## Presentation on theme: "Congratulations! You are about to learn about FUNFORMS, the first advance in the general written numbering system in 1000 years, or more. FUNFORMS © Copyright."— Presentation transcript:

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 © Copyright 2005

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

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

FUNFORMS © Copyright 2005 Yes, everyone knows what that means.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.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

FUNFORMS © Copyright 2005 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

FUNFORMS © Copyright 2005 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…

FUNFORMS © Copyright 2005 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

FUNFORMS © Copyright 2005 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

FUNFORMS © Copyright 2005 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. positivenegative

FUNFORMS © Copyright 2005 UNITY POINT 16 8 4 2 32 64 128 2561 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

FUNFORMS © Copyright 2005 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 2424 2323 2 2121 2525 2626 2727 2828 2020 Unity point As you have seen in the counting that we did earlier, number values double at each successive position going down the staff.

FUNFORMS © Copyright 2005 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. 2424 2323 2 2121 2020 Unity Point 1/22 -1 1/42 -2 1/82 -3 1/162 -4 Fractional Numbers Whole Numbers

FUNFORMS © Copyright 2005 COUNTING COUNTING WITH FUNFORMS Funforms is simple. It is based on the concept of pairs. After number one we come to a pair of one's. 2 1

FUNFORMS © Copyright 2005 COUNTING COUNTING WITH FUNFORMS 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.

FUNFORMS © Copyright 2005 COUNTING COUNTING WITH FUNFORMS 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

FUNFORMS © Copyright 2005 COUNTING 16 8 4 21 8 4 21 8 4 21 8 4 21 8 4 21 12345

FUNFORMS © Copyright 2005 COUNTING 16 8 4 21 8 4 21 8 4 21 8 4 21 8 4 21 678910

FUNFORMS © Copyright 2005 COUNTING 16 8 4 21 8 4 21 8 4 21 8 4 21 8 4 21 1112131415

FUNFORMS © Copyright 2005 COUNTING 16 8 4 21 8 4 21 8 4 21 8 4 21 8 4 21 1617181920

FUNFORMS © Copyright 2005 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.

FUNFORMS © Copyright 2005 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.

FUNFORMS © Copyright 2005 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

FUNFORMS © Copyright 2005 MANIPULATING FUNFORMS 16 8 4 2 1 35 + 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.

FUNFORMS © Copyright 2005 CALCULATION/CLEARING PHASE 16 8 4 2 1 35 + 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

FUNFORMS © Copyright 2005 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. 35 + 8 SUM =

FUNFORMS © Copyright 2005 6591 + 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

FUNFORMS © Copyright 2005 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

FUNFORMS © Copyright 2005 CALCULATION/RESOLVING PHASE 16 8 4 2 1 161 - 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

FUNFORMS © Copyright 2005 16 8 4 2 1 161 - SUBTRACTION MANIPULATING FUNFORMS

FUNFORMS © Copyright 2005 MANIPULATING FUNFORMS 16 8 4 2 1 161 - SUBTRACTION

FUNFORMS © Copyright 2005 MANIPULATING FUNFORMS 16 8 4 2 1 161 - SUBTRACTION The FUNFORM figure is now in its simplest form and nothing further needs to be or can be done. 15 REMAINDER =

FUNFORMS © Copyright 2005 22544 - SUBTRACTION MANIPULATING FUNFORMS 2 1 32 64 4 16 8 128 256

FUNFORMS © Copyright 2005 22544 - SUBTRACTION MANIPULATING FUNFORMS 2 1 32 64 4 16 8 128 256

FUNFORMS © Copyright 2005 SUBTRACTION MANIPULATING FUNFORMS 2 1 32 64 4 16 8 128 256 22544 -

FUNFORMS © Copyright 2005 SUBTRACTION MANIPULATING FUNFORMS 2 1 32 64 4 16 8 128 256 22544 -

FUNFORMS © Copyright 2005 22544 - SUBTRACTION MANIPULATING FUNFORMS 2 64 81 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. =

FUNFORMS © Copyright 2005 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.

FUNFORMS © Copyright 2005 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.

FUNFORMS © Copyright 2005 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.

FUNFORMS © Copyright 2005 610 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

FUNFORMS © Copyright 2005 610 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

FUNFORMS © Copyright 2005 610 X MULTIPLICATION MANIPULATING FUNFORMS 16 8 4 2 1 32 New Temporary Unity Point SPACE FLAG SPACE FLAG

FUNFORMS © Copyright 2005 610 X MULTIPLICATION MANIPULATING FUNFORMS 16 8 4 2 1 32

FUNFORMS © Copyright 2005 610 X MULTIPLICATION MANIPULATING FUNFORMS 2 1 16 8 4 32 60=

FUNFORMS © Copyright 2005 226xMULTIPLICATION MANIPULATING FUNFORMS 16 8 4 2 1 32 64 128 256 SPACE FLAG New Temporary Unity Point

FUNFORMS © Copyright 2005 226xMULTIPLICATION MANIPULATING FUNFORMS 16 8 4 2 1 32 64 128 256 SPACE FLAG New Temporary Unity Point

FUNFORMS © Copyright 2005 226xMULTIPLICATION MANIPULATING FUNFORMS 16 8 4 2 1 32 64 128 256 SPACE FLAG New Temporary Unity Point

FUNFORMS © Copyright 2005 226xMULTIPLICATION MANIPULATING FUNFORMS 16 8 4 2 1 32 64 128 256

FUNFORMS © Copyright 2005 226xMULTIPLICATION MANIPULATING FUNFORMS 16 8 4 2 1 32 64 128 256

FUNFORMS © Copyright 2005 226 x MULTIPLICATION MANIPULATING FUNFORMS 16 8 2 1 32 64 4 128 256 132 =

FUNFORMS © Copyright 2005 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.

FUNFORMS © Copyright 2005 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

FUNFORMS © Copyright 2005 MANIPULATING FUNFORMS 16 8 4 2 1 32 64 1/2 1/4 The same is true for "fiveness" written at various positions. FRACTIONS

FUNFORMS © Copyright 2005 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

FUNFORMS © Copyright 2005 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 5/83/4 + 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.

FUNFORMS © Copyright 2005 MANIPULATING FUNFORMS 8 4 2 1 1/2 16 32 1/4 1/8 ADDING FRACTIONS 5/83/4 +

FUNFORMS © Copyright 2005 MANIPULATING FUNFORMS 8 4 2 1 1/2 16 32 1/4 1/8 ADDING FRACTIONS 5/83/4 + 1 3/8 =

FUNFORMS © Copyright 2005 MANIPULATING FUNFORMS 8 4 2 1 1/2 16 32 1/4 1/8 ADDING FRACTIONS 7/8 + 1 1/2

FUNFORMS © Copyright 2005 MANIPULATING FUNFORMS 8 4 2 1 1/2 16 32 1/4 1/8 ADDING FRACTIONS 7/8 + 1 1/2 = 2 3/8

FUNFORMS © Copyright 2005 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.

FUNFORMS © Copyright 2005 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. 1/21/2 x Unity Point ½ x ½ is shown. ½ can be thought of as space-flag. SPACE FLAG ACSENDING FROM UNITY POINT

FUNFORMS © Copyright 2005 MANIPULATING FUNFORMS 8 4 2 1 1/2 16 32 1/4 1/8 MULTIPLYING FRACTIONS Multiplicand appearance, beginning at unity point. 1/21/2 x New Temporary Unity Point SPACE FLAG

FUNFORMS © Copyright 2005 MANIPULATING FUNFORMS 8 4 2 1 1/2 16 32 1/4 1/8 MULTIPLYING FRACTIONS 1/21/2 x = 1/4

FUNFORMS © Copyright 2005 MANIPULATING FUNFORMS 4 2 1 1/2 1/4 8 16 1/8 1/16 MULTIPLYING FRACTIONS 3/43/4 x SPACE FLAG Unity PointNew Temporary Unity Point

FUNFORMS © Copyright 2005 MANIPULATING FUNFORMS 4 2 1 1/2 1/4 8 16 1/8 1/16 MULTIPLYING FRACTIONS 3/43/4 x SPACE FLAG New Temporary Unity Point

FUNFORMS © Copyright 2005 MANIPULATING FUNFORMS 4 2 1 1/2 1/4 8 16 1/8 1/16 MULTIPLYING FRACTIONS 3/43/4 x

FUNFORMS © Copyright 2005 MANIPULATING FUNFORMS 4 2 1 1/2 1/4 8 16 1/8 1/16 MULTIPLYING FRACTIONS 3/43/4 x 9/16 =

FUNFORMS © Copyright 2005 MANIPULATING FUNFORMS DIVISION DIVISION is just serial subtraction, while keeping score.

FUNFORMS © Copyright 2005 MANIPULATING FUNFORMS DIVISION16-4 164 ÷= ? I believe that a learner should be asked… How many times can the divisor be subtracted from the number to be divided?

FUNFORMS © Copyright 2005 12 8 40 16 MANIPULATING FUNFORMS-4 164 ÷ ? 1 2 34 = DIVISION How many times can the divisor be subtracted from the number to be divided?

FUNFORMS © Copyright 2005 357 ÷ 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

FUNFORMS © Copyright 2005 357 ÷ DIVISION MANIPULATING FUNFORMS 16 8 4 2 1 32 FLAG New Temporary Unity Point RECORD FLAG AT NEW UNITY POINT

FUNFORMS © Copyright 2005 357 ÷ DIVISION MANIPULATING FUNFORMS 16 8 4 2 1 32

FUNFORMS © Copyright 2005 357 ÷ 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

FUNFORMS © Copyright 2005 357 ÷ DIVISION MANIPULATING FUNFORMS 16 8 2 41 32 5 = The final results of our “recording staff” provides us with the answer.

FUNFORMS © Copyright 2005 639 ÷ DIVISION MANIPULATING FUNFORMS 16 8 2 4 1 32 64 Serial Subtraction Recording Staff Unity Point FLAG SPACE

FUNFORMS © Copyright 2005 639 ÷ DIVISION MANIPULATING FUNFORMS 16 8 2 4 1 32 64 RECORD FLAG AT UNITY POINT Unity Point FLAG SPACE

FUNFORMS © Copyright 2005 639 ÷ DIVISION MANIPULATING FUNFORMS 16 8 2 4 1 32 64 New Temporary Unity Point RECORD FLAG AT NEW UNITY POINT

FUNFORMS © Copyright 2005 639 ÷ DIVISION MANIPULATING FUNFORMS 16 8 2 4 1 32 64 RECORD FLAG AT NEW UNITY POINT New Temporary Unity Point

FUNFORMS © Copyright 2005 639 ÷ DIVISION MANIPULATING FUNFORMS 16 8 2 41 32 64 = 7

FUNFORMS © Copyright 2005 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.

FUNFORMS © Copyright 2005 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

FUNFORMS © Copyright 2005 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.

FUNFORMS © Copyright 2005 MANIPULATING FUNFORMS 8 4 1 21/2 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

FUNFORMS © Copyright 2005 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 PointNew Temporary Unity Point FLAG SPACE FLAG SPACE FLAG New Temporary Unity Point

FUNFORMS © Copyright 2005 MANIPULATING FUNFORMS 8 4 2 1 1/2 16 32 1/4 1/8 DIVIDING FRACTIONS Recording Staff 10 = 25 ÷ 2 1/2

FUNFORMS © Copyright 2005 GLOSSARY PAGE 1 NAVIGATION PAGE 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. CLICK FOR PAGE 2 FUNFORMS TERMS

FUNFORMS © Copyright 2005 GLOSSARY PAGE 2 NAVIGATION PAGE 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. CLICK FOR PAGE 1 MATHEMATICAL TERMS

FUNFORMS © Copyright 2005 FUNFORMS NAVIGATION PAGE Introduction The Basics The Basics Negative and Positive Values Negative and Positive Values Unity Point Unity Point Counting Manipulation Addition Subtraction BACK Multiplication Fractions Adding Fractions Adding Fractions Multiplying Fractions Multiplying Fractions Division Dividing Fractions Dividing Fractions GLOSSARY PAGE GLOSSARY PAGE