Two-Digit Subtraction With Regrouping

The hard case: when the top number's ones digit is smaller than the bottom number's ones digit. This is where most kids get stuck.

What we're learning

• What to do when the ones column "doesn't have enough" to subtract
• The regrouping (or "borrowing") move — at the bundles-of-ten level
• The procedure for two-digit subtraction with regrouping

The problem

52 − 27. Look at the ones column: 2 − 7. We can't subtract 7 from 2 — it would give a negative number, which K-2 math doesn't deal with yet.

What do we do?

Adults remember this as "borrow from the tens column" — usually with no understanding of WHY. We'll do better.

Use the bundles

Get the bundling materials again — popsicle sticks rubber-banded into tens, plus loose ones.

Build 52: 5 bundles + 2 loose ones.

Now we want to take away 27 (which would be 2 bundles + 7 loose ones).

Try to take away 7 loose ones. You only have 2 loose ones! Not enough.

Here's the move: unbundle one of the tens. Open up the rubber band on one bundle. Now you have:

• 4 bundles (was 5, used one)
• 12 loose ones (the original 2 plus the 10 you just freed)

Now take away 7 loose ones. You have 5 left.

Then take away 2 bundles. You have 2 bundles left.

Final count: 2 bundles + 5 loose = 25.

That unbundling-of-a-ten IS what "borrowing" means.

Now on paper

   4 12        ← cross out the 5, write 4. Cross out the 2, write 12.
   5  2
 − 2  7
 ------
   2  5

Step by step:

1. Look at the ones: 2 is smaller than 7. We can't subtract. We need to regroup.
2. Cross out the 5 in the tens place; write 4 above it (one less bundle).
3. Cross out the 2 in the ones place; write 12 above it (the original 2 plus 10 from the unbundled ten).
4. Subtract the ones: 12 − 7 = 5. Write 5.
5. Subtract the tens: 4 − 2 = 2. Write 2.

Read the answer: 25.

Try it together

Use the bundles for the first few. Then transition to paper-only.

• 43 − 18 (3 < 8, regroup)
• 70 − 24 (0 < 4, regroup; the top number's ones digit is 0)
• 81 − 49 (1 < 9, regroup)
• 60 − 35 (0 < 5, regroup)
• 92 − 67 (2 < 7, regroup)

The second and fourth examples are special: the top number's ones digit is 0. After regrouping, the ones become 10 (not 1+10). Pause on these; they often confuse kids.

Always check the answer by adding it back: 25 + 27 = 52 ✓.

Watch for

• Subtracting bottom from top within a column when easier. Some kids will look at 52 − 27, see that 7 > 2 in the ones column, and write 5 in the answer (because 7 − 2 = 5). They'd also write 2 in the tens column (5 − 2 = 3, no wait... they'd say 3, but the right answer is 25). Catch this every time. The bundles diagnostic: have them physically try to take 7 ones from a pile of 2 — they'll see they can't, and the unbundling motion makes physical sense.
• Forgetting to decrement the tens. They cross out the 2, write 12, subtract 12 − 7 = 5 — but forget to also cross out the 5 in tens column and write 4. Result: 35 instead of 25. The unbundling has TWO parts. Practice both moves together.
• Borrowing when they don't need to. Some kids will see two two-digit numbers being subtracted and reflexively borrow even when the ones can be subtracted directly. Slow them down: "Can we subtract the ones? Yes? Then no regrouping. Just go."

Where this is going

The third lesson in this module: mixed story problems — situations that need either addition or subtraction (sometimes both), where the kid has to figure out which. That's the integration test for everything in the module.