Three Entry Points in One Fractions Unit
Ms. Alvarez gives a 12-item pre-assessment before her Grade 4 fractions unit. Results: 6 of 22 students miss only the two items on comparing fractions with unlike denominators; 9 students answer all like-denominator items correctly but miss every item involving unlike denominators; 5 students miss the like-denominator items as well, including two who label 3/4 as "three and four." She has 10 instructional days, one 60-minute block per day, and a district benchmark on Day 12 that covers the full unit. Her teammate, who teaches the other Grade 4 class, plans to teach straight through the textbook sequence and says pre-tests mostly just confirm what she already knows. Ms. Alvarez has one paraprofessional for 25 minutes a day and no additional planning time.
1. What does the pre-assessment pattern actually tell Ms. Alvarez, and why is the teammate's plan a poor fit for her data?
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The data does not describe one class with a range of scores; it describes three different instructional needs:
- 5 students with a part-whole misconception (reading 3/4 as "three and four" is a notation/language confusion, not a comparison error). These students need concrete modeling of what the denominator means before any comparison work is meaningful.
- 9 students who own like-denominator comparison and are blocked only by the unlike-denominator procedure. This is a targeted, 2–3 day fix, not a full reteach.
- 6 students who already compare unlike denominators and need extension, not repetition.
The teammate's plan assumes a single entry point. If Ms. Alvarez teaches the textbook sequence to the whole group, the 6 ready students spend several days on material they already know, which is where off-task behavior usually starts, and the 5 students with the notation misconception get procedural instruction layered on top of a broken foundation. The pre-test's value is not "confirmation" - it is the grouping information that determines the lesson format.
2. Ms. Alvarez is deciding between two Day 1 plans: (A) whole-group reteach of like denominators for everyone, then move on; (B) a shared 10-minute launch, then three differentiated station rotations. Recommend one and defend it using the specific numbers, including the paraprofessional constraint.
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Recommend B, with the constraint that station content must be designed so the paraprofessional can run one station from a scripted routine rather than requiring content expertise.
Reasoning with the numbers:
- Plan A spends Day 1 on material that 17 of 22 students already control, and it still does not address the unlike-denominator gap for the 9. That is a full day of instruction aimed at 5 students.
- Plan B lets the 5 students work at the concrete/pictorial level (fraction strips, folding, naming) with the teacher, where the misconception lives, while the 9 work an error-analysis task on unlike denominators and the 6 work a comparison/ordering extension with justification.
- The paraprofessional's 25 minutes can cover the rotation that is procedural and self-checking (e.g., the extension task with an answer key), leaving the teacher with the group that needs diagnostic judgment.
A defensible variation: run the 10-minute launch as a shared anchor for all 22 (what a denominator means) so the 5 are not the only students hearing it, then rotate. What is not defensible is spending the block on content 77% of the class has demonstrated.
3. How should the Day 10 check for understanding be built so that it actually informs the Day 11 reteach decision before the Day 12 benchmark?
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It should be built as a parallel mini-assessment, not a repeat of the pre-test, with one item per skill level so the results map directly onto the three groups:
- One item naming/representing a fraction from a model (probes the notation misconception).
- One like-denominator comparison with justification.
- One unlike-denominator comparison, plus a spot-the-error item where a wrong strategy is shown.
- One transfer item in a non-routine context (e.g., comparing two recipes) so students who only memorized the procedure are exposed.
Design notes that matter:
- Keep the item count small enough to score the same day; a check that returns Friday cannot shape Thursday.
- Have students justify in writing or with a drawing, because a correct answer with a wrong reason is the failure mode most likely to hit the benchmark.
- Record results on the same three-group sheet used on Day 1 so movement is visible. If the 5 notation students have moved but the 9 have not, Day 11 targets the 9, not the whole class again.