IN THE MATTER OF: Student performance — Calculating the Pythagorean Scale and Interval Ratios.
FACTS: Student (age 13) correctly identified 1:2 for a halved string, doubled middle C to 523.26 Hz, and produced a Pythagorean C scale with values closely matching expected frequencies (C 261.63, D ~294.46, E ~331.07, F 348.84, G 392.45, A ~441.45, B ~496.71, C 523.26 Hz). For interval ratios the student attempted decimal simplifications but several simplified fractions are incorrect or missing.
ISSUES: Are calculations mathematically accurate and are interval ratios correctly expressed as exact Pythagorean fractions? Do tasks meet ACARA v9 learning objectives for Years 8–10?
FINDINGS: 1) Correct: Q1a–c (1:2, frequency doubles to 523.26 Hz) and octave limits. 2) Correct: G = 392.445 Hz (2/3 string => 3/2 frequency). 3) Scale frequencies: student values are accurate within rounding. 4) Interval ratios: required exact Pythagorean fractions (use powers of 3/2 reduced into the octave) — correct interval ratios are: C:D = 9/8, C:E = 81/64, C:F = 3/4, C:G = 2/3, C:A = 16/27, C:B = 128/243, C:C = 1/2.
ORDER (recommended actions): 1) Recompute interval ratios using the known Pythagorean fractions rather than decimal guesswork. 2) Follow rounding rules only after forming the exact decimal from the exact fraction, then convert back. 3) Show algebraic steps (helpful: use (3/2)^n and divide/multiply by 2 to fit the octave).
ACARA v9 ALIGNMENT: This task develops ratio and proportional reasoning, manipulation of fractions and decimals, and rounding—meeting and often exceeding Year 8–10 Number and Algebra expectations (real numbers, ratio, and problem solving).
Signed, with affectionate firmness — Your proud, exacting teacher (Tiger Mother cadence — do it again, cleaner, show the work!).
FACTS: Student (age 13) correctly identified 1:2 for a halved string, doubled middle C to 523.26 Hz, and produced a Pythagorean C scale with values closely matching expected frequencies (C 261.63, D ~294.46, E ~331.07, F 348.84, G 392.45, A ~441.45, B ~496.71, C 523.26 Hz). For interval ratios the student attempted decimal simplifications but several simplified fractions are incorrect or missing.
ISSUES: Are calculations mathematically accurate and are interval ratios correctly expressed as exact Pythagorean fractions? Do tasks meet ACARA v9 learning objectives for Years 8–10?
FINDINGS: 1) Correct: Q1a–c (1:2, frequency doubles to 523.26 Hz) and octave limits. 2) Correct: G = 392.445 Hz (2/3 string => 3/2 frequency). 3) Scale frequencies: student values are accurate within rounding. 4) Interval ratios: required exact Pythagorean fractions (use powers of 3/2 reduced into the octave) — correct interval ratios are: C:D = 9/8, C:E = 81/64, C:F = 3/4, C:G = 2/3, C:A = 16/27, C:B = 128/243, C:C = 1/2.
ORDER (recommended actions): 1) Recompute interval ratios using the known Pythagorean fractions rather than decimal guesswork. 2) Follow rounding rules only after forming the exact decimal from the exact fraction, then convert back. 3) Show algebraic steps (helpful: use (3/2)^n and divide/multiply by 2 to fit the octave).
ACARA v9 ALIGNMENT: This task develops ratio and proportional reasoning, manipulation of fractions and decimals, and rounding—meeting and often exceeding Year 8–10 Number and Algebra expectations (real numbers, ratio, and problem solving).
Signed, with affectionate firmness — Your proud, exacting teacher (Tiger Mother cadence — do it again, cleaner, show the work!).