Corrected calculations and simple explanation (for a 13‑year‑old)
Q1a. The monochord division 1:2 is correct — that is the length ratio.
Q1b. Frequency ∝ 1/length. Cutting the string in half halves the length, so frequency doubles: 261.63 Hz × 2 = 523.26 Hz. (Correct.)
Q1c. When string length is halved, pitch goes up one octave (frequency doubles). Correct answer: pitch doubles.
Octave limits: C to C' = 261.63 Hz to 523.26 Hz. Correct.
2/3 division: If the string is shortened to 2/3 of original length, frequency is multiplied by 3/2 (because freq ∝ 1/length). So 261.63 × 3/2 = 392.445 Hz → 392.45 Hz. (Student answer 392.45 is correct.)
Pythagorean C scale (exact interval ratios and frequencies)
- C = 1/1 → 261.63 Hz
- D = 9/8 → 261.63 × 9/8 = 294.33 Hz
- E = 81/64 → 261.63 × 81/64 = 331.13 Hz
- F = 4/3 → 261.63 × 4/3 = 348.84 Hz
- G = 3/2 → 261.63 × 3/2 = 392.45 Hz
- A = 27/16 → 261.63 × 27/16 = 441.51 Hz
- B = 243/128 → 261.63 × 243/128 = 496.70 Hz
- C' = 2/1 → 523.26 Hz
Interval ratios (root C compared to each note) — simplified exact fractions
- C : D = 8 : 9
- C : E = 64 : 81
- C : F = 3 : 4
- C : G = 2 : 3
- C : A = 16 : 27
- C : B = 128 : 243
- C : C' = 1 : 2
Notes: The student’s frequency column is excellent — those numeric frequencies match Pythagorean ratios. The incorrect simplified fractions (4/5, 3/5, 13/25, etc.) are mistaken conversions from decimal approximations back to fractions. Use the exact fractional ratios above (they come from the Pythagorean construction: repeated 3:2 fifths reduced into the octave).
Teacher report (ACARA v9‑aligned) — Amy Chua cadence
You did the arithmetic. That is good. But you must learn to think before you write fractions from decimals. Your frequencies are correct — excellent discipline. Your errors are in converting decimal approximations back to simplified interval ratios. Remember: the Pythagorean scale uses exact ratios (1:1, 9:8, 81:64, 4:3, 3:2, 27:16, 243:128, 2:1). Practice converting fractions ⇄ decimals and simplifying fractions. This activity meets ACARA v9 expectations for proportional reasoning, number sense and measurement: students should use ratio and multiplicative thinking to solve real problems and interpret results in context (music). For next tasks: show your algebraic steps, give exact fraction first, then decimal approximation to the specified rounding rule. I expect precision and neat work. Do it again until flawless.
Parent feedback — Amy Chua cadence
You got the right numbers but relaxed when converting them into clean fraction ratios. That is lazy. Drill these ratios: 9/8, 81/64, 4/3, 3/2, 27/16, 243/128. Test your child daily on converting these fractions to decimals and back. Encourage careful steps: exact fraction → decimal → round if asked → convert back only when necessary. Insist on full written working — no shortcuts.
Overall summary — ACARA v9 mapping (Years 8–10)
These lessons integrate The Arts (Music: pitch, scales, historical context) with Mathematics (Number and Algebra: ratios, fraction↔decimal conversion, rounding, proportional reasoning). For Year 8 this demonstrates multiplicative reasoning and ratio application. For Years 9–10 it extends to precise fractional representations, rounding rules and handling rational approximations. The student’s numerical frequencies meet expectations; refining the fraction conversions will move performance from competent to excellent — meeting or exceeding ACARA v9 outcomes.