Align numbers by place value and operate on each place separately. With regrouping (trading), 10 ones become 1 ten or 10 tens become 1 hundred. 27 + 15: ones (7 + 5 = 12, regroup to 1 ten + 2 ones), tens (2 + 1 + 1 = 4), result is 42.
Use base-ten blocks to show regrouping. Write in vertical format. Practice repeatedly.
Not aligning place values; regrouping errors; confusing addition and subtraction regrouping.
You already understand three-digit place value — that in the number 347, the 3 means 3 hundreds, the 4 means 4 tens, and the 7 means 7 ones. Multi-digit addition and subtraction is built entirely on that foundation. The standard algorithm is just a systematic way of applying place value one column at a time, from smallest to largest.
When you add 27 + 15, you start with the ones: 7 + 5 = 12. But "12 ones" is a problem — the ones place can only hold a single digit. So you regroup: you trade 10 of those ones for 1 ten, leaving 2 ones behind. Now the tens place: 2 tens + 1 ten + the regrouped 1 ten = 4 tens. Result: 42. Every step of this process is just place value — you're applying the rule that 10 of any place value equals 1 of the next larger place.
Subtraction regrouping works in the opposite direction. In 53 − 28, you need to subtract 8 ones from 3 ones — but 3 < 8. So you borrow (or "trade"): take 1 ten from the tens column and convert it into 10 ones. Now you have 13 ones and only 4 tens remaining. Then 13 − 8 = 5 ones, and 4 − 2 = 2 tens. Result: 25. You're trading a larger unit for 10 smaller units, which is the reverse of what addition regrouping does.
The vertical format (stacking numbers on top of each other) is a visual aid for keeping place values aligned. Ones above ones, tens above tens, hundreds above hundreds. If you misalign — adding a hundreds digit to a tens digit — you get nonsense. Alignment is the entire basis of the algorithm. Base-ten blocks make this concrete: a tens rod really does contain 10 ones blocks, and you can physically snap them apart (borrowing) or click them together (carrying). Once you've done that physically, the written symbols on paper represent the same trade.