Twenty-Five Out of Twenty-Eight Say the Product Is Greater

The procedure is largely intact; the concept of magnitude is not.

• Item 2 at 26/28 correct confirms students can multiply across accurately on small, proper fractions. Speed and accuracy are not the bottleneck.
• Item 3 is the informative one: 12/20 on 4 × 3/5 shows that when a whole number meets a fraction, some students keep applying the denominator-to-denominator habit reflexively. They are pattern-matching a procedure, not checking whether an answer is sensible - 12/20 is smaller than 1, and multiplying 4 by something near 0.6 cannot give less than 1. No one flinched at an impossible answer. That self-correction in nine cases happened after the fact, probably from a classmate or a second look, not from reasoning.
• Item 5 is the clearest signal: if 5/6 × 3/4 is greater than 5/6, students believe multiplication always makes things bigger. That is an overgeneralization from whole numbers that survives drill because drill never asks about size. Twenty-five of twenty-eight share it, so this is a class-level gap, not an individual error.
• Item 6 confirms transfer is weak and shows two distinct error types: 5/7 comes from adding numerator and numerator, denominator and denominator under the pressure of a story; 9/8 is the answer to 3/4 × 3/2, i.e., misreading '2/3 of the recipe' as '1 1/2 times the recipe.' So some students also misparse the multiplicative relationship.

Instructional implication: an item analysis on accuracy alone would say 'item 5 low, reteach item 5.' The cross-item pattern says the class needs a magnitude-and-meaning intervention, not more items like 2.

Cut: most of the item types like Item 2. They are already at 93% and practicing them again buys nothing. I would also cut silent individual work for the first 25 minutes - the misconception lives in talk, and it needs to be spoken to be revised.

Open (5 min): the disagreement. Put Item 5 on the board with the class's own split: 25 said greater, 3 said less. 'Twenty-five of you agree with each other. That is not the same as being right. Find someone who disagrees and settle it with a picture, not with a rule.'

Build (15 min): two representations, one per pair.

• Fraction-of-a-fraction area model: a square, shade 5/6, then shade 3/4 of the shaded region. Students see the answer is a part of the shaded part, so it must be smaller.
• Number line: mark 5/6, then take 3/4 of the distance from 0 to 5/6. The result sits left of 5/6.

Generalize (10 min): a claim students write themselves. 'When is multiplying by a fraction going to make the product smaller, and when bigger?' Answer they should reach: multiplying by a number less than 1 shrinks (of a number), by 1 keeps it, by more than 1 grows it. This is the idea that repairs Item 6 too - '2/3 of the recipe' is less than one recipe.

Estimate-then-compute (8 min): four items only. 7/8 × 2/3 (less than 7/8), 4 × 3/5 (about 2.4, so not 12/20), 5 × 2/3, and one '2/3 of 3/4 cup.' Students write the estimate with a reason before the exact answer. Requiring an estimate first is the habit that catches 12/20.

Exit ticket (7 min): two prompts, no computation. (1) Is 8/9 × 4/5 greater or less than 8/9? Draw or explain. (2) A student says 6 × 2/5 = 12/25. What is wrong, and what should they have noticed? Success criterion: 24 of 28 give a magnitude reason on prompt 1.

I would tell students the exit ticket is the review, not a quiz - the goal is to surface the reasoning, and I will retake the question the next day.

What I would say, directly but without dismissing the instinct. 'The worksheet will raise item-2 scores we already have at 93%, and it will make the 25 students who said the product is greater faster at being wrong. Their errors are not from underpractice; they are from a rule that has no magnitude check attached. Thirty more items rehearse the rule and give the misconception another 30 chances to feel confirmed, because on proper-fraction items like 2/3 × 4/5 nobody notices they are holding a false belief.'

What I would offer A shared review protocol that uses the worksheet differently rather than throwing it out - teams often need the artifact, not the plan:

1. Keep six items from the worksheet, chosen so that each one would trip a 'multiplication always grows' belief: 4 × 3/5, 7/8 × 1/2, 2/3 of 3/4, 5 × 2/5, 1/2 × 9/10, and one where the fraction is greater than 1 (3/2 × 4).
2. Print the rest as a home thinking set with one instruction: before each computation, write bigger or smaller than the first number, and why.
3. Have each team give the same two-prompt exit ticket so we can compare results across classrooms - the estimate-first item is the one that distinguishes a rehearsed class from a reasoning class.

The compromise to name out loud: if the team's concern is pacing-guide coverage, cover the guide with the six estimation items rather than thirty rote ones. Same number of minutes, and the evidence we collect will actually tell us whether the unit can close.