Multiplication: Equal Groups and Arrays

What you will learn

• Read $a \times b$ as ”$a$ equal groups of $b$”.
• Show multiplication as repeated addition and as an array.
• Use the fact that the order of the numbers does not change the answer.

Before you start: you should be able to count in steps. See Skip Counting in 2s, 5s and 10s.

The idea

If you have $4$ bags with $3$ apples in each bag, you could count the apples one by one, or add $3 + 3 + 3 + 3$. Multiplication is a shortcut for adding equal groups. We write it $4 \times 3$ and say “four times three” or “four groups of three”.

Equal groups

The groups must all be the same size. $3 + 3 + 3 + 3 = 12$ is the same as $4 \times 3 = 12$.

The first number tells you how many groups, and the second tells you how many in each group. The answer is called the product.

Arrays

An array is objects set out in neat rows and columns. It makes it easy to see the groups, and to count the total in equal steps.

The order does not matter

Turn the array on its side: $3$ rows of $7$ becomes $7$ rows of $3$. The counters are the same, so the total is the same.

So $3 \times 7 = 7 \times 3$. If a multiplication looks hard one way, try it the other way round.

Two special cases

Any number times $1$ stays the same: $1 \times 9 = 9$. Any number times $0$ is zero: $0 \times 9 = 0$, because nothing is in any group.

Worked examples

Example 1: Repeated addition

Write $6 + 6 + 6 + 6$ as a multiplication and work it out.

1. There are $4$ sixes.
2. That is $4 \times 6$.
3. Count in sixes: $6, 12, 18, 24$.

Answer: $4 \times 6 = 24$.

Example 2: Using an array

A baker puts cakes on a tray in $5$ rows of $8$. How many cakes are there?

1. Rows times cakes in each row: $5 \times 8$.
2. Count in eights: $8, 16, 24, 32, 40$.

Answer: $40$ cakes.

Example 3: Swapping the order

Work out $9 \times 2$ by swapping the order.

1. $9 \times 2$ is $9$ groups of $2$, which is a lot of groups.
2. Swap: $2 \times 9$ is just $9 + 9$.
3. $9 + 9 = 18$.

Answer: $18$.

Equal groups matter. $3 + 3 + 4$ is not a multiplication, because the groups are not the same size.

Why it matters

Multiplication is used everywhere things come in equal amounts: packs, rows, shelves, wheels on cars, months in years. It is the base for division, area, and every times table you will learn.