Extension Activity — Interpreting the Graphs of Sound Waves (14-year-old) — Student printable (Tiger Mother cadence)

Extension Activity — Interpreting the Graphs of Sound Waves

Age: 14

Required: Open the graph at http://bit.ly/intervalgraph. Use DESMOS controls to turn graphs on/off as instructed. Work carefully; I expect neat answers and clear reasoning.

Instructions (Follow exactly — do not skip steps)

1. Turn on graphs for C4 and C5 (the first and last options). Find where both graphs intersect the x-axis at the same point.
   1. Count the number of zeros (points where graph crosses x-axis) between 0 and that intersection for C4 (include the intersection). Write number: ______
   2. Count the zeros between 0 and that intersection for C5 (include the intersection). Write number: ______
   3. Calculate the ratio: (C4 zeros) : (C5 zeros). Write ratio in simplest whole-number form: ______. How does this compare to the C:C frequency ratio you recorded in the C Scale Ratio Table? (Explain in 1–2 sentences):
      
   4. Refer to Handout 2 (C Scale Note Pairings Observations). What did you think the pair C:C sounded like? (Brief comment):
      
2. Turn off C and C (C4 and C5). Turn on graphs for C4 and D4.
   1. Count the zeros between 0 and the first intersection for C4 (include intersection). Write number: ______
   2. Count the zeros between 0 and the first intersection for D4 (include intersection). Write number: ______
   3. Calculate the ratio: (C4 zeros) : (D4 zeros). Write ratio in simplest form: ______. Compare this to the C:D frequency ratio you recorded in the C Scale Ratio Table. (Explain briefly):
      
   4. From Handout 2, what did you think C:D sounded like? (Brief comment):
      
3. Reflection: Based on the two investigations above, explain in 2–4 sentences how the graphs (zeros, frequency of oscillation) relate to how notes sound together. Use the terms frequency, period, ratio, and harmony/dissonance where appropriate:
   

Quick Tips (Do these or redo the task)

• Zoom in on a small x-range so you can count zeros cleanly.
• A higher frequency means more zeros in the same x-interval.
• Integer ratios (like 1:2, 2:3) usually sound more consonant; non-integer ratios often sound more dissonant.

Teacher Feedback — 100-word comments (one per task)

Task 1 — C4 & C5 intersection (100 words)

You followed instructions precisely and showed careful counting — excellent. Notice how the higher note (C5) completes more cycles in the same time window; you recorded that cleanly. Next time, label the x-range you used and show a tiny calculation converting zeros to cycles per unit time — this makes your reasoning airtight. Also compare the observed ratio to the frequency ratio from your table and explicitly state whether they match and why integer ratios give strong alignment. Good diagrams or arrows on a printed screenshot would take this work to the next level. Keep precision high; your logic is solid.

Task 2 — C4 & D4 intersection (100 words)

Your counts for C and D showed attention to detail. I liked that you compared the zero counts to your frequency table; that link is crucial. For refinement: explain any small mismatches by referencing the scale used in DESMOS (sample window or rounding). Interpret your ratio in musical terms — is it a simple fraction (consonant) or a more complex fraction (more dissonant)? Describe the listening result from Handout 2 and connect it to the ratio: this demonstrates understanding of how math explains sound. Finally, state whether phase alignment influenced where zeros coincide and sketch that briefly next time.

Task 3 — Reflection on graphs and sound (100 words)

Your reflection shows you understand that number of zeros relates to frequency and that frequency ratios influence consonance. Improve by explicitly using terms: period (time between zeros), frequency (cycles per second), and harmonic relationship. Discuss why simple integer ratios (like 1:2, octave) produce steady reinforcement of wave peaks, creating pleasant consonance, while non-integer ratios create beating or roughness. If you can, include a sentence predicting the result for a 3:2 ratio (perfect fifth) and relate it back to your graphs. Strong conceptual understanding — tighten the technical language for full credit.

Extended Rubrics — Exemplary and Proficient Outcomes

Use the rubric below to assess student work. Each criterion is rated separately.

Criterion 1: Procedure and Accuracy

• Exemplary: All graphs correctly toggled and zeros counted accurately with labeled x-range. Calculations of ratios are simplified and matched to frequency ratios with explicit explanation of any differences. Includes annotated screenshots or sketches showing intersection points.
• Proficient: Graphs toggled correctly, zero counts and ratios correct, comparison to frequency table present and logically explained. Minor omissions (like unlabeled x-range) do not impede understanding.

Criterion 2: Conceptual Understanding

• Exemplary: Clearly explains how frequency, period, and ratio determine the number of zeros and perceived consonance/dissonance. Uses terms frequency, period, harmonic, consonance, dissonance correctly and predicts results for other intervals.
• Proficient: Demonstrates correct relationship between number of zeros and frequency and links ratio to perceived sound. Uses most technical vocabulary appropriately but may lack extension or prediction.

Criterion 3: Communication & Presentation

• Exemplary: Answers are neat, well-structured, with clear sentences, labeled diagrams/screenshots, and concise justification. Reflection connects evidence to claims convincingly.
• Proficient: Answers are clear and complete with correct reasoning. Diagrams may be minimal but sufficient. Reflection is present and mostly well-argued.

Criterion 4: Extension & Insight

• Exemplary: Makes connections to harmonics, explains phase effects or beating, and suggests further experiments (e.g., change amplitude, vary time window). Demonstrates higher-order reasoning.
• Proficient: Provides reasonable extension (e.g., predict other intervals) but may not fully explore harmonic complexities or experimental design.

ACARA v9 — Curriculum Mapping (Year 9 level connections)

Suggested alignment to Australian Curriculum v9 learning areas (phrased to match curriculum language):

• Science (Physical World / Waves): Investigate properties of waves (frequency, amplitude, period) and their applications; analyse how wave characteristics determine sound pitch and timbre; plan and conduct investigations using digital tools to collect and interpret data.
• The Arts — Music: Explore how pitch relates to frequency and how intervals (octave, fifth, etc.) are perceived; analyse and describe musical intervals and their consonance/dissonance using evidence from listening and graphical representations.
• Mathematics (Number and Algebra / Ratios & Graphs): Use ratios and proportional reasoning to compare frequencies; interpret and analyse periodic graphs; relate algebraic description of wave functions to counts of zeros and cycles.

Teachers can map these descriptions to specific ACARA v9 content codes for Year 9 in their planning documents.

Printable teacher note: Expect students to submit the DESMOS screenshot (cropped) showing the counted zeros and a one-paragraph justification linking counts to frequency ratios. Use the rubric above for marking.