Funforms

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Funforms is a binary, place order tally mark system. It is easily learned. It offers the learner the benefits of learning and understanding a working numerical system after one has the capacity for introspection and deeper understanding. Funforms gives the learner a new position from which to examine assumptions and hypotheses accepted before s/he could fully understand the subject being learned.

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Joel S. Steinberg, M.D.:

Joel S. Steinberg, M.D. Physician/psychiatrist Clinical faculty Case Western Reserve U. Symbols and cognition; an old interest 20 years of developing this system Began with a friend and colleague .

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Thinking takes place with pictures and symbols. Main symbols used in human thinking Words Numbers

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Numerals can be merely representational A good example is Roman numerals. Numbers are best when they facilitate computation Our own number system is arbitrary in design. Learning it can be an obstacle .

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The system I am about to explain to you is ultimately simple in design. Stark Iconic and ideographic Easily learned and written Operations become transparent Continuity between fractions and whole numbers becomes obvious

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Funforms is a new mathematical numerical notation learning system. "Fun" stands for fun damental fun ctional and fun . "Forms" stands for form ulae. Funforms are fundamental formulae that are fun to work with.

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Congratulations! You are about to learn about FUNFORMS, the first advance in the general written numbering system in 1000 years, or more. FUNFORMS FUNFORMS © Copyright 2005

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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. FUNFORMS FUNFORMS © Copyright 2005

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Eventually, my goal is that certain students will have the opportunity to be enriched by learning FUNFORMS and seeing how it “works”. FUNFORMS FUNFORMS © Copyright 2005

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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

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XVI Yes, everyone knows what that means. The problem is Roman Numerals are not easily manipulated. A student learning them learns little or nothing about the facts about how numbers work. INTRODUCTION

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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

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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…

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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

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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

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INTRODUCTION There are a few additional things that you’ll need to know to be able to use FUNFORMS. Positive Values Drawn to the RIGHT of the staff. Negative Values Drawn to the LEFT of the staff. positive negative

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UNITY POINT 16 8 4 2 32 64 128 256 1 INTRODUCTION 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

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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 2 4 2 3 2 2 2 1 2 5 2 6 2 7 2 8 2 0 Unity point As you have seen in the counting that we did earlier, number values double at each successive position going down the staff.

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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. 2 4 2 3 2 2 2 1 2 0 Unity Point 1/2 2 -1 1/4 2 -2 1/8 2 -3 1/16 2 -4 Fractional Numbers Whole Numbers

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COUNTING COUNTING WITH FUNFORMS 123

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COUNTING COUNTING WITH FUNFORMS 123 Funforms is simple. It is based on the concept of pairs . After number one we come to a pair of one's. 2 1

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COUNTING COUNTING WITH FUNFORMS 123 Then we come to a pair of pairs . 4 2 1 Funforms is simple. It is based on the concept of pairs . After number one we come to a pair of one's.

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COUNTING COUNTING WITH FUNFORMS 123 Funforms is simple. It is based on the concept of pairs . 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 1

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123 COUNTING LETS BEGIN COUNTING! LETS BEGIN COUNTING!

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COUNTING 16 8 4 2 1 16 8 4 2 1 16 8 4 2 1 16 8 4 2 1 16 8 4 2 1 1 2 3 4 5

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COUNTING 16 8 4 2 1 16 8 4 2 1 16 8 4 2 1 16 8 4 2 1 16 8 4 2 1 6 7 8 9 10

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COUNTING 16 8 4 2 1 16 8 4 2 1 16 8 4 2 1 16 8 4 2 1 16 8 4 2 1 11 12 13 14 15

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COUNTING 16 8 4 2 1 16 8 4 2 1 16 8 4 2 1 16 8 4 2 1 16 8 4 2 1 16 17 18 19 20

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MANIPULATING FUNFORMS HERE ARE THE RULES NECESSARY TO 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.

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MANIPULATING FUNFORMS HERE ARE THE RULES NECESSARY TO 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.

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MANIPULATING FUNFORMS 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

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16 8 4 2 1 3 5 + ADDITION MANIPULATING FUNFORMS

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MANIPULATING FUNFORMS 16 8 4 2 1 3 5 + ADDITION ADDITION is carried out by simply combining (or “coalescing”) whatever number values are to be added from the individual figures or glyphs and then simply "clearing" them by following the already learned rules.

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16 8 4 2 1 3 5 + ADDITION MANIPULATING FUNFORMS

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CALCULATION/CLEARING PHASE 16 8 4 2 1 3 5 + ADDITION 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. MANIPULATING FUNFORMS 8 SUM

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16 8 4 2 1 ADDITION MANIPULATING FUNFORMS The FUNFORM figure is now in its simplest form and nothing further needs to be or can be done. 3 5 + 8 SUM =

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65 91 + ADDITION MANIPULATING FUNFORMS 2 1 32 64 4 16 8 128 256 The FUNFORM figure is now in its simplest form and nothing further needs to be or can be done. Simply add the remaining flags for the answer. SUM 156

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MANIPULATING FUNFORMS 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. SUBTRACTION

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CALCULATION/RESOLVING PHASE 16 8 4 2 1 16 1 - SUBTRACTION negative Converting a number to its negative counterpart simply means drawing it to the left of the staff. MANIPULATING FUNFORMS Any one flag at one level is the same as 2 flags at the preceding level

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16 8 4 2 1 16 1 - SUBTRACTION MANIPULATING FUNFORMS

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MANIPULATING FUNFORMS 16 8 4 2 1 16 1 - SUBTRACTION

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MANIPULATING FUNFORMS 16 8 4 2 1 16 1 - SUBTRACTION The FUNFORM figure is now in its simplest form and nothing further needs to be or can be done. 15 REMAINDER =

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225 44 - SUBTRACTION MANIPULATING FUNFORMS 2 1 32 64 4 16 8 128 256

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225 44 - SUBTRACTION MANIPULATING FUNFORMS 2 1 32 64 4 16 8 128 256

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SUBTRACTION MANIPULATING FUNFORMS 2 1 32 64 4 16 8 128 256 225 44 -

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SUBTRACTION MANIPULATING FUNFORMS 2 1 32 64 4 16 8 128 256 225 44 -

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225 44 - SUBTRACTION MANIPULATING FUNFORMS 2 64 8 1 32 4 16 128 256 181 The FUNFORM figure is now in its simplest form and nothing further needs to be or can be done. =

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16 8 4 2 1 1 - 16 ADDITION AND SUBTRACTION (overview) MANIPULATING FUNFORMS 15 Looking 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.

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16 8 4 2 1 ADDITION AND SUBTRACTION (overview) MANIPULATING FUNFORMS 1 + 15 16 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.

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MULTIPLICATION MANIPULATING FUNFORMS 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.

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6 10 X MULTIPLICATION MANIPULATING FUNFORMS 16 8 4 2 1 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

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6 10 X MULTIPLICATION MANIPULATING FUNFORMS 16 8 4 2 1 If 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

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6 10 X MULTIPLICATION MANIPULATING FUNFORMS 16 8 4 2 1 32 New Temporary Unity Point SPACE FLAG SPACE FLAG

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6 10 X MULTIPLICATION MANIPULATING FUNFORMS 16 8 4 2 1 32

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6 10 X MULTIPLICATION MANIPULATING FUNFORMS 2 1 16 8 4 32 60 =

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FUNFORMS

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22 6 x MULTIPLICATION MANIPULATING FUNFORMS 16 8 4 2 1 32 64 128 256 SPACE FLAG FLAG New Temporary Unity Point

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22 6 x MULTIPLICATION MANIPULATING FUNFORMS 16 8 4 2 1 32 64 128 256 SPACE FLAG FLAG New Temporary Unity Point

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22 6 x MULTIPLICATION MANIPULATING FUNFORMS 16 8 4 2 1 32 64 128 256 SPACE FLAG FLAG New Temporary Unity Point

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22 6 x MULTIPLICATION MANIPULATING FUNFORMS 16 8 4 2 1 32 64 128 256

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22 6 x MULTIPLICATION MANIPULATING FUNFORMS 16 8 4 2 1 32 64 128 256

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22 6 x MULTIPLICATION MANIPULATING FUNFORMS 16 8 2 1 32 64 4 128 256 132 =

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MANIPULATING FUNFORMS 16 8 4 2 1 32 64 FRACTIONS 1/2 1/4 Consider the following FUNFORM figures. In the first figures, 24 is repeatedly halved by moving it up one position at a time.

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MANIPULATING FUNFORMS 16 8 4 2 1 32 64 FRACTIONS 1/2 1/4 3 of 8 24 3 of 4 12 3 of 2 6 3 of 1 3 3 of 2 -1 3/2 3 of 2 -2 3/4 As it passes unity point, it becomes fractional, at least in part. Unity Point

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MANIPULATING FUNFORMS 16 8 4 2 1 32 64 1/2 1/4 The same is true for "fiveness" written at various positions. FRACTIONS

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MANIPULATING FUNFORMS 16 8 4 2 1 32 64 FRACTIONS 1/2 1/4 5 of 1 5 5 of 2 10 5 of 8 40 5 of 2 -2 5/4 5 of 2 -1 5/2 Fractional at Unity Point Unity Point

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MANIPULATING FUNFORMS 8 4 2 1 1/2 16 32 1/4 1/8 In adding fractions, there is no need to seek the lowest common denominator. ADDING FRACTIONS 5/8 3/4 + 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.

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MANIPULATING FUNFORMS 8 4 2 1 1/2 16 32 1/4 1/8 ADDING FRACTIONS 5/8 3/4 +

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MANIPULATING FUNFORMS 8 4 2 1 1/2 16 32 1/4 1/8 ADDING FRACTIONS 5/8 3/4 + 1 3/8 =

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MANIPULATING FUNFORMS 8 4 2 1 1/2 16 32 1/4 1/8 ADDING FRACTIONS 7/8 + 1 1/2

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MANIPULATING FUNFORMS 8 4 2 1 1/2 16 32 1/4 1/8 ADDING FRACTIONS 7/8 + 1 1/2 = 2 3/8

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MANIPULATING FUNFORMS 8 4 2 1 1/2 16 32 1/4 1/8 MULTIPLYING FRACTIONS Multiplying fractions is done just like multiplying whole numbers , attending to how the flags on the multiplicand relate to unity point.

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MANIPULATING FUNFORMS 8 4 2 1 1/2 16 32 1/4 1/8 MULTIPLYING FRACTIONS To multiply fractions, it is again wise to say to oneself how the multiplicand looks, beginning at unity point. 1/2 1/2 x Unity Point ½ x ½ is shown. ½ can be thought of as space - flag . SPACE FLAG ACSENDING FROM UNITY POINT

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MANIPULATING FUNFORMS 8 4 2 1 1/2 16 32 1/4 1/8 MULTIPLYING FRACTIONS Multiplicand appearance, beginning at unity point. 1/2 1/2 x New Temporary Unity Point SPACE FLAG

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MANIPULATING FUNFORMS 8 4 2 1 1/2 16 32 1/4 1/8 MULTIPLYING FRACTIONS 1/2 1/2 x = 1/4

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MANIPULATING FUNFORMS 4 2 1 1/2 1/4 8 16 1/8 1/16 MULTIPLYING FRACTIONS 3/4 3/4 x SPACE FLAG FLAG Unity Point New Temporary Unity Point

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MANIPULATING FUNFORMS 4 2 1 1/2 1/4 8 16 1/8 1/16 MULTIPLYING FRACTIONS 3/4 3/4 x SPACE FLAG FLAG New Temporary Unity Point

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MANIPULATING FUNFORMS 4 2 1 1/2 1/4 8 16 1/8 1/16 MULTIPLYING FRACTIONS 3/4 3/4 x

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MANIPULATING FUNFORMS 4 2 1 1/2 1/4 8 16 1/8 1/16 MULTIPLYING FRACTIONS 3/4 3/4 x 9/16 =

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MANIPULATING FUNFORMS DIVISION DIVISION is just serial subtraction , while keeping score.

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MANIPULATING FUNFORMS DIVISION 16 - 4 16 4 ÷ = ? I believe that a learner should be asked… How many times can the divisor be subtracted from the number to be divided ?

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12 8 4 0 16 MANIPULATING FUNFORMS - 4 16 4 ÷ ? 1 2 3 4 = DIVISION How many times can the divisor be subtracted from the number to be divided ?

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35 7 ÷ DIVISION MANIPULATING FUNFORMS 16 8 4 2 1 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

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35 7 ÷ DIVISION MANIPULATING FUNFORMS 16 8 4 2 1 32 FLAG FLAG FLAG New Temporary Unity Point RECORD FLAG AT NEW UNITY POINT

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35 7 ÷ DIVISION MANIPULATING FUNFORMS 16 8 4 2 1 32

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35 7 ÷ DIVISION MANIPULATING FUNFORMS 16 8 4 2 1 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

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35 7 ÷ DIVISION MANIPULATING FUNFORMS 16 8 2 4 1 32 5 = The final results of our “recording staff” provides us with the answer.

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63 9 ÷ DIVISION MANIPULATING FUNFORMS 16 8 2 4 1 32 64 Serial Subtraction Recording Staff Unity Point FLAG FLAG SPACE SPACE

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63 9 ÷ DIVISION MANIPULATING FUNFORMS 16 8 2 4 1 32 64 RECORD FLAG AT UNITY POINT Unity Point FLAG FLAG SPACE SPACE

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63 9 ÷ DIVISION MANIPULATING FUNFORMS 16 8 2 4 1 32 64 New Temporary Unity Point RECORD FLAG AT NEW UNITY POINT

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63 9 ÷ DIVISION MANIPULATING FUNFORMS 16 8 2 4 1 32 64 RECORD FLAG AT NEW UNITY POINT New Temporary Unity Point

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63 9 ÷ DIVISION MANIPULATING FUNFORMS 16 8 2 4 1 32 64 = 7

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MANIPULATING FUNFORMS 8 4 2 1 1/2 16 32 1/4 1/8 5 1/2 2 1/2 ÷ DIVIDING FRACTIONS Division for numbers that are not whole powers of two is (again) just serial subtraction while keeping score.

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MANIPULATING FUNFORMS 8 4 2 1 1/2 16 32 1/4 1/8 5 1/2 2 1/2 ÷ DIVIDING FRACTIONS Serial Subtraction Recording Staff Unity Point FLAG SPACE FLAG New Temporary Unity Point

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MANIPULATING FUNFORMS 8 4 2 1 1/2 16 32 1/4 1/8 5 1/2 2 1/2 ÷ DIVIDING FRACTIONS DIVISOR The FUNFORM figure is now a lower value than our divisor giving us our remainder.

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MANIPULATING FUNFORMS 8 4 1 2 1/2 16 32 1/4 1/8 5 1/2 2 1/2 ÷ DIVIDING FRACTIONS 2 = REMAINDER AFTER PREVIOUS SUBTRACTION PROCESS 1/2 R Recording Staff

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MANIPULATING FUNFORMS 8 4 2 1 1/2 16 32 1/4 1/8 25 2 1/2 ÷ DIVIDING FRACTIONS Serial Subtraction Recording Staff Unity Point New Temporary Unity Point FLAG SPACE FLAG FLAG SPACE FLAG New Temporary Unity Point

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MANIPULATING FUNFORMS 8 4 2 1 1/2 16 32 1/4 1/8 DIVIDING FRACTIONS Recording Staff 10 = 25 ÷ 2 1/2

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96                    1 48                    2 24                    4 12                    8 6                    16 __         3        __       32 __         1        __ 64 Total 96

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__ 87 [odd]            __        1 __        43 [odd]            __        2 __        21 [odd]            __        4 10 [even]            “8“ __        5   [odd]            __        16 2   [even]                     "32“ __        1   [odd ]           __ 64 Total 87

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COUNTING 16 8 4 2 1 3 5 16 8 4 2 1 16 8 4 2 1 TEMPLATE ITEMS

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FUNFORMS INFORMATION PAGE FUNFORMS WEBSITE http://www.funforms.com CLICK HERE TO RETURN TO PREVIOUSLY VIEWED SLIDE PLAY PRESENTATION OVER GLOSSARY PAGE NAVIGATION PAGE

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GLOSSARY PAGE 1 NAVIGATION PAGE FLAG The horizontal mark used for denoting the presence of that particular numerical value along a staff. FUNFORMS A new place order, tally mark binary numerical system that uses glyphs and is easily manipulated. GLYPH A symbol that conveys [numerical in this case] information nonverbally. RECORDING STAFF A 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. STAFF The 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 POINT The first position on the staff, which is marked by a flag extending to the right. It represents the numerical value one. POSITIVE VALUES All flag values displayed to the RIGHT of the FUNFORMS staff. NEGATIVE VALUES All flag values displayed to the LEFT of the FUNFORMS staff. FUNFORMS CLICK FOR PAGE 2 FUNFORMS TERMS

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GLOSSARY PAGE 2 NAVIGATION PAGE DENOMINATOR The divisor of a fraction. DIVIDEND A number to be divided by another number. E.g. in 10 divided by 2 , “ 10 ” is the dividend. DIVISOR The number by which a dividend is divided. E.g. in 10 divided by 2 , “ 2 ” is the dividend. EXPONENT A mathematical notation indicating the number of times a quantity is multiplied by itself. FRACTION A part of a whole number. MULTIPLICAND The number that is multiplied by the multiplier. E.g. in 10 times 2 , “ 2 ” is the multiplicand. MULTIPLIER The number by which a multiplicand is multiplied. E.g. in 10 times 2 , “10” is the multiplier. WHOLE NUMBER Any of the natural numbers (positive or negative) or zero. FUNFORMS CLICK FOR PAGE 1 MATHEMATICAL TERMS

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FUNFORMS NAVIGATION PAGE Introduction The Basics Negative and Positive Values Unity Point Counting Manipulation Addition Subtraction BACK Multiplication Fractions Adding Fractions Multiplying Fractions Division Dividing Fractions FUNFORMS GLOSSARY PAGE

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