Grid Method
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The grid method (also known as the box method) of multiplication is an introductory approach to multi-digit multiplication calculations that involve numbers larger than ten. Because it is often taught in
mathematics education In contemporary education, mathematics education, known in Europe as the didactics or pedagogy of mathematics – is the practice of teaching, learning and carrying out scholarly research into the transfer of mathematical knowledge. Although rese ...
at the level of
primary school A primary school (in Ireland, the United Kingdom, Australia, Trinidad and Tobago, Jamaica, and South Africa), junior school (in Australia), elementary school or grade school (in North America and the Philippines) is a school for primary e ...
or
elementary school A primary school (in Ireland, the United Kingdom, Australia, Trinidad and Tobago, Jamaica, and South Africa), junior school (in Australia), elementary school or grade school (in North America and the Philippines) is a school for primary ed ...
, this algorithm is sometimes called the grammar school method. Compared to traditional
long multiplication A multiplication algorithm is an algorithm (or method) to multiply two numbers. Depending on the size of the numbers, different algorithms are more efficient than others. Efficient multiplication algorithms have existed since the advent of the de ...
, the grid method differs in clearly breaking the multiplication and addition into two steps, and in being less dependent on place value. Whilst less ''efficient'' than the traditional method, grid multiplication is considered to be more ''reliable'', in that children are less likely to make mistakes. Most pupils will go on to learn the traditional method, once they are comfortable with the grid method; but knowledge of the grid method remains a useful "fall back", in the event of confusion. It is also argued that since anyone doing a lot of multiplication would nowadays use a pocket calculator, efficiency for its own sake is less important; equally, since this means that most children will use the multiplication algorithm less often, it is useful for them to become familiar with a more explicit (and hence more memorable) method. Use of the grid method has been standard in mathematics education in primary schools in England and Wales since the introduction of a
National Numeracy Strategy The National Numeracy Strategy arose out of the National Numeracy Project in 1996, led by a Numeracy Task Force in England. The strategy included an outline of expected teaching in mathematics for all pupils from Reception to Year 6. In 2003, t ...
with its "numeracy hour" in the 1990s. It can also be found included in various curricula elsewhere. Essentially the same calculation approach, but not with the explicit grid arrangement, is also known as the partial products algorithm or partial products method.


Calculations


Introductory motivation

The grid method can be introduced by thinking about how to add up the number of points in a regular array, for example the number of squares of chocolate in a chocolate bar. As the size of the calculation becomes larger, it becomes easier to start counting in tens; and to represent the calculation as a box which can be sub-divided, rather than drawing a multitude of dots.Long multiplication and division
/ref> At the simplest level, pupils might be asked to apply the method to a calculation like 3 × 17. Breaking up ("partitioning") the 17 as (10 + 7), this unfamiliar multiplication can be worked out as the sum of two simple multiplications: : so 3 × 17 = 30 + 21 = 51. This is the "grid" or "boxes" structure which gives the multiplication method its name. Faced with a slightly larger multiplication, such as 34 × 13, pupils may initially be encouraged to also break this into tens. So, expanding 34 as 10 + 10 + 10 + 4 and 13 as 10 + 3, the product 34 × 13 might be represented: : Totalling the contents of each row, it is apparent that the final result of the calculation is (100 + 100 + 100 + 40) + (30 + 30 + 30 + 12) = 340 + 102 = 442.


Standard blocks

Once pupils have become comfortable with the idea of splitting the whole product into contributions from separate boxes, it is a natural step to group the tens together, so that the calculation 34 × 13 becomes : giving the addition :: so 34 × 13 = 442. This is the most usual form for a grid calculation. In countries such as the U.K. where teaching of the grid method is usual, pupils may spend a considerable period of time regularly setting out calculations like the above, until the method is entirely comfortable and familiar.


Larger numbers

The grid method extends straightforwardly to calculations involving larger numbers. For example, to calculate 345 × 28, the student could construct the grid with six easy multiplications : to find the answer 6900 + 2760 = 9660. However, by this stage (at least in standard current U.K. teaching practice) pupils may be starting to be encouraged to set out such a calculation using the traditional long multiplication form without having to draw up a grid. Traditional long multiplication can be related to a grid multiplication in which only one of the numbers is broken into tens and units parts to be multiplied separately: : The traditional method is ultimately faster and much more compact; but it requires two significantly more difficult multiplications which pupils may at first struggle with . Compared to the grid method, traditional long multiplication may also be more abstract and less manifestly clear , so some pupils find it harder to remember what is to be done at each stage and why . Pupils may therefore be encouraged for quite a period to use the simpler grid method alongside the more efficient traditional long multiplication method, as a check and a fall-back.


Other applications


Fractions

While not normally taught as a standard method for multiplying fractions, the grid method can readily be applied to simple cases where it is easier to find a product by breaking it down. For example, the calculation 2 × 1 can be set out using the grid method : to find that the resulting product is 2 + + 1 + = 3


Algebra

The grid method can also be used to illustrate the multiplying out of a product of
binomial Binomial may refer to: In mathematics *Binomial (polynomial), a polynomial with two terms * Binomial coefficient, numbers appearing in the expansions of powers of binomials *Binomial QMF, a perfect-reconstruction orthogonal wavelet decomposition ...
s, such as (''a'' + 3)(''b'' + 2), a standard topic in elementary algebra (although one not usually met until
secondary school A secondary school describes an institution that provides secondary education and also usually includes the building where this takes place. Some secondary schools provide both '' secondary education, lower secondary education'' (ages 11 to 14) ...
): : Thus (''a'' + 3)(''b'' + 2) = ''ab'' + 3''b'' + 2''a'' + 6.


Computing

32-bit CPUs usually lack an instruction to multiply two 64-bit integers. However, most CPUs support a "multiply with overflow" instruction, which takes two 32-bit operands, multiplies them, and puts the 32-bit result in one register and the overflow in another, resulting in a carry. For example, these include the umull instruction added in the ARMv4t instruction set or the pmuludq instruction added in
SSE2 SSE2 (Streaming SIMD Extensions 2) is one of the Intel SIMD (Single Instruction, Multiple Data) processor supplementary instruction sets first introduced by Intel with the initial version of the Pentium 4 in 2000. It extends the earlier Streamin ...
which operates on the lower 32 bits of an
SIMD Single instruction, multiple data (SIMD) is a type of parallel processing in Flynn's taxonomy. SIMD can be internal (part of the hardware design) and it can be directly accessible through an instruction set architecture (ISA), but it should ...
register containing two 64-bit lanes. On platforms that support these instructions, a slightly modified version of the grid method is used. The differences are: # Instead of operating on multiples of 10, they are operated on 32-bit integers. # Instead of higher bits being multiplied by ten, they are multiplied by 0x100000000. This is usually done by either shifting to the left by 32 or putting the value into a specific register that represents the higher 32 bits. # Any values that lie above the 64th bit are truncated. This means that multiplying the highest bits is not required, because the result will be shifted out of the 64-bit range. This also means that only a 32-bit multiply is required for the higher multiples. : This would be the routine in C: #include uint64_t multiply(uint64_t ab, uint64_t cd) This would be the routine in ARM assembly: multiply: @ a = r0 @ b = r1 @ c = r2 @ d = r3 push @ backup r4 and lr to the stack umull r12, lr, r2, r0 @ multiply r2 and r0, store the result in r12 and the overflow in lr mla r4, r2, r1, lr @ multiply r2 and r1, add lr, and store in r4 mla r1, r3, r0, r4 @ multiply r3 and r0, add r4, and store in r1 @ The value is shifted left implicitly because the @ high bits of a 64-bit integer are returned in r1. mov r0, r12 @ Set the low bits of the return value to r12 (ac) pop @ restore r4 and lr from the stack bx lr @ return the low and high bits in r0 and r1 respectively


Mathematics

Mathematically, the ability to break up a multiplication in this way is known as the
distributive law In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmetic, ...
, which can be expressed in algebra as the property that ''a''(''b''+''c'') = ''ab'' + ''ac''. The grid method uses the distributive property twice to expand the product, once for the horizontal factor, and once for the vertical factor. Historically the grid calculation (tweaked slightly) was the basis of a method called
lattice multiplication Lattice multiplication, also known as the Italian method, Chinese method, Chinese lattice, gelosia multiplication, sieve multiplication, shabakh, diagonally or Venetian squares, is a method of multiplication that uses a lattice to multiply two mult ...
, which was the standard method of multiple-digit multiplication developed in medieval Arabic and Hindu mathematics. Lattice multiplication was introduced into Europe by
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...
at the start of the thirteenth century along with Arabic numerals themselves; although, like the numerals also, the ways he suggested to calculate with them were initially slow to catch on.
Napier's bones Napier's bones is a manually-operated calculating device created by John Napier of Merchiston, Scotland for the calculation of products and quotients of numbers. The method was based on lattice multiplication, and also called ''rabdology'', a wor ...
were a calculating help introduced by the Scot
John Napier John Napier of Merchiston (; 1 February 1550 – 4 April 1617), nicknamed Marvellous Merchiston, was a Scottish landowner known as a mathematician, physicist, and astronomer. He was the 8th Laird of Merchiston. His Latinized name was Ioann ...
in 1617 to assist lattice-method calculations.


See also

*
Multiplication algorithm A multiplication algorithm is an algorithm (or method) to multiply two numbers. Depending on the size of the numbers, different algorithms are more efficient than others. Efficient multiplication algorithms have existed since the advent of the de ...
*
Multiplication Table In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system. The decimal multiplication table was traditionally taught as an essenti ...


References

*
Rob Eastaway Rob Eastaway is an English author. He is active in the popularisation of mathematics and was awarded the Zeeman medal in 2017 for excellence in the promotion of maths. He is best known for his books, including the bestselling ''Why Do Buses Come ...
and Mike Askew, ''Maths for Mums and Dads'', Square Peg, 2010. . pp. 140–153. {{reflist


External links


Long multiplication − The Box method
''Maths online''.
Long multiplication and division
BBC GCSE Bitesize Mathematics education Elementary arithmetic Multiplication Primary education