Abstract:
Arithmetic Measurement System quantifies arithmetic operations just by measurement and without arithmetic calculation or even counting. Playing cards may be annotated with size-coded marks or card tags to represent face value. For example, a tag representing a face value 5 is half the size of one representing a value 10 using a linear tag scale, or about 7/10 th  the size using a log 10  tag scale. Users align linear tags in a contiguous row to add values or overlap tags to subtract values. Users align logarithmic tags in a row to multiply values or overlap tags to divide values. Integrated and separate measures with corresponding linear or logarithmic tag-scales quantify a total value of aggregate or net tag row size. Product developers may use the Arithmetic Measurement System of tags and measures to provide novel educational math manipulatives and playing card games for players unskilled in arithmetic.

Description:
COPYRIGHT NOTICE 
   A portion of the disclosure of this patent document, including Figures, contains material which is subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent document or the patent disclosure, as it appears in the Patent and Trademark Office patent file or records, but otherwise reserves all copyright rights whatsoever. 
   FIELD OF INVENTION 
   This invention relates generally to math education and card games. More particularly, but not by way of limitation, this invention relates to an arithmetic measurement system of tags and measures in which players with minimal math and reading skills learn to associate, add subtract multiply and divide fractions, decimals, percentages and degree values to determine a group value through play, and advanced players compete and strive for mastery. 
   BACKGROUND OF THE INVENTION 
   Prior Art 
   We are surrounded by basic math. We deal in fractional, percentage and decimal and even degree-based measurement systems everyday, often without any real understanding. 
   The Nation&#39;s Report Card by the National Assessment of Educational Progress (NAEP) shows very little improvement in U.S. student math scores since 1973. Meanwhile other counties forge ahead. The 2003 OECD Programme for International Student Assessment (PISA) shows many industrialized nations such as Japan, Korea and Canada with significantly higher achievement in mathematics literacy than US students. 
   Numerous games in prior art utilize playing cards for basic math education. These games would not be suitable for the purpose of the present invention as described herein below. 
   Some educational card games deal with just one measurement or number system such as fractions, and show no relationship to other measurement or number systems. Most math games assume a level of competence or familiarity with the underlying math concepts and are unsuitable for rank beginners. 
   Also most educational card games are limited to one game and a single math operation such as addition. 
   Some math games use costly equipment such as game boards. These are too expensive for many users. 
   Other educational card games suit serious-minded players only who want to learn. These offer few fun-in-learning opportunities and are unlikely to generate spontaneous play amongst average students. 
   Objects and Advantages 
   The subject invention is an arithmetic measurement system adaptable to any measurement or number system. In a preferred embodiment it encompasses the addition and subtraction of four measurement or number systems, namely fractions, decimals, percentages and degrees. In other embodiments the arithmetic measurement system supports multiplication and division also. 
   It offers a visual method to compare values and interrelationships and to determine a group value without any need to understand or use the underlying math concepts. Rank beginners become familiar with these mathematical concepts and their relationships simultaneously during play and without prerequisite study. 
   The arithmetic measurement system offers many different card games. It should appeal to novices, intermediate and advanced students alike. It may be customized for any measurement or number system and for any audience in any language and for any level of math proficiency. 
   This game system is inexpensive. In its basic embodiment it comprises a single deck of playing cards. 
   Players start with images and graduate to symbols and numbers at their own pace. Gradually through play, players learn to associate specific numerical values with specific pictures. With practice players start to comprehend arithmetic and should in time become skilled in addition, subtraction, multiplication and division. 
   And the best part is that learning occurs semi-automatically while having fun playing cards. 
   Accordingly, several objects and advantages of the arithmetic measurement system are as follows: 
   (a). A first object is to provide specially marked playing cards that allow anyone even without formal math and reading education to:
         acquire a basic understanding of arithmetic operations including addition, subtraction, multiplication and division;   acquire a basic understanding of equivalent and comparative fractions, decimals, percentages and degree values and their interrelationships just through play;   convert between basic fractions, decimals, percentages and degrees at will;   add, subtract, multiply and divide whole numbers, fractions, decimals and percentages to determine a group value;   review cards in hand without exposing them to opposing players;   evaluate potentially winning combinations visually;   compile a target value proficiently and without recourse to formal mathematics.       

   (b). A second object is to provide a game system for one or more players of any age group, educational and skill level that:
         resembles a standard playing card deck and so has ready acceptance and broad appeal;   can be used to play variations of popular card games;   offers a wide range of new games limited only by the imagination of developers and players;   offers fun-in-learning opportunities in both home and school environments;   appeals to children and induces them to play spontaneously and practice these concepts amongst themselves with little or no adult supervision or direction;   intrigues and challenges even the most advanced players.       

   (c). A third object is to provide an arithmetic measurement system that is inclusive:
         has a wide range of interchangeable equivalent indicia including colors, pictures, symbols and numerals that make play and learning accessible to most players;   provides for novices with little or no math education or understanding of the mathematical concepts involved and with only limited reading ability;   is low cost and customizable for any audience including minority groups, players with learning disabilities or special needs, special interest groups, underprivileged people and developing nations worldwide regardless of demographic;   is language-independent and easily adapted for worldwide use with local character and symbol sets.       

   Still further objects and advantages will become apparent from a consideration of the ensuing description and drawings. 
   SUMMARY 
   In accordance with the present invention an arithmetic measurement system comprises a set of specially marked playing cards and a measure. The cards offer fun, interesting, and effective ways to learn and practice arithmetic. Even players unskilled in arithmetic, and even counting, may add subtract multiply or divide face values just by measurement.
     3. Peripheral indicia denote face value and suit. Overlapped cards may be viewed, melded and aggregated in hand without exposure. Specific peripheral indicia or tags are coded to denote face value in accordance with a predetermined measurement system.   2. A measure is calibrated in accordance with the predetermined tag scale to quantify individual and group face values. It quantifies the length of a row of contiguous card tags, one from each card, or the difference between two overlapping card tags, or the difference between two rows of one or more card tags, directly into a group value in any measurement or number system. Players do not need to know measurement or number system or the underlying math. Addition, subtraction, multiplication and division may be supported.   3. Measures may be integrated into the card indicia or be separate. Slot measures and Template measures may be shared. Clip measures help players hold align and cards in hand.   4. Informational indicia on card faces and backs denote hints and strategies to speed learning and play. They may include pictographs, mathematical equations, notes, and segment charts and data sheets.   

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     In the drawings, closely related Figures have the same number but different alphabetic suffixes. The arithmetic measurement system can better be understood from the following detailed description of a preferred embodiment of the invention and by referring to the illustrative drawings in which: 
       FIG. 1  illustrates a Face of a Round-shaped Number-6 Card; 
       FIG. 1A  illustrates a Round-shaped Card Suit Marker; 
       FIG. 1B  illustrates a Round-shaped Number-6 Card Segment Layout; 
       FIG. 1C  illustrates a Round-shaped Number-6 Card Legend; 
       FIG. 1D  illustrates a Almost-square Number-6 Card Tag Detail; 
       FIG. 2  illustrates a Face of a Rectangular Number-6 Card; 
       FIG. 2A  illustrates a Face of a Rectangular Number-3 Card; 
       FIG. 2B  illustrates a Face of a Rectangular Number-5 Card; 
       FIG. 2C  illustrates a Face of a Rectangular Number-10 Card; 
       FIG. 3  illustrates a Meld of Round-shaped Cards; 
       FIG. 3A  illustrates a Tag Detail; 
       FIG. 3B  illustrates a Measure; 
       FIG. 3C  illustrates a Composite Curved Measure; 
       FIG. 4  illustrates a Meld of Round Cards with Asymmetric Tags; 
       FIG. 4A  illustrates a Asymmetric Tag and Curved Measure detail; 
       FIG. 4B  illustrates a Face of a Round Number-3 Card; 
       FIG. 4C  illustrates a Face of a Round Number-5 Card; 
       FIG. 4D  illustrates a Face of a Round Number-10 Card; 
       FIG. 5A  illustrates a Rectangular Card Back—1 st  Alternative; 
       FIG. 5B  illustrates a Rectangular Card Back—2 nd  Alternative; 
       FIG. 5C  illustrates a Rectangular Card Back—3 rd  Alternative; 
       FIG. 6  illustrates a Round Card Back; 
       FIG. 7A  illustrates a Hand-held Clip Measure—Plan View; 
       FIG. 7B  illustrates a Hand-held Clip Measure—Cross Section; 
       FIG. 7C  illustrates a Using a Hand-held Clip Measure; 
       FIG. 8A  illustrates a Hand-held Clip Measure 1 st  alternative—Plan View; 
       FIG. 8B  illustrates a Hand-held Clip Measure 1 st  alternative—Cross Section; 
       FIG. 8C  illustrates a Hand-held Clip Measure 2 nd  alternative—Plan View; 
       FIG. 8D  illustrates a Hand-held Clip Measure 2 nd  alternative—Cross Section; 
       FIG. 9A  illustrates a Meld of Aligned Rectangular Cards; 
       FIG. 9B  illustrates a Using an Integrated Measure—Rectangular Cards; 
       FIG. 9C  illustrates a Hand-held Clip Measure—Rectangular Cards; 
       FIG. 9D  illustrates a Using a Hand-held Clip Measure—Rectangular Cards; 
       FIG. 10  illustrates a Small Hand-held Clip Measure—3-D View; 
       FIG. 10A  illustrates a Small Hand-held Clip Measure—Plan View; 
       FIG. 11A  illustrates a Using a Small Hand-held Clip Measure—Round Cards; 
       FIG. 11B  illustrates a Group of Aligned Tags—Round Cards; 
       FIG. 11C  illustrates a Using a Small Hand-held Clip Measure—Close-up View; 
       FIG. 12A  illustrates a Transparent Template Measure—Round Cards; 
       FIG. 12B  illustrates a Using a Transparent Template Measure—Round Cards; 
       FIG. 13A  illustrates a Slot Measure; 
       FIG. 13B  illustrates a Using a Slot Measure; 
       FIG. 14A  illustrates a Inclined Slot Measure; 
       FIG. 14B  illustrates a Using an Inclined Slot Measure; 
       FIG. 15  illustrates a Template Measure; 
       FIG. 16  illustrates a set of Factored Tags; 
       FIG. 17  illustrates a Using a Card Back Measure &amp; Custom Rulers—Round Cards; 
       FIG. 18A  illustrates a Set of Logarithmic Tags; 
       FIG. 18B  illustrates a Multiplication and a Division using Logarithmic Tags; 
       FIG. 18C  illustrates a 10×10 Multiplication using Logarithmic Tags. 
   

   DRAWINGS—REFERENCE NUMERALS 
   
       
         20  Segment (share, portion, fraction) 
         22  Segment Layout (map, plan) 
         23  Segment Bar Chart 
         24  Segment Boundary 
         26  Suit Marker 
         28  Card Number 
         29  Information Area 
         30  Fractional Value 
         31  Fractional Hint 
         32  Percentage Value 
         33  Percentage Hint 
         34  Degree Value 
         35  Degree Hint 
         36  Decimal Value 
         37  Decimal Hint 
         40  Number-3 Card Tag 
         42  Number-5 Card Tag 
         43  Number-6 Card Tag 
         45  Number-10 Card Tag 
         49  Integrated Measure 
         50  #3 Segment Ring 
         51  #4 Segment Ring 
         52  #5 Segment Ring 
         53  #6 Segment Ring 
         54  #8 Segment Ring 
         55  #10 Segment Ring 
         56  #12 Segment Ring 
         57  #20 Segment Ring 
         58  Publicity Area 
         60  Body 
         61  Clip 
         62  Card Platform 
         63  Grip 
         64  Clamp 
         65  Magnifying Lens 
         66  Percentage Measure 
         67  Fraction Measure 
         68  Decimal Measure 
         69  Degree Measure 
         70  Hinge 
         71  Finger Hole 
         72  Card Slot 
         73  Inclined Slot 
         74  Composite Curved Measure—Progress 
         75  Composite Curved Measure—Required 
         76  Double Percentage Measure 
         77  Double Fraction Measure 
         78  Double Decimal Measure 
         79  Double Degree Measure 
     
  
   DETAILED DESCRIPTION 
   A preferred embodiment of this invention is a single deck of playing cards. Playing cards are economical, widely accepted and easily customized. They are readily available in a variety of shapes, sizes, colors and materials. Round or rectangular-shaped playing cards suit different audiences and markets. 
   Round-shaped cards may be considered more eye-catching than rectangular cards. Their appeal may generate spontaneous play. A round card is a natural choice for pie-shaped segments. It permits a symmetrical and pleasing design. However curved edges complicate tags and meld value measurement. 
   Rectangular cards are more popular and much cheaper than round cards. Straight edges simplify tag alignment and measurement. The poker-sized format (62 mm×88 mm/2½ in×3½ in; ISO 216 B8) is preferred for this application, although any size/format may be used as a matter of design choice. 
   Deck: The deck comprises 52 number cards. Each number card belongs to a designated suit. A card number designates individual face value and rank. Ranks depend upon intended use. Jokers are optional. 
   Face Value or Rank: In a preferred embodiment face values are expressed as portions. The card number is a divisor, so portion size and card number are inversely proportional. A high card number defines a small portion size, etc. Practical considerations limit minimum portion size, for example one-twentieth of one whole unit. Advanced decks may use multiple portions, for example three one-fifths. 
   Suit: A preferred embodiment uses the standard playing card deck arrangement of four 13-card suits. This familiar format should ease acceptance, use, play and thus learning. Also this format can be used to play many popular non-math card games. However, non-standard suits are also contemplated herein. 
   Color-coded Indicia: Regardless of suit all number cards of the same face value preferably share a unique and distinguishable color code. Thus color designates rank, value or portion size. The information contained on all card backs is also preferably color-coded with the same color-coding scheme as the faces. 
   Color-coding preferably befits each specific application. The actual card set, color perception and economic factors are involved. Colors may be designed to be easily discernible and visually differentiated to avoid confusion and so speed play. Colors are preferably consistently reproducible and the production process preferably cost effective. If possible, colors can have associative qualities also. 
   Game Objective Game developers may set their own specific game objectives. A common math game objective is to meld several number cards together from the same suit to make one whole unit. 
   Example: A meld of two number-5 cards (one-fifth each), one number-10 (one-tenth), one number-3 card (one-third) and one number-6 card (one-sixth) totals one whole unit. This example meld will be used throughout for ease of reference. 
   A preferred embodiment for the general market is a rectangular-shaped card with integrated measure. Separate measures are desirable with round-shaped cards. Both card shapes are viable and will be covered in sufficient detail to allow game developers to offer customized game sets for their specific applications. 
     FIG. 1  shows the face of a sample round-shaped number-6 card. 
   The number-6 card is all about the number six. The rank of a number-6 card is one-sixth part share portion of one whole unit. All indicia on a number-6 card are colored pink, for example. 
   To appeal to the widest possible audience the number-6 card has a plurality of interchangeable indicia types designating its face value. Preferably, the card has seven interchangeable indicia types and color-coding to designate its one-sixth face value. Some players may focus on images. Others may use one of four measurement systems. Still others may use only color. 
   Suit Marker— FIG. 1A   
   A suit marker  26  is a unique suit identifier like the ubiquitous             ♦ ♥           suits. It is visible from the card edge. Players may slide or fan cards open to see all suit markers without exposing any card. Preferably, marker  26  is easily recognizable and distinguishable from other suit markers.
   Markers may be icons, symbols, patterns or images, preferably well known. For example, a game developer could use/•%° to designate ‘fraction’, ‘decimal’ ‘percentage’ and ‘degree’ suits respectively if desired. 
   Segment Layout— FIG. 1B   
   A segment layout  22  depicts five unpopulated one-sixth segments that together with populated segment  20  comprise one whole unit. Radial segment boundary markers  24  may delineate unpopulated segments. 
     FIG. 1B  also shows an annular-shaped developer&#39;s information area  29 . This is reserved for copyright, trademarks, patents, slogan, branding, sponsor information, or the like, as desired. 
   Legend— FIG. 1C   
   The card face has a plurality, preferably five, numerical interchangeable indicia denoting equivalent value. A card number  28  on the card, preferably at center, contains the numeral six. This identifies the card as a number-6 card. The numerical value of segment  20  is expressed in a plurality, preferably four, prevalent measurement systems. Players may use a measurement system of choice or switch between systems at will during play. 
     FIG. 1C  shows the segment size expressed numerically as a fractional value  30  (⅙); as a percentage value  32  (16.67%); as a number of degrees of arc or degree value  34  (60°) and as a decimal value  36  (0.1667). These indicia are prominently displayed in a large legible color-coded font. They offer endless passive and active associative learning opportunities during play. 
   Informational indicia or educational hints may be paired with each measurement value. For example, fractional hint  31  (6×⅙=1) is paired with fractional value  30  showing that six one-sixth portions equal one whole unit. Similarly percentage hint  33  (6×16.67%=100%), -degree hint  35  (6×60°=360°) and decimal hint  37  (6×0.1667=1.0) elaborate percentage  32 , -degree  34  and decimal value  36  respectively. 
   A different (e.g., smaller black colored) font is preferred for hints. Hints are used only on demand. They should not interfere visually with the normal information flows during play. 
   Card Tag— FIGS. 1D ,  3 ,  3 A,  4 ,  7 A, 11 B 
     FIG. 1D  shows an enlarged view of a sample number-6 card tag  43  for round-shaped cards. The dotted lines are for illustration only. Tag  43  designates a value of one-sixth share of one whole unit. A minimum of one tag  43  on the card face is required.  FIG. 1  shows six tags  43  around the edge of the card face, one on each segment boundary  24 . This symmetrical design reinforces card number  28 . 
   Card tags are size-coded using a predetermined tag scale. Players may use tags coded with a linear tag scale to compare face values without understanding or even seeing other indicia. For example, players may correctly deduce that a number-6 portion is half the size of a number-3 portion (not shown). A simple ruler or custom measuring device may speed such comparisons, see  FIG. 3   
   The peripheral location of card tags allows players to guard cards from prying eyes. Players may slide or fan open overlapping cards in hand to see any color-coded card tag (or suit marker) without exposure. 
   Players may also ascertain absolute card values when the tag scale is known. The tag scale divided by card number  28  sets the nominal tag dimensions for each number card. For example,  FIG. 1D  uses a 25.4 mm/1.0 in tag scale on a 76.2 mm/3.0 in diameter round card. Tag  43  measures 4.23 mm/one-sixth of one inch. Similarly a number-3 tag  40  (not shown) measures 8.47 mm/one-third of an inch on this tag scale. 
   Designers may select an appropriate tag scale and tag shape for their intended audience. As stated, preferably tags are color-coded to denote face value. Tags may be scaled in one or two dimensions. 
   In this round card example, tag  43  is scaled in two dimensions and has an almost-square shape. The outer and inner edges are curved with the same radius of curvature as the round card. The radial tag-height and distance between the outer and inner curved edges equals the tangential tag-width and distance between the two straight sides. 
   Each tag  43  straddles (is bisected by) a radius of the round card. The straight sides of tag  43  are parallel to and equidistant from the radius in question. The outer edge of each number-6 tag  43  is located on the circumference of the card face. The center point of the inside edge of tag  43  may be located on the perimeter of segment layout  22  if desired. Preferably scale the diameter of segment layout  22  accordingly. 
   Tag shape facilitates card alignment during meld value assessment. The outer edge of tag  43  on an overlapping card may be aligned exactly with the inner edge of any tag on an overlapped card, see  FIG. 11B . Radial tag size on an overlapped card sets the radial offset between adjacent cards, see  FIG. 7C . Almost-square tags and asymmetrical tags, see  FIG. 3A , may also be aligned side by side using the straight tag edges. Here tag-width sets the offset between adjacent cards, see  FIGS. 3 &amp; 4 . 
   Thus players may offset overlapping cards in a meld and align one tag from each card side by side to form a contiguous row of tags. The row size of adjacent tags in a group of correctly aligned cards is directly proportional to the group value, as determined by the tag scale. 
   When a contiguous row of tags measures one tag scale, then the group value is one whole unit (1/1, 100%, 360° and 1.00). So players with only limited math skills, but who can measure a row of tags, may meld cards to any value without knowledge of the other indicia or underlying math. 
   Rectangular Card Face— FIG. 2   
     FIG. 2  shows the face of a sample rectangular card. It has similar indicia to the round-shaped card above. Card number  28  at center is surrounded by fractional value  30  and hint  31 , percentage value  32  and hint  33 , degree value  34  and hint  35  and decimal value  36  and hint  37 . 
   The number-6 card in  FIG. 2  capitalizes on its rectangular shape, preferably as follows:
         two informational indicia, segment layout  22  and segment bar chart  23 , depict portion size in relation to whole unit size in easy to understand and use chart formats (pictographs);   segment layout  22  is grouped with fractional value  30  and degree value  34  at top;   segment bar chart  23 , percentage value  32  and decimal value  36  are grouped at bottom;   a first tag edge of tag  43  aligns with the origin of integrated measure  49 ; a second tag edge registers face value on integrated measure  49 ; preferably all rectangular card faces have four such integrated measures  49 ;   tags are scaled in one dimension only along a card edge; a 50.8 mm/2.0 in tag scale allows a 6 mm/0.25 in wide suit marker  26  on poker-sized cards; smaller tag scales allow wider suit markers  26 .
 
 FIGS. 2A ,  2 B and  2 C depict sample number-3, -5 and number-10 rectangular card faces.
 
Group Value
       

   As stated the total size of a row of contiguous tags, one from each card in a group of correctly aligned cards, is directly proportional to the total face value of the group, as defined by the tag scale:
         when the tag scale is known, players may measure total tag row size with a linear measuring device, such as a ruler, and convert this distance into a group value by hand or calculator; however this method is tedious, prone to error and slow;   alternatively a measure may be calibrated to convert tag row size directly into a group value in any prevalent measurement system and without recourse to math; the tag scale need not be declared even;   players may use an integrated measure on any card; if so a separate ruler or measure is unnecessary;   integrated measures on rectangular card edges are straight; they are easy to access and use;   integrated tags and measures on round card edges or inside card faces may be more problematical in some instances.       

   Game designers may choose a tag shape, scale and measure appropriate to card shape and application. 
   Measures 
   Actual measure design depends upon the individual application. The illustrations include some examples of straight and curved measures, double measures and composite curved measures as follows:
           FIG. 2  shows integrated measure  49       FIG. 3A  illustrates straight and curved measures     FIGS. 3B ,  5 A,  6 ,  7 A,  9 C,  10 A,  13 A,  14 A and  15  show different applications of percentage measure  66 , fraction  67 , decimal  68  and degree measure  69       FIGS. 3C ,  13 A,  15  show composite curved measures  74  and  75       FIG. 12A  illustrates measures  76 ,  77 ,  78  and  79   
As stated measures may be scaled according to a linear scale or a logarithmic scale.  FIG. 18C  shows a measure with logarithmic graduations.
 
Rectangular and Round-Shaped Card Backs
       

   The card back design is not abstract. Informational indicia and segment charts show relationships between card number  28  and fractional value  30 ; and between different sized segments. The object is to provide instant game hints and suggest winning solutions using colors and cookie-cutter patterns. 
   Rectangular Card Back— FIG. 5A   
   All rectangular card backs are preferably segment charts and measures also. Players may align a row of tags with any data row or measure on any card back to measure a group size. A sample rectangular card back in  FIG. 5A  uses the same 50.8 mm/2.0 in tag scale and color-coding scheme as the card face. 
   It comprises in a preferred embodiment:
         several (12 are shown) segment data rows; segment sizes denote relative and absolute values; each data row starts with a card number  28  (row number, divisor) at left (these may or may not be consecutive); a fraction or segment bar chart  23  occupies the center column (same tag scale as tags and measure  49  on the card face); a fractional value  30  (the reciprocal of card number) ends each row;   each segment bar chart  23  contains one whole unit divided into an appropriate number of color-coded equal sized segments as defined by card number  28  (divisor);   each colored segment is the same size and color as a tag on the face of a corresponding number card; so segments in row/card numbers 3, 5, 6 and 10 match card tags  40 ,  42 ,  43  and  45  respectively;   two integrated measures  68  and  66  at top, and two integrated measures  67  and  69  at bottom, use a 50.8 mm/2.0 in tag scale and align horizontally with segment bar charts  23 .       

   The card back reveals that two number-5 tags  42  and one number-10 tag  45  equal 50% of a whole unit. A number-6 tag  43  and a number-3 tag  40  total 50% also. So these two groups are equal and total one whole unit together. Thus players may find inspiration and identify winning strategies during play. 
     FIG. 5B  has the same data as  FIG. 5A  but in a different sequence. The data rows are grouped into radix-2, radix-5 and radix-3 number groups to speed segment comparisons and conversions. 
     FIG. 5C  shows a card set with ranks 1 through 13 suitable for both popular and math card games. Players may learn to associate card number  28  and portion sizes semi-automatically in play. 
   Round card backs are similar. However absolute segment length is somewhat arbitrary. 
   Round Card Back— FIG. 6   
   The round playing card back has the same basic features as a rectangular card back. It may also include informational indicia. Color-coded concentric segment rings surround a central publicity area  58  reserved for branding, logo, etc. Radial angle, not segment length, denotes value. Round card backs and faces may share the same color-coding scheme. Integrated measures  49  use the same tag scale as the card face. 
   The radial width of each ring shown is 3.175 mm/0.125 in on a 76.2 mm/3 in card. Together the eight rings measure 25.4 mm wide or one tag scale. Each segment ring is divided into a set of equal sized segments as defined by card number  28 . Again segment length does not equal tag size. Instead segment length equals ring circumference (varies based on its position in the radial sequence) divided by card number  28 . 
   Players may find winning strategies by studying segment relationships. For example,  FIG. 6  shows #3 ring  50 , #6 ring  53  and #12 ring  56  divided respectively into three, six and 12 equal-sized segments. It is evident that four one-twelfths equal two one-sixths or one one-third. Multicolored drawings speed comprehension. 
   Radix-2 and radix-5 segment relationships are also evident. For example, two quarters on #4 ring  51  equal four one-eighths on #8 ring  54 ; these in turn equal two one-fifths on #5 ring  52  plus one one-tenth segment on #10 ring  55 . Similarly one one-tenth on #10 ring  55  equals two one-twentieths on #20 ring  57 , etc. 
   Four radial and four tangential integrated measures convert a row of contiguous tags directly into a group value. The edges of the eight segment rings graduate four radial measures  66 ,  67 ,  68  and  69  in 3.2 mm or 0.125 in increments. Developers may calibrate the four tangential measures  49  as desired. 
   Separate Measures 
   Separate measures may be designed for hand-held or tabletop use and for round-shaped or rectangular-shaped cards. They help players organize cards, align tags and measure a row of correctly aligned tags. They may include informational indicia about equivalent and comparative values and game strategies also. 
   Hand-held Measure with Card Clip—Round-Shaped Cards,  FIGS. 7A  &amp; B 
   A hand-held measure with card clip or clip measure is a simple tool small enough to fit comfortably in one hand. It has two main functions. It helps players organize hold, align and meld cards in hand. It measures a row of correctly aligned tags and converts tag row size directly into a total group value. 
   A clip measure is a compound lever comprising a calibrated clip and card platform joined at a fulcrum by a hinge to form a card slot between the clip and the card platform. A clip measure may be manufactured in one piece using an injection molding process, or the like. Clear acrylic or polycarbonate customarily used for rulers and drafting tools may be used. 
     FIG. 7A  shows a plan view in the xy-plane of a sample clip measure.  FIG. 7B  shows a yz cross-section. 
   A body  60  is preferably the shape and size of a round playing card. A card platform  62  supports cards during melding and measurement. A card slot  72  is located between platform  62  and a card clip  61 . Slot  72  capacity depends upon card set and game objectives. A hinge  70  attaches clip  61  to body  60  and allows limited movement of clip  61  back and forth in a direction perpendicular to platform  62 . 
   Clip  61  has a card clamp  64  on its free end and facing inwards. Preferably clamp  64  aligns with the center of body  60 . A grip  63  is located on the outer surface of clip  61 . Light thumb pressure on grip  63  moves clip  61  with clamp  64  towards cards resting on platform  62  gradually closing hinge  70  and slot  72 . Thus cards inside slot  72  are held secure between clamp  64  and platform  62 . 
   Body  60  and/or clip  61  may be calibrated appropriately. Measure  66  converts tag row size into percentage value directly. Body  60  has fraction decimal and degree measures  67 ,  68  and  69  for meld measurement without clip  61 . Measures may be molded into body  60  and clip  61  during injection molding or the like. This may eliminate costly measure calibration, inscription or printing after molding. 
   Preferably clip  61  and platform  62  interleave to reduce product costs. Players may manipulate cards inside slot  72  through a finger hole  71  in platform  62 . Cards inside slot  72  form a bridge between clamp  64  and platform  62 . Equal and opposing forces on adjacent edges of platform  62  and clamp  64  hold cards secure. Clip  61  may be extended to increase leverage and grip by moving clamp  64  beyond the center of body  60 . 
   Hand-held Clip Measure—Rectangular Cards,  FIG. 9C   
     FIG. 9C  shows a similar clip measure intended for rectangular cards. Hinge  70  attaches clip  61  to body  60 . Slot  72  holds aligned cards secure between transparent clip  61  and platform  62 . A contiguous row of aligned tags inside slot  72  is visible through clip  61 . Clip  61  is calibrated with measures  68 ,  67 ,  66  and  69  drawn using a 50.8 mm/2.0 in tag scale. Preferably the scale zero aligns with start of a 1 st  card tag, see later. 
   Thus players may read group value directly in a prevalent measurement system. 
   OPERATION 
   Preferred Embodiment 
   A preferred embodiment uses rectangular cards, see  FIGS. 2 , A-C. There are four 13-card suits. 
   Each suit has the same set of radix-3, -4 and radix-5 number cards comprising two one-third, two one-quarter, two one-fifth, two one-sixth, three one-twelfth and two one-twentieth cards. 
   The overall (this preferred deck does not contain all number cards) color-coding scheme is:
         red primary for radix-3: number-3, -6 and -12 are colored red, pink and brown respectively;   blue primary for radix-4: number-4 and -8 cards are colored bright blue and dark blue respectively;   green primary for radix-5: number-5, -10 and -20 cards are colored bright-, dark- and light green.       

   A typical math game objective is to be the first player to meld one whole unit from cards of the same suit. This deck offers 1,012 (253 per suit) unique ways or combinations to make one whole unit using seven or fewer cards. Two jokers add another 5,576 unique combinations making a total of 6,588 winning hands. 
   Here is how a game might be played based upon the popular card game of Rummy. 
   The Deal: 
   
       
       
         
           deal and play are clockwise; the turn to deal passes to the left after each hand; 
           dealer shuffles and the player at his or her right cuts the deck before the deal; 
           dealer deals six cards, one at a time, face down to each player in clockwise rotation; 
           stack the remaining cards face down on the playing surface to form a stock; 
           turn over the top card, place it face-up next to the stock to start a discard pile. 
         
       
     
  
   After the deal, players sort and group their cards into suits ready for play. Preferably players conceal their hand at all times and use only card edges to review overlapped cards. 
   The Play 
   The player seated to the left of the dealer begins by drawing one card either the top card of the discard pile on view if desired or the top card of the stock. The player completes his or her turn by discarding one unwanted card face up on top of the discard pile. Players may not draw the top card of the discard pile and discard it in the same turn. 
   In turn the next player seated to the left draws a card and discards one card. When the last card of the stock has been drawn, the next player may either select the top of the discard pile or draw the top card of the new stock, after turning the discard pile over without shuffling to form a new stock. 
   A Winning Hand— FIG. 9A   
   Depending upon the game, a winning hand contains at least one whole unit using four, five or six (or seven with the discard) cards of the same suit or Jokers. Typically a joker may duplicate any other card within a meld. Preferably joker value is agreed in advance. 
   A player may only declare a winning hand during his or her turn. After drawing from either the discard pile or stock as usual, the winner should lay down a winning meld face up on the table for review by the other players. The winner may make a final discard(s) to end the game or meld all seven cards without discard. 
     FIG. 9A  shows the same sample meld of five rectangular cards with a group value of one whole unit. From back there is a number-5, -10, -5, -3 card and a number-6 card at front. The end of the number-6 card tag at front registers a group value of one unit on the integrated measure on the 1 st  number-5 card at back. 
   Using a Card Back Measure to Measure a Row of Contiguous Tags— FIGS. 9B &amp; 17   
   Players may use an integrated measure on any rectangular card back to measure group value, see  FIG. 9B . This meld measures 1/1 or 360° at bottom, and 1.0 or 100% at top. A meld of round cards may be measured with an integrated measure in like manner, see  FIG. 17 . For clarity only the tags are shown. The tag group measures one whole unit on a tangential measure, or 360° on a radial measure, on the round card back. 
   Using a Clip Measure—Rectangular Cards,  FIGS. 9C  &amp; D 
   Hold body  60  in the palm of one hand. Align thumb on grip  63 . Use other hand to slide a 1 st  card inside slot  72 . Align bottom edges of 1 st  card and clip measure to automatically align 1 st  tag with the measure zero. 
   Feed remaining cards into slot  72  one at a time. Slide each successive card under clip  61  and offset it from its predecessor to give good tag visibility. Align bottom edge of each successor tag with top edge of its predecessor. Alternately release and clench grip  63  to adjust or secure meld alignment. As desired insert finger(s) through hole in platform  62  to grip or adjust card backs inside slot  72 . 
     FIG. 9D  shows a correctly aligned meld inside the clip measure. Simplified card faces improve legibility. Read the group value directly on any one of four measures on clip  61  (1/1, 100%, 1.00 or 360°). 
   Using a Clip Measure—Round Cards,  FIG. 7C   
   Load a clip measure for round cards in like manner.  FIG. 7C  shows the correct alignment for round cards with almost-square tags. The group value is 100%. 
   Subtraction 
   Align the ends of two tags (not the start of one tag with the end of a predecessor tag as for addition):
         select 1 st  card with the larger face value and tag size;   place 2 nd  card with the smaller face value and tag size over 1 st  card;   offset 2 nd  card so 1 st  card tag is visible;   align the ends of the two selected tags together, one from each card;   the start of the smaller tag registers their difference on the 1 st  card measure;   if necessary, compute the sign mentally.       

   Subtraction works for card groups also. Overlay one group of aligned cards and tags on top of a second larger group of aligned cards and tags. Read group difference on the integrated measure on the bottom card. 
   Note: Transparent card stock may be suitable for subtraction in a non-competitive learning environment. 
   Thus a group value in a prevalent measurement system may be found directly just by aligning and measuring a contiguous row of tags, one from each card. Players need not use or even see the other indicia. Any player may obtain a group value without using or understanding the underlying math. 
   Players may start with images and graduate to symbols and numbers at their own pace. Gradually through play, players learn to associate specific numerical values with specific pictures. With practice players start to comprehend equivalencies and should in time become skilled at converting between and determining a group value in all four measurement systems. 
   Associative Learning 
   Players learn differently and at their own pace. Passive association starts gradually in play and is semi-automatic. Familiarity breeds confidence that leads to understanding and perhaps eventually to proficiency. 
   Some players may use measurement systems interchangeably. A player may start with the decimal system. However during play she will be continually exposed to and become familiar with equivalent values in other prevalent measurement systems and to equivalent image and graphic representations also. 
   Association need not be passive. A particular game may be played in decimals, another in fractions, etc. Say-the-name games help players learn to associate sounds with symbols. There are almost endless opportunities for fun and associative learning limited only by the imagination and abilities of players. 
   Novices may play with number cards even if they know nothing about fractions decimals percentages or degrees. For example novices may:
         play variations of matching card games usually played with a standard deck;   order and rank cards visually by segment size or tag size;   associate colors with segment and tag sizes, and with numbers and symbol sets.       

   And the best part is that learning occurs semi-automatically while having fun playing cards. 
   DESCRIPTION 
   Additional Embodiments 
   The subject invention is a game system. Actual games depend upon intended audience and use. Game developers may vary the range, size and number of cards and suits. They may offer generic card sets for multiple games. They may provide custom card sets for specific games and niche markets, for example radix-10 only. They may also offer game accoutrements such as game boards, dice and spinners. 
   Card Deck Variants 
   The card deck in a preferred embodiment above offers a variety of math games for learners, intermediate and advanced players. However, the one-twentieth card and even the one-twelfth card may be unsuitable for younger audiences. 
   Also the addition of a one-half and one-tenth card, even a one-unit card, could greatly simplify learning and speed play, especially for beginners. 
   The range illustrated in  FIG. 5C  matches a standard 52-card deck. It offers all existing playing card games and fraction decimal percentage and degree learning also. This deck has four suits of 13 individual number cards in ranking (portion size) order: one whole unit (1/1), one-half, one-third, one-quarter, one-fifth, one-sixth, one-seventh, one-eighth, one-ninth, one-tenth, one-eleventh, one-twelfth and one-thirteenth of a unit. 
   Novices may use this deck with regular             ♦ ♥           suits to play only existing card games at first. Unintended and involuntary associative math learning could begin automatically.
   A game set may contain several different decks sharing a common back design, for example one deck each of radix-3 cards, radix-4 cards and radix-5 cards. This would allow individual radix-based games or mix&#39;n match games as desired. As stated, number cards need not be limited to a single portion. Multiple portions such as three one-fifth&#39;s may be accommodated easily. Also tag shape and size may vary. 
   Tag Shape and Size Variations 
   Curved edges on round cards complicate tags and meld value measurement. Tags on overlapped round cards are easily obscured. A round shape limits the maximum tag scale to say 38 mm/1.5 in on a 76 mm/3.0 in diameter card; compare to 51 mm/2.0 in even to 76 mm/3.0 in on a rectangular card. 
     FIG. 3  shows the sample meld of round cards using almost-square tags and a 25.4 mm/1.0 in tag scale. Tags  42 ,  45 ,  42 ,  40  and tag  43  measure 5.1 mm, 2.5 mm, 5.1 mm, 8.5 mm, 4.2 mm/one-fifth, one-tenth, one-fifth one-third and one-sixth of an inch respectively in radial height and tangential width. This group of tags measures 25.4 mm/1.0 in; the group value is one whole unit (1/1, 100%, 360° or 1.00). 
     FIG. 2  shows a rectangular-shaped tag  43 .  FIG. 3A  shows some alternative tag shapes with or without integrated measures. The juxtaposition of tag and measure gives immediate face-value context and speeds user understanding. Also as stated players may align a row of tags, one from each card, and measure aggregate value on an integrated measure. 
     FIG. 3B  shows decimal, fractional degree and percentage measures  68 ,  67 ,  69  and  66 . Instant readout avoids tedious or challenging conversion math; it speeds play and enhances fun and learning too. 
     FIG. 3C  shows composite curved measures  74  and  75 . These show progress towards, and required value to complete one whole unit in four different measurement systems. 
   Round-shaped Cards with Asymmetric-Shaped Card tags— FIG. 4   
     FIG. 4  shows asymmetric-shaped tags scaled using a 30.48 mm/1.2 in integrated measure 49 on a 76 mm/3 in diameter card. (Note: here tag scale equals radius of segment layout  22 , see  FIG. 1B ). These give good overlapped card tag and measure visibility for right-handed players working left to right. A y-axis mirror image, located top right of the card face, could service left-handed players working from right to left. 
     FIG. 4A  shows an expanded view of asymmetric tags  42 ,  45 ,  40  and  43  with integrated measure  49  on the number-5, -10, -3 and number-6 cards in the group. 
     FIGS. 4B , C &amp; D show the faces of sample number-3, -5 and -10 card faces respectively. 
   Tag Value Variants— FIG. 16   
   In a preferred embodiment each number card has an invariant face value and a multiple number of color- and size-coded tags. However, tag alignment and meld value measurement require just one tag per card. Furthermore invariant face values may be unnecessarily restrictive for some advanced applications. 
     FIG. 16  shows a number-6 card with just one almost-square tag  43  designating a nominal face value of ⅙, etc. Using tag  43  size as a base, other tag sizes are factored up or down as labeled to designate premium or discounted face values. Note: game designers may label factored tags or publish a coding scheme. 
   Factored tags may be suitable for time-based applications such as life or money games. Any factor may be applied depending upon the intended audience and application.  FIG. 16  shows four tags factored down in size to 0.50, 0.75, 0.80 and 0.90 of a regular tag  43 . These represent 50%, 25%, 20% and 10% discounted face values respectively. Similarly, five tags are factored up in size to 1.10, 1.20, 1.25, 1.50 and 2.00 of a regular tag  43  and represent 10%, 20%, 25%, 50% and 100% premium face values respectively. 
   Almost-square tags may be impractical for small segment sizes. So for small segments, preferably:
         standardize radial tag-height independent of face value;   scale tangential tag-width only;   fix the radius of segment layout  22   FIG. 1B  (e.g., equal to card radius less radial tag-height).       

     FIG. 16  shows 5-% and 1-% tags scaled in width only with the same radial tag-height as a regular tag  43 . 
   Multiplication and Division—Logarithmic Tags 
   Multiplication and division is possible by simple tag alignment also using tags and measures scaled on a logarithmic scale. Since log(x)+log(y)=log(xy), the sum of two logarithmic tags equals the log of their product. Similarly log(x)−log(y)=log(x/y) and the difference between the log of a dividend tag and the log of a divisor tag is the log of their quotient. 
   Preferably label logarithmic scales with linear values for direct use. Any convenient log base may be used including log 10 (x) or log e (x) for natural logarithms. Exponential tags and measures may be used also. 
     FIG. 18A  shows tags representing face values of 1 through 13 scaled using a logarithmic base  10 . It is evident from  FIG. 18B  that a #3 tag+a #4 tag=a #12 tag. So 3×4=12, 12/3=4 and 12/4=3. Similarly a #12 tag−a #6 tag=a #2 tag. So 12/6=2, 12/2=6, and 6×2=12. Also a #6 tag=a #2 tag+a #3 tag. 
     FIG. 18C  illustrates a 10×10 multiplication. The 76 mm/3.0 in measure displays linear (x) values. The scale is log 10 (x). Two log 10 (10) tags align end to end. The end of the 2 nd  tag indicates a product of 100 on the measure on the 1 st  card. Thus 10×10=100. Note each card has just one logarithmic tag on an otherwise regular number-10 card ( 1/10 th  face value). Actual game cards will likely differ. 
   Mixed Tags 
   Game developers may offer card sets with several tag types for advanced applications. For example code each edge of a rectangular card for a different purpose. Preferably provide unequivocal play instructions. 
   Card Back Variants— FIGS. 5 &amp; 6   
   There are many potential data sheets and card back variants depending upon game design. Card sets and backs may be customized. Backs may match the card deck or be a superset of several different decks in a collection. Segment data rows or ring sets may depict any winning combination of size- and color-coded segments. For example, two number-5 segments, a number-10, a number-3 and a number-6 segment. 
   Another option is to show segments from one radix in one half of a segment row or ring and use another radix in the second half. Or group radix-3 solutions, then radix-4, then radix-5. And so on. 
   Measure Variants 
   Rulers—Regular and Custom 
     FIGS. 3 &amp; 17  show regular metric and imperial rulers. Custom rulers may be calibrated using a linear or logarithmic tag scale to quantify face values in any prevalent measurement or number system. The custom rulers shown in  FIG. 17  use a 25.4 mm/ 1.0 in tag scale. 
   Hand-held Clip Measure— FIGS. 8A-D   
     FIGS. 8A  &amp; B show plan and sectional views of an alternative clip measure with four curved measures. The platform and clip interleave on three sides. Players may manipulate card backs through finger hole  71 . It may be made in plastic or die stamped in metal, for example. Some players may find the triangular-shaped measure in  FIGS. 8C  &amp; D easier to use. The shape is synergistic with round cards and pie-shaped segments. 
   Small Hand-held Clip Measure 
     FIG. 10  depicts a 3-D view of a credit-card sized clip measure for round-shaped cards. It may be made in one piece in clear polycarbonate, acrylic or similar material. It converts a tag row directly into a total value in a prevalent measurement system. Players may manipulate card backs through a finger hole  71  in platform  62 . A similar design not shown is suitable for rectangular cards. 
   Hinge  70  attaches clip  61  to body  60  above platform  62 . Clamp  64  and platform  62  grip cards inside slot  72  as hinge  70  closes. An integrated magnifying lens  65  above measure  68  improves legibility and speeds use. Preferably measure  68  is molded on the inside surface of clip  61  to reduce potential parallax errors. 
     FIG. 10A  shows a plan view of a small clip measure with curved measures  66 ,  67 ,  68  and  69 . Players may switch interchangeably between different measures at will. 
   Using a Small Clip Measure 
     FIGS. 10 &amp; 11A : using the same meld of cards shown in  FIG. 3 , proceed as follows:
         hold small clip measure in one hand between finger and thumb using side grips  63 ;   with the other hand manipulate and measure melds inside slot  72 ;   slide a number-5 card inside slot  72  onto platform  62  and under clamp  64 ;   align a tag  42  with measure  68  zero;   press on lens  65  to secure the number-5 card under clip  61  and measure  68 ;   release lens  65 , slide a number-10 card inside slot  72  on top of the first card;   align the outer curved edge of any tag  45  with the inner edge of selected tag  42  on the number-5 card;   press on lens  65  to secure both cards;   complete the meld in like manner with the number-5, -3 and number-6 cards, see  FIG. 11A .       
     FIG. 11B  shows a close-up of the contiguous row of five card tags  42 ,  45 ,  42 ,  40  and  43 . 
     FIG. 11C  is a close-up plan view of the measure and tags showing a group value of 100%, 1/1, 1.0 and 360°. 
   Developers may select clip and font sizes suited to hand size, dexterity and eyesight of their target audience. Note that although children generally have smaller hands and better eyesight than adults do, young children may not have developed the required motor skills to handle very small clip measures. 
   Measure with Card Slot—Slot Measures 
   A measure with card slot or slot measure encloses a whole card group. It is intended for tabletop use but may be hand-held. Slot measures offer little privacy in a competitive game situation. An appropriately sized slot may accommodate any card shape and meld size. Tags may be scaled in either or both dimensions. The slot may be a hole or an indent. Slots may be calibrated inside and outside to show meld value. 
   Slot Measure for Round Cards—Tags Scaled in Radial Height,  FIG. 13A   
   A slot measure comprises a solid body  60  containing a calibrated card slot  72 . It may be made from any material customarily used for rulers and drafting tools. Slot  72  is an appropriately sized indented slot or hole with semi-circular or card-shaped ends. Slot  72  sides are parallel. It may be calibrated for right-handed or left-handed, or right and left-handed, players. It may be single or double-sided. 
   Preferably slot  72  width just exceeds one card diameter. This allows for easy card placement and removal and reduces card wear and tear. Body  60  thickness and slot  72  depth depend upon card thickness and capacity. The minimum slot depth should hold a stack of six to eight cards, for example. Preferably an indented slot measure has a finger hole  71  to facilitate card manipulation and removal. 
   To accommodate a meld of one whole unit, the minimum slot length should just exceed one card diameter plus one tag scale. It is not essential to calibrate a minimum-length slot. Any group of cards that total one whole unit (whose tags total one tag scale) plus one extra card would fill the slot exactly. Preferably the slot is calibrated with several prevalent measurement systems however to promote learning. 
   The measure in  FIG. 13A  is intended for right-handed players working from left to right. It has decimal measure  68  in the upper left and fraction measure  67  in the lower left corner. Two measures are inverted. Players may rotate the measure 180° to use percentage measure  66  or degree measure  69 . Two internal measures help to reduce potential parallax errors due to peripheral measures. Composite curved measures  74  and  75  show group value progress towards and needed to complete one whole unit. 
   Players may orient the slot measure at any angle such as at 3, 6, 9 and 12 o&#39;clock. Left-handed players may find it easier to work from the right. The other side of the measure (not shown) has upright measures  68  and  67  on the right and inverted measures  66  and  69  on the left. 
   Using a Slot Measure 
     FIG. 13B  shows using a slot measure with the same meld of five cards as used in  FIG. 3 :
         lay the slot measure horizontally on a tabletop or suitable flat surface;   select measure  66 - 69  and rotate body  60  accordingly;   feed successive cards into slot  72  with one hand;   align the outer edge of each successive tag with the inner edge of its predecessor;   use other hand to hold body  60  and to secure the row of aligned cards and tags;   read total meld value on selected measure;   remove cards from slot measure by turning it over and emptying it, or push the cards out of the slot from the other side using finger hole  71 .       
   Opaque cards obscure measures  74  or  75  and potentially render them useless. 
   Proceed as follows:
         select either measure  74  or  75  but work from the opposite end of the slot;   align cards and tags inside the slot to form a contiguous row of aligned tags as above;   take any extra card and align its leading edge with the end of the row of aligned tags;   use the trailing edge of the extra card to read meld value directly on selected measure.
 
Inclined Slot Measure—Tags Scaled in Width,  FIG. 14A 
       

   Inclined slot measures are intended for use with tags scaled in width only (e.g., for small segments). An inclined slot automatically offsets adjacent cards in two directions. The offset exposes successive tags in the alignment and measurement process. Tag-width sets the offset between adjacent round cards in a group of aligned cards. 
   Preferably developers should design tags, slot inclination and measure placement together. 
   Body  60  contains an inclined slot  73  calibrated with measure  68 . Any other measure may be used. Slot  73  may be calibrated top and bottom. The slot measure may be calibrated on both sides. 
   Using an Inclined Slot Measure— FIG. 14B   
     FIG. 14B  shows an inclined slot measure with the same meld of five cards used in  FIG. 3 . Place cards inside the slot in sequence from left; offset and align selected tags  42 ,  45 ,  42 ,  40  and  43  side by side at 12 o&#39;clock. 
   Template Measure 
   A low cost template measure may be inscribed on paper or suitable transparent material. Players may meld cards on top of a template measure or overlay a transparent template measure over a pre-aligned meld. 
     FIG. 12A  shows a round card-sized transparent template measure with four double measures  76 ,  77 ,  78  and  79 . Place it over pre-aligned melds to read meld values, see  FIG. 12B . Each double measure shows group face value achieved and needed to complete one whole unit. 
     FIG. 15  shows a template measure with a notional slot calibrated with measures  66 ,  67   68  and  69  outside a notional slot, and with composite curved measures  74  and  75  inside. 
   Alternative Graphic Representations 
   Pie charts, bar charts, histograms, frequency diagrams, lines, dots, images, icons or any suitable graphical representations may be used to depict rank and size relationships. Bar charts are synergistic with rectangular-shaped cards and pie charts with round-shaped cards. 
   Playing Piece Variants 
   Some applications may require more durable playing pieces than regular playing cards. Plastic card stock or tiles may be preferred for child use. 
   Virtual Playing Pieces 
   A virtual deck of playing card images may be electronically stored in any computing device, cell phone or online system for individual or group play. 
   Conclusions, Ramifications and Scope 
   These specially marked playing cards offer many ways for players of any age, math proficiency and cultural background to acquire an understanding of fractions decimals percentages and degrees, and their interrelationships, and to determine a group value just through play. 
   While my above description contains much specificity, this should not be construed as a limitation on the scope of the invention, but rather as an exemplification of some preferred embodiments thereof. 
   Although round-shaped or rectangular-shaped cards are preferred any card shape may be used. 
   A preferred embodiment uses seven interchangeable indicia types and color-coding to denote face value. Developers may use any number of indicia types, and any number of indicia, to denote face value. Color-coding is optional. Also shape-coding may be desirable for certain applications, and so on. 
   Further, the indicia need not be limited to visual components only. Auditory, olfactory, tactile and even taste indicia may be utilized for specific applications, especially for special needs applications. 
   The indicia may be expanded to use other symbol sets, for example Greek, Cyrillic, Katakana and Kanji. 
   The indicia may be expanded to use words and language and not just symbols. For example, the fraction ¼ may be written as the word quarter in English, un quart in French, un quarto in Spanish, and so on. 
   The game system is not limited to single portions. Multiple portions and mixed numbers may be supported. 
   This arithmetic measurement system is not limited to numbers. fractions, decimals, percentages, and degrees. Rather it has general applicability for all measurement or number systems and conversions. It may be adapted to any measurement system including but not limited to weights and mass, distance and length (land and nautical), capacity and volume, area, speed, time, temperature, pressure, energy and work, power, torque, circular measure, computer storage, fuel consumption, currency conversion and any other unit conversion. 
   Global markets may require international measurement system conversions. This system may be adapted to U.S. gallons, quarts, pints, cups and ounces. It may be used with international liquid and dry measures of volume including metric, U.S., British Imperial, Japanese, Thai, Old Russian and cooking units. 
   As noted this game system is not limited to addition and subtraction but rather it has general math applicability including multiplication, division, logarithms and exponentiation. 
   Playing pieces may be electronically tagged or coded. An electronic measuring device could sense, identify and count electronic tags and so replace mechanical measures. This would greatly simplify card alignment, counting and meld value assessment especially for physically challenged or handicapped players. 
   Also microelectronic processor and storage components may be embedded in the playing pieces similar to the so-called smart card. They may be programmed to store, manipulate process and retrieve machine-readable data. This would greatly expand the range and complexity of potential applications. 
   Accordingly, the scope of the invention should be determined not by the embodiment(s) illustrated, but by the appended claims and their legal equivalents.