w %^y^ IMAGE EVALUATION TEST TARGET (MT-3) !.0 I.I 1.25 1^ 1^ 1 2.2 t 1^ 12.0 1= U III 1.6 ^^ e /a /: :> > /a V CIHM/ICMH Microfiche Series. CIHM/ICMH Collection de microfiches. Canadian Institute for Historical Microreproductions Institut Canadian de microreproductions historiques 1980 Technical Notes / Notes techniques The Institute has attempted to obtain the best original copy available for filming. Physical features of this copy which may alter any of the images in the reproduction are checked bblow. D D V Coloured covftrs/ Couvertures de couleur Coloured maps/ Cartes gdographiques en couleur Pages discoloured, stained or foxed/ Pages ddcolor^es, tachetdes ou piqudes Tight binding (may cause shadows or distortion along interior marcj;*^)/ Reliure serr6 (peut causer de I'ombre ou de la distortion le long de la marge int^rieure) L'Institut a microfilm^ le meilleur exemplaire qu'il lui a 6t6 possible de se procurer. 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The following diagrams illustrate the method: Les cartes ou les planches trop grandes pour dtre reproduites en un seul cliche sont filmdes d partir de Tangle sup6rieure gauche, de gauche d droite et de haut en bas, en prenant le nombre d'images n^cessaire. Le diagramme suivant illustre la mdthode : 1 2 3 1 2 3 4 5 6 CUTHBERT'S 7 PRIMARY NUMBER-WORK; AND COMPANION TO THE (;OMMOX-SENSE ARITHMETICAL CALCULATOR. BY W. X. CUTHBERT, TORONTO. (( Seeing is Beliemng." TORONTO: THE COPP, CLARK COMPANY, LIMITED. I89a, 1 i ,1 ! Entered according to Act of the Parliament of Canada, in the year one thousand eight hundred and ninetysix, by T.ik Copp, Clark Companv, Limitrd. Toronto Ontario, in the Office of the Minister of Agriculture. PREFACE. land nto, This book is published as a Companion to the Common-Sense Arith- metical Calculator, an invention in the form of a Calculator of new design (a full description of which is given — with illustrations— in the introduovury part of the book). The merits of the Calculator are manifold : (1) As an adjunct in the primary department of a Public School it forms an invaluable aid in teaching in a rational loay the first lessons in Arithmetic ; (2) It makes good use of Number-Forms {" Picture- X umbers") in representing Numbers, that simple, but admirable means of developing in the minds of children an appreciation of number ; (3) By giving a clear percep- tion of Numbers, it develops the observing and reasoning faculties, culti- vates the memory, and sharpens the intellect ; (4) It presents Number on a sound psychological basis, and thus laj s the foundation of a good Mathematical Education by giving the child a proper conception of Number ; the arbitrary signs, by which numbers are known, are made intelligent to children, by a comparison with real objects. Believing that the Calculator will prove of practical value in the school-room, and a boon to teachers, and especially so to those engaged in rural schools, I have been induced to place both the Calculator and the accompanying book of explanation in the hands of the publishers. I am indebted to Dr. McLellan, Principal of the Ontario School of Pedagogy, and to Wm. Scott, Esq., M.A., Vice-Principal of the Toronto Normal School, for many suggestions during the progress of the book. The book is designed for Teachers using the Calculator, and is not intended as; a book to be placed in the hands of the pupils. W. NELSON CUTHBERT. Toronto, April 13th, 1896. [iii] n ill i' I M < ii .1' ' III' 1 lit ■ ! I INTRODUCTION. Tills little book, as stateil in the preface, is iiiteudecl as a Companion to the Common-Sense Arithmetical ('alenlator, an invention consisting of two series of eleven rectanguhir number-tablets, each (on which hemi- spheres or half-balls are glued) placed one below the other at the top of the framework. Immediately l)eneath these number-tablets a board is reversibly mounted so as to revolve vertically, thus permitting of either side being turned, at pleasure, towards the class. To the revolving board are aliixed several appliances and devices to be used as an aid in the p/tyc/ioluytcal teaching of Number. The whole is enclosed in a portable frame. The devices shown in Kigur«s A and B nuiy be briefly described as follows : — (1) 'I'he Fundamental Number- Forms (Fig. A') from one to ten, as represented by dots in suitable symmetrical groupintj, each group being adjacent to the arbitrary sign or symbol by which it is known, thus giving the child the idea of the number which the sign represents ; (2) A Small (Calculator ( Fig. A^) consisting of ten balls (movable) on wires for teaching in Synthetic' Analytic N anther- Form (by objects other than dots) the numbers up to ten ; (3) An Automatic Numeral-Frame (Fig. B^) wherein all the Number-Forms from zbro to ten are arranged on rect- angular number-tablets (by hemispheres or half-balls arranged in symmetrical grouping and iixed on these tablets) in two horizontal rows, the lower one of which is movable between two grooved pieces of wood so as to admit (by moving the said row — the lower series of eleven number-tablets— of Number-Forms successively in the same direction) ' See top of Ualoulator (" Pioture-.Nuinbers"). •■' See top of Calculator (Tableta). h-3 ■ See lop (jf Calculator (Ten Ualls). VI INTKODUCTIOX. f : of the formation in numher-forin of Ai.i, iirK cumhina-TIons which can poawihly ])o mado in Addition with anij two of the said nunibcr-forina, thus forming in number-form, all t/o' additton comblnntions on Numbers from ONK to rwKNTV ; (4) A correspondent Automatic Mechanical Schedule (Fig. B^) wherein all tlie number-symbols from Zero to Ten are arranged in two horizontal rows, the lower one of which is movable in a rabbeted groove so as to admit (by moving the said row of Number- Symbols successively in the same direction) of the formation, in symbol- form, (jf the said combinations ; (o) A Larger Calculator (Fig. B^) con- sisting of twenty balls (movable) on wires, for illustrating, in symmetrical number-form, the numbers to twenty, together with their comhlnaiioiiH and separations (additions and sahtractions). All the combinations possible on the Numeral- Frame and Schedule maybe performed upon this larger calculator. Both calculators, having movable balls, have been placed lowest on the calculator for the convenience and benefit of the children who should be required to go to the calculator and form for themselves any of the said numbers and the combinations thereof ; (6) A movable circular disc ( Fig. B^) of flexible material, cut in such a manner that it may be readily transformed into a rectangle, for the purpose of teaching, by ocular demonstration, the area of the circle. The circle of course does not belong to primary work ; but the disc has been added to the appliances for the proper explaining of more advanced work involving mensuration. In entering upon school-life, all children have some idea of Numbers ; some, however, have the sounds (words) that stand for numbers without any real conception of the numbers themselves. It is, therefore, necessary to teach; thoroughly, the numbers by means of objects of some kind, as clear ideas of Numbers should be given. The Common-Sense Calculator will be found to furnish a means by which this may be successfully accomplished. / ( ' See bottom of Calculator (Small Tablets). '- See bottom of Calculator (Twenty Balls). •■' See bottom of Calculator (Circle). OUT I iHi\irr's )ers ; Lhout [fore, 30ine ^ense be k-enty I Common-sense Arithmetical Calculator FlQ. A. Fig. A represents one face of the Conniion -Sense Arithmetical Calculator, and Fig. B represents the reverse face. [vii] OUTHBKRT! !'; ; I ,■! i H Common-sense Arithmetical Calculator FlQ. B, TOR BACK y,lEW OF CALCULATOR. X^l^ [ix] Il, '1' v\\\ 'III II I i i i' i OOMF^ANION TO The Common-Sense Arithmetical Calculator The different devices and appliances found on the Calculator (as sho^vn in Fi.nires A and B) will be referred to under the following names ri. The Fundamental Number-Forms. Fkj. a. J 2. The Smaller Calculator or Ball-Frame. [3. Tlie Automatic Numeral-Frame. Fig. B. '.">. The Automatic Numeral-Frame. 4. The Automatic Mechanical Schedule. 5. The Larger Calculator or Ball-Frame. 6. The Circle. [xl M I tor or (as owing i I EXPLANATION OF THE SEVERAL DEVICES FOUND Ul'ON THE Common-Sense Arithmetical Calculator, AND THE METHOD PURSUED IN USING THEM. 1 • • • 3 • • • 4 • • • • 5 ••• • • • 6 • • • • • • r • • 6 • • • • • • • • 9 ••• • • • • • • • /O ••• • • • ••• • • The above cut represents the Fnndamentnt Number- Forms from One to Ten arrau<^ed in syniiuotrical grouping by dots painted on a slated back -ground, so as to admit of partition by a chalk-line which tnay be erased at pleasure, and also the Arabic Number- Syriiboh which represent these Number-Forms. In order to he clear and to be 'taken in at siijlit," these intuitions must be symmetrically arranged. They must be repeatedly brought before the mind until the instant the "Picture" or Number-Ftn'm comes up in the mind, so does the word (and symbol) and vice versa. These Fundamental Number-Forms and the sixfu-six comhiaations on Nuuibers from One to Twenty, as formed by the Automatic Numeral-Frame, are all-important, as they lay the ForNDATiON for the subse(|uent study of Arithmetic. The Fundamental Numbers, and the Combinations on Numbers to Twenty may justly bo called the Foundation of Mathematics. Each of the Fundamental Numbers is hero represented (in Num- ber-Form) as a whole, and the pupils get a symmetrical picture of [1] 2 PRIMARY Nl MHER-WORK. * the Numbor. The pupils soo also the relatUm of the . \ C If d Ik /me acGi ^V'^' Have some pupil in the class go to the Larger Calculator and move out upon the wires the lowest three balls of the upper ten on calculator, — these balls when moved out will be found to be immediately above the cross-bar of the calculator which equally divides the space (back-ground) behind the calculator, and which corresponds, here, to the chalk line on the floor, — so that the three balls occupy positions on the calculator similar to the positions occupied by the pupils upon the floor, thus : — Tub Larger Calculator. Now, of the three pupils, have the one standing farthest away from the chalk line, cross over the line and take up a position upon tho other side of the line, thus : — AllITHMETICAL CALCULATOR. V I 1 I 4 Have the pupil at the calculator malfe the corresponding change there, by moving, to one side, the upper ball in tlie group of three balls (^*^) and then moving out one of the two balls, on the top wire in the lower ten, so that it shall occupy a position immediately below the said cross-])ar, thus : — >n Have the pupils in the class express orally what they have observed: — **Two AND One aue Thkee." (2 + 1 = o and One are Three : • :• Three andNclkinqdre Three 5 ■>■ =5 The teacher draws, on the black-board, the skeleton scheme and the pupils copy the skeleton and " till it up" ns above indicated ; or better, have the pupils get the "Primary Number- Work" Exercise Book which accompanies ti:is book. Before beginning the exercise on the number Three, by placing the three pupils on the floor, it might be well for the teacher to indicate (by lightly-drawn chalk lines and crosses), on the floor, the way in which the pupils nmst move in order to give the two combin- ations of Three. This will save the teacher placing the pupils at each move : — FLOOR- PLAN. X X'C I^^Tlie X 's represent the stations the pupils assume, and the arrows and dotted lines tlie several courses taken in making the qh«nge9, • AUITUAIETICAL CALCULATOR. 9 :crci90 nviug, 1 con- l, and Wo must have tho conscious *'rec(*gnii:ioii of Three (%*) '"vs one WHOLE (unity), and of the three tliin<^s as intlividuals and of the unity as made up of the three things, l)y cnmparingy relaihuj, mea- suring, counting, etc." All the Fundamental Nttmbers should bo dealt with in a similar way. Take the number Seven, for instance, nnu by pur^suiiig the method as tmtlined, the rehdion of the secend eomhinat ioiiH (pairs of adiUiuis or related parts) which make up that number may be clearly brought before the minds of tiiO pupils : — (a) (b} (e) (d) #• •• •• #^# c» me and ted ; or xercise placing Icher to |)or, the •omhin- \upils at Imd the ng the («) 3 + 4 = 7 (/>) 2 + 5 - 7 (c) 1 + G - 7 (d)0 + 7 =--7 These are all tlie pairs of addends which make up the number Seven. The strokes seen represent chalk lines which may bo erased when no longer needed. Use Number-Form Seven, Funda- mental Numbers on Calculator (Fig. A). The same ideas may be brought clearly before the pupils' minds by using the Smaller Calculator (Fig. A) with the ten balls. It is intended as a means of illustrating all the Fundamental Numbers. Fig. a 10 PRIMARY NUMHKH-WORK. iili'l •II 1 , The abovo cut ropresonts the Smaller Calculator, which consists of six wires on which move ten balls of various colours, the whole being fixed in a ball-frame, as seen in Fig. A. The significance of the numbers from One to Ten, and the SYMBOLS (fkjuues) used to represent them may be taught, ohject- ivelyj by means of the balls above referred to. "Objects (and measured things) aid the mind in its work of constructing numerical ideas." For the sake of variety the numbers three and seven, already referred to, may be more clearly presented to the minds of the pupils by using the Smaller Ball-Frame, as follows : — I I I The above is the Smaller Ball-Frame with the number three represented upon it, by moving out three balls upon the wires as indicated. AKITIIMETICAL CALCULATOR. 11 insists of le whole and the t, ohject- cts (and iimerical already 3 of the jr three (v^ires as ANALYTIC-SYNTHETIC PROCESS. (Analysis and S3nithesis of the Number Three.) ADDITIONS. (a) ^-<^ 3 + = 3 + 3 = 3 2 + 1 = 3 1 + 2 = 3 Move out upon the wires three, halls as indicated in (a) above ; then place the edge of the card-board, held in the hand, immediately 12 PHIMARY NnMHKIt-WO») above ; and, placing the card-board between the two balls and the o»ie ball as indicated (the line drawn between the wires indicating the position of the card-board), and turning it so as to hide the one ball and reveal the two balls, ask the pupils, as before, how many balls they see. They answer : — "Two Balls." Then, turning the card-board the opposite way so as to hide the two balls, and reveal the one ball, ask che pupils how many balls they see now. They answer : — "One Ball." Now, have the pupils make an oral statement, as before, of what they saw as you turned the card-board, revealing and hiding the parts of the number, alternately : — "Two balls and one ball are three bails" (2-f 1 = 3), and vice versa : — "One ball and two balls are three balls" (1 + 2 = 3). This may be illustrated also by moving the balls upon the wires, as follows : — Iio wires so as to liey soo. hide the f. Tliuy AKITIIMK'I'K Al, t ALCHLATOK. 18 the card- ree (three = 3); and 3=3). ted in (h) s and the ndicating le the one low many the two they see ^hat they parts of [ee bails " \ee balls " le wires, (a) + 3-3 3+0 = 3 (6) 1 + 2 = 3 2 + 1 = 3 Having no balls out upon the wires, ask the pupils how many balls are out upon the wires. They answer : — "No Balls." Then, move out, together, three balls, as indicated in (a) above ; '310W, ask the pupils how i?iany balls are out upon tlie wires. They '^mswer : — i "Three Balls." [ow, have them make the oral statement, as you move the balls : "No Balls and Three Bali^ are Three Balls." il i^! 14 PRIMARY NUMBER-WORK. tn i|! f'' I '■ ; Leave the balls as fchey siaiid, and ask the pupils how many you must add to the three balls to make three l)alls. They answer : — "No Balls." From this have them make the statement : "Three Balls and No Balls are Three Balls." Again, move out upon the wires one ball as indicated in (6), and ask the pupils how many balls you moved out upon the wires. They answer : ♦'OxNE Ball." Then, move out two more balls on another wire as indicated, and ask the pupils how many you m.oved out the second time. They answer : "Two Balls." The three balls will then occupy such positions as to indicate the two parts of three, that is, one and two (• and • •). Now, move the one ball to the left, from the position into which it had previously been moved, so as to occupy the position in which they are in (b). Have the pupils make the statement : — "One Ball and Two Balls are Three Balls." Now, move the balls back to the left side of the ball-frame, as indi- cated by the rings in (6), so as to occupy their original position, and then reverse the operation, by moving out the two balls first, etc. The statement will then be : — "Two Balls and One Ball are Three Balls." SUBTRACTION. (a) t< ; ti ii 3-3=0 3-0 = 3 I I many you swer : — 1 AKITHMKTI(!AL (CALCULATOR. IT) LLS." , and ask the tiey answer : ited, and ask 'hey answer : i icate the two »w, move the I id previously ,ey are in (6)- lLLS." fame, as indi- position, and alls first, etc. LLS." .:i if>) ^ ^ \ 1 99 .....Y' 1 1 ^ g j 1 00 I oo 1 5 s. 1 3-2 = 1 3-1 = 2 Move out upon the wires three balls, as in («), so "s to form the number three ; then ask the pupils hov/ many balls are out on the wires. They answer : — ' ' Three Balls. ' Now, move the balls back to their former positions as indicated by the rings in (a), and the pupils will see that: — " Three balls from three balls leaves no balls," (3-3=0) ; also, by moving the balls back into their former positions, so as to form the number three, again [see (a)], and leaving them as they stand upon the wires, it may be shown that : — " No balls from three balls leaves three balls," (3-0 = 3). Again, move out upon the wires three balls, so as to form the number three, as in (/>) ; now, ask the pupils how many balls are out upon the wires. They answer : — "Three Balls." Now, move two of the balls back to the places indicated by the rings [see (6)], and the pupils will see that : *' Two balls from three balls leaves one ball," (3 - 2 = 1). Now, move the balls back, so as to form the number three as before ; then, move away to the left the lower ball forming the number three, so as to occupy the place indicated by the ring, and the pupils will seo that : — " One from three loaves two," (3-1 = 2). I 16 PUIMAUV NUMBER- WOKK. I 1 Thoso snhtrdctions may ho porforiiUMl, also, l)y using the card- board to partitloii off the parLs U)V subtracting. The ADDITIONS and subtractions slumld bo taught together, as the two o[)erations go hand in hand ; because they are inverse operations. Tliis analysis and synthesis, or " partitum and re-comhination " of ni lubcrs is of vit) :— 6 + 1 = 7; 1 + 6 = 7; 7 - 6 = 1, and 7 - 1 = 6 ; therefore we have the following : — Seven. Additions. 7+0=7 + 1=7 6 + 2 = 7 4 + 3 = 7 3 + 4 = 7 2 + 5 = 7 1 + 6 = 7 0+7 = 7 i:)ubt mictions. 7-7 = 7-6 = 1 7-5 = 2 7-4 = 3 7-3=4 7-2=5 7-1 = 6 7-0=7 In teaching the Subtractions, put out upon the wires the requisite number of balls to represent the number under consideration, and, by sliding aw.ay one or more, as the case may require, the sul)trac- tions may be taught ; however, instead of sliding away the balls, a piece of card-board may be held in the hand (as described) and a part of the number (that is one of tlie pair of related addends) may be covered up by placing the card-board between the related parts of the number, e.g., • • (seven), so as to hide, first the • • (2), • • and then the 5*# (^) 5 "^ ^^^^ ^^^7 ^^^^ pupils will readily see that $•0 and • • make seven ; have them tell you orally what they see as you change the position of the card-board to form the combina- tions, as " Tiro ami five ure severi" ; " Five and two are seven" etc. ; or eto, ■^^ Three and four are seven , and " Four and three are aevetit' TJ.I 1,1 '1 1 1 ' ' I 1 ' I ' Ijllj 20 PlUMAliY NUMHEll-WOIlK. Thus it will bo soon tliab, in thr. Addition Jind Subtract i<»n stjigos, there Jiro four tiiinos to h'tnn ahtnii every 2Mi>' *>/ ) 3 + 4 = 7 (c) 7 - 4 = 5 id) 7-3 = 4 Two Additions. ((a ) Four and three are seven, (h) Three and four ai-e sereii. I (<•) Four from seven leaves three. \^(d) Three from seven leaves four, Two Subtractions. Addition and Subtraction are inverse operations ; and hence should be taught together. * ' They have their origin in the opera- tion of counting with an unmeasured unit, with the idea of aggre- gation — of more or less." So also it will be seen that, in the Multiplication and Division stages, there are four thin(;s to learn about every pair of factors which compose a number; e.g.. Four and Three composing Twelve. (a) 4t X 3 = IS {b) 3x4 = /^' (c) 12 ^ i= 3 (d) 12 -f 3 = 4 Two (a) Four times three are twelve. (6) Three times four are twelve. Multiplications. I (g) Four into twelve three times. Xd) Three into twelve four times. Two Divisions. Multiplication and Division, being inverse operations, should go hand in hand together, and should follow the other two rules. This atmlysis and synthesis of a number gives a clear perception of that number. The Fundamental Numbers should be all taken up in logical ARITHMETICAI, CALCULATOR. 21 ids which Two tractions. and hence L the opera- 5a of aggre- tid Division ir of factors order and analyzed, aa outlined, care being taken fa hare each one thoroughly underdvod before the next is taken up. Pupils should have practice in counting })ackward and forward as soon as they have learned the Fundamental Numbers ; afterwards, when they have learned the symbols (figures) which represent numbers, and the numbers from One to Twenty, they should write out these numbers, and count backward and forward from One to Twenty. The Numbers from One to Twenty may })e divided into three divisions, as follows : — (1; Numbers from One to Five. — Numbers .the sum of a pair of which does not exceed ten. (2) Numbers from Six to Ten. — Numbers the sum of a pair of which exceeds ten. (3) Numbers from Eleven to Twenty. — Numbers formed by Ten (as a unit) and some Fundamental Number. The Numbers from One to Five and from Six to Ten may be taken up, by means of the Fundamental Number-Forms and the Smaller Ball-Frame ; then the Numbers from Eleven to Twenty may be taken up, by means of the Automatic Numeral-Frame (Fig. B), the Automatic Mechanical Schedule (Fig. B), and the Larger Calculator or Ball-Frame (Fig. B). For Figures, see p. viii. Two iDivisions. |s, should go rules. \)erce'ption of tv— Fio. B. (h 1 ■- 1 • • • • X • • • • X ♦ • • • • • • • • • • • • • • • • • GUTHBERTJ $ ARITHMETICAi. CALCULATOR ••• X • * *•• • • • • • • • •- • • • • • • • • • • • • • • I kU The a 1 'TOM AT 'IC NUM RRAL-F RAMB. M in logical 09 I'UIMAUY Nl'MUKU-WOUK. The above lepnjseiits tho Automatic Niuneral-Fraiiio whorein are arranguil (by liumisidieros or half-liuils in syimnotiical grouping) all tho Nntnhcr- Forms from Zkro to Tkn, in two horizontal rows. Those in tho lower row are fixed upon rectangular tal)lets movable between two grooved pieces of wood, between which, as these tablets are moved forward successively in the same direction, they may be made to assume different relative positions with regard to tho Ni(in})er-Fonns in the upper row which is stationary, being in one piece and spaced oil* into correspondent divisions with tho lower row of movable tablets. The pressing fx •-.« e « • •• « 9 « e « • e • « » O -7 « • « e « « « liW Fig. (a). Fig. (a) represents the Automatic Numeral-Frame with the Num- ber-Forms inserted therein, in the order in which they are used to form the Fird Set of the Series of Combinations possible to form with the Numeral-Frame, by mcjving the lower row of taljlets. The order in which the Number-Forms in the Numeral-Frame are inserted therem is as follows : One, Three, Five, Seven, Nine, Two, Four, Six, Eight, Ten, Zero. The Number-Forms, as thus arranged, form the First Set of the Series. The upper row remains stationary ; but the lower row^ is movable between two grooved pieces of wood, between which, as the frames are pressed forward successively, in the pame direction and in the order in which they are seen above, the Number-Forms may be made to assume different relative positions with regard to the Num- ber-Forms ill the ui)per row. All the different Combinations in Addition, which can possibly be made by any two of the Number- Forms, may thus be represented. In order to move the lower row of Number-Forms, the tablet on the extreme left-hand side is removed, by pushing it upward (by tlie thumb-catch at back of tablet,— see back view of Calculator, page ix. ) into the upper groove, so as to free it at the bottom, when it may be removed from the Numeral-Frame. The remainder of the lower row is then shifted to the left, and the tablet removed is then inserted, at the other end of the said row, so as to fill the vacancy caused by the moving forward of the said row, thus : — 24 rUIMAUY NUMHEK-WORK. A f1 ••• • « • * W0 a « « J • •! ••• CUTH BERTS ARITHMETICAL CALCULATOR ••• :•: • • • • x! ••• >i*\, Fig. (h). □^ Fig. (/)) represents the Numor;vl-Fraino witli the tablet, represent- ing the Nuinbor-Fonu "One" removed and ready to be inserted at the other end of the row. fi ••• I i :< • • .V. f\ CUTHBERTS ARITHMETICAL CALCULATOR IJ iitz • • dW Fig. (c). Fig. (c) represents the Numeral-Frame with the remainder of the lower series of tablets moved one place to the left, and the tablet representing the Number-Form "One," ready for insertion at the other end of the series of tablets. =fl 1 ^ dW ler of \d the lion at AUITHMETIC'AL CALCULATOR. 25 f\ • • ••• X • • • • I f\ GUTHBERTS ARITHMETICAL CALCULATOR ♦•• ••• • • # • ••• ••• Fio. (d). M Fig. (d) represents the Numeral-Frame with the remainder of the lower series of tablets moved one place to the left, and the tablet representing the Number-Form "One '' inserted at the other end of the series of tablets, thus forming the Second Set of the Series of Combinations. By removing the tablet containing the Number-Form "Three," as it stands in Fig. (d), shifting the remainder of the row and in- serting it at the other end of the series, the Third Set of the Series of Combinations may be formed, as follows :— 4\ 1 ^ 1 ^1 • ••• :< • • X • • X • • • • • • • • • • • • • • • • • • GUTHBERTS ARITHMETICAL CALCULATOR ;•: ••• • • ••• ••• • • • • •• • • • ♦ • • • • • • • • • • • • • • ■ • V ^ llL-> M FiQ. (e). Fig. (e) represents the TJiird Set of the Series of Combinations, 2G rUIMAUV KUMIiKk-VVOltK. foriuod hy tlio roinov.-il jiiul ro-insurtion of tho tahlot ropruacntiii tho Nuiul)ur-F(>rm Throo. Thu reinuiiiiiig Eight Sots comploting tho Sorios aro forniod in n similur way, hy roinoving tho tahlot to tho oxtromo loft in tho lowor row, shifting tho roinaindor of tho said row as hoforo, and in- sorting tlio romovod tablot at tho othor ond of tho row, in the place vacated by the shifting of tho remainder of tho lower row. The foUowing Figures will illustrate the remaining Eight Sets of tho Series. #A W^-\. V V V • » • • ••• fl CUTHBERTS ARITHMETICAL CALCULATOR ••• • • V :•: Fig. (/) represents the Fourth Set of the Series. [iW I I I fiiil ffV— (4^ 1 11 1 • 1 • • • • • « a ••• • « • • • a a a • a • • •a a 8 a a a « a a • a • a a a a a a • a a a • a a a CUTHBERTS ArITHMET IGAL CALCULATOR a • • •• • B • • S s fr a '» a • a • a • a a • • a » a a a a •a» •:•: a a • 1 \L tW FlQ. (j/X Fig. (7) represents the Fifth Set of the Series. I uiL .:»ffi"iii9mgr-?^ formed in ;i left in the j )i'o, and in- : n the place • vv. I ''ht Sots of ^ OR :•: \U R ••t « « U f^ i AUIIIIMETICAL CALCULATOU. 27 ff— ^.-^ 1 - -1 • • • • :< • • ••• • • • • • • • • — -1^ •• • • • • • « • • • • • • k GUTHBERTS ARITHMETICAL CALCULATOR • • • • • • • • • • 1 * * • • • • • • ••• • • — • V :•; ••• • • • •• • • • • 1 v.. M Fig. (h) represents the iSixAh Set of the Series. ffh (h 1 ' 111 • »^^ .•• :••. • • • • « • • • • • • • • • • • • • • • ••• • • CUTHBERTS ArITHMETICAI i CA LCULATOR • • • • • • • • • • • • • • • • • • • %• :•: • • • t I • • 1 ■■ , \L u Fia. (i). Fig. (i) represents the Seventh Set of the Series. ^ (^ 11 1 • .% • • • • V * • • * • • • • • • • • • • • * • • ••• • • Cuthberts $ Arithmetical calculator • t • • • • • • • • • ••• 1 * * V • • •• • •! • • • • B B^ M Fig. (j). Fig. (j) represents the Eighth Set of the Series. i!l!!|iiii ! it: i!i'!i'5ii'i'':'M li 1 liiil!! 28 PRIMARY NUMBER-WORK. A f^ 1 - . — II • 1 ••• • • 1 1 • • ••• « « • • • • • • • • • • • • • 1 • • • • • • • • • • ••• • • GUTHBERTS ARITHMETICAL CALCULATOR • • • • • • • • • ••• ••• •*• • • • • • <• • • • • • • 1 L « iW FiO. (t). Fig. (k) represents tl; Ninth Set of the Series. (fW- ^ 1 11 1 « ••• • • • •- • • • • • •- • « • « • • • • • • • • CUTHBERTS $ Arithmetical calculator • •♦• ••• • • • • • « • • <» '• • • • • • • • • • • • • 1 IW |(W Fig. (0. Fig. (I) represents the Tenth Set of the Series. 4^ • • « • • • • f] Guthberts Arithmetical calculator y •_• ••• ♦ • ••• ft • • • • ft ft ft ft • ft • • ft • ft • ft FiQ. (mX Fig. (w) represents the Eleventh Set of the Series. ■H t\ roR • • • • M f] OR R ;•: :•: AUITllMKTirAL TAI-CULATOU. 29 This completos tlic Sorios. iii;ikiii<,' in Jill Kleven clianges, and OnK HlJNDUEl) AND T\VKNTY-()NK COMBINATIONS. 1 I Of these Combinations, One Ilunclred and Twenty are significant, 1 and One (0 + 0) is insignificant, inasmuch as it represents nothing or zero. Of the Significant Combinations, taking into account the Combin- ations with Zero, there are, in all, Sixty-Five Combinations, and no more. The other Fifty-Five Combinations, shown on the Numeral- Frame, are simply related forms. (See the Table of Combinations on Numbers from One to Twenty, page 38). The next Device upon the Calculator is The Automatic Mechani- cal Schedule whereon are painted all the Number-Symbols, from Zero to Ten, in two horizontal rows. Those in the lower row are painted upon blocks, movable in a rabbeted groove, within which, as they are pressed forward, successively in the same direction, they may be made to assume different relative positions with regard to the Number-Symbols in the upper row. All the different Com- binations in Addition, which can possibly be made by any two of the said Number-Symbols, may thus be represented. The said Schedule is provided at the ends with stops, each of which is made to open outward on a pivot, so as to admit of the removal or reception of the movable blocks in the lower row with the view of forming New Combinations of Number-Symbols, in Addition, as described above. Th ; Schedule is correspondent to the Numeral-Frame ; the former makes the Combination in Symbol-Form and the latter in Number- Form. The Schedule may be better explained by the following Figures: — 30 PRIMAHY NUMREU-VVOHK. The Automatic Mechanical Schedule. Sex 1 OF THE SERiEs OF Combinations, The A'jtomatic Mechanical Sciiedcle. The above cut represents the Automatic Mechanical Schedule with tlie Number-Symbols painted upt»n it (as described in the foregoing paragraph) in two rt)ws, the upper r(nv being statio ry and the lower one movable in a groove- The Symbols as thus arranged f<^rm tlie First Set of the Series of Eleren Combiiiatiois which it is possible to form, with the said Schedule, by moving the lower row of })l()cks. The lower row, moved eleven times, gives the Eleven Sets of the Series of Combinations, thus forming One Hundred and Twenty-One simple additions^ the greatest number possible with any two of the said symbols. Fio. (a). AlilTHMETinAL CALCULATOR. tn Fig. (a) represents tlic Sehod'.iio with tlio st()[)s pushed huck, so as to admit of the rcmovid, from tlie h)\ver horizontal row, of bh)ck marked 1, preparatory to the pushing of the remainder of the blocks, in the said row, one place to the left. Fio. (6). Fig. (h) represents the Schedule with block marked 1 removed and the remainder of the lower row of blocks ready for shifting one place to the left, block marked 1 being ready in position for insertion at the other end of the said lower row. \\ Fio. (c). 32 IMtrMAKY NUMUKK WORK. Fig. (*■) ruprosents the Schedule with tlie renisvinder of the h)wer row of blocks moved one place to the left ; the stop on the left of the Schedule closed and Ijlock marked 1 ready for insertion in the place vacated by the movement forward of block marked zero (0). Set 2 of the Sekies of Combinations. Fig. (d). Figure (d) represents the Schedule with the lower row moved one place to the left ; block marked 1 inserted in the place previously occupied by block marked 0, and the stop in the right of the Sched- ule closed, so that a New Set of the Series of the Eleven Combina- tions for Simple Additions is formed. The next Set of the Series of Combinations is formed in a similar way, pushing back the stops, removing block marked 3, closing the stop on the left of the Sch».dule, and moving the remainder of the lower row of blocks t)ne place to the left, as before ; then inserting block marked 3 in the place vacated by block marked 1, and then closing the stop on the right of the Schedule. This will give another Set of the Series of the Eleven Combinations of the Number- Symbols for Simple Additions. The remaining Nine Sets of the Series of Coinhinations, as formed by the Schedule to complete the Eleven Sets of the Series, may be represented by the following Figures, each of which represents a New Set, after the shifting of the lower row of blocks one place to the left, rts already explained. ARITHMKTICAL CALCULATOF 33 The Remaining Nine Sets of the Schedule Combinations. Set 3 of the Series op Combinations, Fio. (e). Set 4 of the Series of Combinations. Fio. (/•). mi 34 PRIMARY NUMBER- WORK. Skt 6 OF THE Series of Comr, NATIONS. Fio- (g). Set 6 of thc Series of Combinations. Fro. (A). ARITIIMKTICAI. rAI-CULATOH. 35 Set 7 ok thk Series of Com bin a j ions. Fio. (ix Set 8 of the Series of Combinations. Fio. ox 36 IMUMAHY NUMnKK-WORK. Skt J) OF TlIK SkiuES of C OM HI NATIONS. .4 Fio. (k). Set 10 OF the Series op Combinations. F'tf (0. AHITHMETICAL rALCULATOR. 37 Set 11 OF THE Series of Combinations. Fio. (m). Sums of Schedule Combinations. Table of Sums. I • I ■} SETS, 1 2 3 ^ 5 6 7 8 f /O // Sums 2 ^ 6 8 /^ J J 7 ^ n ; 6 8 W 12 5 7 f // /J 3 ^ W f2 l^ 7 7 // /J /5 5 6 8 1^ 16 f n n /:i- n 7 ^ 10 12 m // n IS n /^ f /^ n )H- 6 ^ 6 a 10 n 2 J i 7 ? 11 8 10 n /^ V 5 7 ^ n /J 6 n J^ 16 6 ' 7 ? /; /J n « 12» 13» 14^ 15« 16^ 17* 18" 19' 20» The foregoing Table shows :— «6 Combinations with numbers to 20. These are seen in the upper portion to the spacing. The 55 related forms with numbers to 20. These are seen below the si)acing. Then the Calculator makes + 0, which is an insignificant Com- bination (inasnmch as it stands for nothing), and it is not shown on the Table of Combinations. The G5 Combinations (primary) may be arranged as follows into two classes : — (i) Combinations to Ten. 1 1 1 2 1 2. 1 2 3. 1 2 3. V 2' 1' 3' 2' 4' 3' 2' 5' 4' 3' 6' 5' 4' 3' 7' 3' 6' 4' 01234 01234.0 12 345 8' 7' 6' 5' 4' 9' 8' 7' 6' 5' 10' 9' 8' 7 6' 5 35 Combinations, to 10. (ii) Combinations of Ten with the Intermediate Units to 20. 1 2 3 45 2 3456 3 456. 4 5 67. 10' 9' 8' 7'. 6' 10' 9' 8' 7' 6' 10' 9' 8' 7 ' 10' 9' 8 7 ' 5 6 7 6 7 8. 7 8. 8 9. 9 . 10 10' 9' 8' 10' 9' o' 10' 9' 10' 9' 10' 10 30 Combinations, 11 to 20. (35 -t- 30 = 65 Combinations.) The upper part of the foregoing Table gives the Combinations from One to Ten, and the lower part, the Combinations from Eleven to Twenty, both Numbers inclusive in each case. Leaving out of account the Combination "zero plus zero " (which is insigni- ficant, inasmuch as it represents the symbol zero, and is not shown in Table, though it is formed by the Schedule and the Numeral- Frame), we have, in all, Sixty-Five Comhinations. The other I'l i' 40 '•'^^>'AKV NUMUEH-WOUK. Sevkn = '^« P^T Nmuoral-Frume 7 + 0-1 'J + 1 I and :i + 4 2 + 5 1 +6 1 ^^ + 7 1 f^ufc of these eiald f relatod f„„,„. ^'"'''^ ^''l«™tos the fundan.ent j L,„ the ^f f«L:;h:L'c!s,:!:'L"r -^"-'^ ••- h-'-" «... •«'.ed".«i.otl<„„^„„„„, '^^-^^'.own ... the af„re=ai,l Tab " wl..ch „,ake up that „,„„her " ''" ''"°*^ "" "». Co,„bi„ad„™ Fio. B. (P. v,n.) Tea Larger Calcci^tor. AUITIIMr.TICAL CALCULATOU. II The nhove cut roprosents the Ljvrj^or Ball- Fnimo juul consists <>f twelve wires, iirraiii^aid in i^roups of throe eucii, on which move TWKNTY HALLS, the whnlo Immiii^ tixcd in a franie and attuched t(» the Calculator, for the imrpose of ilJiistr = 13 13 - 1) = 4 13 - 4 = Q L 18 PRIMARY NUMHKK-WOKK. / #^ Fig. (?■). 8 + 5 = 13 5 + 8-13 8 - 5 13 - 5 = 8 1o AlUTUMEllCAL CALCULATOU. 49 4» — -©- Fia. (/). 7 + r. = 13 6 + 7 - 13 13 - 7 - G 13 - <•) = 7 ADDITIONS. 10 + 3 = 13 9 + 4 = 13 8 + 5 ^ 13 7 + (> = : 13 6 + 7 = : 13 5 + 8 = : 13 4 + 9 = = 13 3 + 10- 13 50 PKIMAUV NCMliKU-WOKK. SUHT FACTIONS. j; 10 3 13 - 9 - 4 13 - 8-5 13 - 7 -^ (5 13 - (»- 7 13 - 5 =^ 8 13 - 4=9 13 - 3 = 10 Thirteen will be recognized, too, as being thirteen ouch, or as being composed of thirteeii primary units. The plan followed in the Fundamental Numbers is to be followed in i)roceeding from Meren to Twenty. The Larger Ball-Frame (on which the balls are movable) may be used to form all the Combinations on Numbers from One to Twenty; but all these Combinations are formed by the Numeral-Frame (in which the half balls are fixed) and Automatic Schedule, as the lower row or series, in each, is pressed forward to form New Combinations. In teaching any of the Numbers, use the Number-Form first ; and when the pupils understand thoroughly the Number as 2>resentexl objectively, then teach the corresponding Arabic character (figure) ; but keep the tivo ideas separate at first. Dr. J. A. McLellan says : — "Giving the symbol as soon as the idea is mastered is justified by common sense, as well as by psy- chology There is variety and therefore interest ; dealing with the objects too long becomes monotonous ; symbols open up a new field. There comes also a feeling of power, of advance, etc. There is economy of time and power for both teacher and pn j)il. It affords means of self-instmction. In short, the justification is on the same ground for the child as for the race. The human mind always economizes by means of some condensed symbol as soon as the idea is familiar. It is worse than useless to be always going back to beginnings ; this would render progress extremely slow." — See Dr. McLellan's remarks on Arithmetic in his "Applied Psychology." See, also, " Psychology of Number," by McLellan and Dewey. Twei Tun and sh»)\ A HlTll M KTir \L CA LCU LATOU. 51 K<)llf)\viiii» tin; Tsihle <>f C/oiM))iiiati<)n.s (in Nunihers from One to Twenty, tlie pupils should be required to -irrile out in their Exercise liooks afl the diff'erent Onnh! not ions for oaoh ; c.f/., the number Thirteen— whoso cotnhinoiions are Ten and Three (10 + ^5) ; Nine and FoiR (9 + 4) ; Ekjht and Five (8 + 5) ; Seven and Six (7 + (>)— should be written : — • • • • • • • • • • • •! • • * < • • • • • • • • /O -»- J / = /_^.j'_/4-^ These Combinations may be formed either on the Ijiirge Ball- Frame, the Numeral-Frame (at the top of the Calculator), or various forms may be used on the black-bss-l»arj in the centre of the upper half of the enclosed space, the ten halls may he considered as the Unit Ten (a One Ten or a Ten Unit). The intermediate units may then bo com- bined with the said Unit Ten, and thus the Combinations to Twenty may be tivught. But before considering Ten as a Unit (Ten Unit or a One Ten) in this way, on the Calculator, let us develo]) Ten as a Unit (Multiplex Unit) by means of crayons of chalk. **Va7'iety u the spice of life." Take a box of crayons (a box of whole ones if possible, as broken crayons will not be so handy to tie into bundles) and tie (or fasten with rubber bands) the crayons into bundles of ten each, until you have tied up ten bundles ; now, with these and ten single crayons be- sides, you can teach, under standitigly, the Numbers to One Hundred. In order to develop Ten as a Unit and how to represent it in symbol-form, it is necessary to first teach the number Eleven (11). Now^ take one of your bundles of ten crayons and lay it out on the table before your class ; then place one single crayon on the right of it, as seen from where the pupils stand. You have before the class one ten and one unit, thus :— l\ n y 1 TEN and 1 unit (10 + 1) [a Ten-Unit (Derived Unit) and a Primary unit]. ai f( AUI Bepreseut this on tn ^^^^_^ 1 53 ten 1 •. i,.i„lv iH.ieh of tl.c ones d « , ■„ „t»i.a« j::^r;ff...^^^^^^^^^^ ..- for the tcn-buudle, or bun U ^^^ ^,,,,). ! • \. f the 1 'B stands for the 8in=, J .«f . oN E ten and no which of tue ■* ^^g^^e left onk f\,P ahi'^le crayon, ana yo ]S{ow remove the 8Ui„i« units, thus :— 1 TEN and no (0) units. (10 + 0) . nt^ih that when you have 1 ten a !!rn:-:orS^:UacU-hoard,t.us:-- and no (0) units, you represent ten 1 units Ten P,^e««. crayons be»iac*ej^ - ' ree itmp, thus . ,;.b,.,uu. (»«.-«'»«• "'"' """ have ONE TEN and three 54 PRIMARY NUMBER-WOHK. 1 TEN and S units. (10 + 3) Represent this on the Wack-board, aa before, in symbol-form, ten units 1 3 thus : — Thirteen (13 = 10-f.3). Teach the intermediate units with ten as a unit, in this way, tak- ing them up in logical order (after ten is developed as a unit) and when you arrive at 1 ten and 10 units, thus :— I I TEN mi\ 10 nnith (10 + JO) AUITllMKTIt'AI. CAU'fl'ATOIl. sn you ,h..uUl explain U, the pupiU that the >.: •'r;^'; ""\7;,^ "b:'l't; ll :l .U.eu r^Unting it as a>o.„.h ...N-.n.. ". the tens place, thus :— 2 TENS and no (0) imits (20 + 0). 2 tens is simply a shorter way of writing 1 ten and 1 ten. = 10-1-10 + .= 20 + = 20 Twenty (A. S. twentig) = twa tens = two tens or two ten-units. 56 PRIMARY NUMBER-WORK. The Larger CALCuiiATOR. By moving up the "slides " at sides of upper half of Calculator, so as to draw together the ten movable balls at top, a ten-unit may be made. As it stands we have : Ten -f- One = Eleven. Any of the inter- mediate Kuits may now be added until twenty is reached ; when, by moving up the lower slides two trn-units may be shown, and TWENTY revealed as two tens. Similarly with the Smaller Calculator with slides, '^en may be s^own to be two fives. Thus on the two calculators may be shown Ten-Units and Five-Units. Multiplex Unity as Units. Teach in this way all the other tens as nnits^ that is 3 tens (30) ; 4 tens (40) ; 5 tens (50) ; 6 tens (GO) ; 7 tens (70) ; 8 tens (80) ; 9 tens (90), with the intermediate units between the tens. Now, from 9 tens and 9 units you can easily develop One Hundred as a unit (A Hundred-Unit). Lay out on the talile before the class 9 bundles of ten crayons each, and beside them place 9 r.'^igle crayons ; now add another ARITHMETICAL CALCJULATOK. 57 M single crayon to the nine crayons, and you will have 9 tens and 10 units. Bub these 10 units are equal to another bundle of ten crayons — a ten-unit- -which may be tied into a ten-bundle ; now add this ten-bundle to the 9 tens and it makes 10 ten-bundles and leaves no units. But ten tens, or ten ten- bundles may be tied together, as were the ten single crayons, so as to form a Unit. You have now a unit of the next higher order — One Hundred (1 Hundred) which is not written in the tens' place, but in the hundreds' place, when it is represented in symbol-form ; so that out of your 9 tens and 9 units plus 1 unit, you have now 1 Hundred, iio tens, and no units: H, T. u. 10 Carry on this operation until the pupils know hoiv to express any number in the Hundreds, the Tens and the Units. Bundles of sticks of equal size and length, fastened together into bundles of tens, hundreds, etc., with rubber bands, will be found to answer even better than the chalk ; but the chalk is always to hand, whereas the sticks require to be prepared, specially. Have pupils practise making "pictures " of such numbers as : — 5G ; 127 ; 347 ; 604 ; 502 ; 35, etc. ; e.g., 35— 3 TENS and 5 units = 30-f-5 = 35, etc. Pupils taught in tlw way will not say that Thiri^ -Five is Three and Five, or that One Hundred aid Twenty-Seven is One, Two, and Seven, ii! :1 1. ii 58 PRIMARY NUMBER-WORK. Ten may be taught as a unit by using (as mentioned before) the Larger Calculator or Ball-Frame. The Combinations with Ten as a unit are all formed to Twenty, as each Number-Form, in the lower series of the Automatic Numeral-Frame, comes under the Number-Form Ten in the upper series as the said lower series of blocks is pressed forward. By removing the blocks in the lower series, and then inserting the said blocks at the right-hand side, so that they may each be moved immediately and successively under the Number-Form Ten, the whole Ten Combinations may be formed in logical (natural) order — 10 -rl, ]0-f2, etc. ; but the blocks must be inserted in logical order to accomplish this — 1, 2, 3,4, etc., instead of 1, 3, 5, 7, etc. (the order in which they are — for a p^crpose — arranged u[)on the Cal- culator). Next, the teaching of the intermediate numbers between the tens may be taken up. As in going from eleven to twenty, so proceed from twenty to thirty ; thirty to forty ; forty to fifty, etc., keeping clear the idea that we have always a certain number of tens plus a certain number of units ; e.g.^ 99 = 9 tens plus 9 units. A thorough umlerstanding of the Fundamental Numbers will make the teaching of the tens and the intermediate units an easy task to accomplish. Counting backward and forward past ten may be left off until after the Combinations from One to Twenty are mastered, as the pupils who are able to count readily to twenty, will resort to this means in finding out the Combinations, and instead of making these Combinations a test of Sight and Sound (Seeing and Hearing) they will be finding them out by counting. Permit no counting in giving the Combinations. To know the Number Thirteen, a pupil must know it : (1) ^Y Sight (in Number-Foiin and in Sijmbol-Form), thus : — (Q) Fundamental Forms. (6) ^c) :•• 9 t f • I • • = ••• •-.• I ♦« _ • •»• I • •• = /(? +5 r t > li AKITHMETICAL CALCULATOR. 59 ••• I • • • » 5 -*■ Q 9 9. Related Forms. (h) •• • •..• ■=• 6 si-^ z « • 5 f /(? ••• • f t 3 See Niimeral-Frnme and Schedule Seta ; also, the Table of Sums of Schedule Sets. The foregoing Combinations of Thirteen will be seen by referring to the aforesaid Sets, as follows : — ('0 (fe) (c) (d) (e) (/) (9) Set. No. 3 4 5 (> 7 8 9 10 Combination. No. 5 it a n 10 4 9 3 8 2 7 (2) By Sound or Sense (in Word-Form, either spoken or written), thus : — (a) Nine and Four are Thirteen. (6) Ten '• ((•) Seven " {d) Eight " {e) Five " (/) Six '' ((/) Three *' (/i) Four (. Three Six Five Eight Seven Ten Nine u ii ti, i(. u u Dr. McLellan says : — " It seems plain that, if the child is led by clear intuitions to think the relations as presented in these Number- Forms, the ' mental experiences' will blend into a lasting conception of the number." The Numeral-Frame gives one dejinlte Nnmlter-Fonn for each niiniher; l)ut these may be varied if desirable, as in the case of the numV)er Three in the exami)le given, when gohig over the Funda- mental Numbers. A rurirtij of fonnsmi\.y be made upt)n tlie Larger liHll-Fmiao, on which thu balls are monUde iov that purpose. 60 PRIMARY NUMBER-WORK. Pupils must ktioir the jxdrs of addewh or reUded parU which form any numl)er l)ef<)io tliey know tliat iiinnber. They must know all the combinations, (1) by additions of paira of addends (re-combina- tions or '• irholhuf"), (2) subtraction or resolution of a number into its several combinations (partition or '"'' parting ^^). This involves the Foru thincs to be taught in connection with the additions and subtractions of every pair of addends (in com- bination) which form any number, e.g.: • • Five as made up of the addends three and twa The Fol'r Ideas. (1) Three and Two are Five. 3 4-2 = 5 (2) Two and Three are Five. 2 + 3 = 5 (3) Two/*7>?/i Five /cares Three. 5-2 = 3 (4) Three fror.i Five leaves Two. 5-3 = 2 These Combinations will all be found on the Numeral-Frame, as the lower row or series of tablets containing the Number-Forms are pressed forward to form new Combinations. In connection with all the Combinations made by the Numeral-Frame these four ideas should be taught. The Schedule must be worked so as to correspond to the Numeral- Frame ; the former making in Symbol-Form what the latter makes in Number-Form. By the Numeral-Frame and Schedule the pupils must know : — (1) The Combinations in Number-Form. (2) The Combinations in Symbol-Form in connection there- with ; and (3) The Symbol -Combinations in abstraction. AIUTUMETICAI. CALCULATOU. 61 Wlien the pupils know the Coiiilnn.itions in jibstraction, thi-y are ready to take up work prescribed in Cuthhert's "Desk-Work in the Simple Rules," Nos. 1 to 4, in Addition and Subtraction. When the ])upils have proceeded a little farther, the n»ultii)lica- tions and divisions must be taught. (We have here the " nuiltiplex " unit). The four ideas in connection with these combinations should be taught ; e.(/., (1) Six times Seven are Forty-Two. 6 X 7 - 42 (2) Seven times Six are Forty-Two. 7 X G = 42 (3) Seven into Forty-Two goes Six times. 42 ^ 7 = 6 (4) Six into Forty-Two (joes Seven times. 42 -^ 6 = 7 These may be taken up to a limited extent on the Larger Ball- Frame ; e.g., The Larger BallFramij. 62 PHIMAUY NrMHKU-WOUK. The Number Twenty on the Larger Ball-Frame. Fio. (a). 2x10 = 20 20-!- 10= 2 / FlO. (b). 4x5 = 20 20^5= 4 AHITIIMETICAL CALCUf.ATOH. G:5 Kio. (c). 5x4 = 20 20-=-4= 5 I, V\i 10x2-'20 20-~2 = 10 01 rniMAUV NCMHKK-WOHK. MULTIPLICATIONS. 2xJ0=20 4x 5 = 20 6x 4 = 20 10 X 2 = 20 DIVISIONS. 20^ 2=10 20^ 4= 5 20-- 5= 4 20-1-10= 2 The Number Sixteen. ARITHMETICAL CAU'ULATOU. 05 Fig. (c). 4x4 = 16 16-f4=4 (16 PKIMAHY NUMUKK-WORK. ^10. (d). 8 X 2 - 16 10 -^ '2 = 8 MULTIT^LICATIONS. 2x8 - ]<) 4 X 4 = 16 4x4= 16 8 X 2 = 16 DIVISIONS. 16 ^ 2 = 8 16 -f 4 - 4 16 -f 4 = 4 16 -•- 8 = 2 Two Eights are Sixteen (2x8= 16). Eight Twos are Sixteen (8x2 = 16). Four Foirs are Sixteen (4x4= 16). AlUTHMFTICAL CALCULATOU. iU Tilt! Fours in Sixteen arc. Fodi (1(5 -=-4-4). The Twos in Sixtken are Eioht. (10 -r 2 = 8). Tlie Eir.HTS in Sixteen are Two (16 -T- 8 = 2). Eight is the Half of Sixteen, etc. As soon as the pupils have mastered (<«) the additions, (6) the subtractions, (c) the multiplications, and (d) the divisions (exact), in c 6^^' See examples given on Larger Ball-Frame. In teaching Combinations in Addition from the Numeral- Frame, it is well to have the pupils (1) repeat orally the different Combinations in each Set of the Series, as the eye scans the said Frame from end to end of the Series ; (2) write out in their Exercise Books, in Number-Form, the different Combinations in each series ; (3) write out on their Exercise Books the Corrtsj^onding Com^jiiui- 68 I'lUMAUY NUMHEIl-WOUK. Hans in Syinbol-Form, from the Scliednlu, whicli should ho nrrantjid to correfipotid with tho Nuiiionil-Fi'.-inu! ; tho lust nit'iitioiicd exorciso may bo convoniontly dono hy tho piipiJH in thuii I'limiiry Numhor- Woi'k Exerciso Books, as per tho following scheme : - Scheme. The foregoing is the scheme which tho pupils have in their Exercise Books, ready to be ''filled in " from the Numeral-Frame. The following represents the scheme " filled in " by the pupils, from Set Seven of the Series of Combinations formed by tho Numeral-Frame : — Scheme. / 5 6 7 7 2 H- 6 6 10 k 6 8 10 / 3 5 7 7 z r? *7 15 17 *? 3 7 II 15 11 2 It is well to revei'se the Board upon the Calculator so that the Schedule Combinations are not seen by the class while the Scheme is being "filled in" from the Numeral-Frame; the pupils, of course, must make the Additions (that is, the s«ms of the 'pairs of addends) mentally, themselves. They should also be required to put their work down as follows : — 1 -f 4= 5 3 -h 6 = 9 5 4- 8 = 13 7 + 10 = 17 t AHITIIMFrriCAr- CALCULAmU. 60 !i\ J 1> + 2 + 4 + ^m4 /-A\ I.I = One times Five is Five. = Two ( n Ten. = Three ' i it Fifteen. = Four ' i u Twenty. = Five ' ' ( ( Twenty-Five. = Six ( (( Thirty. = Seven ' ( it Thirty-Five. = Eight i i( Forty. = Nine ' ( u Forty-Five. = Ten ( ( I Fifty. = Eleven ' i n Fifty-Five. = Twelve ' i i i Sixty. k'/ ARITHMETICAT, CALCULATOR. 73 r^\ k'/ For other forms of writing the Tables see Exercise Books 5 and 6 in "Desk-Work in the Simple Rules." The study of Number may be taken up in the order in which I have taken up the explanation of the diflferent Devices found upon the Calculator :— (1) The Fundamental Numbers ; (2) Coml)ination8 with Ten (as a unit) plus each of the Fundamental Numbers ; (3) The Numeral-Frame and Schedule Combinations in conjunction. After pupils have become familiar with the Schedule Combina- tions in the Arabic or Symbol-Form, they may be given desk-work in the wechaniial operations from the exercises in ** Desk- work in the Simple Rules " Nos. i, 2, 3 and 4. Before these Exercises are taken up for drill and practice, the additions and subtractions of the Fundamental Numbers should be thoroughly understood. "Making these partitions and re-combinations, and expressing the process in words and figures, affords good self-instruction work." Dr. McLellan says:— "From the beginning, arithmetic should supply useful examples for desk- work." By means of "Desk-Work in the Simple Rules," all the Junior Division of the school may be employed with suitable desk-work. In the lower classes of our Public Schools we must aim at RAPIDITY and ACCURACY in the mechanical operations. "All at Work ; and Always at Work.'' Plenty of drill in the mechanical operations begets rapidity and accuracy; therefore the work prescribed in "Desk- Work in the Simple Rules" will be found helpful in connection with the work laid down in this book. (i) Pupils should be required to recite orally (1) from the Numeral-Frame ; and (2) from the Schedule in connection, all the COMBINATIONS " AT SIGHT. " (ii) Pupils should be able to write out (1) in Number-Form ; (2) in Word-Form ; and (3) in Symbol-Form, all these Combinations. 74 THIMAKY NUMJJKU-WOKK. Little problems in mental ariMniietic should acconi})finy the mechanical work. "In mentJil work, rapidity, correct language, and logical order of thought and statement must be constantly aimed at.'' In giving practical problems in mental work, suit the problem to the rule in hand : — Additi(m — e.y. , John has 3 marbles in one pocket and 4 in another ; how many marbles has he in both pockets ? Subtraction — ^'.y., Tom had 9 marbles and lost 4 of them ; how many has he left I Afterwards, when the pupils have become accjuainted with both, use a combination of the two; e.g., Jim got 4 candy-sticks from his father and 5 from his mother ; he gave 6 away to his sister ; how many had he left ? Similarly in the other two rules. In teaching the Fundamental Numbers as they are represented upon the Calculator, perhaps drawing the chalk through the number, so as to make the partitions, would be preferable to hiding each addend (of the ])air of addends forming the number) with a piece- of card-board, as already described. Both methods may be employed to advantage. Rapid mental wcjrk must be given to assist the mechanical work ; 2-1-5 X 7 + i-i-o X (i-f ,S-^7 X 9 + 1), etc., equals what ^ In introducing Addition and Subtraction, perform the work by means of the balls, having the pupils to assist you. Exercises in Nos. 1 to 11 in " Desk- Work in the Simple Rules," are intended to su[)ply an ample amount of " desk -work " for the pupils, in the mechanical operations ; but, in tin earlier stages, the pupils should have ])ractice in making and varying the number- forms ; writing out all the Combinations <»f each number, and in the Numeral-Frame and Schedule Combinations. Teach thoroughly the Fmidamental Nttmhcrs, as_on those the con- T \ \}l A H I 'I'll M KTIO A r. VA LCU I. AT( ) I! . /:) \ i^ stnu'Mon of 3L(fhem(itu-s (li'lH'iidii. Following tlio Tabic of Combi- nations on Numbers from One to Twenty, have the [jupiLs write out the Combinations of some of the numbers each clay, until all the numbers with their Combinations are understood. Pupils should be recjuired to learn these Numbers and their Com- binations so thoroughly that they can rendiltj repeat orallij, fnnn memory, all the difJerent Combinations that make each of the Numbers from One to Twenty. Teach Numbers higher than Ten on the basis of ten as a vuiit. Children may be led to see the meaning of Twenty (Two Tens, 10-1-10), Thirteen (Three and Ten, 3 + 10), etc. All the Combinations with Ten, in going from Ten to Twenty, may })e nicely shown by removing all the tablets from the lower series in the Numeral-Frame (Fig. B) and then inserting the block 9 I "P<-'ii it, and moving it with the Number-Form "ten" along, so as to come under eacii number, in the upper series in regular order ; e.g., 1 + 10, 2 + 10, 3 + 10, 4 + 10, 5 + 10, + 10, 7 + 10, 8 + 10, 9 + 10, 10+10. The "ten-tablet" should be moved along so as to do this in re<'u- lar or natural order, the numbers in the upper series being jiiaced there out of their regular or natural order : — 1, 3, 5, 7, etc. When a pupil knows a number, he knows it in all its combinations in addition or subtraction. When he has learned by tuition, that 8+5 are 13, he thinks of the eight and the live as wholes ('-'complex units"'), and of the number thirteen as a whole (" lenity ") ; he will alst) know that each of these numbers is composed of a certain number (as the case may be) of single or primary units (ones). A clear idea of unity is necessary to Number-Teaching. "The idea of a unit can begin only from analysi.". (;f a wliolo ; it is completed only by rehiting the part to the wliole, so that it is linally conceived at once in its isolati(»n and its unity to the whole." 76 PRIMARY NUMHKR-WORK. A unit is any measitring part by wliich the whole (juantity is numerically defined. It is a unit only whot it in emplo\it'd to measure some greater like magnitude. Similar objects, symmetrically arranged, present the best example for the conscious recognition of number as being a unity made up of related parts. — " Units constituting the defined unity." The objective idea has been kejit in view in constructing the Common-Sense Akithmktical Calculatoii, and it is hoped that it may prove useful as an aid to the busy teacher in placing " Number " in an attractive and intelligible form before his pupils. "Objects are not aiumber. No numerical concept or idea can enter into consciousness till the mind orders the objects — that is, compares and relates them in a certain way." In a dozen apples (12), the unit may be a half dozen (6), which m,easures 12 twice; or it nuiy be a third of a dozen (4), which measures 12 three times ; or it may be a quarter of a dozen (3), which measures 12 four times. Again, twelve, the unity in the foregoing, may be considered the unit in measuring eggs, oranges, lemons, etc. Ten balls may be marked off upon the Calculator into unit-groups of 5's or 2's, which are derived measuring parts with regard to the unity ten. Ten may be counted out as composed of ten primary measuring parts or units (ones) ; or it m^^y be regarded as a unit to measure 20, 30, 40, 50, 60, 100, 500, 1000, etc. We have the primary units in the derived unit, and the derived units {units of the same scale) in the measured unity or quantity. Twelve should be measured out by 2's, 3's, 4's, 6's (derived units) ; and in I's {prinuiry units). The twenty balls upon the Larger Calculator may be taken as a unit to measure 40, 60, 80, 100, etc. Five times over it is 100. How many times, then, does twenty as a unit measure 100 ? The twenty balls may be reganled as a unity, and analyzed, as on pages 61, 62, ()3, 64. For more minute and extended exj)lanation of the unit, unity, stages of measurement, etc., see "Psychology of Number." } i \) AHITHMKIICAL (A I.("UFw\T()H. / t 11/ EkcIi of the Coinbiiiatidiis, aftor luiviiig l)een taii<^ht, ititKitivelii, should l)t! mastered " «/ utiiht" in the Avuhic characters (figures). The teacher Jiiay accouii)lish this l)y pointing, |)roniisciiously, to the ditferent Combinations, wliich are contained in each iSetof the Series of Schedule Combinations. The "Hundred" Tables. Teach i.upils to write out the numbers 1 to 100 ; 1(X) to 200 ; 200 to 300 ; 300 to 400 ; 400 to 500 ; 500 to 000 ; (iOO to 700 ; 700 to 800 ; 80se sum is less than ten (numbers from 1 to 5) ; afterwards take those jxdm of addends whose siim is either ten or more tluiti ten (numbers from 6 to 10) ; e.g.. 7H PRIMARY NUMHKR-WORK. 5 + U 10 Iciids t(. 15 4- T) 20; 25 | 5 ;{0 ; .'{5 4 5 = UK etc. 7 + 7 - 14 loads tu 17 + 7 = 24 ; 27 + 7 - 'M; W + 7 = 44, etc. + {) - 18 leads to 1{» + 1) - 28 ; 2!) + 1» - IW ; liO + \) - 48, etc. ; hocauso {> + 9 — 18, and 18 tuiiulnato.s in 8 ; therefore all Coiubiiia- tions w ith 9 + 9 will also terminate in 8, etc. When using these " terminations," liave the pupils count by 2's, I'i's, 5's, etc.,a3 mentioned before ; also have them name all the num- bers between 1 and 100, whicli tehminatk in a given number, say 7; these numbers will then be : — 7, 17, 27, 37, 47, 57, G7, 77, 87, 97. It is well to illustrate, by soiae such means as the following, the fact, say, that there is n(» other number between 18 and 28 that tevminates in 8 ; e.g., 18 ^l9 20 21 22 2:i 24 25 20 27 1,28 Thus it will be seen tliat 28 is the next number after 18 that terminates in S. Pupils will by the foregoing see this for themselves. Give plenty of oral drill on the " terminations '' in connection with the Numeral-Frame and Schedule Combinations ; also in re- l)eating numbers, between 1 to 100, that terminate in a given number, s^ay in 7, as in example given. For "terminations," see No. 1 "Desk-Work in Simple Rules." The Numeral-Frame at the top of the Calculator is a silent teacher of Number, as a pupil will never look at it without getting an idea on number, just as he never sees a good Map on the walls of a school-room without learning some fact in Geography, or having some fact engraven on the memory. < 1 ARITHMKTIC'AL CALOIT.ATOR. 79 Ah (ihdrdd'iou must nUiuuih'hj /«/,>■ /Ac jihwi- <