Proportional Reasoning — Rates and Ratios
Warm-up
The road-trip fuel gauge: "We burned of a tank covering 210 km. Can we reach the city 350 km away on what's left?" Pairs sketch, argue, decide. (Full tank ≈ 560 km; remaining ≈ 350 km — EXACTLY on the edge, which is the fun.)
Grade 8 proportional reasoning graduates to multi-step rate problems, unit conversions, and scale — the tools that turn one measured relationship into every answer it implies.
Explore
Rate-network stations: (1) The gears/labour problem: 3 painters take 8 days for a fence — how long for 4 painters? (INVERSE proportion — more workers, LESS time: painter-days → 6 days. First meeting with a proportionality that runs backwards.) (2) Scale maps: 1:50,000 — a 7 cm map segment is how many real km? (3.5 km.) (3) Unit-conversion chains: 90 km/h into m/s ( m/s) by multiplying with cleverly-oriented "1"s.
Station 1's debrief separates the worlds: DIRECT (more x, proportionally more y; constant ) versus INVERSE (more x, proportionally less y; constant ).
Formalize
Formalize the two proportionality types side by side:
The diagnosis question before ANY setup: "if I double this, does that double — or halve?" Painter-days, speed-time (fixed distance), and sharing problems run inverse; cost-quantity, distance-time (fixed speed), and recipes run direct. Wrong diagnosis, perfectly executed, is still a wrong answer.
Practice
Practice: two direct proportions (one via unit rate); two inverse (workers-time, speed-time); one map-scale; one conversion chain; one mixed diagnosis set where the ONLY task is labelling direct/inverse/neither with a sentence.
Exit ticket: "6 pumps drain the pool in 10 hours. How long for 4 pumps — and what kind of proportionality?" (Inverse; pump-hours → 15 h.)
Exit ticket
Practice: two direct proportions (one via unit rate); two inverse (workers-time, speed-time); one map-scale; one conversion chain; one mixed diagnosis set where the ONLY task is labelling direct/inverse/neither with a sentence.
Exit ticket: "6 pumps drain the pool in 10 hours. How long for 4 pumps — and what kind of proportionality?" (Inverse; pump-hours → 15 h.)
The job: 8 crew members can strike the stage in 6 hours. The venue needs it done in 4. How many crew?
Step 1: Diagnose: double the crew → half the time? Yes — inverse. Constant: crew × hours crew-hours (the job's true size).
Step 2: Solve for the deadline: → crew.
Step 3: Sanity: fewer hours demanded MORE people ✓ (direct-proportion habits would have shrunk the crew — absurd under a deadline).
Step 4: The honest caveat, worth saying at 14: crew-hours assumes workers don't trip over each other — nine women can't strike a stage in 5.3 hours if there are only six exits. Inverse proportion is a MODEL with a validity range; naming its limits is part of using it.
The plan: a deck blueprint at scale 1:40. On paper, the deck is 12.5 cm by 8 cm.
Step 1: Convert each dimension: real paper. Length: cm m. Width: cm m.
Step 2: Real area: m².
Step 3: The area trap, sprung on purpose: paper area is cm²; real area is NOT cm². Area scales by : cm² m² ✓ — matching step 2.
Step 4: The law once more (third sighting this year — m²↔cm², the pizza, now scale): lengths scale by ; areas by ; volumes will scale by . One exponent per dimension. It never stops being tested and never stops being true.
The claim: a cheetah sprints at 1.7 km per minute. Highway limit: 110 km/h. Who's faster?
Step 1: Convert the cheetah to km/h by chaining: km/h.
Step 2: Compare: 102 < 110 — the highway (barely) outruns the cheetah.
Step 3: Convert both to m/s for the sprinter's frame: cheetah m/s; cars m/s.
Step 4: The chain audit: units cancelled like fraction factors at every link (min with min, km surviving) — a chain whose units survive wrong is a chain set up wrong. Dimensional bookkeeping is the physics-class superpower being installed a year early.