The column method and the grid method, both worked out on your own numbers
Long multiplication is not one calculation, it is a handful of small ones added together. This shows both standard ways of laying that out on whatever numbers you enter — the column method, which stacks partial products, and the grid method, which splits both numbers by place value first. Neither is better; they are the same arithmetic arranged differently.
Write the larger number on top and multiply it by each digit of the bottom number in turn, starting from the units. Each digit produces one partial product, and each partial product is shifted one place further left than the last, because that digit is worth ten times more.
That shift is the step people get wrong. Multiplying by the tens digit does not give you the answer to that digit alone — it gives you the answer multiplied by ten, which is why a zero goes on the end before you add.
Add the partial products and you have the answer. The whole method depends on knowing the times tables well enough that each partial product is one step rather than a calculation of its own.
Split both numbers by place value — 47 becomes 40 and 7, 23 becomes 20 and 3 — and multiply every part of one by every part of the other. Four easy multiplications instead of one awkward one.
It is the same arithmetic as the column method, laid out so that nothing is hidden. It is slower to write but far easier to check, which is why it is usually taught first and why it stays useful as a way of finding where a mistake crept in.
It also explains why the column method works. The partial products in the column layout are just the grid rows, already added along each row.
Estimate first and compare. 47 × 23 is roughly 50 × 20, which is 1,000, so an answer of 1,081 is believable and an answer of 108 or 10,810 is not. That one check catches every misplaced zero.
Check the last digit too. The units digit of the answer is always the units digits multiplied: 7 × 3 = 21, so the answer must end in 1. It does.
For a stronger check, cast out nines: add the digits of each number until one digit is left, multiply those, and the result should match the same process applied to the answer.
Plenty of multiplications are faster split than stacked. 47 × 23 can be done as 47 × 20 plus 47 × 3 — 940 plus 141 — which is 1,081 without writing anything down.
Numbers near a round one are easier still. 47 × 99 is 47 × 100 minus 47, and 25 × 16 is 25 × 4 × 4, which is 100 × 4.
The algorithm is the fallback that always works. Knowing when you do not need it is what makes arithmetic quick.
Multiply the top number by each digit of the bottom number in turn, starting from the units, shifting each partial product one place further left, then add the partial products together.
Because that digit stands for tens, not units. Multiplying by the 2 in 23 is really multiplying by 20, so the partial product is ten times bigger and shifts one place left.
Split both numbers into their place values — 47 into 40 and 7 — multiply every part by every part, then add all the products. It is the same arithmetic as the column method with nothing hidden.
Estimate with rounded numbers to check the size, and check that the last digit of your answer matches the two units digits multiplied together.
Yes — every partial product is a times-table fact. If those are slow, long multiplication is slow no matter how well the layout is understood.