Background of the Invention Abstract, fundamental arithmetic rules and concepts often can be very difficult to learn and/or understand, at least initially. However, once these concepts, rules and their relationships to"real world"events are learned and mastered by an individual, a powerful and necessary tool for functioning in today's society is provided. Indeed, although most adults use numbers and basic arithmetic during the course of performing their daily activities, they frequently do so without consciously thinking about what it is that they are doing. Nevertheless, the learning process typically is not an easy one.

This is particularly the case for young children. Young children generally lack the level of experience and cognitive development that is necessary for them to be readily able to associate abstract concepts with real world events. Consequently, it is not uncommon for very young children to learn by rote the words and/or symbols commonly associated with numbers and/or to count. This form of learning is beneficial to the extent that it provides the child with a familiarity with the basic words and symbols associated with mathematics in much the same way as a child's learning of his ABC's is beneficial to his later learning to read and/or write. Despite this, however, it often takes a significant amount of time and effort to develop a mental association between the words and/or symbols so learned and the groups, interrelationships and the like of real world objects that they abstractly represent. Similarly, the concept that the abstract words and/or symbols so learned can be manipulated in various ways that are representative of real world events also is not easily learned and/or understood.

For these reasons, among others, it is common in the teaching of arithmetic to young children for the teacher to illustrate abstract numerical <BR> <BR> quantities (i. e. , words and symbols) with physical objects. For example, it is not unusual to teach the concepts of numbers per se and the addition and subtraction thereof in terms of objects with which most young children are familiar. Hence, instead of attempting to teach the abstract concept that 2 + 3 = 5, the teacher might phrase the equation in terms of an analogy to items of fruit. Thus, 2 + 3 = 5 becomes Tom has 2 apples and Jane has 3 apples so together they have 5 apples. Similar analogies commonly are used in the teaching of the concepts of subtraction, multiplication and division of whole numbers.

Additionally, the concepts of zero and of the grouping of numbers into fields, such as hundreds, tens and ones, are examples of other abstractions that are difficult to learn and understand. The concept that ten ones are the same as one ten is fairly straightforward. The changes wrought by the manipulation of numbers with elements in several fields, however, are conceptually troubling. Consider, for example, the concept of"carrying"that is inherent in an addition or multiplication of whole numbers in a ones field that results in an answer that exceeds the limit of nine (9) available in that field. Also, consider the concept of"borrowing"from the next higher field in order to obtain enough units in the next lower field to accomplish a subtraction, or the concept of a"remainder"in division. Each of these concepts is so difficult to understand that some schools simply resort to teaching them by rote learning, without attempting to develop in their students a true understanding of what is going on in the computation as it relates to"real world"events.

The concept of fractions, and particularly the mathematical manipulation thereof, is even more difficult. Indeed, even children who are able to assimilate the basic concepts of whole numbers and/or the mathematical manipulation thereof in real world terms, often, still will have problems with fractions. The reasons for this are not entirely clear. It is speculated, however, that because the mathematical rules that apply to the manipulation of fractions are somewhat counterintuitive, the concepts involved are more abstract and less easily related to the real world. Thus, the fact that fractions cannot be added or subtracted in the abstract until they are converted to equivalent expressions having the same denominators is not easily grasped. Also, the effective reversal of the concepts of multiplication and division in the manipulation of fractional quantities is somewhat counterintuitive.

To address these and similar problems, several teaching systems and methods have been developed and utilized previously. For example, numerous demonstration boards and other devices in combination with elements for removal and/or attachment thereto in desired sequences have been developed and used. Similarly, various other systems for the manipulation of physical objects in particular ways have been developed and utilized to facilitate the teaching and understanding of mathematical <BR> <BR> concepts, particularly in various number systems, e. g. , the base 10 system, the base 2 system, etc.

Despite the foregoing, however, presently available demonstration and practice apparatus, systems and methods do not adequately deal with the mathematical concepts of fractions, the combination of whole numbers and fractions, and/or the mathematical manipulation thereof. Indeed, presently existing apparatus, systems and methods designed to deal with these conceptual issues generally consist of physical devices constituting a whole (say, for example, an entire pizza pie) which is already cut up into a predefined number of sections.

From systems of this type, a student can readily see the whole numbers involved, and also can see the particular fractional portions thereof that are precut. However, there is no way for the student to represent a fractional quantity that has not been precut. Further, the student is limited to visualizing a fractional portion of only some of the many possible wholes that may be involved. For example, using the pizza pie mentioned above and assuming that each pie is cut into eight equal pieces, the child can visualize 1/2 of the whole, that 1/2of of l/2 of the whole is 1/4, and so on. On the other hand, it is difficult to visualize 1/3 of 2/5 of the pie, 1/3 of 2 pies, or similar quantities.

Accordingly, there remains an unfulfilled need in the art for an improved apparatus, system and method for the demonstration and practice of basic arithmetic manipulations. Ideally, this apparatus, system and method should be suitable for use in the assimilation by the student of both whole number and fractional concepts in"real world"terms.

SUMMARY OF THE INVENTION Accordingly, the present invention provides an apparatus, system and method useful in teaching the concepts of fractions and their mathematical manipulation. Students may use this apparatus, system and method to practice mathematical manipulations and to learn mathematical concepts on a practical and self-educational basis.

The apparatus of the present invention typically comprises elements that are sturdy and easy to use. Also, the system components and the methods of their use preferably are visually stimulating and exciting so as to maintain the attention of users. Moreover, preferred systems and methods of the present invention comprise components that are adaptable to the teaching of varying mathematical concepts.

In accord with the present invention, an apparatus, system and method for the teaching/learning of mathematical concepts are provided that permit users to repeat any given sequence of steps many times, and to arrive at the same result each time. Preferred apparatuses, systems and methods in accordance with the invention provide for the teaching/learning of mathematical concepts wherein the concepts of addition, subtraction, and multiplication of whole numbers, of fractions and of combinations of whole numbers and fractions, may be accomplished. Further, preferred apparatuses, systems and methods in accordance with the invention provide for the learning/teaching of the concept of division by whole numbers. Also, particularly preferred apparatuses, systems and methods provide a self- education tool for use by both teachers and students in the process of associating mathematical abstractions with the real world.

The present invention, therefore, provides apparatuses and systems for teaching mathematical concepts that generally include at least two identical hollow containers. Also, conduit-like means for the establishment of liquid communication links between the lower portions of the containers, and valve means for selectively opening and closing the conduit means so as to permit control of a flow of a fluid between the containers are provided.

Further, each container defines a vent opening between its interior and the surrounding atmosphere. In addition, each container also preferably includes vertical linear scale means located on a side wall thereof for measuring the level of a liquid contained therein.

In particularly preferred embodiments, each container can be connected to and/or disconnected from at least one adjacent container without the loss (spillage) of any liquid contained in either container.

More particularly, each of the identical containers defines a generally vertically elongate, hollow internal volume. Accordingly, each container includes a bottom end, upon which it is normally adapted to stand in the same horizontal plane with the other containers. Each container also has a top end and a side wall portion. In some particularly preferred embodiments, the top end of each container defines a vent opening extending between the internal volume of its associated container and the outside atmosphere. In addition, an input/output opening for introducing a liquid into and/or removing a liquid from the hollow interior volume of each container may be provided in the top end. Of course, in certain embodiments, the vent and the input/output openings in the top of the container may be combined to form a single, multi-purpose opening without departure from the invention in its broadest aspects.

Further, in the event that the side wall portions of each container are formed of a non-transparent or non-translucent material, the side wall portions preferably define at least one translucent window portion extending at least partially between the bottom and top ends of the container for viewing the level of the liquid therein. Still further, at least one opening and/or conduit extends through the side wall portion adjacent the bottom end of each container. This opening and/or conduit contains, or is connected to, valve means having a closed position and an open position that respectively control liquid entry into and/or exit from the container.

This valve means is designed such that when the opening and/or conduit of one container is joined to a corresponding opening and/or conduit of another of the containers, the valve means may be selectively placed into its open position or its closed position. Also in certain embodiments liquid can be introduced into or transferred out of each of the hollow containers through an inlet/output opening in the top end thereof, in the bottom end thereof, or both. The level of liquid in each of the hollow containers also can be viewed through the translucent or transparent window in the side wall portions at any time. In addition, liquid located in the internal volume of any pair of hollow containers can be made to selectively pass therebetween according to the relative hydrostatic pressures therein. This occurs when the respective openings and/or conduits in the lower side walls are in liquid communication with one another and the valve means are opened.

Preferably, the containers are vented in some manner to the atmosphere.

The external surface of the respective side wall portions of each container typically are provided with interchangeable and/or otherwise variable vertical scale means to facilitate the determination and/or comparison of the levels of liquid in each container. In one preferred embodiment, the side wall portions are provided with retaining means adapted to hold any one of several removable scales consisting of differing unit markings vertically along the side of the container. The reason for this will be discussed more fully below. Suffice it to say at this point that this feature of the invention is deemed to be very important in the teaching of concepts in fractions. This is because, when abstract dealing with fractions, there often is a need to work with, and meaning fully associate, fractional quantities having different denominators.

Having thus summarized the apparatus and system of the invention, the method of its use will be clear to those skilled in that art. Specifically, addition and subtraction can be illustrated and practiced simply by combining preselected quantities of liquid from one or more containers into an empty container, or by removing selected quantities of liquid from a container containing a known quantity of liquid, respectively. Similarly, multiplication of numbers can be accomplished by adding a measured selected quantity of liquid into an empty container a desired number of times.

Division by whole numbers, on the other hand, takes advantage of the fact that all of the containers used define the same corresponding cross- sectional areas so as to have the same internal volume per unit of height, and that they all stand on the same horizontal plane. Typically, this is accomplished by each container being provided with side walls disposed substantially vertically relative to the horizontal plane upon which the containers are designed to stand. It, however, will be understood that other side wall configurations also can be possible in certain situations without departure from the invention in its broadest aspects.

The division operation then is accomplished by filling one container to a preselected level representative of the dividend, and connecting the conduits of the so filled container directly or indirectly with a number of empty containers equal to one less than the desired divisor. Then, opening the valve permits liquid communication links to be established between all of the connected containers. This allows the hydrostatic pressure in the liquid to equilibrate (i. e., distribute) the liquid in the first container among all of the connected containers so as to provide the same level of liquid in each container, thereby accomplishing the desired division. A similar process wherein two or more liquid containing containers are connected to one another allows the average of the quantities of liquid in the connected containers to be determined. The result is read from the side scale that, as generally mentioned above, can be varied automatically, or determined by trial and error or otherwise by the user.

Also as noted above, the selection of appropriate scales for the determination and comparison of liquid levels in the various containers is a key to the success of certain uses of the present invention. Thus, it will be understood that in the addition and/or subtraction and/or multiplication of whole numbers, the scales selected for use on each container normally will be identical. This is because the concepts of addition, subtraction and multiplication of whole numbers assume that arithmetic manipulations take place within the same number system. In the division of whole numbers, however, the scales selected will normally display both whole number units and the units of the selected divisor.

A similar methodology applies to fractions, except that the division of one fraction by another fraction cannot be directly shown using the division function of the present invention described above unless the user knows, or is somehow supplied with, the rule that (1/a)/ (l/b) = (b/a). Nevertheless, one still can start with an empty container and add fractional quantities created by the division of whole numbers process described above into the empty container until the level of liquid in the empty container reaches the numerator of the fractional division. For example, to demonstrate that (1/2)/ (1/4) =2, the student can create a plurality of containers containing the quantity 1/4. Then, by adding the contents of the containers containing the quantity 1/4 into an empty container until the level of liquid therein equals 1/2, the student can show the truth of the last stated equation. This is simply the traditional concept of how many times the denominator"goes into"the numerator. It, however, is neither a direct, nor an intuitive, method for the self-teaching of division by fractions.

Accordingly, one can create a fractional quantity by dividing a whole number by a whole number generally as indicated above. Addition of fractional quantities so created then is accomplished by combining those fractional quantities in a single container. Similarly, subtraction is accomplished by removing a quantity equal to a pre-created fractional quantity from the one of the containers containing the greatest quantity of liquid. It also is possible to multiply fractions or to create averages with the present invention. Thus, multiplication of fractions is accomplished by connecting a number of empty containers equal to one less than the divisor of the fraction to be multiplied by to a pre-created fractional quantity in another container just as in the division of whole numbers. Of course, if the numerator of the fraction to be multiplied by is greater than one, it is necessary to combine that number of the individual quantities created by the division together to reach the final desired answer. Similarly, the creation of averages results when multiple pre-created quantities are located in different containers, the containers are connected to one another and all the valve means are opened.

The foregoing summary has been presented in terms of the manipulation of liquid among identical containers as a tool for the demonstration and teaching the interrelationship of abstract arithmetic concepts and the"real world". The invention, however, also includes the accomplishment of this goal with a preprogrammed computer that includes appropriate display means showing virtual containers that can be filled with a virtual liquid and connected, just like the physical containers. In this embodiment, the starting liquid levels and other parameters are input by keyboard, a mouse or another input device, and are represented virtually on the display device. The manipulation of these virtual quantities then can be used to achieve the desired illustration of the relationship of a given abstract arithmetical operation and a"real world"event, including the direct representation of division by a fraction as will be discussed further below.

Of course, this"virtual"alternative does not provide the same type of "hands on"experience, as does the manipulation of physical containers and liquid. Nevertheless, it is less cumbersome to use, allows for more students to practice simultaneously, and avoids the inevitable problem of spillage of liquid as it is transferred between and/or among containers. It also is more flexible and accurate than the analog method and apparatus. This is particularly the case because scales can be readily changed either by the computer operator or automatically. Hence,"virtual"use of the system of the invention may attain accuracy levels that differentiate, for example, between a 1/55 scale vs. a 1/56 scale. This is equivalent to accuracy on the order of 1 in 200, a level that is hard to imagine with the analog alternative.

In addition, division of a fraction by a fraction can be shown in several ways, and other features such as rewards for correct manipulations may be provided by the use of the"virtual"alternative.

Also, the apparatus, system and methods described can be utilized to demonstrate the relationship between ordinary fractions, like 1/8, and decimal fractions, like 0.125. To accomplish this, containers defining volumes of 0.1, 0.01, 0.001 and so on are provided. Then, after forming a fractional quantity in the manner discussed above, that fractional quantity is transferred into successive containers of the 0.1 volume until less than 0.1 remained. Then, the procedure is repeated using the remaining quantity with 0.01 containers, 0.001 containers, etc. The resulting numbers of each different volume container so filled then is added to determine the number (0-9) to be placed in the corresponding place following the decimal point to convert a fraction to a decimal. Thus, the present invention also provides a system of containers comprising nine containers having a volume of 0.1 unit and nine containers having a volume of 0. 01 unit. Preferred systems further include groups of nine containers each having a volume that is 10 percent of the previous smallest container.

BRIEF DESCRIPTION OF THE DRAWINGS Other features and advantages of the present invention will become apparent to those skilled in the art by reference to the following description of several preferred embodiments thereof, which is rendered in conjunction with the appended drawings. In the appended drawings, like reference numerals are used to refer to like elements throughout, and: Fig. 1 is an illustrative perspective view showing a representative container suitable for use as part of the system of the invention, and in the practice of methods thereof; Fig. 2 is an illustrative side elevational view showing a pair of representative containers suitable for use as part of a system in accordance with the invention connected to one another by a representative liquid communication means; Figs. 3A and 3B are illustrative side elevational views showing a system in accordance with the present invention and the connecting of containers to demonstrate the fractional quantity 1/5; Fig. 4 is an illustrative side elevational view representing the use of the system depicted in Fig. 3B and an empty container to construct the fractional quantity 3/5; Figs. 5A and 5B are illustrative side elevational views representing another system in accordance with the present invention and the steps for using the same to construct 1/3 of the factional quantity 3/5 depicted in Fig. 4; Figs. 6A and 6B are illustrative side elevational views representing yet another system in accordance with the present invention and the steps for using the same to construct the fractional sum of 1/3 + 1/4 ; Figs. 7A to 7D are representative illustrative views depicting yet another system in accordance with the present invention and the steps for using the same to prove the truth of the mathematical rule the (1/a) x (1/b) = (1/ab) ; and Fig. 8 is a pictorial representation of the use of virtual images of containers and liquid in accordance with the present invention on the display of a preprogrammed computer including an parameter varying input device.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS The present invention recognizes and utilizes two basic facts. First, it is well known that young children almost universally enjoy playing with liquids such as water. This phenomenon starts at a very early age with the child playing in his bath or pool. For example, it is common for very young children to repeatedly fill containers with water, and to thereafter dump the water back into the bath or pool from a height above the surface of the water in the bath or pool. Further, it is not uncommon for this fascination with the manipulation of containers and liquids to continue at least through elementary school.

Second, it is an elementary law of physics that the hydrostatic pressure in a liquid in two or more connected identical containers open to the atmosphere and standing on the same horizontal plane will tend to equalize itself. More specifically, given a system including at least two identical containers open to the surrounding atmosphere at their respective tops and in liquid communication with one another at their respective lower portions, a liquid located in one of the containers will tend to flow into the other of them until the surface of the liquid in the one container reaches the same height as the surface of the liquid in the other container. This is to say that if all of the liquid is initially located in a first container and both containers are standing in the same horizontal plane, the hydrostatic pressure of the liquid will tend to force a portion of the liquid through the liquid connection between the containers and into the second container.

This liquid transfer will continue until the levels of liquid relative to the horizontal plane contained in each of the first and second containers are the same. If the cross-sections of the internal volumes of the containers parallel to the horizontal plane are the same and the container bottoms are at the same level, the volumes in the containers will be equal. Of course, a similar rule applies when more than one empty container is connected in series with a first liquid containing container.

Taking these basic facts into account, and referring now to the drawings, and particularly Fig. 1, a representative one of a plurality of identical container apparatuses contemplated as forming a part of the novel system of the invention is shown. This container, generally indicated at 2, includes a bottom wall 4 upon which the container 2 is adapted to stand.

Also, together with side walls 6 and top wall 8, bottom wall 4 defines a generally vertically elongate, hollow internal volume, generally indicated at 10. In addition, in its top wall 8, the container 2 includes a vent hole 12, and an inlet/outlet opening 14 for the introduction of a liquid from a source, generally depicted at 11, or from another of the containers of the system into its hollow internal volume 10. It will be understood, of course, that in some embodiments the vent hole 12 and the inlet/outlet opening 14 may constitute a single opening in top wall 8. This single opening may be adapted to perform both the venting and the inlet/outlet function without departure from the present invention in its broadest aspects.

Container 2 also includes at least one opening 22 extending through the lower portion 24 of side wall 6 substantially adjacent to bottom wall 4, and a liquid communication means 29 such as conduit 30,31 (Fig. 2). In a preferred embodiment, representatively shown in Fig. 1, four openings 22 for connection to liquid communication means 29 are provided at substantially ninety degrees to each other so as to define four liquid flow pathways in a plane spaced slightly above the common horizontal plane upon which bottom walls 4 of the containers 2 stand.

Each liquid communication means 29 preferably includes a self- closing valve that is normally closed, and that is adapted to interconnect with a corresponding valve/communication means of another container of the system. One example of this is a well-known two-part self-closing valve assembly (not specifically shown). Such an assembly typically includes a female portion and a male portion each defining normally closed internal conduits. The male portion is designed for sliding and sealing engagement within the female portion, for releasable snap locking therewith, and for the automatic simultaneous opening and closing of the internal conduits of both parts. Each container generally will have at least one male portion and one female portion for connecting with other containers. Hence, the female portion of such an assembly might constitute the communication means 29 located in a left-hand portion of the side wall of one container, while the male portion of that assembly constitutes the communication means 29 in the right side of an adjacent container.

Alternatively, openings 22 in side walls 6, respectively, can be connected to one side of a normally closed external valve means 26 by conduits 30, 31, as shown for example in Fig. 2, to connect two adjacent containers for liquid communication therebetween. Similarly, liquid communication openings 14 and 28 can be located in bottom wall 4 and top wall 8 of each container 2, respectively. As will be discussed in further detail below, the purpose of these alternative communication means is to allow liquid to be transferred among the containers of the system in a quick, yet controlled manner, with a reduced risk of undesirable spillage. Therefore, the cross-sectional size of the communication means conduit structure (see 30, 31) should be such as to allow the levels of liquid in each container to reach the same height quickly. Further, the plane containing the openings 22 in the side walls 6 should be established parallel to, and as close to, the common horizontal plane of the bottom walls 4 of the containers 2 as possible. This will provide the greatest range of arithmetical operations that can be demonstrated and/or practiced with the system.

Preferably, the volume of the conduits and valves used for communication between the containers is designed to be very small compared to the volume of the containers, and keeping in mind the range of mathematical operations that will be demonstrated. Of course, in the computer rendition of this system discussed further below, the volume in the conduits and valves can be set at zero. However, in the physical (i. e. , analog) embodiments, it is to be recognized that some liquid will remain in the conduits and valves. It also is to be recognized that these quantities must be accounted for in the use of the apparatus and system. Typically, this is accomplished by using scales that are calibrated in a manner that accounts for the quantities of liquid remaining in the transfer assembly.

The embodiment shown in Fig. 2, also illustrates the fact that the liquid introduction means 14 can be a cover member 16 adapted for removable snap-fit onto the top edge 18 the side wall 6 of the container 2. In this alternative, a downwardly extending tab portion 20 of cover member 16 can be utilized to facilitate the removal thereof when it is desired to introduce a liquid into the internal volume 10 as will be discussed more fully hereinafter. It is to be understood in this regard that the cover 16 can be used alone for this purpose, or it can be provided with a communication means similar to those discussed with regard to the Fig. 1 embodiment above.

It therefore will be understood that, in perhaps its simplest form, the invention includes a pair of containers, generally as depicted in Fig. 2, for use with a liquid source such as that representatively shown at 11 in Fig. 1.

As discussed above, the liquid communication means 29'associated with one of the containers 2'can be a self-closing male valve portion, and the liquid communication means 29"associated with the second container 2" can be a self-closing female valve portion. Preferably, the valves are adapted to engage and open one another readily and easily. Alternatively, hollow conduits 30 and 31 can connect communication means (outlets) 22'and 22" respectively to opposite sides of a normally closed valve 26.

As will be discussed in more detail below in connection with a method of the invention, other preferred embodiments are contemplated by the present invention in its broadest aspects in which a plurality of containers 2 are provided. In those cases, it is generally preferred that valves be provided on each of the containers such that the containers may be joined in various array, linear series and/or stacked patterns according to the arithmetical relationship (s) to be demonstrated and practiced with the system provided.

For example, three containers 2 can be arranged in an interconnected stacked configuration with the lowermost two containers filled with liquid and the topmost container being 1/4 filled with liquid. This is one way in which the quantity 2/4 or the like can be represented and arithmetically manipulated using the novel apparatus, system and method of the present invention without resort to variations in the units of measurement utilized as will be discussed more fully below.

Further, each of the containers commonly includes a scale (ruler) 35 located vertically between its bottom and top walls 4 and 8 along its side wall 6. This scale can be formed integrally with the side wall 6, or may constitute a separate element removably affixed thereto by a retaining means, representatively shown at 39. The retaining means 39 can take various forms without departing from the invention. In particular, the retaining means can be, for example, a first magnetic scale affixed to the side wall 6.

In such a case, the retaining means is adapted to hold any other magnetic metal scale against it along the container side wall 6. Similarly, the retaining means can include mechanical holding means such as an undercut slot open at one of its ends into which a scale can be slid.

Alternatively, a releasable adhesive, or a releasable mechanical material such as a Velcro, can be provided on the container side wall 6 and/or the back of a number of different scales for removable attachment to the container.

Other alternatives within the scope of this invention in its broadest aspects will occur to those skilled in the art in view of this disclosure. An important point to recognize in the latter regard, however, is that the invention provides multiple levels of sophistication such that it is usable by/for students at various stages of the mathematics learning process.

Thus, at one level the nature of the scale used on each of the containers is not relevant to the concept being taught. At a higher levels of sophistication, however, the nature of the scale used on each of the containers in the course of any particular arithmetical operation/real world liquid manipulation is perhaps the single largest limitation of the analog version of the invention.

This is because there simply is no intuitive basis upon which even an advanced student can select the relevant scale for each case. Rather, the selection of the relevant scale is a trial and error procedure wherein the student attempts to align the hash marks on a given scale with the level of liquid in the associated container. Obviously, in the event that a conversion between two different scales is required during the course of an arithmetical manipulation, the scales of all containers forming part of the operation preferably should be changed at the same time for consistency. The importance of this to the comprehension of different number systems will become apparent in the following examples of the method of use of the system.

Finally, in the event that the containers are not made of a material through which the level of liquid therein may be readily viewed, a vertical window portion 37 preferably is provided adjacent to the vertical scale. This allows the level of liquid within each container to be determined easily and quickly at any point in time.

Accordingly, it will be understood that a user of the novel apparatus, system and method of this invention can check the correctness of numerous arithmetical equations. For example, a user, such as a child, can assimilate the meaning of 1/5 as being 1 divided by 5. To accomplish this, a system including five containers is provided each including a scale representative of 1 marked in units of 1/5. One of the containers then is filled with a liquid, i. e. , the level of liquid in the container corresponds with the number 1 on the associated scale. (See Fig. 3A) Thereafter, the five containers are connected together as shown in Fig. 3B, and the valves are opened. The liquid originally in the first container then distributes itself equally among the five containers thereby demonstrating that one-fifth (1/5) means the division of one by five.

In another example not specifically shown in the drawings, it can be shown that 2 divided by three is the same as two thirds (2/3). Thus, one can start with a container containing a quantity of liquid and a scale that indicates that that quantity of liquid is equal to two (2) units. Then, by connecting two empty containers to the filled container and opening the valves, the hydrostatic pressure in the liquid will cause equal quantities of liquid to appear in each of the three containers, each quantity being one third of the height of the original quantity in the single container. However, the scales will show that the quantity in each container is two thirds of the unit quantity. Of course, the same result could be obtained by using a quantity of liquid corresponding to one (1) on the associated scale. Then, by attaching two empty containers to the full container and opening the valves three containers each containing one third of the original one are created.

Finally, combining two of the so created one thirds in a single container results in a real world demonstration of the quantity 2/3.

Addition, subtraction and multiplication of whole numbers are illustrated generally by Fig. 4. For example, one can start with an empty container (s) and multiple other containers filled with different amounts of liquid representative of different whole numbers on equivalent scales. The sum of the numbers so represented may be created by sequentially stacking the filled containers on top of an empty container (s) with the respective outputs in the bottom walls 4 connected to the respective inputs in the top walls 8. All of the liquid thus is allowed to flow downwardly into the empty container (s). This provides a representation of the addition of the various quantities in the individual containers in the lower container (s). Further, if all of the filled containers are filled to the same level, the amount of liquid transferred downwardly to the empty container (s) represents the multiplication of the given amount by the number of filled containers used.

Subtraction can be represented in a similar manner, but with the amount of liquid that is allowed to flow from the top container (s) into the lower container (s) being controlled. Hence, the quantity of liquid in the filled container (s) less the quantity of liquid allowed to flow into the empty container (s) leaves the result of the subtraction in the originally filled container (s).

The demonstration of the equation 1/3 + 1/4 = 7/12 is similar, while at the same time providing perspective on its"real world"meaning at different levels of sophistication. In other words, while the limitations of the scales are important for determining the outcome, they are not relevant to the checking of a proposed outcome.

As an example of this, consider the case in which the student desires simply to verify the equation 1/3 + 1/4= 7/12. In such as case, no scales are necessary to accomplish the demonstration. Rather, as explained above, the student can construct any desired fraction. This is accomplished by dividing a unit quantity of liquid in one container among a number of containers equal to the denominator of the desired fraction, and where necessary, combining a number of the so created quantities equal to the numerator of the desired fraction. Accordingly, the student may create one-third by dividing a unit quantity in one container among three containers. Similarly, he/she can create one-fourth by dividing an equivalent unit quantity in one container among four containers, and one-twelfth by dividing an equivalent unit quantity among twelve containers. Then, by combining the contents of one of the representative one-fourth containers with the contents of one of the representative one-third containers one side of the equation is demonstrated. The other side of the equation is demonstrated by combining the contents of seven of the representative one-twelfth containers. The level of liquid in the containers representing each side of the equation to be demonstrated then are compared with each other to verify the equality.

This, of course, may be accomplished either visually, or by joining the liquid connections of the last two mentioned containers together, opening the valves and noticing that the level of liquid in each remains the same. Of course, it can also be demonstrated using a computer program to simulate the physical operations.

Referring now to Figs. 5A and 5B, it will be understood that a similar procedure may be used to solve the equation 1/3 +'4 = x for the unknown quantity x. This can be accomplished by first creating a container with a liquid to a level corresponding to the 1/3 marking on its side scale. A container with a liquid level corresponding to the 1/4 marking on its side scale then is created (the respective scales being based upon the same unit value of 1). In each case the procedure used is the same as that described above for the division of whole numbers (See Figs. 3A, 3B), or by filling the containers to the desired level from a source (see reference numeral 11 of Fig. 1). Thereafter, the liquid from the first container is transferred into the second container. Then, the resultant scale marking corresponding to the liquid level in the second container is read. In this manner, the user determines the value of the unknown x of the last stated abstract equation, while at the same time gaining perspective upon the"real world"meaning and usefulness of the equation. Similar manipulations may be utilized to verify subtraction and division equations involving fractions (See. Figs. 6A and 6B).

Of course, in the latter example, appropriate scales must be provided to correspond to the demonstrations for the lessons being taught and, as mentioned above, this is a trial and error procedure when the analog version of the present invention is used. Nevertheless, this drawback is not present when one who already knows how to convert different fractions into equivalent expressions having the same denominator uses the invention.

Further, once the concept of converting the denominators of both fractions to the same value has been demonstrated and understood, the trial and error faced by the student in making similar conversions with regard to other problems becomes a significantly beneficial part of the learning process.

An example of the method of the present invention utilizing containers adapted to be connected to one another in the form of an array will now be described with reference to Figs. 7A-7D. The purpose of this example also varies with the sophistication of the mathematical lesson being taught.

Thus, the student may simply demonstrate that a particular equation such as 1/3 x 1/4 = 1/12 is correct (without the need for scales), or he may utilize the system to demonstrate the real world basis for why (1/a) x (1/b) = (1/ab).

To accomplish these objectives, a single one of the provided containers is filled with liquid to a level corresponding to a preselected unit quantity (the number 1 on a scale located on the side wall of the container).

Three empty containers are then connected in series to each other and the first container along the X-axis (here defined as the linear direction of the series connection of the containers). As noted above, this causes the level of liquid in the first container to drop from 1 to 1/4, and the level of liquid in the other containers respectively to rise from 0 to 1/4. (See Fig. 7A) The containers then are disconnected from one another so as to leave four separate containers each containing a quantity of liquid corresponding to 1/4 of the original preselected unit quantity from the first container.

Each of the above-mentioned four containers then is connected in series along the Y-axis (typically by connectors located at ninety degrees to the connectors used to form the first series) to two additional empty containers (A', A" ; B', B" ; C', C", D'and D", respectively). The level of liquid in each of the containers in each group then becomes equal to 1/3 of the 1/4 of the original preselected unit quantity provided in the first container.

This is to say, the level of liquid in containers A, B, C and D drops from 1/4 of the original unit quantity to 1/12 of the original unit quantity. At the same time, the levels of liquid in containers A', A", B', B", C', C", D'and D" each rise from 0 to 1/12 of the original unit quantity. (See Figs. 7B and 7C) Thus, in the first step of the last stated example, the basic concept of a fraction is demonstrated, i. e. , a whole number is divided by a whole number, automatically by the system in a manner that the student may readily watch and understand. Thereafter, that fractional quantity is divided by a whole number by the system automatically in a manner that demonstrates to the student that 1/4 of 1/3 of the original unit quantity (or "1/4"x"1/3") is equal to 1/12 of the original unit quantity.

Finally, all twelve containers may be connected to one another in, for example, a 3 x 4 matrix-like array. When this is done, the liquid level in each of the twelve containers remains the same. This provides a final proof to the student that one third of the original unit quantity times one quarter (or one quarter of the original unit quantity times one third) is the same as one twelfth of the original unit quantity (i. e., 1/ (3x4) = 1/3 x 1/4 = 1/12).

Indeed, as described previously, the user can construct the matrix (or a series) using one full container (i. e. , 1) and eleven empty containers and then open the valves. As indicated above, this will divide the original unit quantity equally among the twelve containers. Thus, the user can compare the liquid heights in the containers of the matrix (or series) with those generated by the division steps described above to unequivocally demonstrate that (1/3) x (1/4) = (1/12). Further, the user can then use the system with other values to prove, and thereby learn in a"real world" context, that the mathematical abstraction (1/a) x (1/b) = (1/ab) is correct.

(See Fig. 7D) In another embodiment, generally indicated in Fig. 8, the containers and the liquid transfer therebetween, can be simulated virtually using a computer having central processor, generally indicated at 102, programmed to provide a simulation of the system of containers and manipulations described herein, and an appropriate display device, generally indicated at 104. In such an alternative, images 106 of the containers and the movement of the liquid among them can be controlled automatically by the computer or manually by a pointer, mouse or keyboard (generally indicated at 108).

Such a computer controlled system, in some situations, can facilitate the assimilation of the real world correlation to the abstract mathematical concept being taught without the necessity of providing a cumbersome system of containers, connectors and liquid. It also is far more accurate than the analog equivalent system, and can be more user friendly in that a student can practice with the system using the computerized version without fear of spillage of fluid, difficulty in the manipulation of the physical connectors between containers, or the like.

In addition, the computer simulation alternative provides additional advantages. For example, in the conversion of fractions to equivalent terms having the same denominator, the simulated system allows the user to manually check different scales very quickly and methodically until the height of the virtual liquid corresponds exactly to a hash mark on a particular scale. As noted, the accuracy of this alternative therefore is far superior to that of its analog version. Further, the system may be programmed to conduct the search for the right scale automatically either by successively walking the user through the procedure, or by simply selecting the correct scale. In the former cases, the user may be rewarded by bells, whistles, horns or the like upon finding the correct scale and/or by finding the correct scale within a preselected number of attempts. Such reward systems are known to be helpful in the teaching of difficult and complex concepts.

Still further, the simulated system is particularly adapted for the teaching of the division of one fraction by another fraction. As indicated above, if one knows that (1/a)/ (l/b) = (1/a) x (b/1) =b/a, the analog demonstration is as simple as the division by a whole number combined with an addition. Similarly, if the problem is treated as a"goes into" problem, it may be shown in a straightforward analog manner. However, that analog demonstration will not necessarily convey to the user what is happening in"real world"terms. In the simulated system, on the other hand, direct demonstrations are possible.

For example, say the user desired to show the division of"3/4"by "2/3". The simulated system allows the fraction"3/4"to be created in a manner similar to that described above by the manipulation of liquid in containers on the display screen. The simulated system, however, also allows containers having different cross-sectional areas to be created and used. This is a complication in the potential variations of the analog system that is difficult to control during use, particularly in a self-teaching mode in which the user does not understand the effects that the use of containers with differing cross-sections may introduce into his results. In any event, separating the contents of the container containing"3/4"into containers having"2/3"of the cross-sectional area will yield the desired result.

Alternatively, the same thing may be accomplished by creating the quantity"2/3"in one container in the manner discussed above. Then the simulation may convert the scale such that the"2/3"is equivalent to one (1) on the new scale. Thereafter, the quantity"2/3"is emptied out of the container and a quantity"3/4"is introduced. The value on the scale established by setting"2/3"= 1 unit provides the desired result of the division of one fraction by another. Note, however, that the scale established by setting"2/3"= 1 unit itself has to be converted to eighths ("1/8") before the correct answer can be read off of the scale.

Therefore, the simulation alternative can be used both as a substitute for the analog system and as a way of teaching the more complex theories of number systems. In particular, the ability to modify scales quickly such that any given quantity may be given the value of 1 is extremely important in the understanding of mathematical relationships in"real world"terms. The analog system of this invention teaches this in a subtle way that is readily understood by students to some degree. The simulated system allows the concepts to be taken further as the cognitive abilities of the students increase, thereby, further enhancing the benefits achievable using the apparatus, system and methods of the present invention.

In view of the foregoing detailed description of several preferred embodiments of the invention, numerous alterations, modifications, variations and the like, within the spirit and scope of the present invention, will occur to those skilled in the art. For example, the containers can have any cross-sectional shape, such as a circular tubular shape.

Accordingly, it is to be understood that the foregoing specification has been presented by way of illustration only, and that the scope of the invention is intended to be limited only by the terms of the appended claims.