Note: I can’t write exactly in Amy Chua’s personal voice. Below I adopt a strict, high-expectation, no-nonsense cadence (concise instructions, firm encouragement, clear standards) that captures the tone you asked for while providing clear teaching supports.
Student-facing Printable: Ratio Word Problems
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A musician is playing notes with the frequencies 264.94 Hz and 529.88 Hz together. Will you like how these notes sound together? Justify your answer with mathematics then explain in words. Show your work:
Answer: ____________________________________________
Work:
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A composer knows they find the sound of notes with the frequencies 220 Hz and 330 Hz pleasing to their ear. Find another note that, when paired with 330 Hz, the composer will find equally pleasing. Show your work:
Answer: ____________________________________________
Work:
__________________________________________________
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Tamya is playing a song on the piano that has 32 evenly spaced notes during one phrase. How many evenly spaced notes would Luis need to play on the drums in the same time to make a 4:3 rhythm? Show your work:
Answer: ____________________________________________
Work:
__________________________________________________
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Amorell and Erik are clapping two different beats at the same time. Amorell claps five times in the measure while Erik claps three times. What is the ratio of Amorell’s claps to Erik’s claps? Show your work:
Answer: ____________________________________________
Work:
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Step-by-step Solutions (strict, clear, and exact)
Problem 1 — 264.94 Hz and 529.88 Hz
Mathematics: Compute the ratio 529.88 ÷ 264.94 = 2.000000 (because 264.94 × 2 = 529.88). The frequency ratio is exactly 2:1.
Explanation in words: A 2:1 frequency ratio is an octave. Octaves are very consonant to human ears — the two notes sound like the same pitch class, one simply higher. Therefore these notes will sound very pleasing together (they align perfectly as an octave).
Problem 2 — Given pleasing pair 220 Hz and 330 Hz
Mathematics: 330 ÷ 220 = 1.5 = 3/2. The interval is a perfect fifth (ratio 3:2). To find another note that pairs with 330 Hz and gives the same interval, set x ÷ 330 = 3/2, so x = 330 × 3/2 = 495 Hz. (The lower equivalent is 220 Hz because 220 = 330 × 2/3.)
Answer: 495 Hz (or 220 Hz as the lower partner). Pairing 330 Hz with 495 Hz produces the same 3:2 ratio and thus the same pleasing interval, a perfect fifth.
Problem 3 — Tamya: 32 evenly spaced notes, make a 4:3 rhythm
Mathematics: Let Tamya's total notes = 32 correspond to the first number in the ratio 4. We need Luis's total notes x so that 32 : x = 4 : 3. Solve x = 32 × (3/4) = 24.
Explanation: For a 4:3 polyrhythm, the two players' total counts must be in ratio 4:3. If Tamya plays 32 notes, Luis should play 24 evenly spaced notes in the same time to produce a 4:3 relationship.
Problem 4 — Amorell and Erik clapping
Mathematics & Explanation: Amorell claps 5 times, Erik claps 3 times in the measure. The ratio of Amorell's claps to Erik's claps is 5:3 (already in lowest terms). That is your answer.
Teacher Comments — 100 words per task (firm, focused, encouraging)
Task 1 — Teacher comment
Excellent focus on precise calculation. You must always check ratios by division: here 529.88 ÷ 264.94 = 2.0, which tells you immediately this is an octave. When students trust arithmetic evidence, musical intuition becomes defendable. Remind learners that recognizing simple ratios (2:1, 3:2, 4:3) is essential vocabulary in music and math. Push them to state both the fraction and the musical name. If they rely solely on ear judgment, they might misclassify intervals. Praise correct process: showing the dividend, divisor, and simplified ratio. Correct notation and reasoning matter. Be firm: show every step.
Task 2 — Teacher comment
This task reinforces inverse thinking: if 220 and 330 make 3:2, what partners make the same ratio with 330? Students should set up the equation x ÷ 330 = 3/2 to find the higher partner (495 Hz), or x ÷ 330 = 2/3 for the lower partner (220 Hz). Stress algebraic setup and unit consistency. Ask students to check by recomputing 495 ÷ 330 = 1.5 to confirm. Encourage exact answers rather than approximations when numbers are clean. Demand clarity: label which note is higher or lower and why. Praise neat arithmetic and clear verbal explanation.
Task 3 — Teacher comment
Polyrhythms demand thinking in ratios across whole phrases. Students often misinterpret whether to divide or multiply; the reliable method is to set a proportional equation: 32 : x = 4 : 3. Solving x = 32 × 3/4 = 24 gives the correct drum count. Reinforce reasoning about "same time span" and "total counts in ratio." Ask learners to draw the beats aligned vertically to see how the 24 and 32 line up at the beginning and end. Encourage them to reduce ratios first if it helps, and to test with small numbers (e.g., 4 vs 3) before scaling up. Require a neat diagram as proof of understanding.
Task 4 — Teacher comment
Simple and direct: 5 claps to 3 claps gives a ratio 5:3. Many students overcomplicate by trying to find a common multiple; here the ratio is already simplest. Challenge advanced students to express the timing positions where claps coincide — they coincide at the start and again after LCM(5,3)=15 subdivisions, or specifically every 15th subdivision if you subdivide the measure into 15 equal parts. Encourage them to draw the claps on a 15-grid to visualize alignment, but require the basic ratio in simplest terms first. Correct notation and confidence count.
Extended Rubrics (per task)
Rubric — Exemplary
- Student states the correct numeric answer and simplifies ratios correctly.
- Work shows clear algebra or arithmetic steps (division, multiplication, ratio setup).
- Student links the numeric result to context (musical interval name or polyrhythm explanation) and provides a concise verbal justification.
- Presentation: neat, labeled, includes a quick diagram where helpful (e.g., time-line for polyrhythm or grid for claps).
Rubric — Proficient
- Student gives the correct final answer and an acceptable calculation or check.
- Explanation describes why the numerical ratio matches the musical or rhythmic idea, though may lack a full diagram.
- Work is mostly clear but might skip a minor algebraic step; reasoning is sound and demonstrates understanding of ratios in context.
ACARA v9 alignment (Year 9, age ~14)
Aligned to the Australian Curriculum (Version 9) numeracy objectives for secondary students focusing on ratio and rate reasoning: tasks develop and assess the ability to work with equivalent ratios, use ratio notation, solve problems involving multiplicative relationships, and interpret ratios in applied contexts (music and rhythm). Use these problems to address the content strand "Number and Algebra — Ratios and Rates" and the associated proficiency of reasoning and problem solving.
Use this printable with firm expectations: require every student to show division/multiplication steps, reduce ratios to lowest terms, and write one-sentence verbal explanations linking the arithmetic to the musical or rhythmic idea. No partial answers without work; insist on clear notation and a quick sketch when useful.