Equivalent Fractions and Fraction Benchmarks

Way 1 — the fraction strip: shade 2 of 3 equal parts. Now slice EACH third into 4 slivers: 12 slivers total, and the shaded region — untouched — now counts as 8 of them. Same paint, finer grid: 2/3 = 8/12.

Way 2 — the number line: 2/3 and 8/12 land on the SAME point between 0 and 1. One address, two zip-code formats.

Way 3 — the arithmetic, now explainable: multiplying top and bottom by 4 (2×4)/(3×4) is exactly the "slice each part into 4" move — the ×4 on the bottom makes pieces smaller; the ×4 on top keeps the same amount claimed. Never "multiply by 1 because the rule says" — always "re-slice without repainting."

The check question: is 6/10 equivalent to 3/5? Reverse the slicing (merge pairs): yes. Is 4/6 equivalent to 6/8? Reduce both: 2/3 vs 3/4 — different addresses. Equivalence is a same-point claim, checkable every time.

Step 1: Sort against the benchmarks 0, 1/2, 1 before any calculation: 3/8 — half of 8 is 4, and 3 < 4 → BELOW half. 5/9 — half of 9 is 4.5, and 5 > 4.5 → ABOVE half (barely). 7/12 — half is 6, so above half (a bit). 5/4 — top beats bottom → ABOVE ONE.

Step 2: Order the easy skeleton: 3/8 < {5/9, 7/12} < 5/4.

Step 3: Break the middle tie with a finer benchmark or small reasoning: 5/9 is 1/18 above half (5/9 − 1/2 = 1/18); 7/12 is 1/12 above half. 1/12 > 1/18, so 7/12 sits higher. Final: 3/8 < 5/9 < 7/12 < 5/4.

Step 4: Tally the work: zero common denominators for the skeleton; one tiny comparison for the tie. Benchmark-first is how numerate adults actually think — common-denominator-everything is the long way, taught next month for when precision demands it.

Drill: place 4/7, 9/8, 1/3, 6/13 on a number line in under a minute, benchmarks only.

The stuck point: a student agrees 2/5 and 3/7 are both below half but can't decide between them — "fifths and sevenths don't match."

Step 1: Validate the instinct: unlike pieces genuinely can't be counted against each other. The solution isn't to squint harder — it's to re-slice BOTH into a common grain.

Step 2: Find the common grain: fifths and sevenths both re-slice into thirty-fifths (5 × 7). Re-slice: 2/5 = 14/35 (each fifth became 7 slivers); 3/7 = 15/35 (each seventh became 5).

Step 3: NOW count: 14 < 15, so 2/5 < 3/7 — by a single sliver (1/35). No benchmark could see a gap that thin; this is exactly when the common-denominator tool earns its keep.

Step 4: Reflect on tool choice like a craftsperson: far-apart fractions → benchmarks (fast); whisker-close fractions → common denominators (exact). Fluency = owning both AND knowing which the moment calls for.

Exit: compare 5/8 vs 7/11 by whichever tool you choose — then defend the choice, not just the answer.