Handout — Ratio Word Problems
Student age: 14. Read and do every step neatly. Show all working or lose marks.
Instructions (Student-facing)
Drawing upon what you learned in the lesson, complete the following word problems. Write answers neatly. No steps, no credit.
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Problem 1:
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.
Space for work and answer:
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Problem 2:
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 steps.
Space for work and answer:
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Teacher's Guide (Amy Chua cadence — clear, firm expectations)
Be direct. Demand correct notation, units, and justification. If students do not show algebra or clear arithmetic, return the paper for correction.
Quick model solutions (show these to students only after they attempt the problems)
Problem 1 — Model solution
Step 1: Write the ratio 264.94 : 529.88.
Step 2: Divide both numbers by 264.94 to simplify:
264.94 ÷ 264.94 = 1 and 529.88 ÷ 264.94 = 2 so the ratio simplifies to 1 : 2.
Step 3: Interpret musically — doubling frequency = an octave. 529.88 is exactly twice 264.94, so these two notes form an octave, a very consonant interval. Conclusion: Yes, you will likely find the notes pleasing; explain that octaves sound very stable because one waveform repeats every two cycles of the other.
Problem 2 — Model solution
Step 1: Simplify the known pleasing pair 220 : 330. Divide both by 110 to get 2 : 3.
Step 2: To find a note x that pairs with 330 to give the same interval (2 : 3), set up the proportion 330 : x = 2 : 3 (330 is the lower note, x the higher):
330 * 3 = 2 * x → 990 = 2x → x = 495.
Step 3: Check by simplifying 330 : 495. Divide both by 165 → 2 : 3. Musical interpretation: the 2:3 ratio is the perfect fifth — another strongly consonant, pleasing interval. State the unit: 495 Hz.
Curriculum mapping (ACARA v9)
Mapped to Australian Curriculum v9 — Mathematics, Number and Algebra: Ratio and rate. Relevant learning goals: understand and simplify ratios, solve problems using proportional reasoning, and apply ratios in real contexts (here: sound frequencies and musical intervals). Use this to justify assessment alignment and success criteria.
Teacher feedback — 100-word comment for Task 1 (use for marking or parent notes)
Well done on attempting Task 1 — but don’t stop at the answer. You calculated the ratio correctly; now justify every step. Explain why dividing both numbers by 264.94 yields 1:2, and connect that to the musical concept of an octave (frequency doubling). Use exact arithmetic or show the calculator method to avoid rounding errors. If you wrote “I like the sound” that's not enough: state that octaves are highly consonant because waveforms align every second vibration. For full marks, add a brief sentence about temperament (equal temperament vs. pure tuning) and how real instrument tuning might slightly alter exact ratios.
Teacher feedback — 100-word comment for Task 2 (use for marking or parent notes)
You found 330 paired with 495 gives the same 2:3 ratio — good. Now explain steps: simplify 220:330 to 2:3, then set proportion 330:x = 2:3 and solve for x = 495. Show algebra, check by dividing both by 165. Discuss whether the new note is higher or lower and how intervals sound (a perfect fifth? Wait 2:3 corresponds to a perfect fifth), historically very consonant. For excellence, convert ratio to cents or explain where 495 Hz sits relative to standard A=440 tuning. Try also finding the lower partner and explain the listening result in detail.
Extended rubric (use for scoring student work)
Criteria
- Mathematical accuracy (simplification & arithmetic)
- Correct use of ratio notation and units (Hz)
- Logical step-by-step working shown
- Contextual interpretation (musical meaning: octave, perfect fifth)
- Presentation: neatness, labels, and justification
Exemplary (A)
All arithmetic is correct and each simplification step is shown clearly. Ratio notation is precise (e.g., 1:2 or 2:3) and units (Hz) are included. Student explains why the ratio corresponds to a musical interval (octave or perfect fifth), references how frequency doubling or 2:3 alignment produces consonance, and gives an advanced remark (e.g., about temperament or cents). Presentation is flawless — neat, labelled, and complete; student anticipates and checks answers (reverse-check or divide to verify).
Proficient (B)
Correct final answers and valid method shown but might be missing one minor justification detail. Arithmetic is accurate, ratio simplified correctly, and the student states the musical interval and that it is consonant. Units are included. Explanation is clear but lacks the advanced extension (no mention of temperament or cents). Work is legible and logically ordered; minor calculation checks may be absent.
Developing / Needs improvement
Errors in simplification or proportion, missing units, or no explanation of musical meaning. Steps are incomplete or unclear. Return to student for corrections and insist on re-submission with full steps.
Teacher tips & tasks to extend learning
- Ask students to compute the interval in cents and compare to equal-tempered values.
- Have students listen to synthesized pairs (264.94 & 529.88; 330 & 495) and describe subjective consonance.
- Extend to other ratios (3:4, 4:5) and ask which sound more or less consonant and why using waveform alignment.
Cut here for student handout copies. Insist on neatness. No partial credit for sloppy work.