The grid method (also known as the box method or matrix method) of multiplication is an introductory approach to multi-digit multiplication calculations that involve numbers larger than ten. Compared to traditional long multiplication, 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 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. 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 UK 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 UK 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 21/2 × 11/2 can be set out using the grid method
to find that the resulting product is 2 + 1/2 + 1 + 1/4 = 33/4
Algebra The grid method can also be used to illustrate the multiplying out of a product of binomials, such as (a + 3)(b + 2), a standard topic in elementary algebra (although one not usually met until secondary school):
Thus (a + 3)(b + 2) = ab + 3b + 2a + 6.
… excerpt ends here. Continue reading the full article.
