Keep-Change-Flip or Reason From Quantities?
Ms. Alvarez teaches a 6th-grade mathematics class of 27 students in a departmentalized school. The district pacing guide allots two days for dividing fractions; tomorrow the team gives a common quiz and moves to decimal operations. On an exit ticket, 22 of 27 students correctly computed 3/4 ÷ 1/2 = 1 1/2 using keep-change-flip. When asked to draw a model, only 7 did so accurately, and 12 wrote a story such as 'I have 3/4 of a pizza and eat 1/2' that does not match the quotient. One student insists the answer must be less than 3/4 'because division makes smaller.' The team leader says the quiz data are strong enough to proceed; a parent has asked why homework looks different from her own schooling. Ms. Alvarez has one 45-minute period before the quiz and a document camera but no additional planning time.
1. In the 45-minute period, should Ms. Alvarez reteach the division algorithm, move to the quiz as planned, or do something else? Justify the decision using the exit-ticket evidence and describe the next instructional move.
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The evidence separates procedural fluency from meaning: 22 of 27 students can execute keep-change-flip, but only 7 of 27 can model the operation, and 12 of 27 attach an action story that does not fit a quotitive interpretation. That pattern is not strong evidence that the class is ready to move on; it is evidence that the algorithm is masking an unformed concept. The 45 minutes should not be spent re-demonstrating keep-change-flip, which most students already can do. Instead, use a contrasting-cases routine: give pairs two stories for 3/4 ÷ 1/2-one where 3/4 is shared into 1/2-sized portions and one where 1/2 is taken from 3/4-and ask them to match the expression, draw a bar model, and explain why the quotient 1 1/2 means one and a half half-cups. Then pose 3/4 ÷ 2 to challenge the claim that division always makes smaller. End with a two-item exit ticket that requires a model and a story. Keep the common quiz but add one modeling item or score it separately, so the team sees conceptual data, not just algorithm accuracy.
2. How should Ms. Alvarez respond to the team leader's claim that the quiz data are strong enough to proceed, and to the parent's concern?
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To the team leader, she should not frame the issue as 'my students cannot divide fractions' but as 'our common quiz currently measures only one kind of evidence.' She can bring the tally: 22 correct procedures, 7 correct models, 12 mismatched stories. Propose adding a modeling item to the common quiz and sharing results at the next team meeting; this respects pacing while testing whether the pattern is grade-wide. To the parent, avoid implying that the parent's schooling was wrong. Explain that the goal is for the child to know why 3/4 ÷ 1/2 = 1 1/2, not just to flip and multiply; show the bar model and a simple story. Invite the parent to ask the child to explain one homework model. This is a judgment about communication: the teacher needs allies, not a curriculum debate. She should also document the team conversation and the added item so the instructional decision is visible in her practice evidence.