Multiplication as an array

There's another nice picture of multiplication — besides repeated addition. It's called an array or rectangular array.

What an array is

An array is an arrangement of dots (or apples, cubes...) in a rectangle, with the same number in every row.

Look at the 3 × 4 array:

• 3 rows.
• 4 dots in each row.
• 12 dots in total.

And that's exactly 3 × 4 = 12.

Why arrays are useful

The array shows you multiplication two ways at once:

• If you look at the rows: you have 3 rows with 4 dots each → 3 × 4.
• If you look at the columns: you have 4 columns with 3 dots each → 4 × 3.

Both views show the same 12 dots.

The biggest discovery: the order doesn't matter

From the array you can see immediately:

3 × 4 = 4 × 3 (both equal 12)

That means: when you multiply, it doesn't matter which number comes first. This rule always holds.

What does that give you? You practise fewer problems. One half is enough — you get the other half for free.

• 2 × 5 and 5 × 2 are the same: 10.
• 3 × 7 and 7 × 3 are the same: 21.
• 4 × 6 and 6 × 4 are the same: 24.

Arrays are everywhere

They hide in everyday things:

• A chocolate bar — say 4 rows and 6 columns of squares. 4 × 6 = 24 in total.
• Eggs in a carton — 2 rows of 5 = 10 eggs.
• Chairs in a classroom — rows × seats per row.

When you see a rectangular pattern, it's worth pausing to ask: how many rows, how many in each?

Summary

• An array is a rectangle of dots with the same number in every row.
• It shows multiplication two ways at once (rows and columns).
• 3 × 4 = 4 × 3 — the order doesn't matter in multiplication.
• Arrays hide in chocolate bars, egg cartons, rows of chairs.