Eighty-one times-table facts, forty-five to learn

1Only 45 of the 81 are new

3 × 7 and 7 × 3 are both 21. Order does not change the product - multiplication is commutative. Count each such pair once and the 81 facts collapse to 45.

Beyond that, the one-times table just gives back the number. Doubling covers the twos. The fives alternate between 0 and 5 in the ones column, and in the nines the two digits always add to nine. The facts with no handle on them at all are fewer still.

This recount matters because it gives a child a number they can see the end of. Eighty-one sounds endless. Strip out the tables that carry their own pattern and what is left is a countable pile.

2What is left is the big ones

The last to stick are usually in the sixes, sevens and eights - and specifically where those tables meet each other: 6 × 7, 7 × 8, 8 × 6. No pattern to lean on, large products, and neighbouring answers that are easy to confuse.

So a child rarely fails at times tables as a whole. What usually happens is that a handful of specific facts are missing. Starting the chant again from the top reaches each of them exactly once per pass.

3In Japan they are learned as sound

Japanese schools teach the tables as a chant. The readings are clipped so the whole line scans: 7 × 8 becomes shichi-ha gojuuroku. It enters the head as a line of verse, not as a calculation.

The strength of this is that recall costs nothing. The weakness is that the facts may only come out in order. A child who freezes at 56 ÷ 7 is often running through the seven times table from the start. Getting at a fact out of sequence is a separate skill, and needs separate practice.

4Keep the tables apart

If the gaps are a handful of specific facts, then a record that does not go down to that level tells you nothing you can act on. "Weak at multiplication" does not say what to repeat.

That is why this drill keeps multiplication in 11 kinds - one per table, plus two that involve a two-digit number. "Never gets the sevens" survives as a fact in the record. The full breakdown is in the 42 kinds.

2 × 13 and 13 × 2 are counted as different kinds too. Same product, but one is "two, thirteen times" and the other "thirteen, twice", and the working a child actually does is not the same. They are identical to anyone who has the commutative law - and whether the child has it is exactly the question.

5What comes after knowing them

After the tables come two-digit problems. This drill only asks those whose product stays below 100 - 2 × 13 and the like. The aim is to thicken the range a child can still reach mentally, before written methods take over.

Division is laid out the same way. Since 56 ÷ 7 is the seven times table read backwards, a child missing that fact stops in both places. One gap, two places it shows up.