Start. A fictional drawing tool makes 18 cm of line in 3 minutes at a constant rate. Write 18 cm / 3 min = 6 cm/min. Label both units. Predict 30 cm in 5 minutes and 12 cm in 2 minutes. Ask, “What assumption lets us use the same rate?” The assumption is constant output over the times used, with no start-up delay or stopping.
Route 23A · Read a table and test constancy
A fictional copying tool records: 2 minutes → 14 labels; 4 minutes → 28 labels; 6 minutes → 42 labels. Calculate labels/min and predict the output at 5 minutes. A second table is 2 → 14, 4 → 26, 6 → 36. Explain why one constant-rate model fits the first table but not the second. Avoid claiming either is measured performance.
Teacher next move: first is 7 labels/min, giving 35 labels; second rates are 7, 6½ and 6 labels/min. A linear-looking increase is not automatically proportional.
Route 23B · Put units on an unlabeled answer
Three fictional calculations have lost their units: 240/4 = 60, 15/3 = 5 and 9/6 = 1½. Choose one context for each from these pairs: centimetres and minutes; dollars and metres; metres and seconds. Write the complete rate, describe what one unit of the denominator means, and reverse one rate with its units. Do not compare the three bare numbers as though they measure the same thing.
Teacher next move: accept a consistent assignment. If 9 m in 6 s gives 1½ m/s, the inverse is 2/3 s/m. Division order changes the meaning, not just the appearance of the answer.
Route 23C · Design a fair comparison
Paper model A produces 45 marks in 9 minutes. Model B produces 32 marks in 8 minutes. Decide which has the greater constant rate. Find a time at which both predictions are whole numbers and calculate both outputs. Write one limitation of the comparison if A pauses after every ten marks.
Teacher next move: A 5 marks/min; B 4 marks/min. At 2 minutes, for example, A 10 and B 8. The pause would require a different time model; the simple average given may not describe each interval.
Close: say the answer with “per” and both named units.