Long multiplication
National Curriculum: this is in the Year 5 programme of study, part of key stage 2 (Years 3 to 6). Schools have to teach the key stage 2 programme by the end of Year 6, and can introduce content earlier or later within it.
Children in Year 5 are 9 or 10, and turn 10 during the school year.
What it is
Long multiplication multiplies by a two-digit number in two rows, then adds them. Each row is built digit by digit in columns, carrying into the next column exactly as short multiplication does.
- First row: 342 × 7 = 2394.
- Second row: 342 × 20 = 6840. Writing 0 in the ones column first is what makes this row 342 × 20 rather than 342 × 2.
- Add the two rows: 2394 + 6840 = 9234.
- 342 × 27 = 9234.
The zero at the end of the second row is the part worth understanding. That row is not 342 × 2, it is 342 × 20, so every digit sits one place further to the left and the 0 is written in the ones column to hold that place. A child told to put a zero down without being told what it is for leaves it out under pressure, and the answer that comes out, 3078 here, is far too small with nothing on the page looking wrong.
The method uses three earlier skills at once: table recall for each digit of a row, carrying, and column addition to total the two rows. When long multiplication keeps going wrong, it is worth finding which of the three failed before working through more whole calculations.
When children learn it
Long multiplication is a Year 5 requirement: multiplying numbers of up to four digits by a two-digit number. It follows short multiplication, the same column layout with a single-digit multiplier, from Year 4. Year 6 asks for the same method again, with a two-digit multiplier throughout, alongside long division.
What it looks like when it goes wrong
These patterns come up again and again in children's answers on this skill. Each one points at a specific missing idea, not carelessness.
- The second row's zero is missing. 342 × 27 answered as 3078. The second row was written as 684 rather than 6840, so 342 × 20 was counted as 342 × 2 and the answer is short by 6156. Every step above it can be correct, which is why this one is caught by an estimate rather than by rereading the working.
- A carry never reaches the next column. In 342 × 7, the 2 × 7 = 14 puts 4 down and carries 1. If that 1 never joins the next column the row reads 2384 instead of 2394, and the total is ten short. A second version of the same slip: the carries pencilled above the top number are left there from the first row and get added again in the second.
- The two rows drift out of line. Both rows are right and the total is wrong, because the rows were added with the digits out of step by a place. The check is the second row's last digit: the 0 belongs in the ones column, directly under the ones digit of the row above it.
How to help at home
- Before they start, ask what the second row multiplies by. Twenty is the answer that predicts a correct zero; two predicts a missing one.
- Estimate first: 342 × 27 is about 340 × 30, so about 10,000. An answer of 3078 fails that check before any of the working is looked at.
- Check the tables the calculation needs before doing the calculation. If 7 × 8 or 6 × 9 takes thinking time, every row is being built out of facts that are still being worked out.
- When an answer is wrong, have them read the two rows back before redoing the whole thing. A wrong row is a table or carrying problem; two right rows and a wrong total is a column addition problem.