The Exit Ticket That Complicates the Pacing Guide
Ms. Okafor teaches seventh-grade math at a campus where the district pacing guide allots six days to proportional reasoning before a common unit test. On day three, a five-item exit ticket shows that 22 of 28 students correctly set up proportions, but only 9 can explain why cross-multiplying works, and 6 consistently reverse the ratios in word problems. The test is scheduled for Tuesday, and Ms. Okafor has one 45-minute intervention block on Thursday. Her coach is observing Friday and expects to see the scheduled lesson on scale drawings. Ms. Okafor worries that slowing down will put her behind before the next unit, which builds directly on proportional reasoning. She has the exit tickets in hand and 20 minutes of planning time before tomorrow.
1. What is the most defensible instructional decision for days four and five, and how would you justify it to a coach who expects the scheduled lesson?
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Read the data before choosing a response. Three different results are hiding in one set of 28 papers:
- 22/28 (79%) can set up a proportion - procedural knowledge is largely intact.
- 9/28 (32%) can justify - conceptual knowledge is weak.
- 6/28 (21%) reverse ratios - a specific, patterned error, not general confusion.
Decision rule: cancel the whole lesson only if the error is foundational and widespread. Here it is not. The correct move is to keep the scheduled scale-drawing lesson but restructure it so the same content carries the missing reasoning.
Day 4 (observed lesson): open with a 10-minute ratio-table/number-line talk where students explain why the unit rate stays constant; then the scale-drawing task with a required one-sentence justification per answer; close with a two-item exit ticket that asks for reasoning, not just a number. Tell the coach in advance: "You will see the scheduled lesson; the modification is the responsive piece I want feedback on."
Day 5: error-analysis stations using the six reversed-ratio papers (names removed), plus a teacher-led group for those six students.
Thursday block: pull the 13 students who set up correctly but could not justify for a 15-minute justification interview in triads.
What to avoid: reteaching the entire unit (wastes days and does not target the gap) or unilaterally postponing a common test (a team decision, not an individual one). Pacing guides are sequences, not permission to ignore evidence.
2. Design a short formative assessment that separates students who understand the reasoning from students who memorized a procedure, and explain how you would act on the results.
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Design principle: a selected-response item with numbers can be solved procedurally, so the probe must require explanation, comparison, or transfer.
Four-part probe (10 minutes):
- Two worked examples: one student cross-multiplies correctly, one sets up the ratio reversed. Prompt: "Who is right, and how do you know without recalculating?"
- Explain: "Why does multiplying both sides by the same quantity keep the statement true?" (accept a number-line or ratio-table explanation).
- Non-routine transfer: a recipe or map problem with a distractor that rewards matching numbers.
- Error repair: give a reversed setup and ask the student to fix it and name the mistake.
Scoring rubric (3 levels):
- Procedural only - correct answers, no mechanism, or explanation restates the steps.
- Partial - references equivalence or unit rate but cannot connect it to the operation.
- Conceptual - explains equivalence of ratios, uses a representation, spots the reversed setup.
Action rules set in advance: if fewer than half reach conceptual, add one more reasoning day rather than proceeding; if the reversed-ratio group is unchanged, move to small-group intervention with tape diagrams. Interview four or five students for two minutes each to validate the written scores - written explanations systematically underestimate English learners and students with writing IEPs.
3. How would you address the six students who reverse ratios without holding back the other 22?
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Diagnose before reteaching. Ratio reversal usually has one of three causes, and the fix differs:
- Language/comparison confusion ("ratio of boys to girls" vs. "boys:girls") - the student reads the order incorrectly.
- Missing referent - the student does not anchor the first quantity to a unit.
- Rote rule - the student always puts the larger number first.
Targeted small-group sequence (15 minutes, during station time):
- Sorting task: eight word problems with two candidate setups each; students sort into correct and reversed and must say the phrase "for every ___ there are ___" out loud.
- Tape diagram or double number line for two of the sorted problems so the order is visible, not verbal.
- Three fresh items with a required label on each quantity.
Keep the rest of the class moving: the other 22 work in pairs on scale drawings with a justification sentence, then check against a posted key; a station rotation keeps the teacher free for six minutes of direct instruction.
Progress monitor: a two-item exit ticket the next day. If reversal persists, check reading of comparison language and coordinate with the EL specialist - persistent reversal is often a comprehension issue, not a math one.
Do not seat the six together permanently or assign extra worksheets; both increase practice of the error or stigmatize the group.