Unit 1 felt like discovery. We used a monochord and Pythagoras' ratios. Middle C, 261.63 Hz — we halved the string. The octave emerged. Then we applied 2:3 splits to build the Pythagorean C scale. The student calculated each frequency, adjusted octaves, and filled the chart accurately. Standards: apply proportions, convert ratios to frequencies, use decimal arithmetic.
Unit 2 sharpened precision. We computed interval ratios between C and each scale note. We followed specific rounding rules, converted decimals back to fractions, and simplified results. The student handled truncation, rounding to tenths and hundredths, and repeating-decimal conversion. Standards: interpret measurement precision, manipulate rational and repeating decimals, and justify rounding choices.
Parent report (exemplary): She is thrilled. He is proud. I am impressed. At 13, our learner moved from conceptual discovery to rigorous computation. Progression clear: conceptual understanding of pitch and ratio in Unit 1 → procedural fluency with decimals and fraction conversion in Unit 2. Confidence increased. Accuracy improved. Work shown, reasoning explained. Next: apply these interval ratios to tune an instrument. Bravo. I recommend continuing with listening exercises, practical monochord building, and reflections; the student's curiosity and resilience promise deeper musical-mathematical insight with hands-on projects forthcoming next term ahead.