L5 · Proportionality Clinic (Inverse Proportionality: Recognition Only)
“I can sort any relationship into three bins - directly proportional (constant quotient), inversely proportional (constant product), or neither - and state the evidence for my verdict, while handling every direct-proportion skill in the unit without notes.”
Painters painting one fence: 2 workers take 12 h; 3 workers take 8 h; 4 workers take 6 h; 6 workers take 4 h. Step 1: Try the direct test - quotients time/workers: 12/2 = 6, 8/3 ≈ 2.67, 6/4 = 1.5, 4/6 ≈ 0.67. Not constant: NOT directly proportional. Step 2: Try the product test - workers × hours: 2 × 12 = 24, 3 × 8 = 24, 4 × 6 = 24, 6 × 4 = 24. Constant! The total work, 24 worker-hours, never changes. Step 3: Name it: workers and time are inversely proportional, with k = ab = 24. The verbal test matches: double the workers (2 → 4) and the time halves (12 → 6). Say it plainly - and note the boundary: this unit asks you to RECOGNIZE this pattern and name its constant product. Solving for missing values in inverse relationships is not a Grade 8 skill, and you will not be asked to do it here.
Why isn't workers-vs-time directly proportional, even though the numbers are perfectly orderly?
One fixed 120 km trip: 40 km/h takes 3 h; 60 km/h takes 2 h; 120 km/h takes 1 h. Direct, inverse, or neither? -> Inverse: products 40 × 3 = 60 × 2 = 120 × 1 = 120 (the constant distance, km), so k = ab = 120.
"Rectangles all have area 36 cm²: widths 3, 4, 6 pair with lengths 12, 9, 6. Products: 3 × 12 = ___, 4 × 9 = ___, 6 × 6 = ___. The pairs are ___________ proportional." -> 36; 36; 36; inversely
10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.