One Place at a Time

Where it usually goes wrong

Crossing out a digit and writing a small number above it is followed as a rule to imitate rather than a fact about the number — so the small 1 above a carried column and the small 12 above a borrowed one are copied correctly on a hundred worksheets without ever being read as an actual value, which is why a borrow across a zero (4002 − 1568) so often stalls: the rule says "cross out the zero" and gives no number to cross it out INTO.

Try this

• Start at the ones place of 4002 − 1568. 2 is not enough to take 8 away, so the tens place lends 1 — but tens is a 0. Watch what the small number above tens becomes before ones can even borrow from it.
• Step the slider from ones to tens to hundreds. At each place, read the small number as what that place is really worth right now, not as a mark next to a crossed-out digit.
• Switch to addition and step through 5487 + 3768. A carry moves the same direction — into the next place left — but adds instead of borrows; compare the two small-number stories place by place.

Where this goes next

Every place in a multi-digit sum or difference is worked the same way: what a place is worth by the time the algorithm reaches it, not what was originally printed there. Multiplying multi-digit numbers carries the identical small number into a third row for the same reason.