Subtraction with
Regrouping

You know the drill. Students are doing subtraction: 52 minus 18. They get to 2 minus 8 and hit a wall. So they reach for the most common instruction: "borrow from the tens." One student writes "4" above the 5 with a little slash through it. Another crosses out everything and starts over. A third just guesses.

The problem isn't that students can't follow instructions. The problem is that "borrow" describes something that doesn't actually happen in mathematics. When you borrow money from a friend, you have to pay them back. In math? Nothing gets paid back. Something gets exchanged. And that changes everything.

The Language Problem

"Borrow" creates three specific misunderstandings:

• It implies obligation. If you borrowed something, you owe it. Students add extra steps because they think they need to "give back."
• It suggests movement between columns. Like the digit literally jumped across the room. It didn't. A value was decomposed into a different unit.
• It disconnects from place value reasoning. Borrowing sounds like a procedural rule. Exchanging sounds like making sense of units.

Walk Through 52 − 18 Step by Step

Set up the problem visually. Start with what 52 looks like as composed units: five groups of ten and two ones.

Now we want to subtract 18. That means taking away one ten and eight ones. We have five tens, so taking away one leaves four. Easy. But the ones column: we only have two, and we need eight. This is where the exchange happens.

After the exchange: 4 tens and 12 ones. Four plus twelve equals the original 52, just broken differently. Now subtract 8 from 12 and you get 4 ones. Subtract the ten from the five tens and you get 4 tens. Answer: 34.

Why This Approach Sticks

When students understand regrouping as composing and decomposing units, rather than following a mysterious procedure called "borrow," a few things change:

• They can explain it to someone else. Procedural tricks resist explanation. Unit reasoning does not.
• They don't break when faced with zeros. Problems like 502 − 18 feel impossible with "borrow" rules. With exchange, it's just a chain: exchange a hundred for ten tens, then one of those tens for ten ones.
• It connects to multiplication and division later. Understanding that a group of larger units can be decomposed into many smaller ones is the backbone of proportional reasoning.