Ratios and Percentages
Warm-up
Show a paint-mixing photo: "This pink used 2 cups red to 3 cups white. I want a BIGGER batch of the SAME pink. What do I mix?" Collect proposals on the board — 4:6, 6:9, 20:30 — and one deliberate wrong answer to test: "3 red and 4 white (I added one of each)."
The add-one proposal is the perfect foil: it FEELS fair and produces a different pink. Scaling, not adding, preserves the relationship — that single idea is the unit.
Explore
Ratio-table workshop: pairs build tables for the paint (red 2, 4, 6, 10, 1 / white 3, 6, 9, 15, 1.5), discovering the legal moves — double a column, add two columns, halve — and the illegal one (add the same number to both entries).
Then percents join as "ratios out of 100": stretch the paint table until white hits 100 — red lands at … choose friendlier data: a juice mix 1:4 concentrate-to-water → concentrate is of the drink → 20 out of 100 → 20%. The table is the bridge from ratio to fraction to percent, all one relationship in three outfits.
Formalize
Formalize: a ratio compares quantities multiplicatively (part:part like red:white, or part:whole like red:batch). Equivalent ratios come from scaling both parts by the same factor. A percent is a part:whole ratio scaled to 100:
The part:part vs part:whole distinction does real work: red:white is 2:3, but red is (not !) of the batch. Every ratio error worth catching this year is one of these two confusions — adding instead of scaling, or part:part read as part:whole.
Practice
Practice: complete three ratio tables (one with a fractional scale factor); convert between ratio, fraction, and percent forms; two story problems — a recipe rescale and a "what percent of the class" question requiring the part:whole read.
Exit ticket: "A team's win:loss ratio is 3:2. What PERCENT of games did they win?" (Wins are = 60% — the part:whole trap in one line.)
Exit ticket
Practice: complete three ratio tables (one with a fractional scale factor); convert between ratio, fraction, and percent forms; two story problems — a recipe rescale and a "what percent of the class" question requiring the part:whole read.
Exit ticket: "A team's win:loss ratio is 3:2. What PERCENT of games did they win?" (Wins are = 60% — the part:whole trap in one line.)
The recipe (serves 4): 2 cups flour, 3 eggs, 1.5 cups milk. Needed: 10 servings.
Step 1: Find the scale factor: .
Step 2: Scale EVERY ingredient by the same 2.5: flour cups; eggs → 7 or 8 real eggs (round and note it); milk cups.
Step 3: Verify one ratio survived: flour:milk was ; now ✓ — same pancake, bigger stack.
Step 4: The egg wrinkle is worth discussing: continuous quantities scale exactly; countable ones round. Real recipes are ratio tables with a tolerance.
Step 1: Read 35% as a ratio: 35 per 100.
Step 2: Build the table toward 80: per 100 → 35; halve: per 50 → 17.5; per 10 → 3.5. Assemble 80 = 50 + 10 + 10 + 10: .
Step 3: Cross-check by the fraction route: and ✓.
Step 4: The benchmark route for mental math: of 80 is 8; → ✓ three routes, one answer — students should own the benchmark route for restaurants and the decimal route for precision.
Pitcher A: 3 cups concentrate, 7 cups water. Pitcher B: 4 cups concentrate, 10 cups water. Which tastes more strongly of juice?
Step 1: Resist the count trap ("B has more concentrate") — more concentrate in more water may be weaker. Compare part:whole fractions: A is concentrate; B is .
Step 2: Compare vs : common denominator 70 → vs . A is stronger — barely.
Step 3: Percent both for the taste-test report: A = 30%; B ≈ 28.6%.
Step 4: Debrief the moves: chose part:WHOLE (taste is concentrate per total drink), compared exactly (they were too close for benchmarks), reported in percent (the lingua franca of comparisons). Ratio literacy is exactly this pipeline: choose the right comparison, compute it honestly, report it readably.