Speed comes less from calculating faster than from not calculating what you can see.
Last-digit checking
The fastest error check available. The final digit of a product depends only on the final digits of the factors.
47 times 23 must end in 1, since 7 times 3 = 21. If your answer ends in anything else, it is wrong - and you know within a second.
On multiple choice this often eliminates several options outright without doing the multiplication at all.
Digit sums (casting out nines)
The digit sum of a product equals the digit sum of the product of the digit sums, mod 9.
123 times 456: digit sums 6 and 15 (which reduces to 6). 6 times 6 = 36, reducing to 9. So the answer's digit sum must reduce to 9. 56088 has digit sum 27, reducing to 9. Consistent.
Catches most arithmetic slips in a couple of seconds.
Divisibility rules
- 2: last digit even.
- 3: digit sum divisible by 3.
- 4: last two digits divisible by 4.
- 5: ends in 0 or 5.
- 6: divisible by 2 and 3.
- 8: last three digits divisible by 8.
- 9: digit sum divisible by 9.
- 11: alternating digit sum divisible by 11.
These turn an intimidating division into an easy factorisation - see mental division.
Magnitude first
Before computing, know roughly where the answer sits. 47 times 23 is near 50 times 20 = 1000, so an answer of 108 or 10810 is immediately wrong.
This is the habit that prevents the worst errors - the ones that are off by a factor of ten and would look absurd to anyone reading them.
Recognising factors
Spotting that 91 = 7 times 13, or that 51 = 3 times 17, converts problems instantly. Numbers that look prime often are not, and the common composites in the 50-100 range are worth knowing.
How to build it
Volume, with feedback. Short daily sessions on the mental math drill beat long infrequent ones, and paying attention to why an error happened matters more than the count of errors.