The times tables

The times tables – the techniques that shrink them

How do you learn the times tables without memorising a hundred facts? A walk through a technique for every table from 1 to 10 – doubling, the five rule, the digit rule for nines and the square numbers – and the few facts that genuinely have to be practised.

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What this covers

How many facts the times tables actually contain
Commutativity – why half of them disappear on their own
A technique for every table from 1 to 10
The doubling chain that covers 2, 4 and 8
The nine times table, with the finger trick and the digit rule
The square numbers as anchor points
One off a square – the trick that takes the hardest ones

What this covers (cont.)

The few facts left over, and how to practise them
Why the techniques work

Order is free

The first thing that halves the work is something most people know but few use deliberately: 3 × 8 and 8 × 3 are the same fact. Learn one and you have the other.

A technique for each table

Every table has its own way in. Some are plain rules, some are shortcuts, and some are both.

The ten tables

Table	The technique	Example
× 1	The number is unchanged	1 × 7 = 7
× 2	Double it	2 × 7 = 7 + 7 = 14
× 3	Double it, then add the number itself	3 × 7 = 14 + 7 = 21
× 4	Double it twice	4 × 7 = 7 → 14 → 28
× 5	Take half of the ten times table	5 × 7 = 70 ÷ 2 = 35
× 6	The five times table, plus the number itself	6 × 7 = 35 + 7 = 42
× 7	The fives plus the twos	7 × 8 = 40 + 16 = 56
× 8	Double it three times	8 × 7 = 7 → 14 → 28 → 56
× 9	The ten times table, minus the number itself	9 × 7 = 70 − 7 = 63
× 10	Put a zero after it	10 × 7 = 70

Note that none of the rows says “remember this”. Every one says “do this”. That is the difference between a table you can lose and one you can rebuild in two seconds.

The doubling chain: 2, 4 and 8

Three of the tables are really one technique, applied once, twice or three times. If you can double, you have all three.

7 through the doubling chain

7The starting point
14 — that is 2 × 7Doubled once
28 — that is 4 × 7Doubled twice
56 — that is 8 × 7Doubled three times

It is the same chain whichever number you start from. And it gives three answers for the price of one skill – worth knowing when you decide what to spend practice time on.

The fives, and the sixes that follow from them

The five times table is half of the ten times table. 5 × 8 is 80 divided by two, so 40. That always works, and it is faster than counting up in fives.

6 multiplied by an even number ends in that same even number – and the tens are half of it.

See for yourself: 6 × 2 = 12, 6 × 4 = 24, 6 × 6 = 36, 6 × 8 = 48. The units digit is the number you multiplied by. The tens digit is half of it.

The nines have the most checks

The nine times table has a reputation for being hard, but it is the best equipped of them all. The simplest route is to multiply by ten and subtract the number once: 9 × 7 = 70 − 7 = 63.

Three ways to check the nines

The check	How it works	9 × 7 = 63
The digits	The digits of the answer always add up to 9	6 + 3 = 9
Tens up	The tens digit is one less than the number you multiplied by	7 − 1 = 6
Units down	The units digit is what is missing up to 10	10 − 7 = 3

The three hang together, and not by accident: the tens digit rises by one at each step while the units digit falls by one, so the sum stays at 9. That is why 9, 18, 27, 36 … is a staircase running both ways at once.

The square numbers as anchors

The square numbers – 3 × 3, 4 × 4, 6 × 6, 7 × 7, 8 × 8, 9 × 9 – are worth learning as a short list of their own. Not because they come up so often in themselves, but because they work as anchors you can step out from.

If a product sits one step either side of a square, the answer is that square minus one.

Examples: 6 × 8 sits around 7. The answer is 7 × 7 − 1 = 49 − 1 = 48. And 7 × 9 sits around 8: 8 × 8 − 1 = 64 − 1 = 63. It works everywhere: 4 × 6 = 5 × 5 − 1 = 24.

What is actually left

Work through the techniques above and strikingly little remains to memorise. Most of the facts are now covered by at least one rule, and several of them by two or three.

Why the techniques work

This is the part that keeps the rules from becoming new things to remember.

And if you want to go further

Key points

The times tables look like a hundred facts. In practice they are a handful.
Order does not matter, and that alone removes nearly half of them.
The 1, 2, 5 and 10 tables are rules, not memory work – they take most of the rest.
2, 4 and 8 are the same technique repeated: double once, twice or three times.

Key points (cont.)

The nine times table has several checks that catch an error immediately.
The square numbers are anchors you can step out from.
A few facts remain at the end. Those are worth practising deliberately.

Questions this answers

How many multiplication facts do you actually have to learn?
Why do the times tables halve themselves?
What is the simplest technique for each table from 1 to 10?
Why is the seven times table the hardest?
How does the finger trick for the nine times table work?
What does it mean for a product to sit one off a square?

Questions this answers (cont.)

Which facts do you genuinely have to memorise?
How do you practise the last ones that will not stick?
Is it better to understand the techniques than to memorise?

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