Start. Draw two equally long bars. Mark 1/2 of one and 1/3 of the other. Partition both into sixths. Add 3/6 + 2/6 = 5/6. Ask why 1/2 + 1/3 is not 2/5. Fractions count equal-size parts of a named whole; adding the denominators would change that part size.
Route 27A · Compare picture and exact notation
Use equal whole bars to show 3/4 + 1/6. Find a common partition, calculate exactly, and say whether the sum is below or above one whole. Then show how much is missing from one whole. A ruler drawing can support the reasoning without pretending to measure an exact physical length.
Teacher next move: 9/12 + 2/12 = 11/12; missing 1/12. Equal whole bars matter; unequal wholes cannot be combined as though the fractions refer to the same amount.
Route 27B · Reconstruct a missing amount
A paper length model uses 5/8 m in one section and an unknown length in a second. Together they use 1⅜ m. Find the second length, write an equation and check by adding. Give the answer as an exact fraction and a finite decimal. Explain the meaning of the metre unit in both forms.
Teacher next move: 11/8 − 5/8 = 6/8 = 3/4 m = 0.75 m; check 5/8 + 6/8 = 11/8. Convert 1⅜ before subtracting; a mixed number is a sum, not a product of its written parts.
Route 27C · Decide which calculation is efficient
Calculate 7/10 + 3/5 − 1/2 using exact fractions. Show a second method using finite decimals. Choose which representation you find easier to check and explain why both methods have the same exact result here. Give one fraction for which a short rounded decimal would lose exactness.
Teacher next move: 7/10 + 6/10 − 5/10 = 8/10 = 4/5 = 0.8. Finite decimals 0.7 + 0.6 − 0.5 are exact. For 1/3, 0.33 is an approximation, not an equal value. Full recurring-decimal notation is reserved for the later mapped sequence.
Close: name the whole, the common part size and the exact answer.