BY JEFF PARKER

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The Workbook exercises can be treated as educational blackline masters.Such that any page(s) can be photocopied and distributed to a class foreducational purposes only. This copyright also applies to the stored pdf fileson the CD as well. The copyright is not transferable, nor can it be onsold.

Cyclic Addition: Create Number from Numerals“Workbook of Mathematics with just Number”© Copyright Jeff Parker 2011

CYCLIC ADDITION: CREATE NUMBER FROM NUMERALS

Contents

Introduction 1

1 Object Count in a Circle 3

2 Create Circular Addition Sequences 5

3 Circular Addition Counting 7

Cyclic Addition

4 Counting 49

5 Place Value 61

6 Wheels 93

7 Advanced Place Value 97

8 Count + Place Value 111

9 Remainder 167

10 Hierarchy 213

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Introduction to Cyclic Addition Workbook

This Cyclic Addition Workbook is the practical for the Guidebook. The Guidebook isformally titled “Cyclic Addition: Create Number from Numerals” with a subheadingof ‘Mathematics with just Number’.

The Workbook is a step by step development of Cyclic Addition Mathematics. Thesesteps in order are ‘Object Counting’, ‘Circular Addition’, ‘Counting’, ‘Place Value’,‘Wheels’, ‘Move tens to units’, ‘Remainder’, ‘7×Multiple’ and ‘Hierarchy’.

The Workbook is whole Number. Completely.

The Workbook exercises are tailored to suit the beginner in Primary, the beginnerCounter, the older Primary with ‘intermediate’ exercises, even the year 6-7 Primarycan have a go at the ‘advanced’ exercises.

The Workbook has been put into a pdf format so that the teacher/student can print anypage with instant exercises. These are introduced with text and have an increasingMathematical skill level as they progress from beginner to advanced Cyclic Addition.

The Workbook right at the very start introduces a wheel of Number. This might be anew concept to begin with but allows very perfect relationships between Number,Pattern, Order, Sequence, Operation +, ×, ̶ , ÷, all in a Circle. The circle of Numbers called the ‘Wheel’ is used consistently and constantly throughout the Workbook. Asthe skill level rises so to does the use of the circular Wheel.

Number, perhaps for the first time, is given a universe of ‘Mathematics with justNumber’. Hence the title of the Guidebook. Number to a large degree serves otherstrands of the Mathematics curriculum. So a whole view of Number is given to a‘Mathematics’ shield to enable being taught under this subject.

Before you venture into a Mathematical realm of Number. Contrast the numerals of aNumber with letters of a Word. Words have a sound and spelling. Number a sequenceand a size. Words form sentences. Number a Count. Full Stops, Comma and GrammarCycle, Wheel and Circle. A Paragraph a ‘common multiple’. A story An infiniteMathematical beauty.

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1 Object Count

The aim of the circle of objects is to develop and discover a Number sense of object.

Pick an object, either 2 or 3 dimensional and form a circle with objects. Place oneobject at the top of the circle, leave a space, then three objects, a space, then twoobjects, a space, then six objects, a space, then four objects, a space, then five objectsand a space before circling around to the one object.

The object should form an arc of a circle with each group like the ring of stars on theright-hand page.

Each object should roughly be the same size and shape.

The groups of objects themselves should be able to be seen as a number of likeobjects.

Firstly, with each of the six groups of objects, see, sound and attach a number to eachand all six.

Secondly, add two objects as two Numbers together, in clockwise sequence to beginaddition.

Thirdly, add any two objects, as two Numbers, spaced apart by objects in between.

Fourthly, begin adding objects, in a clockwise sequence starting at any one of the sixgroups of objects.

When there are more than two numbers being added together, train the youngMathematician to add the first two numbers, receive a total, then add the third numberto the total to receive the addition of three numbers, then add the fourth number to thetotal, to receive the next total, then add the fifth number, to receive the cumulativetotal then add the sixth number, to receive the whole cycle total.

The sequence of addition starting at 1 object is like :-

The first number chosen around the circle 1 + the number clockwise to the first 3 = 4The first total 4 + the next number in sequence 2 = 6The second total 6 + the next number in sequence 6 = 12The fourth total 12 + the next number in sequence 4 = 16The fifth total 16 + the last number in sequence to form a complete circle 5 =21 = Cycle Total

This circular counting technique and method prepares the young Mathematician forfuture Cyclic Addition steps. These steps also use circle and sequence.

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1 Object Count

Count the number of stars in each group all around the circle.

___ ___ ___ ___ ___ ___

Count any two groups of stars in clockwise sequence.

1+3= ___ ___ ___ ___ ___ ___

Count any two groups with a group in-between.

1+2= ___ ___ ___ ___ ___ ___

Count any two groups that add to 7. These are spaced three groups apart.

1+6=7 ___ ___ ___ ___ ___ ___

Count any three groups in clockwise sequence.

1+3+2=___ ___ ___ ___ ___ ___

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2 Create Circular Addition Sequences

Write down all the sequences that are formed by starting at any number, moveclockwise around the circle, and stopping at any number. These Circular Additionsequences must be 1 to 5 numbers in length.

Write down the Circular Addition Sequence and write a total for all numbers in eachsequence. Use order by starting with 1 number, then 2 numbers in sequence, then 3numbers, then 4 numbers and then 5 number sequences. There are only 30 sequences.

The whole table of Circular Addition 1 to 5 number sequences is shown in chapter 3“Maths with just Number”. There are extra sequences in the table from repeating 1number and 2 number sequences. And the special 6 number sequences used nextchapter are included. The table also combines Circular Addition Sequences with thesame total. This Total is called ‘common multiple’.

The next workbook exercise is to Count with these Circular Addition Sequences.‘Common multiples’ from 1 to 12 are beginner level. ‘Common multiples’ from 13 to30 are intermediate level. As addition becomes more complex and requires managingwhere one is within a cycle of Counting.

Each workbook exercise groups Circular Addition Sequences by their ‘commonmultiple’. This aids pattern making and following various ways to make the ‘commonmultiple’ each cycle of Counting. Note during the Counting the factors of a ‘commonmultiple’.

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2 Create Circular Addition Sequences 1 to 5 numbers

Circular Addition Sequence Total

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3 Counting with Circular Addition 1 to 5 number Sequences

Begin with ‘common multiples’ 1 to 12. See table in “Maths with just Number”.Some sequences although only counted once are used in Circular Addition for up to 5different ‘common multiples’. For example Counting with ‘1’ is used with ‘commonmultiple’ 1, 2, 3, 4 and 5. Likewise with all one number Circular Addition sequences.The two number sequences are used in 2 different ‘common multiples’. For examplecounting with ‘3 2’ is used with ‘common multiple’ 5 and 10, and not with 15 as sixnumber sequences are left to Cyclic Addition.

The reference and guide book “Maths with just Number” has Mathematical Countsfor all 30 Circular Addition Sequences. These can be used to emphasise what to lookfor within a ‘common multiple’. The book also has circular templates for all 30sequences. These might be of help to the beginner to train Counting in a Circular way.

Follow the text within the guide book “Maths with just Number” to gain insight andto steer the teacher, child and student in a methodical and Mathematical way.

Once a ‘common multiple’ like the 6 with Circular Addition Sequences of ‘5 1’, ‘3 3’,‘2 2 2’, ‘1 3 2’ and ‘6’ has been given a go Counting. Stop and review the knowledgeof each Count. Note the ‘common multiple’ 6 appears every cycle with thesesequences.

As all sequences are circular look for the hops over numbers to increment by the‘common multiple’ in this case 6. Hop over any three counts along with ‘2 2 2’ and ‘13 2’, any two counts along with ‘3 3’ and ‘5 1’, every count with just the ‘6’. Thisperfectly presents the nature of ‘6’ at this stage of Cyclic Addition. Again to repeatthis way of Counting is Circular so that every Count has circular qualities given to it.

Discuss how factors of, in this case the ‘common multiple’ 6, are treated with 2×3=6,and 3×2=6 and the 3 unique counts with sequence ‘1 3 2’ showing 3’s and 2’s withinthe 3 Counts.

How far to Count ? “Maths with just Number” has roughly 30 count numbers in aCount. If the patterns mentioned above are simply seen move on to the next ‘commonmultiple’ and then perhaps the intermediate Counts with longer Circular AdditionSequences.

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Counting with Circular Addition 1 to 5 Number Sequences

Circular Addition, by counting with a circle of numbers over and over again, shows how to make amultiple, how to resist & strengthen a multiple, & how to create patterns using that multiple.

Circular Addition automatically spaces numbers apart by the multiple counted each cycle. Thecircular addition loops over & over to intertwine numbers belonging to the count sequence. A littlepractice is required for 4 and 5 number sequences to apply a circle with the given straight linesequence.

The common multiple is illuminated every cycle of circular addition. As the counting revolvesaround the circular addition sequence presenting the multiple each and every cycle.

The counting sequence may have unusual numbers in between those of the common multiple.These resist the spacing in between the common multiples. This resistance is evenly spaced everycycle of counting.

Each count with a particular number from a circular addition sequence is exactly spaced by thecommon multiple. Often the lesson counts ask the student of circular addition to hop over counts todiscover this even spacing.

Multiples or factors of a circular addition number are also discovered whilst counting. The number12 has factors of 1, 2, 3, 4, 6, & 12. The counts with circular addition highlight all of these factors.Within each cycle of multiples of 12 the same factors are created for the length of the count.

Often circular addition counts with sequences that have more than 1 number. The 2, 3, & 4 numbersequence are chosen from the 6 number circular sequence ‘1 3 2 6 4 5’. These sequences makeperfect multiples from 1 to 12. The actual numberness of a number from 1 to 12 is revealed like asecret code being unlocked. The nature of how a number & its multiples fit within all number isdiscovered.

Mastering the ‘common multiples’ 1 to 20 with Circular Addition builds a permanent way ofreceiving multiples as and when they are presented.

Start a count by following the pictures under the circular sequence. The 1 has candles, the 2 hasbells in groups of 2, the 3 has egg timers in groups of 3 and so forth. Practice seeing each group ofpictures makes the counting easier.

A teachers’ note, its probably wiser to have a go with as many different Counts as possible or ablerather than to just count on and on with a single Count. Remember this is pattern makingMathematics with whole Number. Complete whole ‘Natural’ Number.

The guide book “Mathematics with just Number” introduces what to look for with most commonmultiples from 1 to 20 applying this ‘Circular Addition’.

There are two skill levels with Circular Addition. The common multiples 1 to 12 are at ‘beginner’level. The common multiples 13 to 20 are at ‘intermediate’ level. As managing a cycle of countingwith the longer 4 and 5 number sequences is a skill requiring a higher degree of Mathematicalmastery.

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Common Multiple 1

Count with just number 1 continuously. Note how the place value of the 10’s increases by 1 eachrow. And the sequence of the units every 10 counts is the same each row. Count by 1’s, start at 1add 2 ( 1+1=2 ) write 2 next to 1, add 1 again ( 2+1=3 ) write 3 next to 2. The start is 1 2 3… .Follow the counting number by number. This Circular Addition Count has every whole number tothe end of the Count.

Note the action from 9 to 0 units in the count above. The tens place value increases by 1. Movingdown a row from any number the tens place value also increase by 1. For example 4+10=14,14+10=24. The 4 stays in the units every +10.

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Common Multiple 2

Simply add 2 continuously, like the introduction example, to the previous count. Start at 2. Notethe pattern of 2 4 6 8 0… in the units all the way along the Count.

Count by 1’s starting at 1 and notice the multiples of 2 along the way. Match the 2’s in this countwith the 2’s above. Note how 1+1=2 or 2 ones = 2

Look at the previous Count with Circular Addition Sequence 1. Hop over every odd number tohighlight the evens. Spacing of 2 is found every two counts. 1, 1+2=3, 3+2=5, 7, 9…

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Common Multiple 3

Count by 1’s starting at 1 and notice the multiples of 3 along the way. Match the 3’s in this countwith the 3’s below. Note how 1+1+1=3 or 3 groups of 1=3.

Notice the multiples of 3 are found every third number.

The next Count simply add 3 continuously to the previous count. Start at 3.

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Common Multiple 4

Count by just the 2 again. Like the introduction. Notice the multiples of 4 every second count.Note this whole count is with even numbers. Note how 2+2=4.

Every second count shows multiples of 4. The numbers in between resist the multiple. For examplein between the multiples of 4 ( 4 8 12 16 20 24) are the counts 2 6 10 14 18 22 these arespaced 4 apart, are all even, showing that 2 × 2 = 4 and resists the multiples of 4 continuously inthe same pattern all the way along the count.

Count with the circular sequence 1 3. Start at 1 add 3 ( 1+3=4 ) then add 1 again ( 4+1=5 ) thenadd 3 again ( 5+3=8 ) and continue counting in a circular pattern. Notice the multiples of 4 everysecond count. The circular addition of 4 appears by hopping over any one count.( For example 1+4=5, 5+4=9, 13, 17, 21 ….)

Count by the ‘1 3’ circular sequence again. This time start at the 3 add 1 ( 3+1=4 ) then add the 3again ( 4+3=7 ) then the 1 again ( 7+1=8 ) and continue in a circular pattern. Again notice themultiples of 4 every second count. The circular addition of 4 can be found by hopping over anyone count. ( For example 3 hop over the 4 to the 7 then hop over the 8 to the 11, 15, 19, 23….)

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Count by 1’s starting at 1 and notice the multiples of 4 along the way. Match the 4’s in this countwith the 4’s below. Show how 1+1+1+1=4 or 4 groups of 1=4.

Add 4 over and over continuously. How do the previous counts help to see multiples of 4 clearly ?

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Common Multiple 5

Count by 1’s starting at 1 and notice the multiples of 5 every fifth count along the way. Match the5’s in this count with the 5’s below.

The next count to highlight the multiples of 5 is with the circular sequence 3 2 . Start at 3 add 2 (3+2=5 ) then add 3 again ( 5+3=8 ) and add 2 again ( 8+2=10 ). Continue until the end number isreached.

Notice the circular addition of 5 hopping over any one count. (For example 3 hop over 5 to 8, hopover 10 to 13 these are all 5 apart. 3+5=8, 8+5=13, 13+5=18, 23, 28, 33….)

Count with the circular sequence 3 2 again this time starting at 2 add 3 ( 2+3=5 ) then add 2 (5+2=7 ) then add 3 again ( 7+3=10 ).

Notice the units pattern is the same counting each group of 10. The circular addition of 5 showsthat in between the 2, 7, 12, 17, 22, 27… and the multiples of 5 ‘5, 10, 15, 20, 25, 30….’

Simply add 5 to the previous count. Start at 5 add 5 and continue until the end of the count.

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Common Multiple 6The next count is with 3 continuously. Showing 3+3=6 or that 3×2=6 two groups of 3 = 6.

The next count is with 2 continuously. Showing 2+2+2=6 or 2×3=6 three groups of 2 = 6.

Note the multiples of 6 are shown every third count. The other multiples of 2 resist the multiples of6. Move along 3 counts from any count to +6.

The next count with multiples of 6 is with the circular sequence 5 1 . Start at 5 add 1( 5+1=6 ) then add 5 again ( 6+5=11 ) then add 1 again ( 11+1=12 ). Continue until the end countmarker. The multiples of 6 appear every second count.

Notice the circular addition of 6 hopping over any one count. (For example 5 hop over 6 to 11, hopover 12 to 17 shows 5+6=11, 11+6=17, 17+6=23 and so on.)

The ‘5 1’ sequence has 2 unique numbers therefore, it has 2 unique counts. The same circularsequence to count with again ‘5 1’ this time start at the 1. Add 5 ( 1+5=6 ) then add the 1 again( 6+1=7 ) then add 5 ( 7+5=12 ) continue this way until the end of the count.

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The next count is with the circular sequence ‘1 3 2’ . A three number sequence. The process ofcounting is exactly the same as with a two number sequence. Start at 1 add 3( 1+3=4 ) add 2 ( 4+2=6 ) add 1 again ( 6+1=7 ) add 3 again ( 7+3=10 ) add 2 again( 10+2=12 ) continue this way until the end marker. The multiples of 6 are found every third count.

The count presents an odd, then an even then a multiple of 6 for the whole count. Notice thecircular addition of 6 hopping over any 2 consecutive numbers. (For example 1 hop over the 4 & 6to 7, hop over the 10 & 12 to 13. This shows 1+6=7, 7+6=13, 13+6=19….)

The next count with the circular sequence ‘1 3 2’ starts at the 3. Add the 2 ( 3+2=5 ) add 1 (5+1=6 ) then add 3 again ( 6+3=9 ) add 2 again ( 9+2=11 ) add 1 again ( 11+1=12 ). Continue withthis circular pattern.

Note the multiple of 3, an odd then the multiple of 6 all the way along the count.

The next count with ‘1 3 2’ starts at 2. Add 1 ( 2+1=3 ) add 3 ( 3+3=6 ) add 2 again( 6+2=8 ) add 1 again ( 8+1=9 ) add 3 again ( 9+3=12 ). Continue this way until the end count isreached.

Note the count presents a multiple of 2, a multiple of 3 and then a multiple of 6 all the way alongthe count. These circular addition patterns show the nature of 6.

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Count by 6 over and over until the end count. Start at 6 add 6 ( 6+6=12 ) then add 6 again( 12+6=18 ).

How do the multiples of 6 act with factors of 6 in the above counts for common multiple 6 ?

Common Multiple 7

The sequence used to discover multiples of 7 is with the whole six number “cyclic” additionsequence ‘1 3 2 6 4 5’ . We will count with this sequence next chapter. Until then merelyremember the first 6 numbers adding by 7 continuously.

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Common Multiple 8

Counting by multiples of 4 one can notice 4+4=8, two groups of 4 = 8 ( 2×4=8 ).Continuous counting with 4 shows this clearly. Note the circular addition of 8 with a hop over anyone count.

The multiples of 8 appear every second count.

The next count to highlight the multiples of 8 is with the circular sequence ‘2 6’. Start at 2 add 6 (2+6=8 ) then add 2 again ( 8+2=10 ) then add 6 again ( 10+6=16 ). Continue with this countingpattern until the end count. Notice the circular addition of 8 with a hop over any one count. Forexample 2 hop over 8 to 10 then hop over 16 to 18 stepping up by 8 each hop.

The next count with the same circular sequence 2 6 starts at the 6. Add 2 ( 6+2=8 ) add 6 again (8+6=14 ) add 2 again ( 14+2=16 ). Continue this pattern until the end count.

Notice with these two count sequences using 2 6 the pattern shows an even then a multiple of 8,an even and then next multiple of 8.

The next count is a repeated sequence of ‘1 3’. There are 2 unique counts with this sequence.

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Note Counting with ‘1 3 1 3’ the pattern of odd, multiple of 4, odd, multiple of 8 and so on.

Count again with 2 continuously. Every fourth multiple of 2 = multiple of 8. Showing 2+2+2+2=8.

Contrast the sequence ‘2 2 2 2’ with ‘1 3 1 3’. This count has all evens. Hop over any three countsto find a +8. For example 2+8=10, 10+8=18, 26, 34, 42…

Lastly with the 8 is to count with just the 8 continuously.

Do the multiples of 8 look stronger with the counting of common multiple ‘8’ ?

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Common Multiple 9

Count with circular addition number 3. Note that 3+3+3=9 or three groups of 3 = 9.

This count sequence has multiples of 9 every third count. With each count add the numerals of anycount together to find a pattern with multiples of 9. The first count is 3, the second 6, the third 9,the forth 1+2=3 like the first count, the fifth 1+5=6 like the second count, & the sixth count 1+8=9like the third count. The pattern of the ‘sum of numerals’ follows 3 6 9 then 3 6 9 again and soon for the whole count.

The next count with the multiple of 9 is with circular addition sequence 4 5 . Start at the 4 add 5 (4+5=9 ) then add 4 again ( 9+4=13 ) then add 5 again ( 13+5=18 ). Continue with this countingpattern. Showing multiples of 9 every second count.

Like the circular sequence ‘3’ the ‘4 5’ has a simple pattern each cycle. The example count has asum of numerals in each number of 4 then 9 then 1+3=4 then 1+8=9 then 2+2=4 then 2+7=9. Thispattern 4 9 runs through the whole count.

The next count with the same circular addition sequence ‘4 5’ beginning at 5 has similar qualities.Start at 5 add 4 ( 5+4=9 ) then add 5 again ( 9+5=14 ) then add 4 again ( 14+4=18 ). Continue thepattern until the end of the count.

Note the circular addition of 9 with a single hop over any number. For example 5 hop over the 9 tothe 14, hop over the 18 to the 23. These numbers are all 9 apart. 5+9=14, 14+9=23. Note also thesum of numerals in this sequence of 5, 9, 1+4=5, 1+8=9, 2+3=5, 2+7=9…

The next count is with the circular addition sequence ‘5 1 3’ . A three number sequence. Thereare 3 starts. Look for the patterns of sum of numerals in all three count sequences. The first starts

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with 5. Add 1 ( 5+1=6 ) add 3 ( 6+3=9 ) then add 5 again ( 9+5=14 ) then add 1 again ( 14+1=15 )then add 3 again ( 15+3=18 ). Continue with the sequence until the patterns are seen clearly. Noteall with three counts the multiples of 9 appears every third count.

Note the circular addition of 9 is presented by hopping over 2 consecutive numbers. For example 5hop over 6 & 9 to 14, hop over 15 & 18 to 23 gives numbers 9 apart. Likewise with 6 hop over 9 &14 to 15, hop over 18 & 23 to 24 also gives numbers 9 apart. And the pattern each cycle is amultiple of 1, a multiple of 3 and a multiple of 9.

The next count with ‘5 1 3’ starts at 1. Add 3 ( 1+3=4 ) add 5 ( 4+5=9 ) then add 1 again( 9+1=10 ) then add 3 again ( 10+3=13 ) then add 5 again ( 13+5=18 ). Continue until the endlimit.

Look for the sum of numerals pattern 1 4 9 throughout the whole count. Some numbers addtogether to more than 9, merely add the numerals twice to receive the single numeral pattern. Forexample 19, 1+9=10 then again 1+0=1.

The next count with the circular sequence ‘5 1 3’ starts at 3. Add 5 ( 3+5=8 ) then add 1( 8+1=9 ) then add 3 again ( 9+3=12 ) add 5 again ( 12+5=17 ) add 1 again ( 17+1=18 ). Continueuntil the end of the count.

Note the sum of numerals pattern 3 8 9 every cycle of counting. And the pattern each cycle of amultiple of 3, a multiple of 1 and a multiple of 9.

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Count by just 9’s. Consider the multiple of 3. What is the sum of numerals within each number ?For example count 189 the sum of numerals is 1+8+9=18 then add then numerals from 18 suchthat 1+8=9. The sum of numerals with this count is always 9 !

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Common Multiple 10

Count using the single number circular sequence 5. Add 5 continuously. Note the pattern of unitsplace value and the multiple of 10 every second count. Observe how 5+5=10 or 2 groups of 5 = 10.

Count using the circular sequence 6 4 . Start at 6 add 4 ( 6+4=10 ) add 6 again( 10+6=16 ) add 4 again ( 16+4=20 ) and so on.

Note the pattern of units every two counts. And the hop over any one count to show the circularaddition of 10. The multiples of 10 appear every second count.

Count with the same sequence 6 4 again. Start this time at 4 add 6 ( 4+6=10 ) add 4 again( 10+4=14 ) add 6 again ( 14+6=20 ). Continue the count in this circular pattern.

Note as with the previous count the pattern of units every two counts. Hopping over one countshows the circular addition of 10. And the multiples of 10 appear every second count.

The next count is with ‘2 2 2 2 2’ continuously. Show 2+2+2+2+2=10 or 5 groups of 2=10.

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Count with the three number circular sequence ‘4 5 1’. Start at 4 add 5 ( 4+5=9 ) add 1( 9+1=10 ) continue with this pattern of three numbers.

Count again with ‘4 5 1’ starting this time with the 5 add 1 ( 5+1=6 ) add 4 ( 6+4=10 ) continuein this pattern until the end count. Note the multiple of 5, then a multiple of 2 then a multiple of 10every cycle.

Count with the same sequence starting at 1 add 4 ( 1+4=5 ) add 5 ( 5+5=10 ) continue with thispattern until the end count.

With all three counts using the circular sequence ‘4 5 1’ note the pattern of units place valueevery three counts. And the circular addition of 10 every 2 hops. For example with the last countstarting at 1 hop over 5 & 10 to 11 hop over 15 & 20 to 21. Then from 21 to 31 to 41… these areall 10 apart.

Count with ‘3 2 3 2’ has 2 unique counts. The first starting at 3 the next starting at 2. Note how thiscontributes new numbers in-between multiples of 10.

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Hop over any three counts to receive +10. For example start at 2, hop over 5, 7 and 10 to 12, thenhop to 22, then to 32 and so on. The spacing of +10 is shown all the way along the count.

Lastly Count by just 10’s. What happens to the units each count ? What changes in the tens placevalue every count ?

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Common Multiple 11

Count using the circular sequence ‘3 2 6’ . Start the first of three unique counts with 3. Add 2 (3+2=5 ) add 6 ( 5+6=11 ) continue with this pattern until the end count.

Count again with ‘3 2 6’ starting at 2 add 6 ( 2+6=8 ) add 3 ( 8+3=11 ). Continue with thispattern until the end count.

Count again with 3 2 6 this time starting at 6 add 3 ( 6+3=9 ) add 2 ( 9+2=11 ) and continue withcircular addition of 11.

Look for the circular addition in the above 3 counts of 11 every two hops. For example in thecount above moving from the 6 hopping over the 9 & 11 to 17 then hop over the 20 & 22 to 28then to 39 and so forth. All these numbers are 11 apart.

Counting with circular addition 11 we can also use the circular sequence ‘5 1 3 2’ . This is a fournumber sequence and consequently has 4 unique counts. Start with the 5 add 1 ( 5+1=6 ) add 3( 6+3=9 ) add 2 ( 9+2=11 ) continue counting with this pattern.

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The next count with the same sequence starts at 1 add 3 ( 1+3=4 ) add 2 ( 4+2=6 ) add 5( 6+5=11 ) and continue until the end count.

The next count again with the same sequence starts at 3 add 2 ( 3+2=5 ) add 5 ( 5+5=10 ) add 1 (10+1=11 ) and continue the pattern until the end count.

The final count with the circular addition of 11 starts at the 2 add 5 ( 2+5=7 ) add 1( 7+1=8 ) add 3 ( 8+3=11 ) continue until the end count

Notice the circular addition of 11 by hopping over any three counts. For example using the lastcount, start at 2 hop over the 7 & 8 & 11 to the 13, then hop over the 18 & 19 & 22 to the 24, hopto 35, hop to 46 and so on. These are all 11 apart. The 11 is a prime so there are no multipleswithin every cycle of 11.

Count by 11’s. A simple way is to increase the units by 1 and the tens place value by 1. Note thespecial counts of 99 & 209.

What happens to the units and tens every count to 99 ? What pattern is shown for the whole count?

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Common Multiple 12

Now the 12 has multiples of 1, 2, 3, 4, 6 & 12. And the circular addition sequences show thisperfectly.

The first sequence is just the 6. Add 6 continuously. Notice the multiples of 12 every second count.This count shows 6+6=12 or 2 groups of 6=12.

The second sequence is just the 4. Add 4 continuously. Note the multiples of 12 every third count.Note also the circular addition of 12 by hopping over any 2 counts. Observe how 4+4+4=12.

The third sequence is just the 3. Add 3 continuously. Note the multiples of 12 every forth count.Note once again the circular addition of 12 by hopping over any 3 counts. Observe how3+3+3+3=12 or 4 groups of three =12.

The next sequence is a three number circular sequence ‘2 6 4’. Start at the 2 add 6( 2+6=8 ) add 4 ( 8+4=12 ) and continue in this pattern. Note a multiple of 2, then a multiple of 4then a multiple of 12 every cycle. Follow the factors of 12 when counting or reviewing.

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With the same sequence ‘2 6 4’ start at the 6 add 4 ( 6+4=10 ) add 2 ( 10+2=12 ) and continuewith the same pattern. Look for the multiple of 6, multiple of 2 & multiple of 12 every cycle ofcounting.

Again with the same sequence ‘2 6 4’ start at 4 add 2 ( 4+2=6 ) add 6 ( 6+6=12 ) and continue inthe same pattern. Look for the multiple of 4, then a multiple of 6 then a multiple of 12 every cycleof counting.

Note the circular addition of 12 by hopping over any 2 counts. For example with the last countstart at 4 hop over the 6 & 12 to the 16, hop over the 18 & 24 to the 28, hop to 40, hop to 52 and soon. These counts are all 12 apart.

The last sequence to count with is the circular ‘1 3 2 6’, a four number sequence. Start at 1 add 3( 1+3=4 ) add 2 ( 4+2=6 ) add 6 ( 6+6=12 ). Continue to add these numbers in this pattern. Lookfor a multiple of 1, then a multiple of 4, then a multiple of 6 then a multiple of 12 every cycle ofaddition.

The next count starts at 3 add 2 ( 3+2=5 ) add 6 ( 5+6=11 ) add 1 ( 11+1=12 ) and continue in thesame pattern for the whole count. Notice the three odds and then the multiple of 12.

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The next count starts at 2 add 6 ( 2+6=8 ) add 1 ( 8+1=9 ) add 3 ( 9+3=12 ) and continue in thesame pattern until the end count. Notice a multiple of 2, then a multiple of 4, then a multiple of 3,then a multiple of 12 every cycle of four counts.

The last count with the sequence 1 3 2 6 starts at the 6 add 1 ( 6+1=7 ) add 3 ( 7+3=10 ) add 2 (10+2=12 ) and continue in the same pattern until the end marker. Look for a multiple of 6, thenand odd, then an even, then the multiple of 12 every cycle of addition.

Considering all the Counts above for common multiple 12, and all the patterns made with CircularAddition what does every count with 12 show ? Which 2 factors of 12 prove a multiple of 12 everycount?

This is the last ‘Beginner’ Circular Addition Sequence. As mentioned in “Mathematics with justNumber” the work of Circular Addition is with Common Multiples 1 to 20.

The guide book shows knowledge on what to look for with the ‘Intermediate’ Circular AdditionCounting. Apply this to the practical lessons ‘Common Multiple’ 13 to 20 that follow. Refer to“Mathematics with just Number” for common multiples from 13 to 20. This guide book gives hintsand tips on how to move within the Count sequence to gain the full mastery of the CircularAddition Counting.

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Common Multiple 13

Remember all Circular Addition Sequences have a length. Either 1, 2, 3, 4 or 5 numbers. Toreceive addition of the common multiple jump over the Count sequence by its length. For a 4number sequence like ‘4 5 1 3’ hop over any 3 consecutive numbers in the Count and add 13 to theCount. For example from 4 hop over 9, 10 and 13 to land on 17. Thus 4+13=17. This highlightsthe circular nature of the common multiple.

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Common Multiple 14

The ‘Common Multiple’ 14 is not used with Circular Addition. Instead the Mathematician waitsfor Counting with Cyclic Addition Sequence ‘2 6 4 12 8 10’ and the 14’s are presented with aRemainder. This Mathematics is presented later in this ‘workbook’.

Common Multiple 15

Look for how the factors of 1, 3, 5 & 15 are used in the Counts below. Note 5×3=15 and 3×5=15.

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Note the factors of 16 are 1, 2, 4, 8 and 16. 2×8=16, 4×4=16 and 8×2=16.

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Remember hop over any 3 consecutive numbers in a Count with ‘2 6 4 5’. Hop over any 4consecutive numbers in a Count with ‘5 1 3 2 6’. Both receive a +17.

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Note factors of 1, 2, 3, 6, 9 and 18. Look for 2×9=18, 3×6=18, 6×3=18 and 9×2=18.

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Common Multiple 19

Hop over any 4 consecutive Count numbers from ‘6 4 5 1 3’ and receive +19 along the way. Forexample start at 6 hop over 10, 15, 16 and 19 to land on 25. 6+19=25 The Count is circular so thisworks all the way along these 5 Counts right to the end.

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Common Multiple 20

Remember factors of 1, 2, 4, 5, 10 and 20. 10×2=20, 4×5=20 and 5×4=20. Look carefully at theCounts to receive how these factors are made. Also note +20 every cycle of Counting with any ofthe ‘Common Multiple’ 20 circular addition sequences below.

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4 Cyclic Addition Counting – Beginner Level

The first step with Cyclic Addition using 6 number circular sequences is Counting.Each Count starts by choosing a number from the 6 and continuously counting clock-wise around the Cyclic Addition Sequence.

The Workbook Counting exercises make up 10 pages in all. Each page deals with aparticular ‘common multiple’. The ‘common multiples’ range from 1 to 10.

Upon each page is space to count all 6 starting points. Each Count is for only 2 cycles.The first Count starts from the top of the Cyclic Addition Sequence wheel. The firstthree counts are given to show how to start that particular ‘common multiple’. Thesecond Count starts from the next number around the wheel and also has three countsgiven. Likewise the other counts, six in all, have a starting point as a guide.

For Example the ‘common multiple’ 1 wheel, being the circular ‘ 1 3 2 6 4 5 ’,starts with 1, add the next number around the wheel +3 = 4, the 4 is written next to the1, then +2 the next number = 6, write the 6 next to the 4 and continue for 12 counts inall or 2 whole cycles around the wheel. Then begin a new Count from 3, +2=5 writethe 5 next to the 3, 5+6=11, write the 11 next to the 5, and continue like the previouscount for a whole 2 cycles around the wheel.

Continuing with the same ‘common multiple’ 1 follow the same pattern of Countingfor the other 4 Counts. The third Count starts at 2. The fourth Count starts at 6 thenext number around the wheel. The fifth Count starts at 4. The final Count for this‘common multiple’ starts at 5.

The exercise trains the beginner Mathematician to use a circle of numbers. The aim isto Count in sequence, 1 number at a time, all the way around the wheel twice for eachCount.

The other 9 pages of exercises follow the first page exactly. Introducing a new‘common multiple’ from 2 to 10 each page.

There can be a discussion on what type of patterns appear for a particular ‘commonmultiple’. For example by applying the first page ‘common multiple’ 1, what patternsare shown with all 6 counts. What numbers are missing from 1 to 21 ?

For example by applying ‘common multiple’ 2, what patterns of even numbers areshown in the 6 counts ?

For example by applying ‘common multiple’ 3, what patterns are shown with the‘sum of digits’ in each Count ?

For example by applying ‘common multiple’ 4, what patterns show each number to bea multiple of 4 all the way along the count. All Counts are both even and the ‘tens’numeral shows a pattern with certain even numbers ? What is this pattern ?

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For example by applying ‘common multiple’ 5, what patterns show all counts are amultiple of 5. Does the units numeral have a pattern ? Does the tens and hundredsnumeral matter ?

For example by applying ‘common multiple’ 6, what patterns show a multiple of 6 ?The 6=3×2 so how do these Counts show both multiples of 3 and 2 at the same time ?

For example by applying ‘common multiple’ 7, what patterns show a multiple of 7 ?After each Count try joining the Counts all the way along with the wheel‘ 7 21 14 42 28 35 ’. Example 21+14=35, 35+42=77, 77+28=?, This might helpjoining the units and tens with basic addition together.

For example by applying ‘common multiple’ 8, what patterns show a multiple of 8.More difficult to spot than just multiples of 4×2=8. How are the units, tens andhundreds brought together ?

For example by applying ‘common multiple’ 9, what patterns show a multiple of 9 ?Use the ‘sum of digits’ and show that all Counts have a special quality. Why does thisalways work ?

For example by applying ‘common multiple’ 10, what patterns show a multiple of 10?What is the difference between ‘common multiple’ 1 counting and ‘commonmultiple’ 10 ? What pattern does the units numeral have in all counts ?

This recognition of pattern making with Counting is possible as Cyclic Additioncombines addition with multiplication. So properties of both can be seen together.

Why is the 6th Count on each page the same for all 6 counts ? Why is the 12th count,the last Count, on each page the same for all 6 counts ? How many counts form 1cycle or revolution around the wheel ? Does it matter where the start is ?

The 6 number circular Cyclic Addition Sequences are designed perfectly to present allmultiples of a ‘common multiple’. The wheel of 6 numbers resists and balances theuse of numbers with that ‘common multiple’. By Counting with the wheel allconsecutive numbers are unique and presents perfect patterns of a ‘common multiple’.Its completeness and wholeness are unmatched.

Errors and duplication are avoided by simply using the wheel correctly.

This concludes the first beginner step of Cyclic Addition.

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5 Cyclic Addition Place Value ̶ Intermediate Level

The second step with Cyclic Addition following Counting is Place Value. Followingeach Count this step assigns a Set of numbers chosen from the Cyclic AdditionSequence wheel to each Place Value position.

Typically the units first, then the tens and if the Count is high enough the hundredsplace value position as well.

A way to understand the actions within Place Value is with the first ‘commonmultiple’ 1. Like simply observing the units numeral in a Number, then the tens andthen hundreds. Each numeral from 1 to 9 has a Place Value Set that adds to eachnumeral.

The first Place Value example in the workbook shows this. Build each Place Value upby applying addition within the Cyclic Addition Sequence wheel. Let’s go step bystep through this example. Count 1 has a Place Value of 1 unit. Count 4 has a PlaceValue of 1+3=4 units. Count 6 has a Place Value of 4+2=6. Count 12 has 2 units and1 ten. Note the _ underscore is the print character used to mark the tens and hundredsPlace Value positions. Count 16 has 5+1=6 units and 1 ten. Count 21 has 1 units and 2tens. Count 22 has 1+1=2 units and 2 tens. Count 25 has simply 5 units and 2 tens.Count 27 has 3+4=7 units and 2 tens. Count 33 has 3 units and 3 tens. Count 37 has6+1=7 units and 3 tens. Count 42 has 2 units and 4 tens. Count 43 has 3 units and2+2=4 tens. Count 46 has 3+3=6 units and 4 tens. Count 48 has 4+4=8 units and 4tens. Count 54 has 4 units and 3+2=5 tens. Count 58 has 2+6=8 units and 5 tens. Thefinal count 63 has 3 units and 6 tens. The exercise uses just the ‘common multiple’ 1wheel to assign a Place Value Set to each Count Number.

The next 5 pages in the workbook exercises have a Count with the same 6 numberwheel for 3 cycles. The Count is given to enable concentration on the new step PlaceValue. ‘Common multiple’ 1 exercises all have unique starts producing uniqueconsecutive counts mastered in the previous step Counting. Two of the pages havestarting answers and the rest of the page left blank to fill in. The other three pages relyon the student to work with part of an answer and fill in the rest. Basically followingthe first example with units and tens Place Value Sets.

There is a blank count with the ‘common multiple’ 1 wheel for again three cycles tohave a go at doing both steps Counting and Place Value from scratch.

‘Common multiple’ 2 wheel with a 3 cycle count on each page is the next exercise.Again like ‘common multiple’ 1 the first page is an example of Counting and PlaceValue all with multiples of 2. Notice how the wheel is used for the units and tensPlace value positions. Look at Count 12 the total of 2+6+4=12 overlaps into the tens.The Count 24 has 12+12=24. The two 12’s are placed in the units, but include the 2tens. So a Place Value Set can both overlap into the tens and later hundreds, so longas the rule that Place Values come from the ‘common multiple’ 2 wheel applies. Let’scontinue with the Count. Count 32 has Place Values 10+2=12 units and 2 tens. Count42 has 2 units and 4 tens. Count 44 has 4 units and 4 tens. Count 50 has 10 units and 4tens. Note carefully how the 10 units is applied to convert the 5 in the 50 to =4+1 ten.So the 4 represents 4 in the tens Place Value position and the 10 is in the units Place

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Value position. This mastery of both the Wheel selection of Place Values and thepositioning of the Place Values within a Count is a main feature of Place Value.

Let’s continue from Count 54 has 8+6=14 in the units and 4 tens. Count 66 has 6 unitsand 4+2=6 tens. Count 74 has 6+8=14 units and 6 tens. Count 84 has simply 4 unitsand 8 tens. Count 86 has 2+4 units and 8 tens. Count 92 has 12 units and 8 tens.Count 96 has 4+12=16 units and 8 tens. Count 108 has 8 units and 10 tens tointroduce the hundred Place Value position. Count 116 has 10+6 units and 10 tens.Only with correct application of units 16 can the remaining 10 tens be seen.

The whole purpose of this circular wheel ‘ 2 6 4 12 8 10 ’ is Mathematicallyworking with multiples of 2. Place Value Sets adding to the Count always apply thisfact.

There are 3 other workbook exercises for ‘common multiple’ 2. The first 2 the Countsare both given with a starting piece of Place Value Sets for each Count. The Student isrequired to master multiples of 2 and Place Value positions of a number at the sametime. The third page is blank with the wheel at the top of the page. Again like‘common multiple’ 1 this is to train using both, firstly the Count then Place Value,Cyclic Addition steps with the given wheel.

The next 16 pages have 2 pages each for ‘common multiples’ 3 to 10. The first pageof each ‘common multiple’ is an example of both a Count with Place Value Setsgiven. The second page of each ‘common multiple’ is blank. Requiring the student toboth Count and apply Place Values with a particular ‘common multiple’.

Note how the wheels with each increasing ‘common multiple’ overlap the units PlaceValue position with the tens. This purpose is to master, as mentioned before, the‘common multiple’ with just the wheel of 6 numbers.

There are generally more than 1 answer to every Count with any wheel. Encouragingthe addition again of numbers within the wheel. When the wheel is strong with thestudent the Cyclic Addition remains also strong and accurate.

The principle of finding the units Place Value first then the tens strengthens thepositions of all numerals in any Count. Most Addition is generally carried out fromthe rightmost numeral to the next left then the next left and so forth.

Place Value is liken to Counting in the way that both use Addition. Both use theWheel. Both require a knowledge of sequence within the wheel. Both applyMultiplication by using only a ‘common multiple’. This is unique to Cyclic Addition.It fosters unified Mathematical Number. As all Cyclic Addition basics are the samefurther on down the track with advanced ‘secondary schooling’ Steps.

Follow are hints on what to look for with each ‘common multiple’. Take into accountthe Counting hints for each ‘common multiple’ in the previous Cyclic Addition Step.

‘Common multiple’ 3 requires how to form the units of any numeral from 1 to 9 and0. As its odd. It has 3 numbers in the wheel units only and 3 with units and tens.Though the example answers seem simple, building of numerals from right to left

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requires consistent Mathematical logic and dealing with alternatives. The alternativessplit and separate wheel numbers by there multiple of the ‘common multiple’. So each‘common multiple’× ‘ 1 3 2 6 4 5 ’ = the Wheel.

‘Common multiple’ 4 note how the Place Values build and interlock the multiple of 4in the units with a particular ten. Thus forming a multiple of 4 or equalling the Count.

‘Common multiple’ 5 in some ways is trickier than the others. As there are manychoices of Place Value Sets that give the units of 5 or 0. What is left over in the tensmust also be matched to equal the Count. Look for a units that leaves a ten of either a5 or 0. Note the example answers.

‘Common multiple’ 6 is again both even and a multiple of 3. 2×3=6. Both multipleshave to be used in both units and tens.

As the child/student becomes more familiar with the sequence of numbers around thegiven ‘common multiple’× ‘ 1 3 2 6 4 5 ’. This allows simpler Multiplication withstarting at one number, then two numbers in sequence, then any two numbers from thewheel, even the same number twice, and later the wheel’s Place Value Sets arebrought together to form up to 5 numbers. This stems toward Advanced CyclicAddition. Working with 1 or 2 Place Value numbers in a Set is just the basics.

‘Common Multiple’ 7 like the 3 is odd. One should experiment with making variouscombinations of numbers from the wheel to make the same units Place Value. Theexample answers have a 9 in the units 3 times with 3 different answers.

‘Common multiple’ 8 is like the 4. Working with just the wheel of only 6 numbers ofa ‘common multiple’ in this special circular sequence allows perfect pattern makingof other multiples of the ‘common multiple’. If a student is adept enough he/she canapply a 3 number Place Values Set. Like the Count 472 has 24+24+24=72 units and40 tens. Like Count 168 can equal 24+16+48=88 units and 8 tens. Watch how theunits gives either a multiple of 4 in the tens where a multiple of 8 might be needed.

‘Common multiple’ 9 is like 3×3. Odd number and just notice how the Count alwayshas a ‘sum of digits’ equal to 9. As the student becomes proficient at applying PlaceValue Sets to a position, creativity with Number is brought to the forefront. Thelimitation of 1 to 10 times the ‘common multiple’ in the units is let go to higher orderand higher truths and higher Mathematical use of the wheel. For example,remembering our 5 number limit to any one Place Value Set, Count 288 can equal9+27+18+54=108 units and 18 tens.

In sequence Place Value Sets are encouraged where possible as they can be used overand over again with any ‘common multiple’.

‘Common multiple’ 10 is simply like the 1’s in the tens and a trailing 0 for the unitsposition. The trick or trait of the tens is simply place the units 0 with the tens at thesame time. So for example Count 370 equals 10+60 units and presents the tens at thesame time, with 30 tens presenting 3 hundreds. The nature of the ‘common multiple’is better understood.

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The Place Value Step in Cyclic Addition is also a confirmation that the Count belongto the ‘common multiple’. Or simply a multiple of the ‘common multiple’. So to allCounts by using the Place Value Step.

The Place Value Step also allows the Mathematician to find the ‘other multiple’ suchthat the ‘other multiple’× ‘common multiple’ = Count. Simple whole numberDivision is available at the Place Value Step in Cyclic Addition. By matching thenumbers in a Place Value Set to their position around the‘common multiple’× ‘ 1 3 2 6 4 5’ and adding the units, tens and hundreds togetherto form the ‘other multiple’.

In closing of this Introductory Place Value Step, the use of the Cyclic AdditionSequence wheel is illuminated with knowledge. The basis for future Steps withinCyclic Addition are prepared by the action of Place Value.

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6 Cyclic Addition Wheels ̶ Intermediate Level

These two pages of Cyclic Addition Wheels show that a wheel can have almost any‘common multiple’.

The two pages that follow present ‘common multiple’ 1 to 21 Wheels.

All of these 21 Cyclic Addition Wheels can be used for Beginner Counting andIntermediate Place Value steps.

Note how the Wheels are always written in the circular form of‘common multiple’ × ‘ 1 3 2 6 4 5 ’. Each Wheel has 6 starts and Counting can befor practicality sake between 1 and 7 cycles. Although, as the Wheels are circularthere is no Mathematical limit to how far a child/student counts with any of them.

A Wheel of 6 numbers can be made up by hand. Start at the top with the ‘commonmultiple’. Pick a number from 1 to 69. Then multiply this ‘common multiple’ by 3 tomake the next number clockwise. Then multiply the ‘common multiple’ by 2 to makethe next number again clockwise. Add the first 3 numbers to make 6×’commonmultiple’. Add the first 2 numbers to make 4×’common multiple’. Then add the‘common multiple’ to the 4× ‘common multiple’ to make 5×’common multiple’.

The 6 number Wheel should look like one of the following 2 pages of templates.

Check the Wheel before Counting with it. Each Wheel has multiples of‘ 1 3 2 6 4 5 ’ in a circular form. Add the 1+2=3 the number in-between. Add2+4=6 the number again in-between. Add 4+1=5 the number again in-between. Somoving around the circle the student creates the ‘common multiple’ב 1 3 2 6 4 5’.

For exampleThe ‘common multiple’ 3× ‘ 1 3 2 6 4 5 ’ = ‘ 3 9 6 18 12 15 ’.The ‘common multiple’ 5× ‘ 1 3 2 6 4 5 ’ = ‘ 5 15 10 30 20 25 ’.The ‘common multiple’ 12× ‘ 1 3 2 6 4 5 ’ = ‘ 12 36 24 72 48 60 ’.The ‘common multiple’ 14× ‘ 1 3 2 6 4 5 ’ = ‘ 14 42 28 84 56 70 ’.The ‘common multiple’ 21× ‘ 1 3 2 6 4 5 ’ = ‘ 21 63 42 126 84 105 ’.

This forms the foundation of Multiplication with Addition. Exactly the same processfor each Wheel created.

Once the Wheel is created the student can immediately start Counting with it. Andstart forming Place Values with each Count, once they’re confident with Countingwith higher ‘common multiples’.

Recommend using 11mm lined A4 paper to present a Count and Place Value.Numerals are encouraged to be neat add able to be in-line for the Place Value step ofCyclic Addition. This yields stronger circle and stronger Mathematics.

Making a Wheel is part of Cyclic Addition and at least drawing a circle in the middleof the Wheel of 6 numbers helps to identify the Counting action as circular. Alsohelping the recognition of a multiple in a certain space around the wheel. This aids asa building block for future Advanced Cyclic Addition Steps including Place Value.

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7 Cyclic Addition Place Value ̶ Advanced Level

The next step in Cyclic Addition Mathematics is the creation of a perfect Place ValueSet for each place value position.

The Counting remains the same. However the pattern making of a Count’s PlaceValues brings together the circle and Circular Addition Sequences for 1 to 5 numbers.

How does it work ?

Circular Addition creates 30 sequences all made from the circular ‘ 1 3 2 6 4 5 ’.These 30 sequences are each placed into a wheel of there own. Flip through the 12Workbook pages and notice how the 30 circular addition sequences are placed intothere own circle.

The next step is to use rotation with a circle for between 1 and 5 numbers. Lookcarefully at the first 2 pages. The answers are given for all circles beginning with 1.Notice first there are 5 circles starting with 1. Each circle forms vertical Place ValueSets. Each Place Value Set has a total in bold at the top of the Set.

For Sets with just 1 number there are only 5 Place Value Sets possible. Remember aPlace Value Set has only between 1 and 5 numbers. For circles with 2 numbers like‘ 1 3’ there are a possible 8 Place Value Sets. For circles with 3 numbers like‘ 1 3 2 ’ there are a potential 10 Place Value Sets. For circles with 4 numbers like‘ 1 3 2 6 ’ there are 11 Place Value Sets. For circles with 5 numbers like‘ 1 3 2 6 4 ’ there are again 11 possible Place Value Sets. There are 45 Place ValueSets with a circle starting at ‘1’. Since there are 6 starts. This brings the whole totalpossible complete range of Place Value Sets to 45×6=270. Just for this Wheel of 6numbers.

All well and good, how is a Place Value Set created ?

Start at any unique number and Count / Apply Circular Mathematics for 1 to 5numbers moving around and around and possible around the circle. For a circle with 1number a 4 number Place Value Set is created by rotating 4 times. For a circle with 2numbers a 4 number Place Value Set is created by rotating completely twice. For acircle with 3 numbers a 5 number Place Value Set requires only a 1 and part circle tomake the 5. For a circle with 4 numbers a 2 number Place Value Set is only part of thecircle. For a circle with 5 numbers a 1 number Place Value Set is not needed as thesewould repeat the 1 number circles.

The way the circles form Place Value Sets prevents repetition. The circles again with‘1’, ‘1 3’, ‘1 3 2’, ‘1 3 2 6’ and ‘1 3 2 6 4’ have all answers given. Look at howthey’re constructed. Each circle has a number of groups that match the number ofnumbers in a circle. These groups have unique starts. 1 group of 5 for ‘1’. 2 groups of4 for ‘1 3’. 3 groups of 3 and 1extra =10 for ‘1 3 2’ with no repeats from any othercircle. 4 groups of 2+2+3+4=11 for ‘1 3 2 6’ again no repeats. 5 groups of1+1+2+3+4=11 for ‘1 3 2 6 4’ likewise no repeats.

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The only rule the student needs to seek a simple Advanced Place Value Set is pick a 1to 5 number circular addition sequence from the Wheel and treat it as a circle. Startfrom anywhere in the sequence and stop anywhere. Again 1 to 5 numbers only.

To practise making Place Value Sets the other 5 starts × 5 circles = 25 circles on 10pages are given as a workbook exercise. Remember this is just for the ‘commonmultiple’ 1 Wheel.

The Workbook exercise starts at circle ‘3’. Fill in the blanks and total the Place ValueSets. Find the Place Value Set first and then total all numbers in that Place Value Set.Those circles starting with ‘3’ have totals for the Place Value Sets given. Likewisewith circles beginning with ‘2’. The circles starting with ‘6’ are a little morechallenging and those starting at ‘4’ and ‘5’ become progressively harder.

This workbook exercise prepares the young Mathematician to discover how to form aPlace Value Set that perfectly fits with a Count for units, tens and hundreds positions.

The Guidebook “Mathematics with just Number” Chapter 5 has a complete list of the270 ‘common multiple’ 1 Place Value Sets sorted by total. These show acompleteness working with the circle and ‘common multiple’ to create AdvancedPlace Value.

Look carefully at the Guidebook table of 270 Place Value Sets. Pick a total and askthe student to find all possible Place Value Sets from the ‘ 1 3 2 6 4 5 ’ Wheel.Then when confident with Advanced Place Value with this wheel practise on anothersimple ‘common multiple’ Wheel.

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102

15 18 15 17 13 15 21 7 9 15 19

20 20 18 20 12 14 20 8 10 16 20

3

2

6

4

3

2

6

4

5

103

2 4 6 8 10

8 10 16 18 8 14 16 22

12 14 20 12 18 22 6 12 16 18

2

2

6

2

6

4

104

17 19 17 23 11 17 21 7 13 17 22

18 18 12 18 8 14 18 3 9 13 18

2

6

4

5

1

2

6

4

5

105

15 21 15 15

6

10 16 20 10 14 20

6

6

4

6

4

5

106

16 16 16 16

19 19 19 19 19

1

6

4

5

13

6

4

5

107

4 16

9 9

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4

4

5

1

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108

13 13 13 13

15 15 15 15 15

13

4

5

13

24

5

109

9

5

6

5

15

135

110

11

17

13

2

5

13

2

6

5

111

8 Cyclic Addition Count + Place Value ̶ Advanced Level

The next step in Cyclic Addition is to apply advanced Place Value Mathematics to aCount. This directly follows the pervious exercise of creating a Place Value Set froma circular addition sequence.

The two steps of Counting and Place Value are given to higher Laws and moreintricate pattern making with Number and the ‘common multiple’.

The beginner practise of Counting is exactly the same in these workbook exercises.What follows each Count is applying Advanced Place Value to every Count made.

The workbook exercises use Mathematics from ‘common multiple’ 1 to 21. Eachpage of Mathematics has the Count, where given, in bold and plenty of room to createPlace Value Sets underneath the Count.

As with previous workbook exercises we start with the ‘common multiple’ 1 whichhas 7 pages. All 6 starts for the ‘ 1 3 2 6 4 5 ’ Wheel are included. The 7th page isblank to have a go from scratch. The ‘common multiples’ 2 to 7 have 3 pages ofexercises each. The first two pages are given a start and the third is blank. The‘common multiples’ 8 to 21 have two pages each. The first page a start again is givenand the second page is again left blank to have a go from nothing but the wheel.

This practical workbook exercise focus’s on combining each Count with Place Value.For the first 3 pages of ‘common multiple’ 1 the Count is completely given and allthat is needed is to fill in the Place Values. The rest of the ‘common multiples’ 1 to 21have the first cycle of Counting given and all the Place Values are partly given.Requiring only to fill-it-in where there is a line underneath the Count.

Perhaps use of the Guidebook “Mathematics with just Number” table of Place ValueSets can be used for the 6 starts with ‘common multiple’ 1 and also the blank withonly the Wheel ‘ 1 3 2 6 4 5 ’ given. All of the Place Value Sets for ‘commonmultiple’ 1 are presented in the Guidebook Chapter 5, all 270 of them. Using this as avaluable reference tool will enable the student to discover not only the workings of aPlace Value Set, but also choosing a Place Value Set for each and every Count.

Once ‘Common multiple’ 1 is mastered the use of the blank page, even though theCount is repeated, allows practise with a new selection of Place Value Sets.

The whole Workbook exercise is to shape and mould the boundaries of each students’way that a Count number is built. This is accomplished with Advanced Place Value.

There is, as far as the author has checked and proofed the Workbook, only onesolution to every Count. That’s all just one. Complete answers are on the CD.

From the previous Workbook chapter the student is encouraged to focus on the Wheelto make a Place Value Set. Here is the same, but rather requires the student to exploremaking mini-wheels within the wheel to generate a Place Value Set that fits theWorkbook exercise. Not only does the Wheel remain the tried, true and tested way of

112

applying ‘common multiple’ 1, but the inner workings of the wheel are put to the testwith every Count, with every Place Value Set.

Some basics with the Workbook exercises show the following. All the Counts followthe first cycle of counting. All the Place Value Sets in a Count add to the Count. Allplace value positions are indicated upon the exercise. Whether they be units first orfollowed by tens. In fact the tens place value is always marked as so “_”. An endCount to the 27 counts or 4 and half cycles each page is also given to mark the end ofeach Count. This is to aid accuracy.

The Place Value Sets are designed to explore the Wheel, both its circular nature andalso the workings of the ‘common multiple’. Just like in Intermediate CyclicAddition.

The Place Value Step is performed immediately after Counting. As the Count has justbeen created with Addition and Circle with the Wheel, the energy of what Number theCount represents is continued over into the Place Value Step. This is a fundamentalaction of Cyclic Addition and the next Step Remainder relies upon this very fact.

Once ‘common multiple’ 1 is mastered. Use the around the Wheel whole numberDivision to apply the other ‘common multiple’ Wheels.

For those getting stuck with the higher ‘common multiples’ use the reference pages inthe Guidebook to make a multiple of the working ‘common multiple’.

For example using Wheel ‘ 3 9 6 18 12 15 ’ with Count 63. There is a tens of 3 bydeduction. Leaving 33. 33=11×3. From the Guidebook only one solution is possible.3+2+6=11. Note the position around the Wheel. Follow the position for ‘commonmultiple’ 3. Simply 9+6+18=33. Then fill-in the Workbook.

For example using Wheel ‘ 4 12 8 24 16 20 ’ with Count 84. What’s given24+_+_+_+24=84. Look at their relative positions around the wheel. Find6+_+_+_+6=21 a little easier. Look at the patterns in the Guidebook. 6+4+3+2+6=21then multiply all by 4 or note the relative positions around the Wheel are the same.Look carefully for the sequence as well as the numbers that make up the Count. Thesequence is just as important as the numbers hence making ‘a Wheel’.

For Example using Wheel ‘ 5 15 10 30 20 25 ’ with Count 50. Simple? What’sgiven _+15+10+_+_=50 so using the multiples of 5. 25=15+10 so 25 is left. Lookcarefully at the circular addition sequence ‘ 5 15 10 ’ now wheel around thissequence to fit ‘15 10’ as the second and third numbers in the Place Value Set. Fromthe Laws of creating a Place Value Set the first number is either 10 or 5. Try 5 andloop around the wheel to fill-in ‘5+15’ at the fourth and fifth Place Value numbers.

For Example using Wheel ‘ 6 18 12 36 24 30 ’ with Count 102. What’s given_+36+30+_+_=102. Look for the circular addition sequence within the Wheel. Thenumber 36 and 30 are not in sequence therefore must be wrapping around the wheel.Does the answer use the circular sequence fit 12+36+30+6+18=102 yes.

113

For Example using Wheel ‘ 7 21 14 42 28 35 ’ with the last Count 665. What’sgiven _+35+_= units 5 and four tens Place Values with 7+21+_+_=? To maintainunits 5 the units Place Values must be 35+35+35=105. The tens =56_ now that’s8×7=56. How to create the 4 number Place Value set. 7+21=4×7 so repeating thenumbers 7+21 as they’re in sequence, 7+21+7+21=56.

There are only 270 Place Value Sets per Wheel. And only one answer to the fill-it-inWorkbook exercises. So reduce the possibilities, follow the sequence, add the PlaceValues try and try again and seek out how the ‘common multiple’ is built.

114

1 4 6 12 16 21 22 25 27

2 6 3 5 4

1 6 5 5

2

2 _ _ 5

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5 6 1 2 6 6 2

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4

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115

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116

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117

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118

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119

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120

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190

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122

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192

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285

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288

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470

30 5

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465

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133

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558

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134

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540

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139

48 80 120 128 152 168

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792

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141

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143

50 60 90 110 170 210

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930

40 60 10

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144

145

11 44 66 132 176 231

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990

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146

147

36 60 132 180 240 252

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148

149

26 104 156 221 234 273

65 52

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1248

26 65 13 39 39

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151

84 140 210 224 266 294

70 14 14 14 28 70

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1386

14 14 56 56 84

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152

153

60 135 150 195 225 315

15 90 60 75 60

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154

155

80 96 144 176 272 336

48 16

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34 51 102 51 68 34 85

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159

54 90 198 270 360 378

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_ _ _

160

161

38 152 228 323 342 399

19 57

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38

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_ _ _

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38

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1824

114 57 76 38

76 19

114

_ 19 _ _

57_ 57_ 114_ 95_ 57_ 76_ 19 _

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_

162

163

120 200 300 320 380 420

40 100 80 100 80

20 80 40

40 40 60

_

40 _ _

_

120 100 80 60 100 60

40 80 20

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_ _ _ 40 _ 80 _ 60_

_ 60_ _ _ _ _

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1980

80 100 60 20

20

100 80

100

40 40_ 120_ _

_ _ _ _ 60_ 120_ 60_ 80_

100_ 40_ _ _ _

164

165

84 189 210 273 315 441

105 42 126 105

63

105 84 21

_

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126 84 84 105 84

42

84

42_ 84

_ _ _ 105 105_ _

21_ _ _ _ 42_

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1974

21 42 84 84 63

126 63

42_ 42 42

21 42_ _ 84_ 21 21_ 63_ 105_

_ _ _ _ _ _ _ 21_

105_ _

166

167

9 Cyclic Addition Remainder ̶ Advanced Level

The final Step of Cyclic Addition is Remainder. Actually the Remainder presentswhat’s called a ‘7×Multiple’. The Remainder is the joining mechanism between theCount and a higher order 7×Multiple.

The Remainder is found by manipulating each Place Value Set and making aRemainder for each Set. Then place all the Remainders into the ‘units’ add them up toequal the final Remainder. Subtract the Remainder from the Count to reveal the7×Multiple. Then continue with the next Count.

So Cyclic Addition Steps follow the order of Count, Place Value, Move tens to units,Remainder, 7×Multiple then returning to the next Count. The workbook templatemakes this step by step process of discovering Cyclic Addition easier and moremethodical.

The Remainder is always a number from the Counting Wheel of 6 numbers. TheRemainder is found for each Place Value Set. Typically units then tens and possiblyhundreds.

The Remainder Step for the ‘tens’ and ‘hundreds’ positions have a small step called‘Move tens to units’. Merely find the remainder for ‘tens’ first, match this number tothe Wheel and rotate around the Wheel one number clockwise. For the ‘hundreds’Remainder rotate two numbers clockwise. Add all the Remainders up to find a finalRemainder. Record these steps on the Workbook exercise in turn.

How to find a Remainder from a Place Value Set ?

Two ways. The first add all the numbers in a Place Value Set together and take awaywhat is left over from the 7×Multiple. The 7×Multiple is a multiple of 7 × ‘commonmultiple’.

The second way is find the pattern of a Place Value Set on the Wheel, eliminate allthe Place Values that add to 7 × ‘common multiple’ and receive what is left over. Thiswill be a single number from the Wheel.

Once the Remainder is found apply the simple formula of Count ̶ Remainder = 7×Multiple. Always even if the Remainder = zero.

Let’s follow the steps with the Workbook exercises for Remainder. ‘Commonmultiple’ 1 goes for exactly 4 cycles. Two cycles to a page. So each Wheel is given 2pages. There is a Count and all Cyclic Addition Steps in a template for ‘commonmultiples’ 1 to 21 and a blank.

For ‘common multiple’ 1 the first page with all Steps is given. Note the Remainder inthe ‘units’ comes from the Place Value Set in the units. The tens from the tens PlaceValue Set. The Place Values are converted to a Remainder by the Mathematics of theWheel leaving a Remainder from each Place Value Set. The Remainder in the ‘tens’is moved around the Wheel as shown.

168

The Remainder units and tens form a pattern of two numbers from the wheel. Theseare converted to a single number Remainder.

To convert a Remainder pattern of 1 to 5 numbers to a single number Remainderrequires practise. Hence the Cyclic Addition Step. The Guidebook “Mathematics withjust Number” in Chapter 6 shows all 270 Place Value Sets and the resultantRemainder. As well as simple beginner counts and advanced counts with all CyclicAddition Steps.

For a simple starting example use the Wheel with two number Place Value Sets.Following the table in the Guidebook is helpful. A Pattern with two of the samenumbers rotate around the Wheel two numbers to find a Remainder. This Patternworks all the way around the Wheel. So to with every Remainder Pattern. In fact the45 patterns with 6 of each type = 270 possible Place Value Sets. Continuing with twonumber Patterns. A Pattern with two numbers next to each other move around theWheel 3 numbers from the second number clockwise to find a Remainder. A Patternwith two numbers spaced two apart the Remainder is the number in-between both.Again follow the Guidebook Patterns. A Pattern of two numbers spaced three apart oropposite each other on the Wheel leaves a zero Remainder as they both add to a7×Multiple.

For another Pattern example let’s use 3 number Place Value Sets. Three of the samenumber Pattern gives a Remainder of the next number around the Wheel. Threenumbers in sequence like ‘ 1 3 2 ’ also gives a Remainder of the next number aroundthe Wheel. Three numbers all spaced two apart like ‘ 1 2 4 ’ gives a zero Remainderas they add to a 7×Multiple.

Go through the Guidebook 45 Patterns to learn how to seek out the Remainder fromeach Place Value Set. Tips include eliminate the numbers opposite each other aroundthe wheel and use the simpler two and three numbers left over from the Pattern tomake a Remainder. Other wise just add the Place Value Set and work out manuallyhow to find both the 7×Multiple and the Remainder.

The Guidebook has 7 Wheels with a complete 4 cycle Count and all other Steps.Follow these practicals carefully to master the Mathematics of applying the Wheel atevery stage. Whether it’s the Count around the wheel, finding Place Value SetPatterns that equal the Count, applying Remainder Patterns to find a single numberaround the Wheel, or following the final Remainder to the next Count Number aroundthe Wheel. The Wheel is, as given in the Workbook, used in every Step of CyclicAddition.

The Remainder sequence for the first 6 Counts form a cycle. This Cycle repeats itselfwith exactly the same sequence next cycle or every 6 counts. This makes it handy toboth check the next Count number in sequence around the Wheel. And to check theaccuracy of each Count. A correct Remainder will reveal both a true Count and7×Multiple. If the Count is chosen incorrectly the Remainder sequence within a cyclewill be flawed.

169

The 7×Multiples ? What is their relevance with Cyclic Addition. The 7×Multiples asimply a higher order of the same ‘common multiple’. They are declared in sequenceas the student Counts along the Cycle.

The next Chapter ‘Hierarchy’ discusses how the 7×Multiple works within thecomplete Cyclic Addition. Once again the Guidebook Chapter 7 ventures into theboth theoretical and practical realms of Cyclic Addition with higher order Wheels andfurther Pattern Making to match a student’s infinite potential.

170

Count Cycle #1 1 4 6 12 16 21

Place Value Sets 1 3 2 6 3 2

1 2 1 4 6

2 3 5 3

2 1 1_

3

Remainder Units 1 4 6 5 2 4Remainder Tens 1_Move tens to units 3

Remainder 1 4 6 5 2 -

7 x Multiple (7=7x1) - - - 7 14 21

Count Cycle #2 22 25 27 33 37 42

Place Value Sets 4 5 4 5 6 2

5 1 5 4 3 4_

3 4 2 5 2

1_ 5 6 4 6

1_ 1_ 5 2_

1_

Remainder Units 5 1 3 2 3 2Remainder Tens 1_ 1_ 1_ 1_ 2_ 4_Move tens to units 3 3 3 3 6 5

Remainder 1 4 6 5 2 -

7 x Multiple (7=7x1) 21 21 21 28 35 42

171

Count Cycle #3 43 46 48 54 58 63

Place Value Sets 1 6 2 6 6 2

3 4 6 4 6 1

5 6 2 1 6 1_

1 3_ 6 3 4_ 3_

3 2 4_ 2_

3_ 3_

Remainder Units 6 2 4 - 4 3Remainder Tens 3_ 3_ 3_ 4_ 4_ 6_Move tens to units 2 2 2 5 5 4

Remainder

7 x Multiple (7=7x1)

Count Cycle #4 64 67 69 75 79 84

Place Value Sets 1 5 4 6 4 4

1 1 5 4 5 6

1 3 4_ 5 1 4

1 2 2_ 3_ 3 6

6_ 6 3_ 6 4

5_ 5_ 2_

1_ 4_

Remainder Units 4 3 2 1 5 3Remainder Tens 6_ 5_ 6_ 6_ 6_ 6_Move tens to units 4 1 4 4 4 4

Remainder

7 x Multiple (7=7x1)

172

Count Cycle #1 6 10 22 30 40 42

Place Value Sets 6 8 4 12 10 2

2 12 8 10 4_

6 10 10

10

Remainder Units 6 10 8 2 12 2Remainder Tens 4_Move tens to units 12

Remainder 6 10 8 2 12 -

7x Multiple (14=7x2) - - 14 28 28 42

Count Cycle #2 48 52 64 72 82 84

Place Value Sets 10 2 8 10 2 12

2 6 8 2 8 12

4 4 8 6_ 10 2_

12 4_ 4_ 2 2_

2_ 4_ 2_

2_

Remainder Units - 12 10 12 8 10Remainder Tens 2_ 4_ 4_ 6_ 6_ 6_Move tens to units

Remainder

7x Multiple (14=7x2)

173

Count Cycle #3 90 94 106 114 124 126

Place Value Sets 10 8 6 6 4 6

8_ 6 12 10 6 8

4_ 8 2 4 10

4_ 2_ 6 6 2

6_ 10 4 8_

8_ 10_ 2_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (14=7x2)

Count Cycle #4 132 136 148 156 166 168

Place Value Sets 2 2 8 12 2 4

12 6 4_ 4 4 12

8 2 10_ 6_ 4_ 10

10 6 2_ 10_ 2

4_ 10_ 6_ 2_ 2_

6_ 2_ 12_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (14=7x2)

174

Count Cycle #1 9 15 33 45 60 63

Place Value Sets 9 12 6 18 15 9

3 15 12 18 6

3 15 12 18

9 15 3_

Remainder Units 9 15 12 3 18 12Remainder Tens 3_Move tens to units 9

Remainder 9 15 12 3 18 -

7x Multiple (21=7x3) - - 21 42 42 63

Count Cycle #2 72 78 96 108 123 126

Place Value Sets 15 15 12 18 15 6

3 3 12 12 3 18

6 12 12 18 15 12

18 15 6_ 6_ 9_ 9_

3_ 3

3_

Remainder Units - 6 15 6 12 15Remainder Tens 3_ 3_ 6_ 6_ 9_ 9_Move tens to units 9 9 18 18 6 6

Remainder

7x Multiple (21=7x3)

175

Count Cycle #3 135 141 159 171 186 189

Place Value Sets 9 6 3 6 9 3

6 9 9 18 9 6

18 6 12 12 9 3_

12 12_ 15 15 9 9_

9_ 12_ 12_ 15_ 6_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (21=7x3)

Count Cycle #4 198 204 222 234 249 252

Place Value Sets 6 3 15 15 12 6

18 9 12 12 15 6

6 3 15 15 3 9_

18 9 12_ 12 9 6_

15_ 18_ 6_ 18_ 3_ 9_

18_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (21=7x3)

176

Count Cycle #1 24 40 60 64 76 84

Place Value Sets 24 20 20 24 24 12

20 4 16 12 8

16 4 4_ 20

20 12 4

8 4_

Remainder Units 24 12 4 8 8 16Remainder Tens 4_ 4_Move tens to units 12 12

Remainder 24 12 4 8 20 -

7x Multiple (28=7x4) - 28 56 56 56 84

Count Cycle #2 108 124 144 148 160 168

Place Value Sets 12 8 8 24 12 8

20 20 24 4 8 24

4 4 8 12_ 12 16

12 12 24 8 12_

20 8_ 8_ 12_

4_

Remainder Units 12 16 8 - 12 20Remainder Tens 4_ 8_ 8_ 12_ 12_ 12_Move tens to units 12 24 24 8 8 8

Remainder

7x Multiple (28=7x4)

177

Count Cycle #3 192 208 228 232 244 252

Place Value Sets 16 16 8 4 12 8

20 20 24 12 8 24

4 12 16 4 24 20

8 16_ 20 12 20_ 12_

24 8_ 20_ 8_

12_ 8_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (28=7x4)

Count Cycle #4 276 292 312 316 328 336

Place Value Sets 16 16 12 12 4 12

20 20 8 8 12 4

4 16 12 4 8 16_

16 24_ 4_ 12 20 4_

20 24_ 16_ 4 12_

16_ 12_ 8_

4_ 20_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (28=7x4)

178

Count Cycle #1 20 45 50 65 75 105

Place Value Sets 5 25 5 10 10 5

15 5 15 30 20 30

15 10 15 25 20

5 10 5 5_

15 15

Remainder Units 20 10 15 30 5 20Remainder Tens 5_Move tens to units 15

Remainder 20 10 15 30 5 -

7x Multiple (35=7x5) - 35 35 35 70 105

Count Cycle #2 125 150 155 170 180 210

Place Value Sets 15 10 20 10 25 25

15 15 25 30 15 5

15 10 10 20 10 25

15 15 10_ 10 30 5

15 10_ 10_ 10_ 15_

5_

Remainder Units 5 15 20 - 10 25Remainder Tens 5_ 10_ 10_ 10_ 10_ 15_Move tens to units 15 30 30 30 30 10

Remainder

7x Multiple (35=7x5)

179

Count Cycle #3 230 255 260 275 285 315

Place Value Sets 30 30 5 25 25 10

20 15 15 25 5 30

5 10 10 25 15 25

15 20_ 25 20_ 10 25_

10 5 30

15_ 20_ 20_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (35=7x5)

Count Cycle #4 335 360 365 380 390 420

Place Value Sets 30 20 25 30 15 15

20 10 5 20 25 10

25 30 15 25 5_ 20

10 10_ 20 5 30_ 25

25_ 5_ 15_ 30_ 15_

15_ 15_ 20_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (35=7x5)

180

Count Cycle #1 30 36 54 66 102 126

Place Value Sets 18 12 12 24 12 6

12 24 6 30 36 24

18 12 30 30

12 6 6

6 18 6_

Remainder Units 30 36 12 24 18 24Remainder Tens 6_Move tens to units 18

Remainder 30 36 12 24 18 -

7x Multiple (42=7x6) - - 42 42 84 126

Count Cycle #2 156 162 180 192 228 252

Place Value Sets 36 36 18 36 36 24

24 24 12 36 24 24

30 30 18 12_ 12 24

6 12 12 36 18_

6_ 6_ 12_ 12_

Remainder Units 12 18 18 30 24 30Remainder Tens 6_ 6_ 12_ 12_ 12_ 18_Move tens to units 18 18 36 36 36 12

Remainder

7x Multiple (42=7x6)

181

Count Cycle #3 282 288 306 318 354 378

Place Value Sets 24 36 6 6 24 18

30 24 18 18 30 12

6 30 12 30 36 18

18 18 30 6 24 12

24 18_ 24_ 18 24_ 18

18_ 18_ 30_

6_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (42=7x6)

Count Cycle #4 408 414 432 444 480 504

Place Value Sets 30 6 36 24 12 6

6 18 24 30 12 6

18 36 12 18 12 6

24 24 36 12 12 6

30 30 24 18_ 12 24_

6_ 30_ 30_ 18_ 6_ 24_

24_ 36_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (42=7x6)

182

Count Cycle #1 21 35 77 105 140 147

Place Value Sets 21 7 35 28 42 7

28 7 35 28 21

35 7 42 14

21 28 35

14 7_

Remainder Units 21 35 28 7 42 28Remainder Tens 7_Move tens to units 21

Remainder 21 35 28 7 42 -

7x Multiple (49=7x7) - - 49 98 98 147

Count Cycle #2 168 182 224 252 287 294

Place Value Sets 14 42 14 35 21 35

42 28 21 7 14 7

28 7 14 28 42 35

14 21 21 35 14_ 7

7_ 14 14 7 7_ 21_

7_ 14_ 14_

Remainder Units - 14 35 14 28 35Remainder Tens 7_ 7_ 14_ 14_ 21_ 21_Move tens to units 21 21 42 42 14 14

Remainder

7x Multiple (49=7x7)

183

Count Cycle #3 315 329 371 399 434 441

Place Value Sets 28 42 21 28 21 14

21 21 42 35 35 42

14 14 28 7 7 35

42 42 14_ 21 21 7_

21_ 21_ 14_ 28 35_ 28_

28_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (49=7x7)

Count Cycle #4 462 476 518 546 581 588

Place Value Sets 28 35 28 42 35 21

14 7 35 42 7 14

42 14 21 42 21 42

28 21_ 14 7_ 28 21

35_ 21_ 42_ 21_ 14_ 42_

14_ 35_ 7_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (49=7x7)

184

Count Cycle #1 48 80 120 128 152 168

Place Value Sets 16 8 40 40 24 16

8 32 40 8 16 40

24 40 40 32 8 8

40 24 24

8 8_ 8_

Remainder UnitsRemainder TensMove tens to units

Remainder 48 24 8 16 40 -

7x Multiple (56=7x8)

Count Cycle #2 216 248 288 296 320 336

Place Value Sets 8 24 48 48 8 32

24 16 32 24 24 16

16 48 8 16 40 48

8 16_ 24 48 8 16_

16_ 16 16_ 24_ 8_

16_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (56=7x8)

185

Count Cycle #3 384 416 456 464 488 504

Place Value Sets 24 32 16 48 48 40

8 40 24 8 32 8

24 8 16 24 8 24

8 16 40_ 16 40_ 32

32_ 32_ 48 24_

32_ 16_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (56=7x8)

Count Cycle #4 552 584 624 632 656 672

Place Value Sets 32 16 40 24 24 24

40 40 8 24 40 16

8 8 16 24 8 48

32 24 32_ 8_ 24 24

40 16 24_ 40_ 16_ 48_

8_ 48_ 8_ 24_ 8_

32_ 16_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (56=7x8)

186

Count Cycle #1 36 81 90 117 135 189

Place Value Sets 9 27 27 54 45 45

27 27 18 36 9 9

27 27 27 27 45

18 18 9_

36

Remainder UnitsRemainder TensMove tens to units

Remainder 36 18 27 54 9 -

7x Multiple (63=7x9)

Count Cycle #2 225 270 279 306 324 378

Place Value Sets 36 18 45 27 9 36

27 54 9 18 27 36

18 36 54 54 18 36

54 18 36 27 54 9_

9_ 54 45 18_ 36 9_

9_ 9_ 18_ 9_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (63=7x9)

187

Count Cycle #3 414 459 468 495 513 567

Place Value Sets 9 9 54 54 9 9

54 36 36 36 27 27

36 45 18 45 18 45

45 9 54 9_ 9 9

27_ 18_ 36 27_ 27_ 27

18_ 27_ 18_ 45_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (63=7x9)

Count Cycle #4 603 648 657 684 702 756

Place Value Sets 54 36 18 36 9 36

45 45 54 18 27 45

9 9 27 54 9 9

27 18 18 36 27 36

18 27_ 36_ 54_ 9_ 27_

45_ 27_ 18_ 54_ 36_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (63=7x9)

188

Count Cycle #1 50 60 90 110 170 210

Place Value Sets 30 10 30 10 50 40

20 50 60 30 10 50

20 50 20

50 10 60

50 40

Remainder UnitsRemainder TensMove tens to units

Remainder 50 60 20 40 30 -

7x Multiple (70=7x10)

Count Cycle #2 260 270 300 320 380 420

Place Value Sets 30 60 40 40 30 60

20 30 50 40 60 40

60 20 10 40 40 20

40 60 60 20_ 50 60

10 10_ 40 20_ 40

10_ 10_ 20_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (70=7x10)

189

Count Cycle #3 470 480 510 530 590 630

Place Value Sets 50 40 50 10 40 10

20 50 10 30 50 30

60 40 30 50 60 20

40 50 20 10 40 60

30_ 30_ 40_ 30 40_ 10

40_ 50_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (70=7x10)

Count Cycle #4 680 690 720 740 800 840

Place Value Sets 20 20 40 40 60 60

20 10 50 60 30 60

20 30 30 40 20 60

20 20 60_ 60_ 60 60

50_ 10 30 40_

10_ 60_ 60_ 20_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (70=7x10)

190

Count Cycle #1 11 44 66 132 176 231

Place Value Sets 11 22 22 33 44 55

22 11 33 55 11

33 33 11 55

33 66 11_

Remainder UnitsRemainder TensMove tens to units

Remainder 11 44 66 55 22 -

7x Multiple (77=7x11)

Count Cycle #2 242 275 297 363 407 462

Place Value Sets 11 33 44 66 44 22

33 22 55 44 55 44

22 44 11 55 11 55

55 55 33 22 33 11

11 11 44 66 44 33_

11_ 11_ 11_ 11_ 22_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (77=7x11)

191

Count Cycle #3 473 506 528 594 638 693

Place Value Sets 22 44 66 44 55 66

55 22 44 55 11 44

11 66 22 66 22 33

33 44 66 44 55_ 55_

22 33_ 33_ 55

33_ 33_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (77=7x11)

Count Cycle #4 704 737 759 825 869 924

Place Value Sets 55 11 55 55 33 22

11 66 11 11 22 66

33 44 33 44 44 11

55 55 66_ 55 44_ 33

33_ 11 66_ 33_ 22

22_ 55_ 66_

11_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (77=7x11)

192

Count Cycle #1 36 60 132 180 240 252

Place Value Sets 24 12 12 72 48 60

12 36 36 48 48 12

12 24 60 48 60

60 48 12_

48

Remainder UnitsRemainder TensMove tens to units

Remainder 36 60 48 12 72 -

7x Multiple (84=7x12)

Count Cycle #2 288 312 384 432 492 504

Place Value Sets 72 36 48 12 12 36

48 24 60 36 48 60

12 12 12 24 60 12

36 24_ 24 72 12 36

12_ 24_ 48 36_ 36_

24_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (84=7x12)

193

Count Cycle #3 540 564 636 684 744 756

Place Value Sets 60 12 24 72 36 12

12 36 72 48 36 60

36 24 36 60 36 12

24 12 24 24 36 60

48 12_ 48_ 48_ 12_ 12

36_ 36_ 48_ 60_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (84=7x12)

Count Cycle #4 792 816 888 936 996 1008

Place Value Sets 48 36 36 12 12 24

60 24 24 36 36 72

12 36 72 12 60 48

72 36_ 36 36 12 24

36_ 36_ 72_ 12_ 36 12_

24_ 72_ 60_ 60_

24_ 12_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (84=7x12)

194

Count Cycle #1 26 104 156 221 234 273

Place Value Sets 13 26 26 65 65 78

13 13 78 13 13 52

39 52 39 39 13

26 26 52 13_

78 65

Remainder UnitsRemainder TensMove tens to units

Remainder 26 13 65 39 52 -

7x Multiple (91=7x13)

Count Cycle #2 299 377 429 494 507 546

Place Value Sets 13 52 13 26 39 52

26 65 39 78 26 65

78 13 65 26 13 13

52 52 13 78 39 26

13_ 65 39 26 39_ 13_

13_ 26_ 26_ 26_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (91=7x13)

195

Count Cycle #3 572 650 702 767 780 819

Place Value Sets 26 65 39 52 26 65

78 65 26 65 26 13

52 65 65 78 26 39

26 65 13 52 26 52

39_ 39_ 39 52_ 26 13_

26_ 65_ 52_

26_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (91=7x13)

Count Cycle #4 845 923 975 1040 1053 1092

Place Value Sets 26 26 39 65 78 52

52 65 26 39 39 65

65 13 78 26 26 39

13 39 52 78 65_ 26

39 52_ 78_ 52 26_ 78_

65_ 26_ 78_ 13_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (91=7x13)

196

Count Cycle #1 84 140 210 224 266 294

Place Value Sets 70 28 70 14 14 14

14 28 14 42 42 56

28 56 28 84 70

28 70 84 56 14

28 56 70 14_

Remainder UnitsRemainder TensMove tens to units

Remainder 84 42 14 28 70 -

7x Multiple (98=7x14)

Count Cycle #2 378 434 504 518 560 588

Place Value Sets 28 84 70 14 56 42

84 56 14 42 84 42

56 14 84 28 56 42

70 28_ 56 84 84 42

14_ 28_ 70 28_ 28_

28_ 14_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (98=7x14)

197

Count Cycle #3 672 728 798 812 854 882

Place Value Sets 42 28 14 70 28 28

84 84 42 14 84 70

56 14 28 42 42 14

70 42 14 56 70_ 42

42_ 28_ 70_ 70 28

28_ 56_ 70_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (98=7x14)

Count Cycle #4 966 1022 1092 1106 1148 1176

Place Value Sets 28 14 70 14 14 56

84 42 14 42 42 70

42 56 28 84 28 42

28 70 14_ 56 84 28

84 42_ 84_ 70 56_ 14_

70_ 42_ 84_ 42_ 84_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (98=7x14)

198

Count Cycle #1 60 135 150 195 225 315

Place Value Sets 15 45 15 15 90 60

45 75 45 75 60 75

15 30 15 75 30

60 75 15_

15

Remainder UnitsRemainder TensMove tens to units

Remainder 60 30 45 90 15 -

7x Multiple (105=7x15)

Count Cycle #2 375 450 465 510 540 630

Place Value Sets 75 60 30 60 75 30

15 60 90 75 15 45

45 60 45 45 90 30

90 60 30_ 30 60 45

15_ 60 30_ 30_ 30

15_ 45_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (105=7x15)

199

Count Cycle #3 690 765 780 825 855 945

Place Value Sets 60 90 60 45 60 15

30 45 75 45 75 30

90 30 15 45 15 90

60 15_ 30 45 45 60

45_ 45_ 60_ 45 60 75_

60_ 60_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (105=7x15)

Count Cycle #4 1005 1080 1095 1140 1170 1260

Place Value Sets 30 75 30 30 30 60

90 15 90 90 90 75

60 75 45 60 60 90

75 15 30 15 75 60

45_ 60_ 90_ 45 15 75

30_ 30_ 45_ 90_ 90_

45_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (105=7x15)

200

Count Cycle #1 80 96 144 176 272 336

Place Value Sets 32 64 80 96 32 48

48 32 64 80 96 32

64 16

80 48

32

16_

Remainder UnitsRemainder TensMove tens to units

Remainder 80 96 32 64 48 -

7x Multiple (112=7x16)

Count Cycle #2 416 432 480 512 608 672

Place Value Sets 48 16 80 64 48 16

32 96 80 32 32 80

96 64 80 96 48 16

80 80 80 32_ 48_ 80

16_ 16 16_ 32_

16_ 16_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (112=7x16)

201

Count Cycle #3 752 768 816 848 944 1008

Place Value Sets 16 96 32 80 32 48

48 96 80 16 16 96

32 96 16 48 48 64

96 48_ 48 64 32 48_

80 32_ 64_ 16 32_

48_ 32_ 80_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (112=7x16)

Count Cycle #4 1088 1104 1152 1184 1280 1344

Place Value Sets 96 64 64 96 16 32

64 80 80 64 48 96

32 96 16 16 80 64

96 64 32 48 16 32

80_ 80_ 48_ 96_ 32_ 16_

48_ 80_ 96_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (112=7x16)

202

Count Cycle #1 17 68 102 204 272 357

Place Value Sets 17 34 85 51 102 51

34 17 85 68 34

17 17 85

51 51 17

34 17_

Remainder UnitsRemainder TensMove tens to units

Remainder 17 68 102 85 34 -

7x Multiple (119=7x17)

Count Cycle #2 374 425 459 561 629 714

Place Value Sets 68 34 85 34 102 102

85 68 17 102 68 68

17 85 85 68 85 34

34 17 17 17 34 102

17_ 51 85 34_ 34_ 68

17_ 17_ 34_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (119=7x17)

203

Count Cycle #3 731 782 816 918 986 1071

Place Value Sets 102 17 102 85 34 17

68 102 102 17 102 51

51 68 102 34 68 34

17_ 85 17_ 102 85 102

34_ 51_ 17_ 34_ 17 17

17_ 34_ 68_ 85_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (119=7x17)

Count Cycle #4 1088 1139 1173 1275 1343 1428

Place Value Sets 68 68 102 51 68 34

85 85 68 51 85 102

17 17 85 51 17 17

68 51 17 51 68 51

51_ 68 51 51 85 34

34_ 85_ 85_ 51_ 102_ 102_

51_ 17_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (119=7x17)

204

Count Cycle #1 54 90 198 270 360 378

Place Value Sets 18 18 72 108 72 108

18 72 90 72 108 72

18 36 90 72 90

108 108

Remainder UnitsRemainder TensMove tens to units

Remainder 54 90 72 18 108 -

7x Multiple (126=7x18)

Count Cycle #2 432 468 576 648 738 756

Place Value Sets 54 54 54 36 18 72

36 72 90 108 108 90

72 90 18 72 72 54

90 18 54 18 36_ 54_

18_ 54 36_ 54 18_

18_ 36_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (126=7x18)

205

Count Cycle #3 810 846 954 1026 1116 1134

Place Value Sets 90 108 90 36 18 36

18 54 18 108 90 108

72 36 54 72 18 90

90 108 72 90 90 90_

54_ 54_ 36_ 72_ 54_

36_ 36_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (126=7x18)

Count Cycle #4 1188 1224 1332 1404 1494 1512

Place Value Sets 72 36 54 90 18 90

18 36 36 18 54 18

54 36 90 36 36 36

36 36 18 18_ 108 108

108 18_ 54 90_ 18 72_

90_ 54_ 108_ 18_ 90_ 54_

36_ 36_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (126=7x18)

206

Count Cycle #1 38 152 228 323 342 399

Place Value Sets 19 19 38 19 114 57

19 57 114 114 114 38

76 19 76 114 19

57 95 57

19 38

19_

Remainder UnitsRemainder TensMove tens to units

Remainder 38 19 95 57 76 -

7x Multiple (133=7x19)

Count Cycle #2 437 551 627 722 741 798

Place Value Sets 38 57 114 38 95 114

114 114 76 114 19 76

57 76 95 76 57 38

38 95 38 95 19_ 114

19_ 19 114 19 38_ 76

19_ 19_ 38_ 38_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (133=7x19)

207

Count Cycle #3 836 950 1026 1121 1140 1197

Place Value Sets 76 76 114 76 95 19

95 76 38 95 57 57

57 76 114 19 38 38

38 76 76_ 76 114 114

57_ 76 95 76 19

57_ 76_ 76_ 95_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (133=7x19)

Count Cycle #4 1235 1349 1425 1520 1539 1596

Place Value Sets 114 57 76 19 57 95

76 38 95 57 38 76

95 95 19 95 114 95

19_ 19 57 19 38_ 38_

57_ 57_ 38 114_ 95_ 57_

19_ 57_ 114_ 19_ 38_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (133=7x19)

208

Count Cycle #1 120 200 300 320 380 420

Place Value Sets 40 100 80 100 40 100

40 60 100 20 120 20

40 40 20 120 80 100

60 80 100 20_

40 40

Remainder UnitsRemainder TensMove tens to units

Remainder 120 60 20 40 100 -

7x Multiple (140=7x20)

Count Cycle #2 540 620 720 740 800 840

Place Value Sets 100 20 80 120 100 20

20 80 40 60 120 60

60 100 120 40 80 40

40 20 80 120 100 120

120 40_ 40_ 40_ 40_ 60_

20_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (140=7x20)

209

Count Cycle #3 960 1040 1140 1160 1220 1260

Place Value Sets 80 60 80 120 100 60

40 100 100 120 20 80

120 20 20 120 120 100

80 60 60 20_ 80 20

40 20_ 80 20_ 100 60_

60_ 60_ 80_ 20_ 80_ 40_

20_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (140=7x20)

Count Cycle #4 1380 1460 1560 1580 1640 1680

Place Value Sets 40 20 80 80 100 80

120 60 100 100 40 120

80 100 80 120 120 80

100 20 100 80 80 120

40 60 40_ 120_ 100 80

100_ 20_ 40_ 120_ 60_

100_ 40_ 60_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (140=7x20)

210

Count Cycle #1 84 189 210 273 315 441

Place Value Sets 21 105 21 42 63 126

63 84 63 105 42 84

105 21 84 105

21 63 105 126

42 21

Remainder UnitsRemainder TensMove tens to units

Remainder 84 42 63 126 21 -

7x Multiple (147=7x21)

Count Cycle #2 525 630 651 714 756 882

Place Value Sets 126 84 105 126 84 84

84 105 21 84 21 84

105 21 63 21 63 84

126 126 42 63 42 42_

84 84 42_ 42_ 126 21_

21_ 21_

21_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (147=7x21)

211

Count Cycle #3 966 1071 1092 1155 1197 1323

Place Value Sets 105 84 105 21 21 21

21 105 42 63 63 42

126 42 126 42 42 126

84 126 84 84 21 84

63_ 84 105 105 105_ 63_

63_ 63_ 63_ 42_

21_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (147=7x21)

Count Cycle #4 1407 1512 1533 1596 1638 1764

Place Value Sets 21 42 42 84 42 84

126 126 126 42 126 105

84 84 63 126 84 63

105 42_ 42 84 105 42

21 42_ 126_ 84_ 21 21_

105_ 42_ 42_ 126_ 126_

Remainder UnitsRemainder TensMove tens to units

Remainder

7x Multiple (147=7x21)

212

213

10 Mastering Cyclic Addition with higher tiers ̶ Hierarchy

The workbook so far highlights 5 Cyclic Addition steps. These in order are Counting,Place Value, Move tens to units, Remainder and 7×Multiple.

This part of the workbook merely adds a creative and Mathematical emphasis to thisfoundation. These new emphasis are added 1 at a time, but can be used with anyCyclic Addition Count.

Each emphasis is in turn presented one question at a time so as to strengthen theMathematics of Cyclic Addition.

The guide book “Mathematics with just Number” in Chapter 7 Hierarchy has bothplenty of explanatory theory and ample practicals at the end of the Chapter. Theauthor recommends to read this chapter before attempting these advanced CyclicAddition final chapter Workbook questions.

1. Count with first tier ‘ 1 3 2 6 4 5 ’ starting at 1 for 7 cycles. Perform the usual 5Cyclic Addition steps each count. Note and record the remainder pattern for a cyclebetween the 7×Multiples without remainder. Map this Remainder pattern whilecounting along for the purpose of moving from the last Remainder to the next Count.

Count with the first tier ‘ 1 3 2 6 4 5 ’ starting at 6 for 7 cycles. Follow theinstructions as the previous paragraph guides. Map the Remainder pattern whilecounting. As before, move from the last Remainder to the next count for each of the42 Counts.

Count with the first tier ‘ 1 3 2 6 4 5 ’ starting at the other 4 starts for just 1 cycleeach. Note the Remainder pattern formed with a whole cycle of counting.

From this first tier Cyclic Addition count what can be said about the RemainderPattern of all six counts with any first tier Cyclic Addition Sequence.

2. Count with the first tier and second tier of ‘common multiple’ 1 i.e. CyclicAddition Sequences ‘ 1 3 2 6 4 5 ’ and ‘ 7 21 14 42 28 35 ’. Without copyingthe guide book make as many unique second tier Counts from 147 to 294 even if thecount excludes 147 and 294. Map each count on the left page with the CyclicAddition Counts on the right.

Explain the Mathematics that only 14, no less no more, counts can be made with asecond tier Count.

Count with the first and second tier of a ‘common multiple’ of your choice. Pick from7 to about 147 in multiples of 7. Use the mapping technique applied with ‘commonmultiple’ 1. Make all 14 counts for at least 2 cycles each count sequence. How can thestudent prevent duplication and omission of counts with the same ‘common multiple’.

3. Count with the first, second and third tier of ‘common multiple’ 1. i.e. CyclicAddition Sequences ‘ 1 3 2 6 4 5 ’ and ‘ 7 21 14 42 28 35 ’ and‘ 49 147 98 294 196 245 ’. Again without copying the guide book make as many

214

unique third tier Counts from 1029 to 2058 even if the count excludes 1029 and 2058.Map, again like the second tier Counting in the previous question, all the third tierCounts. Prove and show they are all unique consecutive numbers within the range of1029 to 2058.

Count with a third tier Cyclic Addition Sequence of your choice applying the first twotiers where needed to construct all possible counts for 2 cycles. Use the ReferencePages in the book ‘Mathematical Laws and Practicals’ Chapter 13. Use the mappingtechnique applied with ‘common multiple’ 1 to form all 21 counts.

4. Use a Remainder Multiple to act upon the Count number to emphasise the sequenceof numerals in a Count number. Find out where abouts around the wheel of ‘ 1 3 2 64 5 ’ is the Remainder. Read the second edition ‘Mathematical Laws and Practicals’Chapter 29. Perform Cyclic Addition with a tier that is commensurate with thestudent’s talent and skill level. With each Count find the Remainder’s position aroundthe Cyclic Addition Sequence and apply that ‘multiple’ against the Count. Oftenrevealing how higher tiers show further knowledge of a ‘common multiple’.

5. Have a go at a 5 tier Count. Start with a simpler ‘common multiple’ at tier 1 joinwith tier 2, count for a cycle or so, join the count to tier 3, count for another cycle orso, join with tier 4, count for a cycle or so and join with tier 5 and count. Again usethe Reference Pages in the book ‘Mathematical Laws and Practicals’ Chapter 13 as atool to navigate in-between tiers from 1 to 7.

6. Find the ‘other multiple’ such that with each Count the ‘other multiple’ × ‘commonmultiple’ = Count. While applying the Place Value Cyclic Addition step match all thePlace Values to the position around the Cyclic Addition Sequence. Remember theCyclic Addition Sequence is always in the form of‘common multiple’× ‘ 1 3 2 6 4 5 ’. Find the units first then tens and then hundredsif the count is high enough. Join units, tens and hundreds to make the ‘other multiple’.This can be done Count by Count and is also a further check of the Counts accuracy.

7. Use Circular Addition to navigate between counts within a cycle of each other. i.e.Counts between 1 and 6 counts apart in a Cyclic Addition Count sequence. Start witha tier 1 simple ‘common multiple’ count. Start with identifying the count betweencounts all the way along the Count. This shows Circular Addition for 1 count. Nextfind the difference between 2 counts i.e. spaced two apart. Using ‘common multiple’1 look for ‘ 1+3, 3+2, 2+6, 6+4, 4+5 and 5+1’. Then look for counts with a differencethree numbers apart. Again using ‘common multiple’ 1 look for ‘1+3+2, 3+2+6,2+6+4, 6+4+5, 4+5+1 and 5+1+3’. Continue this pattern for 4, 5 and 6 numberspacing in a Cyclic Addition Count.

The aim is to strengthen the wheel like nature that connects the Cyclic AdditionSequence to the Count sequence. Read second edition ‘Mathematical Laws andPracticals’ Chapter 30.

8. Explore Place Value Sets. Find all possible Place Value Sets for a Number. Startwith the Number 144 and all the ‘common multiples’ that can count with a CyclicAddition count to 144. Use the Guide Book ‘Mathematics with just Number’ Chapter5 as a start. Remember no cheating. Notice in the same chapter of the guide book all

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Place Value Sets for ‘common multiple’ 1. There is a giant table of 270 possible PlaceValue Sets. These, and only these, are for ‘common multiple’ 1. Simply multiply anyof these by the ‘common multiple’ to generate a Place Value Set for a Cyclic AdditionCount.

We now come to the finish of the Workbook. For those with a desire to use CyclicAddition with higher order the author recommends using the Reference Pages foundin the pdf book “Mathematical Laws and Practicals” on the CD.

As the Cyclic Addition Hierarchy is infinite so to is the latent ability of every teacher,parent and child.

Common Multiple 1

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

101 102 103 104 105 106 107 108 109 110

111 112 113 114 115 116 117 118 119 120

121 122 123 124 125 126 127 128 129 130

131 132 133 134 135 136 137 138 139 140

141 142 143 144 145 146 147 148 149 150

1

Common Multiple 2

1 2 1 2 3 4 5 6 7 8 9 10

3 4 11 12 13 14 15 16 17 18 19 20

5 6 21 22 23 24 25 26 27 28 29 30

7 8 31 32 33 34 35 36 37 38 39 40

9 10 41 42 43 44 45 46 47 48 49 50

11 12 51 52 53 54 55 56 57 58 59 60

13 14 61 62 63 64 65 66 67 68 69 70

15 16 71 72 73 74 75 76 77 78 79 80

17 18 81 82 83 84 85 86 87 88 89 90

19 20 91 92 93 94 95 96 97 98 99 100

21 22 101 102 103 104 105 106 107 108 109 110

23 24 111 112 113 114 115 116 117 118 119 120

25 26 121 122 123 124 125 126 127 128 129 130

27 28 131 132 133 134 135 136 137 138 139 140

29 30 141 142 143 144 145 146 147 148 149 150

2

11

Common Multiple 3

1 2 3 1 2 3 4 5 6 7 8 9

4 5 6 10 11 12 13 14 15 16 17 18

7 8 9 19 20 21 22 23 24 25 26 27

10 11 12 28 29 30 31 32 33 34 35 36

13 14 15 37 38 39 40 41 42 43 44 45

16 17 18 46 47 48 49 50 51 52 53 54

19 20 21 55 56 57 58 59 60 61 62 63

22 23 24 64 65 66 67 68 69 70 71 72

25 26 27 73 74 75 76 77 78 79 80 81

28 29 30 82 83 84 85 86 87 88 89 90

31 32 33 91 92 93 94 95 96 97 98 99

34 35 36 100 101 102 103 104 105 106 107 108

37 38 39 109 110 111 112 113 114 115 116 117

40 41 42 118 119 120 121 122 123 124 125 126

43 44 45 127 128 129 130 131 132 133 134 135

31

1

1

Common Multiple 4

1 2 3 4 1 2 3 4 5 6 7 8

5 6 7 8 9 10 11 12 13 14 15 16

9 10 11 12 17 18 19 20 21 22 23 24

13 14 15 16 25 26 27 28 29 30 31 32

17 18 19 20 33 34 35 36 37 38 39 40

21 22 23 24 41 42 43 44 45 46 47 48

25 26 27 28 49 50 51 52 53 54 55 56

29 30 31 32 57 58 59 60 61 62 63 64

33 34 35 36 65 66 67 68 69 70 71 72

37 38 39 40 73 74 75 76 77 78 79 80

41 42 43 44 81 82 83 84 85 86 87 88

45 46 47 48 89 90 91 92 93 94 95 96

49 50 51 52 97 98 99 100 101 102 103 104

53 54 55 56 105 106 107 108 109 110 111 112

57 58 59 60 113 114 115 116 117 118 119 120

2

2

13

11

1

1

4

Common Multiple 5

1 2 3 4 5 1 2 3 4 5 6 7 8 9 10

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

11 12 13 14 15 21 22 23 24 25 26 27 28 29 30

16 17 18 19 20 31 32 33 34 35 36 37 38 39 40

21 22 23 24 25 41 42 43 44 45 46 47 48 49 50

26 27 28 29 30 51 52 53 54 55 56 57 58 59 60

31 32 33 34 35 61 62 63 64 65 66 67 68 69 70

36 37 38 39 40 71 72 73 74 75 76 77 78 79 80

41 42 43 44 45 81 82 83 84 85 86 87 88 89 90

46 47 48 49 50 91 92 93 94 95 96 97 98 99 100

51 52 53 54 55 101 102 103 104 105 106 107 108 109 110

56 57 58 59 60 111 112 113 114 115 116 117 118 119 120

61 62 63 64 65 121 122 123 124 125 126 127 128 129 130

66 67 68 69 70 131 132 133 134 135 136 137 138 139 140

71 72 73 74 75 141 142 143 144 145 146 147 148 149 150

53

2

11

1

1

1

Common Multiple 6

1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28 29 30

31 32 33 34 35 36

37 38 39 40 41 42

43 44 45 46 47 48

49 50 51 52 53 54

55 56 57 58 59 60

61 62 63 64 65 66

67 68 69 70 71 72

73 74 75 76 77 78

79 80 81 82 83 84

85 86 87 88 89 90

6

15

13

2

3

32

2

2

Common Multiple 7

1 2 3 4 5 6 7

8 9 10 11 12 13 14

15 16 17 18 19 20 21

22 23 24 25 26 27 28

29 30 31 32 33 34 35

36 37 38 39 40 41 42

43 44 45 46 47 48 49

50 51 52 53 54 55 56

57 58 59 60 61 62 63

64 65 66 67 68 69 70

71 72 73 74 75 76 77

78 79 80 81 82 83 84

85 86 87 88 89 90 91

92 93 94 95 96 97 98

99 100 101 102 103 104 105

2

5

3

4

13

2

6

4

5

1

6

Common Multiple 8

1 2 3 4 5 6 7 8 1 2 3 4

9 10 11 12 13 14 15 16 5 6 7 8

17 18 19 20 21 22 23 24 9 10 11 12

25 26 27 28 29 30 31 32 13 14 15 16

33 34 35 36 37 38 39 40 17 18 19 20

41 42 43 44 45 46 47 48 21 22 23 24

49 50 51 52 53 54 55 56 25 26 27 28

57 58 59 60 61 62 63 64 29 30 31 32

65 66 67 68 69 70 71 72 33 34 35 36

73 74 75 76 77 78 79 80 37 38 39 40

81 82 83 84 85 86 87 88 41 42 43 44

89 90 91 92 93 94 95 96 45 46 47 48

97 98 99 100 101 102 103 104 49 50 51 52

105 106 107 108 109 110 111 112 53 54 55 56

113 114 115 116 117 118 119 120 57 58 59 60

13

1

3

2

6

2

2

2

2

4

4

Common Multiple 9

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18

19 20 21 22 23 24 25 26 27

28 29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45

46 47 48 49 50 51 52 53 54

55 56 57 58 59 60 61 62 63

64 65 66 67 68 69 70 71 72

73 74 75 76 77 78 79 80 81

82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99

100 101 102 103 104 105 106 107 108

109 110 111 112 113 114 115 116 117

118 119 120 121 122 123 124 125 126

127 128 129 130 131 132 133 134 135

1353

3

3

4

5

Common Multiple 10

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

101 102 103 104 105 106 107 108 109 110

111 112 113 114 115 116 117 118 119 120

121 122 123 124 125 126 127 128 129 130

131 132 133 134 135 136 137 138 139 140

141 142 143 144 145 146 147 148 149 150

55

6

4

3

2

3

2

1

4

52

2

2

2

2

Common Multiple 11

1 2 3 4 5 6 7 8 9 10 11

12 13 14 15 16 17 18 19 20 21 22

23 24 25 26 27 28 29 30 31 32 33

34 35 36 37 38 39 40 41 42 43 44

45 46 47 48 49 50 51 52 53 54 55

56 57 58 59 60 61 62 63 64 65 66

67 68 69 70 71 72 73 74 75 76 77

78 79 80 81 82 83 84 85 86 87 88

89 90 91 92 93 94 95 96 97 98 99

100 101 102 103 104 105 106 107 108 109 110

111 112 113 114 115 116 117 118 119 120 121

122 123 124 125 126 127 128 129 130 131 132

133 134 135 136 137 138 139 140 141 142 143

144 145 146 147 148 149 150 151 152 153 154

155 156 157 158 159 160 161 162 163 164 165

13

2

53

2

6

Common Multiple 12

1 2 3 4 5 6 7 8 9 10 11 12

13 14 15 16 17 18 19 20 21 22 23 24

25 26 27 28 29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45 46 47 48

49 50 51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70 71 72

73 74 75 76 77 78 79 80 81 82 83 84

85 86 87 88 89 90 91 92 93 94 95 96

97 98 99 100 101 102 103 104 105 106 107 108

109 110 111 112 113 114 115 116 117 118 119 120

121 122 123 124 125 126 127 128 129 130 131 132

133 134 135 136 137 138 139 140 141 142 143 144

145 146 147 148 149 150 151 152 153 154 155 156

157 158 159 160 161 162 163 164 165 166 167 168

169 170 171 172 173 174 175 176 177 178 179 180

4

4

41

3

2

6

3

3

3

3

15

1

56

62

6

4