Abstract:
The devices for teaching elementary fractional concepts each include a board having a raised periphery and central depression representing unity. Each embodiment includes a series of fractional pieces of different sizes for removable installation therein, with the number of pieces of each size having a total span equal to the span of the central depression. Thus, each piece represents the denominator of a fractional portion of the central recess. The pieces may be stacked in several layers, thus demonstrating the equivalency of different sizes of fractional pieces subtending the same area of the recess. The devices may be formed in rectangular or circular configurations, and may represent the basic abstract concept of fractions, monetary values in different systems, time and divisions thereof, etc. Separate boards may be provided for each embodiment, or inserts may be provided for placement within recesses of correspondingly configured boards.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
       [0001]    This application claims the benefit of U.S. Provisional Patent Application Ser. No. 60/844,091, filed Sep. 13, 2006. 
     
     BACKGROUND OF THE INVENTION 
       [0002]    1. Field of the Invention 
         [0003]    The present invention relates generally to educational tools and devices used for teaching various concepts and skills. More specifically, the present invention relates to devices for teaching elementary fractional concepts encompassing boards or trays, each board or tray having a central recess therein. A series of fractional pieces having sizes proportional to the fractional values represented may be installed in and removed from the central recess, and different sizes of pieces representing different fractional denominators may be placed in a series of layers within the recess. 
         [0004]    2. Description of the Related Art 
         [0005]    The teaching of most mathematical concepts, including the concept of fractions, is traditionally accomplished by means of blackboard and chalk, pencil and paper, and/or textbooks and workbooks. None of these means provide a solid, three-dimensional, hands-on experience for the student. The result is that many, if not most, students have at least some degree of difficulty in grasping the subject, particularly younger children first learning about fractions. Somewhat the same points apply to teaching younger children about basic money and coin values, in that the basic unit of currency is subdivided into smaller units (e.g., dollars and cents) for just about every currency in the world. At least when learning about the monetary system, the student may actually handle the units (coins, etc.) while gaining some familiarity with their relative values. However, most monetary systems use coins that are often not proportional in size and weight to the values represented. This results in further difficulties for the student learning about basic monetary values, whether in one&#39;s native currency or when learning about a foreign currency for a trip abroad. 
         [0006]    As a result, a few devices and systems for teaching fractions and monetary values have been developed in the past. Simulated currency (“play money”) has often been used in the past for teaching the value of currency and more particularly paper or folding money, with some development of simulated coins being done in the past as well. A major problem with simulated coins is that they represent the actual sizes of the coins represented, rather than providing any components having proportional sizes to the values of the coins represented. This can make it difficult for the student to grasp the relative values of the coins represented. Others have developed boards with various removable overlays representing different monetary or fractional values. The plethora of such devices attests to the continuing need for tools for teaching fractional concepts. 
         [0007]    Thus, devices for teaching elementary fractional concepts solving the aforementioned problems are desired. 
       SUMMARY OF THE INVENTION 
       [0008]    The devices for teaching elementary fractional concepts comprise a series of embodiments, each embodiment including a board having a relatively large central recess defined by a raised periphery. The central recess represents unity, or a value of one unit. Each embodiment further includes a series of fractional pieces of different sizes proportional to their fractional values and representing the denominator of a fraction. For example, each piece of a series of four fractional pieces subtends one fourth of the area of the central recess, with each of a series of ten pieces subtending one tenth of the area, etc. Multiple layers of the fractional pieces may be installed atop one another in order to clearly show the equivalent values of different fractions, e.g., five 1/10 pieces may be placed atop two ¼ pieces, subtending exactly the same half board area for both and clearly showing their equivalence. Thus, the student can readily see the sizes of the pieces relative to one another and to the size of the central recess of the board to grasp the concept of the fractional quantity represented. 
         [0009]    The devices for teaching elementary fractional concepts may be applied to abstract fractional numerical values, and may also be applied to a special case for teaching basic monetary values as well. In the case of the monetary teaching device, the central depression may include a representation of a basic unit of currency, e.g., one dollar, with fractional pieces representing various coin values and subtending areas of the central depression proportional to their values. The same principle may be applied to various currencies of different nations as well. The currency teaching device may be constructed as a stand-alone device, or may be formed as an insert for removable installation within the central depression of the basic fractional teaching device. 
         [0010]    The devices may be formed in rectangular configurations with the fractional pieces extending from one edge to the other and having a lateral span or width proportional to their fractional values, or may be formed in circular configurations with each of the fractional pieces subtending a circular sector. In the case of the circular configurations, one embodiment may include a set of selectively positionable clock hands, with the periphery of the board marked in hours and minutes. The fractional pieces may be marked to represent the minutes corresponding to the fractional value of the piece, e.g., a piece covering ⅙ of the circular area could be marked to indicate ten minutes, etc. In any event, all embodiments utilize the common theme of the central depression and multiple values of fractional pieces removably installable therein in multiple layers to show the equivalence of various fractional values. 
         [0011]    These and other features of the present invention will become readily apparent upon further review of the following specification and drawings. 
     
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0012]      FIG. 1  is an exploded perspective view of a first embodiment comprising a rectangular board and fractional pieces for teaching fractions according to the present invention. 
           [0013]      FIG. 2  is an exploded perspective view of a removably installable insert for the board of  FIG. 1  and fractional pieces for use therewith, with the insert and pieces representing various U.S. monetary values. 
           [0014]      FIG. 3  is a perspective view of a group of fractional pieces representing different values of smaller denomination coins for use with the monetary teaching insert of  FIG. 2 . 
           [0015]      FIG. 4  is an exploded perspective view of a removably installable insert for the board of  FIG. 1 , with the insert and fractional pieces representing various Euro coin monetary values. 
           [0016]      FIG. 5  is an exploded perspective view of another alternative embodiment, comprising a circular board and recess with fractional pieces in the form of arcuate circular segments. 
           [0017]      FIG. 6  is an exploded perspective view of a removably installable insert for the board of  FIG. 5  and fractional pieces for use with the insert, with the insert and fractional pieces representing various U.S. coin values. 
           [0018]      FIG. 7  is an exploded perspective view of another alternative embodiment, comprising a circular board and recess with selectively positionable clock hands therein and a series of fractional pieces representing periods of time. 
           [0019]    Similar reference characters denote corresponding features consistently throughout the attached drawings. 
       
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       [0020]    The present invention comprises a series of embodiments of a device for teaching elementary fractional concepts, e.g., abstract arithmetical fractions, money and small change, time and fractions of an hour, etc. Each of the embodiments includes a board, or insert for a basic board, and a series of groups of fractional pieces, with each group having pieces of like size representing fractional pieces having the same denominator. The various pieces of each group are used to form various strata within the central recess of the board, with the student readily able to see the correspondence in value between different pieces when so overlaid within the board recess. 
         [0021]      FIG. 1  provides an exploded perspective view of a first embodiment  100 , comprising a rectangular board  102  having raised opposite first and second or major peripheral edges  1   04  and  106 , and raised opposite third and fourth, minor or end peripheral edges  108  and  110 . The raised peripheral edge components  104  through  110  form a continuous periphery defining a central recess  112  therein, with the recess  112  subtending a lateral span or major width  114  representing unity and extending between the inner edges of the two end peripheral edges  108  and  110 . The first and second major peripheral edges  104  and  106  preferably include a series of numerical designators  116  thereon, representing arithmetical, monetary, or other values or quantities, as desired. 
         [0022]    In the example of  FIG. 1 , the numerical designators  116  comprise a series of evenly spaced numbers representing fractional divisions of the major width or span  114  of the central recess  112 , with other numbers indicating the factors of non-prime numbers as applicable, e.g., the fractional division  12  has its factors 2, 3, 4, and 6 in adjacent brackets. The numbers along the first or upper periphery  104  are used to represent the numerator of any fractional quantity being represented, and the numbers along the second or lower periphery  106  are used to represent the denominator. While a total of twenty divisions are shown along each of the peripheries  104  and  106 , with each division being equal to 1/20 of the total span  114  of the central recess  112 , it will be understood that any practicable number of divisions may be provided as desired. 
         [0023]    A series of relatively thin, flat, planar fractional pieces is provided for removable placement within the central recess  112  of the board  102 . Each series comprises a group of identically sized pieces each having a width less than the span  114  of the board recess  112 , with all of the pieces in any one group having a total width equal to the board recess span  114 . 
         [0024]    In the example of  FIG. 1 , three different series of fractional pieces are shown. The first set or group comprising the largest pieces, e.g., pieces  118   a,    118   b,  and  118   c,  comprises a total of four pieces (three of which are shown at least partially in  FIG. 1 ), with each piece having a width  120  subtending one quarter of the span  114  of the central recess  112 . The next group or set of pieces comprises a series of ten intermediate pieces, e.g.,  122   a,    122   b,    122   c,  etc., each having a width  124  subtending one tenth of the central recess span  114 . A third set of pieces comprises a series of twenty smaller pieces of which five are shown in  FIG. 1 , i.e.,  126   a  through  126   e.  Each of these smaller pieces has a width  128  subtending one twentieth of the span  114  of the central recess  112 . 
         [0025]    Thus, each piece represents the denominator of a fraction, with the denominator being equal to the number of pieces within each group or set. One or more of the smaller pieces  126   a,  etc. could be subdivided to form even smaller pieces, if so desired, but this may not be practicable. Alternatively, one or more of the smaller fractional pieces  126   a,  etc. could be marked with lines indicating the division of the pieces into still smaller components, while still maintaining a practicable size for handling the pieces. It will be seen that the groups of pieces representing quarter, tenth, and twentieth fractions of the unit represented by the length of the board recess  112  may be adjusted as desired, with the number of pieces and corresponding fractional denominator for each group or set being any practicable integer as desired. 
         [0026]    A single group of identical fractional pieces may be used to form a single complete stratum or layer of pieces within the central recess  112 , completely filling the central recess, if so desired, with other groups of identical pieces being placed thereon to form multiple layers or strata of fractional pieces. Alternatively, fractional pieces of different groups may be placed within the same stratum to indicate the equivalency or relative value of a series of smaller pieces compared to a single one or smaller number of larger pieces. In the example of  FIG. 1 , it will be seen that the width  120  of a single larger fractional piece  118   c  subtends ¼, or 25%, of the total span  114  of the board recess  112 , thus requiring a series of four such pieces to fill the recess  112  completely. In turn, two intermediate width pieces  122   e  and  122   f  are shown disposed above the larger fractional piece  118   c.  As each of the pieces  122   e  and  122   f  has a width  124  equal to one tenth of the total span  114  of the board recess  112 , two such pieces  122   e  and  122   f  subtend only ⅘ of the width  120  of the larger ¼ piece  118   c.  The series of five 1/20 width pieces  126   a  through  126   e  will be seen to have a total width equal to 5/20, or ¼, of the total span  114  of the board recess, or in other words equal to the span or width  120  of a single ¼ piece. The various fractional pieces may be mixed and matched in various layers or strata to show the equivalencies of various fractions, generally as described above. 
         [0027]    It will be noted that the various fractional pieces  118   a  through  126   e,  etc., are provided in two different colors or shades. For example, a first surface  130  of each of the pieces could be shaded in red, with the opposite second surface  132  colored black. The use of red and black are particularly suitable for embodiments of the present invention representing monetary fractions, i.e., subtraction and addition of monetary values or profit and loss. However, other colors may be used as desired. 
         [0028]      FIG. 2  of the drawings illustrates a second embodiment  200  of the present fractional teaching devices, comprising an insert  202  which may be used with the board  102  of  FIG. 1  and a series of fractional pieces for teaching monetary values. The insert  202  essentially serves the same function as the board  102 , i.e., the insert includes a pair of raised, opposed major peripheral edges  204  and  206  that define a central recess  212  therebetween. The recess  212  has a major width or span  214  equal to the width or span  114  of the board  102 , with the recess  212  of the insert  202  fitting into the recess  112  of the board  102  and the two opposed major peripheries  204  and  206  being positioned over the respective peripheries  104 ,  106  of the board  102  when the insert  202  is placed thereover. 
         [0029]    The purpose of the insert  202  is to provide different numerical designators along the two upper edges or peripheries  204 ,  206  of the device, which may be used to indicate different fractions or functions for the use of the device. In the example of the insert  202  of  FIG. 2 , the numerical designators  216   a  and  216   b  are graduated in hundredths, to represent percentile values or increments. This permits the insert  202  to be used for the teaching of relative monetary values found in coin or other divisions of a basic monetary unit, e.g., nickels, dimes, quarters, etc., with a dollar being the basic monetary unit. Accordingly, the recess  212  of the insert  202  may include a representation  213  of the basic monetary unit, e.g., a dollar bill, thereon. Two different numerical designators  116   a  and  116   b  are shown in  FIG. 2 , with their values increasing in opposite directions. This facilitates the concept of making change, i.e., subtracting the remainder of the change from a dollar from the amount due in a transaction. The inventive concept of  FIG. 2  may be expanded to cover other than U.S. monetary systems as well, as exemplified in  FIG. 4  and discussed further below. It will also be noted that the information provided on the insert  202  could be applied to a board similar to the board  102  of  FIGS. 1 and 2 , if so desired. 
         [0030]    A series of relatively thin, flat, planar fractional monetary pieces is provided for removable placement within the central recess  212  of the insert  202 . These monetary pieces are quite similar to the pieces  118   a  through  126   e  shown in  FIG. 1  of the drawings and discussed further above, differing only in that each of the fractional monetary pieces of  FIG. 2  includes a representation of a coin thereon corresponding to the fractional value of the piece. In the example of  FIG. 2 , three different series of fractional pieces are shown. The first set or group comprising the largest pieces, e.g., pieces  218   a  through  218   d,  comprises a total of four pieces, with each piece having a width  220  subtending one quarter of the span  214  of the central recess  212  of the insert  202  and representing the value of a quarter coin, or 25 cents. A 25-cent coin symbol or representation  221  may be provided upon each of the ¼ width monetary fractional pieces. 
         [0031]    The next group or set of pieces comprises a series of ten intermediate pieces, e.g.,  222   a,    222   b,    222   c,  etc., each having a width  224  subtending one tenth of the central recess span  214  of the insert  202 . As there are ten such pieces, a dime representation  225  or coin having a value of one tenth of the monetary unit represented by the span  214  of the insert  202  may be provided on each piece. A third set of pieces comprises a series of twenty smaller pieces of which five are shown in  FIG. 2 , i.e.,  226   a  through  226   e.  Each of these smaller pieces has a width  228  subtending one twentieth of the span  214  of the central recess  212  of the insert  202 . A nickel representation  229 , or other coin having a value of one twentieth of the base currency represented, may be provided on the surface of each of the smaller monetary fractional pieces  226   a,  etc. 
         [0032]    One or more of the smaller pieces  226   a,  etc. could be subdivided to form even smaller penny or one cent pieces, if so desired, but this may not be practicable. Alternatively, one or more of the smaller fractional pieces could be marked with a plurality of smaller coins having a total value equivalent to that represented by the width of the fractional monetary piece. 
         [0033]    An example of the above is illustrated in  FIG. 3  of the drawings, with a series of three fractional monetary pieces  326   a,    326   b,  and  326   c  being shown. Each of the pieces  326   a  through  326   c  has a width  328  equal to the widths  228  of the pieces  226   a  through  226   e  of  FIG. 2 , i.e., 1/20, or 5/100, of the total span or width  214  of the insert recess  212 . However, rather than displaying a single coin thereon, the pieces  326   a  through  326   c  of  FIG. 3  have representations of a series of smaller coins having a total value equal to that of one of the equivalent pieces  226  of  FIG. 2 . The coin representations  329  on the pieces  326   a  through  326   c  are grouped to indicate different values and remainders, with the first piece  326   a  having all five of the one-cent coin representations  329  grouped together. 
         [0034]    The next piece  326   b  shows only four such coins grouped together, with the fifth coin separated. This piece  326   b  also has one fifth of its width shown in a different color, e.g., black, to indicate the relative lessening of value of the piece. Finally, the piece  326   c  has two coins grouped separated, with a black pattern subtending two fifths of its surface. In this manner, the values of the smallest coins may be represented, without the need to subdivide the 1/20 width pieces into narrower pieces that are impracticable to handle. The relative percentages of the areas covered by each color, and the groupings of coins, may be reversed on the opposite sides of the pieces  326   a,    326   b,  etc., in order to obviate need for a duplicate set of such multiple coin fractional monetary pieces. 
         [0035]      FIG. 4  of the drawings illustrates an alternative insert and fractional piece configuration  400  for teaching the relative coin values of a monetary system other than the U.S. monetary system. The Euro monetary system is shown in the example of  FIG. 4 , but it should be noted that the monetary teaching embodiments of the present invention may be applied to virtually any monetary system using a base unit of currency that is divided into smaller fractional units. The insert  402  of  FIG. 4  is quite similar to the insert  202  of  FIG. 2 , having opposite first and second major peripheries  404  and  406 , each of which is marked with appropriate fractional divisions  416   a  and  416   b  for the monetary units used. The span or major width  414  of the insert recess  412  is essentially equal to the major span or width of the underlying board  102  into which the insert  402  is placed for use. However, rather than placing a representation of a U.S. monetary unit on the recess  412 , a representation  413  of a one euro coin is shown. 
         [0036]    Accordingly, the various monetary fractional pieces provided with the insert  402  are sized and marked to correspond with various euro coins that are fractions of the basic euro denomination, which is the one-euro coin. In this example, five larger fractional monetary pieces  418   a  through  418   e  are shown, with each having a width  420  subtending ⅕ of the span or width  414  of the insert  402 . The pieces  418   a  through  418   e  are each marked with a twenty-cent euro coin representation  421 , with each such coin having a value of ⅕ of the one-euro coin shown in the representation  413  in the insert recess  412 . A series of ten intermediate size fractional monetary pieces is also provided, with three such pieces  422   a  through  422   c  being shown in  FIG. 4 . Each intermediate piece has a width or span  424  equal to one half of the width  420  of one of the larger ⅕ euro pieces  418   a  through  418   e,  with two such intermediate pieces, e.g., pieces  422   b  and  422   c,  subtending a total width equal to that of one of the larger ⅕ euro pieces. Accordingly, each of the 1/10 pieces is marked with a ten-cent euro representation  425 . 
         [0037]    In addition to the above pieces, a series of twenty smaller pieces is provided, with each of the smaller pieces representing 1/20 of a euro. Four such pieces  426   a  through  426   d  are illustrated in  FIG. 4 , with each having a width or span  428  equal to one fourth of the width or span  420  of one of the larger pieces  218   a  through  218   e.  Accordingly, each of the smaller pieces, e.g., pieces  426   a  through  426   d,  is marked with a five-cent euro representation  429 . It will be seen that other groups of pieces having sizes corresponding to other euro coin denominations may be provided as desired. Smallest denominations may be represented in the manner shown for the pieces  326   a  through  326   c  of  FIG. 3 . Also, while color is not indicated for the euro fractional monetary embodiment  400  illustrated in  FIG. 4 , it will be recognized that opposite surfaces of the various fractional pieces may be colored, shaded, marked, or designated differently from one another if so desired, similarly to the manner shown in  FIGS. 1 and 2  and discussed further above. 
         [0038]      FIG. 5  illustrates a fourth embodiment  500  of the present device for teaching fractional mathematical concepts, wherein the board  502  has a circular configuration and the fractional pieces are each in the form of a circular sector subtending an arc equal to a fraction of the circumferential span of the circular board. The board  502  has much the same structure as the board  102  of  FIGS. 1 ,  3 , and  4 , in that the board  502  includes a raised peripheral edge  504  forming a continuous periphery defining a central recess  512  therein. The recess  512  subtends a circumferential span  514  representing unity, and extends around the central recess  512  and concentric to the circumferential periphery  504 . The circumferential peripheral edge  504  may include a series of numerical designators thereon representing arithmetical, monetary, or other values or quantities as desired, generally as shown on the opposed peripheral edges  104  and  106  of  FIG. 1 . Examples of such are illustrated in the embodiments of  FIGS. 6 and 7 , and discussed further below. 
         [0039]    A series of relatively thin, flat, planar fractional pieces is provided for removable placement within the central recess  512  of the board  502 , with each of the pieces being in the form of a circular sector. Each series of pieces comprises a group of identically sized pieces each having an arcuate width or span less than the circumferential span  514  of the board recess  512 , with all of the pieces in any one group having a total width equal to the board recess span  514 . 
         [0040]    In the example of  FIG. 5 , three different series of fractional pieces are shown. The first set or group comprising the largest pieces, e.g., pieces  518   a  and  518   b,  comprises two pieces, with each piece having an arcuate span or width  520  subtending one half of the circumferential span  514  of the central recess  512 . The next group or set of pieces comprises a series of four intermediate pieces, with two such pieces  522   a  and  522   b  shown in  FIG. 5 . Each of the intermediate pieces has a width  524  subtending one quarter of the central recess span  514 . 
         [0041]    A third set of pieces comprises a series of eight smaller pieces of which four are shown in  FIG. 5 , i.e.,  526   a  through  526   d.  Each of these smaller pieces has a width  528  subtending one eighth of the span  514  of the central recess  512 . Thus, each piece represents the denominator of a fraction, with the denominator being equal to the number of pieces within each group or set. One or more of the smaller pieces  526   a,  etc. could be subdivided to form even smaller pieces, if so desired, but this may not be practicable. 
         [0042]    Alternatively, one or more of the smaller fractional pieces  526   a,  etc. could be marked with radial lines indicating the division of the pieces into still smaller components, while still maintaining a practicable size for handling the pieces. It will be seen that the groups of pieces representing half, quarter, and eighth fractions of the unit represented by the circumferential span  514  of the board recess  512  may be adjusted as desired, with the number of pieces and corresponding fractional denominator for each group or set being any practicable integer as desired. 
         [0043]    A single group of identical fractional pieces may be used to form a single complete stratum or layer of pieces within the central recess  512 , completely filling the central recess, if so desired, with other groups of identical pieces being placed thereon to form multiple layers or strata of fractional pieces. Alternatively, fractional pieces of different groups may be placed within the same stratum to indicate the equivalency or relative value of a series of smaller pieces compared to a single one or smaller number of larger pieces. In the example of  FIG. 5 , it will be seen that the width  520  of a single larger fractional piece  518   a  subtends ½, or 50%, of the total circumferential span  514  of the board recess  512 , thus requiring two such pieces to fill the recess  512  completely. In turn, two intermediate width pieces  522   a  and  522   b  are shown disposed above the larger fractional piece  518   a.  As each of the pieces  522   a  and  522   b  has an arcuate width  524  equal to one fourth of the total arcuate span  514  of the circular board recess  512 , two such pieces  522   a  and  522   b  are equal in arcuate width to one of the larger ½ pieces  518   a.  The series of four ⅛ width pieces  526   a  through  526   d  will be seen to have a total width equal to 4/8, or ½, of the arcuate span  520  of one of the half width pieces  518   a  or  518   b,  with eight such ⅛ pieces being used to cover the entire area of the circular recess  512  of the board  502 . The various fractional pieces may be mixed and matched in various layers or strata to show the equivalencies of various fractions, generally as described above. 
         [0044]    It will be noted that the various fractional pieces  518   a  through  526   d,  etc., are provided in two different colors or shades. For example, a first surface  530  of each of the pieces could be shaded in red, with the opposite second surface  532  colored black. The use of red and black are particularly suitable for embodiments of the present invention representing monetary fractions, i.e., subtraction and addition of monetary values or profit and loss. However, other colors may be used as desired. 
         [0045]      FIG. 6  of the drawings illustrates a fifth embodiment  600  of the present fractional teaching devices, comprising a circular insert  602  which may be used with the circular board  502  of  FIG. 5  and a series of fractional pieces for teaching monetary values. The insert  602  essentially serves the same function as the circular board  502 , i.e., the insert includes a raised peripheral edge  604  forming a continuous periphery defining a central recess  612  therein. The recess  612  subtends a circumferential span  614  representing unity, and extends around the central recess  612  and concentric to the circumferential periphery  604 . The circumferential peripheral edge  604  may include a series of numerical designators  616  thereon representing arithmetical, monetary, or other values or quantities as desired, generally as shown on the opposed peripheral edges  104  and  106  of  FIG. 1 . 
         [0046]    The purpose of the insert  602  is to provide different numerical designators along the circumferential upper edge or periphery  604  of the device, which may be used to indicate different fractions or functions for the use of the device. In the example of the insert  602  of  FIG. 6 , the numerical designators  616  are graduated in tenths. This permits the insert  602  to be used for the teaching of relative monetary values found in coin or other divisions of a basic monetary unit, e.g., nickels, pennies, etc., with a dime being the basic monetary unit. Accordingly, the recess  612  of the insert  602  may include a representation  613  of the basic monetary unit thereon. The inventive concept of  FIG. 6  may be expanded to cover other coins or other than U.S. monetary systems as well, as exemplified in  FIG. 4  and discussed further above. It will also be noted that the information provided on the insert  602  could be applied to a board similar to the board  502  of  FIG. 5 , if so desired. 
         [0047]    A series of relatively thin, flat, planar fractional monetary pieces is provided for removable placement within the central recess  612  of the insert  602 , with each of the pieces being in the form of a circular sector. These monetary pieces are quite similar to the pieces  518   a  through  526   d  shown in  FIG. 5  of the drawings and discussed further above, differing only in that each of the fractional monetary pieces of  FIG. 6  includes a representation of a coin thereon corresponding to the fractional value of the piece. In the example of  FIG. 6 , two different series of fractional pieces are shown. The first set or group comprising the largest pieces, e.g., pieces  618   a  and  618   b,  comprises two pieces, with each piece having a width  620  subtending one half of the circumferential span  614  of the central recess  612  of the insert  602  and representing the value of a nickel, or 5 cents. A 5-cent coin symbol or representation  621  may be provided upon each of the ½ width monetary fractional pieces. 
         [0048]    The next group or set of pieces comprises a series of ten intermediate pieces  622   a  through  622   j,  each having an arcuate width  624  subtending one tenth of the central recess circumferential span  614  of the insert  602 . As there are ten such pieces, a penny representation  625  or coin having a value of one tenth of the monetary unit represented by the circumferential span  614  of the circular insert  602  may be provided on each piece. It will be seen that if a larger value coin is represented in the recess  612  of the insert  602 , then more groups of fractional monetary pieces having different arcuate widths or spans and corresponding values from one another may be provided. 
         [0049]      FIG. 7  of the drawings illustrates a sixth embodiment  700  of the present invention, in which a circular board  702  (or alternatively, insert for a circular board) is used to teach time increments. The circular board  702  has a general configuration much like the circular board  502  of  FIG. 5 , i.e., having a continuous circular raised periphery  704  defining a circular central recess  712  therein, with the central recess subtending a circumferential span  714 . However, the board  702  of  FIG. 7  further includes an hour hand  706  and minute hand  708 , extending from a central pivot point  710  and independently positionable thereon. A conventional clock motor drive (not shown) could be used to drive the two hands  706  and  708 , if so desired, but the intent of the time teaching embodiment  700  of  FIG. 7  is to allow the user or teacher to position the two hands as desired, to demonstrate various time increments and their indication by means of the hands. The raised circumferential periphery  704  preferably includes a series of numerical designators  716  in accordance with conventional clock or timepiece markings. Alternatively, a circular insert similar to the insert  602  of  FIG. 6  (but including clearance for the clock hands  706  and  708 ) could be provided with some alternative numerical designators, e.g., twenty-four hour clock markings, etc. 
         [0050]    A series of relatively thin, flat, planar fractional pieces is provided for removable placement within the central recess  712  of the board  702 , with each of the pieces being in the form of a circular sector. Each series of pieces comprises a group of identically sized pieces each having an arcuate width or span less than the circumferential span  714  of the board recess  712 , with all of the pieces in any one group having a total width equal to the board recess span  714 . Each piece further includes a truncated relief at the apex thereof, for clearance for the central clock hand pivot  710 . 
         [0051]    In the example of  FIG. 7 , three different series of fractional pieces are shown. The first set or group comprising the largest pieces, i.e., pieces  718   a  through  718   d,  comprises four pieces, with each piece having an arcuate span or width  720  subtending one quarter of the circumferential span  714  of the central recess  712 . The four quarter-hour pieces  718   a  through  718   d  may be marked with appropriate indicators  721  to indicate their value or represented time span, e.g., 15, to indicate the 15-minute time increment represented by each quarter hour piece. The next group or set of pieces comprises a series of five intermediate pieces  722   a  through  722   e.  Each of the intermediate pieces has an arcuate width or span  724  subtending one tenth of the central recess span  714 , or a time span of one tenth of an hour or six minutes. 
         [0052]    Accordingly, each of these pieces  722   a  through  722   e  may be marked to show the time increment  725  represented, i.e., six minutes. It will be seen that a total of ten such pieces may be provided to cover the entire central recess  712  of the clock board  702 , if so desired. A third set of pieces comprises a series of six smaller pieces  726   a  through  726   f.  Each of these smaller pieces has a width  728  subtending one sixth of the span  714  of the central recess  712 . Thus, each piece represents a time span of five minutes, with six such pieces equaling thirty minutes or half an hour, and twelve such pieces (if provided) equaling one hour and completely covering the clock board recess  712  when placed thereon. The corresponding time period may be indicated by an appropriate numerical designator  729  placed upon each of the pieces  726   a  through  726   f.    
         [0053]    As in the other embodiments of the present invention discussed further above, the various fractional time pieces could be further subdivided into smaller increments, or alternatively marked with one or more radial lines to indicate smaller time increments. Other fractional time increments may be provided as desired, e.g., one third hour pieces representing twenty minutes, one twentieth hour pieces representing three minutes each, etc. as desired. As in the case of the other embodiments of the present invention, a single group of identical fractional pieces may be used to form a single complete stratum or layer of pieces within the central recess  712 , completely filling the central recess, if so desired, with other groups of identical pieces being placed thereon to form multiple layers or strata of fractional pieces. Alternatively, fractional pieces of different groups may be placed within the same stratum to indicate the equivalency or relative value of a series of smaller pieces compared to a single one or smaller number of larger pieces. 
         [0054]    It will be noted that the various fractional pieces  718   a  through  726   f  are provided in two different colors or shades. For example, a first surface  730  of each of the pieces could be shaded in red, with the opposite second surface  732  colored black. Other colors may be used as desired. The provision of different colors on opposite surfaces of the various fractional time pieces serves to facilitate an understanding of such concepts as “half past the hour,” “quarter to the hour,” etc. 
         [0055]    In conclusion, the devices for teaching elementary fractional concepts provide a series of tools that greatly facilitate the ability of the student to visualize the concepts being taught. The various embodiments include devices for teaching abstract fractional concepts, as well as monetary and time concepts. Some concepts, e.g., the basic abstract fractional concept, are adapted for a linear, rectangular presentation, others, e.g., the teaching of time using a conventional clock configuration, are better suited to a circular presentation or configuration. However, it will be seen that either configuration as disclosed herein may be adapted to the teaching of virtually any fractional concepts. Accordingly, the devices for teaching elementary fractional concepts will prove to be most useful in the classroom and as educational toys for teaching such concepts. 
         [0056]    It is to be understood that the present invention is not limited to the embodiments described above, but encompasses any and all embodiments within the scope of the following claims.