Bead-on-tile apparatus and methods
10839719 ยท 2020-11-17
Inventors
Cpc classification
International classification
G09B1/04
PHYSICS
Abstract
Apparatus and methods based on applied cognitive science, where children play the lead role in storylines staged upon a rule-enforcing apparatus and by so doing, become self-enlightened about denumerability, rank-wise denumerability, addition, subtraction, multiplication, division, and other change-of-state processes encountered in mathematics and the quantifiable sciences.
Claims
1. An apparatus for providing instruction, comprising: at least one instruction tile having a plurality of instruction sites that are each located at a discrete location within a predefined area on the instruction tile and a saturation state instruction site that is located on the instruction tile remote from the predefined area; and a plurality of instruction pieces configured to be positioned on the plurality of instruction sites and the saturation state instruction site in a predetermined order; and a stencil having means for revealing the next instruction site or saturation state instruction site on the instruction tile in the predetermined order.
2. The apparatus of claim 1, wherein each instruction site and the saturation state instruction site has a recess that defines the location of the instruction site and the location of the saturation state instruction site on the instruction tile.
3. The apparatus of claim 1, wherein the instruction pieces are edible.
4. The apparatus of claim 1, wherein the instruction relates to displaying intermediate and final results of state machine emulation of multistate computing for at least one of mathematics and quantifiable sciences.
5. The apparatus of claim 1, wherein each instruction tile has an edge and wherein edges of adjacent instruction tiles are positioned adjacent one another to form an instruction board.
6. The apparatus of claim 5, wherein edges of a plurality of the instruction tiles are positioned adjacent one another to form a tessellation.
7. The apparatus of claim 5, wherein the edge of a first instruction tile adjoins the edge of a second instruction tile to form a tessellation that defines the instruction board.
8. An apparatus for providing instruction in at least one of mathematics and quantifiable sciences, the apparatus comprising: an instruction board formed from a plurality of instruction tiles, each of the instruction tiles having a predetermined number of instruction sites that are each located at a predetermined discrete location within an area on the instruction tile defined by a plurality of borders and a single saturation state instruction site that is not located within the area on the instruction tile defined by the plurality of borders; and a plurality of instruction pieces configured to be received on the instruction tiles at the instruction sites and the saturation state instruction site in a predetermined order; wherein the instruction pieces are manipulated on the instruction sites and the saturation state instruction site on the instruction tiles to perform a change of state operation that provides the instruction in the at least one of mathematics and quantifiable sciences.
9. The apparatus of claim 8, wherein each of the instruction sites and the saturation state instruction site has a recess formed on the instruction tile.
10. The apparatus of claim 8, wherein each of the instruction tiles has at least one edge, and wherein the edges of adjacent instruction tiles abut to define a tessellation that forms the instruction board.
11. The apparatus of claim 10, wherein the adjacent instruction tiles define a tessellation on the instruction board that is configured to perform at least one of addition, subtraction, multiplication and division.
12. The apparatus of claim 8, wherein each of the instruction tiles has at least one channel formed thereon for sliding the instruction pieces on the instruction tile, and wherein the channel is disposed between a border of the area and an edge of the instruction tile and the saturation state instruction site is disposed within the channel.
13. The apparatus of claim 8, wherein each of the instruction sites and the saturation state instruction site has printed indicia that corresponds to the location of the respective instruction site and to the location of the respective saturation state instruction site.
14. The apparatus of claim 8, further comprising a stencil having means for revealing the next instruction site or saturation state instruction site on the instruction tile in the predetermined order.
15. A method for providing instruction in at least one of mathematics and quantifiable sciences, the method comprising: providing an instruction board formed from one or more instruction tiles wherein each instruction tile comprises a plurality of instruction sites at discrete locations within a defined area on the instruction tile and a saturation state instruction site at a location remote from the defined area on the instruction tile; providing one or more instruction pieces configured to be received on the instruction sites and the saturation state instruction site of the instruction tile in a predetermined order; and manipulating at least one of the instruction pieces from the saturation state instruction site on a first instruction tile to at least one of the instruction sites on a second instruction tile to perform a change of state operation relating to the at least one of mathematics and quantifiable sciences.
16. The method of claim 15, wherein the instruction sites and the saturation state instruction site have recesses formed in the instruction tiles.
17. The method of claim 15, wherein the instruction tiles have at least one edge and wherein the edges of adjacent instruction tiles abut one another to form a tessellation that defines the instruction board.
18. A method for SM-table multiplication comprising the steps: a) providing a multiplier value in canonical form having a tile count; b) providing a multiplicand value in canonical form having a tile count; c) providing an instruction board comprising at least a multiplier row having a plurality of tiles and a product row having a plurality of tiles, the multiplier row and the product row each having a tile count equal to the tile count of the multiplier value added to the tile count of the multiplicand value; d) creating an SM-table having four rows using an S-value list of 6, 3, 2, 1 from a top row to a bottom row, setting a 1M M-value in the bottom row to the multiplicand value, and by using addition, generating a 2M M-value, a 3M M-value and a 6M M-value in a second row, a third row and a fourth row, respectively, above the bottom row; e) zeroing all of the tiles in the product row; f) duplicating the multiplier value into the multiplier row; g) setting a multiplier focus tile at a rightmost tile in the multiplier row; h) determining whether a value in all of the tiles in the multiplier row at and to the left of the multiplier focus tile are zero and if the value in all of the tiles in the multiplier row at and to the left of the multiplier focus tile are zero terminating the method, otherwise continuing the method with step i); i) setting a focus row of the SM-table to the top row of the SM-table; j) determining whether the value in the multiplier focus tile is less than the S-value in the focus row of the SM-table and if the value in the multiplier focus tile is less than the S-value in the focus row of the SM-table go to step m); k) reducing the value in the multiplier focus tile by the S-value in the focus row of the SM-table; l) adding the M-value in the focus row of the SM-table into the product row beginning at the tile in the product row that is aligned with the multiplier focus tile and into all of the product tiles to the left; m) determining whether the focus row of the SM-table is at the bottom row of the SM-table and if the focus row of the SM-table is at the bottom row of the SM-table go to step o); n) shifting the focus row of the SM-table down one row in the SM-table and repeating step j); o) shifting the multiplier focus tile one tile left; and p) repeating step h); whereby the product row provides the product value.
19. A method for SM-table quotient auto-generation comprising the steps: a) providing a divisor value in canonical form; b) providing a dividend value in canonical form; c) providing an instruction board comprising at least a quotient row, a dividend row, and a subtrahend row, each row having a plurality of tiles and an additional tile above the order of magnitude of the dividend value; d) creating an SM-table having four rows using an S-value list of 6, 3, 2, 1 from a top row to a bottom row, setting a 1M M-value in the bottom row to the dividend value, and by using addition, generating a 2M M-value, a 3M M-value and a 6M M-value in a second row, a third row and a fourth row, respectively, above the bottom row; e) zeroing all tiles in the quotient row; f) duplicating the dividend value into the dividend row; g) setting a dividend focus tile at a leftmost non-zero valued tile in the dividend row; h) aligning the tile bearing the leftmost non-zero valued tile of the 1M M-value in the SM-table with the dividend focus tile established in step (g), thereby setting a quotient focus tile to be aligned with a rightmost tile of the 1M M-value in the SM-table, which also establishes a rightmost tile in a partial dividend field of tiles; i) setting a focus row of the SM-table to the top row of the SM-table; j) comparing the M-value in the focus row of the SM-table to the value in the partial dividend field of tiles and if the M-value in the focus row is greater go to step n); k) adding the S-value in the focus row of the SM-table to the value in the quotient focus tile; l) duplicating the M-value on the focus row of the SM-table from the rightmost tile to a leftmost tile in the subtrahend row, beginning at the subtrahend tile aligned with the quotient focus tile; m) subtracting the subtrahend row from the partial dividend field within the dividend row, beginning at the subtrahend tile aligned with the quotient focus tile; n) determining whether the focus row of the SM-table is at the bottom row of the SM-table and if the focus row of the SM-table is at the bottom row of the SM-table go to step p); o) shifting the focus row of the SM-table down one row in the SM-table and then returning to step j); p) determining whether the quotient focus tile is aligned to the rightmost tile of the quotient row and if the quotient focus tile is aligned to the rightmost tile of the quotient row terminating the method, otherwise continuing the method with step q); q) shifting the dividend focus tile one tile to the right; r) shifting the quotient focus tile one tile to the right; and s) repeating step i) to step p); whereby the dividend row provides a remainder value and the quotient row provides a quotient value.
Description
BRIEF DESCRIPTION OF THE DRAWINGS
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
LEXICON OF MEANS AND TERMS OF ART
(37) To ensure the invention is articulated as coherently as possible and its fabrication is readily apparent to any person having ordinary skill in the arts relevant to the invention, the following defined terms apply to the Claims and the Specifications. All terms used in their capitalized form are used to aid readability, and all capitalized forms shall be read and understood as the lowercase forms, and visa versa.
(38) STORYLINE defined. Storylines are narratives that guide a child through a story that requires the resolution of a particular dilemma or mystery. For example, in the Game of Adding Up Mommy's Grocery List, the child is given a sequence of grocery item costs that must be setup on the Candy Board and accumulated one by one. Storylines can revolve around mysteries such as solving primitive algebra problems where the mystery is the unknown values of its variables.
(39) BEAD defined: Beads (24) are preferably spheroidal objects, and include ellipsoid, ovulate, single and double crescent cross-section profiles; in disk form when viewed in plan view, as well as single and multi-punctured toroidal forms. Beads also include multi-faceted prismatic objects and flattened forms stylizing silhouettes of said prismatic objects or art objects, such as Monopoly-type pieces and such, as well as naturally occurring objects such as pebbles, seeds, nuts or parts thereof. Beads also include bead composites whereby beads are interconnected into a rigid unity. For example, three beads interconnected in a row creates a rod, and two beads creates a pod, both of which are shaped on the underside so as to socket into a given cluster of three and two bead sites respectively. For example, a pod can represent paired electrons on a given tile, where said tile represents a particular atomic element or bond state. From the viewpoint of child safety, candies in the form of M&Ms, Smarties, Skittles and similarly shaped edibles are the preferred embodiments for all beads used by or accessible to children. All figures, disclosed herein, depict this focus on bead safety. Beads are characterized, and may be distinguished one from the other, by any combination of and any distribution over the extents of their physical body, via the following features: (a) size, shape, reshaping, plasticity, remoldability, divisibility into sub-beads and aggregation into bead composites, (b) color and color patterning, including typographic symbols and glyphs, (c) material, (d) texture, (e) electro-sensitive properties, (f) non-passive sound and light emitting properties, including the indication of state changes resulting from movement, position and bead site location within tiles, (g) magnetic properties, (h) sonic properties, and (i) all known and yet to be discovered properties and sensor means that permit tracking of bead movement, including the movement of beads into out-of-bounds regions, the confirmation of correct and incorrect bead placement, and means capable of discerning said bead's proximity to one or more other beads, tiles, bridging tiles, chips and stencils. Furthermore, beads are not limited to a top-side with a single-facet display purpose, but can be re-oriented in-situ or otherwise, and thereby take on multiple top-side display purposes. For example, the spin up and spin down states of an electron. While not essential to the unadorned embodiment of a bead, all bead manifestations may transmit and receive broadcast signals. Beads include icons reproduced on computer-proctored display devices and other reprographic analogues, whereby said icons aim to conceptualize bead function and emulate physical beads, as characterized by properties (a) through (i) above, including any multi-faceted character.
(40) TILE defined: A tile (27, 28) is a planar object comprising of any finite number of edges in any combination of edge straightness and curvature which demark the extents of said tile's boundaries. A tile includes (a) a tile with bead sites, and thus is capable of state representation, such as a Digit-Square, as depicted in
(41) BEAD SITE defined: Bead sites (11, 17) are preferably indicia bearing recessed cavities in the substrate of the tile which conform with and are compatible to the physical profiles of all beads pertinent to the bead site and placeable within the bead site. As groups, bead sites are located in plateau-region layouts on a tile and laid out to optimize their didactic function based on a pertinent principle of pedagogy, such as the principle of subitization and super-subitization, as applied to a given problem domain in the quantifiable sciences. Bead sites may be characterized by any of the character features (a) through (i) that characterize beads. While not essential to the unadorned embodiment of a bead site, all bead site manifestations may transmit and receive broadcast signals. Bead sites include bead-spaces reproduced on computer-proctored display devices and other reprographic analogues, whereby said bead-spaces aim to conceptualize bead site function, and emulate bead-space to icon correlation and virtualized rubber-banding.
(42) BRIDGING TILE defined. Bridging tiles are special purpose tiles capable of corralling a series of two or more tiles into a unity whether overlaid or underlaid across said tiles and thereby interlocking said plurality of tiles into a schema of tessellation. Bridging tiles include underlay base-mats capable of creating a variety of game board embodiments of the invention. For example, for the purposes of emulating chemical bonds, such as hybridized and co-ordinate bonding, bridging tiles in either overlay bond bridges or underlay base-mat bridges employ a socket profile that mates with the non-bridging tiles that represent the chemical elements so bonded within the given molecule being emulated. Bridging tiles have all the properties and full feature set of tiles, as defined under TILE, including proximity behavior. Bridging tiles can interbridge other bridging tiles, i.e. be nested. Bridging tiles include multi-prong stanchions that when affixed to a base mat of tiles creates vertically layered schemas of tessellation incorporating a plurality of tiles and schemas of tessellation at other levels in the structure. Bridging tiles can serve as power and communication pathways into all tiles they interconnect.
(43) SCHEMA OF TESSELLATION defined. A schema of tessellation is the architecture that embraces a problem domain, and is manifested in a unified object called a singular tessellation embodiment. Such notional and physical tessellations comprise a plurality of tiles, including bridging tiles, chips and stencils, chosen to address one or more specific problem sets in the study of mathematics and the quantifiable sciences. A singular tessellation embodiment could have, but does not need to have, physical connectivity one tile to another in the form of edge-to-edge adjacency, vertex-to-vertex adjacency, or any adjacency combination thereof, either with or without the use of bridging tiles. Schemas of tessellation include storyline specific singular tessellation embodiments. While not essential to the unadorned singular tessellation embodiment, all singular tessellation embodiments may transmit and receive broadcast signals. All schemas of tessellation are capable of replication on computer-proctored display devices and other reprographic analogues, whereby said virtual singular tessellation embodiments aim to emulate physical singular tessellation embodiments. Several custom multi-row and multi-column Game of Candy singular tessellation embodiment configurations are specifically addressed in the detailed description.
(44) CHIP defined: Chips (26) are preferably disc-shaped objects that preferably are contoured or have edges which enable them to be fixed in location across a tessellation of interconnected tiles or more simply on a single tile, as depicted in
(45) STENCH, defined: Stencils (30 . . . 39) are preferably rigid sheets, which conform to some restricted region of a tile or a tessellation of interconnected tiles, as depicted in
(46) PLOSIVE-STATE defined. A plosive-state is an operation-lock-up state that may occur during an operation on the focus tile, and in an alternate embodiment, on the multi-state focus bit, i.e. a mit, in a plosive-state enabled, multi-state computer. A plosive-state causes the operation on the focus tile or mit to lock-up, which prevents said operation-in-progress from executing further. For example, during addition of beads onto the focus tile, a TEN state, as depicted in
(47) PLOSIVE-STATE EQUILIBRATION defined. Plosive-state equilibration is the process of resolving operation-in-progress lock-up conditions, and such resolution is dependant on the particular schema of tessellation, the particular focus tile on which the lock-up has occurred, the particular plosive-state and the particular operation-in-progress. For example, during an addition operation when a plosive-state TEN lock-up occurs on a radix-10 focus tile, the process of equilibration under addition-in-progress ripples into a co-ordinate tile one rank of higher order of magnitude, by Western conventions on the leftside edge of the focus tile. Additive ripple increments the bead count of said co-ordinate tile. Mathematicians call this Carry. In the parlance of children playing a Game of Candy, equilibration under addition causes a Packaging operation to reset the state of the focus tile from the plosive-state TEN, where ten candies await packaging, to its plosive-state conjugate 0. Similarly, during radix-10 subtraction, when the plosive-state 0 lock-up occurs in the focus tile, equilibration ripples into a co-ordinate tile one rank of higher order of magnitude. Subtractive ripple decrements the bead count of said co-ordinate tile. Children relate to this as Unpackaging, which notionally tears open a packet of candy and creates in the focus tile a bead state of TEN unpackaged beads, the plosive-state conjugate to 0.
DETAILED DESCRIPTION
(48) The invention's target users are children aged three and above. When combined with a need for rigor, this makes the kid-friendly description of the invention lengthy.
(49) Lesson #1, The Stencil Game
(50) The first concepts young children must acquire about bead patterns come by way of the bead pattern Stencil system of
(51) The bead pattern Stencil system enforces a correct understanding of what the bead sites on a Digit-Square represent, how beads correlate to count and number, and how the sites are populated with the correct order of beads from bottom right to top left, and produce valid bead patterns on the Digit-Square, as depicted in
(52) The first game involves cycling through the stencils in count sequence on a single Digit-Square. The parent shows the child the sequence of bead sites starting at the lower right corner, moving leftwards, and upwards in ascending order. Insert the 1 Stencil and take a candy from the Tray and socket it into the 0 bead site, and speak aloud Fill zero and you have one. Proceed with the Stencil for 2 by placing it into the Digit-Square and take a second candy from the Tray and socket it into the 1 bead site, and speak aloud Fill one and you have two. and so on, all the way to the TEN pattern, as depicted in
(53) As the child repeats the game, moving beads one by one from the Tray and socketing them into the Digit-Square using ascending stencil value placement, i.e. 1 Stencil, 2 Stencil, and so forth, he acquires (a) the typographic number symbol, (b) an escalating bead count, (c) placement in the correct bead site, and (d) the overall bead pattern on the Candy Board. This process regiments a child's capacity to subitize three candies in a row and thereafter super-subitize one, two and three rows of bead patterns all the way to nine. The TEN pattern auto-subitizes because the Digit-Square is saturated. The means of communicating with toddlers is: first the pattern, next the pattern embeds as a subitized unity, and finally, the actual numeric count becomes embedded in due course. Digits correlate at-a-glance with a symbol 5, and the spoken word five, which a child sees at a glance as the 3,2 bead pattern, as depicted in
(54) Even if the toddler is too young to speak unerringly, by finger movement of candy alone, he can master the task of bead placement in the proper order to create the corresponding pattern, as he progresses through the Stencil sequence from 0 to 9, and lastly, TEN.
(55) Once the child masters a single Digit-Square, next comes the Game of Setup the Candy Board, where the child sets up values on multiple Digit-Squares joined side by side in what is explained to the child as the Bag, Packet and Candy ranking system. Proper language is vital so that learning via acquisition takes root. In due course, a three digit glyph sequence 243 takes on the meaning 2 Bags, 4 Packets, 3 Candies, something relatable to children because that is what 243 represents in real life, not abstractions like hundreds, tens and ones. Abstraction comes later, as the child's brain matures.
(56) The final task of Lesson #1 uses a two-row multi-rank Candy Board, as depicted in
(57) Setting up a two-row Candy Board is the initial task in all Games of Candy involving addition and subtraction, so mastery of this is essential.
(58) Lesson #2, Game of Packaging and Unpackaging
(59) Lesson #2 has two purposes. Demonstrate how each rank presents a higher containment of candy, namely candy, packets, bags and boxes, and how each correlate one to the other. More importantly, Lesson #2 details how the plosive-state equilibration method resolves lock-up conditions under various mathematical operations on the Candy Board. Take note that the plosive-state equilibration aspect of Lesson #2 can be deferred and Lesson #3 and #4 addition can proceed as long as the summations do not create a plosive-state lock-up condition, i.e. carry. Similarly, Lesson #6 subtraction can also be played as long as the numbers do not create a plosive-state lock-up condition, i.e. borrow.
(60) Because candy comes in boxes, bags, packets and individual pieces, candy inside real-life packaging is the simplest vehicle to describe the rank-wise denumeration of candy in conjunction with the process of Packaging and Unpackaging.
(61) The storyline starts with one packet of ten candies above the Packet-Rank Digit-Square and one bead socketed into the 0 bead site of the Packet-Rank Digit-Square, as depicted in
(62) Next, the parent sweeps all the candy into the Tray, so all Digit-Squares are zeroed out, as depicted in
(63) The parent demonstrates the Packaging process by picking up the ten candies, putting them back into the improvised packet opened earlier and places the filled packet in the Candy-Rank Digit-Square but maintains a hold on the packet, as depicted in
(64) The next game in this lesson proceeds as follows. Focus returns to the Candy-Rank, and the parent slides one candy from the Tray into the 0 bead site in the Candy-Rank Digit-Square. After each candy placement, the parent points to the bead in the Packet-Rank and picking up the improvised packet explains and shakes the packet so the association is clear, and then speaks the number represented on the Candy Board, i.e. 1 Packet, 1 Candy. This is repeated until 1 Packet, TEN Candies, as depicted in
(65) The parent opens a second empty packet, such as a mini ziplock bag, and asks the child to fill the second packet with the ten candies saturating the Candy-Rank Digit-Square. Next, the filled packet is placed on top of the Candy-Rank Digit-Square, as depicted in
(66) This exercise should be repeated one more time so the child understands the process that incrementing to the TEN bead pattern triggers a Packaging operation where all candies are removed from the Candy-Rank Digit-Square, in this case by physical packaging, and one bead in the Packet-Rank is slid into its corresponding next bead site in the Packet-Rank Digit-Square.
(67) Unpackaging arises during subtraction and can only proceed when there are zero beads in the Digit-Square on the right. Hence, sliding a bead from the Digit-Square into the Tray on the left triggers the Unpackaging of ten beads into the Digit-Square on the right, namely zero is the plosive-state that may require equilibration during subtraction operations, and which creates a TEN conjugate state as a result.
(68) Lesson #3, Game of One Digit Addition
(69) Once Lessons #1 and #2 are mastered, the child can launch into the adventure aspect of the Game of Candy, starting with single digit numbers with low values then graduating to larger values that cause Packaging to be triggered.
(70) Because addition is a binary operation, even on paper, the Game of Addition requires at least two rows of Digit-Squares, as depicted in
(71) For example, because the process must be relatable to children, a storyline for single digit 7 plus 6 addition with Packaging would proceed as follows. Katie has seven candies for the church and they are located on the top row, i.e. Addend, as depicted in
(72) The parent then demonstrates how to create the packet via a process called Addition and Packaging. Starting with the candy on the 6 bead site, to the immediate right of the visible 7 bead site on the top row, one by one, the candy is added to Tommy's candy in the bottom row. Finally, the candy on the 3 bead site in the top row is slid down to the unoccupied 9 bead site in the bottom row, which forms a plosive-state TEN bead pattern, as depicted in
(73) Reverend Michaels thanks the children, and tells them that the three Candies are their reward for their charity, which they must share together, as depicted in
(74) Thus, the storyline concludes with a two-candy reward for the child, too. Such rewards turn stories into candy motivated adventures, and where arithmetic is an auto-acquired dividend.
(75) Lesson #4, Game of Adding Up a List of Numbers
(76) Technically, the addition process can proceed in any piecemeal Digit-Square order, the only essential requirement is that every bead in the Addend row must be moved down to the Result row until the Addend row is completely zeroed out. However, the most efficient process of addition mimics the traditional addition process of pencil-on-paper arithmetic where Packaging performs carry, and the digits in the result reveal their final values in the customary leftward sequence ones, tens, hundreds and so on, in their normalized canonical form.
(77) The traditional approach for adding up one or more numbers in a list on the Candy Board executes as follows.
(78) Step (A): Setup the first number on the list in the Results row, i.e. bottom row.
(79) Step (B): If all numbers in the list are exhausted, addition is complete and the Result row contains the answer, so stop. Otherwise, setup up the next number on the list on the Addend row, i.e. top row. Set the focus of addition on the Candy-rank, i.e. the rightmost rank.
(80) Step (C): Move all the beads in the Addend Digit-Square on the focus rank into the Result Digit-Square being sure to perform a Packaging operation whenever it is triggered.
(81) Step (D): If the Addend row is completely devoid of beads, go to Step (B). Otherwise move the focus left one rank and go to Step (C).
(82) Atypical storyline is as follows. Each year on the day after Christmas, the Church hands-out all the candy accumulated in the Church pantry over the preceding twelve months. In January, the parishioners donate a combined amount of 8 Bags, 6 Packets. This number is setup in the bottom row, as depicted in
(83) This story shows the child how simply moving beads on the Candy Board allows a long list of numbers to be added up in a logical and orderly manner. Employing a similar storyline, parents can make a customized Game of Adding Up Mommy's Grocery List using shopping receipts. Every game is end-capped by the use of plosive-state equilibration if normalizing the final value into canonical form is required.
(84) Lesson #5, Game of BPC Mystery
(85) An excursion into experimenting with primitive algebra is worthwhile to season the child's mind about the ins and outs of solving storyline mysteries.
(86) In this lesson, the Candy Board configuration is a three-row board with three ranks and three mystery values. The storyline goes: Daddy brings home candy and puts it all in the pantry. He knows he added 4 Bags and 6 Candies to the pantry, but he forgets how many packets resulted. In the top row of the Candy Board, the child is told to setup 4 beads in the Bag-Rank Digit-Square and 6 beads in the Candy-Rank Digit-Square, and then told to place the [P] guide chip in the Packet-Rank Digit-Square, as depicted in
(87) Before Daddy came home the pantry had 3 Packets, 5 Candies, but Mommy isn't sure how many bags of candy were there originally. The child is told to setup the middle row with 3 beads in the Packet-Rank Digit-Square and 5 beads in the Candy-Rank Digit-Square, and to place a [B] guide chip in the Bag-Rank Digit-Square, as depicted in
(88) The storyline continues. Mommy, Daddy and Tommy go into the pantry to count everything, because that might help solve what are now two mysteries. Tabby the house cat runs out with candy in its mouth, and disappears. Mommy, Daddy and Tommy see that all the individual pieces of candy have been stolen, leaving 7 Bags and 6 Packets on the shelf. In the bottom row representing the current pantry inventory, the child is told to place 7 beads in the Bag-Rank Digit-Square and 6 beads in the Packet-Rank Digit-Square and a [C] guide chip in the Candy-Rank Digit-Square, as depicted in
(89) With 4P6+B35=76C setup on the Candy Board, the storyline dilemma presents three mysteries called [B], [P] and [C]. The parent and child work through the three mysteries.
(90) First, the focus is on solving the [C] chip mystery. Since 6 candies came home with Daddy and there were already 5 in the pantry, adding these two values together and treating the middle row as the Result row, a plosive-state TEN results, as depicted in
(91) The focus resumes with the Packet-Rank. We now know there were 4 packets, shown in the middle row, before Daddy added more packets, and we know after Daddy added his full packets the pantry now has 7 Packets. The parent and child can walk through the process of adding beads one by one until the middle row Packet-Rank bead pattern/count equals the Bottom row pattern/count of 6. From the Packet Tray, one by one beads are added to the middle row, as well as placed on the right-side of the board, as depicted in
(92) Focus turns to the Bag-Rank. Daddy brought home 4 Bags which got added to the pantry creating a total of 7 Bags. The child might see the solution because he has applied it to solve the Packet mystery. One by one, beads are added to the 4 Bags Daddy brought home, with another bead added on the right-side of the board, as depicted in
(93) Unlike prior art, such games are possible on the Candy Board because rigor is enforceable. From time to time, the parent should shepherd the child on an adventure in BPC algebra, so the child understands many mysteries are not as intractable as they first appear. Also, the child will start decoding the basis behind algebraic reasoning whenever the storyline has a mystery element. The desired goal of the Game of BPC is for the child to play out his reasoned hunches on the Candy Board. There is no wrong answer, just an immersive learning experience. The element of intrigue and surprise at a solvable outcome becomes a huge part of the sense of accomplishment a child derives from the Game of BPC.
(94) Lesson #6, Game of in-Situ Subtraction/Addition
(95) In-situ games emulate a closed system where candy is conserved. Hence, a misbehaving child eating candy always creates incorrect results and unmasks cheating. Every game is end-capped by the use of plosive-state equilibration if normalizing the final value into canonical form is required.
(96) Initial games should be single digit subtraction with no unpackaging. For example, on a two-row Candy Board the storyline states that Tommy possesses 6 candies in Tommy's row, the top row. The bottom row is Katie's and currently shows she has zero candies. However, Tommy owes Katie 4 candies. The storyline mystery is how many candies will Tommy have leftover when he gives 4 candies to Katie. Performing the story results in four candies sliding from Tommy's row into Katie's row. This leaves 2 candies in Tommy's row, i.e. the end result for Tommy after subtraction is done.
(97) The need to apply the Unpackaging process comes into play when Tommy has 3 Packets, 2 Candies, as depicted in
(98) The Game of In-Situ Addition is also playable in this format. For example, next scene in the story goes: Katie gives Tommy 6 Candies for his Birthday. And, backwards and forwards the storyline can go, and at every scene in the story the child is performing arithmetic, sometimes packaging and other times unpackaging.
(99) In an enhanced version, a three-row in-situ game has Katie on the top row, Tommy on the bottom row and Mr. Timpkins, the grocer, on the middle row. Setup each row with a given number of candies and play the game of commerce between two customers and Timpkins Grocery Store.
(100) Lesson #7, Game of Multi-Digit Subtraction
(101) A two-row Candy Board configuration is the minimal layout for subtraction. Although either row can be subtracted from the other, the Minuend in the top row and Subtrahend in the bottom row convention not only mimics traditional pencil-on-paper layout, it is also the most efficient layout for the Game of Division, as detailed later in Lessons #11 and #14.
(102) Whenever the Minuend is greater than the Subtrahend, a positive result remains in the top row when subtraction concludes. However, when the Minuend zeros out, the bottom row yields the result and means subtraction has generated a negative value. Thus, the two-row Candy Board acts like an accounting ledger, where negative numbers are recorded in the Debit register, i.e. what is owed, and positive numbers are recorded in the Credit register, i.e. what is left-over, namely the remainder.
(103) After setting up two numbers in the two rows, subtraction entails sliding bead pairs of similar rank from both rows simultaneously. Technically, the subtraction process can proceed in any piecemeal rank/Digit-Square sequence, the only essential requirement is that every bead in either the Subtrahend, bottom row, or Minuend, top row, must be moved into the Tray until one row becomes completely zeroed out.
(104) At the start of the training process for young children, parents should always focus on one Digit-Square subtraction, then two Digit-Squares. This should be done without the need for any Unpackaging operations, i.e. the Minuend digit is always bigger than the Subtrahend digit. Once the child has mastered three Digit-Squares, training should proceed with problems that include Unpackaging.
(105) Traditional pencil-on-paper subtraction is limited to right to left processing, using the just-in-time borrow technique from accounting practice. Plosive-state equilibration provides the Candy Board with the flexibility to model four borrow techniques as well as deploying them in any rank order, not merely right to left. For a child, the preferred technique is always the simplest.
(106) For example, consider the problem of a customer using a $100 note to pay a $83.53 debt. There are four distinct ways of handling what conventional arithmetic calls borrow when performing a subtraction operation.
(107) The first technique is the cashier's full breakout method for making change as depicted in
(108) The second technique is the just-in-time version of the cashier's method, $100, as depicted in
(109) The third technique is the accountant's just-in-time borrow method as used in traditional pencil-on-paper subtraction. Consider 37 minus 9 emulated on the Candy Board, as setup in
(110) Once children understand Credit/Debit concepts, they can adopt the accountant's just-in-time borrow technique as an alternative to the process of unpackaging surplus which presumes the Minuend has surplus amounts and hence never takes on negative values. Caution is called for because one-sided-one-function candy can't model negative values, only dual Credit/Debit accounting columns can.
(111) The fourth technique called gratuitous borrow is a tweak on the accountant's just-in-time borrow method. The gratuitous borrow process is as follows. Starting with the uppermost digit in the Minuend, in every Minuend Digit-Square where 0 appears add one bead to both Minuend and Subtrahend Digit-Squares, Repeat this, rank by rank going left until you hit the rightmost rank in which case processing stops. Thereafter, during right to left subtraction and in the event it is necessary, one bead in the top row can always be unpackaged into ten beads on the right. For pencil-on-paper use, this amounts to redundant work, and so never finds use. However, because of the Candy Board's multi-row architecture, gratuitous borrow is the simplest among all the borrow/unpackage methods. Consider the problem 1000 minus 0999. After applying gratuitous borrow, this becomes 1110 minus 0TENTEN9 something the Candy Board handles with ease, but which traditional mathematical techniques abhor because the TEN digit is stymied by a rigid pencil-on-paper, dual-digit 10 canonical form.
(112) For example, consider the problem setout in
(113) Furthermore, using gratuitous borrow sets up the Candy Board so subtraction on each rank can proceed in any piecemeal sequence without qualm.
(114) Step (A): Setup the Minuend value in the top row and the Subtrahend value in the bottom row.
(115) Step (B): If the Subtrahend is completely zeroed out then stop, subtraction has finished.
(116) Step (C): If the Minuend has completely zeroed out then stop, subtraction has finished and take note that the Subtrahend holds a negative value.
(117) Step (D): Set the focus on the higher of the uppermost digit in the Minuend or the Subtrahend.
(118) Step (E): Move the focus one rank to the right. If the focus rank is now the rightmost rank, then go to Step (F). Otherwise, if the Minuend focus Digit-Square is 0, then add one bead to both Minuend and Subtrahend Digit-Squares. Go to Step (E).
(119) Step (F): if the Subtrahend focus Digit-Square is non-zero, then go to Step (H).
(120) Step (G): Shift the focus one rank to the left. If the focus rank has run past the uppermost digit in both the Minuend and Subtrahend, or has run off the compute board then subtraction is complete, so stop. Otherwise, go to Step (F).
(121) Step (H): In the focus rank, from both the Minuend and the Subtrahend, simultaneously slide one bead out of top and bottom Digit-Squares while beads remain in both Digit-Squares. If the Subtrahend Digit-Square becomes 0, go to Step (G).
(122) Step (I): If the Minuend Digit-Square is 0, check the value of the Minuend Digit-Square left of the focus rank. If this value is 0 or the left rank has run off the board, then subtraction is complete so stop and note that the result is a negative number. Otherwise, in the Minuend, perform an Unpackaging operation from the rank left of the focus rank, thereby setting up a TEN bead pattern in the focus rank of the Minuend. Go to Step (H).
(123) Once subtraction is mastered, the best games going forward involve storylines combining adding in and subtracting out, such as Game of Add-up the Grocery List, Redeem Coupons and Pay the Bill, because this puts the whole process of Packaging, Unpackaging and right-to-left rank-wise bead processing into play. As the child's mastery improves the numbers should get larger as well.
(124) Lesson #8, Introducing Symbol Chips into Calculations
(125) Once the child has mastered the various Games of Addition and Subtraction, the digit glyphs 0 through 9 would have been subitized into his mind as proxies for the bead pattern and bead count. As age and maturity dictate, a reboot of the Game of Adding up the Grocery List using the numeric chips, as depicted in
(126) For example, $2.74, the first number on the grocery list, is placed in the top row as a sequence of numeric chips and off to the rightside of the bottom row, the [+] guide chip is placed, as depicted in
(127) Using red-circle chips and the [] guide chip, subtraction can be performed. As depicted in
(128) Lesson #9, Game of Algebra
(129) Encouraging the child to think and reason plays a huge part in self-learning and cultivating mastery and accomplishment. Once adept at the Game of Subtraction, the child should be more astute about the ins and outs of what he is doing on the Candy Board, and so should be better equipped to decode the mystery behind algebraic reasoning.
(130) Consider the algebraic scenario, 743 minus 5PC equals B21, as depicted in
(131) The storyline is as follows. The Smith pantry initially contains 7 Bags, 4 Packets, 3 Candies. The child places the pantry inventory setup in the top row, as depicted in
(132) Once the Game of Algebra is setup, the parent shows the child how the beads on the Candy Board can solve these three mysteries. Starting with the Candy-Rank, the parent removes candy from the top row one by one, and deposits it into the middle row atop the [C] guide chip, as depicted in
(133) The focus turns to the [P] guide chip in the Packet-Rank. The process the parent uses for the [P] guide chip is exactly the same line of reasoning that solved the problem of the [C] guide chip. Once two beads are moved from the top row into the middle row atop the [P] guide chip, the child should observe that the top and bottom rows now have matched up number of packets, as depicted in
(134) The focus turns to the [B] guide chip in the Bag-Rank, bottom row. The child should be able to solve this without much prompting because it entails direct subtraction of the middle row from the top row. From the 7 beads in the top row and the 5 beads of the middle row, one by one beads are removed from the Candy Board. This countdown represents the 5 Bags Daddy took to work. As depicted in
(135) Lesson #10, Game of Doubling and Sharing
(136) The Game of Doubling is best played on a one-row Candy Board because it compels the child to in-situ add an equal number of beads into the Digit-Squares as were originally there before doubling. This exercises the child's power of memory and subitization. Mimicking multiplication, the Game of Doubling proceeds right to left. As an equal number of beads are slid into the focus Digit-Square, packaging operations may be triggered that require resolution before more beads can be slid into the focus Digit-Square. This game is the lead in for children to recognize how 5 plus 5 equals 0, package 1, 6 plus 6 equals 2, package 1, and so forth.
(137) The Game of Sharing is played on a multi-row Candy Board that reflects the number of batches that will be created. For example, dividing candy among two people requires a two-row Candy Board, for three people, a three-row board, and so forth.
(138) Consider the Game of Sharing where a larder of candy must be divided between two people, as depicted in
(139) Mimicking division, the Game of Sharing always proceeds left to right, namely share the Boxes first, Bags second, Packets third, and the Candy last. During each step, a remainder of either zero or one bead is created. For example, 5 Boxes divides into 2 Boxes a piece with 1 Box remaining undivided. This remainder box can be unpackaged into TEN Bags, which can then be shared along with whatever Bags are already in the candy larder.
(140) The Game of Sharing Executes as Follows.
(141) Step (A): Setup the Dividend value in the bottom row and zero out the top row, as depicted in
(142) Step (B): Slide beads from the bottom Digit-Square into the Digit-Square above until the bead pattern in the top Digit-Square is one bead greater than or equal to the bead pattern in the bottom Digit-Square, as depicted in
(143) Step (C): If the bead pattern in the top Digit-Square is one bead greater than the bead pattern in the bottom Digit-Square, as depicted in
(144) Step (D): If the top row is less then the bottom row, then go to Step (B). Otherwise, slide beads from the top row into the Digit-Square on the bottom row until the Digit-Square in the top row has a bead pattern one bead greater than or is otherwise equal to the Digit-Square in the bottom row, as depicted in
(145) Once the child has mastered the concept of divvying by two, he should tackle sharing among three on a three-row Candy Board. The process is similar to division by two, except there are two upper rows. Similar to the divide by two rule, all three rows must show the same bead pattern. With three-way division, remainders of zero, one or two beads are created, which are subsequently unpackaged into the Digit-Squares right of the focus rank in one or both the top and middle rows as the circumstances dictate.
(146) Divide by ten is demonstrated to the child as the right to left process of shifting all the beads right by one rank. Divide by five should also be mastered, namely in-situ double the bead pattern, or in the alternative duplicate the bead pattern into the top row and then add the top row into the bottom row. Thereafter, divide the doubled-up value by ten. These divide by five and divide by ten skills are used to calculate square roots via iterative improvement in Lesson #15, Game of Magic Twins.
(147) Lesson #11, Game of Simplified Division
(148) The Game of Simplified Division involves repetitive subtraction of the divisor, and is best played with problems where individual digit values in a quotient don't exceed four.
(149) Using super-subitization alone, division requires a child to recognize by inspection when a Digit-Square's bead pattern in the top row is numerically greater than or equal to the bead pattern in the bottom row. If the child shows any hesitation, several games involving the question of which is greater, top or bottom Digit-Square should be played, as covered in Lesson #1, the Stencil Game. The Game of Divide by Two Sharing where beads are slid into the top row until they are greater than or equal to the bottom row, trains a child to compare by inspection, namely super-subitize, when top is greater or equal to the bottom.
(150) The Game of Simplified Division uses a three-plus-one row Candy Board, as depicted in
(151) In Division, the rule of deciding to do a subtraction is contingent on making a proper comparison where the partial dividend in the top row must always be greater than or equal to the divisor. If the child botches the comparison and subtracts the divisor from the partial dividend, the dividend is prematurely zeroed out while beads remain in the bottom row, i.e. the subtrahend. On the Candy Board this error is detected and easily rectified, as follows. Reduce the quotient by one bead, duplicate the divisor into the zeroed-out partial dividend and subtract the remaining beads in the bottom row from the top row.
(152) The Quotient is an incremental count of the number of times the Dividend minus Divisor operation has been performed before the comparison step indicates the partial dividend in the top row is less than the Divisor in the middle row.
(153) The Game of Simplified Division is best illustrated to a child through the vehicle of a storyline. The Game of Church Christmas Pantry storyline in Lesson #4 concluded with twelve months of repeated additions that yielded 8 Boxes, 2 Bags, 7 Packets, 4 Candies in the Church's pantry on Christmas Day. The day after Christmas, the Church has the tradition of Alms Giving, namely providing candy to needy people. Reverend Michaels has all the candy stores in the Church pantry brought to the front of the Church and asks Katie to calculate how many Boxes, Bags, Packet and Candies are to be given to each of the 67 needy people, so that every person receives an equal share.
(154) The Game of Simplified Division Executes as Follows.
(155) Step (A): Setup the Dividend in the top row, the Divisor in the middle row and zero out the bottom and Quotient rows. The leftmost non-zero rank of the Dividend row is the focus rank. Place [Dividend], [Divisor] and [] guide chips one rank left of the leftmost digit of the Dividend, and place the [Quotient] guide chip one rank to the left of the lowest ranked digit of the Divisor, as depicted in
(156) Step (B): Comparison. Left to right, one rank/Digit-Square at a time perform a comparison. If the top row Digit-Square is greater than the middle row Digit-Square, as depicted in
(157) Step (C): Subtraction. Duplicate the middle row into the bottom row aligned just right of the [] guide chip, as depicted in
(158) Step (D): Shift Right. The focus of division shifts one rank to the right. Move the [Quotient] guide chip in the Quotient row one rank right. If the [Quotient] guide chip is now in the rightmost rank, reset the [Quotient] guide chip left of the leftmost digit in the Quotient row, as depicted in
(159) Step (E): Underflow Error Rectification: Subtract one bead from the Quotient. Duplicate the Divisor into the zeroed out top row and subtract the remaining beads in the bottom row from the top row Go to Step (D).
(160) Doing simple division before multiplication ensures that the child will not be intimidated by multi-digit multipliers and the massive numbers multiplication creates. Division ensures that the child knows how every massive number can be smashed into smaller chunks.
(161) Lesson #12, Game of Simplified Multiplication
(162) Simplified multiplication comprises the repetitive addition of the Multiplicand by a count given in the Multiplier. To speedup the addition process, rank shifting comparable to the Game of Simplified Division is used, except the shift is from right to left. The Multiplier is the smaller of the two numbers. The best way to ease children into understanding the process is to begin with single digit numbers, preferably in the range zero to three.
(163) Multiplication, as a practical tool, is best related to the child through hands-on examples.
(164) Household bathroom tiles provide a visual way of explaining how multiplying one length of tiles by another length of tiles quantifies the total tile count inside a rectangular area. Hence, the Game of Tiles makes a good vehicle for animating multiplication. Money also provides solid examples, and should be explained in terms of dollars and decimal units of the dollar which appear as metallic disks called coins, namely dimes/deci and cents/centi. Money introduces a child to the concept of thousands and hundreds, and counting from zero to ninety-nine rather than 9 Tens, 9 or 9 Packets, 9 Candies.
(165) The Game of Simple Multiplication is best played on a four-row Candy Board. The smaller of the two numbers is the Multiplier, and it is setup on the top row because the beads on this row will be discarded into the top Tray one by one as multiplication unfolds. The Multiplicand is setup on the second row from the top because this number remains unchanged and is duplicated into the third row in the appropriate right rank. Initially zeroed out, the third row holds a duplicate of the Multiplicand shifted left as multiplication unfolds. Initially zeroed out, the bottom row holds the accumulated partial products and the final Product, once the Multiplier has zeroed out.
(166) Consider the storyline. In the Smith home, if a new kitchen countertop requires 3 times 4 granite tiles with each tile costing $7.64, how much money will Mrs. Smith be spending?
(167) The Game of Simple Multiplication Executes as Follows.
(168) Step (A): Setup the Multiplier on the top row and the Multiplicand on the second row. Place [Multiplier] and [Multiplicand] guide chips to their immediate right, off the board. Zero out the third row and place an [+] guide chip to its immediate right, off the board. Zero out the bottom row. The [+] and [Multiplier] guide chips mark the focus rank to their left and these chips shift left as the higher ranks of the Multiplier are processed. The focus begins in the rightmost rank of the top row.
(169) Step (B): If the top row is completely zeroed out, as depicted in
(170) Step (C): If the Multiplier Digit-Square in the focus rank is 0, as depicted in
(171) Step (I)): Slide one bead from the Multiplier Digit-Square into the Tray. Adhering to right to left Digit-Square order, duplicate the Multiplicand from the second row into the third row starting on the Digit-Square to the immediate left of the [+] guide chip, as depicted in
(172) Step (E): Add the third row to the bottom row, as depicted in
(173) Setting Up the 632M-Board
(174) The synonymous terms 632M-Board and 632M-Table are used interchangeably. Radix-10 numeric representation will be the vehicle for describing the method, which finds generalized use in any desired radix system.
(175) Illustrating a pencil-on-paper breakdown of the 632M method,
(176) The 632M method is efficient because it uses rote additions/subtractions operations where the net speed penalty is 1.4 operations per step, on average, and never exceeds 2 operations. In
(177) Multiplication can be reduced to 1.2 additions per step if the tweaks detailed in the description of Lesson 413, Game of 632M Multiplication, are applied. Furthermore, the method of 632M is open to obvious adaptation. Certain digits repeated in a multiplier may give a better M-value selection, such as 532M, for example, whenever 5's outnumber 6's by two to one and 9's are scarce. Similarly, for 742M and 732M, which have an overhead of four additions to setup the M-Table, but otherwise super-subitize over radix-10 as well as 632M does, and are optimal for radix-11, as well. Similar extensions of the method apply to other radixes. For example, using a nine M-value 50/40/30/20/10/632M-Table with its setup overhead of nine additions, radix-60 arithmetic requires no more than 3 operations per step. Similarly, for radix-100, with its overhead setup cost of thirteen rote additions, a thirteen M-value SM-table requires no more than 1.5 operations per radix-10 digit step with an average of 1.15, making it a better means for slaying huge numbers. Similarly, radix-1000 reduces the average to 1.07 operations per radix-10 digit.
(178) The 632M-Board preferably has an S-value field, as depicted in
(179) The NI-value field is mandatory and the width of the NI-value fields of the 632M-Board ought to be one rank wider than the 1M value to accommodate all potential 6M values, not depicted in the
(180) The Setup of the 632M-Board Executes as Follows.
(181) Step (A): Setup a series of S-values from top to bottom rows, namely 6, 3, 2, 1 in the S-value field of the M-Table.
(182) Step (B): Setup the 1M value on both the bottom and next row up, as depicted in
(183) Step (C): Add the bottom row into the next row up, which yields 2M, as depicted in
(184) Step (D): Duplicate the 2M value into the row above it, and duplicate 1M into the top row, as depicted in
(185) Step (E): Add the top row downwards into the row beneath, which yields 3M in that row, as depicted in
(186) Step (F): Duplicate the 3M value into the top row and the bottom row, as depicted in
(187) Step (G): Add the bottom row into the top row, which yields 6M in the top row, as depicted in
(188) Step (H): Finally, Setup the 1M Value in the Bottom Row, as Depicted in
(189) As will become clear from Lessons #13 and #14, detailed below, an 632M-Board detached from the Candy Board facilitates both rank shifting and duplication of M-value presets onto the Candy Board in the partial product row during multiplication and divisor/subtrahend row during division.
(190) Once the child has mastered digit glyphs, numeric chips can be substituted for the Digit-Square bead patterns on the 632M-Board, as depicted in
(191) Lesson #13, Game of 632M Multiplication
(192) For the Game of 632M Multiplication, the Candy Board is configured as two central rows with dual Trays top and bottom, with guides chips and the 632M-Board positioned, as depicted in
(193) Technically, because addition is commutative, rank-wise repetitive additions of 632M based multiplication can proceed in any Digit-Square sequence. However, to mimic pencil-on-paper arithmetic conventions, a right to left process is used.
(194) The solution for 372 times 137 is depicted in
(195) The Game of 632M Multiplication Executes as Follows.
(196) Step (A): Setup the Multiplier in the row above the Tray. Zero out the top and bottom rows below the Tray. Setup the four 632M Multiplicand M-values on the 632M-Board, and the S-values 6, 3, 2, 1 using guide chips. Align the rightmost rank of the 632M-Board with the rightmost rank of the Candy Board. On the right edge of the Candy Board, setup the [Multiplier] and [+] guide chips for the Multiplier row and top row respectively, as depicted in
(197) Step (B): If the Multiplier row is completely zeroed out, the Game of Multiplication has concluded. As depicted in
(198) Step (C): 6S/6M Step. If the Multiplier Digit-Square bead pattern is 6 or greater, as depicted in
(199) Step (D): 3S/3M Step. If the Digit-Square bead pattern is 3 or greater, as depicted in
(200) Step (E): 2S/2M Step. If the Digit-Square bead pattern is 2 or greater, as depicted in
(201) Step (F): 1S/1M Step. If the Digit-Square bead pattern is 1, as depicted in
(202) Step (G): If one or more beads remain in the Multiplier focus Digit-Square to the immediate left of the [Multiplier] guide chip, then an error has occurred and the game needs to start afresh because it is possible not enough beads were slid off during 632M processing. Otherwise, shift the [Multiplier] and [+] guide chips and the 632M-Board one rank to the left, as depicted in
(203) An alternative means of error correction is one where after each Digit-Square of the Multiplier is zeroed, the child duplicates a backup value of the bottom row, which can be used to reboot the process in the event the child makes an error later in the multiplication process chain.
(204) One optimization tweak occurs when no digit in the multiplier exceeds five. In such a case, the 6M value is never used, and so never needs to be added onto the 632M-Board. Rather, it might make more sense to create a 4M or 5M value depending on which digit occurs with greater frequency in the multiplier. In fact, if the multiplier is a long digit sequence, creating every pertinent M-value above 3M on-the-fly makes sense because each in-situ added NI-value to a just-in-time 98765432M-Board involves no burdensome overhead, and reduces every step to one rote addition.
(205) Optimization of the Multiplier row includes four tweaks called 10-1, 10-2, 10-3 and 10-6. These substitute for multiplier digit values 9, 8, 7 and 4 respectively, and only make sense when the multiplier two-digit pattern reduces the overall number of steps. This occurs for x9, x8, x7 and x4 patterns where x=1, 2, 5 and 9. For x8, x7 and x4, four steps reduce to two steps, and in the case of the x9 pattern one step. For example, 99, 59, 29 or 19 are the two digit patterns that make the 10-1 tweak optimal.
(206) With x9 pattern, the 10-1 tweak proceeds as follows. Add one bead to the Digit-Square left of the [Multiplier] guide chip. For x=1, 2, 5, and 9, the x-value 1 becomes 2, 2 becomes 3, 5 becomes 6 and 9 becomes TEN, respectively. Next, 1M is placed in the top row and subtraction is performed. In the 99 scenario, TEN0 is generated, the only operation is to ripple the Packaging operation up the Multiplier row.
(207) With the x8 pattern, the 10-2 tweak mimics the 10-1 tweak except 2M is placed in the top row and a 2M subtraction is done.
(208) With the x7 pattern, the 10-3 tweak mimics the 10-1 tweak except 3M is placed in the top row and a 3M subtraction is done.
(209) With the x4 pattern, the 10-6 tweak mimics the 10-1 tweak except 6M is placed in the top row and a 6M subtraction is done.
(210) These optimal two-digit patterns occur 16% of the time and reduce the average operations per multiplier digit from 1.4 to 1.2. However, at the start of game play, if the rightmost Multiplier Digit-Square shows 9 8, 7 or 4, as will happen 40% of the time, then zero out the rightmost Multiplier Digit-Square, duplicate the 1M value into the bottom row but shifted left one rank, namely multiply by ten, and proceed with the 1M, 2M, 3M or 6M subtraction pertinent to 9 8, 7 or 4, respectively. This execution sequence avoids generating a negative number at the start. Tweaks add speed but can make for errors. Every game is end-capped by the use of plosive-state equilibration if normalizing the final value into canonical form is required
(211) Lesson #14, Game of 632M Division
(212) The Game of 632M Division uses a two-plus-one row Candy Board with Trays including guides chips and the 632M-Board positioned, as depicted in
(213) The [Quotient] guide chip prevents focus-rank errors. The [Dividend] and [] guide chips aid the digit comparison and M-value duplication processes. These guide chips shift right as division unfolds. The 632M-Board is setup based on the Divisor value, in this example 1M is 137, as depicted in
(214) The solution for 51,077 divided by 137 is depicted in
(215) The Game of 632M Division executes as follows.
(216) Step (A): Zero out the Quotient in the row above the Tray. Setup the Dividend in the top row below the Tray, and zero out the bottom row. Setup the four 632M Divisor M-values on the 632M-Board, along with the S-values, depicted as chips. Align the rightmost rank of the 632M-Board with the rank of the uppermost digit in the Dividend on the Candy Board. This is the initial focus rank, so set the [Quotient] guide chip one rank left of the focus rank. Rank-align the [Dividend] and [] guide chips with the 6, 3, 2, 1 S-value chips on the 632M-Board.
(217) Step (B): 65/6M Step. Visually inspect the Dividend Digit-Square sequence with the 6M value on the 632M-Board. If the Dividend Digit-Square bead pattern is greater than or equal to the 6M bead pattern then slide 6 beads into the Quotient Digit-Square to the right of the [Quotient] guide chip and duplicate the 6M bead pattern on the bottom row, as depicted in
(218) Step (C): 3S/3M Step. Visually inspect the Dividend Digit-Square sequence with the 3M value on the 632M-Board. If the Dividend Digit-Square bead pattern is greater than or equal to the 3M bead pattern then slide 3 beads into the Quotient Digit-Square to the right of the [Quotient] guide chip and duplicate the 3M bead pattern on the bottom row, as depicted in
(219) Step (D): 2S/2M Step. Visually inspect the Dividend Digit-Square sequence with the 2M value on the 632M-Board. If the Dividend Digit-Square bead pattern is greater than or equal to the 2M bead pattern then slide 2 beads into the Quotient Digit-Square to the right of the [Quotient] guide chip and duplicate the 2M bead pattern on the bottom row, as depicted in
(220) Step (E): 1S/1M Step. Visually inspect the Dividend. Digit-Square sequence with the 1M value on the 632M-Board. If the Dividend Digit-Square bead pattern is greater than or equal to the 1M bead pattern then slide 1 bead into the Quotient Digit-Square to the right of the [Quotient] guide chip and duplicate the 1M bead pattern on the bottom row, as depicted in
(221) Step (F): Alignment. Assuming the above 632M compare and subtract steps (B) through (E) have been done, rather than Step (A), at this point the Dividend value should be less than 1M. If it is not, what has happened is a comparison was botched. For example, 2M was possible but not done. In this event, alignment, as below, will immediately bounce the process back to Step (B), the hardwired 632M compare and subtract cascade, namely Steps (B) through (E). Expect the TEN pattern to occur as the Quotient Digit-Square is filled with more beads, and hence, a Packaging operation on the Quotient row is triggered. Otherwise, an Alignment While-loop executes as follows. While Dividend Digit-Square bead pattern is less than the 1M bead pattern on the 632M-Board, shift the focus rank, all guide chips, and the 632M-Board one rank to the right. If right edge of the Candy Board is left of the right edge of the 632M-Board, then division is complete, so reset the guide chips, as depicted in
(222) Step (G): Precision step. If a numerical precision is specified and the partial dividend is greater than or equal to that required precision, then go to Step (B). Otherwise, division is complete.
(223) Once the game is complete, as depicted in
(224) Once the child understands division, the parent should show how the remainder in whole numbers can be further processed into its decimal fraction form. This is accomplished by re-initializing the division process as follows. Move the remainder value all the way up to the leftmost rank of the top row and restart the division process. Being sure to account for zero formation right of the decimal point, i.e. for small remainders like 2/2385. The new Quotient contains the decimal fraction right of the decimal point. For example, based on the problem depicted in
(225) Storylines performed on the Candy Board make arithmetic a living art, not merely a rote do-the-process discipline. For example, stories can insert modifications on-the-fly. Consider the church Christmas pantry division story of Lesson #11. Suddenly, Mrs. Michaels calls out Stop the division. I have found eight bags of candy in the Deacon's Office. They need to be added to the inventory. The division process is currently at the Bag-Rank, so the extra 800 can be added-in, and division resumes without a hitch. No student ever encounters such dynamic, brain-training storyline intrusions in a formal mathematics class tackling static exercises in division.
(226) Lesson #15, Game of Magic Twins
(227) The Candy Board can solve problems using iterative improvement. Mastering division, the child has already used iterative improvement because the Remainder termination condition stops the iterative process to whatever level of decimation is desired.
(228) The Game of Magic Twins introduces children to the overt process of iterative improvement for solving square root mysteries. and requires children to understand decimal fractions because solving the Magic Twin problem will throw numbers off to the right of the decimal point. Each iterative step yields a Magic-Twin correction value which gets smaller and smaller until a termination precision is reached.
(229) Consider the storyline. For Uncle Fred's Birthday dinner, Mommy wants to put two ribbons from one corner of the dining room table to the opposite corner to create an X pattern. The table is 8 feet by 4 feet. Mr. Timpkins sells ribbon on a standard roll in 10-foot lengths for $32.50, but he also sells cut-to-length ribbon at $3.50 per foot. What is the length of ribbon Mommy needs to do the job? Can Mommy save money if she can avoid buying two 10-foot rolls of ribbon? If so, how much money? Katie's job is to help Mommy save money so she can spend the savings on candy instead.
(230) In the storyline, Mr. Pythagoras tells Katie that the length of ribbon needed is a Magic Twin and she needs to multiply 8 by 8 and add this value to another multiple 4 times 4 to generate the target value, namely 80, so the Game of Magic Twins can begin.
(231) The Game of Magic Twins uses a three-row Candy Board with the number of ranks two greater than the required precision. For example, if there are six Digit-Square ranks, and the square root of 26 is required, the initial guess will be 050000 and the process continues until 052915 is produced in the top row, namely the square root is estimated as 5.2915.
(232) The top row holds the Multiplier. The Multiplicand and Multiplier are always equal because they are Magic Twins. The middle and bottom rows hold a variety of intermediate values.
(233) All square roots can be mapped into the range 2-100 because all numbers equal or exceeding 100 are a factor of ten of some number in the range 2-100. The child should be shown 0 and 1 are the Magic Twin of themselves, and consequently, all target values in the range 0 to 2 should be scaled up by factors of ten until they exceed 2. Thereafter, the Magic Twins process solves all target values in the range 2 through 100.
(234) The process uses divide by 2, doubling and shifting to calculate correction values. Worst case, it produces four significant decimal places within four iterations. A rule of thumb is used to establish the initial guess of the square root value. For generating the correction factor another rule of thumb applies. When the target value is less than 10, a correction divisor 5 is used. For 10 through 50, a correction divisor of 10 is used. And, if above 50, a correction divisor of 20 is used.
(235)
(236) The Game of Magic Twins Executes as Follows.
(237) Step (A): Setup initial guess. A rule of thumb establishes an initial estimate of the square root. If target value is less than 5 then 2. If less than 10 then 3. If less than 20 then 4. If less than 30 then 5. If less than 40 then 6. If less than 60 then 7. If less than 80 then 8. Otherwise, 9. Setup the initial estimate in the top row, one rank right of the leftmost Digit-Square, as depicted in
(238) Step (B): Setup the 632M-Board using the value in the top row, as depicted in
(239) Step (C): Multiply the top row with itself using the M-values on the 632M-Board forming the Product in the bottom row, as depicted in
(240) Step (D): Duplicate the 1M value from the 632M-Board in the top row. Duplicate the target value, namely 80 in this example, into the middle row so it aligns with the uppermost digit of the value in the bottom row, as depicted in
(241) Step (E): if said residual value appears in the bottom row, place a [] guide chip in the leftmost Digit-Square of the middle row, as depicted in
(242) Step (F): If the target value is greater than 50, use the Game of Sharing division by two method, so that said residual value is split half in the middle row and half in the bottom row, as depicted in
(243) Step (G): The iteration correction value is generated by shifting all middle row of beads one rank to the right, namely divide by 10, as depicted in
(244)
(245) Thereafter, the Uncle Fred's Birthday storyline is concluded using skills mastered in prior lessons.