Start. A fictional paper display has 8 segments, each 5/8 m long. Total length is 5 m. Stock comes in whole 2 m packs at $3 each. Calculate three packs, 6 m, $9 and 1 m leftover. Write the assumptions: every segment may use joined lengths; all stock is usable; no fees; prices fictional. If joins were forbidden, a total-length calculation alone would need a cutting plan. Ask why the numerical total does not prove every physical arrangement works.
Route 29A · Formulate a question and check a limit
A paper diagram uses six long sections of 3/4 m and four short sections of 1/2 m. Stock is sold in whole 1 m units at $1.60 each. Joins are allowed and every length can be used. Budget is $12. Write a useful question about whether the diagram can be supplied, choose the operations, calculate the exact needed length and whole-unit purchase, and interpret the result with leftovers and budget. Then review what would need checking if joins were forbidden.
Teacher next move: 6 × 3/4 + 4 × 1/2 = 13/2 m = 6½ m; buy 7 m for $11.20, leftover 1/2 m, budget spare $0.80. Without joins, examine individual pieces and stock cuts rather than claiming the same calculation is sufficient.
Route 29B · A rate with a rational duration
A fictional plotting model draws at 3/5 m/min for 2½ minutes. Find the length. Compare with a second model drawing at 2/3 m/min for 2¼ minutes. Explain why their outputs can match even though their rates differ. Write the constant-rate/no-delay assumption and one change that would invalidate the simple prediction.
Teacher next move: (3/5) × (5/2) = 3/2 m; (2/3) × (9/4) = 3/2 m. Equal output does not imply equal rate when duration differs. A start-up delay must be subtracted from productive time if that is the stated model.
Route 29C · Build an efficient expression
A fictional stock ledger begins with 4½ m, allocates three sections of 2/3 m, then returns 1/4 m of usable stock. Write one expression for the final usable amount. Calculate exactly and check using a second method. A note says “subtract every written number first.” Explain why that advice loses the meaning of the three equal allocations and the return.
Teacher next move: 9/2 − 3 × 2/3 + 1/4 = 11/4 m = 2¾ m. The return adds stock; multiplication groups equal allocations. Reject a reordered expression only when its meaning or operation structure changes.
Close: “My answer fits ___ because ___. My model assumes ___. I would check ___ next.”