Partial-sums addition adds one place value at a time and writes down the result of each, then adds those results. There is no carrying, because a partial sum is allowed to be larger than nine. A tens column that comes to 110 is simply written as 110.
How the method works
- Line the numbers up by place value.
- Add the largest place value first and write the answer in full.
- Work down through the places, writing each partial sum on its own line.
- Add the partial sums. That total is the answer.
A worked example
268 plus 457
268 + 457 -------- hundreds 200 + 400 = 600 tens 60 + 50 = 110 ones 8 + 7 = 15 -------- 600 + 110 + 15 = 725
A four-digit example
1,847 plus 2,596
thousands 1000 + 2000 = 3000 hundreds 800 + 500 = 1300 tens 40 + 90 = 130 ones 7 + 6 = 13 -------- 3000 + 1300 + 130 + 13 = 4443
Three of the four partial sums spill past their own place value, and none of that matters. The spilling is handled once, at the end, when the partial sums are added.
Why it works
Addition can be regrouped freely. Writing both numbers in expanded form and adding matching places is only a reordering of the same sum:
268 + 457 = (200 + 60 + 8) + (400 + 50 + 7) = (200 + 400) + (60 + 50) + (8 + 7) = 600 + 110 + 15 = 725
Carrying in the standard algorithm does the same tidying up, but does it early and in small pieces. Partial sums leaves the tidying to the last line, which is why a student can see the size of the answer from the very first partial sum.
Common mistakes
Adding bare digits
Writing 6 + 5 = 11 for the tens column instead of 60 + 50 = 110 loses ninety nine from the answer. Write the zeros.
Misaligned columns
When numbers have different lengths, a digit in the wrong column is added to the wrong place value. Line the numbers up on the right before starting, or write the expanded forms out.
Rushing the final line
The last step is itself an addition of three or four numbers. It deserves the same care as the rest, and it is where most wrong answers actually come from.
Check yourself
356 + 287
300 + 200 = 500 50 + 80 = 130 6 + 7 = 13 500 + 130 + 13 = 643
4,275 + 1,968
4000 + 1000 = 5000 200 + 900 = 1100 70 + 60 = 130 5 + 8 = 13 5000 + 1100 + 130 + 13 = 6243
89 + 76
80 + 70 = 150 9 + 6 = 15 150 + 15 = 165
Why is a partial sum allowed to be bigger than nine?
Because it is a real quantity, not a digit. 60 + 50 really is one hundred and ten, and writing it that way keeps the value visible. The place value system only needs each digit to be under ten in the final answer, and the last addition takes care of that.
See also
Sources
- Partial-sums addition is a standard algorithm in the Everyday Mathematics curriculum. Examples on this page are our own.