GB2111268A - Abacus - Google Patents

Abstract

A board has spaced vertical rods screw threaded therein to accommodate nut or disc shaped counters the rods and counters being generally incremented in size from the right to the left. The sets of counters are of differing colours, each rod being of the corresponding colour to its associated counters to a height equal to the base being used e.g. each rod may be coloured to a height of ten counters and be long enough to accommodate 30 counters. As shown decimal point marked blocks and a division board may be provided. A further array of rods on a smaller board can be placed below the main array to enable long addition, multiplication and division to be carried out. <IMAGE>

Description

SPECIFICATION Number concepts board Introduction As outlined in the claim this invention goes beyond the scope of conventional abacus' to allow the accumulation of any amount of counters in any place value position in any number base system. The full range of arithmetical calculations can be performed, and physically displayed, fully in any number base as the calculation progresses to the solution. The entire conceptual development and structure of any number base place value system can be made manifest using the invention. The invention is designed as a remedial, re-teaching or initial teaching device leading to standard numerical written calculations. It can also be used purely as a numerical calculating device and as an aid to initial computer studies involving varied number base systems.To be used effectively a teacher requires a separate teaching booklet that is being written separately. At Annex A is the photographic record.

The invention is based upon a rectangular piece of white faced chipboard as pictured in Figure 1 of Annex A. Any similar laminated board would serve provided the surface could be written upon with wipe clean spirit based and water based pens.

A series of dowel rods which can be of wood, metal or plastic material are screw-threaded and fitted into screw-threaded rnceptj9ies in the board in the form of the array shown in Figure 1 , Annex A.

One set of thirty nut or disc shaped counters is provided for each column of rods and each column counter set is slightly incremented in size. The smallest size is used on the right hand column of rods and the incremented sizes are used on successive columns of rods, which are similarly incremented in diameter and height where necessary to match the counters, to the left; each successive column taking a slightly larger set of counters. See Figure 2 of Annex A.

Each column of rods is painted a different colour at the base to a height equal to 10 stacked counters of the type designed to go over a particular rod. This is illustrated at Figure 3, Annex A. All the counters designed to go over any particular rod are painted the same colour as that rod.

A further 2 dowel rods are inserted into the board on the extreme left of the columns mentioned in para 3 and beside the 3rd row from the top. The right hand of these is cut off at a height equal to the height of 9 of the smallest counters and the other of the height of the next 9 smallest counters. These are used to take divisors. This quantity of 2 dowel rods can be increased in theory to any limit as can the quantity of associated counters.

The height of each dowel rod is sufficient to take 30 counters.

Rods of heights sufficient to take 100 counters can be substituted for those mentioned in para 3. In theory there is no height limit.

Several separate arrays of rods can be substituted for those mentioned in paras 2 and 6. The following requirements are examples of different number base systems: Coloured to a height Number base of x counters Normal height of rod 2 x = 2 10 counters height 3 x= 3 12 4 x= 4 16 5 x= 5 20 8 x= 8 32 12 x=12 48 ,, " 20 x=20 60 ,, 60 x=60 120 The amount of counters required varies dependent upon the height of the rods. In theory any number base system can be used on the board.

A facility is given on the board to copy in conventional numerals the number and place value of the counters on the rods. This can be seen in Figure 4 of Annex A. Arrows are used to show the progression of a calculation.

The following are also printed on the board:

a. BASE D Where the base being used is written-in the blank space; illustrated here by the variable y.

b. LIMIT Y-1 =z This shows the maximum number of (or limit) of counters permitted in any place value position after a calculation has been completed; y is the base and z the limit.

During calculations it is often necessary to exceed this limit to give a physical illustration of the processes involved. -- c. The 4 arithmetic symbols are printed on the board; 3 being covered to show which operation is being used. See Figure 4, Annex A.

Decimal points can either be written directly on to the board or indicated by decimal point marked blocks. See Figure 5, Annex A.

A cover is provided to limit the visual display of rods when counting, dividing or rounding off. A division board is also provided. See Figure 6 and 7 of Annex A which also shows a calculation in progress.

The size of the board is fairly arbitrary. Two basic sizes are most commonly required: a. Instructional Number Concepts Board 1.4 x .6 metres.

b. Student Number Concepts Board .36 x .24 metres.

A counter holder is provided. See Figure 8, Annex A.

The array of rods described in paras 3 and 1 6 is shown in Annex A to be limited to 5 columns.

separate smaller board. This is placed below the main array when long division, long multiplication, and addition sums involving more than 2 numbers are calculated on the board. In theory there is no limit to the number of additional rows of rods (or counters).

1 7. The array of rods described in paras 3 and 16 is shown in Annex A to be limited to 5 columns This enables calculations to be performed within the limit of place values 1 or to 1 or+4 where 1 or is the place value of the extreme right hand rod and r is any positive or negative integer. The number of columns of rods can be extended to meet the requirements of calculations involving a greater number of place value positions. In theory there is no limit to the number of columns of rods and their equivalent place values or in the associated number of counters.

Claims (1)

1. CLAIM

   The invention is a type of abacus but differs from the conventional Chinese (Suan Pan) and Japanese (Soroban) versions in that firstly the place value rods are open ended and thus allow the accumulation of any number of counters in any place value position, and secondly that the discrete operations in any arithmetical calculation are physically and continuously displayed as the arithmetical calculation progresses to the solution. Unlike the conventional abacus there are no mental processes that are not replaced by a physical process. Thirdly it also incorporates a decimal point (or points when more than one number is involved) that can be placed in any position and thereby enables any arithmetic calculations to be performed over any theoretical range of place values and fourthly with any theoretical number base value.In general the range and bases are: Range Ar+i (i=o), Ar+i (i=1), Ar+i Ar+imax where A is the base of the number system being used and A can be any positive integer greater than unity Ar+ is the place value of the extreme right hand rod.

   r is any positive or negative integer (includes zero).

   i is incremented from its initial 0 value successively by +1 to indicate the place value of each successive rod from the Ar+o place value to the Ar+i max place value position.

   Ar+i is the place value of any particular rod.

   Ar+imax is the extreme left hand rod, and max+l is the number of rods.

   a. As claimed in Claim 1 the invention can be used to physically perform the following calculations in any base.

   (1) One to one correspondence ie counting.

   (2) Addition.

   (3) Subtraction.

   (4) Multiplication.

   (5) Division.

   (6) Rounding off ie approximating numbers to the limit of any place value position.

   (7) Any combination of the above sub-paras 1 a (1 ) to 1 a (b) in problems falling within conventional arithmetic e.g. percentage calculations.

   b. As claimed in Claim 1 the invention makes physically manifest the following arithmetic and mathematical concepts: (1) One to one correspondence.

   (2) Place value in any base and change of place value and number as being dependent upon place value position.

   (3) Unity as the whole.

   (4) Unity as the basic counting unit.

   (5) Fractions as being less than unity.

   (6) Integers greater than 1 being both accumulations of unity and greater than unity.

   (7) Decimal fractions.

   (8) The function, position and role of the decimal point.

   (9) Displacement of numbers to higher or lower place values when multiplied by numbers of the general form AX where A is the base and x is any positive or negative integer.

   (10) The purpose of the zero ie the indication of empty place value.

   (1 1 ) The magnitude of any number.

   (12) The infinite and infinitesimal (oo, -cc).