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Long multiplication is the standard method taught in UK primary and secondary schools for multiplying large numbers without a calculator. It breaks the problem into a series of smaller multiplications and additions, leveraging our knowledge of single-digit multiplication (times tables) to handle any size of number.
There are two main methods taught in UK schools: the traditional column method and the grid (box) method. Both give the same answer and both are accepted in GCSE non-calculator examinations. The column method is more compact; the grid method makes the place value structure explicit and is often easier for students who are new to the process.
The column method multiplies each digit of the bottom number by the entire top number, then adds the partial products. Using 347 × 28 as an example:
The grid method makes place value visible by splitting each number into its components. For 347 × 28:
Before performing long multiplication, always estimate the answer. This catches errors and gives you a target range. For 347 × 28: round to 350 × 30 = 10,500. The exact answer (9,716) is close to this, confirming no major error. If your answer were 97,160 or 971, the estimate would immediately flag the mistake.
GCSE Mathematics in England requires students to perform calculations without a calculator on the non-calculator paper (Paper 1 in both AQA and Edexcel specifications). Long multiplication is specifically tested and students are expected to show their full working. An answer without working may receive no marks even if the answer is correct, as the examiners assess method marks separately from answer marks.
Understanding long multiplication also reinforces understanding of place value, the distributive property of multiplication (a(b+c) = ab + ac), and builds the foundation for algebraic expansion (FOIL method for brackets).
Long multiplication works with decimals too. The strategy:
When multiplying with negative numbers, the sign rules are:
Apply the sign rule first, then multiply the absolute values using long multiplication. For −347 × 28: the sign is negative (one negative, one positive), and 347 × 28 = 9,716, so the answer is −9,716.
The lattice method (also called the gelosia method) is an alternative approach that was used in medieval Europe and is still taught in some UK schools as an enrichment activity. It uses a grid with diagonal lines dividing each cell:
Lattice multiplication handles carries automatically within the diagonal sums, making it less error-prone for some learners, though it requires more setup time than the column method.
Long multiplication is the foundation of expanding brackets in algebra. Multiplying (3x + 4)(2x + 7) follows exactly the same structure as long multiplication:
This is the FOIL method (First, Outer, Inner, Last), which is simply long multiplication applied to algebraic expressions. Students who understand long multiplication find algebraic expansion much more intuitive.
Long multiplication is a method for multiplying large numbers by breaking them down into a series of smaller, manageable steps. Each digit of the second number multiplies the entire first number, and the resulting partial products are added together. It relies on times table knowledge for single-digit multiplications and careful attention to place value for correct alignment.
The grid (or box) method splits each number into its place value components (hundreds, tens, units) and organises all partial products in a table. For 347 × 28: split into (300+40+7) × (20+8), creating a 2-row by 3-column grid. Each cell contains the product of its row and column value. All six cells are then added for the final answer. The grid method makes the distributive property visually clear.
In the column method, you write the larger number on top and multiply by each digit of the second number separately. For a 3-digit second number (e.g., 286), you create three partial products: 347×6, 347×80, 347×200, adding the appropriate zero placeholders (none, one, two). Then add all three partial products. Always estimate first: 347 × 286 ≈ 350 × 300 = 105,000. Exact answer: 99,242.
Count the total decimal places in both numbers, then ignore the decimal points and multiply as whole numbers. Place the decimal point in the answer by counting the same total number of decimal places from the right. Example: 3.47 × 2.8 — total decimal places = 2+1 = 3. Multiply 347 × 28 = 9,716. Place decimal 3 from right: 9.716. Check: 3.47 × 2.8 ≈ 3.5 × 3 = 10.5. Answer 9.716 is close to 10.5, so correct.
Mathematically, this follows from the properties of the number system. Think of it as direction: a positive number moves right; a negative number moves left; "negative of" reverses direction. So −3 × −4 = "the negative of 3 lots of −4". Three lots of −4 = −12. The negative of −12 is +12. In abstract algebra, this follows from the distributive law and the definition of additive inverses.
Lattice multiplication uses a grid with diagonal lines in each cell. The digits of the first number head the columns; the second number's digits head the rows. Each cell contains the two-digit product of its row and column digit (tens in upper-left triangle, units in lower-right). Diagonal bands (running top-right to bottom-left) are summed with carries to produce the final answer. It automatically handles carries, which some students find reduces errors.
Yes, showing working is essential in GCSE maths. Questions worth 2 or more marks allocate method marks (M marks) separately from accuracy marks (A marks). If you show correct partial products but make an arithmetic error in the final addition, you can still earn the method marks. A bare answer with no working earns zero marks if wrong. Always show estimation, full partial products, and the final addition clearly labelled.