Patterns in the Times Table

The table is full of patterns

The times table looks like a lot to learn. It is much less than it looks.

Most of the grid follows a handful of rules. Each rule comes from the way multiplication works. Learn the rule and the numbers follow.

$\times$	$1$	$2$	$3$	$4$	$5$
$1$	$1$	$2$	$3$	$4$	$5$
$2$	$2$	$4$	$6$	$8$	$10$
$3$	$3$	$6$	$9$	$12$	$15$
$4$	$4$	$8$	$12$	$16$	$20$
$5$	$5$	$10$	$15$	$20$	$25$

Multiples of 2 are all even

Look down the $2$ column. Every answer is even.

Here is why. Multiplying by $2$ is the same as doubling:

$$2 \times 7 = 7 + 7$$

Doubling puts everything in pairs. Nothing is left over. A number that splits into pairs is even, so a double is always even.

Multiples of 5 end in 0 or 5

Look down the $5$ column: $5$, $10$, $15$, $20$, $25$.

The last digit goes $5$, $0$, $5$, $0$ over and over.

Two fives make ten. So every second five lands on a whole ten and ends in $0$. The ones in between sit halfway and end in $5$.

$$5 + 5 = 10 \qquad 10 + 5 = 15 \qquad 15 + 5 = 20$$

Multiples of $10$ are simpler still. They all end in $0$, because each step adds a whole ten and never touches the ones.

The table matches across the diagonal

Find $3 \times 4$ in the grid. Now find $4 \times 3$. Same answer.

$$3 \times 4 = 12 \qquad 4 \times 3 = 12$$

Swapping the two numbers never changes the answer. Three rows of four dots is the same picture as four rows of three, turned on its side.

This halves the work. Learn one half of the table and you have the other half too.

Why patterns are worth more than the facts

A fact you have forgotten is gone. A pattern you understand rebuilds it.

Say you forget $4 \times 6$. Two patterns get you there.

Times $4$ is double, then double again:

$$6 \rightarrow 12 \rightarrow 24$$

Or use the fact next door. You know $5 \times 6 = 30$, and $4 \times 6$ is one group of six less:

$$30 - 6 = 24$$

Both give $24$. A pattern turns one fact you know into a fact you need.

Odd and even in the addition table

The addition table has patterns too.

Adding	Answer
even $+$ even	even
odd $+$ odd	even
even $+$ odd	odd

An even number splits into pairs. An odd number splits into pairs with one left over.

So odd plus odd gives two leftovers. Those two make a pair of their own, and nothing is left. The answer is even.

$$3 + 5 = 8$$

Even plus odd keeps the single leftover, so the answer is odd.

Worked examples

Common mistakes

Practice problems