How this instrument works
Multiplying a two-digit number by a single digit works because of an idea called the distributive property: a number like 34 is really 30 plus 4, so 34×6 is the same as (30×6) plus (4×6). Handle each of those two multiplications on its own, and adding the results gives the same answer as multiplying the whole number at once.
The two pieces are called partial products. The ones digit's contribution is the simplest: multiply it by the single-digit factor directly. The tens digit's contribution needs one extra move — multiply the tens digit by the factor, then multiply by ten again, since that digit was never really standing for itself; it was standing for that many tens all along.
This sheet keeps the multiplier down to one digit specifically so both partial products stay easy to see: nothing here is hidden inside a single large multiplication step. The same distributive idea scales up to multipliers with several digits of their own, each contributing its own row of partial products before everything is added together. Readers wanting the equivalent bookkeeping for the other operations will find carrying and borrowing worked through on this site's long addition and long subtraction sheets.
- Enter the two-digit number's digits into Number: tens digit and Number: ones digit.
- Enter the multiplier into Single-digit multiplier.
- Read Partial product from the ones digit for the ones digit times the multiplier, on its own.
- Read Partial product from the tens digit for the tens digit times the multiplier times ten.
- Read Total product for the sum of those two partial products — the full answer.
Worked example — 34 times 6, plus two more products
Take 34 times 6. The ones digit contributes 4×6=24 on its own. The tens digit contributes 3×6=18, then that gets multiplied by ten to restore the place value the 3 actually represents, giving 180. Add the two partial products together: 24+180=204, exactly what 34×6 gives when worked out any other way.
Ten times five keeps one partial product at zero: the ones digit is 0, so 0×5=0, while the tens digit gives 1×5×10=50, for a total of 50. Seventy-two times three splits into 2×3=6 from the ones digit and 7×3×10=210 from the tens digit, adding up to a total of 216.
Questions
Why does the tens digit get multiplied by ten again?
Because that digit was never worth just itself — in 34, the 3 stands for three tens, not three units. Multiplying 3×6 gives 18, but that 18 is really 18 tens, so multiplying by ten again turns it into 180, the number that actually belongs in the total.
What are partial products?
They are the separate pieces a multiplication splits into before being added back together. Multiplying 34 by 6 produces two of them here — 24 from the ones digit and 180 from the tens digit — and adding 24+180 gives the same 204 that multiplying 34 by 6 directly would.
Does this method work for a two-digit multiplier as well?
The same idea extends further, though this sheet keeps the multiplier to one digit to keep both partial products visible at a single glance. A two-digit multiplier would add a second row of partial products, one for each of its own digits, before everything gets totaled.
Why can one partial product be much bigger than the other?
Because the tens partial always carries an extra factor of ten that the ones partial does not. Even a small tens digit, once scaled up by ten, tends to outweigh the ones digit's contribution — in 34×6, 180 dwarfs 24 even though 3 is smaller than 4.
Is this the same idea as the distributive property?
Yes, exactly that. Writing 34 as 30+4 and multiplying each part by 6 before adding the results is the distributive property in action — this sheet just gives each of those two multiplications its own labeled output so the working stays visible.
What if the multiplier is zero?
Both partial products come out at zero, since any digit times zero is zero, and the total follows the same way. It is a useful boundary check: the total should always read zero whenever the multiplier does, whatever the two-digit number itself happens to be.