The times tables – the techniques that shrink them

The times tables are presented as a wall: ten by ten, a hundred facts, memorise them. It is one reason they sit badly with a good many adults too.

But a hundred is the wrong number. Once you subtract what follows from rules rather than from memory, a handful is left. This episode goes through where they go, one table at a time.

How big are the times tables really?

1. 100 factsEverything from 1 × 1 to 10 × 10. The way they are presented.
2. 55 leftOrder means nothing: 7 × 8 and 8 × 7 are the same fact.
3. 21 leftThe 1, 2, 5 and 10 tables are rules you can apply, not facts you have to hold.

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.

It sounds obvious, but it is worth saying out loud while practising. If 7 × 4 gives you trouble, you can fetch it from the four times table instead – and the fours are one of the easy ones.

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

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

1. 7The starting point
2. 14 — that is 2 × 7Doubled once
3. 28 — that is 4 × 7Doubled twice
4. 56 — that is 8 × 7Doubled three times

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.

The answer also always ends in 0 or 5: in 0 when you multiply by an even number, in 5 when you multiply by an odd one. That is a free check on the answer.

The six times table hangs straight off it: multiply by five, then add the number once more. 6 × 8 = 40 + 8 = 48.

The sixes also have a pattern of their own when you multiply by an even number, and it is one of the prettiest in the whole table:

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.

On top of that come three checks that catch an error immediately:

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 finger trick is the same rule done with your hands: hold up ten fingers and fold down the one matching the number you are multiplying by. For 9 × 7 you fold the seventh. That leaves six fingers on the left and three on the right – 63.

Who came up with the finger trick, we do not know, and we do not claim to. It is one of those things that has travelled from classroom to classroom without anyone putting their name on it.

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.

And then there is one more trick, which takes several of the very hardest facts at once:

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.

This is not a curiosity. 6 × 8 and 7 × 9 are among the facts people get wrong most often, and here both fall out of the same rule.

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.

What tends to sit worst at the end is a small group in the middle of the table – typically 6 × 7, 6 × 8, 7 × 8 and 7 × 9. They are a long way from the easy rules and have no conspicuous patterns. Two of them are admittedly taken by the square trick above.

These few are the ones worth practising deliberately. And the point there is to retrieve them from memory, not to read them again: pose yourself the fact, wait until the answer arrives, and check. That is harder work than reading a table, and that is precisely why it sticks better. Spread the sessions across several days rather than doing everything in one – spaced repetition is one of the most widely used techniques in study skills, and the times tables are close to an ideal use for it.

Why the techniques work

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

None of them is a trick in the real sense. They fall out of two things: that multiplication can be split up (7 × 8 is the same as 5 × 8 plus 2 × 8), and that the decimal system makes ten an easy number to multiply by. That is why the fives are half the tens, and why the nines are the tens minus one round.

The square trick has a simple explanation too: 7 × 9 is 8 − 1 multiplied by 8 + 1, and when you expand that, everything vanishes except 8 × 8 minus 1. The same holds for any other pair sitting equally far from the same square.

Grasp that and you have not learned ten rules. You have learned two, and ten ways of using them.

And if you want to go further

This episode is about learning the times tables. If you would rather do the arithmetic without them, there is a whole system for exactly that – one short rule per digit, and no times table at all. That is the subject of the Learn something new episode on the Trachtenberg system.

The content is a practical walk-through you can practise yourself.

Main topics

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

Questions this episode 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?
• 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?

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.
• 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.