Congratulations! You are about to learn about FUNFORMS, the first advance in the general written...

Congratulations! You are about to learn about FUNFORMS, the first advance in the general written numbering system in 1000 years, or more.

FUNFORMS© Copyright 2005

My goals today are to teach you enough about FUNFORMS that you will either choose to teach yourself the rest, or contact me for further instructions.


Eventually, my goal is that certain students will have the opportunity to be enriched by learning FUNFORMS and seeing how it “works”.



• You could easily ask, “Well don’t we already teach an alternate numbering system?”

• Yes, we do teach Roman numerals.

Everyone knows what

means, don’t they?

XVI

INTRODUCTION


• Yes, everyone knows what that means.Yes, everyone knows what that means.

• The problem is Roman Numerals are The problem is Roman Numerals are not easily manipulated. A student not easily manipulated. A student learning them learns little or nothing learning them learns little or nothing about the facts about how numbers about the facts about how numbers work.work.

INTRODUCTION


• FUNFORMS is a numbering system that is easily manipulated. It is a tally mark place order system, likely the first one.

• All of you recognize what this figure means:

INTRODUCTION

• It means 16, just like the Roman numeral that we just saw


How about if we wrote “16” like this?

INTRODUCTION

You would still understand that it represented the number value 16, wouldn’t you?

Or this…


• That brings us to FUNFORMS. In FUNFORMS the glyphs [numerals] are written vertically.

• There are specific positions where a horizontal mark or flag can be written.

• These flags reside on a vertical backbone structure called a staff.

INTRODUCTION

flags

staff


INTRODUCTION

16

8

4

2

32

64

128

Number values double at each successive position going down the staff.

Not surprisingly, number values halve at each successive position going up the staff.

256

1


INTRODUCTION

There are a few additional things that you’ll need to know to be able to use FUNFORMS.

Positive ValuesPositive ValuesDrawn to the RIGHT of the staff.

Negative ValuesDrawn to the LEFT of the staff.

positivenegative


UNITY POINT16

8

4

2

32

64

128

256

11INTRODUCTION

By convention, the first position on the staff, which can be marked by a flag extending to the right, represents the number one. This position is called UNITY POINT


INTRODUCTION

16

8

4

2

32

64

128

256

1

All potential positions ("points") below unity point have a whole number value that corresponds to a whole number power of 2

24

23

22

21

25

26

27

28

20Unity pointAs you have seen in the

counting that we did earlier, number values double at each successive position going down the staff.


INTRODUCTION

16

8

4

2

1

All potential positions ("points") above unity point are fractional in nature and represent the value of whole number negative powers of 2.

24

23

22

21

20 Unity Point

1/2 2-1

1/42-2

1/8 2-3

1/16 2-4

Fractional Numbers

Whole Numbers


COUNTING

COUNTING WITH FUNFORMSCOUNTING WITH FUNFORMS


COUNTING


• Funforms is simple. It is based on the concept of pairs.

• After number one we come to a pair of one's. 2

11


COUNTING


• Then we come to a pair of pairs. 4

2

11


• After number one we come to a pair of one's.


COUNTING



• After number one we come to a pair of one's.

• Then we come to a pair of pairs.

• Next is a pair of pairs paired.

• That is what each new position going down the staff stands for.

8

4

2

11


COUNTING

LETS BEGIN COUNTING!LETS BEGIN COUNTING!LETS BEGIN COUNTING!LETS BEGIN COUNTING!


COUNTING

16

8

4

2

11

16

8

4

2

11

16

8

4

2

11

16

8

4

2

11

16

8

4

2

11

11 22 33 44 55


COUNTING

16

8

4

2

11

16

8

4

2

11

16

8

4

2

11

16

8

4

2

11

16

8

4

2

11

66 77 88 99 1010


COUNTING

16

8

4

2

11

16

8

4

2

11

16

8

4

2

11

16

8

4

2

11

16

8

4

2

11

1111 1212 1313 1414 1515


COUNTING

16

8

4

2

11

16

8

4

2

11

16

8

4

2

11

16

8

4

2

11

16

8

4

2

11

1616 1717 1818 1919 2020


MANIPULATING FUNFORMS

HERE ARE THE RULES NECESSARY HERE ARE THE RULES NECESSARY TO MANIPULATE FUNFORMSTO MANIPULATE FUNFORMS

You have already learned that numerical values double each time a flag (or a series of flags) moves down one position (or set of positions) on the staff.



HERE ARE THE RULES NECESSARY HERE ARE THE RULES NECESSARY TO MANIPULATE FUNFORMSTO MANIPULATE FUNFORMS

Similarly, numerical values halve each time a flag (or a series of flags) moves up one position (or set of positions) on the staff.



No more than one flag can be at any one position (except temporarily during manipulation). That is, there is either one flag at any given point (position), or there is no flag there.

flags


16

8

4

2

11

33 55++

ADDITIONADDITION




16

8

4

2

11

33 55++

ADDITIONADDITION

ADDITION is carried out by simply ADDITION is carried out by simply combining (or “coalescing”) whatever combining (or “coalescing”) whatever

number values are to be added from the number values are to be added from the individual figures or glyphs and then individual figures or glyphs and then

simply "clearing" them by following the simply "clearing" them by following the already learned rules.already learned rules.


16

8

4

2

11

33 55++

ADDITIONADDITION



CALCULATION/CLEARING CALCULATION/CLEARING PHASEPHASE

16

8

4

2

11

33 55++

ADDITIONADDITION

Remember, no more than one

flag is allowed at any point (EXCEPT

temporarily during calculations).

Remember, Any two flags at one position (point)

are the equivalent of one flag at the

next position down.


88

SUMSUM


16

8

4

2

11

ADDITIONADDITION


The FUNFORM figure is now in its simplest form and

nothing further needs to be or can

be done.

33 55++ 88

SUMSUM

==


6565 9191++ADDITIONADDITION


2

11

32

64

4

16

8

128

256



be done.

Simply add the remaining flags for

the answer.

SUMSUM

156156



Now, let's shift our attention to negative numbers and subtraction. Converting a number to its negative counterpart simply means drawing it to the left of the staff. Conversely, converting a negative number to its minus equivalent would involve writing it on the right side of the staff.

SUBTRACTIONSUBTRACTION


CALCULATION/RESOLVING CALCULATION/RESOLVING PHASEPHASE

16

8

4

2

11

1616 11--


negative

Converting a number to its negative counterpart simply means drawing it to the left

of the staff.


Any one flag at one level is the

same as 2 flags at the

preceding level


16

8

4

2

11

1616 11--





16

8

4

2

11

1616 11--




be done.

1515

REMAINDERREMAINDER

==


225225 4444--SUBTRACTIONSUBTRACTION


2

11

32

64

4

16

8

128

256




2

11

32

64

4

16

8

128

256

225225 4444--


225225 4444--SUBTRACTIONSUBTRACTION


2

64

8

11

32

4

16

128

256

181181



be done.

==


16

8

4

2

11

11--1616

ADDITION AND SUBTRACTIONADDITION AND SUBTRACTION(overview)(overview)


1515Looking back, pay particular attention to 16 – 1, please. Note that whenever there is a long gap between a minus number value (in this case –1) and a positive one (16), exactly what you see here happens. Each intervening point becomes filled with a flag beginning one position up from the first positive flag after the gap.

You should only have to see this operation take place once to be able to apply it each time the situation presents itself. I think of this like unzipping a zipper.


16

8

4

2

11

ADDITION AND SUBTRACTIONADDITION AND SUBTRACTION (overview)(overview)


11++1515 1616

Similarly, if you added 1 to 15, you would begin with 2 flags at the one position, which would become 1 flag at the two position (where there was already a flag), and the flags would “tumble” down sequentially, leaving just one flag at the sixteen position.

I think of those events mechanically like a zipper closing or like a Jacob’s ladder.


MULTIPLICATIONMULTIPLICATION


Multiplication amounts to serial addition. It is done by writing the multiplicand at each position that a flag exists on the

multiplier (using the formulaic qualities of Funforms), coalescing the intervening glyphs, and then clearing the resultant

figure by the rules already learned.


66 1010XX



16

8

4

2

11

In the writing processes, it helps to think about the

multiplicand in terms of its spaces and flags beginning at

unity point.

Unity point

For example “10” could be thought of as “space-flag-

space-flag”.

SPACE

FLAG

SPACE

FLAG

32


66 1010XX



16

8

4

2

11If 10 were to be

multiplied by 6, one would write space-flag-space-flag first at the position of one of the

two flags that make up 6, and then by writing space-flag-space-flag

at the level of the other flag making up 6.

32

New Temporary Unity Point

SPACE

FLAG

SPACE

FLAG


66 1010XX



16

8

4

2

11

32


SPACE

FLAG

SPACE

FLAG


66 1010XX



16

8

4

2

11

32


66 1010XX



2

11

16

8

4

32

6060==


FUNFORMS


2222 66xxMULTIPLICATIONMULTIPLICATION


16

8

4

2

11

32

64

128

256

SPACE

FLAG

FLAG





16

8

4

2

11

32

64

128

256




16

8

2

11

32

64

4

128

256

132132==



16

8

4

2

11

32

64

FRACTIONSFRACTIONS

1/21/2

1/41/4

Consider the following FUNFORM figures.

In the first figures, 24 is repeatedly halved by

moving it up one position at a time.



16

8

4

2

11

32

64

FRACTIONSFRACTIONS

1/21/2

1/41/43 of 8

24243 of 4

12123 of 2

663 of 1

333 of 2-1

3/23/2

3 of 2-2

3/43/4

As it passes unity point, it becomes

fractional, at least in part.

Unity Point



16

8

4

2

11

32

64

1/21/2

1/41/4

The same is true for "fiveness" written at various positions.

FRACTIONSFRACTIONS



16

8

4

2

11

32

64

FRACTIONSFRACTIONS

1/21/2

1/41/45 of 1

555 of 2

10105 of 8

40405 of 2-2

5/45/4

5 of 2-1

5/25/2

Fractional at Unity Point

Unity Point



8

4

2

1

1/21/2

16

32

1/41/4

1/81/8

In adding fractions, there is no need to seek

the lowest common

denominator.

ADDING FRACTIONSADDING FRACTIONS

5/85/8 3/43/4++

5/8 (red) + 3/4 (orange) is added exactly like you would add any

Funform figure.

You do not need to know that they have fractional

values to correctly add them.



8

4

2

1

1/21/2

16

32

1/41/4

1/81/8


5/85/8 3/43/4++



8

4

2

1

1/21/2

16

32

1/41/4

1/81/8


5/85/8 3/43/4++ 113/83/8==



8

4

2

1

1/21/2

16

32

1/41/4

1/81/8


7/87/8 ++ 111/21/2



8

4

2

1

1/21/2

16

32

1/41/4

1/81/8


7/87/8 ++ 111/21/2 == 223/83/8



8

4

2

1

1/21/2

16

32

1/41/4

1/81/8

MULTIPLYING FRACTIONSMULTIPLYING FRACTIONS

Multiplying fractions is done Multiplying fractions is done just like just like multiplying whole numbersmultiplying whole numbers, attending to , attending to

how the flags on the multiplicand relate to how the flags on the multiplicand relate to unity point. unity point.



8

4

2

1

1/21/2

16

32

1/41/4

1/81/8


To multiply fractions, it is again wise to say to

oneself how the multiplicand looks,

beginning at unity point.

1/21/2 1/21/2xx

Unity Point

½ x ½ is shown. ½ can be thought of as space-flag.

SPACE

FLAG

ACSENDING FROM UNITY POINT



8

4

2

1

1/21/2

16

32

1/41/4

1/81/8


Multiplicand appearance, beginning at unity point.

1/21/2 1/21/2xx

NewTemporaryUnity Point

SPACE

FLAG



8

4

2

1

1/21/2

16

32

1/41/4

1/81/8


1/21/2 1/21/2xx == 1/41/4



4

2

1

1/2

1/41/4

8

16

1/81/8

1/161/16


3/43/4 3/43/4xx

SPACE

FLAG

FLAG

Unity Point




4

2

1

1/2

1/41/4

8

16

1/81/8

1/161/16


3/43/4 3/43/4xx

SPACE

FLAG

FLAG




4

2

1

1/2

1/41/4

8

16

1/81/8

1/161/16


3/43/4 3/43/4xx



4

2

1

1/2

1/41/4

8

16

1/81/8

1/161/16


3/43/4 3/43/4xx 9/169/16==



DIVISIONDIVISION

DIVISION is just DIVISION is just serial subtractionserial subtraction, while , while keeping score.keeping score.



DIVISIONDIVISION

1616 -- 44

1616 44÷÷ == ??I believe that a learner should be asked…

How many times can the divisor be subtracted from the number to be divided?


12128844001616


-- 44

1616 44÷÷ ??11223344==

DIVISIONDIVISION

How many times can the divisor be

subtracted from the number to be divided?


3535 77÷÷

DIVISIONDIVISION


16

8

4

2

11

32

Serial Subtraction

Recording Staff

The results of the repeated subtractions are simply recorded

off to one side using a “recording staff” and cleared at the end of

the operation.

Unity Point

FLAG

FLAG

FLAG


3535 77÷÷

DIVISIONDIVISION


16

8

4

2

11

32

FLAG

FLAG

FLAGNew

TemporaryUnity Point

RECORD FLAG AT NEW UNITY POINT


3535 77÷÷

DIVISIONDIVISION


16

8

4

2

11

32


3535 77÷÷

DIVISIONDIVISION


16

8

4

2

11

32

OUR ORIGINAL MATH PROBLEM Serial Subtraction

NOTE: Unity point marker does not change from

original position.

RECORD FLAG AT UNITY POINT

FLAG

FLAG

FLAG


3535 77÷÷

DIVISIONDIVISION


16

8

2

4

11

32

55==

The final results of our “recording staff” provides us with the

answer.


6363 99÷÷

DIVISIONDIVISIONMANIPULATING FUNFORMS

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8

2

4

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32

64

Serial Subtraction

Recording Staff

Unity Point

FLAG

FLAG

SPACE

SPACE


6363 99÷÷


16

8

2

4

11

32

64

RECORD FLAG AT UNITY POINT

Unity Point

FLAG

FLAG

SPACE

SPACE


6363 99÷÷


16

8

2

4

11

32

64




6363 99÷÷


16

8

2

4

11

32

64

== 77



8

4

2

1

1/21/2

16

32

1/41/4

1/81/8

551/21/2 221/21/2÷÷

DIVIDING FRACTIONSDIVIDING FRACTIONS

Division for numbers that are Division for numbers that are not wholenot whole powers of two is (again) just serial powers of two is (again) just serial subtraction while keeping score.subtraction while keeping score.



8

4

2

1

1/21/2

16

32

1/41/4

1/81/8

551/21/2 221/21/2÷÷


Serial Subtraction

Recording Staff

Unity Point

FLAG

SPACE

FLAG




8

4

2

1

1/21/2

16

32

1/41/4

1/81/8

551/21/2 221/21/2÷÷


DIVISOR

The FUNFORM figure is now a lower value than

our divisor giving us our remainder.



8

4

1

2

1/21/2

16

32

1/41/4

1/81/8

551/21/2 221/21/2÷÷


22==

REMAINDER

AFTER AFTER PREVIOUS PREVIOUS

SUBTRACTION SUBTRACTION PROCESSPROCESS

1/21/2RRRecording Staff



8

4

2

1

1/21/2

16

32

1/41/4

1/81/8

2525 221/21/2÷÷


Serial Subtraction

Recording Staff

Unity Point


FLAG

SPACE

FLAG

FLAG

SPACE

FLAG




8

4

2

1

1/21/2

16

32

1/41/4

1/81/8

DIVIDING FRACTIONSDIVIDING FRACTIONSRecording Staff

1010==2525 ÷÷ 221/21/2


FUNFORMS INFORMATION PAGE

FUNFORMS WEBSITEFUNFORMS WEBSITEhttp://www.funforms.comhttp://www.funforms.com

CLICK HERE TO RETURN TO PREVIOUSLY VIEWED SLIDE

PLAY PRESENTATION OVERPLAY PRESENTATION OVER

GLOSSARY PAGEGLOSSARY PAGE

NAVIGATION PAGENAVIGATION PAGE


GLOSSARY PAGE 1


FLAGThe horizontal mark used for denoting the presence of that particular numerical value along a staff.

FUNFORMSA new place order, tally mark binary numerical system that uses glyphs and is easily manipulated.

GLYPHA symbol that conveys [numerical in this case] information nonverbally.

RECORDING STAFFA special vertical line used to record the results of repeated subtractions during the division process. These results are later cleared at the end of the operation and used to determine the final answer.

STAFFThe vertical backbone structure used to place flags. Flag values increase in a doubling fashion [arithmetically progressive powers of 2] as they move toward the bottom of the staff and decrease by halving [in whole powers of 2] as they move toward the top of the staff.

UNITY POINTThe first position on the staff, which is marked by a flag extending to the right. It represents the numerical value one.

POSITIVE VALUESAll flag values displayed to the RIGHT of the FUNFORMS staff.

NEGATIVE VALUESAll flag values displayed to the LEFT of the FUNFORMS staff.

CLICK FOR

PAGE 2

FUNFORMS TERMSFUNFORMS TERMS


GLOSSARY PAGE 2


DENOMINATORThe divisor of a fraction.

DIVIDENDA number to be divided by another number. E.g. in 10 divided by 2, “10” is the dividend.

DIVISORThe number by which a dividend is divided. E.g. in 10 divided by 2, “2” is the dividend.

EXPONENTA mathematical notation indicating the number of times a quantity is multiplied by itself.

FRACTIONA part of a whole number.

MULTIPLICANDThe number that is multiplied by the multiplier. E.g. in 10 times 2, “2” is the multiplicand.

MULTIPLIERThe number by which a multiplicand is multiplied. E.g. in 10 times 2, “10” is the multiplier.

WHOLE NUMBERAny of the natural numbers (positive or negative) or zero.

CLICK FOR

PAGE 1

MATHEMATICAL TERMSMATHEMATICAL TERMS


FUNFORMS NAVIGATION PAGE

IntroductionIntroduction

The BasicsThe Basics

Negative and Positive ValuesNegative and Positive Values

Unity PointUnity Point

CountingCounting

ManipulationManipulation

AdditionAddition

SubtractionSubtraction

BACK

MultiplicationMultiplication

FractionsFractions

Adding FractionsAdding Fractions

Multiplying FractionsMultiplying Fractions

DivisionDivision

Dividing FractionsDividing Fractions

GLOSSARY PAGEGLOSSARY PAGE

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