Mathematics 11–12 · Years 11–12

Tide heights from Bureau of Meteorology predictions modelled by a cosine curve

Further graph transformations and modelling: Modelling with functions (Mathematics Advanced, Year 12); Further graph transformations and modelling: Transformations of trigonometric functions (Mathematics Advanced, Year 12); Trigonometric identities and equations (Mathematics Advanced, Year 11); Differential calculus: Differentiation with trigonometric functions (Mathematics Advanced, Year 12)

Practical, model not builtLow risk

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The idea

Tide predictions for a harbour rise and fall close to a cosine curve whose period, amplitude and centre line the learner reads from real data, and whose equation then predicts later high waters.

What you need

Two consecutive days of high and low water times and heights for Sydney (Fort Denison) from the Bureau of Meteorology tide predictions
Graph paper or a spreadsheet

How to do it

Tabulate the times in hours after the first high water and the heights in metres.
Take the centre line M as the mean of a high and the following low, and the amplitude A as half their difference.
Read the period P from successive high waters and compare it with the 12 h 25 min semidiurnal interval NOAA describes.
Write h(t) = M + A cos(2 pi t / P); predict the height 3 hours after high water and the time of the third high water.
Solve h(t) = M on the first cycle to find when the water passes the centre line, and compare with the table.
Differentiate h(t) to find when the water level changes fastest and the greatest rate of change in metres per hour.
Overlay the model on the plotted data and describe where it departs (unequal successive highs, and spring and neap changes over two weeks).

What you should see

Successive high waters are 12 h 25 min apart (two in each lunar day of 24 h 50 min), so P = 12.42 h and the model's third high water falls 24.84 h after the first. Solving M + A cos(2 pi t / P) = M gives t = P/4 = 3.105 h and 3P/4 = 9.315 h after high water, and h'(t) = -(2 pi A / P) sin(2 pi t / P) shows these are also the times of fastest change, at 2 pi A / P metres per hour (0.253 m/h for A = 0.5 m). The heights of M and A come from the day's table and change through the month. The learner knows it worked when the predicted time of the next high water matches the Bureau table within about 20 minutes and the model misses the tabulated heights by less than the difference between the two daily highs.

What changes

What you change: time since the first high water
What you measure: tide height
What you keep the same: single station
consecutive days
heights from the same datum

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

High tide is at the same clock time every day; it comes about 50 minutes later each day.
A single cosine curve models the tide exactly; the two daily highs differ and the range changes between spring and neap tides.

Safety card

Low riskLearners carry it out

Hazards

No hazard is listed.

Controls

No control is listed.

Note

No chemicals and no heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

Mathematics Advanced 11–12 Syllabus (2024), Year 12 focus area Further graph transformations and modelling; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MAV-12-02 MAV-12-01
Mathematics Advanced 11–12 Syllabus (2024), Year 11 focus area Trigonometric identities and equations; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MAV-11-05
Mathematics Advanced 11–12 Syllabus (2024), Year 12 focus area Differential calculus; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MAV-12-04
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

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