subject
whole to interconnect all topics perfectly is a masterful Curriculum. The Number Mandala is just a small, significant and mandatory component of Mathematics.
The second stage is Cyclic Addition as a Complete Number System. Presented as a collection
of solutions to the inadequacies of Current Day Base 10 Place Value Number. Incorporating
the previous chapters Sphere #1 to Sphere #6 to solve these problems. Sub-Topics include
Properties of a Number System. The Cyclic Addition ToolKit. Major Cyclic Addition Laws
in brief. Operations +× ̶ ÷ and order. Scale of Number from a Cylinder. The role of Circle
within Number Mandala. Patterns, multiples and factors with Whole Number. Creativity and
purpose with Cyclic Addition. Uniting Rational, Whole and Exponential Numbers.
Advantages and improvements to plain ‘ol Number.
The Solution: Cyclic Addition as a Complete Number System
What are the properties of a Complete Number System with Cyclic Addition ? Cyclic
Addition preserves all existing qualities of our Current Day Base 10 Place Value Number.
Although Cyclic Addition may not use 1,234=1000+200+30+4= (1×103)+ (2×102)+ (3×101)+
(4×100)=1,234 the base 10 Zero in the same way, one can still navigate learning from the
‘Current Day’ Number to Cyclic Addition (Place Value) Number by applying existing
teaching methods. Cyclic Addition uses the exact same 10 numerals (1, 2, 3, 4, 5, 6, 7, 8, 9,
0) as does ‘Current-Day’. Cyclic Addition also perfectly preserves Operations +× ̶ ÷. Again
Cyclic Addition has Laws and a Sequence to follow with Operations +× ̶ ÷ however this can
be accomplished with ‘Current Day’ teaching methods. Current Day trading (or carrying) 10
ones for 1 ten is shown in many ways with Cyclic Addition. Current Day word names for a
Number are still applied to Cyclic Addition Number, there is again a transition of
Mathematics with Number that supersedes Current Day weaknesses. Cyclic Addition also
perfectly preserves Rational Number, Decimal Number, Exponential Number and Fibonacci
Number. Again Cyclic Addition applies Law and Order that may appear new, however
existing teaching methods can achieve a result of Numerical mastery with Cyclic Addition
Mathematics. Current Day Number and its Numerals are universal and eternal, Cyclic
Addition must preserve this universality and must work perfectly for all time. There is no
solution without these conditions placed upon a new Cyclic Addition Number System
A great parallel of Cyclic Addition with Current Day Number is how to start teaching
Number of Objects or a Count of Objects. Most have real life experience teaching children
how to Count from a plethora of ‘Current-Day’ text book examples. Cyclic Addition starts
with a very simple Object Count with Circle, Cycle, Sequence, (6) Numbers, groups of
Objects, and progressing to Counting multiple groups of Objects with Circular Addition. This
prepares the child to receive qualities of beginner Cyclic Addition at a very early age.
A simple Brief on the Cyclic Addition ToolKit
Cyclic Addition, as it is new, uses a simple ToolKit of 7 Tools to steer the Teaching of
Number. The ToolKit comprises of (WPOSCCC) Wheel, Pattern, Operation +× ̶ ÷,
Sequence, Circle, Common Multiple and Cylinder. Let’s look at each of the 7 Tools.
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When a Teacher or student investigates Whole Number with Cyclic Addition invariably the
start is with the Cyclic Addition ToolKit Wheel. Every Wheel is completely Circular and
conforms to the Sequence formula of ‘Common Multiple’ב1 3 2 6 4 5’×7(n-1) where the
Common Multiple is a single selection from integers 1 to 69 and n=Tier of the Hierarchy.
Normally one starts by forming a Count Sequence with the Common Multiple 1 Wheel
‘1 3 2 6 4 5’. When working with a Wheel the Common Multiple is stable and continuous,until one completes a Wheel and moves onto another Wheel. This second Wheel might be a
higher Tier than the first or a completely new Common Multiple.
The 6 number Circular Wheel is used to construct and build Cyclic Addition Mathematics.
There are 5 Steps to Cyclic Addition, discussed further on, these are Counting, Place Value,
Move Tens, Remainder and 7×Multiple. The Wheel interacts mathematically and uniquely
with each of these 5 Steps. These 5 Steps and their corresponding mathematical Laws to
make practical the Cyclic Addition are found in this chapter.
The Numerical and geometrical ToolKit Patterns formed by the Wheel are second to none.
Cyclic Addition leads Mathematics with Pattern making and Number. Basically there areseveral types of Patterns. The Count Sequence formed with the Wheel has Patterns of the
Common Multiple, Remainder Patterns and presentation Patterns of the 7×Multiple. Each
Count within a Count Sequence has Place Value Patterns and with larger Number, 3 to 8
digit, the Sequence of numerals within a Number also presents Patterns. A Wheel, from the
first Tier, can combine all possible Count Sequences to form a Cylinder of Number, unique to
that Wheel. This Cylinder shape and managing Number on the Cylinder requires mastery of
the Wheel, Circle, Spiral and Line all geometrical features of the Cylinder. The Cylinder, as
presented in a whole Sphere of this book, is completely full of numerous Patterns. Some of
these Patterns are Circular, some form a diamond, some have a special symmetry, other form
part of the Cyclic Addition Count Sequences that run in Spirals all the way from top to the
bottom of the Cylinder. The language of a Common Multiple consistently given to a Cylinder
also has Pattern. This is the pinnacle of Pattern Making with Whole Number.
The ToolKit Operations +× ̶ ÷ play a consistent and stable role with the mathematics of
Whole Number. Operation × is with one Wheel presenting all multiples of a certain Common
Multiple onto one or four Cylinders. Operation + is with the Cyclic Addition Step Counting.
As one Counts only Addition around the Wheel is used. Operation ̶ is with the Cyclic
Addition Step Remainder. Using the universal formula of ‘Count ̶ Remainder = 7×Multiple’.
Operation ÷ is with positions of Place Values around the Wheel. The Place Values are a Step
in Cyclic Addition. Thus all the Operations are formed from the 5 Steps of Cyclic Addition.
Note importantly the previous Count, the Wheel Count and the next Count all share the sameCommon Multiple and are all produced with Addition, thus Operations +× are together. The
examination of Patterns on a Cylinder is with equality and all Operations +× ̶ ÷.
The ToolKit Sequence has at least three forms. The first is the Cyclic Addition Step:
Counting used to construct a Count Sequence. This Sequence forms a Spiral of Counts on the
Cylinder belonging to the same Wheel. The second is the Sequence of numerals forming a
Count Number. Usually read from left to right, although conversely Operations + ̶ are
performed from right to left. The third is a Sequence of Counts forming a 6 number Ring
around the Cylinder. This Sequence connects Count Sequences together. And most
importantly a Sequence of action with the Cyclic Addition 5 Steps. Again these 5 Steps are
performed on a Count in Sequence. The first Counting, the second Place Value, the thirdMove Tens, the fourth Remainder and the fifth 7×Multiple.
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The ToolKit Circle and its portrayal in Number Mandala emphasises that all Whole Number
is created by, for and with the Circle. The Wheel and all the Mathematics of 5 Steps of Cyclic
Addition are performed with the Circle. Counting is in a direction of clockwise or anti-
clockwise around the fixed Wheel. The Wheel remains stationary and the Mathematics
Circles around it. Place Value applies mini-Wheels within the Wheel, to build each Place
Value position, typically units, tens and hundreds. Each Place Value Set, in each position, is asimple 1 to 5 numbers from the Wheel, again only Circular numbers from the Wheel. Move
Tens to Units moves Place Values outside the Units to the Units via a rotation of a Place
Value around the Wheel. Move Tens is always in a clockwise direction following the
movement of reading a Number from left to right. Remainder uses Circular Patterns from the
Wheel, every Wheel, to calculate a final Remainder. The Mathematics of all steps prior to
Remainder are put to use calculating a Remainder. The 7×Multiple is confirmed by its
presence on the next higher Tier Wheel.
The Circle in this Number Mandala plays a perfect role with all Spheres of Cyclic Addition
Mathematics. Rational Number and their Pure Circular Fractions are always completely
Circular. Exponential Number is connected seamlessly with Pure Circular Fractions to givethem a special role in connecting the Wheel to the ‘Common Multiple’×7(n-1) . Even
Fibonacci Number also connects to Pure Circular Fractions. All Wheels no matter the Tier of
Common Multiple are always Circular. A way to begin Cyclic Addition is with mini-Wheels
onto NumberGrids being perfectly Circular. The Count Sequence from the Wheel becomes a
Spiral around the Cylinder. Navigating with Number around the Cylinder requires Circle,
Spiral and Vertical. Patterns with Number across multiple Cylinders show equality of Circle.
The ToolKit Common Multiple is given to every Wheel and every Cylinder that is
constructed with a single Wheel. This stability and constancy of the Common Multiple,
throughout Number and Pattern exploration, allows perfect knowledge to be gleaned.
Common Multiple unites Number for the whole Cylinder or Tier of Cylinders. The Common
Multiple can be simply a number from 1 to 69 and the Tier n. Or ‘Common Multiple’×7(n-1)
for example Common Multiple 2 for Tiers 1, 2, 3, 4 and 5 are exactly the same as Common
Multiple 2, 14, 98, 686 and 4802. This terminology is used throughout Number Mandala.
The Common Multiple may have a Pattern of numeral Sequence that proves the Common
Multiple and possibly higher Tiers. The Common Multiple may have a Pattern at the end of
Cycle to declare the 7×Multiple. The Common Multiple for higher Tiers shows numerals
within a Number that put the lower Tier to a higher Place, Position and Order amongst all
Cyclic Addition Mathematics. Thus when one reaches for a Cylinder and Wheel to perform
Cyclic Addition upon it one invariably is choosing a Common Multiple and Tier.
The Common Multiple 1 and its higher Tiers 7, 49, 343, 2401… are used more
predominantly than others for a couple of reasons. These Common Multiples act as a
benchmark to reference against other Common Multiples. The Common Multiple 1 and
higher Tiers aid the confirmation of a Tier n. This reference tool helps guide the
mathematician to familiar territory when starting a new Common Multiple. Thus it is
recommended to begin with a very strong and familiar Common Multiple 1 and Tier n. This
concentration of energy upon the 1’s is strongly connected to the dominance of the 1’s from
our early learning as children Counting by 1’s. In the long run every Count on a Cylinder
shows ‘Common Multiple’בOther Multiple’=‘Count’ once the role of the other multiple is
discovered the reliance on the 1’s dissipates.
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The ToolKit Cylinder is a Cylinder covered in Number. Ordered by 6 Spirals in a clockwise
rotation and 6 Spirals in an anti-clockwise rotation. Thus forming 12 Count Sequences or
Spirals in any Tier 1 Common Multiple. One usually reads a Cylinder with the Spiral as a
guiding direction to interpret all the Cylinder’s Pattern and Knowledge of the Common
Multiple. The Wheel is highly connected with the Cylinder. In fact each Ring or Circle of 6
Numbers sliced through the Cylinder has a Remainder of exactly the whole Wheel. This featmakes for perfect Patterns shown in the Sphere Cylinder. This cross meshing of Count
Sequences binds and ropes the Counts into the strongest form of Whole Number known to
mankind. Higher Tiers, Tier 3 and above, have 4 complete Cylinders for every Common
Multiple. There is perfect cross Cylinder Mathematics that interweaves Rings of all 6 Counts
on one to other Cylinders. All this numerical Patterning is with the fact that every Cycle of 6
Counts on a Cylinder is perfectly unique.
The Cylinder, for Tier 2 and above, although it appears to be created from just 1 Wheel is
actually made from all the lower Tiers. Tier 3 Wheel and matching 4 Cylinders are made
from Tier 1 Counting connecting to Tier 2 without Remainder, followed by Tier 2 Counting
connecting to Tier 3 without Remainder and then continuing onto the Cylinder. So basicallyhigher Tier Cylinders use lower Tier Counting to generate more complete Count Sequences
or Spirals. In fact a Tier 3 and above Common Multiple set of 4 Cylinders have exactly 6 of
the same Count followed by a unique Count around the Wheel. All intermeshed Spirals to
presents the perfection and Pattern of the Common Multiple with Cyclic Addition
Mathematics.
When ever there is a problem or confusion with Cyclic Addition, bring the ToolKit into
action and promote Order and Law over chaos. Bringing truth to Whole Number has
ramifications across all strands of Mathematics and many other Subjects. The ToolKit is
simple and can universally be applied to Cyclic Addition and Number Mandala.
Cyclic Addition Laws and the 5 Steps
The Laws book pdf on the CD-Rom was created about 6 years ago. To permanently record all
the accumulated Cyclic Addition actions in one consolidated work of Mathematical Laws for
Number. This provided an avenue to communicate with English the works of Cyclic
Addition. However to teach Cyclic Addition from a text required the last two pdf text ‘A
New Invention’ and ‘A Prophetic Design’. These two pdf text together with the practical
Cylinder and this text show a readiness to be incorporated into a Mathematics Curriculum.
These Laws thus are a guide and formation of Cyclic Addition Mathematics, rather than the
be all end all as the previous Laws book attempted to be. Those wanting to fathom the originsof the Laws are welcome to the older pdfs on the CD-Rom.
The Structure of these Laws include (1) Cyclic Addition 5 Steps and (2) the Cyclic Addition
Cylinder. So that the foundation, and established wisdom of the 5 Steps remains perfect and
true to Law amongst all Cyclic Addition Mathematics.
Laws Step 1: Counting
The word ‘Counting’ is a continuous incrementing Addition from Number around the Wheel.
The Wheel is a 6 number Circular Sequence.
The Wheel is always in the form of ‘Common Multiple’ב1 3 2 6 4 5’×7 (n-1) . n=Tier number.
A Cycle of Counting is a 6 Number Count Sequence. Usually finishing at an end of Cycle.A Number with Cyclic Addition has arithmetical qualities with Operations +× ̶ ÷.
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A Number has place value positions describing each numeral from right to left.
A Wheel has a certain ‘Common Multiple’ from 1 to 69. All Wheel members contribute to
the study of a Common Multiple.
A complete Count with a single Common Multiple forms a Cyclic Addition ‘Cylinder’.
A Count Sequence of any Cycle of 6 numbers is unique to the Wheel and Cyclic Addition.
Counting with a Tier 1 Wheel presents all other multiples to the length of the Count wherethe ‘Common Multiple’בOther Multiple’=Count. This is with all 5 Steps of Cyclic Addition.
At any moment during a Count a ‘Pattern’ can be formed. Patterns include the Count
Sequence itself, 6 Remainders each Cycle, a Common Multiple, or 7×Multiple formed from 2
Cylinder Counts.
Cyclic Addition is not here to question an established 1400 year old invention of Number,
rather to perfect Number with Cyclic Addition Mathematics.
All 12 Tier 1 Count Sequences for a Common Multiple are generated by the Wheel. There
are 6 Clockwise and 6 anti-clockwise Count Sequences. This is shown best by the Cylinder.
All Counts from 1 Wheel share the same Common Multiple. This perfects Patterns.
An end of Cycle of Counts is in most cases with a 7×Multiple=Count, without Remainder.
A Count Number has its place, position and Order amongst all Counts by its position on aCylinder or Tier of Cylinders.
The Student of Cyclic Addition may review any Cylinder Count Sequence. Recommending
the review of an entire Spiral of Counts is performed at one duration of time. Aiding memory.
The Circle and Sequence of a Wheel are preserved by any of the 5 Steps of Cyclic Addition.
The 7 Cycle Count limit on the Cylinder is a guide to allow all Counts to be performed with
equal eye without prejudice or preference. All Counts require a go to receive the whole
Cylinder.
A Count can be with multiple Tiers. Begin at a lower Tier and connect the Count without
Remainder to the next higher Tier. Then apply Cyclic Addition with the higher Tier to all 5
Steps. Move to a higher Tier Wheel and all Cyclic Addition Mathematics acts upon this
higher Tier Wheel.
One can begin Cyclic Addition with Circular Addition using mini-wheels. These 30 mini-
wheels are created from the original Wheel ‘1 3 2 6 4 5’. Then grouped into like common
multiple. All of these mini-wheels train the beginner Counter with Circle, Cycle, Sequence,
Common Multiple all with Whole Number. See 2 recent pdf books on the CD-Rom.
Counting brings forth Mathematics of magnitude, scale, ratio, proportion, quantity and
estimation, exactness and preciseness.
Counting is the beginning of exploring other strands of Mathematics.
Laws Step 2: Place Value
Place Value builds a Set for each place value position in a Whole Number Count. This Set ismade from Circular Mathematics of the Counting Wheel.
A Place Value Set is 1 to 5 numbers counting clockwise with a mini-wheel. A mini-wheel is a
partial 1 to 5 number Sequence from a 6 number Wheel. A Place Value Set can begin at any
number around a mini-wheel and finish at any number. See Workbook pdf Chapter
‘Advanced Place Value’ on the CD-Rom.
A Place Value Set matches its total with the units of a Count. Then what remains is given to
another Place Value Set in the tens. Then, if the Count is high enough, a hundreds Place
Value Set. All Place Value Sets mesh with Addition to equal the Count.
There are only 30 mini-wheels to any Cyclic Addition 6 number Wheel. Any of these 30
mini-wheels can be used to create a Place Value Set.
There is only a potential 270 clockwise Place Value Sets possible for any 1 Wheel.
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These laws provide a foundation to act creatively with any Wheel in this Step 2. All the while
this Step reinforces and preserves the Circle and Sequence of the Wheel.
Note illustrations and tables of Place Value Sets and how they’re derived in pdf book ‘Laws
within a Number Universe’ on the CD-Rom.
There are about 3× as many Place Value Sets between 10× to 20בCommon Multiple’
compared with 1× to 9× ‘Common Multiple’. So connecting any possible Place Value Set toa Count is a Science and Art form. This choice leads to actively participating with the Wheel
members in a creative and Circular way. A brilliant way to build a Count.
Place Value Sets can and often do overlap into higher place value positions. For example
Count 525 with Wheel ‘7 21 14 42 28 35’ can use two Sets ’21 14 42 28’=105 in units and
‘7 21 14’=42 in tens. Note the overlap of 105 into tens and hundreds, and the overlap of 42
into hundreds.
Creating a Place Value Set can be viewed as Circular Addition with just 1 to 5 numbers.
Traditional Names for place value positions like units, tens and hundreds are completely
preserved. Merely applying Cyclic Addition Mathematics to Number.
The Place Value Step selects Wheel members that Add to the Count. This action joins the
numerals forming a Number with Mathematics. A highly sought after goal accomplished bysimple Wheel Mathematics.
A Place Value Set with Common Multiple 1 Wheel ‘1 3 2 6 4 5’, like the Sets adding to 11
above, prepare one to act, with exactly the same Wheel member locations, with another
Common Multiple. For example any of the 17 possible Place Value Sets to make 11, can be
then used to make 22=11×2 with Common Multiple 2, and 33=11×3 with Common Multiple
3 and so on.
This awesome flexibility to choose a Set is then available to any Wheel and any Tier.
Laws Step 3: Move Tens
The Move Tens Step makes use of Traditional Names for place value positions units, tens and
hundreds. As these are familiar to all and ideal to involve them with this Step.
The tens Place Values are matched to the Wheel and then rotated one number clockwise and
literally placed in the units position.
The hundreds Place Values and matched to the Wheel and rotated two number clockwise and
placed in the units position. Higher Place Values rotate around the Wheel clockwise one
number for each place value position left of units.
This acknowledges each Place Value’s position within a Count with Wheel Mathematics.
As all Wheels are in the form of ‘Common Multiple’ב1 3 2 6 4 5’×7(n-1) this rotation
Mathematics of Step 3 Move Tens works universally.
The Cyclic Addition Step 3: Move Tens can be combined with Step 4: Remainder, performed
at the same time. The simplest and most accurate method is mathematically encouraged.See Workbook pdf for ‘Remainder’ examples showing a template for this Step 3.
The Sphere #3M and #3N illustrates the Math of this Step.
The Mathematics of this Step 3: Move Tens shows yet further application of the Wheel’s
Circle and Sequence.
The Wheel Sequence is actually the Remainder Sequence of 7×’Common Multiple’. Maths
with the 6 number Remainder Sequence can be viewed as the Step 3:Move Tens function.
Laws Step 4: Remainder
The Cyclic Addition Step 4: Remainder searches for 1 member of the Wheel that shows the
difference between the Count and the nearest 7×Multiple.
This follows the universal formula of ‘Count ̶ Remainder = 7×Multiple’. At the end of Cyclethere is no Remainder, thus the Count = 7×Multiple. See Cylinder.
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By matching Place Values to the Wheel and applying Patterns or Remainder Laws to the
Wheel one receives the single number Remainder, again from the Wheel.
A Place Value Set in the Tens having 1 Remainder can be combined with a Place Value Set
in the Units having another Remainder. Merely apply Step 3: Move Tens after the Remainder
for both has been found. Leaving two Remainders in the Units for simple final Remainder
calculation.The Remainder Laws that apply to all Cyclic Addition Wheels are such.
Two Number Place Values condensing down to a single Remainder.
Consider the space around a 6 number Wheel.
Two of the same number yields a single Remainder found two numbers clockwise.
Two numbers next to each other yields a single Remainder found 3 numbers clockwise.
Two numbers two apart yields a single Remainder found in the middle of both.
Two number three apart, or opposite sides of the Wheel, yields no Remainder as they add to a
7×Multiple.
Three Number Place Values. 3 examples only.
Three of the same number yields a single Remainder found the next number clockwise.
Three numbers in Sequence yields a single Remainder found the next number in rotationclockwise.
Three numbers all spaced 2 apart yields no Remainder as they add to a 7×Multiple.
Practise making other Remainder Patterns with the simplest Wheel ‘1 3 2 6 4 5’.
With more Place Values in a position, search for Patterns that eliminate 7’s. The Laws pdf
has a complete Table of 270 possible Place Value Set and their corresponding Remainder.
This Table Patterns the Set in groups of 6 and in Sequence around the Wheel.
The searching for a Remainder can also be accomplished with Addition of the Place Value
Set and finding the Remainder from the nearest 7×Multiple. The Table above has this
Addition with Pattern and Remainder.
The Workbook pdf has a Chapter Remainder to practically apply many Place Value Sets with
a range of Common Multiples. Try Addition of the Set to form the Remainder as well.
The Step 4: Remainder mathematically acts upon the Wheel to preserve Circle and Sequence.
In a Cycle of 6 Counts, there are 6 unique Remainders. The end of Cycle, typically has a
7×Multiple=Count, with no Remainder. The other 5 Remainders have a Pattern. This Pattern
applies to most Cylinders and Cyclic Addition. It is important to master this Circular Pattern.
This Pattern of 5 Remainders and a 7×Multiple each Cycle of Counts remains the same for
the length of the one Spiral Count on a Cylinder.
Again this Pattern of Remainders, once mastered, prevents errors of double Counting a
Number on the Wheel, missing out a Count member of the Wheel, incorrect Place Value Sets
to Add to the Count, faulty application of the Move Tens Step. So the Pattern of Remainders
each Cycle protects the Count and consequent 7×Multiple. This is to perfect Cyclic AdditionMathematics.
When applying Cyclic Addition to a Whole Number, any Whole Number, search effectively
for that Remainder and Whole Number takes on the role given by the Hierarchy and
Common Multiple.
Remainder Mathematics on the Cylinder is left to the next Topic in this Chapter.
A Remainder has a position around the Wheel ‘Common Multiple’×’1 3 2 6 4 5’. Use the
location around the Wheel as one of ‘1 3 2 6 4 5’. Then apply Mathematics of that number to
the Count. Often the Remainder’s position around the Wheel highlights and illumes
knowledge on how to read numerals in a Number with Cyclic Addition. This may take
practise from current day methods, however it yields that role that a number plays amongst
all. Also the Remainder itself may contribute like knowledge to the Count. This is amysterious part of Cyclic Addition. Where the Count is mapped to its nearest 7×Multiple.
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See 2 pdf text ‘A New Invention’ and ‘A Prophetic Design’ and Sphere #3 for illustrations
and guide.
Laws Step 5: 7×Multiple
This is the final Step culminating in the receive of the 7×Multiple. The next Count follows.
Once the Remainder is determined, and Subtracted from the Count the 7×Multiple appears.This is from the universal formula ‘Count ̶ Remainder = 7×Multiple’.
Actually the 7×Multiple sits transparently underneath the Cylinder Count. Like the many
Patterns of the Cylinder serving and Addition or Subtraction of two Counts, in Pattern,
equalling a 7×Multiple. Consider the unity of these Patterns and Step 5: 7×Multiple.
The 7×Multiple is a Number that can be found by Counting around the next higher Tier
Wheel. For example Counting with Tier 1 Common Multiple 2 Wheel ‘2 6 4 12 8 10’
presents 7×Multiples from the Tier 2 Common Multiple 14 Wheel ’14 42 28 84 56 70’.
Typically the end of Cycle Count has no Remainder and equals a 7×Multiple. The Cylinder
clearly presents this as a Ring of 6 identical 7×Multiples.
The creation of the 7×Multiple unites the lower Tier with the next higher Tier for Cyclic
Addition. In fact the higher Tier Wheel should be present whilst performing Cyclic Additionwith the lower Tier Wheel. See Wheels pdf for the first 7 Tiers of all Common Multiples.
Cylinder Patterns of the 7×Multiple are detailed in Sphere #5 text.
The Count, Remainder and 7×Multiple all share the same Common Multiple.
The 7×Multiple acts as a unseen framework beneath the Count to enhance the Count’s
numerals forming Number, Pattern of the Common Multiple, and a Reference point for the
Count. So one is mathematically informed as to the Count’s place, position and Order
amongst other Counts on the Cylinder, and other Cylinders with the same Count.
The 7×Multiple thus aids scale, ratio, magnitude, size and proportion of the Count.
There is a maximum of 6 unique Counts with Remainder utilizing one 7×Multiple. The
seventh multiple is where the Count = 7×Multiple without Remainder.
All 7×Multiples are shown as an end of Cycle, Count = 7×Multiple, from Tier 2 and above.
The Order and Hierarchy of Cyclic Addition 5 Steps introduces and unites the next higher
Order with the lower order. So the skill in working with the lower Order is presenting the
higher Order in conjunction with the lower Order Cyclic Addition 5 Steps.
Before moving to a higher Order Wheel and Cyclic Addition, make sure of your confidence
in this presentation of the higher Order within the lower Order 5 Steps. This is the test of
Mathematical ability required to shift to a higher Order Wheel.
The range of 7×Multiple in a 7 Cycle Count is about a complete Cycle with the next higher
Tier.
One can introduce oneself with Patterns found in higher Tier Wheels. This shows the starting
point from the Wheel’s Reference Page from the Wheels pdf on the CD-Rom. These Wheelsare in straight line form and can be rewritten in a Circular Form for Cyclic Addition 5 Steps.
The whole search for the 7×Multiple, found on the next higher Tier Wheel, again preserves
the Circle and Sequence.
Note the number of occurrences of a 7×Multiple within a Spiral Count on a Cylinder. This is
akin to the Remainder Pattern each Cycle. This number of a distinct 7×Multiple repeats each
Cycle. There is either 1, 2, 3 or 4 identical 7×Multiple in any complete Cycle of 1 Spiral.
Finding an accurate 7×Multiple is a perfect sign as to all the Mathematics of the other Steps.
Again the Count choice, Place Value Set choice, Move Tens Circle, Remainder Patterning to
derive the 7×Multiple shows mastery of Cyclic Addition 5 Steps with this accuracy.
Thus the Cyclic Addition Mathematics is a journey that continually perfects the higher Order by applying ToolKit Mathematics to the lower Order. The Mathematics of the lower Order is
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adequate preparation to receive the next higher Order. From a glimpse of the Wheels pdf
showing 7 Tiers for each Common Multiple, one asks how to join consecutive Tiers. The
Cyclic Addition 5 Steps, the ToolKit and the Cylinder answers this connection completely.
The Mathematics of Cyclic Addition matures the Student’s Number, discarding guesswork,
gambling and magical tricks with Number, in preference to completeness of mastering the 5Steps with the ToolKit.
After going through a few Cylinders with Cyclic Addition 5 Steps one realises the inherent
perfection of a Number System that is guided by such Laws. A Number System that provides
freedom and creativity to express Number, with beauty and perfection, by natural Laws.
These natural Laws guide and instruct the newcomer to Cyclic Addition, to effectively
challenge all prior beliefs, in how to use the 10 Hindu-Arabic Numerals forming Number.
Imagine finding these 10 numerals in an Order of Cyclic Addition 1000’s of years ago.
Etched into a stone tablet found at a historical monument. The transition to merge Cyclic
Addition Number with current day Number would be simple. Today with competing forces todistract our attention away from the big picture, away from the whole view, the eternal view
of the subject matter, ones powers of truth, reason and discernment are limited by current day
propaganda.
Laws of Cyclic Addition Cylinder
Cylinder Creation and Form
Cyclic Addition and the Mathematical and Numerical ‘Cylinder’ are one.
Cyclic Addition Mathematics is only complete with the mathematical study of the Cylinder.
As mentioned in the Chapter Structure the Cylinder contains 4 visual forms. These are a
Spiral Clockwise, a Spiral anti-clockwise, a Ring and Vertically aligned Number.The Cyclic Addition 5 Steps are applied to any or all of the Spirals forming a Cylinder.
Cyclic Addition recommends that the core activity of using a Cylinder is with any or all of
the 5 Steps of Cyclic Addition. And treat the Pattern making and exploration of a Common
Multiple as a supplementary activity. This allows consistent concentration on a Spiral with
the purpose of Cyclic Addition 5 Steps in mind. Sure Patterns form along the way however to
remain true to the long term use of the Cylinder one should always return to the 5 Steps.
The form of the Cylinder is a 7 Cycle Count with each Spiral down and around the Cylinder.
The paper Cylinder is about 32 Counts or 5 Cycles designed for an A4 piece of paper.
The form of the Cylinder has 6 members to a Ring, for Tier 1. The first, third and fifth Rings
in any Cycle are Vertically Aligned. So to the second, fourth and sixth Rings. These form 12
points Vertically aligned, equally spaced, all the way down the Cylinder.The Tier 1 Common Multiple Cylinder has 12 Counts on 1 Cylinder created with 1 Wheel.
The Tier 2 Common Multiple Cylinder has 28 Counts on 4 Cylinders created with 2 Wheels.
The Tier 3 and above Common Multiple Cylinder has 42 Counts on 4 complete Cylinders
created with 3 Wheels. As per Cyclic Addition Law of creating a Count Sequence.
The Tier 2 Wheel is exactly 7× Tier 1 Wheel. The Tier 3 Wheel is exactly 7× Tier 2 Wheel.
From the formula of any Wheel ‘Common Multiple’ב1 3 2 6 4 5’×7(n-1) . When n=Tier.
Each new Ring of 6 Numbers is the next Count. The smaller Cylinder, Tier 2 and up, has a
Ring of 3 Numbers, where the next Ring down is also the next Count.
The spacing of Counts on any Spiral is generated by the Wheel in either clockwise or anti-
clockwise direction.
Counting all Counts of a Tier with Cylinder(s) presents completeness with the CommonMultiple for that Tier.
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Teaching and Learning the 6 Spheres of Number Mandala
Sphere #1 The Book of Wheels
The Wheels pdf shows all Wheels for every Common Multiple 1 to 69 and presenting 7
Tiers. Each page is a single Common Multiple. This is called a Reference Page used
continuously to manage inter-Tier Cyclic Addition Mathematics. One can draw a Circular Wheel to construct a Count Sequence or simply use the Reference Page. Even both can be
used.
The Wheels Reference Pages contain 3 parts of Cyclic Addition. The top row of 22 numerals
is part of the Pure Circular Fraction 69. The diagonal numbers beginning with the Common
Multiple are exponentials of 7. i.e. ‘Common Multiple’×7(n-1) where n= Tiered Exponential.
Underneath the exponentials are the 6 number Wheels where the first leftmost number match
the exponentials above. Sphere #1 gives 7 examples of Reference Pages to familiarise one
with the simpler Common Multiples 1 to 7.
This Reference Page, or Circular Wheel, is an unmatchable Tool used with all 5 Steps of Cyclic Addition. To move around the Wheel and Add the next member of the Wheel with
Step 1: Counting. To construct a Place Value Set for each position units, tens and hundreds
with Step 2: Place Value. To use hands around the Wheel, to physically move tens to units or
place value sets, condensed to one Remainder number, around the Wheel in a clockwise
motion. To further Pattern final units Wheel members to reveal a single Number Remainder
with Step 3 and 4: Move Tens and Remainder combined. To Subtract the Remainder from the
Count to link the next higher Tier Wheel Number to the Count, with Step 5: 7×Multiple.
Sphere #1B shows Mathematics of Remainder Patterns common to most Common Multiples.
Sphere #1C asks one to construct the new Pure Circular Fractions. Teaching all fractions of
these special circular sequences of Rational Number. These are constructed wholly from one
number. For example Pure Circular Fraction 19 uses a 2 to construct the whole fraction and
remainder. The Fraction can be formed with Operations +× from right to left or
Operations ̶ ÷ from left to right. This teaches Mathematics of Number 2 and the citing of the
fraction Sequence prepares one for Cyclic Addition with the Common Multiple 2.
The Pure Circular Fraction 19 also present perfect exponentials of 2 see Sphere #1D. To
master a number’s exponential Order with the formula ‘multiple ’×Exponential Number (n-1)
One merely perfectly connects all exponentials to the Fraction. This Mathematical Feat of
numerical engineering allows one to put Order to a home and a Reference for exponentials.
Sphere #1F shows Cyclic Addition Count Numbers and matching them to a Pure Circular
Fraction 19. So a special Cyclic Addition Count can form all parts of the Fraction Sequence.
This is demonstrated in further Spheres #1 to show how Whole Number created from Cyclic
Addition Wheels is perfectly able to present Rational Number.
Complete Rational Number has 3 types. A Pure Circular Fraction perfectly presented with
Cyclic Addition. And a fixed length Decimal Fraction which has a denominator of 2n×5m
Where n and m are positive integers. These Fractions always produce a fixed length Decimal
and the Mathematics of multiples of 5 and 2 can be created with a Cyclic Addition Wheel.
Thus are excluded from Cyclic Addition Mathematics. The third type is a mixed Fraction.
These have the qualities of both types of Fraction above. Most numerators show a decimalfixed portion and a repeating or circulating portion in the Fraction. Thus are obsolete and are
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excluded from a study of Number. Little is to be gained by studying Pure Circular Fraction
19 and then denominator 38 or 76. These types are discussed in Rational Number Sphere #1.
Cyclic Addition exploration and discovery into Rational Number is with Pure Circular
Fractions only. These special denominators with any numerator forming a perfectly Circular
Decimal Fraction from 0 to 1 are the purpose of Number Mandala. Other omitted Rational Number Mathematics can be found in a Text book anywhere.
There are perfect and pure Mathematical Skills to be learnt by working with Pure Circular
Fractions. With Denominators 9, 19, 29, 39, 49, 59 and 69 the working number is +1 and tens
only to the denominator i.e. 1, 2, 3, 4, 5, 6 and 7. This proves Circular Mathematics of
creating these Fractions has a major and significant role to play with Rational Number. See
the two pages of fraction sequences formed by Pure Circular Fraction 399 in Sphere #1J.
Consider the Circular Place Value Skills in constructing perfect Fractions. See Sphere #1.
Any fraction formed is Circular, merely move along the Sequence of a Pure Circular Fraction
to find the numerator and begin circling from there. The diagonal staggering of Exponentials,there corresponding remainder from multiples of the denominator and there vertical Addition
to equal the fraction sequence presents perfect Mathematics. Note how the diagonal
Exponentials form with any numerator along the circular Sequence. This Design, Order and
Numeracy is recognised formally as Cyclic Addition Mathematics. Thus also present on a
Reference Page. In fact Pure Circular Fraction 69 is fundamental to Cyclic Addition Wheels.
Remainder Sequences define a Circular Sequence of Number that show the Cyclic Addition
Step 3: Move Tens to Units from another Multiple. See Sphere #1K. These are, from the
authors view of Number Mandala, considered to be lengthy sequences that are difficult to
remember and apply mathematically. Often Patterns moving from one Remainder to the next
is of more use. For example Remainder Sequence 19 one can apply mathematics of creating a
Pure Circular Fraction and produce the same result. So in perspective, these Remainder
Sequences are completely overshadowed by the Cyclic Addition 6 Number Wheel.
Sphere #2 The Wheels within the Wheel
The start of Cyclic Addition for a pre-school child or early primary is with Object Count.
Cyclic Addition constructs Sphere #2A with Circle, Cycle, Sequence, Number, Word name
for number, and a Count in groups, then progressing to Counting in multiple groups around
the Circle. That simple. Teaching all of these qualities from scratch from new. This prepares
the child to receive down the schooling track the 6 number Cyclic Addition Wheel. Thus the
Circle, Wheel and nature of both with Number are familiar in later primary schooling. Of course any Object can replace the pictured 21 Stars either 2D or 3D shapes that have a similar
Counting ability with them.
The Number alone starts with Sphere #2B. This list is derived from the Wheel at the top.
Simply all 60 mini-Wheels, with some repeating Wheels, are contained in the Table. The
Table groups a Cycle Total or Addition of all numbers in the mini-Wheel together. Then
Count. Counting with Sphere #2C shows the regular NumberGrid in 1’s, yes 1’s. The length
of each Row of the NumberGrid matches the Cycle Total of all mini-Wheels above. To make
it work simply start anywhere around a single mini-Wheel and finger Count, write on another
piece of paper, or colour in the journey around and around and around the mini-Wheel.
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This teaches beginner Circle, Cycle, Sequence, Number, Common Multiple and Remainder
Mathematics. The Remainders are the Numbers in between the right most Common
Multiples. This relationship with essentially fancy, elaborate, skip Counting and the
NumberGrid builds a Number found on a Wheel and its Order with the Table and Common
Multiple. Patterns formed from familiar Times Tables are made perfect with these mini-
Wheels onto NumberGrids. Spacing of Number, which Cyclic Addition perfectly achieves, begins here.
For teachers who ask for a believable nexus between Current Day Base 10 Number and
Cyclic Addition Number Sphere #2F was created. There are 5 Tables. The first is used to
form any 4 digit Base 10 Number. Pick at most one number from each of the 4 rows. Simple.
The second Table converts Base 10 to Base 7. Same Pattern as the first Table, however our
Base 10 Number temporarily disappears. For example 1+20+300+4000=(1×70)+ (2×71)+
(3×72)+ (4×73)= Base 10 Number of 1+14+147+1372=1534. The third Table converts all
Numbers in the second Table back to Base 10. Note the Zero as a Place value Marker has
disappeared. So Cyclic Addition converts the linear Table into Circular Wheel in the fourth
and fifth Table. The fifth Table is simply a construction of a multiple of the fourth Table. i.e.Common Multiple 21 Wheels for Tier 1 to 4. From here the Cyclic Addition Wheel performs
the 5 Steps mentioned in Laws above. Simple. Actually there is a lot of hard work to search
for a Whole Number truth this important, however the journey is simple to Teach. Making
the transition from Current-Day to Cyclic Addition a little easier.
Sphere #3 The Creation of Whole Number
This Sphere #3 is the nuts and bolts of the 5 Steps of Cyclic Addition. The labelled
illustrations communicate how to teach these 5 Steps. The Order again for the 5 Steps is
Counting, Place Value, Move Tens, Remainder and 7×Multiple.
Sphere #3A is a 12 way Count down the page. Showing Counting forwards and Counting
backwards for 2 Cycles with the Wheel of ‘1 3 2 6 4 5’. This is the start of the Common
Multiple 1’s. The Cylinder of 1’s shows these Counts all inter-meshed with one another. This
vertical Counting shows especially the spacing of each Sequence. This Count spacing may
seem new to some, however it allows incredible perfection and Pattern to be formed on the
Cylinder of 1’s shown in Sphere #3B.
Sphere #3C is also vertical Counts showing a beginning to Common Multiple 1 Tier 2. This
shows only clockwise Counting. All possible clockwise Counts are created with Tier 1 and
Tier 2 Wheels. 6 Counts are simply Tier 2 alone, 2 others show no Tier 3 end-of-cycle and
the remaining 6 show the other Tier 3 end-of-cycles with the spaced Counting in-between.Sphere #3D is also vertical Counts showing a beginning to Common Multiple 1 Tier 3 or
Common Multiple 49. Note the 6+6+6+3=21 Counts and how they’re formed. The first 6 a
Tier 3 alone. The second and third 6+6=12 use Tier 1 and Tier 2 and Tier 3 Wheels to present
all Clockwise Counts with an end-of-cycle 343 and 686. This spacing of 49’s shows 6 Counts
for every multiple between 1029 and 2058. The next Count from any number is all 6 Wheel
members from ‘49 147 98 294 196 245’. To find the remaining 21 anti-clockwise Counts see
the 4 Cylinders for Common Multiple 49 on the CD-Rom.
This connection between Tier 1, Tier 2, Tier 3 and Tier 4 Number is the original making of
Cyclic Addition complete Counting for Common Multiple 49 or Tier 3 of Common Multiple
1. To move to a higher Tier, according to Cyclic Addition Law, one Counts with the lower Tier first and when there is no Remainder, such that the Count = 7×Multiple, one can begin
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Counting with the next higher Tier. Thus Sphere #3D shows this connection between Tier 1,
2 and 3 of Common Multiple 1. This also represents a seamless Count with two or more
Tiers. All of the Cylinders, both paper and pdf, use only 1 Wheel for set of (1 or 4) Cylinders.
This aids simplification and concentration of Count Sequences with again 1 Wheel.
Step 1: Counting essentially completely links the Wheel with the Cylinder. Every incrementon a Cylinder from one Ring to the next is with a Wheel member. Counting builds other
mathematical talents like scale, ratio, proportion, magnitude and gaining the ability to use all
Counting Numbers rather than a zero dominated or rounded Number. Counting is the thread
of each Spiral Count Sequence, every Spiral and every Count Sequence, without exception.
Counting with Addition and the Wheel trains one to use the Common Multiple perfectly.
Place Value is called as such because Cyclic Addition requires a way to build each Number
from each place value position, typically units, tens and hundreds. Most Wheels have
overlapping numerals into a higher place value position. See the two Steps Counting and
Place Value in Sphere #3I. This building is with Wheel, is with Circle and ‘kind of’ replaces
Base 10 association with numerals in a place value position. The purpose of which is to prepare Wheel Numbers to find a single Number Step 4: Remainder.
Other Mathematics, during the Place Value Step, is accomplished. The Count is confirmed to
belong to the Common Multiple Wheel, as all Place Values in units and tens and hundreds
originate from the Wheel. Thus proving the Count is a multiple of the Common Multiple.
Very valuable Mathematics to see a Number and connect a Common Multiple to it. The Place
Value Step also encourages manipulation of place value positions with Wheel members. The
basics of which enhance traditional carrying or gluing numeral aside numeral Number. This
determines that Number remains as Mathematical Number rather than symbol. One of the
formal primary outcomes of Number Mandala with Cyclic Addition Mathematics.
Sphere #3F is 30 mini-Wheels complete. This is an invaluable Tool to aid visualising how to
perform the Step 2: Place Value. One need only select a mini-Wheel and Count in a
clockwise rotation for 1 to 5 numbers as per Laws. This Circle and Wheel with 30 mini-
Wheels applies mathematical dexterity to form a Place Value Set. With a little practise with
Common Multiple 1 Wheel ‘1 3 2 6 4 5’ one can see how the Wheel reveals many options for
a choice of a Place Value Set. Note Sphere #3G for all Place Value Sets totalling 9, 10 and 11
for Common Multiple 1 Wheel. This Sphere shows mathematical perfection and creativity
with the Wheel to choose a Place Value Set. Look at Sphere #3H and the 45 Patterns of Place
Value Sets. Note the mini-Wheels all start with 1, and show all corresponding possible Place
Value Sets. Visualise the mini-Wheel and Circling around it to create a Place Value Set. The30 mini-Wheels generate 45×6=270 possible Place Value Sets for any 1 Wheel.
Sphere #3J shows all possible Place Value Sets with all Wheels for the Number 144. All
Place Value Sets are unique and show a scale of how to form 144 from a Wheel. This tests
the child’s Circle, Number and mini-Count Total from the Wheel. A perfectly unique and
universal feature of the Cyclic Addition Wheel. As the effort put into simple Wheels can be
transferred to higher Tier or highe