Ratios and Rates
Warm-up
Show two receipts: "Store A: 6 granola bars for \$4.50. Store B: 8 of the same bars for \$5.60. Which store, and how much do you save per bar?" Pairs decide (A: 75¢/bar, B: 70¢/bar → B) and report their comparison strategy.
Every strategy that worked reduced both offers to a common denominator of comparison — per ONE bar. That's a unit rate, and it's the tool of the unit.
Explore
Rate-hunting circuit: stations with real contexts — heart-rate (beats per 15 s → per minute), recipe scaling, speed from a road-trip table (340 km in 4 h), currency-style conversions (3 tokens per 2 tickets). At each, pairs compute the unit rate BOTH directions where sensible (km per hour AND hours per km) and decide which direction answers which question.
Then proportional-or-not sorting: tables where doubling one quantity doubles the other (proportional — constant unit rate) versus tables where it doesn't (a taxi with a base fare: cost = 3 + 2d). The test: does the table pass through "zero gives zero," and is every ratio identical?
Formalize
Formalize: a rate compares different units; the unit rate is the amount per ONE; proportional relationships have constant unit rate :
Solving proportions honestly: scale the known ratio (multiplicative reasoning) or use with one unknown. Cross-multiplication is legal but LAST — students who meet it before scaling lose the sense of what they're doing.
Practice
Practice: four unit-rate computations with direction choices; two proportion solves by scaling; one proportional-or-not classification with defence; one multi-rate problem ("which printer is faster, and by how many pages over an hour?").
Exit ticket: "12 cookies need 3 cups of flour. How much flour for 20 cookies — solved by unit rate, not cross-multiplication." ( cup per cookie → 5 cups.)
Exit ticket
Practice: four unit-rate computations with direction choices; two proportion solves by scaling; one proportional-or-not classification with defence; one multi-rate problem ("which printer is faster, and by how many pages over an hour?").
Exit ticket: "12 cookies need 3 cups of flour. How much flour for 20 cookies — solved by unit rate, not cross-multiplication." ( cup per cookie → 5 cups.)
The recipe: 2 parts blue to 3 parts yellow makes the green. Needed: 5 L total.
Step 1: Read the part:whole structure: parts total → blue is of any batch, yellow .
Step 2: Scale to 5 L: blue L; yellow L. (The numbers landed sweetly because 5 L matches 5 parts — one litre per part.)
Step 3: Now the honest version: 4 L total. Each part is L → blue L, yellow L. Check: ✓ and ✓ — same green.
Step 4: Name the pipeline: parts → fraction of whole → scale. It never breaks, even when the numbers stop being gifts.
The data: 340 km driven in 4 hours.
Step 1: Unit rate, direction 1: km/h — answers "how far per hour?"
Step 2: Direction 2: h/km ≈ 42 seconds per km — answers "how long per km?" Both rates are the same fact, reciprocals of each other; the question picks the direction.
Step 3: Use each: "How far by hour 6 (same pace)?" km. "How long for the remaining 255 km?" h.
Step 4: The proportionality caveat said aloud: these predictions assume CONSTANT rate — real drivers stop for gas. Proportional models are exact about their assumptions; users should be too.
Gym A: \$8 per visit. Gym B: \$30 monthly fee plus \$3 per visit. A student claims "B is proportional too — more visits, more money."
Step 1: Test the claim with the zero-zero criterion: zero visits at A costs \$0 ✓; zero visits at B costs \$30 ✗. B fails — increasing isn't the same as proportional.
Step 2: Confirm with ratios: A: constant ✓. B: 2 visits → $36 → ratio 18; 5 visits → $45 → ratio 9. Not constant ✗.
Step 3: Find the crossover anyway (the useful question): → → . Fewer than 6 visits/month: A wins. More: B wins.
Step 4: The modelling lesson: proportionality is a special, checkable structure — and when it fails, the comparison doesn't die; it just needs the equation instead of the shortcut.