Fifth-Grade Division: The Digit-Skipping Error
Mr. Alvarez gives his fifth graders an exit ticket: 1,248 ÷ 6. Of 22 students, 13 write 208, six write 28, and three leave it blank. He interviews two of the six. Both explain: '12 divided by 6 is 2, then 4 you can't do, so you bring down the 8, and 48 divided by 6 is 8.' Both students correctly solve 24 ÷ 6 = 4 and 600 ÷ 6 = 100 when asked separately. None of the six checks the answer by multiplying. When Mr. Alvarez asks the class whether 28 is reasonable for 1,248 ÷ 6, several students say yes because 'division makes numbers smaller.' The unit test on multi-digit division is scheduled for next Friday.
1. Name the misconception precisely, and explain why 'they don't know their facts' is the wrong diagnosis.
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The misconception: the students treat each digit as its own small division problem, and when a digit is smaller than the divisor they skip it instead of recording a 0 in that place. Their answer, 28, has lost the tens place entirely.
Why it is not a fact-fluency problem: both interviewed students correctly produce 24 ÷ 6 = 4 and 600 ÷ 6 = 100 when the problems are presented separately. Their facts are intact. What fails is the place-value frame that makes 1,248 into 1,200 + 48 rather than 1, 2, 4, 8. The digit 4 represents 40, and 40 ÷ 6 = 6 remainder 4 - it is not 'doesn't go.'
The confirming evidence is magnitude: 6 × 28 = 168, nowhere near 1,248. Students have no habit of estimating or checking, and several actively hold a false rule ('division makes numbers smaller') that suppresses the check that would have caught it. That rule is itself worth a lesson - dividing by a number less than 1 makes a number larger, and the class will meet that case.
Note the blank papers: three students may be avoiding the task rather than holding a different misconception, so interview them separately before grouping them with the six.
2. Design the instructional sequence for the six students, and say what the other 13 should be doing at the same time.
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Sequence for the six (three short sessions):
- Concrete - base-ten blocks. Build 1,248 with flats, rods, and units; share among 6 groups by exchanging a flat for 10 rods when needed. The exchange is the physical version of the step they skipped. Have them say 'this 4 is really 40' each time.
- Representational - area model. Draw a rectangle of area 1,248 with one side 6; partition it into 200 + 8 sections. The missing tens column becomes visibly empty - a zero goes there.
- Abstract with estimation first. Before computing 1,248 ÷ 6, students write a range: '6 × 200 = 1,200, so the answer is a bit more than 200.' That single habit eliminates 28 as a candidate.
The other 13 are not done learning. Give them 3,024 ÷ 6 and 6,012 ÷ 6 (zeros in the quotient and in the dividend), require a written estimate and a multiplication check, and pair them to explain to each other why the zero must be written. Correct answers on 1,248 ÷ 6 do not prove they understand place value - several may simply have had no digit smaller than the divisor.
3. How would you handle next Friday's unit test, and what would you change about future exit tickets?
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The test: keep the date but change the composition. Include a mandatory estimate-and-check box for each division item, and include at least two items with a zero in the quotient. Rationale: if the test only repeats 1,248-type problems, the six students will fail on a skill they are actively re-learning, and the 13 who passed will be certified on evidence that never tested the place-value understanding. Weight the estimate/check portion, since it is the habit that makes the error self-correcting.
Exit tickets going forward:
- Three items, not one - one routine problem, one with a zero in the quotient, one with a zero in the dividend. A single item cannot distinguish a place-value gap from a computational slip.
- Require the check. 'Show 6 × your answer = ' makes magnitude visible in 10 seconds of grading.
- Sort responses into misconception groups before the next lesson, not just right/wrong. The six 28s and the three blanks were two different instructional needs hiding in one score.