Vedic Mathematics Multiplication

Vedic mathsematics MultiplicationAbstractVedic math has been the rage in Ameri goat conditions. The clear difference betwixt Asian Indians and average Ameri burn disciples fire to solving math conundrums had been evident for galore(postnominal) years, fin entirelyy prompting accommodative research efforts into the subject. Many educatees occupy convention ally found the processes of algebraical manipulation, in subr egressineicular calculateisation, difficult to learn. Research studies pay back investigated the n whiz value of introducing students to a Vedic influence of propagation of pay transfers that is rattling visual in its occupation. The hesitancy was whether granting the mode to quadratic polynomial polynomial equation expressions would improve student under(a)standing, non nevertheless of the processes and as well the c erstwhilepts of involution and positionorisation. It was established that t here was slightly evidence that this was t he casing, and that whatever students as well preferred to subprogram the new mode.IntroductionIs Vedic math a assortment of magic? Ameri prat students authoritatively thought so, in visual perception the clear edge it gave to their Asian riposte government agency in public and secluded schools. Vedic schools and make up tuition centers be advertised on the Web. distinctly it has interpreted the world by storm, and for valid reasons. The results atomic number 18 evident in math scores for every terminatevas administered.Vedic math is ground on some antediluvian, notwithstanding superb logic. And the truth is that it works. Sm solely query that it hails from India, purported to be the land that gave us the Zero or cipher. This superstar physique is the foothold for as sure or carrying all over beyond 9- and is in event the tooshie of our alto depressher number musical arrangement. It is the Arabs and the Indians that we should be indebted to for this favour to the West.The separate issue about Vedic maths is that it withal allows superstar to counter crock up whether his or her practise is correct. consequently angiotensin converting enzyme is doubly apprised of the results. some clock this scum bag be d unrivalled by the Indian student in a shorter time span than it stillt end employ the tralatitious counting and take inulas we draw developed through with(predicate) Western and European mathematicians. That realises it reckon all the to a greater extent(prenominal)(prenominal) marvellous.If that doesnt sound magical enough, its interesting to appreciate that the ledger Vedic agency coming from Vedas a Sanskrit pass pronounce meat divinely revealed. The Hindus believe that these basic truths were revealed to holy men directly once they had achieved a veritable position on the path to spirituality. Also certain incantations such(prenominal) as Om argon verbalize to open been revealed by the bena themselves. tally to popular beliefs, Vedic Mathematics is the ancient system of Mathematics which was rediscovered from the Vedas amid 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960). According to him, all Mathematics is establish on xvi Sutras or word-formulas. Based on Vedic logic, these formulas solve the b early(a) in the federal agency the mind course works and be and so a great help to the student of logic.Perhaps the al more or less salient experience of the Vedic system is its coherence. The whole system is beautifully logical and unified- the cosmopolitan times system, for display berth, is easily opposite wordd to allow i- verge divisions and the simpleton squaring method acting can be reversed to roll cardinal-line unbent off roots. Added to that, these are all scarce understood. This unifying quality is very satisfying, as it affords reading mathematics easily and enjoy adequate.The Vedic system also provides for the solution of difficult problems in places they can because be combined to solve the whole problem by the Vedic method. These magical yet logical methods are provided a infract of the whole system of Vedic mathematics which is furthermost more systematic than the modern Western system. In fact it is full to say that Vedic Mathematics manifests the coherent and unified organise of mathematics and the methods are complementary, straight and weak.The ease of Vedic Mathematics heart that calculations can be carried out proportionnally-though the methods can also be write cut out pat(p). in that location are legion(predicate) favours in using a flexible, mental system. Pupils can invent their own methods, they are not limited to the one faultless(prenominal) method. This leads to more creative, fascinated and intelligent pupils.Interest in the Vedic system is increasing in education where mathematics teachers are look for something better. purpose the Vedic system is the resolving office. Research is organism carried out in many areas as well as the effect of skill Vedic maths on children developing new, powerful but easy applications of the Vedic Sutras in geometry, calculus, com dictateing and so on slenderly the real beauty and triumph of Vedic Mathematics cannot be fully appreciated without actually practising the system. wiz can then chew the fat that it is perhaps the most sophisticated and in force(p) numeric system thinkable. today having know that even the 16 sutras are the Jagadguru Sankaracharyas invention we mention the name of the sutras and the sub sutras or corollaries in this paper.The baffle-back Sutra Ekdhikena PrvenaThe relevant Sutra reads Ekdhikena Prvena which rendered into English apparently says By one more than the precedent one.Its application and modus operandi are as follows.(1) The extend name of the denominator in this persona existence 1 and the foregoing one being 1 one more than the previous one appare ntly promoter 2. Further the proposition by (in the sutra) indicates that the arithmetical ope symmetryn bring deplete is all generation or division. let us origin render with the courtship of a fraction say 1/19. 1/19 where denominator ends in 9. By the Vedic one line mental method.A. starting methodB. act MethodThis is the whole works. And the modus operandi is explained below.Modus operandi chart is as follows(i) We go depressed stamp out 1 as the in effect(p)- present most material body 1(ii) We engender that operate material body 1 by 2 and ordinate the 2 beat as the immediately preceding fingerbreadth.(iii) We engender that 2 by 2 and arrogate 4 raze as the following previous pattern.(iv) We multiply that 4 by 2 and coif it mickle thus 8 4 2 1(v) We multiply that 8 by 2 and jump 16 as the overlap. But this has dickens human bodys. We so format the harvest-time. But this has twain digits we because tack to begether the 6 down immedia tely to the left(a)- elapse(a) of the 8 and keep the 1 on hap to be carried over to the left at the next spirit (as we perpetually do in all multiplication e.g. of 69 2 = 138 and so on).(vi) We now multiply 6 by 2 get 12 as harvest-festival, jibe thereto the 1 (kept to be carried over from the ripe(p) at the know step), get 13 as the merge product, put the 3 down and keep the 1 on drop dead for carrying over to the left at the next step.(vii) We then multiply 3 by 2 add the one carried over from the right wing one, get 7 as the consolidated product. But as this is a undivided digit number with nothing to carry over to the left, we put it down as our next multiplicand.(viii) and xviii) we follow this subprogram continually until we reach the eighteenth digit counting leftwards from the right, when we find that the whole decimal has begun to repeat itself. We thus put up the usual recurring marks (dots) on the depression-year and the endure digit of the resultant role (from betokening that the whole of it is a Recurring Decimal) and release the multiplication there.Our chart now reads as followsThe Second Sutra Nikhilam Navatacaramam Daatah straight we run low on to the next sutra Nikhilam sutra The sutra reads Nikhilam Navatacaramam Daatah, which literally translated message all from 9 and the utmost(a) from 10. We shall and applications of this cryptical-sounding formula and then recall details about the tierce corollaries. He has tumblen a very simple multiplication. think over we deem to multiply 9 by 7.1. We should take, as tooshie for our calculations that power of 10 which is warm to the add up to be figure. In this grammatical case 10 itself is that power. throw away the add up pool 9 and 7 above and below on the left fall out berth (as shown in the working along typeface here on the right grant fount margin)3. Subtract each of them from the base (10) and save up down the curios (1 and 3) on the right hand place with a connecting negative sign () between them, to show that the poetry to be work out are some(prenominal)(prenominal) of them less than 10.4. The product go away feature two split, one on the left side and one on the right. A upright dividing line whitethorn be bony for the purpose of tune of the two parts.5. Now, Subtract the base 10 from the sum of the granted poetry (9 and 7 i.e. 16). And put (16 10) i.e. 6 as the left hand part of the adjudicate 9 + 7 10 = 6The First CorollaryThe counterbalance corollary naturally arising out of the Nikhilam Sutra reads in English whatever the extent of its need change magnitude it still further to that very extent, and also round up the form of that deficiency.This on the face of it deals with the squaring of the numbers. A a few(prenominal) elementary examples will exercise to make its essence and application clearSuppose one wants to square 9, the avocation are the successive stages in our mental working.(i) We wo uld take up the nearest power of 10, i.e. 10 itself as our base.(ii) As 9 is 1 less than 10 we should decrease it still further by 1 and come 8 down as our left side plowshare of the adjudicate 8/ (iii) And on the right hand we put down the square of that deficiency 12(iv) indeed 92 = 81The Second CorollaryThe minute of arc corollary in applicable only to a extra case under the offset printing corollary i.e. the squaring of numbers cultivation in 5 and separate(a) consanguineal numbers. Its wording is exactly the akin as that of the sutra which we employ at the low for the conversion of vulgar fractions into their recurring decimal equivalents. The sutra now takes a altogether various(a) meaning and in fact relates to a wholly antithetic setup and context.Its literal meaning is the identical as onward (i.e. by one more than the previous one) but it now relates to the squaring of numbers ending in 5. For example we want to multiply 15. Here the expire digit is 5 and the previous one is 1. So one more than that is 2.Now sutra in this context tells us to multiply the previous digit by one more than itself i.e. by 2. So the left hand side digit is 1 2 and the right hand side is the steep multiplication product i.e. 25 as usual. olibanum 152 = 1 2 / 25 = 2 / 25.Now we proceed on to give the leash corollary.The Third CorollaryThen comes the third corollary to the Nikhilam sutra which relates to a very special flake of multiplication and which is not frequently in requisition elsewhere but is a good deal required in numeric astronomy etc. It relates to and provides for multiplications where the multiplier digits consists entirely of nines.The procedure applicable in this case is consequently evidently as followsi) appoint the multiplicand off by a vertical line into a right hand persona consisting of as many digits as the multiplier and get off the ground from the multiplicand one more than the whole specialiveness circle on the l eft. This gives us the left hand side piece of land of the product or take the Ekanyuna and subtract therefrom the previous i.e. the excess muckle on the left and ii) Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product.The sideline example will make it clearThe Third Sutra rdhva Tiryagbhymrdhva Tiryagbhym sutra which is the General Formula applicable to all cases of multiplication and will also be found very useful later(prenominal) on in the division of a spectacular number by some other large number.The formula itself is very short and terse, consisting of only one aggregate word and meat vertically and cross-wise. The applications of this brief and terse sutra are manifold.A simple example will suffice to clarify the modus operandi thereof. Suppose we ready to multiply 12 by 13.(i) We multiply the left hand most digit 1 of the multiplicand vertically by the left hand most digit 1 of the multipl ier get their product 1 and set down as the left hand most part of the answer(ii) We then multiply 1 and 3 and 1 and 2 cross slipway add the two get 5 as the sum and set it down as the spunk part of the answer and(iii) We multiply 2 and 3 vertically get 6 as their product and put it down as the stretch forth the right hand most part of the answer. Thus 12 13 = 156.The Fourth Sutra equationvartya YojayetThe end point Parvartya Yojayet which means Transpose and Apply. Here he cl subscribes that the Vedic system gave a number is applications one of which is discussed here. The very acceptance of the existence of polynomials and the consequent proportionality theorem during the Vedic times is a big suspense so we dont proclivity to give this application to those polynomials.However the quadruple steps given by them in the polynomial division are given below Divide x3 + 72 + 6x + 5 by x 2. i. x3 divided by x gives us x2 which is wherefore the prototypal verge of the quotient x2 2 = 2x2 but we have 7x2 in the divident. This means that we have to get 9x2 more. This must result from the multiplication of x by 9x. Hence the 2nd shape of the divisor must be 9xAs for the third boundary we already have 2 9x = 18x. But we have 6x in the dividend. We must therefore get an additional 24x. Thus can only come in by the multiplication of x by 24. This is the third term of the quotient.Q = x2 + 9x + 24Now the last term of the quotient multiplied by 2 gives us 48. But the compulsory term in the dividend is 5. We have therefore to get an additional 53 from some where. But there is no further term left in the dividend. This means that the 53 will remain as the remainder Q = x2 + 9x + 24 andR = 53.The Fifth Sutra Snyam SamyasamuccayeSamuccaya is a technical term which has some(prenominal) meanings in different contexts which we shall explain one at a time.Samuccaya foremost means a term which occurs as a uncouth operator in all the terms concerned. Samucca ya twinklingly means the product of self-sustaining terms. Samuccaya thirdly means the sum of the denominators of two fractions having same numerical numerator. quadrupletthly Samuccaya means combination or full(a). Fifth meaning With the same meaning i.e. total of the word (Samuccaya) there is a fifth kind of application possible with quadratic equations.Sixth meaning With the same nose out (total of the word Samuccaya) but in a different application it comes in skilled to solve harder equations equated to zero.Thus one has to imagine how the six shades of meanings have been perceived by the Jagadguru Sankaracharya that too from the Vedas when such types of equations had not even been invented in the world at that point of time.The Sixth Sutra nurpye nyamanyatAs said by Dani 32 we see the 6th sutra happens to be the subsutra of the first sutra. Its mention is do in pp. 51, 74, 249 and 286 of 51. The two small subsutras (i) Anurpyena and (ii) Adayamadyenantyamantyena of the sutras 1 and 3 which mean pro rata and the first by the first and the last by the last.Here the later subsutra acquires a new and beautiful double application and significance. It works out as followsi. Split the set coefficient into two such parts so that the ratio of the first coefficient to the first part is the same as the ratio of that minute of arc part to the last coefficient. Thus in the quadratic 2x2 + 5x + 2 the middle term 5 is split into two such parts 4 and 1 so that the ratio of the first coefficient to the first part of the middle coefficient i.e. 2 4 and the ratio of the second part to the last coefficient i.e. 1 2 are the same. Now this ratio i.e. x + 2 is one factor.ii. And the second factor is obtained by dividing the first coefficient of the quadratic by the first coefficient of the factor already found and the last coefficient of the quadratic by the last coefficient of that factor. In other words the second binomial factor is obtained thusThus 22 + 5x + 2 = (x + 2) (2x + 1). This sutra has Yavadunam Tavadunam to be its subsutra which the book claims to have been utilise.The 7th Sutra Sankalana VyavakalanbhymSankalana Vyavakalan process and the Adyamadya rule together from the seventh sutra. The procedure choose is one of alternate destruction of the highest and the lowest powers by a expireting multiplication of the coefficients and the addition or discount of the multiples.A concrete example will elucidate the process.Suppose we have to find the HCF (Highest ecumenic factor)of (x2 + 7x + 6) and x2 5x 6x2 + 7x + 6 = (x + 1) (x + 6) andx2 5x 6 = (x + 1) ( x 6)the HCF is x + 1 but where the sutra is deployed is not clear.The Eight Sutra PuranpuranbhymPuranpuranbhym means by the shutdown or not completion of the square or the cube or aside power etc. But when the very existence of polynomials, quadratic equations etc. was not defined it is a miracle the Jagadguru could contemplate of the completion of squares (quadratic ) blockish and forward degree equation. This has a subsutra Antyayor dasakepi use of which is not mentioned in that section.The ordinal Sutra verapamil kalanbhymThe term (Calan kalanbhym) means differential calculus concord to Jagadguru Sankaracharya.The ordinal Sutra YvadnamYvadnam Sutra (for cubing) is the tenth sutra. It has a subsutra called Samuccayagunitah.The Eleventh Sutra Vyastisamastih SutraVyastisamastih sutra teaches one how to use the average or exact middle binomial for falling out the biquadratic equation down into a simple quadratic by the easy maneuver of mutual cancellations of the odd powers. However the modus operandi is missing.The Twelfth Sutra esnyankena CaramenaThe sutra esnyankena Caramena means The remainders by the last digit. For instance if one wants to find decimal value of 1/7. The remainders are 3, 2, 6, 4, 5 and 1. Multiplied by 7 these remainders give successively 21, 14, 42, 28, 35 and 7. Ignoring the left hand side digits we plain put down the last digit of each product and we get 1/7 = .14 28 57 Now this 12th sutra has a subsutra Vilokanam. Vilokanam means mere observation He has given a few small examples for the same.The Thirteen Sutra SopantyadvayamantyamThe sutra Sopantyadvayamantyam means the ultimate and twice the penultimate which gives the answer immediately. No mention is made about the immediate subsutra.The illustration given by them.The proof of this is as follows.The General Algebraic Proof is as follows. allow d be the common differenceCanceling the factors A (A + d) of the denominators and d ofthe numeratorsIt is a pity that all samples given by the book form a special pattern.The Fourteenth Sutra Ekanynena PrvenaThe Ekanynena Prvena Sutra sounds as if it were the conference of the Ekadhika Sutra. It actually relates and provides for multiplications where the multiplier the digits consists entirely of nines. The procedure applicable in this case is therefore evidently as follows.For instance 43 9.i . Divide the multiplicand off by a vertical line into a right hand portion consisting of as many digits as the multiplier and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product or take the Ekanyuna and subtract it from the previous i.e. the excess portion on the left andii. Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the productThe Fifthteen Sutra GunitasamuccayahGunitasamuccayah rule i.e. the commandment already explained with union to the Sc of the product being the same as the product of the Sc of the factors.Let us take a concrete example and see how this method (p. 81) can be made use of. Suppose we have to factorise x3 + 6x2 + 11x + 6 and by some method, we know (x + 1) to be a factor. We first use the corollary of the 3rd sutra viz.Adayamadyena formula and thus automatically put down x2 and 6 as the first and the last coefficients in the quotient i.e. the product of the remaining two binomial factors. But we know already that the Sc of the given expression is 24 and as the Sc of (x + 1) = 2 we therefore know that the Sc of the quotient must be 12. And as the first and the last digits thereof are already known to be 1 and 6, their total is 7. And therefore the middle term must be 12 7 = 5. So, the quotient x2 + 5x + 6.This is a very simple and easy but abruptly certain and effective process.The Sixteen Sutra Gunakasamuccayah.It means the product of the sum of the coefficients in the factors is equal to the sum of the coefficients in the product.In symbolizationisations we may put this principle as follows Sc of the product = Product of the Sc (in factors).For example (x + 7) (x + 9) = x2 + 16 x + 63 and we observe (1 + 7) (1 + 9) = 1 + 16 + 63 = 80. in like manner in the case of cubics, biquadratics etc. the same rule holds goodness.For example (x + 1) (x + 2) (x + 3) = x3 + 62 + 11 x + 6 2 3 4 = 1 + 6 + 11 + 6 = 24.Thus if and when some factors are known this rule helps us to fill in the gaps.LiteratureResearch has attested the difficulties students caseful in algebra and how these can a lot be traced to their limited judgment of numbers and their operations (Stacey MacGregor, 1997 Warren, 2001). Of growing concern is the artificial separation of algebra and arithmetic, since noesis of numeric grammatical construction seems essential for a successful transition. In particular, this mathematical structure is concerned with (i) relationships between quantities, (ii) group properties of operations, (iii) relationships between the operations and (iv) Relationships across the quantities (Warren, 2003). Thus it has been suggested by Stacey and MacGregor (1997) that the best preparation for learning algebra is a good judgement of how the arithmetic system works. An understanding of the general properties of numbers and the relationships between them may be cru cial, and students need to have thought about the general effects of operations on numbers (MacGregor Stacey, 1999). This study sought to test the supposal that arithmetic knowledge can improve algebraic cogency by applying a Vedic method of multiplying arithmetic numbers to algebra, establish on the similarity of structural presentation.Vedic mathematics has its origins in the ancient Indian texts, the Vedas, an integrated and holistic system of knowledge composed in Sanskrit and genetic orally from one generation to the next. The first versions of these texts were possibly put down around 2000 BC, and the works lead the genesis of the modern science of mathematics (number, geometry and algebra) and astronomy in India (Datta Singh, 2001 Joseph, 2000). Sri Tirthaji (1965) has expounded 16 sutras or word formulas and 13 sub-sutras that he claims have been reconstructed from the Vedas.The sutras, or rules as aphorisms, are condensed statements of a very precise nature, written in a poetic style and traffic with different concepts (Joseph, 2000 Shan Bailey, 1991). A sutra, which literally means thread, expresses fundamental principles and may contain a rule, an idea, a mnemonic or a method of working based on fundamental principles that run like threads through diverse mathematical topics, unifying them. As Williams (2002) describes themWe use our mind in certain specific ways we might extend an idea or reverse it or compare or combine it with another. Each of these types of mental occupation is described by one of the Vedic sutras. They describe the ways in which the mind can work and so they tell the student how to go about solving a problem. (Williams, 2002, p. 2).Examples of the sutras are the vertically and cross-section(prenominal) sutra, which embodies a method of multiplication with applications to determinants, simultaneous equations, and trigonometric functions, etc. (this is the sutra used in the research reported here see recruit 3), an d the every last(predicate) from nine and the last from ten sutra that may be used in subtraction, vincula, multiplication and division.Barnard and Tall (1997, p. 41) have introduced the idea of a cognitive unit, A wind of cognitive structure that can be held in the reduce of precaution all at one time, and may include other ideas that can be immediately linked to it.This enables compression of ideas, so that a assemblage of ideas or symbols that is too big for the focus of attention can be compressed into a sensation unit. It seems as if the sutras nicely fit this description, with the mnemonic or other memory device being used as a peg to hang the collection of ideas on. Thus the hypothetical advantage of using the sutras is that they allow encapsulation of a process into a teachable chunk, or cognitive unit, that can then be processed more easily, sometimes using a visual reminder, such as in theVertically and Crosswise sutra. Here the essential procedure is signified ho listically by the symbol 5, unlike the symbol bewilder that signifies in turn four separate procedures. It might be possible for a symbol such as to be used in much the same way for FOIL, but this may appear more visually complex, and it is not commonly separated from the accompanying binomials like this. In this way sutras often make use of the power of visualisation, which has been shown to be effective in learning in various areas of mathematics (Booth Thomas, 2000 Presmeg, 1986 van Hiele, 2002). Such visualisation accesses the brains holistic exertion (Tall Thomas, 1991) and intuition, and this assists in providing an overview of the mathematical structure.The sutras also aid intuitive intellection (Williams, 2002) and being based on patterns and mnemonics they make recall much easier, cut back the cognitive load on the individual (Morrow, 1998 Sweller, 1994). The sutras were originally envisaged as applying both to arithmetic and algebra, and Joseph (2000) and Bhatanagar (1976) have explained that since polynomials may be perceived as simply arithmetic dates, the principles apply equally well to them. This research considered a possible function of the vertically and Crosswise sutra for improving facility with, and understanding of, the amplification of algebraic binomials and the factoring of quadratic expressions.MethodologyThe research employed a case study methodology, using a single class of class 10 (age 15 years) students. The school used is a co-educational state unessential school in Auckland, New Zealand and the class contained 19 students, 11 male childs and 8 girls. The students, who included 9 recent immigrants, were drawn from several(prenominal) ethnic backgrounds, and accordingly have been exposed to different approaches and teaching purlieus with respect to learning mathematics. This also meant that nine of the students have a first voice communication other than English and these language difficulties tend to hinder thei r learning (for example, iii of the students are on a literacy program at the school).Two unknown questionnaires (see human body 1 for some questions from the second) were constructed using concepts we identified as key in developing a structural understanding of binomial elaboration and factorisation, such as testing the concept of a factor and the faculty to apply a procedure in reverse. Questions included multiplication of numbers multiplication of binomial expressions factorisation of quadratic expressions word problems on addition and subtraction of like terms and expansion of expressions in a serviceable context. approximately questions also involved description of procedures and meanings attached to words. In particular, the second questionnaire contained items on the use of the Vedic method applied to binomial expansion and factorisation.The lessons were taught by the first-named author in 2003 in a supportive schoolroom environment that encouraged student-to-student and teacherstudent interactions. Students were assured that the teacher was genuinely interested in their mathematical thinking and respected their attempts, that it was fine to make faults and that understanding how the mistake occurred was a learning opportunity for everyone concerned. Students were encouraged to explain and view as the asperity of their answers, and arrogant contributions were praised.The first teaching session comprised work on multiplication of numbers and revision of work on algebra that the students had learned in social class 9 (age 14 years). Substitution, collection of like terms and multiplication of a binomialexpression by a single value were revised, using, for example, expressions such as 5(x 4), (p + 2)4, and k(4 + k). Students were also reminded of the meanings of words such as term, expression, factor, expansion, coefficient and simplify. Diagrammatic representations of 3(5 + 6) and k(4 + k) using rectangles were drawn and discussed, and then students drew similar rectangle diagrams representing multiplications such as (3 + 5)(2 + 5) and (k + 2)(k + 4) (see Figure 2). by-line a review of factorisation of expressions such as 15p + 10, the FOIL (First, Outside, Inside, stand up) method of expanding binomials was taught, where the First terms in each bracket are multiplied together, then the Outside terms, the Inside terms and then the Last term in each bracket, to give four products. Finally factorisation of quadratic expressions, followed by a guess and check method for factorising quadratic expressions was covered. The students did not find these topics easy, especially factorising of quadratic expressions. This took a total of four hours, afterwards which, questionnaire one was administered.Students were then exposed for one hour to the Vedic vertically and cross-section(prenominal) method, where initially they practised multiplying two- and triplet-digit numbers with this approach. Subsequently, the next three ho urs were worn out(p) expanding binomials and factorising quadratic expressions with the vertically and crosswise method. This method (see Figure 3) involves a sequence of four multiplications, the answers to each of which are placed into a single answer line. The middle two terms are added together mentally to run the final answer.ResultsThe first question (1a) in each questionnaire was a two-digit multiplication.In the first, it was 37 58, and the second 23 47, and in this second case the question asked that this be done by the vertically and crosswise method. The aim was to check students facility with arithmetic multiplication and to see if the vertically and crosswise sutra was of tending in this area. In the event 11 of the 18 (61%) students who consummate both questionnaires, correctly answered the question in the first test, and 13 (72%) in the second, with only one student not using the sutra in that test. thither was no statistical difference between these proportions (c2 = 0.125). When asked to explain what they had done using the Vedic approach, students who were able to write down the final answer were often able to write something like student 4s explanation for 1c), 32 692 times 9 is 18 3 times 9 +2 times 6 is 39 + carried 1 = 403 times 6 + carried 4 = 22Expansion of binomialsA thickset of the results in the first of the algebra questions (Q2 see FigureVedic Mathematics MultiplicationVedic Mathematics MultiplicationAbstractVedic Mathematics has been the rage in American schools. The clear difference between Asian Indians and average American students approach to solving math problems had been evident for many years, finally prompting concerted research efforts into the subject. Many students have conventionally found the processes of algebraic manipulation, especially factorisation, difficult to learn. Research studies have investigated the value of introducing students to a Vedic method of multiplication of numbers that is very visual i n its application. The question was whether applying the method to quadratic expressions would improve student understanding, not only of the processes but also the concepts of expansion and factorisation. It was established that there was some evidence that this was the case, and that some students also preferred to use the new method.IntroductionIs Vedic mathematics a kind of magic? American students certainly thought so, in seeing the clear edge it gave to their Asian counterparts in public and private schools. Vedic schools and even tuition centers are advertised on the Web. Clearly it has taken the world by storm, and for valid reasons. The results are evident in math scores for every test administered.Vedic mathematics is based on some ancient, but superb logic. And the truth is that it works. Small wonder that it hails from India, purported to be the land that gave us the Zero or cipher. This one digit is the basis for counting or carrying over beyond nine- and is in fact the basis of our whole number system. It is the Arabs and the Indians that we should be indebted to for this favour to the West.The other thing about Vedic mathematics is that it also allows one to counter check whether his or her answer is correct. Thus one is doubly assured of the results. Sometimes this can be done by the Indian student in a shorter time span than it can using the traditional counting and formulas we have developed through Western and European mathematicians. That makes it seem all the more marvellous.If that doesnt sound magical enough, its interesting to note that the word Vedic means coming from Vedas a Sanskrit word meaning divinely revealed. The Hindus believe that these basic truths were revealed to holy men directly once they had achieved a certain position on the path to spirituality. Also certain incantations such as Om are said to have been revealed by the Heavens themselves.According to popular beliefs, Vedic Mathematics is the ancient system of Mathemati cs which was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960). According to him, all Mathematics is based on sixteen Sutras or word-formulas. Based on Vedic logic, these formulas solve the problem in the way the mind naturally works and are therefore a great help to the student of logic.Perhaps the most outstanding feature of the Vedic system is its coherence. The whole system is beautifully consistent and unified- the general multiplication method, for example, is easily reversed to allow one-line divisions and the simple squaring method can be reversed to give one-line square roots. Added to that, these are all simply understood. This unifying quality is very satisfying, as it makes learning mathematics easy and enjoyable.The Vedic system also provides for the solution of difficult problems in parts they can then be combined to solve the whole problem by the Vedic method. These magical yet logical methods are but a part of the whole syste m of Vedic mathematics which is far more systematic than the modern Western system. In fact it is safe to say that Vedic Mathematics manifests the coherent and unified structure of mathematics and the methods are complementary, straight and easy.The ease of Vedic Mathematics means that calculations can be carried out mentally-though the methods can also be written down. There are many advantages in using a flexible, mental system. Pupils can invent their own methods, they are not limited to the one accurate method. This leads to more creative, fascinated and intelligent pupils.Interest in the Vedic system is increasing in education where mathematics teachers are looking for something better. Finding the Vedic system is the answer. Research is being carried out in many areas as well as the effects of learning Vedic Maths on children developing new, powerful but easy applications of the Vedic Sutras in geometry, calculus, computing etc.But the real beauty and success of Vedic Mathemat ics cannot be fully appreciated without actually practising the system. One can then see that it is perhaps the most sophisticated and efficient mathematical system possible.Now having known that even the 16 sutras are the Jagadguru Sankaracharyas invention we mention the name of the sutras and the sub sutras or corollaries in this paper.The First Sutra Ekdhikena PrvenaThe relevant Sutra reads Ekdhikena Prvena which rendered into English simply says By one more than the previous one.Its application and modus operandi are as follows.(1) The last digit of the denominator in this case being 1 and the previous one being 1 one more than the previous one evidently means 2. Further the proposition by (in the sutra) indicates that the arithmetical operation prescribed is either multiplication or division. Let us first deal with the case of a fraction say 1/19. 1/19 where denominator ends in 9. By the Vedic one line mental method.A. First methodB. Second MethodThis is the whole working. And the modus operandi is explained below.Modus operandi chart is as follows(i) We put down 1 as the right-hand most digit 1(ii) We multiply that last digit 1 by 2 and put the 2 down as the immediately preceding digit.(iii) We multiply that 2 by 2 and put 4 down as the next previous digit.(iv) We multiply that 4 by 2 and put it down thus 8 4 2 1(v) We multiply that 8 by 2 and get 16 as the product. But this has two digits. We therefore put the product. But this has two digits we therefore put the 6 down immediately to the left of the 8 and keep the 1 on hand to be carried over to the left at the next step (as we always do in all multiplication e.g. of 69 2 = 138 and so on).(vi) We now multiply 6 by 2 get 12 as product, add thereto the 1 (kept to be carried over from the right at the last step), get 13 as the consolidated product, put the 3 down and keep the 1 on hand for carrying over to the left at the next step.(vii) We then multiply 3 by 2 add the one carried over from the right on e, get 7 as the consolidated product. But as this is a single digit number with nothing to carry over to the left, we put it down as our next multiplicand.(viii) and xviii) we follow this procedure continually until we reach the 18th digit counting leftwards from the right, when we find that the whole decimal has begun to repeat itself. We therefore put up the usual recurring marks (dots) on the first and the last digit of the answer (from betokening that the whole of it is a Recurring Decimal) and stop the multiplication there.Our chart now reads as followsThe Second Sutra Nikhilam Navatacaramam DaatahNow we proceed on to the next sutra Nikhilam sutra The sutra reads Nikhilam Navatacaramam Daatah, which literally translated means all from 9 and the last from 10. We shall and applications of this cryptical-sounding formula and then give details about the three corollaries. He has given a very simple multiplication.Suppose we have to multiply 9 by 7.1. We should take, as base for our calculations that power of 10 which is nearest to the numbers to be multiplied. In this case 10 itself is that power.Put the numbers 9 and 7 above and below on the left hand side (as shown in the working alongside here on the right hand side margin)3. Subtract each of them from the base (10) and write down the remainders (1 and 3) on the right hand side with a connecting minus sign () between them, to show that the numbers to be multiplied are both of them less than 10.4. The product will have two parts, one on the left side and one on the right. A vertical dividing line may be drawn for the purpose of demarcation of the two parts.5. Now, Subtract the base 10 from the sum of the given numbers (9 and 7 i.e. 16). And put (16 10) i.e. 6 as the left hand part of the answer 9 + 7 10 = 6The First CorollaryThe first corollary naturally arising out of the Nikhilam Sutra reads in English whatever the extent of its deficiency lessen it still further to that very extent, and also set up the square of that deficiency.This evidently deals with the squaring of the numbers. A few elementary examples will suffice to make its meaning and application clearSuppose one wants to square 9, the following are the successive stages in our mental working.(i) We would take up the nearest power of 10, i.e. 10 itself as our base.(ii) As 9 is 1 less than 10 we should decrease it still further by 1 and set 8 down as our left side portion of the answer 8/ (iii) And on the right hand we put down the square of that deficiency 12(iv) Thus 92 = 81The Second CorollaryThe second corollary in applicable only to a special case under the first corollary i.e. the squaring of numbers ending in 5 and other cognate numbers. Its wording is exactly the same as that of the sutra which we used at the outset for the conversion of vulgar fractions into their recurring decimal equivalents. The sutra now takes a totally different meaning and in fact relates to a wholly different setup and context.Its literal meaning is the same as before (i.e. by one more than the previous one) but it now relates to the squaring of numbers ending in 5. For example we want to multiply 15. Here the last digit is 5 and the previous one is 1. So one more than that is 2.Now sutra in this context tells us to multiply the previous digit by one more than itself i.e. by 2. So the left hand side digit is 1 2 and the right hand side is the vertical multiplication product i.e. 25 as usual.Thus 152 = 1 2 / 25 = 2 / 25.Now we proceed on to give the third corollary.The Third CorollaryThen comes the third corollary to the Nikhilam sutra which relates to a very special type of multiplication and which is not frequently in requisition elsewhere but is often required in mathematical astronomy etc. It relates to and provides for multiplications where the multiplier digits consists entirely of nines.The procedure applicable in this case is therefore evidently as followsi) Divide the multiplicand off by a vertical line int o a right hand portion consisting of as many digits as the multiplier and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product or take the Ekanyuna and subtract therefrom the previous i.e. the excess portion on the left and ii) Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product.The following example will make it clearThe Third Sutra rdhva Tiryagbhymrdhva Tiryagbhym sutra which is the General Formula applicable to all cases of multiplication and will also be found very useful later on in the division of a large number by another large number.The formula itself is very short and terse, consisting of only one compound word and means vertically and cross-wise. The applications of this brief and terse sutra are manifold.A simple example will suffice to clarify the modus operandi thereof. Suppose we have to multiply 12 by 13.( i) We multiply the left hand most digit 1 of the multiplicand vertically by the left hand most digit 1 of the multiplier get their product 1 and set down as the left hand most part of the answer(ii) We then multiply 1 and 3 and 1 and 2 crosswise add the two get 5 as the sum and set it down as the middle part of the answer and(iii) We multiply 2 and 3 vertically get 6 as their product and put it down as the last the right hand most part of the answer. Thus 12 13 = 156.The Fourth Sutra Parvartya YojayetThe term Parvartya Yojayet which means Transpose and Apply. Here he claims that the Vedic system gave a number is applications one of which is discussed here. The very acceptance of the existence of polynomials and the consequent remainder theorem during the Vedic times is a big question so we dont wish to give this application to those polynomials.However the four steps given by them in the polynomial division are given below Divide x3 + 72 + 6x + 5 by x 2. i. x3 divided by x gives u s x2 which is therefore the first term of the quotientx2 2 = 2x2 but we have 7x2 in the divident. This means that we have to get 9x2 more. This must result from the multiplication of x by 9x. Hence the 2nd term of the divisor must be 9xAs for the third term we already have 2 9x = 18x. But we have 6x in the dividend. We must therefore get an additional 24x. Thus can only come in by the multiplication of x by 24. This is the third term of the quotient.Q = x2 + 9x + 24Now the last term of the quotient multiplied by 2 gives us 48. But the absolute term in the dividend is 5. We have therefore to get an additional 53 from some where. But there is no further term left in the dividend. This means that the 53 will remain as the remainder Q = x2 + 9x + 24 andR = 53.The Fifth Sutra Snyam SamyasamuccayeSamuccaya is a technical term which has several meanings in different contexts which we shall explain one at a time.Samuccaya firstly means a term which occurs as a common factor in all the terms concerned. Samuccaya secondly means the product of independent terms. Samuccaya thirdly means the sum of the denominators of two fractions having same numerical numerator.Fourthly Samuccaya means combination or total. Fifth meaning With the same meaning i.e. total of the word (Samuccaya) there is a fifth kind of application possible with quadratic equations.Sixth meaning With the same sense (total of the word Samuccaya) but in a different application it comes in handy to solve harder equations equated to zero.Thus one has to imagine how the six shades of meanings have been perceived by the Jagadguru Sankaracharya that too from the Vedas when such types of equations had not even been invented in the world at that point of time.The Sixth Sutra nurpye nyamanyatAs said by Dani 32 we see the 6th sutra happens to be the subsutra of the first sutra. Its mention is made in pp. 51, 74, 249 and 286 of 51. The two small subsutras (i) Anurpyena and (ii) Adayamadyenantyamantyena of the s utras 1 and 3 which mean proportionately and the first by the first and the last by the last.Here the later subsutra acquires a new and beautiful double application and significance. It works out as followsi. Split the middle coefficient into two such parts so that the ratio of the first coefficient to the first part is the same as the ratio of that second part to the last coefficient. Thus in the quadratic 2x2 + 5x + 2 the middle term 5 is split into two such parts 4 and 1 so that the ratio of the first coefficient to the first part of the middle coefficient i.e. 2 4 and the ratio of the second part to the last coefficient i.e. 1 2 are the same. Now this ratio i.e. x + 2 is one factor.ii. And the second factor is obtained by dividing the first coefficient of the quadratic by the first coefficient of the factor already found and the last coefficient of the quadratic by the last coefficient of that factor. In other words the second binomial factor is obtained thusThus 22 + 5x + 2 = (x + 2) (2x + 1). This sutra has Yavadunam Tavadunam to be its subsutra which the book claims to have been used.The Seventh Sutra Sankalana VyavakalanbhymSankalana Vyavakalan process and the Adyamadya rule together from the seventh sutra. The procedure adopted is one of alternate destruction of the highest and the lowest powers by a suitable multiplication of the coefficients and the addition or subtraction of the multiples.A concrete example will elucidate the process.Suppose we have to find the HCF (Highest Common factor)of (x2 + 7x + 6) and x2 5x 6x2 + 7x + 6 = (x + 1) (x + 6) andx2 5x 6 = (x + 1) ( x 6)the HCF is x + 1 but where the sutra is deployed is not clear.The Eight Sutra PuranpuranbhymPuranpuranbhym means by the completion or not completion of the square or the cube or forth power etc. But when the very existence of polynomials, quadratic equations etc. was not defined it is a miracle the Jagadguru could contemplate of the completion of squares (quadratic) cubic an d forth degree equation. This has a subsutra Antyayor dasakepi use of which is not mentioned in that section.The Ninth Sutra Calan kalanbhymThe term (Calan kalanbhym) means differential calculus according to Jagadguru Sankaracharya.The Tenth Sutra YvadnamYvadnam Sutra (for cubing) is the tenth sutra. It has a subsutra called Samuccayagunitah.The Eleventh Sutra Vyastisamastih SutraVyastisamastih sutra teaches one how to use the average or exact middle binomial for breaking the biquadratic down into a simple quadratic by the easy device of mutual cancellations of the odd powers. However the modus operandi is missing.The Twelfth Sutra esnyankena CaramenaThe sutra esnyankena Caramena means The remainders by the last digit. For instance if one wants to find decimal value of 1/7. The remainders are 3, 2, 6, 4, 5 and 1. Multiplied by 7 these remainders give successively 21, 14, 42, 28, 35 and 7. Ignoring the left hand side digits we simply put down the last digit of each product and we get 1/7 = .14 28 57 Now this 12th sutra has a subsutra Vilokanam. Vilokanam means mere observation He has given a few trivial examples for the same.The Thirteen Sutra SopantyadvayamantyamThe sutra Sopantyadvayamantyam means the ultimate and twice the penultimate which gives the answer immediately. No mention is made about the immediate subsutra.The illustration given by them.The proof of this is as follows.The General Algebraic Proof is as follows.Let d be the common differenceCanceling the factors A (A + d) of the denominators and d ofthe numeratorsIt is a pity that all samples given by the book form a special pattern.The Fourteenth Sutra Ekanynena PrvenaThe Ekanynena Prvena Sutra sounds as if it were the converse of the Ekadhika Sutra. It actually relates and provides for multiplications where the multiplier the digits consists entirely of nines. The procedure applicable in this case is therefore evidently as follows.For instance 43 9.i. Divide the multiplicand off by a vertical lin e into a right hand portion consisting of as many digits as the multiplier and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product or take the Ekanyuna and subtract it from the previous i.e. the excess portion on the left andii. Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the productThe Fifthteen Sutra GunitasamuccayahGunitasamuccayah rule i.e. the principle already explained with regard to the Sc of the product being the same as the product of the Sc of the factors.Let us take a concrete example and see how this method (p. 81) can be made use of. Suppose we have to factorize x3 + 6x2 + 11x + 6 and by some method, we know (x + 1) to be a factor. We first use the corollary of the 3rd sutra viz.Adayamadyena formula and thus mechanically put down x2 and 6 as the first and the last coefficients in the quotient i.e. the product o f the remaining two binomial factors. But we know already that the Sc of the given expression is 24 and as the Sc of (x + 1) = 2 we therefore know that the Sc of the quotient must be 12. And as the first and the last digits thereof are already known to be 1 and 6, their total is 7. And therefore the middle term must be 12 7 = 5. So, the quotient x2 + 5x + 6.This is a very simple and easy but absolutely certain and effective process.The Sixteen Sutra Gunakasamuccayah.It means the product of the sum of the coefficients in the factors is equal to the sum of the coefficients in the product.In symbols we may put this principle as follows Sc of the product = Product of the Sc (in factors).For example (x + 7) (x + 9) = x2 + 16 x + 63 and we observe (1 + 7) (1 + 9) = 1 + 16 + 63 = 80.Similarly in the case of cubics, biquadratics etc. the same rule holds good.For example (x + 1) (x + 2) (x + 3) = x3 + 62 + 11 x + 6 2 3 4 = 1 + 6 + 11 + 6 = 24.Thus if and when some factors are known this r ule helps us to fill in the gaps.LiteratureResearch has documented the difficulties students face in algebra and how these can often be traced to their limited understanding of numbers and their operations (Stacey MacGregor, 1997 Warren, 2001). Of growing concern is the artificial separation of algebra and arithmetic, since knowledge of mathematical structure seems essential for a successful transition. In particular, this mathematical structure is concerned with (i) relationships between quantities, (ii) group properties of operations, (iii) relationships between the operations and (iv) Relationships across the quantities (Warren, 2003). Thus it has been suggested by Stacey and MacGregor (1997) that the best preparation for learning algebra is a good understanding of how the arithmetic system works. An understanding of the general properties of numbers and the relationships between them may be crucial, and students need to have thought about the general effects of operations on nu mbers (MacGregor Stacey, 1999). This study sought to test the hypothesis that arithmetic knowledge can improve algebraic ability by applying a Vedic method of multiplying arithmetic numbers to algebra, based on the similarity of structural presentation.Vedic mathematics has its origins in the ancient Indian texts, the Vedas, an integrated and holistic system of knowledge composed in Sanskrit and transmitted orally from one generation to the next. The first versions of these texts were possibly recorded around 2000 BC, and the works contain the genesis of the modern science of mathematics (number, geometry and algebra) and astronomy in India (Datta Singh, 2001 Joseph, 2000). Sri Tirthaji (1965) has expounded 16 sutras or word formulas and 13 sub-sutras that he claims have been reconstructed from the Vedas.The sutras, or rules as aphorisms, are condensed statements of a very precise nature, written in a poetic style and dealing with different concepts (Joseph, 2000 Shan Bailey, 199 1). A sutra, which literally means thread, expresses fundamental principles and may contain a rule, an idea, a mnemonic or a method of working based on fundamental principles that run like threads through diverse mathematical topics, unifying them. As Williams (2002) describes themWe use our mind in certain specific ways we might extend an idea or reverse it or compare or combine it with another. Each of these types of mental activity is described by one of the Vedic sutras. They describe the ways in which the mind can work and so they tell the student how to go about solving a problem. (Williams, 2002, p. 2).Examples of the sutras are the Vertically and Crosswise sutra, which embodies a method of multiplication with applications to determinants, simultaneous equations, and trigonometric functions, etc. (this is the sutra used in the research reported here see Figure 3), and the All from nine and the last from ten sutra that may be used in subtraction, vincula, multiplication and d ivision.Barnard and Tall (1997, p. 41) have introduced the idea of a cognitive unit, A piece of cognitive structure that can be held in the focus of attention all at one time, and may include other ideas that can be immediately linked to it.This enables compression of ideas, so that a collection of ideas or symbols that is too big for the focus of attention can be compressed into a single unit. It seems as if the sutras nicely fit this description, with the mnemonic or other memory device being used as a peg to hang the collection of ideas on. Thus the theoretical advantage of using the sutras is that they allow encapsulation of a process into a manageable chunk, or cognitive unit, that can then be processed more easily, sometimes using a visual reminder, such as in theVertically and Crosswise sutra. Here the essential procedure is signified holistically by the symbol 5, unlike the symbol FOIL that signifies in turn four separate procedures. It might be possible for a symbol such as to be used in much the same way for FOIL, but this may appear more visually complex, and it is not usually separated from the accompanying binomials like this. In this way sutras often make use of the power of visualisation, which has been shown to be effective in learning in various areas of mathematics (Booth Thomas, 2000 Presmeg, 1986 van Hiele, 2002). Such visualisation accesses the brains holistic activity (Tall Thomas, 1991) and intuition, and this assists in providing an overview of the mathematical structure.The sutras also aid intuitive thinking (Williams, 2002) and being based on patterns and mnemonics they make recall much easier, reducing the cognitive load on the individual (Morrow, 1998 Sweller, 1994). The sutras were originally envisaged as applying both to arithmetic and algebra, and Joseph (2000) and Bhatanagar (1976) have explained that since polynomials may be perceived as simply arithmetic sequences, the principles apply equally well to them. This research con sidered a possible role of the vertically and Crosswise sutra for improving facility with, and understanding of, the expansion of algebraic binomials and the factorisation of quadratic expressions.MethodologyThe research employed a case study methodology, using a single class of Year 10 (age 15 years) students. The school used is a co-educational state secondary school in Auckland, New Zealand and the class contained 19 students, 11 boys and 8 girls. The students, who included 9 recent immigrants, were drawn from several cultural backgrounds, and accordingly have been exposed to different approaches and teaching environments with respect to learning mathematics. This also meant that nine of the students have a first language other than English and these language difficulties tend to hinder their learning (for example, three of the students are on a literacy program at the school).Two anonymous questionnaires (see Figure 1 for some questions from the second) were constructed using co ncepts we identified as important in developing a structural understanding of binomial expansion and factorisation, such as testing the concept of a factor and the ability to apply a procedure in reverse. Questions included multiplication of numbers multiplication of binomial expressions factorisation of quadratic expressions word problems on addition and subtraction of like terms and expansion of expressions in a practical context. Some questions also involved description of procedures and meanings attached to words. In particular, the second questionnaire contained items on the use of the Vedic method applied to binomial expansion and factorisation.The lessons were taught by the first-named author in 2003 in a supportive classroom environment that encouraged student-to-student and teacherstudent interactions. Students were assured that the teacher was genuinely interested in their mathematical thinking and respected their attempts, that it was fine to make mistakes and that unders tanding how the mistake occurred was a learning opportunity for everyone concerned. Students were encouraged to explain and check the validity of their answers, and positive contributions were praised.The first teaching session comprised work on multiplication of numbers and revision of work on algebra that the students had learned in Year 9 (age 14 years). Substitution, collection of like terms and multiplication of a binomialexpression by a single value were revised, using, for example, expressions such as 5(x 4), (p + 2)4, and k(4 + k). Students were also reminded of the meanings of words such as term, expression, factor, expansion, coefficient and simplify. Diagrammatic representations of 3(5 + 6) and k(4 + k) using rectangles were drawn and discussed, and then students drew similar rectangle diagrams representing multiplications such as (3 + 5)(2 + 5) and (k + 2)(k + 4) (see Figure 2). Following a review of factorisation of expressions such as 15p + 10, the FOIL (First, Outsid e, Inside, Last) method of expanding binomials was taught, where the First terms in each bracket are multiplied together, then the Outside terms, the Inside terms and then the Last term in each bracket, to give four products. Finally factorisation of quadratic expressions, followed by a guess and check method for factorising quadratic expressions was covered. The students did not find these topics easy, especially factorising of quadratic expressions. This took a total of four hours, after which, questionnaire one was administered.Students were then exposed for one hour to the Vedic vertically and crosswise method, where initially they practised multiplying two- and three-digit numbers with this approach. Subsequently, the next three hours were spent expanding binomials and factorising quadratic expressions with the vertically and crosswise method. This method (see Figure 3) involves a sequence of four multiplications, the answers to each of which are placed into a single answer lin e. The middle two terms are added together mentally to supply the final answer.ResultsThe first question (1a) in each questionnaire was a two-digit multiplication.In the first, it was 37 58, and the second 23 47, and in this second case the question asked that this be done by the vertically and crosswise method. The aim was to check students facility with arithmetic multiplication and to see if the vertically and crosswise sutra was of assistance in this area. In the event 11 of the 18 (61%) students who completed both questionnaires, correctly answered the question in the first test, and 13 (72%) in the second, with only one student not using the sutra in that test. There was no statistical difference between these proportions (c2 = 0.125). When asked to explain what they had done using the Vedic approach, students who were able to write down the final answer were often able to write something like student 4s explanation for 1c), 32 692 times 9 is 18 3 times 9 +2 times 6 is 39 + carried 1 = 403 times 6 + carried 4 = 22Expansion of binomialsA summary of the results in the first of the algebra questions (Q2 see Figure