Subtracting Across Zeros: 500 − 236 and the Check That Was Ignored
Diego, a third grader, is solving this word problem: “The school store began the day with 500 pencils. By noon, 236 pencils had been sold. How many pencils were left?”
His paper shows:
500
- 236
-----
336
In the ones column he wrote 6, in the tens column 3, and in the hundreds column 3. He explains, “Zero minus six is six, zero minus three is three, and five minus two is three.”
Diego then checks his work by adding: 336 + 236 = 572. He frowns at the 572 and says, “That’s not 500, but it’s close. Maybe I should change my answer to 364 so the check works out better.”
The teacher has one short conference slot with him before the class moves on to the next problem set, and she wants to use the failed check as the teaching moment rather than simply marking the problem wrong.
1. What misunderstanding produced the answer 336, and what is the correct difference? Show the regrouping step by step.
Read the worked answer
Diego is treating 0 as if it were larger than any digit, subtracting the smaller digit from the larger one and keeping the sign of the bigger number. That is why 0 − 6 became 6 and 0 − 3 became 3. He never regroups, so the columns are not actually being subtracted.
The correct work regroups 500 into 4 hundreds, 9 tens, and 10 ones:
- Ones: 10 − 6 = 4
- Tens: 9 − 3 = 6
- Hundreds: 4 − 2 = 2
So 500 − 236 = 264. The check confirms it: 264 + 236 = 500. The key idea to name for Diego is that borrowing from a zero means borrowing from the next place that has something to give, then passing the ten along.
2. Diego’s check gave 572, not 500, and he wanted to change his answer to make the check work. What should the teacher say?
Read the worked answer
The teacher should be direct: the check is a test, and a failed test means the answer is wrong - not that the check needs adjusting. An answer is never chosen because it makes the addition come out nicely; it is chosen because the subtraction is correct. Changing 336 to 364 without redoing the subtraction is guessing with extra steps, and it still would not check out (364 + 236 = 600).
The useful follow-up is to ask where the 72 came from. Diego’s total overshot by 72, which points at the tens and ones columns, where his “borrowing” invented value out of nothing. The check should send him back to the columns, not to the answer line.
3. A classmate claims Diego only struggled because the problem had zeros in it. Would 506 − 236 require the same regrouping? Explain.
Read the worked answer
Yes, and this is the point that shows the “zeros are special” explanation is incomplete. In 506 − 236, the ones column is 6 − 6 = 0, so no borrowing is needed there. But the tens column is 0 − 3, which still requires borrowing from the hundreds.
Regrouping 506 gives 4 hundreds, 10 tens, and 6 ones. Then 6 − 6 = 0 ones, 10 − 3 = 7 tens, and 4 − 2 = 2 hundreds, for an answer of 270. Check: 270 + 236 = 506.
The real issue is any column where the bottom digit is larger than the top digit, not whether a zero happens to appear. A quick contrast problem such as 566 − 236, which needs no regrouping at all, would make that difference concrete for Diego.