Exit Ticket Splits Subtraction With Regrouping
On a three-item exit ticket (45 − 18, 62 − 37, 53 − 26), Ms. Boone's 24 third graders sort into four groups: 11 students correct on all three; 7 students answered 33, 35, and 33; 4 students traded correctly on item 1 but wrote 45 − 18 = 37; 2 students got all three right but, when asked how, said "I just know it" and could not show any method or explain a trade. The district pacing guide schedules three-digit addition with regrouping to begin Monday, and Ms. Boone's grade-level team is expected to stay on that calendar.
1. Diagnose the 7 students. What rule are they applying, why is it not carelessness, and what does it reveal about their place-value understanding?
Read the worked answer
All three of their answers come from subtracting the smaller digit from the larger digit in each column: 5 − 4 = 1 and 8 − 3 = 5 would give 15, so actually check: 45 − 18 they wrote 33 → tens 4 − 1 = 3, ones 8 − 5 = 3. 62 − 37 → 35 (3 −... ) tens 6 − 3 = 3, ones 7 − 2 = 5. Yes - always smaller from larger, column by column.
Why not careless: the rule is correct for every single-digit subtraction they have ever done, where the minuend digit is always larger. They are applying a valid generalization outside the conditions that made it valid.
What it reveals: they are treating digits as separate numbers rather than as tens and ones, and they have no reason to trade because they do not believe a ten can be broken into ten ones. Their answers look plausible (33 is close to 27), so nothing triggers a reasonableness check.
2. Plan the next three math blocks for the four groups. Be specific about what each group does and what you do.
Read the worked answer
Group of 7 (teacher-led, 15 min each day): base-ten blocks only, no algorithm at first. Build 45, try to remove 18, hit the wall of not enough ones, physically trade one ten for ten ones, then record what happened. Move to a written record that matches each move (drawing sticks and dots). Only after two days do they see the written algorithm beside the model. The goal is that they ask to trade, not that they memorize a procedure.
Group of 4 (partner work): their error is a recording/sequencing slip, not a conceptual one - they traded but subtracted the untraded number. Give them the same block work plus a checklist (trade, mark the new number, subtract ones, subtract tens) and have them audit each other's written work.
Group of 2 (independent, then explain): accurate but fragile. Have them write explanations and build models for a partner, and solve one problem two ways (blocks and algorithm). Correct answers they cannot explain are at risk of collapsing when numbers get larger.
Group of 11: independent practice with word problems and a missing-number task (□ − 18 = 27) to push reasoning; they also serve as explainers, not as tutors for the 7.
3. Should Ms. Boone follow the pacing guide and start three-digit addition Monday? Justify your answer with the instructional consequences either way.
Read the worked answer
Start the unit on schedule, but restructure what "on schedule" means for the 7.
The reasoning: three-digit addition with regrouping uses the same composing-a-ten concept. For the 11 fluent students, delaying is wasted time and loses the team's coherence; for the 7, marching into three-digit addition without trading will produce more of the same error at greater magnitude, and the gap hardens.
Concrete plan:
- Whole-class launch on Monday as scheduled, using base-ten blocks for everyone - the concrete model serves both the new three-digit work and the 7 students' gap.
- The 7 spend the first 15 minutes of each block on two-digit trading before joining the whole-group application; by mid-week their work merges with three-digit trading, which is the same idea.
- Document the modification and the evidence (exit ticket data) for the team, and propose that the team's next common assessment include a two-digit trading item so the gap is visible to everyone.
Following the calendar is defensible; following it while pretending the data don't exist is not.