Relationships: Decimals, Fractions, Ratios, and Percents
Warm-up
Project a sports headline: "Fraser sank 7 of 8 free throws — a 92% night — leaving her season rate at 0.845, up from 4:5 odds-ish territory…" Ask: how many number-dialects did one sentence use? (Fraction, percent, decimal, ratio.)
The unit's promise: these are ONE quantity wearing four outfits, and translating between all four — fluidly, in any direction — is the finish line of number sense for this grade.
Explore
Grand-conversion carousel: six stations, each holding one quantity in one form with three blanks (e.g., station 1: "3:5 — as a fraction of the whole? decimal? percent?" — careful: part:part vs part:whole must be resolved FIRST). Pairs rotate, fill, and flag any station where the ratio's referent was ambiguous — that flag IS a learning objective.
Then the number-line convergence: all quantities from all stations get placed on ONE line 0 to 1.5, forcing cross-dialect comparison ( vs 0.845 vs 92% vs …).
Formalize
Formalize the translation hub with the fraction as the central form:
The ratio-to-fraction gate is where errors concentrate: a part:part ratio must pass through part:WHOLE before becoming a fraction/decimal/percent of the total. Every conversion pipeline should begin by asking "of WHAT?" — the referent question.
Practice
Practice: complete a four-dialect table for six quantities (two starting from ratios); order a mixed set on a number line; two context problems (a recipe ratio to percent; a survey percent back to "about how many of the 30 students"); one ambiguity hunt (a headline ratio with unclear referent, rewritten both ways).
Exit ticket: "Paint mixes red:white 2:3. What PERCENT of the mix is red — and what percent would 2:3 red:white BECOME if one more part red were added?" (40%; then 3:3 → 50%.)
Exit ticket
Practice: complete a four-dialect table for six quantities (two starting from ratios); order a mixed set on a number line; two context problems (a recipe ratio to percent; a survey percent back to "about how many of the 30 students"); one ambiguity hunt (a headline ratio with unclear referent, rewritten both ways).
Exit ticket: "Paint mixes red:white 2:3. What PERCENT of the mix is red — and what percent would 2:3 red:white BECOME if one more part red were added?" (40%; then 3:3 → 50%.)
The record: wins:losses = 9:6 (no ties).
Step 1: The referent question: percent OF ALL GAMES → need part:whole. Total games .
Step 2: Fraction of whole: .
Step 3: Convert: win rate.
Step 4: The trap answer, named and buried: — a "win rate" over 100% should smell impossible and does. It computed wins per LOSS (a legitimate ratio! — 1.5 wins per loss) but answered a different question. Same numbers, different referents, different truths: the referent question isn't pedantry, it's the whole game.
The report: "65% of the 480 students surveyed prefer later start times."
Step 1: Convert the percent to a count: students.
Step 2: The complement without re-multiplying: prefer current times (or: 35% of 480 = 168 ✓ — two routes, one answer).
Step 3: The precision caveat: had the report said 65% of "about 480," the 312 inherits the "about." Counts computed from percents are only as exact as both inputs.
Step 4: Reverse skill check: "312 of 480 — what percent?" ✓ round trip. Fluency means the pipeline runs both directions without friction.
The phone: battery at . The trip needs 5 hours; the phone burns about 8% per hour. Enough?
Step 1: Convert the battery to percent: .
Step 2: Convert the need: .
Step 3: Compare in the common dialect: — short by 2.5%, about 19 minutes' worth ( of an hour).
Step 4: Decision with options: charge briefly (2.5% ≈ a few minutes on fast charge), or dim the screen (cut the burn rate).
Step 5: Notice what made the problem solvable: three differently-dressed quantities (a fraction, a duration, a rate in %/h) were dragged into ONE dialect before comparing. That drag-to-common-form move is the entire unit, compressed into a phone panic.