Long Multiplication
Once numbers grow past a single digit, you can no longer reach for a
multiplication fact
you have memorised. The trick is to break each number into its tens and
ones, multiply the easy pieces, and add them back up. This is the
grid method, and it is just the
area model
of a big rectangle chopped into smaller rectangles.
Take 23 \times 14. Split 23 into
20 + 3 and 14 into
10 + 4. Now multiply every part of one by every part of the
other — four easy products:
(20 \times 10) + (20 \times 4) + (3 \times 10) + (3 \times 4)
= 200 + 80 + 30 + 12 = 322
Each piece is a little rectangle of the big 23 \times 14
rectangle; add the four areas and you have the whole.
See it built
Lay the parts out in a grid. The columns are headed 20 and
3; the rows are headed 10 and
4. Each cell holds the product of its row and column heading —
then add all four.
On paper people often write it more compactly as column
multiplication. Stack the numbers, lined up by
place value,
and multiply in two passes. First multiply 23 by the
ones of 14 (that is 4):
23 \times 4 = 92. Then multiply by the tens
(that is 10): 23 \times 10 = 230,
which you write shifted one place to the left. Finally add the two rows:
\begin{array}{r} 23 \\ {} \times 14 \\ \hline 92 \\ 230 \\ \hline 322 \end{array}
Notice the four grid products are hiding in here: 92 = 80 + 12
(the ones row) and 230 = 200 + 30 (the tens row). Same parts,
same total.
- Split each number into its place-value parts (tens, ones, …).
- Multiply every part of one number by every part of the other.
- Add all of those products together.
- The grid method and the column method are just two layouts of this — they give the same total.