Everything so far has been somebody else’s data. Today the class generates its own, and finds out how well a smooth mathematical wave describes something that actually happened in this room.
Choose a source
Each group takes one, and no two groups take the same:
- A pendulum, timed by video: horizontal position against time.
- A mass bouncing on a spring: height against time.
- Sound, recorded on a phone and opened in any free audio editor: the waveform of a sung note, or a struck glass.
- A bicycle wheel: mark the tyre, film it rolling slowly, and track the mark’s height above the ground.
- Light through a slot, or a shadow’s length across a period, if the weather cooperates.
- Daylight minutes for our latitude across the year, from a published table — the one source that is not measured today, and worth including because a whole cycle takes twelve months.
Collect properly
- Decide the sampling interval before you start. Too coarse and the peaks vanish; too fine and you spend the period counting.
- Record at least three complete cycles. One cycle can be fitted by almost anything; three cannot.
- Write down the units. A table of numbers whose units nobody noted is a table of nothing.
- Note anything unusual as it happens — a bump, a pause, a miscount. That note is what explains your outlier later.
Then the mathematics
- Plot it. By hand first, roughly, so the shape is in your head before a screen smooths it.
- Is it periodic? Is it sinusoidal, or merely repeating? Say which, and give the evidence.
- Pull out the period, the amplitude, and the axis from your own graph.
- Fit an equation of the form , using the method in From Ratio to Function.
- Test it against a point you did not use. If it misses, say by how much and why.
The extrapolation question
Use your equation to predict a value beyond the data you collected — one cycle further on. Then, where you can, go and measure it.
Two things are worth arguing about afterwards, and they are the real content of the day:
- How far ahead would you trust your model? A pendulum loses amplitude to friction; a singer runs out of breath; daylight is reliable for centuries.
- What in the situation, not in the mathematics, sets that limit?
Afterwards
Post your graph, your equation, and your prediction. We will compare which sources produced clean sinusoids and which did not, and why — What Makes a Model Good is the argument this feeds into, and your data becomes fair game for it.
Curriculum connection
D3.1
collect data that can be modelled as a sinusoidal function (e.g., voltage in an AC circuit, sound waves), through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials, measurement tools such as motion sensors), or from secondary sources (e.g., websites such as Statistics Canada, E-STAT), and graph the data Sample problem: Measure and record distance–time data for a swinging pendulum, using a motion sensor or other measurement tools, and graph the data.
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D2.2
predict, by extrapolating, the future behaviour of a relationship modelled using a numeric or graphical representation of a periodic function (e.g., predicting hours of daylight on a particular date from previous measurements; predicting natural gas consumption in Ontario from previous consumption)
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D3.2
identify periodic and sinusoidal functions, including those that arise from real-world applications involving periodic phenomena, given various representations (i.e., tables of values, graphs, equations), and explain any restrictions that the context places on the domain and range Sample problem: Using data from Statistics Canada, investigate to determine if there was a period of time over which changes in the population of Canadians aged 20–24 could be modelled using a sinusoidal function.
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