Long Multiplication

La méthode, étape par étape

1. 1

   Set out with the larger number on top

   342 × 27

   Columns aligned by place value. Putting the number with more digits on top means fewer rows to add at the end, which is a small saving in work and a large saving in opportunities to make an error.

2. 2

   Multiply by the ones digit

   342 × 7 = 2394

   Work right to left, carrying as in column addition: 7×2=14 write 4 carry 1; 7×4=28 plus 1 is 29, write 9 carry 2; 7×3=21 plus 2 is 23. The carried digits from multiplication are added after multiplying, never before — reversing that order is a common and quiet error.

3. 3

   Write a zero, then multiply by the tens digit

   342 × 20 → place 0, then 342 × 2 = 684 → 6840

   The zero is the whole reason this row works. You are not multiplying by 2, you are multiplying by 20, and the zero records that ten-fold. Students taught to "just put a zero there" without the reason omit it under pressure and lose the entire question.

4. 4

   Add the partial products

   2394 + 6840 9234

   The two rows are the answer split by place value — 342 lots of 7 and 342 lots of 20 — so adding them gives 342 lots of 27. Nothing new happens at this stage; it is ordinary column addition.

5. 5

   Check by estimating

   342 × 27 = 9,234. Check: 300 × 30 = 9,000 ✓

   The estimate confirms the magnitude, which is what a missing zero would destroy. Omitting the zero would give 3,078 — comfortably far from 9,000 to be caught, and the reason this check is worth the five seconds.

The zero is the method

Everything difficult about long multiplication is contained in one question: why does the second row start with a zero? The answer is that the second row is not 342 × 2 but 342 × 20, and the zero is how twenty is written rather than two. Students who can say that never omit it; students who cannot omit it roughly one time in five.

The same logic extends without change. A hundreds digit gets two zeros, a thousands digit three. There is no new rule to learn for bigger multipliers, which is what makes this method worth the effort of understanding rather than memorising.

• Multiply by each place value of the multiplier separately
• Ones row: no zeros. Tens row: one zero. Hundreds row: two
• Carry after multiplying, not before
• Add the partial products with ordinary column addition

This method is only as good as the tables under it

Long multiplication requires roughly a dozen table facts per question, each of which must be right and none of which the method itself checks. A student whose tables are shaky produces working that is structurally perfect and numerically wrong, and no amount of practising the layout improves it.

This is the most common reason long multiplication fails, and it is diagnosable in a minute. If the rows are correctly placed and the digits are wrong, the problem is tables, not method.

When the grid method is the better choice

The grid method — partitioning both numbers and multiplying every pair in a table — takes more space and is considerably harder to get structurally wrong. For students who understand multiplication but keep misplacing rows, it is not a lesser method to be outgrown; it is a legitimate written approach that trades compactness for reliability.

We move students to the column method when they are ready for it, not on a schedule. An accurate grid beats an inaccurate column every time, and exam marks do not distinguish between them.

How we teach long multiplication

We check times tables before teaching the method, always. Teaching a written procedure on top of insecure recall produces a student who can describe long multiplication perfectly and cannot complete one correctly, and that is a demoralising place to spend a term.

We teach the zero as multiplication by twenty from the first example, never as a placeholder rule. It is one extra sentence in the explanation and it removes the most expensive error in the method.