Regrouping Across a Zero
Mr. Bell's third graders are working toward fluent subtraction within 1,000. The day's problem was 503 − 268. Ana used an open number line: 503 − 200 = 303, 303 − 60 = 243, 243 − 8 = 235. Ben used the standard algorithm and recorded 503 as 4 hundreds, 9 tens, and 13 ones, also getting 235. Cara wrote 365 and explained, '8 − 3 = 5, 6 − 0 = 6, 5 − 2 = 3.' Before solving, students were asked to estimate. Cara estimated 500 − 300 = 200, but she did not compare that estimate to her answer. Mr. Bell has 30 minutes and wants Cara to see regrouping across a zero as a sensible exchange rather than a set of rules to memorize. Place-value blocks and number lines are available.
1. Explain Cara's error in terms of place value, and show why her own estimate should have caught it.
Read the worked answer
Cara subtracts the smaller digit from the larger digit in every column, so place value and regrouping disappear. In the tens column she computes 6 − 0 = 6, but the problem requires taking 6 tens from 0 tens, which demands an exchange from the hundreds. In the hundreds column she computes 5 − 2 = 3, but the 5 has already been reduced to 4 by that exchange. Her estimate, 500 − 300 = 200, says the answer is near 200; 365 is 165 away from 200, so it cannot be right. The exact answer, 235, is only 35 from 200, which is reasonable because she rounded 268 up by 32 and 503 down by 3. Comparing an estimate to an exact answer is a habit Cara is not yet using, so the estimate is wasted work for her.
2. Which representation would you use first with Cara, and what questions would you ask? Explain why the standard algorithm alone is not the starting point.
Read the worked answer
Start with place-value blocks for 503: 5 flats, 0 rods, 3 units. Ask, 'Can you take 8 units from 3 units?' She trades one flat for 10 rods, then trades one rod for 10 units, producing 4 flats, 9 rods, 13 units. Now 13 − 8 = 5 units, 9 − 6 = 3 rods, 4 − 2 = 2 flats: 235. Record each trade next to the written algorithm so the crossed-out digits carry meaning. The algorithm alone is not the starting point because Cara can execute digit rules; what she lacks is the meaning of a hundred as 10 tens and a ten as 10 ones. The blocks make the exchange visible, and the number line can follow as a second model: 503 → 303 → 243 → 235.
3. Write the next problem and the response you would listen for.
Read the worked answer
Give 402 − 157. Listen for: 'I can't take 7 from 2, so I trade a ten; but there are 0 tens, so I trade a hundred for 10 tens, then a ten for 10 ones.' The expected answer is 245 (402 − 100 = 302, 302 − 50 = 252, 252 − 7 = 245). Evidence of repair: Cara states the value of each regrouped digit correctly (3 hundreds, 9 tens, 12 ones), checks her answer against an estimate (400 − 160 = 240), and explains why 245 is close to 240. If she writes 355 by subtracting smaller digits from larger ones again, return to the blocks and have her build 402 and physically remove 157 before recording anything.