Multiplication as Arrays

What is an array?

An array is a tidy way to lay out objects in equal rows and columns. You see arrays everywhere: eggs in a box, windows on a building, squares in a chocolate bar, and tiles on a floor.

Because every row has the same number, an array is a perfect picture of a multiplication.

Here is an array of dots:

● ● ● ●
● ● ● ●
● ● ● ●

There are 3 rows, and each row has 4 dots. That is 3 × 4. If you count every dot, you get 12. So 3 × 4 = 12.

Reading an array: rows × columns

To turn any array into a multiplication, ask two questions:

1. How many rows are there? (going down)
2. How many in each row — the columns? (going across)

Then multiply: rows × columns = total.

Array	Rows	In each row	Multiplication	Total
2 by 5	2	5	2 × 5	10
4 by 3	4	3	4 × 3	12
5 by 5	5	5	5 × 5	25
3 by 6	3	6	3 × 6	18

Why arrays show the swap rule

Look at this array of 2 rows of 5:

● ● ● ● ●
● ● ● ● ●

That is 2 × 5 = 10.

Now imagine turning the whole array a quarter turn so the rows become columns. You get 5 rows of 2:

● ●
● ●
● ●
● ●
● ●

That is 5 × 2 = 10.

The dots never moved apart — there are still 10 — so 2 × 5 = 5 × 2. This is the commutative property, and it is the why behind a handy time-saver: if you know one fact, you also know its swap. That nearly halves the times tables you have to learn.

Worked example

A classroom has chairs in 4 rows. Each row has 6 chairs. How many chairs?

1. Picture the array: 4 rows, 6 in each row.
2. Rows × columns = 4 × 6.
3. 4 × 6 = 24 chairs.

You could also turn it the other way — 6 rows of 4 — and you would still get 24. The swap rule never fails.

Arrays grow into area

Here is a powerful idea for later. If you draw an array as a rectangle of squares, the number of squares is the same as the rectangle's area.

A rectangle that is 3 squares tall and 4 squares wide holds 3 × 4 = 12 squares — and its area is 12 square units. So learning arrays now quietly prepares you for Area and Perimeter later.

Try it yourself

• Find three real-life arrays at home (try an egg box, a window, or a tray of muffins). Write the multiplication for each.
• Draw a 3 × 5 array of dots. Count them, then turn your page sideways and check you get 5 × 3.
• Build an array with 12 counters. How many different rectangles can you make? (Hint: there is more than one.)

Where this leads

Arrays are the bridge between Introduction to Multiplication and quick recall of your Times Tables. The more you picture rows and columns, the faster multiplication becomes — and the same idea returns when you study area, factors, and even algebra.