Education and demonstration – Mathematics – Arithmetic
Reexamination Certificate
2002-11-01
2004-06-29
Fernstrom, Kurt (Department: 3712)
Education and demonstration
Mathematics
Arithmetic
C434S188000, C434S208000, C434S209000
Reexamination Certificate
active
06755658
ABSTRACT:
FIELD OF THE INVENTION
The present invention relates to educational game apparatuses and methods for use in teaching the addition, subtraction, multiplication, and division of positive and, especially, negative numbers. The game apparatuses and methods of the present invention enable students to see and understand a theory for adding, subtracting, multiplying, and dividing positive and negative numbers.
DESCRIPTION OF THE PRIOR ART
A description of the prior art is set forth in U.S. Pat. Nos. 1,294,126, 3,094,792, 3,229,388, 3,410,002, 3,414,986, 3,452,454, 3,935,649, 4,177,681, 5,474,455, 6,089,871, and U.S. Pat. No. 6,413,099, which patents are incorporated herein in their entireties by reference.
As evidenced by the above-cited patents, educational game apparatuses and methods exist for teaching mathematical concepts. However, teaching the addition, subtraction, multiplication, and division of positive and, especially, negative numbers usually entails students learning by rote the rules of adding, subtracting, multiplying, and dividing positive and negative numbers without ever understanding the rhyme or reason behind what they are doing. Other students, unfortunately, never learn the rules and, for them, mathematics becomes a dreaded black hole.
SUMMARY OF THE INVENTION
Accordingly, a technique is needed for teaching the addition, subtraction, multiplication, and division of positive and, especially, negative numbers that clearly explains a cogent theory behind the rules.
The apparatuses and methods of the present invention for teaching the addition, subtraction, multiplication, and division of positive and negative numbers solve the above need. More specifically, the present invention is based on the Null Theory of Adding, Subtracting, Multiplying, and Dividing Positive and Negative Numbers (hereinafter referred to as the “Null Theory”). According to the Null Theory, the natural state of a given environment is the null state. In the null state, the environment is in perfect balance and appears to be devoid of any matter. However, the environment is, in fact, composed of a plurality of null units, with each null unit being, in turn, composed of a positive unit and a negative unit. The environment can be disturbed by introducing (i.e., adding) into it one or more positive units or one or more negative units. The environment can also be disturbed by removing (i.e., subtracting) from it one or more positive units or one or more negative units. In the latter case, if there are not enough free positive units available to be removed from the environment, a sufficient number of null units are split to obtain the desired number of positive units to be removed from the environment. When a null unit is split and the positive unit thereof is removed from the environment, a negative unit is left behind in the environment. (Hence, the foregoing explanation of the Null Theory clarifies and visually demonstrates the reason behind the rule that the subtraction of a positive number +X is equal to the addition of a negative number (i.e., −(+X)=+(−X)).) Likewise, if there are not enough free negative units available to be removed from the environment, a sufficient number of null units are split to obtain the desired number of negative units to be removed from the environment. When a null unit is split and the negative unit thereof is removed from the environment, a positive unit is left behind in the environment. (Accordingly, the foregoing explanation of the Null Theory clarifies and visually demonstrates the reason behind the rule that the subtraction of a negative number −X is equal to the addition of a positive number (i.e., −(−X)=+(+X)).)
Another aspect of the Null Theory is that only excess positive units or excess negative units remain in the free state within the environment. For example, if there are 5 free positive units in the environment and if 3 negative units are introduced into the environment (as is the case in the mathematical expression 5+(−3)), the 3 negative units will combine with 3 of the free positive units to form 3 null units, leaving only 2 free positive units in the environment.
With the Null Theory in mind, in one embodiment of the present invention, the game apparatus employed to teach the addition, subtraction, multiplication, and division of positive and negative numbers comprises (a) a plurality of positive units, (b) a plurality of negative units, and (c) a demarcated playing environment or zone. The positive units and the negative units are adapted to reversibly attach to or be associated with one another to form null units, with each null unit comprising at least one positive unit and at least one negative unit and the number of positive units and the number of negative units per null unit being equal. (As used in the specification and claims, the terms “attached to” and “associated with” both mean that the objects in question either can be physically reversibly held together or can be positioned in a manner such that the objects appear to have an affinity for or relationship with one another.) Preferably, each null unit comprises just one positive unit and just one negative unit.
The demarcated playing zone is typically an integral part of a playing surface.
Generally, the game apparatus further comprising a first means for measuring the number of units selected from the group consisting positive units, negative units, and combinations thereof, with the first measuring means desirably being located on the playing surface and, preferably, within the demarcated playing zone. The purpose of the first measuring means is to measure the degree that the demarcated playing zone has been disturbed from the null state. The first measuring means, which can be a scale for weighting the positive and/or negative units, is ideally an axis marked in substantially equal units from 0 to M and in substantially equal units from 0 to N, where M is a positive whole integer, N is a negative whole integer, and substantially each of the positive units is adapted to reversibly attach to or be associated with a unit from 0 to M on the axis on the playing surface, and substantially each of the negative units is adapted to reversibly attach to or be associated with a unit from 0 to N on the axis on the playing surface. While M can be virtually any positive integer, M is typically a whole positive integer from 5 to 50, more typically from 10 to 25, and most typically from 10 to 20. Similarly, while N can be virtually any negative integer, N is commonly a whole negative number from −5 to −50, more commonly from −10 to −25, and most commonly from −10 to −20. Usually, M equals the absolute value of N.
In another preferred embodiment of the invention, the apparatus further comprises a second means for measuring the number of units selected from the group consisting positive units, negative units, and combinations thereof, with the second measuring means desirably being located on the playing surface and, preferably, outside the demarcated playing zone. The purpose of the second measuring means is to act as a check point or zone to ensure that (a) the correct number of positive units and/or negative units are being transported into the demarcated playing zone and (b) the correct number of positive units and/or negative units have been removed from the demarcated playing zone. The second measuring means, which can also be a scale for weighting the positive units and/or the negative units, is ideally an axis marked in substantially equal units from 0 to P, where P is a positive whole integer and substantially each of the positive and negative units is adapted to reversibly attach to or be associated with a unit from 0 to P on the axis on the playing surface. While P can be virtually any positive integer, P is typically a whole positive integer from 10 to 100, more typically from 20 to 50, and most typically from 20 to 40. Usually, P equals M plus the absolute value of N.
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